Keywords: cement-based materials, mechanical and diffusive properties, mu lti -scale homogenization, damage, leaching, coupled chemo-mechanical degradations

 and of double-inclusion type schemes. A real istic multi-scale homogenization approach is then developed and validated for cement pastes and mortars. Simu lations of chemo-mechanical degradations of leached cementitious materials are subsequently carried out by implementing these models into the platfonn ALLIANCES. The numerical resu lts confronted with experimental tests give valuable information in tem1s of the material durabi lity.

RESUME

L'evaluation de la durabili te des materiaux cimcntaires utilisables dans les structures d'entreposage de dechets nucleaires necessite la prise en compte de facteurs .de d~terioration, tels que lcs attaques chimiqucs ou la fissuration, ainsi quc des interactions cntre ces phenomenes. La resistance des materiaux cimentaires a ces degradations est fortement conditionnee par leurs proprietes mecaniques et diffusives macroscopiques. Le premier objectif de Ia these vise a construire unc approche mu lti -echelle et un modele d'endommagement ortl1otrope pour estimer ces proprietes ainsi que leurs evolutions au cours de processus de deteriorations chimiques. Le second est de realiser des simulations de degradat ions couplees ch imi e-mecanique de matcriaux cimentaires Iixivics afin d'ctudier leur comportement a long termc. Apres une synthese non exhaustive consacree aux techniques d' homogeneisation analytique, de nouveaux modclcs, tels que !'estimation par assemblage 111ixte de spheres composites, sont proposes en revisitant les theories d ' Hashin ct Shtrikman ( 1962) ct lcs methodes de type double-inclusion. Unc approche d'homogeneisation multiechelle est ensuite developpee pour les pates de ciment et lcs mortiers et validcc. Des simu lations de degradations couplecs chimie-mecanique de materiaux cimentaires li xivics sont enfin effectuees en implementant les modeles dcvcloppes precedemment au sein de la plateforme numerique ALLIANCES. Les rcsultats numcriques confrontes avcc des essais experimentaux donnent des infom1ations uti les concemant la durabil itc de ccs matcriaux.

Mots des: materiaux cimentaires; proprietcs mecaniques et diffusives; homogcneisation mu lti-echelle; cndommagcment; lixiviation ; degradations couplccs ch imi e-rnecan ique. J, percolation threshold of particulate phaser percolating volume fraction of phaser relative volume fraction of phaser in the inner coating of a doubly coated sphere relative volume fraction of phaser in the outer coating ofa doubly coated sphere calcium concentration in the pore solution (in mol/ L of solution) concentration of aqueous spcciesj in the pore solution (in mol/L or solution) diffusion coefficient of an isotropic phaser (m 2 s- 1 ) effective diffusion coefficient of a macroscopically isotropic material (m 2 s- 1 ) diffusivity oftritiated water in bulk water at 23°C (m 2 s- 1 ) aqueous specie aggressive ionic specie mineral specie density of coated spheres with phase 2 as extemal coating in a MCSA equilibrium ba lance of the dissolution of em number of mineral phases total mass of CSH ratio of the mass of CSHini to the total mass of CSH number of aqueous species in the interstitial solution concentration of e,. (in mol/ L of solution)

NOMENCLAT URE FOR MECHANICS

rad ius of the spherica l core of a composite sphere external radius of a composite sphere calcium concentration in the solid phase (mol L" 1 LIST OF FIGURES l:'igure 11.9: Schematic illustrating the effective inclusion concept employed by Kuster-Toksllz ( 1974) and Shen and Yi (200 I) Bejaoui and Bary (2007) computed with the hydration model of [START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland c. ement pastes[END_REF] and [START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland c. ement pastes[END_REF] (Galle et al. 2004) Table V .6: Mechanical input data adopted for the micromechanically-based damage model in the cases of a HCP sample [START_REF] Hcukamp | Docs calcium leaching increase ductility of cementitious materials? Evidence from direct tensile tests[END_REF] Due to the pluridisciplinary nature of the subject treated, the present manuscript is concerned with well separated physical problems in cementitious materials, such as chemistry, diffusion or mechanics. for conciseness, this introduction gives only a general overview of the thesis, a more specific introduction being associated with each main Part of the document.

J.l I ndustrial and scientific context

The durability of concrete materials employed in many kinds of infrastructures, such as bridges, dams, or nuclear reactors, is an acute issue in Civ il Engineering because they involve important security aspects. The cost of maintenance of these concrete structures is tremendously high. This economical factor consequently motivates scientific research allowing for better assessing how cement-based materials behave with time. Some of the primary mechanisms of deterioration of concrete faci lities include cracking due to external loadings and chemical attacks. The loss of performance and sometimes the complete rnpture of these infrastructures can consequently be originated by mechanical and/or chemical deteriorations of concrete.

It appears therefore a necessity to adopt multi-physical coupled approaches to deal with material durability. Besides its importance for previously mentioned concrete facilities, this topic is also of current concern in the context of nuclear waste storage. As is already the case in the experimenta l site of Bures in the east of France, concrete is very likely to be employed for different applications in future underground strnctures devoted to the storage of nuclear waste. It shou ld be used as engineering barrier but also as outside coating in the waste containers in which vitrified waste may be stored. The life time service of these underground facil ities should be about tens or hundreds of thousand of years, since the radioactivity of nuclear waste can last such a long time scale. The present manuscript is more specifically situated in this context of long-terrn durability of cement-based materials.

T he scope of the thesis is however not only purely industrial but also scientific.

Homogenization techniques have known an increasing success among researchers and their application to cement-based compos ites appears promising in particular for pred icting with accuracy the evolutions of their properties with time. These methods indeed allow for estimating the macroscopic physical properties of a heterogeneous material from the knowledge of the microstructure and the physical characteristics of the elementary phases. In that sense, they constitute deductive approaches, since they may a priori be applied to any material provided that its composition and phase physical properties are known. These approaches that are focused on in the present manuscript furthem1ore represent an outstanding tool to link the microscopic phenomena, like the disso lution of a phase, to the macroscopic behaviour of the materia l and even of the structure.

l.2 Research objectives

The objective of the present manuscript is mainly two-fold. In the context of durability, two of the most essential factors that detennine the service life of concrete facilities are the diffusive and mechanical characteristics of the construction material for the following reasons. The resistance of a structure to loadings is of course conditioned by the mechanical behavior of concrete and its opposition to chemical deterioration largely depends on the opportunity offered to ions to diffuse through the material.

The first objective of the thesis thus consists in building an analytical approach, based on homogenization techniques and on a multi-scale description of cement-based materials, allowing for estimating both their mechanical and diffusive properties and for predicting their evo lution with time. Many empirical laws already exist for computing these quantities but they suffer some limitations. For instance, their validity range is quite restricted or they require many experimental data. To circumvent this difficulty, homogenization models have recently been proposed (e.g. [START_REF] Bernard | A multiscale micromechanics-hydration model for the eaerly-age elastic properties of cement-based materials[END_REF]Constantinides and Ulm 2004;[START_REF] Pivonka | Microscopic effects on chloride diffusivity ofeement pastes -a scale-transition analysis[END_REF]). However, these models are generally only designed for determ ining one specific physical property of cement-based materials, such as their elastic moduli or their macroscopic diffusivity. It is consequently of high interest to develop a unified homogenization approach that is capable of predicting both the mechanical and transport properties of concrete materials on the basis of a real istic microgeomctry. Furthennore, progress in experimental analysis techniques (e.g. [START_REF] Scrivener | Backscattered electron imaging of cementitious microstructures: understanding and quantification[END_REF][START_REF] Garboczi | Shape analysis of a reference cement[END_REF] has lead to a more accurate knowledge of the microstructure of these materials, which is valuable for developing a precise modelling. A damage model also gaining profit from homogenization techniques is proposed to predict the rather brittle behavior of concrete.

The second and final goal of the present manuscript is to model and carry out simulations of degradations coupling chemistry, transport and mechanics of cement-based composites. These simulations are performed with the help of a numerical integration platfonn ALLIANCES that allows for coupl ing different phys ical problems (Montamal ct al. 2006;2007) developed by the French Atom ic Energy Commission (CEA) in collaboration w ith ANDRA and EDF.

The study of such deteriorations is crucial in the context of nuclear waste storage. The concrete material, assumed to be saturated in the nom inal storage phase, undergoes chemical deteriorations in contact with ground water. This degradation process is ruled by the macroscopic diffusion coefficients of chemical species through the saturated cementitious material. Therefore a coupled chem ical-transport approach is necessary to predict the alteration with time of its microstructure. The engineering barrier made of concrete of the underground structures serving for the disposal of nuclear wastes should also act against the propagation of radionuclide. The impact of chem ical alteration of the constrnction material on the global diffusive properties of th is structure element is a key point for assessing its capacity to work as a radionucl ide barrier.

Moreover, the chem ica l degradation process generating a strong augmentation of porosity inside the ccmentitious material affects the overall mechanical behaviour of the underground faci lity. Damage of the structure may then occur because of the combined sol icitations due to external loadings and to chemical deterioration. This nucleation and growth of cracks in the concrete can in turn accelerate the transport phenomena and then enhance the chemical alteration process. Nevertheless, the simulations give some usetill insights for other thematics, such as C0 2 storage or tunnels construction. But in such fields of applications, it is necessary to consider unsaturated conditions, which generally lead to particularly comp licated computations. The assumption of saturated conditions is presently retained, since the degradations of concrete underground structures for the storage of nuclear waste may be envisaged at a long-term. The thc1mal effects are not taken into account, even though nuclear waste can induce very significant elevations of temperatures.

I .3 Outline of the thesis

The manuscript is composed of five Parts. The first one is dedicated to the presentation of cement-based materials and of the degradations of these materials subjected to leach ing. In Reinforced concretes are also commonly used in constructions but phenomena like corrosion are out of our scope.

A good knowledge of the microstrucn1re of cementitious materials and of its impact on tbeir macroscopic properties is a prerequisite for properly predicting their physical deteriorations.

A detailed investigation of HCP microstructurc is fundamental to better understand the mechanisms of degradation of cement-based materials and explain the differences of properties from one material to another. This investigation constitutes forthem1ore the cornerstone for the construction of the homogenization model treated in the following Parts of the thesis. The present Part gathers experimental results and observations that are useful for unders tanding the microstructurc of ccmcntitious mntcrinls, for knowing its basic properties and assessing the material durabil ity. A complex microstructure then forms due to hydration of cement grains, the reaction of cement particles with water initially taking place at the particle surface. The microstructure obtained is usually described as a complex mixture of pores (P), calcium-silicate-hydrates (CSH), unhydrated clinker (UC), portlanditc or calcium hydroxide (CH) and aluminates (AF), such as etrringitc or alumino-ferritc (AFt), monosulfoaluminates (AFm). The chemical composition of these mineral phases is given at the end of the page The relative proportions of these phases in HCP may significantly vary from one paste to another and strongly depend from the minera l composition of 1he cement grains and on 1he water/cement (w/c) ratio. [START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland c. ement pastes[END_REF] proposed the following equations to model the hydration of the cement grains and compute the resulting vol ume fractions of tbe hydrated phases:

MICROSTRUCTURE AND PH

2C 3 S + I0.6H ----? 2CSJI + 2.6CH, 2c,s + 8.6 11 ----? 2CSH + 0.6C H, C 3 A + 3Gypsum + 26 H ----? AFt, 2C 3 A + Aft+ 4 H ----? AFm, C 3 A + CH+ 12 H----? C.AH, 3 ,
C,AF + 2 Cl! + 10 H ----? Hydrogarnet.

(I. I)

Other hydration models exist in the literature such as CEMHYD3D developed by the NIST [START_REF] Bentz | Three-Dimensional Computer simulation of Portland Cement Hydration and Microstructure Development[END_REF]) but the Tennis and Jennings model (2000) is adopted in the ensuing to estimate the volume fractions of the mineral phases composing the microstructure of HCP.

During hydration of cement pa1ticles, it is generally asserted [START_REF] Richardson | The nature of the hydration products in hardened cement pastes[END_REF] that two diffusive layers presently defined as inner and outer layers form successively from cement grains surface. Tbe inner layer is less porous than the outer one, since the first one res ults from higher confinemem conditions and from poorer water accessibility during hydration process. An image provided by scanning electronic microscopy (SEM) is proposed to illustrate the HCP heterogeneous microstructure (sec Fig. I. I).

1 CSH = x CaO -Si0 2 (aq)-x H 2 0 , where x varies between 0 and 2.3; CH = Ca(OH)2;

A Ft 0 6 CaO -Al 23 [START_REF] Bejaoui | Experimental and modeling studies of the link between microstructure and effective diffusivity of cement pastes[END_REF].

Depending on the in itial sizes of cement grains, an anhydrous part of the cement particles remains after hydration has stopped and constitutes an impermeable core surrounded by these two heterogeneous layers.

2.1.J.2 information on the HCP mineral phases

Many phases appear in the mineral composition of HCP and their characteristic sizes can range from the nanometer to micrometer scales. The main solid phases, wh ich volume fractions displayed on Table I.I in CEM I cement pastes can be estimated by means of the Tennis and Jennings model (2000), are presented in depth below. A particular emphasis is put on the morphology inside the HCP microstructure that is an important aspect for the models developed subsequent ly.

Anhydrous CSH;., CSH"' AFt -AFm Hydrog:1rnet a11d J e1111ings (2000).

CSH: they are formed by hydration of the si licates C 3 S and C 2 S and represent about half of the paste in 1cm1s of volume fractions. The quantity of calci um in the CSH is variable. They are thus characterized by their Ca lcium over Si licon (C/S) rati o, which can vary from 0. 7 in very degraded states to generally about l.7 in a sound state. At the beginn ing of tl1e hydration process, a nucleation phase followed by a growing phase of CSH occurs at the surface of cement grains. cs11•" first fonn a layer around the clinker and hydration process is controlled by diffusio n through it. The denser type of CSH, namely CSH;"', is then fonned inside this coating. The CSH 1 "' and the CSH''" have a characteristic size of I 0 to I 00 nanometres, which makes it very difficult to gain precise information about their geometrica l shape. [START_REF] Taylor | Cement Chemistry[END_REF] describes the CSI-1 1 " ' as a honeycomb or reticular network, while the CSH"' rather have a foliated structure [START_REF] Bejaoui | Experimental and modeling studies of the link between microstructure and effective diffusivity of cement pastes[END_REF], as may be seen on Fig. (I.I), A Ft, A Fm and C 4 AH 13 are produced by the hydration o f Ci A, whereas Hyd roga111et is produced by the reaction of c . AF with portlandite. Though thei r volume fraction can be rather scarce (about 7 % in volume), those phases have to be accounted for in leach ing cases or in sulfatic attacks, since they can be partially or totally dissolved [START_REF] Adcnot | Modelling of the corrosion of the cement paste by deionized water[END_REF]. AFt and A Fm arc usually more abundant in standard cement pastes than Hydrogamet and c . AH 13 and have received a closer attention from experimentalists. According to [START_REF] Richardson | The nature of the hydration products in hardened cement pastes[END_REF], AFm is present in mature pastes as large irregu lar plates simi lar to those of CH and Aft occurs as thin hexagonal prism need les of up to JO run in length. Aft is quite easy to observe on the Environmental Scanning Electron M icroscopy (ESEM) since they are characterized by long rods when crystallized in no n-confined environment. However, according to Tenn is and Jenn ings model, AFI that forms rapidly during the hydration process then react with C 3 A to form i\Fm and therefore hardly exists in marure cement pastes.

Unhydrated clinker: it corresponds to the ponion of cement grains that has not reacted with water. The amount of anhydrous residuals present in HCP depends on the w/c ratio and time of hydration. Cement paste micrographs generall y tend to show that cement particles do not have spherical shapes. Thanks to very advanced techniqu es such as X-ray microtomography, cement particles morphology has been rigorously in vestigated. [START_REF] Garboczi | Shape analysis of a reference cement[END_REF], performing rccemly a spherical harmonic coefficient analysis on microtomograph ic images of Portland cement grains, managed to characterize q uanti tatively their real shape (sec Fig. I.3).

They ana lyzed about I 200 patticles with volume ranging from I ftm 3 to I 20000 µ111 3 and concluded that these pa1t icles were definitely non-spherical by plotting their surface area versus their volume. T he curve of surface area versus their volume for cement pa1ticles was indeed clearl y above the spherical one. By filling their experimental curve, the following resu lt was obtained for the volume V over surface area A ratio:

!:_ = -1 -V"~. (1.

3)

The knowledge of the volume over surface area ratio of real cement particle is a precious piece of information for the modell ing of this phase that is further exploited in Part II. The previous discussions high light the complex morphology of all the mineral phases composing HCP. Hence, it is natural to pose the following question: what is the best way to model in a simple and realistic manner these phases? An answer to this question will be prov ided in Part II.

2. 1 .1.3 The complex multiscale porosity of HCP

The HCP described previously (Figs. I. I and 1.2) are complex multi-scale porous media, where the pore size distribution varies by several orders of magnitude (from I nm to more than I 0 µm) . At the micrometer scale, capillary pores are defined as the remaining space situated between hydrated cement grai ns. They are originated from the chemical shrinkage of the hydration products or stem from the interstitial space left by water in excess. The CSH, representing about half of the paste in tenns of volume fractions, furthermore contain a finer porosity, defined as gel poros ity. The CSH"'' are more porous than CSH' "", the CSH;"' resu lting main ly from higher confinement conditions during hydration reactions. According to the Jennings and Tennis model, the gel porosity comprised in CSH 1 " ' is about 26 % and the one contained in CS H" ' arou11d 36 % in tenns of volume fractions. In the present work, the gel pores sizes are assumed to be situated between a few nm and 0.2 µrn and the capi llary pores ones to range from 0.2 µm and a few µm , thoug)l the defin ition of these two domains significantl y varies in the different classifications for the pore structure of HCP proposed in literature (e.g. [START_REF] Baroghel-Bouny | Caracterisation des pates de ciment ct des betons. Methodes, ana lyse et interpretations[END_REF][START_REF] Daimon | Pore strncture of calcium si lica te hydrate in hydrated tricalcium silicate[END_REF][START_REF] Lgarashi | Analysis of cement pastes and mortars by a combination of backscatter-based SEM image analysis and calculations based on the Powers model[END_REF]Powers I 948).

For example, according to Mindess and Young (198 1), the size of the cap illary pores may vary from several orders of magni tude (from I 0 nm to I 0 µm) and the gel pores from a few angstrljms to I 0 nm. It is also noteworthy that some authors differentiate more than two categories of pores in cement pastes (e.g. [START_REF] Daimon | Pore strncture of calcium si lica te hydrate in hydrated tricalcium silicate[END_REF] but we limit for simplicity to two main classes of pores.

The pore structure of HCP, especially the capillary porosity, is the subject of many experimenta l investigati ons [START_REF] Taylor | Cement Chemistry[END_REF]. To measure the total porosity of HCP, the simp lest technique consists in performing total free water porosity measurements (Gall e 200 I). The porosity is practically obtained by measuring the total amount of water removed from saturated pastes after drying them. Other techniques exist but, in the present dissertation, the total porosity is considered to be the value measured by water porosimetry. Two techniques are commonly used to measure the pore size distribution: mercury intrusion porosimetry (MIP) and image analysis on SEM results. In add ition, the breakthrough of microtomography and nanotomography could lead to more precise measurements in the oncoming years [START_REF] Bentz | The visible cement data set[END_REF]Burlion ct al. 2006;llolzer et al. 2003).

MIP is used to measure the pore size distributions by quantifying mercury ab le to penetrate the dried material under successively increased pressures. The abi lity of mercury to pass througJ1 pores of a given size depends on the pressure applied. The volume intrnded per change in pressure provides the data for calcu lating the pore size distrib\ltion [START_REF] Galle | Effect of drying on cement-based materials pore structure as identified by mercury intrusion porosimetry: A comparative study between oven-, vacuum-, and freezedrying[END_REF]). However, [START_REF] Diamond | Mercury porosimetry-an inappropriate method for measurement of pore size distributions in cement-based materials[END_REF] evidenced the fact that MIP is an inappropriate method for the correct measurement of the pore size distribution in cement-based materials. In particular, he argued that the amount of large pores is underestimated, s ince most of them are inaccessible to mercury injected at low pressure. In order to attain certain large pores, mercury may indeed be constrained to penetrate through smaller pores. As a consequence, the quantity of small pores is overestimated. Despite these shortcomings, thi s experimental method gives some insights on the material pore stnictme in terms of connectiv ity. Indeed, the measurement of the critical pore or threshold diameter by mercury porosimetry allows for evaluating the largest scale at wh ich percolation, i.e. the formation of a pore cluster connecting one extremity of the material to another, occurs [START_REF] Diamond | Mercury porosimetry-an inappropriate method for measurement of pore size distributions in cement-based materials[END_REF]. The critical pore diameter is defined as the pore width correspond ing to the highest intrusion of mercury per change in pressure, practically determined the maximum of the dV/dP versus pore diameter curve as plotted in Fig. J.4 (or by the inflection point on the volume intrusion). According to [START_REF] Katz | Prediction of rock electrical conductivity from mercury injection measurements[END_REF], the threshold diameter pcnnits to identify the less constri cted percolating group ing of pores, since this length corresponds to the size of the smallest pore belonging to the percolating ones. It is thus possible to identi fy the less constricted percolating path in the paste that sho uld logicall y inn uence its transport properties (Katz and Thompson 1987). For example, accordi ng to the experiments of Galle et al. (2004),

this diameter is about 20 nm for an o rdinary CEM I paste with a w/e ratio equal to 0.45. This result shows that the gel pores can percolate through the paste. On tbc contrary, the larger capillary pores, that are mostly inaccessible to mercury injected at low pressure, may be isolated o r c9nnccted to each other by a network of gel pores. Values for the threshold diameter can eas ily be found in the very large data collection of MIP tests ava ilable in the open literature (sec Table 1.2).

Furthermore, other experimental observations confirm the fact that the gel pores should percolate through the paste. Recently, Holzer et al. (2005) asserted that the skeleton ization of porosi ty on 30-images obtained wi th the help of nanotomography at resolutions of 20 nm indicated that the pore network is almost completely connected. [START_REF] Bejaoui | Experimental and modeling studies of the link between microstructure and effective diffusivity of cement pastes[END_REF] also performed an accurate investigation of the pores entrapped in the CSH in CEM I pastes. Their TEM images of CSHin• show that they keep an invariant morphology characterized by a very finely divided porosity at the nanometre sca le. On the contrary, it appears clearly on their TEM observations (see Fig. 1.3) thaJ CSI I"' present diverse morphology depending on the w/c ratio. In pa1ticular, the characteristic length of the porosity on their images varies from 10 nm for low w/c pastes to a hu ndred of nm for high w/c ones. As may be observed on the micrographics of [START_REF] Bejaoui | Experimental and modeling studies of the link between microstructure and effective diffusivity of cement pastes[END_REF], the gel pores in CSH' " arc strongly connected for the pastes of medium o r high w/ c rat ios.

The capillary pores can be investigated by image analysis. These techniques have an increasing success in the field of cement-based materials but they arc limited by their resolution, generally about 0.2 µm [START_REF] Scrivener | Backscattered electron imaging of cementitious microstructures: understanding and quantification[END_REF]). Nevertheless, they offer the opportunity 10 measure the size distribution of capillary pores larger than 0.2 µm. lgarashi et al. ( 2004) measured the volume fraction of capillary pores in different types of cement pastes and mortars by image analysis. The morphology of the capillary pores is very difficult to characterize, thoug)l the recent microtomographic images of Rattanasak and Kendall (2005) and of the Visible Cement Data Set [START_REF] Bentz | The visible cement data set[END_REF]) could give some insights on their three-d imensional shapes. To summarize, the present subsection pulling together qualitative but also quantitative results obtained on the cement paste porosity with diverse techniques (MIP, SEM, ... ) indicates that the percolation of poros ity occurs at a scale of several tens of nanometers and that the pore structure of 1 lCP is highly connected at the nanometre scale. It is useful to keep in mind these remarkable insights gained on porosity by experimentalists in the ensuing Pa1ts ded icated to modelling so as to correctly represent the pore structure.

2.J.2 Microstructure of mortars and concretes

Mortars consist in a mixrure of cement pas1es and of sand particles, which size ranges from O. l mm to 5 mm, in order to improve the packing of these grains. Concretes are obtained by further adding aggregates with sizes varying between I mm and about 25 cm. Many srudics have evidenced that an interfacial transition zone (TTZ), which thickness ranges from 15 to 40 iun (Hashin and Monteiro 2002), fom1s between the aggregate particles and the cement. It has been noted that the thickness of the ITZ layers is qu ite independent from the size of the pa1ticle. The !TZ is generally more porous than the cement paste [START_REF] Lutz | Effect of an inhomogeneous interphase zone on the bulk modu lus and conductivity of a particulate composite[END_REF] but is also likely to contain more portlandite and al uminates [START_REF] Sun | Modeling the elastic properties of concrete composites: Experiment, differential effective medium theory, and numerical simulation[END_REF]) because of the wall effect. The volume fractions occupied by the JTZ is difficult to gain experimentall y, because the diverse ITZ layers tend to overlap. But [START_REF] Lu | Nearest-surface distribution functions for polydispersed particle systems[END_REF] developed a statistical model based on a representation of porosity by a polydisperscd system of spheres, which may be employed for estimating the latter quantity. Due to these overlaps, the ITZ is furthermore prone to percolate through the microstrucn1re of mortars or concretes [START_REF] Scrivener | The percolation of pore space in the cement paste/aggregate interfacial zone of concrete[END_REF] and strongly influences their effective properties [START_REF] Wang | Effect of ice forrnation on the elastic moduli of cement paste and mortar[END_REF]).

Microscopic and macr oscopic properties of cement-based materials

For their practical use as a construction material, it is necessary to know some basic properties of concrete materials, such as their Young modulus, strength, permeability or diffusivity.

These properties may s ignificantly vary from one material to another as will be shown subsquently. Modelling approaches aiming at estimating these characteristics are thus usefu l to avoid measuring them for each type of cement or concrete. But the development of model for predicting their properties is often constrained by the lack of data concerning the transport and mechanical features of the residual cl inker, hydrated phases or aggregates. Therefore th is subchapter gathers useful experimental results for the properties of both the cementitious materials and their main constituents at different scales.

2.2.J l11formatio11 011 the elastic properties of the HCP elementary phases

At the microscopic scale, the measurements of the elastic properties of the phases composing the microstructure of HCP are ma inly carried out by nanoindentation and, at the macroscopic scale, by diverse techn iques, such as resonance frequency measurements or by dynamic tests.

The nanoindentat ion tests consist in pushing an indenter with a diamond• tip into the specimen. The characteristic length of the indentation area is on the order of I o• 6 m, and the indentation depth varies in the range of 300-500 nm (Velez et al. 200 l ). The complete F -h response is recorded during the test, where h is the depth of penetration into the specimen and F is the load applied. The Young modu lus can be extracted by analyzing the unloading part of the F -h curve according to a model for the elastic contact problem. From the measurement of ran I: P-rescnlation of ccment~bascd materials P•g< IS the effective modulus E, of the test area, the elas1ic modulus E,. of the phase r is obtained using ihe equation (Damidot et al. 2003):

1 l 1 2 -v, 1 -V, E, Ed +~, (l.4)
where v, and £" are the Poisson ratio and the elastic properties of the diamond indenter and ~ is the Poisson ratio of the measured phase. One of the shortcomings from nanoindentation ' technique is that the Poisson ratio of the material 1ested is not accessible. The en-ors of the measures by indentation thus mainly come from the fact that the Poisson ratio elevated to square in the previous fonnu la has to be estimated. Nevertheless, the Young modu li of the HCP elementary phases (see T able 1.3) measured by nanoindentati on represent crucial pieces of infom1ation for tbe prediction of the mechanical behaviour of cementitious materials.

Different nanoindentation tests measuring the Young moduli of cl inker and CH have been pcrfom1cd (e.g. Boumiz et al. 1997;Damidot et al. 2003;[START_REF] Velez | Detennination by nanoindentation of elastic modulus and hardness of pure constituents of Portland cement clinker[END_REF]) and arc furthennore in quite good agreement with each others. The values displayed on Table 1.3 arc retained as reference values for the micromechanical computations in Pai1 II. The measurements performed on cement pastes exposed to a leaching process, explained in the ensuing chapter, arc also displayed. Nevertheless, it is usefu l to recall that these measures arc made by indenting o n surfaces representative of the CSl-I with a minima l area of 1 µm 2 and therefore enclose not only the CSH but also some porosity (< 1 iim) and probably a certain volume fraction of other hydratio n products, such as CH or AF phases , that is very difficu lt to quantify. These two phases of CSH arc however considered as homogeneous materials with elastic properties, which may be ident ified by nanoindentation technique, and aluminatcs, such as AFt or hydrogarnet, are thus assumed to have similar elastic properties to the ones measured for the 1wo types of CSl-I (Table 1.3). In the ensuing, it is supposed that the entire gel porosity is comprised in the measurements of the CSH" ' and CSJ-ii"' mechanical properties.

Elastic properties o/llCP

Diverse experiments have been performed for measuring the macroscopic e lastic moduli of HCP, for instance by resonance frequencies techniques (see e.g. Wang ct al. 1988;Galle et a l. 2004). Th is type of measurement is also used for testing mortar and concrete [START_REF] Sun | Modeling the elastic properties of concrete composites: Experiment, differential effective medium theory, and numerical simulation[END_REF][START_REF] Wang | Effect of ice forrnation on the elastic moduli of cement paste and mortar[END_REF]). In the resonance method, the test specimen is made to vibrate as a whole in one of its natura l frequency modes (transverse, longitudina l, or torsional). The resonance frequency depends on the geometry of the sample, on its density, its pores distribution and on its elastic properties. As an illustratio n, the Young modul i of different HCP gained by diverse resonance frequency tests are gathered against their total water porosity on A way to increase the stiffness of the cement pastes consists in add ing rigid sand particles or aggregates to these materials thus forming mortars and concretes.

Elastic properties of mortars amt co11cretes

The sand particles, which Young modu lus is measured as 86.7 GPa by [START_REF] Wang | Effect of ice forrnation on the elastic moduli of cement paste and mortar[END_REF], are much stiffer than the cement pastes. One would thus expect that the Young modu lus shou ld significantly increase as the volume fractions of sand aggregates augment. But the experiments of [START_REF] Wang | Effect of ice forrnation on the elastic moduli of cement paste and mortar[END_REF] showed on ly a 25 % increase of the overall Young modu lus of the mortars with 40 % of sand aggregates compared to the plain cement paste.

This result may be d ue to a weakening effect of the JTZ that is more porous than the bulk cement paste [START_REF] Sun | Modeling the elastic properties of concrete composites: Experiment, differential effective medium theory, and numerical simulation[END_REF]. Unfortunately, the elastic moduli of the ITZ layers are very difficu lt to measure experimentally, since they do not really appear as a homogeneous interface but rather as a graded o ne. To address this lack of data, many micromechanical approaches modelling the ITZ as a homogeneous phase have been developed to estimate their effective elastic moduli from an inverse approach (sec e.g. Hashin and Monteiro 2002).

However, [START_REF] Sun | Modeling the elastic properties of concrete composites: Experiment, differential effective medium theory, and numerical simulation[END_REF] noticed a strong dispersion between the Young modulus of ITZ proposed in literature as the following expression:

E,rz:a E11c•• (1.5)
where a is a constant with a value that is identified between 0.2 and 0.8 depending o n the model and on the thickness of the JTZ retained by the authors transition (Hash in [START_REF] Lfashin | An inverse method to determine the elastic properties of the interphasc between the aggregate and the cement paste[END_REF][START_REF] Ramesh | Effect of transition zone on elastic moduli of concrete materials[END_REF][START_REF] Yang | Effect of the Transition Zone on the Elastic Modul i of Mortar[END_REF]) and E 11 c p is the Young modulus of the corresponding plai n paste. It is however emphasized that, even though the JTZ is generally modelled as a homogeneous interphase, it would be more realistic to model it as an inhomogeneous one [START_REF] Lutz | Effect of an inhomogeneous interphase zone on the bulk modu lus and conductivity of a particulate composite[END_REF]. Jn the case of concrete, porous rrz layers still exist but their influence on the physic.al behaviour of concrete seems less important than in mortars probably because the volume fraction of JTZ is more important in the case of mo11ars, since the aggregates have much smaller size. That's why these interphases are generally not taken into account in the models developed to predict the physical properties of concrete (Constantin ides and Ulm 2004).

Tra11.~port properties of cement-based materials

The main transport processes in concrete include diffusion, permeation, and convection.

Diffusion describes the transport for instance of a particular gas (e.g. C~ vapor) in the gaseous phase or dissolved ions as a result of a concentration gradient (Dullien 1992).

Penneation describes the flow of a fluid (e.g. water or air) as a result of gravity or a pressure gradient. Convection (or advection) is the process that describes the transport of a solute (e.g. ch loride or sulfate ions) as a result of the bu lk moving fluid. The transport processes in most concrete structures can be complex and may involve more tban one of the referred transport mechan isms. In most of these processes, water is the principal medium by which aggressive agents (such as chloride or sulfate ions) are transported into the concrete.

However, in the context of the long-tem1 behavior of underground concrete facilities, the material is generally assumed to be in saturated conditions in the nominal storage phase so that the chemical species in pore solution are transported by diffusion through the concrete.

Convection is neglected, since it is assumed that there is no pressure gradient in water. As may be observed in Fig. 1.5, the macroscopic diffusion coefficients of cement pastes strongly vary from one paste to another, with sudden variations occurring for certain w/c ratios. The causes of these variatioos are discussed in details in Part 111.

Concerning the transport properties of mortars and concretes, some authors [START_REF] Nguyen | A separation of sca les homogenization ana lysis for the modelling of ca lcium leaching in concrete[END_REF]) argue that the presence of aggregates in a hydrated cement paste matrix probably bas two opposite effects that compensate each other. On the ooe hand, it increases the tortuosity of the matrix and reduces tbe total space available to diffusion and, on the other hand, the high porosity of ITZ enhances the diffusion process.

As a result of the strong variations between the difTusivities of cement-based material, aggressive agents diffuse more or less rapid ly through the material, which is a key issue that influences the long-tem1 degradation of concrete facilities. The present chapter bas presented some basic features of cementitious materials in a sound state. However, these materials are sensitive to many degradation factors and therefore the evolution with time of their microstructure is strongly conditioned by their environment. The next chapter focuses on some of these deterioration sources that influence the behavior of cementitious materials.

DEGRADATI ONS OF CEMENT-BASED MATERI ALS SUB.I ECTED TO

LEACHING

Various degradation mechan isms may affect the concrete materials under service life. They may be altered for instance by: (i) chemical reactio ns caused by ionic migration between the interstitial solution and ground water; (ii) damage due to external mechanical loadings and to possible precipitation of secondary solid phases generati ng internal pressures; (i ii) thennal expansion due to strong temperature variations; (iv) creep and shrinkage phenomena. The present dissertation only treats the first two deterioration factors that may strongly impact the durabi lity of concrete underground iofrastrnctures, such as the ones for the disposal of longterm nuclear wastes. Jn these facilities, concrete is for instance employed in the engineering ban:ier that serves as one of the protections against the propagation of radionucleides. In this context, the chemical degradation mechanisms due to leach ing and then the damage processes, which may alter the diffusive properties of the material, are presented. In both cases, their effects on the microstruct\1re and the overall behaviour of the concrete are pointed o ut. Finally, the possible interactions between these two degradation processes are investigated.

C hemical deterioration of cementitious materials caused by leaching

The chemical alteration of concrete is a phenomenon that has to be carefully taken into account for pred icting the long-tenn durability of underground structures. In contact with ground water, the material is subjected to chemica l reactions originated by the migration of ions, such as calcium or sulfates, caused by concentration grad ients between the pore solution and the ground water. The materia l composed of a solid skeleton and the interstitial solution is intially in a chemical equilibrium state. The concentration of ions in the pore solutio n is consequently imposed by the mineral composition of the solid part. The movement of ions towards the exterior and inside the material di sturbs thi s equilibrium state and causes dissolution-precipitation reactions in the material to compensate the variation of ions concentrations in the pore solution. This chemical degradation process by contact with pure water is generally called leaching or decalcification, since it principally concems the calcium ions and the minera l phases containing calcium. However, ground water may contai n aggressive chcm1cal species. If the water in contact with the concrete is rich in sulfates or chlorides, we rather speak about extemal sulfate o r chloride attacks but these attacks are out of our scope.

The propagation of the zones affected by leaching is usual ly very low, since pure water leaches on ly 4 cm of a concrete structure in roughly 300 years [START_REF] Adenot | Durabilite du Seton: Caraccerisation et Modelisation des Processus Physiques ct Chimiques de Degradation du Ciment[END_REF]), but may reveal detrimental for long-term ind ustrial applications. The development of an accelerating method using ammonium nitrate as aggressive solution (Cardc 1996) A convenient variable to describe in good approximation the leach ing process is the calcium concentration in the pore solution Cea• Fig. 1.7, adapted from [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi -coated sphere assemblage model[END_REF] The porosi ty measurements with water and MlP for both asymptotically leached and intact pastes allow for a quantitive analysis of the microstruetural changes with leaching (see Table 1.5). To obtain a homogeneously degraded paste, the latter is exposed to a very aggressive ammonium nitrate solution during a number of weeks that depends on the size of the sample (see e.g. llcukamp 2002). On Table 1.5, the total porosity in the material appears almost doubled in the uniformly leached state. Concerning the totally decalcificd state, [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi -coated sphere assemblage model[END_REF] assumed that all the CH, half of the AF phases and 5 % of the CS I I are dissolved and replaced by additional capillary pores. Moreover, the internal porosity of CSH is likely to increase because of its degradation (Heukamp 2002). To qua11tify this porosity, it is proposed to compare the measure of total porosity by MlP with the sum of the initial capillary porosity and the additional one issued from the complete leaching of CH. The difference obtained is then assumed to be attributable to the CSH degradation and the partia l one of AF. Th is empirical method is appl ied to a CEM I paste with w/ c ~ 0.45 and the values of porosity comprised in the CSH collected in Table 1.5 before and aficr leaching indicate that a total volume fraction of supplementary percolating fine gel porosity of 0.04 appears in the CSH during the decalcification process. In the case of mortar and concrete, the degradation main ly takes place in the cemcntitous matrix, the aggregates being more resistant to leaching. The kinetics of degradation of mortars is however quite simi lar to the ones of HCP (Bourdctte 1994).

The leaching experiments presented on Table l.5 (e.g. Galle et al. 2004) For instance, a sharp diminution of the HCP elastic properties was measured by [START_REF] Carde | Caractcrisation ct modclisation de !'alterat ion des propric1es mccaniques due;\ la lixi viation des matcriaux cimentaires[END_REF], where total dissolution ofCJI caused a reduction of63 % in the Young modulus ofa CEM I 42.5 w/c = 0.50 cylindrical sa111ple wi th a 30 mm diameter. Galle et al. (2004) GPa representing a 84 % decay. To obtain a uniformly degraded paste. the sample with a diameter of 11.5 mm was attacked by an ammonium nitrate solution during 5 months. The differences between these measurements [START_REF] Carde | Caractcrisation ct modclisation de !'alterat ion des propric1es mccaniques due;\ la lixi viation des matcriaux cimentaires[END_REF]Galle ct al. 2004;Constantinides and Ulm 2004) may be explained by the fact that the pastes are more or less chemically altered,

given that the diameters of the samples and the leaching time considered vary from one author to another. I lowever these experiments do not account for the possible effects of damage on the chem ical degradation processes that arc investigated below.

I11fl11e11ce of dam age on the leaching process

Chem ical degradations are potentially influenced by the presence of microcracks because they may provide preferential pathways for fluid or ions to pass through. Many studies have reported that cracks can dramatically influence the transport properties of concrete, in particu lar its permeability (e.g. Wang et al. 1997). For example, Wang et al. ( 1997) studied the permeability of cracked concrete and concluded that cracks generally accelerated water penneation in concrete. [START_REF] Jacobsen | Effect of Cracking and Healing on Chloride Transport in OPC concrete[END_REF] investigated the effects of cracking on chloride transport in concrete and found that internal crack ing increased the chloride penetrati on rate 2.5 to 8 times when compared with undamaged specimens. [START_REF] Tognazzi | Couplage fissuration-degradation chimique dans les matcriaux cimentaires: caraeterisation ct modelisation[END_REF] carried out an experimental campaign to investigate the effects of damage induced by compressive or tensile loads on mortar effective diffusivity. First, displacementcontrolled mechanical tests were perfonncd on mortar specimens so as to generate different states of cracking. A visible impact of the damage created inside the material on its diffusivity increasing by 150-200 % was noticed for samples having attained the post-peak regime of the mechanical test. Th is influence may be attributed to the coalescence of cracks. On the contrary, load induced damage docs not appear to influence the diffusion properties substantially at the peak load due to the discontinuous and localized nature of the crack pattern. Further experimental works confinn the visible effect of damage on the material degradation by putting in evidence the existence of chemically deteriorated zones around artificial cracks [START_REF] Tognazzi | Couplage fissuration-degradation chimique dans les matcriaux cimentaires: caraeterisation ct modelisation[END_REF]. It is also noteworthy that Yang ct al. (2006) draw the same type of conclusions in their work on water absorption and electrical conductivity of concrete damaged by tensile loading and freeze and thaw cycling.

The material diffusivity therefore depends on the characteristics of the crack system, like _ its connect ivity and its distribution. However, it must be emphas ized that the influence of cracks on the diffusivity may be less important than the effects of the dissolution due to leaching that may augment the effective diffusivity by a few orders of magnitude [START_REF] Tognazzi | Couplage fissuration-degradation chimique dans les matcriaux cimentaires: caraeterisation ct modelisation[END_REF].

Fulfy C()llpled experime11ts

The experiments entire ly coupling damage and chemical degradation are rare, even though they arc usefu l for assessing the material durability. Le Bellego (2001) and Schneider and CJicn ( 1998;2005) have conducted such tests, called «l ife-time» experiments, where a mortar or concrete beam is simultaneously subjected to mechanical solicitations and to a chemical attack by an aggressive solution of ammonium nitrate. More precisely, the inferior part of the beam is simultaneously sollicitated in traction because of the flexural load [START_REF] Schneider | The Chemomechanical Effect and the Mechanochemical Effect on High-Performance Concrete Subjected to Stress Corrosion[END_REF]2005) In Schneider and Chen (2005), mortar and concrete samples are sub_icctcd to a flexural load such as the macroscopic stress supported by the material varies by 30 to 50 % of its ultimate strength. For a stress equal to 30% of the maximal one, they showed that the influence of creep was low enough to be neglected. These experiments lead to the complete rnpture after on ly a few months.

In Le Bellego (200 I), a displacement is imposed on the center of the upper face of a mortar beam , the lower face being in contact with a 6M NH 4 N0 3 solu1ion. This flexural displacement is small enough so lhal 110 da111age should occur i11 a beam that has not undergone any chem ical attack. However, the maximal stresses inside the beam in flexion attain about 80 % of 1he maximal ultimate strength and, at this level, the creep effects are not negligible. They noticed that a lower streng1h value of the mortar was obtained in this fully coupled e~pcriments compared with residual mechanical tests. Neve11heless, it is difficult to judge if th is• decrease is due to the influence of damage oo the chem ical alteration or rather caused by creep.

CONCLUSI ONS OF PART J

The present Part has collected experimental insights of various types on cement-based materials. First, it was shown that these materials have a complex multi-scale heterogeneous microstructure. The latter is an intricate mixture of different sol id phases and porosity. The relative volume fractions of these different components can significantly change from one cemen t-based materia l •to another, depending for instance on its w/c ratio or its hydration degree. Furthermore, it has been highlighted that strong contrasts may ex ist between the prorerties of these diverse phases. As a result. the macroscopic features of the concrete materials vary a lot depending on its composi tion and its rnicrostructure. I lomogenization models based on a simplified representation of the microstructurc of cementitious materi als will therefore be developed in Parts I1 and TU to predict their effective elastic and diffusive properties.

Some important aspects of the long-tenn deterioration factors, such as chemical alteration and damage, of HCP have been presented. In particular, their consequences on the microstructure and the material physical properties have been underlined. All the experimental results issued from residual resistance tests (Cardc 1996;Galle et al. 2004;Le Bellego 2001) and from lifetime experiments (Le Bellego 2001;[START_REF] Schneider | The Chemomechanical Effect and the Mechanochemical Effect on High-Performance Concrete Subjected to Stress Corrosion[END_REF]2005) evidence the detrimental interactions between chemical deterioration and damage. It shows the importance to thorough ly account for the couplings between these degradation factors to predict the durability of cement-based materials. These experiments will be discussed more in depth in Part V that is devoted to the chcmo-mechanical simulations.

Part II: Prediction of lhe elastic properties of cCnlent 4 bascd Olatcrials part 11

pREDJCTION OF THE EFFECTIVE ELASTIC PROPERTIES OF CEMENT-BASED MATEIUALS

The purpose of the present Part is to elaborate a realistic homogenization model to predict the effective elastic properties of cement-based materials and their evolution with time. In particular, the model proposed should be capable of correctly estimating the stiffness reduction of these materials due to leach ing. The interest of using homogenization techniques is to develop an approach appl icable to any cemcntitious material, provided its composition and the properties of elementary phases are accurately known.

Jn the first chapter of Part U, diverse analytical homogenization schemes are reviewed so as to find the best suited one for predicting the elastic properti es of cementitioos materials.

Micromechanical models aim at estimating the effective macroscopic properties of a heterogeneous material from the knowledge of the geometric and physical characteristics of its various microstructural components. Due to the plethora of ex isting micromechanical schemes (e.g. Ma ct al. 2004;[START_REF] Milton | The theory of composites[END_REF][START_REF] Torquato | Random Heterogeneous Media: Microstructure and Macroscopic Properlies[END_REF]), the review is onl y limited to matrix-inclusion type schemes with a particular emphasis on the double-inclusion type models that have been the subject of some interesting developments in the last decade. The respecti ve advantages and drawbacks of these homogen ization methods presented in a concise way are discussed and the probable connections existing between these different theories arc highl ighted. Some improvements for the scheme developed by Hori and Nemat-Nasser (1993) are also proposed. The validity range of tlfe reviewed schemes is tested by comparing their predictions with numerical resu lts on well-defined cases. On the basis of this comparative study, the Interaction Direct Derivative (JOO) model due to Zheng and Ou (200 1) is chosen for it is versati le and simple of use. The Generalized Self-Consistent Scheme (GSCS) is also found to be of interest for composites with coated inclusions.

The second chapter focuses on tl1e effects of pa1ticle phase shapes on the effective isotropic linear elastic modu li of hardened cement pastes (HCP). Jn most micromechan ical models applied to cement pastes (e.g. [START_REF] Constantinidcs | The effect of two types ofC-S-H on the elasticity of cement-based materials: Results from nanoindentation and micromechanieal modeling[END_REF]Neubauer ct al. 1996), particulate phases are modeled as spheres. However, experimental observations clearly show that certain of them are far from being spherical. An attempt •to develop a more realistic micromechanica l model is proposed by using spheroidal inclusions and including a novel morpbological parameter. The latter is identified on the basis of experimental results issued for example from microtomographic images of Portland cement grains and HCP. In the first Part of the thesis dedicated to the microstructure and the properties of cementbased materials, the main phases cornposing the microstructure of cement and concrete have been identified as well as their mechanical and diffusive characteristics. On the bas is of these results and of the microstructural observations of Part I, the third chapter of the present Part aims at building a multi-scale description with a matrix-inclusion morphology of the microstructure of cement-based materials for predicting the elastic properties of HCP, mortars and concretes, respectively.

JNTRODUCTION TO EFFECTIVE-MEDI UM T HEORI ES (EMTs)

4. 1 Fundamental pr inciples of homogenization methods

J General principles of ltomogenizution metltods

The first step of these methods consists io representing the microstructure of the concerned material by a representative volume element (RY E) composed of a matrix phase, indexed by M, and of P particu late phases, which are assumed to be individually homogeneous. Their compliance tensors are respectively denoted by S,, and S, (r = I , .. . , P). Each particulate phase r consists of inclusion particles v; O = I, .. ., 11, ) that are of same shape but can be of different sizes or orientations. Let V be the domain occupied by the RYE of the heterogeneous material and let V' correspond to the sub-domain of inclusion phase r.

Jn what follows, we designate the boundary surface of V by ()V and the volume fraction of phaser bye,. .

The RYE is subjected to uniform boundary tractions over its surface oV:

(II. I)

where t . denotes the traction vector, n designates the outward normal unit vector to the surface (JV and a• is a constant stress tensor. At the scale of the heterogeneities of the RYE, the equilibrium law and Hooke's constitutive one are written in the following manner: divo{x) = 0, 'Vx e V, (11.2) s(x) = S(x):a(x), 'Vxe V, (11.3) where S(x), o(x) and t(x) are respectively the microscopic compliance, stress and strain tensors at a given point x of the RYE. The operator ':' indicates a double contraction. The macroscopic stress tensor (j is defined as the volume average <a>, of the microscopic one 0 over Vand can be shown to be equal to o-(e.g. Zaoui 1997):

a:: _!_ f o( x )dV = a- (ll.4) v.
The ensuing relation is also obtained from Eq. (II.3) by perfom1ing averaging operations on the entire volume of the R VE:

< 6 >, = s• :< (J >v' (TI.5)
where < o >v and < t >v are respectively the averages of the microscopic stress and strain tensors on the volume of the RYE and where §' is the compliance tensor of the equivalent effective medium (Fig. 11.1). In the case of a macroscopically isotropic material, the latter tensor may be written in tcnns of the macroscopic bu lk and shear modu li of the heterogeneous material, respectively denoted as K• and c• :

s• = ( 3K' f' J + ( 2G' f'
3
where JI and OC are the hydrostatic and the deviatoric operators and where n is the f9m'1h: order identity tensor. The effective Young modulus £' and Poisson ratio v' can simply be deduced from these macroscopic bulk and shear moduli by the intennediary of:

•. 9K" c• • 3K' -2c• /:. = and v = (II 7) 3K' + G' 6K" + 2c• . .
It is easy to show that the macroscopic strain tensor t can be expressed in tem1s of the phase strain averages, tr and tM as:

{I 1.8) " r
The latter relation may be rewritten as a function of the phase stress averages by the intennediary of I looke's law: s• a= s,, :<i + I;t,cs. -s.., ):<i., (11.9) ' where o, =<a >r denotes the volume average of o over a particu late phase V, .. In linear elasticity, this vol ume average (j, over phase r alld the macroscopic stress tensor a arc related by a fourth-order tensor B called stress "localization tensor": " (i,. = B,. :0.

(JI.I 0)

The effective compliance tensor can then be expressed in tenns of the compliance tensors of the various phases and of the localization tensors defined by Eq. (11. l 0) [START_REF] Hill | Elaslic properties of reinforced solids: some theoretical principles[END_REF]:

s• = S., + l::c,(S, -S..,) : B,.

(Ill I) ' lt appears from the latter equation that the knowledge of the localization tensors of inclusion phases I,. allows for determining the effective compliance tensor of the heterogeneous material. Jn the case of random heterogeneous materials, these localization tensors are very difficult to determine and approximations for JIJ, are required. Estimations of these tensors are provided by means of diverse homogenization techniques, some of the widely used ones being presented later. These micromechanical schemes thus permit to predict the effective compl iance tensor of the material. The previoos equation can be further simplified by introducing the compliance increment tensor IBI and inclusion compliance fluctuations tensor IHI , defined as: IHI=S• -S..,, lli, =S,-S.,, (11.12) so that the relation written in Eq. (11.9) results in: IHI = l::c,IHI, : B, .

(11.13) r

In order to derive the localization tensors B, , most of the homogenization methods use as a scaning point the Eshelby problem (t957) presented below.

4./.2 Presentation of the Eshelby problem

• 4. 1.2

.1 Presentation of the Eshelby theorem

In linear elasticity, the Eshelby theorem constitutes the comerstone of most of the effective medium theories (EMTs) studied in the present dissertation and is therefore briefly recalled below. A homogeneous elastic ell ipsoidal inclusion 1, occupying a domain Vi, is enclosed in an infinite li11ear elastic matrix medium, which stiffness tensor is C.,. This single inclusion is further subjected to a stress-free strain, or "eigenstrain" r.• that is un iform inside the inclusion and vanishes outside, undergoes a unifonn deformation ~~: r.U -I:" • t.

(11.15)

I - J • ,
where I::' is called the Eshclby tensor expressed as:

( :!:" ) = -CM,;., J[r .. (y -x)]dv

I IJJ.I 2 I/ , , v,
where the operator f q (y-x) is related to the Green function G;.,J'' (y -x} by: r ,, (y-x} = G,.,,,. (y-x }+ G;.,,,, (y-x}.

(IJ.16) (11.17)

The Eshelby tensor thus links the strain inside the homogeneous ell ipsoidal inclusion surrounded by an infinite linear elastic medium to its stress-free strain. The main limit of this 1iJ!!damental theorem is 1hat it is restricted to ellipsoidal inclusions in its original form [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF]. Tbe Eshelby tensor is generally quite difficult to detennine analytically but haS a very simple expression in the case of spherical homogeneous inclusion embedded in an infinite isotropic matrix [START_REF] Mura | Micromechanics of defects in solids[END_REF]):

t.: where the subscript 'Sph' refers to the fact that the inclus ion is spherica l. The Eshelby tensor of a spheroidal inclusion also presents the benefit of being analytically calculable (e.g.

Torquato 2001) and its expression for a spheroid with aspect ratio r, which revolution axis is oriented along the e3 ax is, is:

I [ 3a ' 9 J ] I: -I: - +ll-2v q " - 22 -4(1 -v) 2(a-I) 4(a-1) / ' :E - 1 [1 -2v+ (Ja-I) l' l-2v+ 3 ' J ] 33 -2(1-v) (a -1)
(a-1) q' I:,, =:E,, = 4(1~v)[ 2(:-1) -(t-2 v+ • 4(:-1) ) ql

I: = I: = 1 [ a ( 1-2v- 3 \,] (1120)
" " 4(1 -v) 2(a-1) (a -1) ) ' ' • I:2l = I,, 1 [-1+2v a +(l -2v+ 3 l q]. 2(1-v) (a-1)
2(a-1))

I = :E = 1 [ 1-2v -(a+ I) _!_l' l-2v-J(a + l)) ] " ss 2(1-v) (a-1) 2 (a-1) q ' I:. 14 = 1 [ a +( l-2v - 3 )q] 2(1-v) 2(a-I) 4(a-I) ' r 312 [ rJa -1-cosh-'(r)J. r <: I
where a = r ' and q = (a-1)

r m[-rJi -a+cos-'(r)] , rSI

(I-a)
Where v is the Poisson ratio of the infinite matrix medium. 1: 1 = 1+31'; Jr+ l+5A 1 +3B, !K, (11.23) where the average Eshelby tensor t: 1 for the spherica l inclusion with its homogeneous interphase will be referred to as inhomogeneous fahelby tensor in the ensuing.

Extension of the

( ) 1-p) {l-g,.,}(l+v,)-4g,"v'+2g"' F. = -2 l -2v I • ' • 3H , I B, 2p 3 H ,, 7 -5v 1 15H 2 (-8-7g,., +5(2+g 1 .,)v)-45 -45v 1 ' // 1 = 3( 1-11, }(I+ 2g,,, +{ I -4g"') v 1 ). H 2 = 3(1-v 1 ) ( 7 (I+ 4g,.,) + 5 ( 1 -8g,.,) v 1 ), H, 2 = ( 200v,2 -300v 1 -63p 2 + 175}(7 -Sv)g,.,2 + 3g," [ 75v(7 -9v 1 )-25v/ (l 3-15v)-7 ( 25+3p 2 -5( 5 +3p 2 }v )]+ {4-5v}(25v

4.J.2.3 Presentation of the equivalent homogeneous inclusion method

first of all, let us consider the situation in which no heterogeneous inclusion is inserted in the infinite linear elastic medium with compliance tensor denoted by S.,, which is subjected to a unifonn far-field stress G-. In such case, the stress field is everywhere equal to G-and the strain field is such as: (11.24)

A heterogeneous elastic inclusion T occupying a volume v, with elasticity tensor C, is now inserted in this virgin medium and we are interested in calculating the average &1 of the strai11

field over V 1 •
The microscopic strain tensor decomposes itself into two parts:

t 1 (x) = t 0 + t,'(x), ':/x, (ll.25)

where t,'(x) corresponds to the perturbation strain tensor caused by the insertion of the heterogeneity. In order to apply the Eshelby theorem, the heterogeneous inclusion is assumed to be substituted by a homogeneous fictive inclusion w ith a stiffness tensor denoted by C., that is subjected to an appropriate uniform eigenstrain e• , as illustrated in Fig. 11.3. The latter must be chosen in such a way that the same average stress and strain fields are obtained in the homogeneous fictive inclusion and in the heterogeneous one: The volume average ll, of the microscopic stress tensor o over the ell ipsoidal inclusion is found as:

(j, =C, :[ I+i::' :S..,:(C, -c.,)r' :S., :a" , which can also be expressed as :

; , =[K+C., :(1 -i:7): (s, -s.,)J 1 :o".

(Il.28) (l l.29)

The average stress tensor over the inclusion can then be written in a co01pact way [START_REF] Zheng | An explicit and un iversally appl icable estimate for the effective properties of multiphase composites which accounts for inclusion distribution[END_REF]):

(11.30)

with fl "' -C •(K-I:")

(1131) I - M • I '
where n:' is ca!Jed eigenstiffuess tensor of the single inclusion 1 embedded in the matrix [START_REF] Zheng | An explicit and un iversally appl icable estimate for the effective properties of multiphase composites which accounts for inclusion distribution[END_REF] and where S, and JHl, arc respectively the compliance and the compliance increment tensors of the inclusion, which can be expressed by means of Eqs.

(11.6) and (11.12). The eigcnstiffncss tensor n:' used by Zheng and Du (200 I) has the following mechanical interpretation: the average stress tensor (1 1 induced by the uniform eigenstrain t ' prescribed over the homogeneous fictive inclusion is linearly related to this cigenstrain by n7 . This tensor can be determined in the case of spheroidal inclusions by means of the expressions of the Eshelby tensor given in Eqs. (11.20). For more complicated cases, the Eshclby tensor usually has to be estimated numerically. The equivalent inclusion method is also particularly difficult to extend to tbe case of non ellipsoidal inclusions (e.g. [START_REF] Zheng | Irreducible structure, symmetry and average of Eshelby's tensor fields in isotropic elasticity[END_REF]. The ensuing derivations are therefore limited to the case of ellipsoidal inclusions. Owing to fundamental basis of micromechanics collected in the present section, the next one is ded icated to the presentation of EMTs that are of interest for predicting the properties of cement-based materials.

Presentation of so me classical matrix-inclusion type EMTs

This presentation does not intend to be exhaustive, since there exist many good monoi,>-raphs on the subject (e.g. [START_REF] Milton | The theory of composites[END_REF]Nemat-Nasser and Hori 1993;Torquato 200 1). It aims at providing sufficient theoretical basis for a correct understanding of the homogenization methods that will be applied to cemcntitious materials subsequently.

4.2.l Dilute approximation

The dilute model provides a simple expression for the localii.ation tensors defined in Eq.

(!LIO). It considers the ideal situation already illustrated on Fig. 11.3 where a single inclusion denoted as V,', having the same shape as phase r, is embedded in an infinite matrix medium subjected to a uniform far-field stress. The detennination of these di lute localization tensors is thus directly achieved by means of the equivalent homogeneous inclusion method described previously [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF][START_REF] Mura | Micromechanics of defects in solids[END_REF]. According to Eq. (ll.30), these tensors can be expressed as follows:

B;'" = [I + Q~ : 111,r'. (11.32) By injecting this expression of the localization tensor in Eq. ( 1Ll3), an estimation designated by H'" is derived for the compliance increment tensor:

lll"" = L H;"' (11.33) '

with H~' = c,Ill, : [I+ n; : IHI, r' .

(fl.34)

The validity range of the dilute estimation is very limited, since all the particles are required to be far enough from each other, so as to be regarded as isolated. In other words, the inclusioos are supposed to be dispersed in such a dilute manner that there is no interaction with the other inclusions and that they are only subjected to the imposed far-field stress <JM.

In addition, it is highlighted that if instead of imposing a unifonn stress at infinity, a far-field displacement is appl ied to the infinite medium, a different estimation is then derived for the compliance increment tensor (Hori and Nemat-Nasscr 1993). Cooscquently, the di lute model is considered as inconsistent by some authors (e.g. Berryman and Berge J 996). More consistent sche!Tles trying to reflect the interaction effects between inclusions are required.

Mori-Tanaka (MT) effective medium approximation may be used for this purpose.

MT approximation

This scheme was primarily proposed by Mori and Tanaka in 1973. Benveniste ( 1987) has proposed an elegant reformulation of this estimate, where it can be seen as an improved dilute model. More precisely, the stress applied at infinity is no longer <J~ but is replaced by the average o., of the microscopic stress <J inside the matrix so that:

(IJ.35}

wh ich allows for taking into accou~t the stress perturbation inside the matrix phase due to the presence of the other inclusions. The average of the microscopic stress tensnr (j on the entire volume of the RYE can be decomposed a.5 a function of microscopic stress tensor 011 each Phase:

(ll.36)

' r , the localization tensors estimated by the MT model are expressed as (Bcnveniste 1987):

BMT = BdU •[(1-"c )1 + "c !Rd 11 J 0 I' , .
L.J r L..J , , ' ' ' - (CT.37) which may also be expressed as follows:

B',,. = JB' "'

• [1-°" c n"' •Ill. • B "' 1 ] -' ,. r • ~~1 •1•1 i (11.38)
The effective compliance increment tensor pred icted by MT model is then expressed as:

l!lm = 111"' 1 : [I-~ !lj 11 : lll1'' r• (11.39)
The MT localization tensors arc also more realistic than the dilute ones, since they account in a certain manner for the interactions between the inclusions and the matrix. However, [START_REF] Berryman | Critique of two explicit schemes for estimating elastic properties of multiphase composites[END_REF] asserted by confronting the MT estimations for porous media to experimental data that MT shou ld not be used for inclusion volume fraction superior than 20-30 %. This discrepancy may be due to the fact that the MT localization tensors arc not able to accurately take into account the interactions between the inclusions themselves and to incorporate the influence of inclus ion spatial distribution.

Generalized self-com•iste111 scheme (GSCS)

The GSCS was originally developed by Christensen and Lo ( 1979) for estimating the effective elastic properties of a two-phase composite with a matrix-inclusion microstructurc.

This implicit homogenization method is based on the three-phase model illustrated on Fig.

11.4. A spherical core of radius r 1 represent ing the particle phase and a coating of thickness (r, -r;) occupied by the matrix arc immerged in the yet unknown effective medium so as to account for the interactions between inc.lusions. The radii r; and r e are chosen so that each coated-sphere has the same composition as the original two-phase material. The rather lengthy derivation of the GSCS is not presently recalled, since it may be found in many articles or manuscripts (e.g. [START_REF] Herve | N-layered inclusion-bused micromechanical modeling[END_REF]. The effective bulk modulus predicted by this method for a two-phase isotropic composite reads (e.g. Hashin and Monteiro 2002):

K • -K c, -u + • . 1/ ( K, -K") + 3( I -c 2 )!(3K" + 4G.,) (11.40)
and coincides with the bulk modulus estimated by the MT scheme. The effective shear modulus is obtained by solving a second order equation:

Ac• 2 +ao•+c=o, (fl.41)
where the expressions of the coefficients A, B and C can be found in Christensen and Lo ( 1979). The GSCS was extended to the case of composites with multicoa1ed inclusions by (-!erve and Zaoui (1993). The expressions of the shear modulus then become lengthy and 1he symbolic toolbox of MATLAB has been used to perform the compulations of 1he GSCS.

Figure Jl. 4: Two-1/ime11sioll(i/ represe11wtio11 i/111slrati11g tire three-phase model on which is basetl the GSCS.

The GSCS in its original form (Christensen and Lo 1979;Herve and Zaou i 1993) cannot be compulcd in 1he case of composites contai ning multi dispersed particulate phases, even though an empirical method was proposed by Huang ct al. ( 1994) for extend ing the GSCS to this case. Some attempts have also been proposed in literature to extend the GSCS to the case of non-spherical inlusions (Riccard i and Mo111heillet 1999) but they appear far too difficult to handle. In the next subsection, another type of models that is more suited for these limiti ng cases is presented.

Presentation of dou ble-inclu sion typ e models

The basic idea used in the homogenization methods presented below is to model the inclusion spatial distribution by an ellipsoidal cell, called double-inclusion, surrounding the inclusion (e.g. Hori and Nemat-Nasscr 1993; Zheng and Du 200 1). Its geometry depends o n how the inclusions are dispersed through the matrix and thus is representative of the spatial distributions of the particu late phases. As a pr<>of of the p<>pu larity of the double-inclusion concept, we have found at least five models using more o r less expl icitly this idea, such as the ones proposed by Kiister-Toksoz ( 1974), Nemat-Nasser and Hori ( l 993), Ponte-Castaneda and Wi llis (1995). S heu and Yi (2001) and [START_REF] Zheng | An explicit and un iversally appl icable estimate for the effective properties of multiphase composites which accounts for inclusion distribution[END_REF]. [START_REF] Hu | The connec1ions between the double•inclusion model and the Ponte Castaneda-Wi llis, Mori-Tanaka, and Kus1cr-Toksoz models[END_REF] revealed that st rong connections exist between the schemes developed by Kiister-Toksoz (1974), Ncmat-Nasser andllori ( 1993) and Ponte-Castaneda and Will is { 1995), even though '.heir derivation may radical ly differ. That's why the notion of double-inclusion type models is

•ntroduccd in this section to deal with the estimates ment io ned. For conciseness, only two of these models, namely the ones of Ncmat-Nasser and Hori ( 1993) and Zheng and Ou (200 I), are presented in details below.

4.3.J /llteractio11 Direct Derivative (TDD) model

The localization tensors 8~0 0 for the !DD estimate due to Zheng and Du (200 I) can be obtained easi ly by introducing the eigenstiffness tensors of all double-inclusions and by modifying s:" in Eq. (IJ.38) as follows:

8' 00 =Bd'' •[ •-" c H •B'''' •n"] I wi th n" =C • (ll-I:") r r • L..J 1 1 • 1 • D t ' Di M • Di ' ' (11.42)
where n ~, denotes the cigenstiffness tensor of the double-inclusion of a given phase i. The subscript 'D' is used here, since the inclusion distribution is involved in these eigenstiffi1ess tensors. Eqs. (11.38) and (U.42) reveal that MT and !DD estimates coincide whenever all inclusion phases v; have the same eigenst ifTness tensors n.~ = n:' as their surrounding double-inclusion. It means that every inclusion v; and its enclosing cell designated by v;

are similar in shape and coaxial in orientation.

Jn the case of a macroscopically isotropic material, the spatial distribution of inclusions is identical in all directions. A spherical double-inclusion is thus usually adopted to model this distribution (Ponte Castaneda and Willis I 995), since no particular direction in space is then privileged. lt implies in particu lar that TDD and MT models coincide for macroscopically isotropic heterogeneous materials containing on ly spherical inclusions. However, if the inclusions arc spheroids and are isotropically distributed inside the effective material, IDD and MT schemes do not coincide. The substitution of B,. in Eq. (U.13) by its expression in Eq. (11.42) yields the following TDD estimate provided that IHI'"' is commutative: (11.43) where IHl"'' and H~' arc expressed in Eqs. , respectively. The tensor H '" is generally commutative in the case of a macroscopical ly isotropic material. The effective stiffness tensor of the material can then be deduced from the knowl egde of this increment tensor:

ll[IDD = [ ll-~ ilf:;i : fi~, r•: 111"'',
(11.44)

The demonstration proposed here shows that the !DD scheme can be seen as an improved dilute model (see Fig. Another model derived by Hori and Ncmat-Nasscr ( 1993) using the concept of doubleinclusion is now presented. Its demonstration is a bit more sophisticated but is useful to understand the physical meaning of the double-inclusion.

Tire Hori and Nemat-Nasser doubfe-inclu~•ion model (DIM)

I General framework

The DIM developed by Hori and Nemat-Nasser (1993) basically consists in the following three-phase configuration: an ellipsoidal inclusion domain denoted as Vm that contains an ellipsoidal heterogeneity V 1 is immersed in an infinite homogeneous medium. The latter model is very flexible, since the shape and the orientation of the double-inclusion and of the heterogeneity as well as the elastic properties of the three phases may be chosen. The stiffness tensors of the inclusion, its coatiug and the embedding infinite medium composing the three- (11.26), the averages of the latter eigenstrains have be taken in such a way that the same average stress and strain fields arc obtained in the homogeneous fictive double-inclusion and in the heterogeneous one:

(1, ==Ct : tt == C 0 : \tt -£: ),

Get == C 0 : f.ct = C 0 : (tc1 -f.~t ), £1 = t 0 + E/ and Cc,= t 0 + ~c1 ', (
'-----------' c) + ..-.------- Co . () . . . . . . •-• . -. -.
. . : s~' : : . . :~ . . : .

. . . . . . .. . . . . -. .._ ______ ~ d) ..-------- .__ ______ _, e)
l'igure 11.6: Sclrematic oft/re origi11af tltree-pltase co11figuratio11 a) used/or the derivation of DIM 011d representation of tire different steps b), c), d) a11d e) employed for its derfration.

In the situation b), the volume average of the perturbation strain tensor t,' may be written as:

-, I ff .

CJ = v Co: r (x-y): t (y)dV,dV,, (II.49)

I v, t'°'
Which can be decomposed as:

I ( ) / '
v J Jc.: rcx -y)dV, :i:;dv,+-1 Jl Jc 0 : r(x-y)dV,.J :E;;,dv, -:

Jc 0 : r(x -y)dv,):r.~.,dv, +-1 J f C 0 : f(x -y):(e"(y)-£~, )dv,dv,

I t' y JI I I / I', ~f
where the operator r(x-y) is defined in Eq. (ll.17 The last double-integral accounts for the fluctuations between the uniform eigenstrain applied in the situations d) and e) and tl1e actual one in situation b) that is not necessarily uniform. lt is pointed out that the integrals between parenthesis correspond either to the Eshelby tensor of the inclusion or of the double-inclusion. Consequently, the expression of the average strain tensor inside the inclusion can be obtained with the aid of these tensors in a compact fonn:

(11.5 1)

with 'J =: f f C 0 : r(x -y) :{ t'(y) -ii~, )dV, dV,.

I V 1 Yo (11.52)
By taking advantage of tl1ese s imple relations:

-' (1 c, J-. c, -.

£ D1= --f. c1 + -t 1 , ' Co1 Co1
(n.53) (lf.54) the volume average of the strain tensor t c1

Eq. ( U.51):

over the inclusion coating can be deduced from

- 0 -• c ( 0 0) ( -• -• ) c r.c1 =t 0 + EDI : f.o + / l:. 01 -r. 1 : t i -to - / 'J. CD/ -c 1 CJ)I -<: 1 (11.55)
Hori and Nemat-Nasser ( 1993) then make the following approximation: the integral tenn in Eq. (lJ.55) is neglected to simplify the ensuing derivations. The relations proposed in Eqs.

01.51) and (fl.55) then simply become:

(IT.56) - 0 -• c ( 0 ") -• t c1 = t 0 + EDI : t c1+ 1 Em-I: 1 :t>r,, C 01 -c 1 with (11.57)
The effects of this approximation made by Hori and Nemat-Nasser ( 1993) will be carefully discussed later. The consistency conditions in Eqs. (11.46-48) then imply that (e.g. [START_REF] Hu | The connec1ions between the double•inclusion model and the Ponte Castaneda-Wi llis, Mori-Tanaka, and Kus1cr-Toksoz models[END_REF]:

(1 158) -(I:~ +(Cc, -C 0 r' :C 0 J: &~, -c ( E~ -r.~): t.&' = t 0 ,
(c 01 -c 1 ) -F' . [ • [IF' . .,,_, . ( 0 0 ) -I c, IFCJ J I . (IF' IF 0 )] (Il.65) UJ, -DI • -DI •"'1 • fi,-fi,,, + (cDl-cl) DI • DI-DI .
By combining these expressions of the eigenstrains witb Eqs. (11.56-57) and the consistency conditions written in Eqs. (11.46-48), the average strain and stress inside the inclusion and its coating can then be related to the applied far-field stress as:

t1 : [SO + :!::~/ : UJ Cl +I;~ : 1[J I]: o•' (11.66)
£c, =[so + :i;~, : u c1 + c ( r.~, -i:n : u, J : .,-,

CIJI -C 1 (II.67) -a -[n-"' 0 • U -"' 0 •U J •a- ' - :i.e.DI • CJ J.e. I • I • ' (11.68) nc1=[n -n~1 : Uc,-c, (n~-nn :u,] :o-. CIJI -c 1 (11 .69)
From the knowledge of these average strains and stresses inside the inclusion and its coating, the volume averages of the strain and stress tensors inside the double-inclusion can finally be computed:

(JI. 70) ooi =(1-n~, :(Uc 1 +c 1 U 1 )]: 0• .
(11.71)

Now that the average stra in and stress inside the double-inclusion are known, different estimates can be developed by making particular choices of the elastic properties of the coatings and of the shapes of the ellipsoids. In the case of a two-phase composite with a Particulate phase I with a volume fraction designated by c, , the double-inclusion is considered as a RVE so that c 01 = l. ln such occurrence, the effective compliance tensor of the material can be estimated by eliminating .,-in the last two equations:

(II. 72)

where the tensors § 0 and I:~, may be chosen in diverse ways [START_REF] Hu | The connec1ions between the double•inclusion model and the Ponte Castaneda-Wi llis, Mori-Tanaka, and Kus1cr-Toksoz models[END_REF]. Given the different possibilities offered by DIM, we only focus below on its application to composites having matrix-inclusion morphology. The simplest choice to obtain an estimation for such materials then consists in letting the reference infinite medium in the three-phase configuration represented in Fig. 11.7 be the matrix of the material, i.e. C 0 = C., .

---------,

Applica1ion of DIM to two-phase composites

In the case of a two-phase composite, let the coating of the inclusion be filled by the matrix of the material so that Cc, =C., and c 01 = l. Noticing that !Hl~' 1 =c 1 !F: for a two-phase composite, Eqs. (11.63) and (U.70-7 1) respectively reduce to:

L\&' = r: and &~, = 0 ' --(S + "'" •IHI""]• - t - M L. l)f • I • (f ' (l L73) (JJ.74) <J = [ n-n~;, : IEl1"] ()' -. (l l.75}
It is interesting to observe from the last equation that, if the modified far-field stress written in Eq. (11.45) is applied instead of O'-, the average of the stress tensor inside the doubleinclusion is then simply equal to a-. By combining the last two equations, the following estimation can be obtained for the effective compliance tensor:

<> WM =[S + "' M •Jll"'' ]•[ll-" " .ur•"' ]-1 i::I f.t L.DI ' I ' :."I)/ ' " 1 >
which is recast in the simplified fonn:

lll m" = IHI' .' u : [ 1-n~, : 1HI~1 J' .

(I l.76)

(I I 77)
The expression of the DIM estimate for the effective increment compliance tensor is very close to the one given by the IDD estimate in Eq. (rl.43). More precisely, the two estimates coiocide whenever 1!11' is invert ible, which is true in the case of a macroscopica lly isotropic composite. [START_REF] Hu | The connec1ions between the double•inclusion model and the Ponte Castaneda-Wi llis, Mori-Tanaka, and Kus1cr-Toksoz models[END_REF] have investigated DIM in some special cases and shown theoretical connections with other micro mechan ica l schemes. They have already demonstrated that, in the case of two-phase composites, DIM leads to the same results as the Kiister-Toksoz (KT), and Ponte-Castaf'ieda and W illis (PCW) models but, to our k11owledge, this connection between OJM and the !OD model has no t yet been evidenced. It also appears by comparing Eqs. (ll.38) and (ll.77) that, if the incl usion and its enclosi ng cell have the same shape aod orientation, D IM yields exactly the same predictions as the MT mode l, as illustrated in Fig. 11.7 [START_REF] Hu | The connec1ions between the double•inclusion model and the Ponte Castaneda-Wi llis, Mori-Tanaka, and Kus1cr-Toksoz models[END_REF]. Hori and Ncmat-Nasser (1993) general ized their DIM to composites conta ining inclusions with n layers. In the next subsection, the ex tension of DIM to inclusions with one layer is treated. The latter representation is of great interest for mortars for example, where an ITZ forms between the sand aggregates and the cement paste. In order to derive an estimate for this configuration, it is necessary to compute the averages of the strain and stress fields inside the domain occupied by the double-inclusion and its matrix coating that may be decomposed as follows:

App/ica1ion o.fDJM to coated inclusions

t=(l-cDl)tM+CDltDt and o=(l -cDl )"OM+Co10m, (11.78) where (,,, and 001 arc given by Eqs. and the averages of the strain £., and stress (j., inside tl1c matrix outer layer are still to be determined. Hori and Nemat-Nasscr ( 1993) showed that in the case of layered inclusions:

- ( M "' )•l/ c, -• Cc1 -• ) &., = &o + :ECOi -I:l)I • --t , + F'CI > 1-c 01 1-cDI ( 1179
)
where I:~Dt designates the Eshelby tensor of the coating of the double-inclusion and where

;:; and £~, are provided by Eq. (I I.63). The coating of the double-inclusion havi11g the same shape and orientation of the double-inclusion, the latter equation reduces to EM = 1: 0 • Furthermore, the coati ng of the double-inclusion having the same properties as the unbounded matrix medium, the average stress inside the coating of the double-inclusion simply writes as: Th is discrepancy is symptomatic of the fact that DTM as proposed by Hori and Ncmat-Nasser ( J 993) fails to accurately account for the interactions between the inclu sion and its coating. [START_REF] Hu | The connec1ions between the double•inclusion model and the Ponte Castaneda-Wi llis, Mori-Tanaka, and Kus1cr-Toksoz models[END_REF] furthennore noticed other shortcomings of DIM for the ca5es in which the infinite reference medium is chosen as the yet unknown effective material, i.e. C 0 = c•.

The only approximation used for the derivation of DIM (Hori and Nemat-Nasser 1993) consists in disregarding the integral term in Eq. (Il.52) and is thus likely to be the cause to these anomalies. This integral term neglected practically means that the eigenstrain is supposed unifonn inside the coating of the inclusion. Th is approximation of a uniform eigcnstrnin field inside the coating is strong, whenever the coating and tbe infinite reference medium do not have the same elastic properties, and may lead to significant shortcomings such as the ones described on Fig. ll.8. Nevertheless, this integral tenn vanishes whenever the eigenstrain is uniform inside the coating of the inclusion. This practically means that, for the particu lar case represented in Fig. 11.7 for which the coating has the same stiffness tensor as the infinite matrix, the DIM of Hori and Necnat-Nasser is exact, since the eigcnstrain is equal to zero everywhere inside the coating of the inclusion.

Other double-inc/11sion type models

l The Kuster-Toksoz (1974) model

The latter model is popular among geophysicians. [START_REF] Berryman | Critique of two explicit schemes for estimating elastic properties of multiphase composites[END_REF] have confronted this method with different experimental data and concluded that it should not be used for volume fractions of inclusions higher than 30 %. Analogously to the Hori and Nemat-Nasser DIM, the starting point of the Kuster-Tokstiz (KT) model consists in embedding the different inclusions modelling the P particulate phases inside a matrix coating with ellipsoidal form and in tum in inserting this pattern, designated as composite inclusion (e.g. [START_REF] Hu | The connec1ions between the double•inclusion model and the Ponte Castaneda-Wi llis, Mori-Tanaka, and Kus1cr-Toksoz models[END_REF], inside an infinite matrix medium. The external geometry of this composite inclusion depends on the spatial distribution of the particulate phases. In the case of two-phase composites, the KT model is based on the same pattern as DIM (Fig. ll.7 left) and, as shown by [START_REF] Hu | The connec1ions between the double•inclusion model and the Ponte Castaneda-Wi llis, Mori-Tanaka, and Kus1cr-Toksoz models[END_REF], the two estimates coincide.

The KT model was developed by comparing two situations illustrated in Fig. Il.9 and asserting that the average strain field is identical inside the composite inclusion represented at the top of Fig. 11.9 and inside the homogeneous elTective inclusion depicted at the bottom of the figure (e.g. [START_REF] Hu | The connec1ions between the double•inclusion model and the Ponte Castaneda-Wi llis, Mori-Tanaka, and Kus1cr-Toksoz models[END_REF]. Recently, [START_REF] Shen | An effective inclusion model for effective moduli of heterogeneous materials with ellipsoidal inhomogeneities[END_REF] derived a11 estimate leading to the same results as KT by using the same philosophy. The only difference is that they assume that the elastic strain energy changes in the infinite matrix due to the respective insertion of the two previous types of double-inclusion are identical in both c~ses (Fig. 11.9).

The demonstration proposed by Shen and Yi (200 I) seems more rigorous than the KT original one and is briefly recalled below.

:1'c elastic strain energy perturbations of the infinite matrix, caused respectively by the insertion of the composite inclusion and of the equivalent homogeneous inclusion (Fig. 11.9), are denoted tJ.U m • = and ti.U' . It is assumed that these two elastic strain energy changes are identical [START_REF] Shen | An effective inclusion model for effective moduli of heterogeneous materials with ellipsoidal inhomogeneities[END_REF]. Tbe quantity ti.U' can be determ ined from the results of the Eshelby problem ( 1957):

(11.83)

where the subscript DJ designates the homogeneous effective inclusion (Fig. 11.9) that 1s analogous to a double-i nclusion. Tbe elastic strain energy tJ.U mkro can be directly related to the average eigenstrains over the different particulate phases, denoted as & ; (re (t,PJ) (e.g.

Mura J 987):

1

~ (f-•) llU 111 ;,.r'f) =-V 01 a : L.J Er 1 2 ,,, (
--------------------' t G- ,-------------------~ 1 M I I I I I I I I I I I I I ~------------------- Figure ll
.9: Schematic illustrating the effective inclusion concept employed by Kmter-Toksiiz (1974) and Shen a11d Yi (2001).

An estimation for the effective stiffness censor can then be derived by assuming the equality between the last two equations (Shen and Yi 200 I):

• •hi • • •I-I • • ilil , [ ' ' I J C = i-l S.
-[n-llJIJ,/ .n MJ-' •(IH!"'' •C .~ .. +IH!d" •""') •S - lf1I ' ~' DI • • Al • J.. DI •~' OJ • M' : [ J -Jlidl/ : n::, J I : JH[di/.
The latter combinations are valid provided that lll"' is commutative with C" , which is true in the case of a macroscopically isotropic composite. The last equation coincides witb the !DD fonnu la in Eq. (11.43), whenever al l double-inclusions have an identical shape.

The Ponte-Castaneda and Willis (PCW) model

The demonstration of the PCW model ( I 995) based on the Hashin and Shtrikman (I 962) variational principles is not recalled here, since their rigorous derivation is quite lengthy and requires notions of statistics. Nonetheless, the comparison of the previous double-inclusion type models with the PCW approach is instructive because it enables us to understand better the double inclusion role. In fact, it geometrically characterizes the cumulative density limction for the distribution of inclusions introduced by Ponte- [START_REF] Castaneda | The effect of spatia l distribution on the effective behavior of composite materials and cracked media[END_REF]. It is defined from a conditional probability density function that represents the probability density for finding an inclusion centred at one point given that there is an inclusion centred at another point x. Imagine that we random ly toss a segment from the centre x of length R with a specified orientation and count the fract ion of times the end point coincides with the centre of another inclusion. With the above interpretation, it becomes obvious that for macroscopically isotropic composites, such as cement-based materials, this probability is the same in all the directions. Consequently, the double-inclusion will be taken as spherical for these materials.

According to [START_REF] Zheng | An explicit and un iversally appl icable estimate for the effective properties of multiphase composites which accounts for inclusion distribution[END_REF], explicit expressions are difficult to obtain with the PCW scheme if the different particulate phases do not have the same distribution. When one cumulative density function is enough for characterizing the distribution of all the inclusions, the PCW estimate coincides with the formula of the KT model and with the predictions of DIM and 100. Another consequence of the similarities of these double-inclusion type models ~ith PCW is that these estimates may be interpreted as rigorous bounds, provided that the inclusion distribution is accurately known.

In The effective elastic properties of the multiphase composite are then computed using the energy balance employed by [START_REF] Shen | An effective inclusion model for effective moduli of heterogeneous materials with ellipsoidal inhomogeneities[END_REF]. To achieve the com~utation of the effective stiffness of the double-inclusions, eigenstrain fields are introduced inside the inclusion domain so that each inclusion has the same elastic properties as its coating.

- -"' -• £'::1: : t,, (I f. 87)

----_t_ ~.._ -------___ t_ ~.._ ---- c@ CM C M ~@J' a) b) -----_ J_ ~-----
, ,

where the expression of E:' is given in Eq. (11.21 ). By neglecting the interactions between the double-inclusions, the average strain tensor inside the inclusion corresponding to a given particulate phaser takes the form:

-W"' S --, (IL.91)

The average perturbation strain tensor inside the inclusion "t; can then be deduced from Eq.

(11.87). Using Eqs. (lf.53-54) and (II. 91 ), the mean perturbation strain tensor inside the inclusion coating i c; is obtained as:

(IJ.92) and the stiffness tensor C 0 , can be predicted by Eq. (ll.95). Tn the ensuing, this estimation will be referred to as the modified DIM.

Accuracy of EMT s

The actua l capacity of pred ictions of the different schemes in literanirc has been a subject of extensive debate [START_REF] Christensen | A critical evaluation for a class of micro-mechanics models[END_REF]). The precise knowledge of the accuracy of each scheme is a crucial issue of micromcehanics, since it would allow for determin ing the cases for which an EMT can be preferentially employed or not.

4.4.J Review on the accuracy of £MTs

Testing the accuracy of an EMT is not a trivial matter, since their predict ions highly depend on the way the microstructure is represented. The two best methods for perfom1ing such tests consist in comparing estimations of micromeehanical schemes with FE simulations of simple mierogeometries (Segurado and Llorca 2002;Zheng and Du 200 I) or in confronting them with experimental data on materials having a simple and well-defined microstructure (e.g. [START_REF] Berryman | Critique of two explicit schemes for estimating elastic properties of multiphase composites[END_REF]. Cement-based materials with their complex microstructure are not su ited for perform ing such experimental comparisons.

For instance, Segurado and Llorca (2002) performed 30 numerica l simulations of materials reinforced by rigid spheres or weakened by spherical voids. The simulated materials contain different volume fractions (up 10 50 %) of spherical inclusions that are monodispersc and random ly di stributed . They concluded that GSCS provided estimations close to the FE results en for 50 % of spheres, while the MT estimate visibly differs from the elastic and shear :,duli obtained numerically for volume fractions higher than 30 %. Another instructive coinparative study has been carried out by [START_REF] Zheng | An explicit and un iversally appl icable estimate for the effective properties of multiphase composites which accounts for inclusion distribution[END_REF] on a 20 matrix weakened by jsOtropically distributed circular voids. On the contrary to Segurado and Llorca (2002), they ooticed that the MT and double-inclusion type shemes provide better estimations than the oscs.

Betr)'lnan and Berge ( 1996) have compared both KT and MT estimates with experimental

•i a 00 porous materials, such as siutere<l glass beads, and have noticed that the prediction of rhc KT or MT estimate with spherical inclusions was only in acceptable agreement with the citperimcntnl measurements for volume fractions of pores less than 20 %. [START_REF] Berryman | Critique of two explicit schemes for estimating elastic properties of multiphase composites[END_REF] managed to obtain some slight improvement by using needled-shaped inclusions but concluded that KT or MT schemes should not be used wheu the inclusion volume fraction is greater than 20-30 %. These results seem to be coherent with the analysis of Segurado and Llorca (2002). These errors may be due to the fact that over a certain volume fraction of inclusions, MT and double-inclusion type estimates do not manage to accur-dtely take into account the interactions between inclusions. It appears however intricate from these diverse results found in literature to affirm which one of the presently reviewed models is the most 1ecurate. Therefore, it is useful to further compare these estimations with recent numerical simulations performed on cementitious materials.

Comparison with 1111merical simulations on cement-based materials

Owing to progress in the computer modelling of cement microstructure, many threedimensional numerical simulations have been carried out for computing the macroscopic elastic properties of cement-based materials and constitute good comparison points. By confronting the predictions of the different EMTs presented previously with different types of simulations, we try to determine which scheme is the best suited for three well-defined cases:

• a concrete composed of a mortar matrix and aggregates modeled as spherical inclusions [START_REF] Wriggers | Mesoscale models for concrete: Homogenisation and damage behaviour[END_REF]; a mortar consisting in a cement matrix rigidified by spherical coated inclusions representing the sand grains surrounding by ITZ (Garboczi and Berryman 200 I);

• a porous medium that can be assimi lated to a cement paste comprising a rigid solid phase and spheroidal porous inclusions [START_REF] Roberts | Computation of the linear elastic properties of random porous materials with a wide variety of microstructurc[END_REF] .

. lhe input elastic parameters for the difTerent simulations and micromechanical estimations ~gathered in Appendix II.A. The three materials treated arc simulated as macroscopically lllOtropic and the inclusions are supposed to be isotropically distributed. Therefore, a spherical double-inclusion is adopted for the computations of the double-inclusion type schemes. [START_REF] Wriggers | Mesoscale models for concrete: Homogenisation and damage behaviour[END_REF] Wriggers and Moftah ( 2006) perfonned advanced FE simulations on geometrical models of concrete generated with the Monte-Carlo's method to compute its mechanical properties. The aggregates are modeled as spheres of different sizes, so as to accurately reproduce the material granu lometry. )) lead exactly to the same results for a macroscopically isotropic material. A frequent probl em with using spherical inclusions is that too many EMTs give exactly the same result even if certain ones may be much more evolved and efficient. The case of non-spherical inclusions will therefore be treated later. The mortar matrix considered as a homogeneous phase in the [START_REF] Wriggers | Mesoscale models for concrete: Homogenisation and damage behaviour[END_REF] is in fact a heterogeneous material that bas been the subject of advanced numerical simulations by Garboczi and Berryman (200 I). (Roberts 11f1d Garboczi 200<1) for an elastic body weakened by oblate spheroidal voids with aspect ratio r = 0.25, as 11f1111ction of porosity.

I Comparison with the simulations of

Comparison with the simulations ofGarboczi and

Conclusion of chapt er 4

Different analytical homogeni1~ition schemes from literature have been presented and examined in a critical way. The respective benefits and drawbacks of these methods are now recapitulated on the basis of well-defined criteria. In order to choose the best suited scheme for the estimati ons of the elastic properties of cement-based materials, the following criteria are adopted:

(i) simplicity of computation and versati lity;

(ii) respect of the rigorous bounds;

(i ii) accuracy. By versatil ity, it is intended that we assess if the model can be extended to the cases of multi phase composites, non-spherical inclusions or diverse distributions of inclusion. The IDD scheme is the most versati le one, si nce it is the only model that is valid for multiphase composites with various inclusion geometries and that can properly take into account the influence of various inclusion distributions. It should furthermore be theoretically applicable for non-elli psoida l inclusions, according to [START_REF] Zheng | An explicit and un iversally appl icable estimate for the effective properties of multiphase composites which accounts for inclusion distribution[END_REF]. Jn addition, it has a simple and explicit structure, as well as MT and other double-inclusion type models. Many authors (e.g. [START_REF] Berryman | Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media[END_REF]) argue that a consistent EMT shou ld respect rigorous bounds such as the Hash in-Shtrikman (HS) or the Willis bounds tha1 have been proved to be S Norris (J 989) found a simple three-phase example with disk-shaped particles for rou .

. h the effective thermal conductivity estimated by MT violates HS bounds. It may thus be ~lous to apply MT to multiphase composites containing more than two phases.

verscly, Ponte- [START_REF] Castaneda | The effect of spatia l distribution on the effective behavior of composite materials and cracked media[END_REF] proved that their estimate may be interpreted i:o;:gorous bounds. The other double-inclusion type schemes that coincide in most cases with the pCW model are also likely to respect rigorous bounds. c confrontation of the different schemes with diverse numerical simulations has revealed t the GSCS is generally more accurate than MT and double-inclusion type estimates in the case of spherical inclusions. ln addition, the GSCS and modified DJM are the only two 1110 dds that do not lead to unexpected results for the case of coated spherical inclusions that is of practical importance for cement-based materials. However, the double-inclusion type nioclels provide the closest estimations to the FE results of [START_REF] Roberts | Computation of the linear elastic properties of random porous materials with a wide variety of microstructurc[END_REF] for the important case of an elastic body weakened by spheroidal voids. The accuracy of an EMT is strongly conditioned by the way in which the phase interactions are accounted for. For instance. the di lute scheme does not take into consideration any interaction between inclusions and frequently appears to be inaccurate in the present study. As already mentioned, MT and double-inclusion type estimates can also be seen as dilute models improved by modifying the far-field stress (see Eq. (11.45)). They on ly incorporate indirectly the interaction bctwc~n inclusions and their immediately surrounding region. The GSCS integrates more precisely these interactions by embedding matrix and inclusions in the yet unknown effective infin ite medium. Table Tl. I aims at weighing the pros and the cons of the various theories in an infonnal way. Even though, the GSCS and mod ifed DIM are more appropriate for materi als with coated spherica l inclusions, the IDD scheme seems to present the best compromise otherwise. It i~ emphasized that this approach is more accessib le and versatile than other double-inclusioll type models (Ponte-Castaneda and Willis 1995). The IDD estimate shows in particular i t~ eflicicncy in the cases of macroscopically isotropic materials containing non-spheri cal inclusions and therefore constitutes a suitable tool for the next chapter dedicated to the influence of inclusion shape on the elastic effective properties of cement pastes.

Micromechanical

INFLUENCE OF IN CLUSION S.H APE ON T HE LINEAR ELASTIC EFFECTIVE PROPERTl ES OF HCP (STORA ET AL. 2006a)

In most micromechanical models applied to HCP in literature, all inclusion phases arc appruximated by spheres for simp licity, though cement paste micrographs indicate that none 'of these phases is really spherical. The present chapter assesses how the predictions of the effective linear elastic properties of HCP change with the inclusion geometry. In particular, i1

is usefu l to detect the cases, for wh ich these properti es vary significantly and for which more rea list ic geometric fonns shou ld be applied . The determination of the Eshclby tensor :r.~1 ol the inclusion phases is usuall y qui te complicated for non-spherical inclusions and it general!> has to be estimated numerically except for some s imple cases. In particu lar, the Eshelb> ::~s why all inclusion phases are approximated by spheroids with different aspect ratios in order to scrutinize the influence of inclusion shapes on the linear elastic effective propenies of c:csnent pastes. The IDD scheme that has been previously identified to be suited for iiccounting for the effects of non-spherical morphologies of inclus ions is chosen for computing these macroscopic propenies.

The chapter is organized as follows. First, a dimensionless morphological parameter is introdm;ed to characterize the inclusion shape. Then, the cases for which this parameter actua lly plays a significant role on HCP elastic macroscopic properties are investigated by successively modeling each particulate phase as spheroids, which shape ranges from needlelike to disk-like. Finally, an attempt is made to approximate the morphological parameter on the basis of experimental results. The spheroids representing the particulate phases in the micromechanica l approach are then detcnnincd through this parameter. The estimations thus obiaincd for the effective Young modulus are compared with the results respectively predicted by a homogenization model with spherical inclusions and by FE simu lations.

Definition of a shape parameter for the inclusions

In the micromcchanical schemes presented in tbe previous chapter, none of the basic input parameters, i.e. the phase volume fractions and elastic properties, takes into consideration the morphology of the particulate phases. The aim of this section is to introduce a relevant parameter approximating the real shapes of inclusions.

We propose to analyze the geometry of an inclusion I by considering the ratio of its volume v, over surface area A 1 . For example, this ratio for an arbitrary spheroid can be wrillen as:

!:J_= 1(b)a, (11.97) A, a
where a is the radius of the circular base of the spheroid and b is either the minor axis for oblate spheroids or the major one for prolate ones. The aspect ratio of the spheroid is designated by r 1 = b I a. The most natural way to nonnal ize V 1 I A 1 is to divide this ratio by one third of the radius R,. of the equ ivalent sphere having the same volume V,:

3V 1• ~, = I ~= --L with "" . (11.98) A,R,.
l q,,,,, = ~_,,, = 0 This shape or morphological parameter c; bas tl1e interesting property that it varies between 0 alld I. This comes from the fact that, among all shapes, the sphere is the one that maximizes the inner volume for a given surface area according to an isoperimetric theorem (P61ya and Szegii 1951 ). The less spherical the inclusion is the lower c; becomes. Therefore the value of c; allows for quantifying the asphericity of a panicle. 1n order to assess the reliability of<;, it is interesting to study if oblate and prolate spheroidal inclusions with the same shape parameter and volume fraction actually have simi lar effects on macroscopic elastic moduli. Hence, computations are performed in which an arbitrary particulate phase r of HCP is approximated respectively by prolate and oblate spheroids, while the other inclusion phases are represented by spheres. The evolution of tbe effective elastic moduli is examined with these two types of spheroids as their parameter <; decreases from l to 0. According to this equation, the variation of the aspect ratio r 1 , and consequently of the shape parameter c;, of the spheroids can be performed in two different ways:

• The maximum axis of the spheroid (= a for an oblate one and b for a prolate one) varies and c,. is kept constant, as il lustrated on Fig. 11.14 (bottom left).

• The maximum axis of the spheroid remains constant and c,. changes, as illustrated on For the present l!PPli~ation, the first possibility seems more appropriate because it keeps a constant volume fraction. But it presents the fo llowing deficiency: the spheroid maximum axis appearing on the equation just above has to augment, as i; decreases, and in particu lar tends to infinity, as <; goes to 0. The second possibi lity is more adapted for inclusion forms tend ing to cracks, but we prefer the first option, since the inclusion phases present inside the The effective elastic properties of HCP arc estimated by means of the IDD estimate. The cement pastes are considered as macroscopically isotropic. Consequently, for each particulate phase of HCP, the correspond ing double-inclusion should be taken as spherical. Nevertheless, for a spherical distribution of randomly oriented spheroids, highl y-concentrated inclusions may overlap. The maximum volume fraction for whi ch these spheroidal inclusions do not overlap is calculated in Append ix H.B. For higher volume fractions, a non-spherical doubleinclusion must be adopted and a volume average over all possible orientations must be perfom1ed. Once its exact shape has been detennined for each particulate phase r, the eigenstiffness tensors of tl1e various double-inclusions denoted by Q~, enclosing the inclus ions V 1 " are calculated from Eq. (Jl.30). The IDD estimate of the compliance increment The input data required for the micromechanical estimations are the volume fractions c, and CM of each phase that can be found respectively on Table 11.2 for the six HCP samples and the bulk and shear moduli, respectively denoted as K1 and G;, reading as follows:

K .= E, andG, E, withie{AFt,AFm,CP, CH, CSH, UC} , (ll.102) ' 3(1-2v,) 2(1+ v 1 )
where the Young modulus E, and the Poisson ratio v, of each phases arc given in Table 1.3.

for sound and degraded HCP. The present input values are the same as those employed by [START_REF] Kamali | Comportement et simulation des materiaux cimentaires en environnements agrcssifs: lixiviation et temperature[END_REF] have a much more pronounce<! impact. These tendencies are even more marked for high volume fractions of initial capil lary pores. In this condition, the parameter seems not to be well appropriate except for values higher than 0.80. But it should be underlined that this example corresponds to an extreme case, since the contrast between the elastic properties of matrix and inclusion is infinite . dotted lines: w/c = 0.25(cCP = 11%) (Storti et al. 2006a). The subscript Spit means that , the modulus has been estimated using the .spherical particle approximation.

T he second example illustrates the case of finite contrast by focusing on unhydrated clinker inclusions, which are stiffer than the CSH matrix. Their volume Ii-actions in the three CEM I pastes arc cue= 10.04 %, 14.46 % and 28 % respectively. The curves obtained when<; varies from O to J for both types of spheroids are presented on Fig. 11. 17; they are relatively close and, for <; higher than 0.5, the Young moduli estimated with oblate and prolate spheroids are almost confounded. For lower va lues, oblate spheroids produce a stronger stiffening effect.

Th is result is not surprising hi view of the evolution of the spheroids with i;: as the latter goes to 0, oblate spheroids tend to infinite disks whereas prolate ones tend to infinite needles, as may be observed on Fig. II.I 4. Both types of spheroids thus tend to radically different geometries. For this particular example,<; turns out to be relevant on a limited range, where it is usefu l for approaching re;1J inclusion shapes. For inclusions whose geometrical shape is very far from spherical, <; is small and other parameters are required to precisely characterize the effects of morphology on elasti c moduli . This remark applies particularly when the contrnst between matrix and inclusion elastic properties 1s high, as in the first example presented above. dolled lines: w/c = 0.25 (Stora et al. 2006a).

1.2s ~.-.. --~---~---~---~--~ " •• .....

S.2 Significance of the shape param eter for H CP elastic properties

The purpose of this section is to Qetermine the particulate phases whose shape parameter may become significant for the estimation of HCP elastic properties. We now 1J1ake the shape parameter vary individually for each inclusion phase, while<; is kept fixed and equal to I for the other particulate phases. The dependence of the elastic properties on <; is thus investigated successively for all inclusion phases composing HCP in both sound and degraded pastes defined in Table U IJ.17).

1.2 .... , .. , .. . • . .. . . . (Stora et al. 2006a).

Unhydra1ed ••••• ••••• Cl' k -_ _ _ _ _ _ _ _ ....
We now focus on the degraded state. On Fig. 11.19 are presented the evolutions of the Young modulus as a function of<; for the three CEM I pastes when CH, A Ft and A Fm are degraded successively. For <; higher than 0.1, the influence of dissolved ettringite and AFm shapes remains negligible because of their scarce volume fractions (see Table !1.2). Conversely, the influence of leached portlandite morphology increase~ considerably compared to the sound state. This is partly due to its high volume fract ions(> 15 %) but it should also be emphasized that its effects are even more sign ificant than the initia l capi llary pores one, although some of thei r concentrations arc higher (CC/' = 11 %, 18.3 % and 23.7 %). As may be observed on Fig. (Stora et al. 2006a).

------_• , : _••••••• ... ........ ..... •• ... .. ....
Because of the straightforward calculation of the Eshelby tensor for spherical inclusions (see Eq. (!1.1 8)), the spherical assumption is easy to use provided tl1at morphology has a negligible effect. In this section, the influence of inclusion shapes 011 the effectiv. e mechanical prope1ties is shown to highly depend on the contrast between matrix and inclllsion elastic parameters, on the volume fractions and on the flatteni11g of the particles. In particular, the morphology of a completely dissolved phase such as portlandite can severely affect the effective properties of leached HCP. Consequently, it is quite natural to wonder if the sphere really constitutes a reasonable approximation. The next section assesses the validity range of this spherical shape assul\\plion and investigates the possibil ity of improving the micromechan ical pred ictions by integrating some significant information about the shapes of HCP particulate phases.

S.3 Approximation of the par ticulate phases by spheroidal inclusions

The 11

.20 as a function of cap illary porosity and are compared to the FE results of [START_REF] Kamali | Comportement et simulation des materiaux cimentaires en environnements agrcssifs: lixiviation et temperature[END_REF].

For all sound pastes, the results of the micromechanical model are in good agreement wi th the FE values, si nce the deviation compared to numerical simu lations is always lower than 12 %.

In view of the fact that FE simu lat ions arc more difficult to perfonn than the present homogenization method, the latter provides an efficient analytical tool for the prediction of sound I ICP elastic properties. Fig11re JJ.20: E110/11tio11s of the effective Yo1111g moduli estimated respectively by the JDD niode/ witfl .;pflel'ict1/ m1d spfl eroidal i11c/usio11s and by FE simulations [START_REF] Kamali | Comportement et simulation des materiaux cimentaires en environnements agrcssifs: lixiviation et temperature[END_REF] for both a) so1111d and b) partially degraded HCP as a function of capillary porosity vo/11111e fraction (Stora et al. 2006a).

Concerning the partially degraded pastes, the FE simulations of [START_REF] Kamali | Comportement et simulation des materiaux cimentaires en environnements agrcssifs: lixiviation et temperature[END_REF] assume the complete leaching of CH and alum inates but do not incorporate the degradation of CSH; we have then adopted the same leaching scenario to be able to compare the different results. The difference between the results in the sound and the degraded pastes suggest a few general comments. The detrimental effect of leaching on Young modulus is sign ificant, though the elastic properties of CSH are kept constant during the degradation process. Karnali (2003) Predicted a reduction of 58 % for the CEM I 42.5 paste with w/c = 0.50 and obtained good agreement with experimental results (Cardc 1996), where total dissolution of CH caused a reduction of 63 % in the Young modu lus of rhis cement paste. The numerical resu lts of [START_REF] Kamali | Comportement et simulation des materiaux cimentaires en environnements agrcssifs: lixiviation et temperature[END_REF] agree quite well with their measures considering the fact that the paste appears more degraded in the experiments than in the simulations. The decrease of the Young modulus is lower for CEM ll/A pastes, varying between 25 % and 40 %. The blended cement pastes sbow better mechanical properties after degradation process than CEM I because of their smaller content of portlandite. This phase appears then to be predominant for leaching problems.

For the leached HCP, the IDD estimations computed with spherical inclusions diverge significant ly from the numerical results for volume fractions of capillary porosity higher than 35 % . It is striking to notice that the predictions follow the same trend as in [START_REF] Berryman | Critique of two explicit schemes for estimating elastic properties of multiphase composites[END_REF] and in Segurado and Llorca (2002): the more porous the material becomes, the more the homogeni7.ation models overestimates the elastic properties.

Though volume fractions and mechanical data of the main phases composi ng IJCP can be determined in a relatively precise way, less is known concerning their morphology. Based on relevant experimental results, an attempt to approach the components real shapes by means of a morphological parameter is made. Although the breakthrough of three-dimensional analysis techniques provides some interesting data, the introduction of realistic particle shapes in homogen ization models is still a difficult task. Due to the difficulty of computing the Eshelby problem in the case of non-ellipsoidal shapes, it is generally impossible to incorporate the phase real morphologies directly in micromechanical methods and approximations of these shapes by simpler geometrical forms are necessary. The dilemma is to integrate the most realistic shapes possible and at the same time conserve a calculable model. An analytical solution is proposed in the present homogen ization approach by introducing sph~roids adequately chosen with the help of the previously mentioned morphological parameter.

Therefore this section aim s at finding some approximate values for i; on the basis of experimental observations.

As an example, the morphological parameter i; of Portland cement particles is calculated with the aid of the results of [START_REF] Garboczi | Shape analysis of a reference cement[END_REF]. If use is made of their fit fonction A= 8 vo. 114 , <; is not dimensionless. Consequent ly, we perform a new fitting of their resu lts with a fixed exponent of2/3 and obtain:

A = 5.95V 2 n .
{CT. I 04)

The dimensionless morphological parameter<; is easily calculable from this new fit function:

(11.105)

This value agrees with the analysis of [START_REF] Garboczi | Shape analysis of a reference cement[END_REF] concluding that cement grains are not spherical. By approximating the particle shape, <; introduces some important infonnation on material microstructure in the elastic moduli estimate, and may be regarded as the most significant microstructural parameter after volume fracti ons.

l C of r for unhydrated clinker can be proposed by assuming that these clinker residuals va u .,

• 1 from a shape similar to the initial cement particle one. The argument for such an n .

mpiion is that the reaction of cement particles with water initially taking place at the idc surface, the morphology of their unhydrated core should not change significantly ing the hydration process. The microtomographic pictures of the evolutiou of cement ides morphology issued from the Visual Cement Database (Bentz ct al. 2002) during ]lydllllion permit to properly assess this assumption and show that 1he latter may be too jdcnlistic. Nevertheless, th is approximation is maintained, since the res idual clinker seems to jnlieril from a shape closer to unhydrated cement particles than spheres. Under this hypothesis, the unhydrated clinker keeps the same shape parameter value as the initial cement grains: <;uc = ~""'"",..,_ : 0.81. As illustrated on Fig. 11.21, this particulate phase is then IPProximated by equivalent oblate and prolate spheroids of respective aspect ratios of rue= o.35 and rue = 3.5. However, the difference between HCP effective Young moduli estimated by means of clinker spheroidal inclusions having a shape parameter of <;uc = 0.81 with respect to the spherical case remains quite negligible except for high volume fractions of clinker. This lllltement leads to the remarkable consequence that, though the shape of anhydrous residua ls is noticeably non-spherical, it may not affect importan1ly the elasric properties.

Furthennore, some approximate val u~5 may be found for CH and AFI shape parameters. By examining carefully micrographs of [START_REF] Brown | Ettringite and thaumasite formation in laboratory concretes prepared using sulfate-resisting cements[END_REF], ettringite is charac1erized by long cylinders, whose height is at leasl ten limes bigger than ils diameter, as represented in and AFt in the previous section are perhaps too approximate, even though they provide better results than the spherical one. The latter values were introduced in order to give an example illustrating the limits of the spherical particulate assumption and more precise quantitative data for the morphological parameter are necessary to build a sound homogenization model

There is presently a lack of quantitative data cooceming the hydrate shapes but better estimations could be obtained by characterizing more accurately their morphologies, for example with the help of the Visual Cement Database and the shape analysis techniques developed recently [START_REF] Bentz | The visible cement data set[END_REF][START_REF] Garboczi | Three-dimensional mathematical analysis of particle shape using Xray tomography and spherical harmonics: Appl ication to aggregates used in concrete[END_REF]. It constitutes a promising field of investigation for future improvement of the homogenization processes.

Nevertheless, even if it were ideally possible to directly integrate the exact shapes of all elementary phases in the homogenization process, it is not guaranteed that micromcchanicat estimations of the effective elast ic properties would be fu lly accurate. Indeed, the fOD effective medium approx imations may fa il to adequately account for interactions between high ly-concentrated porous inclusions (e.g. [START_REF] Berryman | Critique of two explicit schemes for estimating elastic properties of multiphase composites[END_REF].

Conclusions of ch apter S

In the present chapter, the influence of inclusion shapes oo the effective elastic properties of HCP has been examined by successively modeling each particulate phase as spheroids, whose shape ranges from needle-like to disk-like. The validity range of the spherical inclusion approximat ion in the homogenization process applied to both sound and leached HCP has also been assessed. For this purpose, the estimations obtained with both spherical and spheroidal inclusions are compared to the results of Kama Ii (2003) by FE simulations. On the one hand, the spherical particle simplification tums out to be suitable for sound pastes. On the other hand, our work evidences the limits of the spherical assumption for leached pastes. The significant e!Tects observed for the add iti onal capillary porosity formed by leaching of hydrated products shows the necess ity of better approaching the morphologies of dissolved phases such as portlandite.

Furthermore, an analytical way has been proposed to integrate more realistic inclusion morphologies in homogenization estimates. The key point of this method is the introduction of the morphological pa rameter c;, wh ich corresponds to the nomializcd characterizations of the ratios between the volume and surface of inclusions. This parameter appears as the most important one after volume fractions. Approx imate values of<; for particulate phases such as clinker, portlandite and ettringite are proposed on the basis of experimenta l results. These phases are then represented by equ ivalent spheroids. This procedure allows for conveniently coupling experimental observations and micromechanical modeling. The explicit micromechanical estimations thus obtai ned for the effective elastic properties of MCP are shown to give better results than those performed with the spherical approximation. However, they are still insufficient for degraded cement pastes with capillary porosity higher than 40%.

Three reasons arc possible for these shortcomings. [t could first be due to intrinsic insufftciencies of the IDD micromechanical estimate. The latter might fail to properly account for interactions between highly-concentrated inclusions. The application of a generally more accurate scheme, like GSCS, may be of interest. Secondly, tbe convenient use of spheroids detennined by means of the morphological parameter presents the considerable advantage of providing an analytical model but it may have its limits to approach fairly complex shapes.

Moreover, the values proposed for this parameter are very approximate due to the lack of complete morphological information on hydration products. The further development of three-dimensional image analysis methods [START_REF] Garboczi | Three-dimensional mathematical analysis of particle shape using Xray tomography and spherical harmonics: Appl ication to aggregates used in concrete[END_REF]) should give the opportunity to better characterize the morphology of inclusion phases, in particular for portlandite, and might lead to improvement of homogenization methods in the field of cementitious materials durability.

The third reason is that the one-scale representations depicted in ~•ig. 11.15 may be too simple to correctly describe the complex microstructure of HCP, which heterogeneities manifest themselves at different scales. Therefore, the next chapter aims at building a more real istic multi-scale homogenization model for cement-based materials and in particular for HCP. At level I, the CS I I matrix behaves as a heterogeneous material with an inclusion-matrix type microstructure. The CSllex 1 play the role of a matrix phase, surrounding the CSH;"' as inclusion. According to Constantinides and Ulm (2004), the latter phase modeled as spherical inclusions has a characteristic size of I o• 8 to 10• 7 m. However, according to some authors [START_REF] Taylor | Cement Chemistry[END_REF][START_REF] Richardson | The nature of the hydration products in hardened cement pastes[END_REF][START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi -coated sphere assemblage model[END_REF], the distinction between CSH'"' and CSH'" sh?uld be done on ly at a higher scale since these phases fom1 successively from cement grain surfaces, the CSH"" resulting main ly from higher confinement conditions and from water accessibility during hydration reactions. At level U, the main phases are the homogenized CSH, cap illary pores, clinker residuals, and hydrated phases such as CH crysta ls, A Ft and AFm. Precisely, the CSH act as a matrix phase in which the other phases are embedded and play the role of inclusion. The next section is devoted to the construction of another multi-scale model that aims at representing more faithfu ll y the microstructure of cement pastes.

CO NSTRUCTI ON OF

Co11structio11ofa11111lli-scale lro111oge11izatio11 mode/f(}r HCP

The proposed two-scale descrif1tion of HCP is inspired by the one developed by Bary and Bcjaoui (2006) for diffusion. Two microstructura l levels respecting the scales separation condition arc presently distinguished: the first ranging from a few tens of µm to about I 00 ~nn is typical of the sizes of hydrated cement grains, while the second one ranging from 0.2 µm to a few tens of µm corresponds to the scale of hydration products, such as CH or AF, and of capillary pores. These two sca les arc respectively denoted as I and II.

The level I corresponding to the biggest scale of HCP is described first. This level depicted in Constantinidcs and Ulm ( 2004) measured that the stiffness modulus of a un iformly leached OPC type I paste decreased to about 3 GPa representing a 84 % decay. To obtain a unifonnly degraded paste, the sample with a diameter of I 1.5 mm was attacked by an ammoni um nitrate solution during 5 months. Inside this leached sample, Constantinides and Ulm (2004) furthermore asserted that even the CSH;.,, and CSHcxi reach a unifonn degraded state and measured by nanoindentation that their Young moduli dimin ish from 21.7 GPa to 3 GPa and from 29.4 GPa to 12 GPa, respectively. Their bu lk and shear moduli, designated by K csH;,,,, Gcsm.,, for CSH 1 "' and by KcsHw . GcsH., 1 for CSHc", arc immediately deduced from Eq.

(11.102). The input elastic parameters for the two sound phases are then equal to Kc 5111 ., = 18.8

GPa, GCS1 11 ., = 11 .9 GPa and Kcsum = 13.9 GPa, Gcsum = 8.8 GPa, while their degraded elastic properties decrease to K~1 11 ., = 7. 7 GPa, G::S,"'" = 4.8 GPa and K~1 1 ..... = 1.9 GPa, G~s 11 ,,, = 1.2

GPa (the subscript I stands for leached state). All these modu li arc calculated supposing that the Poisson ratio of both phases is not sensitive to leaching and remains equal to 0.24 (Cons1antilliclc.5 and Ulm 2004). This approximation is made due to the lack of experimental data on the Poisson ratio. The input mechanical data required for compuling the multi -scale model are the same as those employed by Kamali (see Table 1.3) except that in the micromechanical model the CSH gel is modelled as two phases, CSHint and CSH""', which elastic moduli are given just above. ln In order to compute the multi-scale model for the case of partially leached pastes simulated by [START_REF] Kamali | Comportement et simulation des materiaux cimentaires en environnements agrcssifs: lixiviation et temperature[END_REF], it is necessary to know the volume fractions and the elastic properties of The mean deviation with respect to the numerical simulations is about 12 % for both estimates. However, the estimations computed with the modified DIM can deviate by more than 30 %. whereas those c~mputed with GSCS do not. Therefore GSCS that is a little more accurate than the modified DIM is chosen in the ensuing computations of the multi-scale homogenization model. Second, the present two-scale description of HCP microstructure may be mo re realistic than the representation proposed by Constantinidcs and Ulm (2004).

Better predictions plotted in

6.2 A multi-scale approach for estimating the macroscopic elastic prop er ties of mortars T he app lication of homogen ization methods to mortars is complicated by the lack of experimental data o n rrz elastic properties. To circumvent this difficu lty, many authors have estimated these characteristi cs of ITZ from experimental values or mortars by inverse analysis employing for instance GSCS (e.g. Ramesh ct al. 1996;Hashin and Monteiro 2002) or DIM [START_REF] Yang | Effect of the Transition Zone on the Elastic Modul i of Mortar[END_REF]) as upscaling methods. Unfortunately, very different values have been identified

for t)1e Yo ung modulus of ITZ from one author 10 another, as pointed out in Eq . ( 1.5). An altemati ve multi-sca le homogenization approach based on the modified DIM presented previously is proposed to pred ict the elastic propert ies of mortars without having to estimate the elastic modu li of ITZ by inverse analysis.

In addi tion, the total volume fract ion occupied by this zone is also difficult to gain Good agreement is found for the different micromechanical predictions with the experimental mcasuremems but the Young modu lus of the uniformly leached mortar is somehow underestimated by the various models. This may be due to an excessive porosity inside ITZ auaining more than 90 % in the present representation. The mod ified DIM with three different types of aggregates gives the closest estimation to the experimental result on sound mortar, While GSCS is the most accurate in the case of uniformly leached mortar. Despite the fact that the modified DIM with three diverse sizes of aggregates provides the best predictions of the Yound modu lus of the sound mortar, the GSCS will be employed in the ensuing Parts for cornlluting the elastic modu li of mortars for two motives. First, it is more coherent to adopt the same scheme for estimating both the effective properties of HCP and mortars. Second, the llnllling of the involved mortars may not always known precisely, so that it is not possible to use the modified DIM with diverse sizes of sand gra ins.

CO NCL USIONS O F PART II

The first goal of the present Part is to expose some of the fundamental aspects of homogenization techniques and review in a non exhaustive manner some EMTs that may be of interest for predicting the linear elastic properties of cement-based materials. By assessing these methods using different criteria and by testing their accuracy on particular cases that have computed numerically, a small set of EMTs, like JOO, GSCS and the modi fed DIM, are outlined to present valuable features for estimating with a good precision the elastic moduli of cement-based materia ls and arc reta ined for the construction of a homogenization model specifically suited for these composites. According to the authors' knowledge, the modifed DIM proposed in the present manuscript by revisiting the double-inclusion scheme of Hori and Nemat-Nasser ( 1993) has not been derived elsewhere.

The influence of inclusion shapes on the micromcchanical estimations of the effect ive elastic properties of' I ICP has then been investigated. This study puts in evidence the fact that the inclusion shapes of phases such as portlanditc have a visible impact on the predictions of the macroscopic Young moduli of leached pastes. However, it also appears possible in most cases to approximate the particle phases of HCP as spherical inclus ions, even though their real shape inside the microstrucn1re of cement pastes (see Figs.

Part Ill PREDICTION OF THE MACROSCOPIC DI FFUSIVITI ES OF CEMENTITIOUS MATERIALS

The macroscopic diffusion coefficients of dissolved chemical species such as chlorides or sulfates in saturated cementitious materials control the degradation process of these materials.

Therefore U1ey represent important data for predicting the serv ice life of cement-based materials, wh ich can serve as engineering barriers for the disposal of nuclear waste.

Homogenization models aim at estimating the macroscopic diffusion coefficients of cementi tious materials from their microstructure and their phase diffusive properties. The present Part is concerned with the construction of a homogenization model for predict ing the effective diffusivities of cement-based materials. As already mentioned in Part I, these materials are complex mu lti-scale porous media, where the pore size distribution varies by several orders of magnitude (from I nm to more than IO µm). On the basis of these observations, a multi-sca le approach is adopted to deal wi th such a porous medium as in linear elasticity (e.g. [START_REF] Constantinidcs | The effect of two types ofC-S-H on the elasticity of cement-based materials: Results from nanoindentation and micromechanieal modeling[END_REF].

Some theoretical connections ex ist between elasticity and diffusivity. An elasticity problem can be decomposed in three diffusion problems (M ilton 2002). Hence, the derivations exhibited in the present Part are generally much simpler than the ones presented in the previous Part. Nevertheless, it does not mean that the development of a homogenization model for predict ing the effective diffusive properties of cement-based materials is a simp le task, s ince the fact that the derivations in diffus ion arc easier than in linear elasticity is compensated by other difficulties. Indeed, much less information seems to be available on the phase diffusivities than on their elastic properties in particular for the CSH. Another difficulty for building a suited homogenization rnodel for concrete materials consists in reproducing the highly non linear changes of their transport propert ies measured experimentally as their total porosity varies (e.g. [START_REF] Oh | Prediction of diffusivity of concrete based on simple analytic equations[END_REF].

The present Part is organized as follows. The first chapter assesses the capacity of matrixinclusion type methods tbat have been focused on in linear elast icity to face th is aspect. ln the second one, other theories [START_REF] Bergman | Calculation of bounds for some average bulk properties of composite materials[END_REF]Bnaggeman 1935;[START_REF] Kirkpatrick | Classical transport in disordered media: scaling and effective-medium theories[END_REF][START_REF] Kirkpatrick | Percolation and conduction[END_REF] By diffusion, we mean mass transfer generated by concentration gradients of aqueous species through saturated materials. The resu lts obtained for diffusion are directly transposable to ocher transport properties such as electrical or thermal conductivity.

REVIEW OF MATRIX-INCL USION T YPE EMTs FOR PREDICTING TH E OJFFUSI VITIES OF POROUS MEDJ A

This chapter aims at assessing if the matrix-inclusion type EM Ts, already presented in Part 11, arc suited for estimating the macroscopic diffusive properties of porous media. The equations necessary for deriving these EMTs in the case of diffusion are quite alike the ones presented in the previous Part except that second-order tensors are now involved instead of fourth-order ones [START_REF] Milton | The theory of composites[END_REF], so that only the most basic or important ones are presently recalled.

Fundamenta l pri nciples of m atr ix-inclusion type EMTs

The computations presented are analogous to the one performed in linear elasticity. However, the basic formulae necessary for applying homogenization methods are recalled for clarity.

I Ge11er11l principles of homoge11izatio11111etlrods

The first seep of these methods consists in representing the microstructure of the concerned material by a R VE, such as the one depicted in The RVE is subjected to unifonn ions flux vector over its surface oV:

J.(x) = .r.n, 'efxe av, (Ill. I)

Where J.(x) denotes the ions flux vector at a given point x of the surface and J~ is a uniform flux vector. The operator ' .' indicates a simple contr~ction. At the scale of the heterogeneities ofche RVE, the equilibrium Jaw and Fick's constitut ive one take the fonn:

div(J(x)) = O, 'ifxe V, J(x) = -D(x). g(x), 'ifx e V, ( 1112
)
(lll.3)

where J (x), D(x) and g(x) designate the microscopic flux vector, diffusivity tensor and concentration gradient of ions at a given point x of the RVE, respectively. The macroscopic flux tensor J is defined as the volume average < J >v of the microscopic one J over V and can be shown to be equal to J -:

J =_!_ f J ( x )dV = J •. Vv (111.4)
The ensuing relation is also obtained from Eq. (Jn.3) by performing averaging operations on the entire volume of the RVE:

g=-D• ' .J '

(111.5)
where J and g arc respectively the averages of the microscopic flux and concentration gradient tensors on the total volume of the RVE. Estimations of these averages by means of the diverse homogenization techniques presented before then allows for predicting the effective diffusivity tensor o• of the material. In the ca~e of a macroscopically isotropic composite, the latter tensor may be written as:

o• =10• 1 ' (111.6) 3
where I is recalled to be the second order identity tensor and D' is a sca lar.

1.2 Prese11tatio11 of the Es/le/by problem

2. I Presentatio11 of the Eshelby theorem

The Eshelby theorem presented in subsection 4. 1.2 is also applicable in diffusion and is therefore briefly recalled below. A homogeneous ellipsoidal inclusion /, occupying a domain VJ and having for diffusivity tensor D,, is enclosed in an infin ite matri x, which diffusivity tensor is denoted as 0 ,.,. The concentration grad ient through this single inclusion subjected to a fictive concentration gradient g•, which is unifonn inside the inclusion and vanishes outside, is unifonn:

(Ill. 7)
where A 1 is called the Eshclby or the depolarization tensor (e.g. Torquato 200 I ) of the inclusion I expressed as:

(A 1 l, = J r ,,(y -x )dV,. , (111.8) ., 'Jbe Eshelby tensor thus links the concentration gradient inside the homogeneous ellipsoidal joclusion surro.unded by an infinite medium to its fictive concentration gradient. The main Jintit of this fundamental theorem is that it is restricted to ell ipsoidal inclusions in its original form (Esbelby 1957). The Eshelby tensor is generally quite difficult to determine analytically but has a very simple expression in the spherical homogeneous inclusion immersed in an infinitely large isotropic matrix (e.g. [START_REF] Torquato | Random Heterogeneous Media: Microstructure and Macroscopic Properlies[END_REF][START_REF] Milton | The theory of composites[END_REF]:

A, ==l t.

( 111.1 0) 3 Tue Eshelby tensor of a spheroidal homogeneous inclusion also presents the benefit of being analytically calculable (e.g. Torquato 200 I) and its expression for a spheroid with aspect ratio r which revolution axis is oriented along the e 3 axis, is:

'

A,=[~ ~ ~ ], 0 0 l-2Q (Ill.II) ~[I+ , . 2 1 _,[1-2 ~1 n(:~~)Jl with Q = p t a n• •(J, : r , )Jl
a1 _ D 1 -D 1 µ, - , D,+ 2D 1 (l[l.J2)
.

-u a simple closed-form solution is found for the Eshelby tensor denoted as A, of a spheri cal isotropic inhomogeneity with a homogeneous interphase enclosed in an infinite matrix, as depicted in Fig. --"' . g, = A, .g' (lll.14) where the Eshelby tensor A.:' for the spherical inhomogeneity with its homogeneous interphase will be referred to as inhomogeneous Eshelby tensor in the ensuing (Duan et al. 2006).

Presentation of the equivalent homogeneous inclusion method

First of all, let us consider the situation in which no heterogeneous inclusion is inserted in the infinite matrix with diffusivity tensor D.., , which is subjected to a uniform far-field flux J• .

In such case, the flux field is everywhere equal to Jand the concentration gradient field is such as: 0 _, J ~ ~ = .., . .

(ID. 15)

A heterogeneous inclusion I occupying a volume V 1 with diffusivity tensor 0 1 is now inserted in this virgin medium and we are interested in calculating the average ii, of the concentration gradient field over V 1 • The microscopic concentration grad ient decomposes itself into two parts:

gi{x) =g 0 +g/{x), "Ix, (Jll.16) where g/{x) corresponds to the perturbation of the concentration gradient caused by the insertion of the heterogeneity. In order to apply the Eshelby theo rem, the heterogeneous inclusion is assumed to be substituted by a homogeneous ficti ve inclusio n with a diffusivity tensor denoted as O., that is subjected to an appropriate unifonn fictive concentration gradient g • . The latter must be chosen in such a way that the same average flux and concentration grad ient fields are obtain ed in the homogeneous fictive inclus ion and in the heterogeneous one:

-J, = O,.g, = Ow(g, -g') and

(Ill. l 7)

By solving g• in (111.17 where 0~ is called the eigendiffusivity tensor of the single inclusion I embedded in the infinite reference medium by analogy with the eigenstiffness tensor in linear elasticity [START_REF] Zheng | An explicit and un iversally appl icable estimate for the effective properties of multiphase composites which accounts for inclusion distribution[END_REF] and where H, is the resistivity increment tensor of the inclusion. The eigcndiffusivity tensor 07 can be determined in the case of spheroidal inclusions by means of the expressions of the Eshelby tensor given in Eq. (TIT. I I). For more complicated cases, the Eshelby tensor usuall y has to be estimated numerically [START_REF] Douglas | Intrinsic viscosity and the po larization of particles having a wide range of shapes[END_REF]. The ensuing derivations are therefore limited to the case of spheroidal inclusions. The next section gathers the basic expressions for the effective diffusive properties predicted by the EMTs presented in Part II.

Basic expressions of EMTs for estimating macroscopic diffusivities

Various EMTs have been presented in depth in the previous section. T he derivations of these methods for predicting the material effective diffusivity that are very simi lar to the ones written previously are not recalled for conciness. However, their final expressions are enlisted below.

Classical matrix-inclusion type estimates

In such type of EMTs, a matrix phase designated by the letter Mis supposed to enclose the other phases modelled as inclusions. As in Part II, the effective resistivity increment tensor H oftl1c heterogeneous material defined as follows is introduced:

lJ = D.-I -D M -I .
(ITl.20)

The effective diffusivity tensor of the material can be immediately deduced from the knowlegde of this resistivity increment tensor:

(Jll.2 1)

By perfonning the same reasoning as in Eqs. ( II.32-34), the dilute model provides a s imple expression for estimating this effective resistivity increment tensor:

H"' =" H JH with H"" = c H (t + 0 "' H ]-' ,L., r1 r r t " r " r • (lll.22) r
It is recalled that the validity range of the dilute estimation is very limited, since all the particles arc required to be far enough from each other, so as to be regarded as isolated. More consistent schemes trying to reflect the interaction effects between inclusions are required.

MT effective medium approximation may be used for this purpose. Following the same path as in Eqs. (11.35-39), the MT estimate of the resistivity increment tensor takes the fonn: The MT estimate and GSCS are seen by comparing the last two equations to provide identica l express ions for the effective diffus ivity of an isotropic two-phase composite with spherical inclusions. It was shown in section 4.4. that GSCS is more accurate than MT in linear elasticity, even though the first one is more intricate to compute. Nevertheless, these two rnodels appear to be quite simi lar in the case of diffusion.

Double-inc/11sio11 type schemes

These methods have been preferred to compute the linear elastic properties of cement-based composites to the classica l MT scheme, because of their better accuracy, versatility and their properties of the multiphase composite are then computed using the energy balance employed by [START_REF] Shen | An effective inclusion model for effective moduli of heterogeneous materials with ellipsoidal inhomogeneities[END_REF]. To achieve the computation of the effective diffusion tensor of the double-inclusions, fictive concentration gradients are introduced inside the inclusion domain so that each inclusion has the same diffusive properties as its coating.

t J~ -------------~@J" i -. -,./ .

I I L _ -----------' a) t J~ ------------- b) ------_t _ ~~--
( ) g, =A, .g, ,

Ill.31

where the expression of A.; 1 is given in Eq. (111.13) for the particular configuration of a• spherical isotropic inhomogeneity embedded in an infinite isotropic matrix. By neglecting the interactions between the double-inclusions, the average concentration gradient inside the inclusion corresponding to a given particulate phaser takes the form:

(111.32)

where A,,, denotes the homogeneous Eshelby tensor (Eq. (ill. I I)) of the spherical doubleinclusion D,. (re [1,Pl). The consistency condition written in an average sense:

(TTT.33)

provides che following relation for the fictive concentration gradient g; in the inclusion:

(IIl.34)

The average perturbation of the concentration gradient inside che inclusion i,' can then be deduced from Eq. (lll.31 ). Using the fact that:

-l ' c ' Jc - where the diffusion tensor D 0 ,. can be predicted by the preceding equation. Jn the ensuing, this estimation will be referred to as the modified DIM. If all tbe involved phases are assumed to be isotropic, simple expressions are provided for the homogeneous and iJJhomogenous Eshelby tensors by Eqs. (lll. l 0) aJJd (111.13), respectively.

In the particular case where only one double-inclusion is considered, the effective diffusion coefficients for this double-inclusion and the compos ite respectively simplify into: which exactly coincide with GSCS fonnula written in Eq. (lll.25). A good number of homogenization methods have been presented in this section but it is now necessary to test their relevance for predicting the macroscopic diffusivities of porous media. For this purpose, attention should be primari ly focused on the principal effects acting on the material effective transport properties. It is pointed out in the next section that these properties are strongly dependent on certain microstrucnJTal factors that have to be taken into consideration in the computation by homogenization.

D =D I Cm +2c, ,

Significa nt microstructu ral para meters from t he viewpoint of diffu sion

Porous media are generally regarded as a mixture of a solid phase, denoted as Di, considered non-diffusive and a diffusive porous phase designated by Di. A very strong contrast exists between the transport properties of these two phases so that it is frequently assumed that D, I D 1 ~ 0 in porous media. This percolat ion threshold physica lly corresponds to the critical volume fraction at which a cluster of the diffusive phase first connects one extremity of the microstructure to another.

According to the scaling law above, the formation of this cluster spann ing the entire microstructure induces sudden variations of diffusivity, which may be denominated as percolation effects. As was already observed io Fig. l.5, the macroscopic diffusivities of HCP strongly vary from one paste to another. Many authors attribute these sign ificant changes to the geometrical percolation of capillary porosity, defined as the remaining space situated between hydrated cement grains (e.g. [START_REF] Richardson | The nature of the hydration products in hardened cement pastes[END_REF]. It physically corresponds to the fonnation of a cluster composed of pores that first spans the material. It appears consequently necessary co assess the ability of the homogenization schemes introduced previously to account for such phenomenon.

At this point, an important clarificati on is necessary about the difforcnces between linear elasticity and diffusion in cement-based materials. According to the previous remarks, the percolation of pores is suspected to have a very strong impact on the macroscopic diffusive properties of porous materials. This phenomenon has not been discussed in the previous Part dedicated to linear elasticity but it seems to have less innuence on the material effective elastic properties as illustrated by the subsequent example. Let a standard CEM I paste with w/e = 0.45, which effective elastic prope1ties and diffusivity have both been measured experimentally (Richet et al. 1997;Galle et al. 2004), be perforated by infinitely long cylindrical water-filled tunnels isotropically distributed. The diffusivity of these idealistic porous percolating paths is assumed to be equal to the diffusion coefficient D,rro = 2.2x 10 9 m 2 1s of tritiated water io bulk water at 23°C [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi -coated sphere assemblage model[END_REF], while their bulk and shear moduli are respectively equal to 2.2 GPa and 0 GPa. The size of the RVE of the perforated paste is taken large enough so that the original cement paste can be regarded as homogeneous. Its macroscopic diffusivity is measured to be Ill.3, the material effecti ve diffusivity appears to be much more affected by the tunnels than its effective elast ic modu li. A major difference between di !Tusion• and elasticity is that the macroscopic elastic properties of a material such as concrete may be more innucnced by the fonnation of the solid skeleton (Bernard ct al. 2003) than by the percolati on of pores, whereas this percolation of porosity is crucial for the effective diffusive properties. According to this example, the pore percolation has to be carefully taken into consideration in diffusion, despite the fact that it has a secondary importance in linear elasticity.

Volunle rraction of porous IUnnels

OlS

Importance of otfter 111icro.l'lr11ct11ral indicators

In linear isotropic porous media, the following equation is generally employed to compute the macroscopic diffusion coefficiem denoted as o• (e.g. [START_REF] Bear | Introduction to modeling of transport phenomena in porous media[END_REF][START_REF] Dullien | Porous Media, Fluid Transport and Pore Structure[END_REF]):

(111.44)

where -r is called the tortuosity and c 2 corresponds to the volume fraction of porous phase.

The tortuosity can geometrically be interpreted in tenus of the ratio of the length of the material over the distance covered by the ions or the fluid to get through the sample (Dull ien 1992). Practically, the closer to I the tortuosity is, the straighter the percolating paths arc. It also gives some information in terms of pore connectivity and percolation. For the extreme case where r = 0, the distance that the ions should cover to get through the sample becomes infinite, which means that the porous phase does not perco late. In addition, the material effective diffusion D' goes to 0, since the porous phase remains occluded. This shows the primary imponance of the percolation of the porous phase on the diffusive propenies of porous media.

The macroscopic diffusivity can also be related to the formation factors, which constitute precious rnicrostructural indicators and arc defined as (Be1Tyman 2005):

I . o• l ' o• ' (111 s) rm - = -- 1111 - = - .4 n.--D, F, ' o,-0 2 F,
The formation factors arc usually measured from electrical resistance data [START_REF] Dullien | Porous Media, Fluid Transport and Pore Structure[END_REF]. The matrix -inclusion type EMTs, such as GSCS, DIM, MT, KT and !DD, studied in this manuscript all present the considerable advantage of being explicit in diffusion. However, ail these approaches except the GSCS are obtained by embedding one single isolate homogeneous or compos ite inclusion into a supposed infini te matrix phase. as pointed out by [START_REF] Lutz | Effect of an inhomogeneous interphase zone on the bulk modu lus and conductivity of a particulate composite[END_REF]. T hey consequently fail to take into account the connectivity of the particulate phase and appear hardly compatible with the percolation theory, since a single inhomogeneity cannot percolate except if it degenerates into an infinite disk or cylinder.

However, though appearing inadequate to percolation problems, these models have up to now not been shown to be unable to mimic the effects due to the transition from a microstructure comprising no or few tortuous percolating clusters of a high-diffusive phase to one where significant paths of this phase have formed.

The 3D simulations of [START_REF] Garboczi | Geomclrical percolation threshold of overlapping ellipsoids[END_REF] demonstrated that the aspect ratio of spheroidal inclusions denoted by r = b I a (where a designates the radius of the circu lar basis of the spheroid and b corresponds to its revolution axis) influences s ignificantly their percolation threshold and consequently their connectivity. For example, the percolation threshold of spherical inclusions corresponds to a critica l vo lume fraction c: : : : : 0.29 (Rintoul and Torquato 1997), whereas for spheroidal inclusions with r = I 00 this threshold decreases to c: ~ 0.01 [START_REF] Garboczi | Geomclrical percolation threshold of overlapping ellipsoids[END_REF]. The shape of the spheroidal inclusion should be in average representative of the diverse pore clusters, since the pore stnicture is simultaneously composed of isolated pores assumed quasi-spherical and of very connected pores forming elongated clusters. In other words, the better the pores connect to each other, the more the Spheroids representing the porous phase shou ld be elongated. Thus the connectivity could indirectly be taken into account by using adequately chosen spheroids.

As an example, an attempc is now made to reproduce percolation effects in a macroscopieaUy isotropic material composed of two isotropic phases by modelling the most diffusive phase, called 2, as a randomly oriented prolate spheroid embedded in the other phase, cal led I . In the ensuing, the number I will always refer to the phase less diffusive and 2 to the most diffusive one so that D 2 > 0 1 • The volume fractions of chc two phases arc respectively denoted by c, and c 2 with c, + c, = l . Two cases, one of high contrast and another one of infinite contras1, can be considered in order to 1est the reliability of EMTs for predicting percolation effects.

The diffusion coefficients, O, and D2, arc supposed to be respectively such as 0 2 I 0 have been developed to address this issue. Some of them deriving from the self-consistency principles [START_REF] Bruggeman | Bercchnung versch icdcner physikalischcr Konstantcn von hetcrngenen Substanzen: I. Dielcctrizitiitskonstanen und Lcitfahigkeiten der Mischk()rpcr aus Isotropcn Substanzen[END_REF][START_REF] Landauer | The electrical resistance of binary metall ic mixtures[END_REF] or from 1he variational ones [START_REF] Hashin | A variational approach to 1he theory of the elastic behaviour of polycrys1als[END_REF] are therefore presently reviewed. By focusing on some classical exactly solvable microstructures [START_REF] Hashin | A variational approach to 1he theory of the elastic behaviour of polycrys1als[END_REF], a novel estimate based on a simple exactly solvable assemblage is also proposed to predict the macroscopic diffusivities of porous media.

111 the present chapter, on ly macroscopically isotropic two-phase materials are considered.

The volume fractions of the two phases respectively called I and 2 are denoted by c, and c 2 .

The diffusion coefficients of the two isotropic phases, 0 1 and D2, arc taken such as D2 > D1 .

Review of b ou ndin g approaches

The HS bounds for a macroscopically isotropic two-phase material are given by: o--0(1 3/J; ) • -( . 3/J. The form and the derivation of these bounds (Beran 1965) in their origina l form can be quite complex but Berryman introd uced a convenient canonical function that enables to express the mai n bounds within a unified framework:

A(y) !!!:[ c, + C2 ]-'-2y. 111 the case of infinite contrasts (Fig. 111.6), the lower bounds all tend to zero and are therefore 1101 represented . Moreover, no percolation thresho ld appears on the upper bounds. None of the bounds reviewed consequently seems to be apt to account for percolation effects. That's why self~consistent (SC) EMTs that present valuable features for the prediction of the diffus ivity of porous media arc discussed hereafter.

Review of SC ty pe schemes

The SC scheme [START_REF] Bruggeman | Bercchnung versch icdcner physikalischcr Konstantcn von hetcrngenen Substanzen: I. Dielcctrizitiitskonstanen und Lcitfahigkeiten der Mischk()rpcr aus Isotropcn Substanzen[END_REF][START_REF] Landauer | The electrical resistance of binary metall ic mixtures[END_REF]) counts among the most popular homogenization models in diffusion (e.g. Oh and Jang 2004; Bary and flejaoui 2006) because of its capacity of capturing sudden variation of diffusivity. That's why many commonly used EMTs (Kirkpatrick 197 1;[START_REF] Kirkpatrick | Percolation and conduction[END_REF][START_REF] Mclachlan | An equation for the conductivity of binary mixtures with anisotropic grain structures[END_REF] for dealing with homogenization problems in diffusion or conduction directly derivate from the SC scheme. On the contrary 'to matrixinclusion type models where the particulate phases are represented as inclusions embedded in a supposed infinite matrix phase, in the SC ones all the phases are modelled as inclusions enclosed in a yet unknown effective medium supposed infini te. The SC estimate in three --sc 0.9 It is straightforward that Eq. (Ill.50) obtained with the classical SC scheme is a particular case of Eq. (ITI.51) for z = 6. Supposing that 0 2 is infin itely bigger than 0 1 in the equation above,

--GEM (c~ • O.S. n • 2) 0.8 ---HS+ 0.7 • -• -• Kirkpairick (z • 4) N -0.6 Q -o.s • Q , 0 .4 , , , , 0.3 ,
the percolation threshold appears to be inversely proportional co the coordination number: 0 • = a+.Ja 2 ~z2 ~~) 2 )D,D 2 with a=o 2 ((~-1}:, -1)-o, ((~-1)c,-1). (111.52) 111e Kirkpatrick model uses the concept of coordination number that allows for taking into account the connectivity of particles. This scheme is based upon resistor networks constituting an idealistic discrete representation of the microshucture of ordered heterogeneous materials.

They are very practical of use in the case of crystall ine materials, where the coord ination number is exactly known. Nevertheless, this number is very difficult to estimate in random heterogeneous materials containing particles that can be of very different sizes and the discrete resistor network on which the SC estimates of Kirkpatrick (1971; 1973) . ( D, 11

• -D'"• ) (Di"' -D'"• ) c, ( D ,"• +( c~• -I) D' "•) + c, ( D ,"• + ( c~• -I) D'"•) = 0.
(Tll.53) 'tihcrc c't is recal led to be the percolation threshold and p is a parameter identified as the e .. Another method that is capable to predict different percolation thresholds consists in deriving the SC scheme by adopting spheroida l incl usions instead of spherical ones, as is generally the case. The percolation threshold estimated with this approach then varies with the aspect ratio of the spheroid. Nevertheless, the respect of HS bounds is not necessarily guaranteed with this methodology.

To summarize the discussions made in this section, a comparati ve table recapitulating the merits of the presently reviewed EMTs is proposed (Table 111 (1962) proved that their bounds written in Eq. (111.46) are realized by two composite spheres assemblages (CSAs) corresponding to idea lized microstructures, where the space is entirely fil led by coated-spheres of sizes ranging from fi nite down to infinitesimal. Neither the s ize nor the spatial distribu1i on of the coated-spheres is specified in CSA. However, all the composite spheres are homothetic and consist of a spherical core of radius '" and a coating of thick ness (re -r;). The radi i r; and re are chosen so that each coatedsphere has the same composi tion as the original two-phase materia l. Though the opportunities offered by the self-consistency principles [START_REF] Bruggeman | Bercchnung versch icdcner physikalischcr Konstantcn von hetcrngenen Substanzen: I. Dielcctrizitiitskonstanen und Lcitfahigkeiten der Mischk()rpcr aus Isotropcn Substanzen[END_REF][START_REF] Landauer | The electrical resistance of binary metall ic mixtures[END_REF] or the variational ones [START_REF] Hashin | A variational approach to 1he theory of the elastic behaviour of polycrys1als[END_REF] for estimating the transport properties of random heterogeneous media with high contrasts have been thoroughly investigated, the possibilities given by the celebrated CSA ofHashin and Shtrikman (1962) may conversely not have been studied sufficiently for such media. This exactly solvable microstructure was initially established to materialize HS bounds and thus prove their optimality. In the present section, by exploiting the idea underlying the original CSA, an assemblage mixing two types of composite sphere is proposed to obtain an estimation model capable of remedying the shortcomings of HS bounds.

The construction of Hashin and Sbtrikman (1962) relies on the fact that the concentration field in a homogeneous medium characterized by a diffusion coefficient D 0 , subjected to a unifom1 concentration gradient on its boundary, is not disturbed by the insertion of a spherical composite inclusion which overall diffusion coefficient is equal to D 0 . In particular, any previously defined coated-sphere, which core and shell are respectively occupied by phases 2 and l, embedded in a homogeneous medium with D 0 = D~s does not perturb the concentration field outside the composite element in question. Consequently, an assemblage exclusively composed of these particular composite spheres realizes the lower bound. The upper one can in tum be attained by a similar cons truction, where the roles of the two phases inside the composite sphere arc switched and the radi i are taken such as (r,/r, )3 =c,.

CSAs are widely used, because they provide exact and straightforward analytical results.

Nevertheless, the two CSAs materializing HS bounds correspond to extreme matrix/ inclusion morphology (Fig. 111.10). In the assemblage achieving the HS lower bound, the low-diffusive phase l is fully connected and plays the role of matrix, whi le the high-diffusive phase 2 occupying the cores of the coated-spheres is entirely disconnected and distant from each other due 10 the systematic presence of coatings made of phase I. The exact opposite situation occurs in the assemblage attaining the HS upper bound. As a matter of fact, this characteristic can even be regarded as an advantage for some special design problems (Torquato ct al. 2005). Nevertheless, actual microstmctures having such an extreme morphology do not abound in Nature. To obtain a more real istic and more versatile microstructurc, it would be desirable to use for instance different types of composite spheres. That's why another microgeometry, called MCSA, is presented below. Slttrikma11 (1962).

The MCSA follows the same constrnction rule as CSA but two types of composite spheres are involved in MCSA instead of one type in CSA. The idea of us ing assemblages mix ing two types of composite spheres is not new. Pham (1997) proposed an estimate on the basis of a MCSA, where the volume fractions of the phases in the composite spheres are fixed but the embedd ing orders of the two phases can be interchanged. However, this assemblage is not exactly solvable, s ince the two types of c-0ated-spheres generally do not have the same effective diffusivity and the diverse perturbations they induce on the concentration fields are tremendously difficult to calculate. G ilom1ini ( 2001) evidenced that a simple MCSA can be constructed so as to realize any compressibility or conductivity of isotropic two-phase materia ls comprised between HS bounds. Analogously, it is possib le to build a MCSA that achieves any macroscopic diffusion coefficient D' of an isotropi c two-phase material verifying the condition D;, 5 $ D' $ o;,s, as shown below.

T he macroscopic diffusion coefficient D' of the material and the phase volume fractions c, and c, are assumed given; the particular example of the diffusion through a porous material with high constrast between the phase diffusivities is considered for the present (Stora et al. 2006b).

The MCSA with a given effective diffusion coefficient D' is determined through a unique parameter f specified in Eq. {Tll.56). Inversely, by considering the macroscopic diffusion coefficient o• as unknown and the parameter fas given, we can establish the quadratic equation: where

•' .

oD +bD +c=O. given above varies monotonically with f. The MCSA estimate can consequently attain any value comprised between HS bounds. The derivation of a MCSA estimate for the effective elastic modul i cou ld also be of interest but is not a simple matter. Whereas the effective diffusivity and bulk modulus of CSA presents the benefit of being exactly solvable, the effective shear modulus of this microgeometry is not exactly computable (e.g. [START_REF] Milton | The theory of composites[END_REF]. Consequently, an exactly realizable MCSA estimate can be obtained in linear elasticity for the effective bulk modulus, but not for the shear modulus.

It is insightful to see how the MCSA estimate in Eq. (lll.58) can be recast by means of the canonical function proposed by [START_REF] Berryman | Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media[END_REF] in Eq. (Ul.47). In two-phase materials, the relation linking/to the transformation parameter y can be derived from Eqs. (JU.47) and

(lll.56-57) (Stora et al. 2006b):

f [ I 2 ( D, -Y ' ]( D, -y) = /3,2 + •C2 D2+2y) D1+2y .
(lII.59)

The inverse relation requ ires solvi ng a quadratic equation with only one positive root which takes the form:

where -e+ ~e2 -4dg y= 2d .

d = 2 + 2(2/ -c 2 )/3,', e = -2D, + D, + 2/3, 2 ( / + c 2 )(D, + D 2 ), g = D, oi[(J -2c 2 )/3, 2 -d.
(III .60)

This function providing the transfonnation parameter y is also monotonic with respect to f

Eqs. show that a one-to-one correspondence exists between j and y. So, the geometric parameter j in the MCSA model can be taken as equivalent to the transfonnation parameter entering the canonica l function defined in Eq. (lll.47). At the same time, since the Properties of the canonical function are such that it leads to excell ent estimates even with rough estimates of the transformation parameter [START_REF] Berryman | Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media[END_REF], we can expect that Eq.

(Ill.58) produces good estimates when f is evaluated from relevant experimental data. It is emphasized that, by construction, these estimates always comply with HS bounds. Besides, lhe parameter j has a simple geometric signification and can easily be related to physica l fac1ors that arc primordial for transport phenomena, as wi ll be shown in the next section.

Signification and interpretation oftlte MCSA parameter

Given the phase volume fractions of a random heterogeneous material, the estimate derived from MCSA for the effective diffusivity depends only on one geometric parameter/reflecting the relative proportion of the two types of composite spheres in the assemblage. The purpose of the present section is to highlight the com1ections existing between the parameter f characterizing MCSA and important physical concepts, such as connectivity and tortuosity.

I A parameter for connectedness

The estimate given by Eq. (lll.58) is mathematically quite similar to the SC model of [START_REF] Kirkpatrick | Classical transport in disordered media: scaling and effective-medium theories[END_REF][START_REF] Kirkpatrick | Percolation and conduction[END_REF]. They are both analytical and each of them only depends on one clearly identified parameter, respectively the coordination number z in the Kirkpatrick approach and the geometric parameter / in the MCSA model. In the Kirkpatrick theory ( 1971Kirkpatrick theory ( , 1973)), the microstructure is considered as a resistor network and the coord ination number z is defined as the number of bonds connected at the same node. Jn the actual microstructure, this quantity is practically identified to the number of contacting neighbors to a given particle.

This parameter thus allows for taking into account the conuectedness of the diverse phases.

Analogously, the geometric parameter/can be interpreted as a parameter for connectedness in the MCSA model. As illustrated in Figs. lll.12, each phase inside the assemblage can be broken into two parts: the first one contained in the cores of the composite spheres is completely isolated, whi le the second one present in the coatings is well connected, since a given coated-sphere is in contact with infinity of neighbors. In practice, the volume fraction of the phase 2 dispersed in the outer layers of the Cl spheres, denoted by c;"', can be identified to the experimental measurement of the connected volume fraction of phase 2, once such measure is available. After some calcu lations detailed in Appendix 111.8, the geometric parameter/can be expressed as a function oftbis volume fraction c;"' :

! -•"'( 1+2P 1 2 /J)(c 1 -cj ) J' -C2 . c 2 + 2P 1 2 p; (c 2 -c;"') (Ill.61)
According to this equation, it is possible in the case of porous media to gain infonnation for f from the knowledge of the percentage of connected pores, which may be accessible via various experimental procedures such as MIP. On the opposite, the coordination number z is far more difficult to estimate for such materials generally containing pores that can be of very different sizes.

link with the formation/actors

The measurement of one of these fonnation factors allows for calculating tbe value of the geometric parameter/ when the volume fractions arc also lrnown. Indeed, the parameter f is related to the formation factors /~ and F 2 in the following manner:

/ =c (2F,+ l ) =l-c l' 2F, + In this section, some connections have been established between the geometric parameter characterizing MCSA and crucial microstructura l parameters, such as the connected volume fractions and the fonnation factors. A detailed comparison with the SC type estimates is carried out in the subsequent section to sbow the original features of the MCSA estimate and discuss its potential application fields.

Compariso11s with other EMTs

The predictions by the MCSA of the effecti ve diffusivity of a porous medium, with phase diffusivities such as 0 Olhcr materials, such as for example porous media having a multi-scale pore structure like rocks or sandstones, where this highly diffusive phase always percolates.

• . . r

I - D 2 /D 0 • lrf' • •--1>, 10, -10 0.8 ---D,ID, •lo' . .... o,ID, •so' 0.6 0.4 0.2 ..... •• , , ••••••••• .. ' •••••••• .... --~ 0.2 0.4 0.6 (

R.4 Conclusions of chapter R

The success of models developed for engineering applications depends to a large extent on their simplicity and, in particular, on the easiness with which the parameters involved in the models can be physically interpreted. Based on simple space filling assemblages composed of two types of composite spheres, the MCSA estimate presently developed needs only one geometric parameter, in addi ti on to the phase volume fractions and diffusivities. This parameter can take into account the phase connectivity and is in direct relation with the formation factors (Eq. (11 1.62)).

Original features of the MCSA, such as its capacity to describe tortuous percolating paths, have been outlined and cou ld be of particular interest for mu lti-scale random porous media.

The MCSA estimate is also capable of capturing pronounced variations of the material diffusivity, while respecting rigorous bounds. It thus appears suited for predicting the macroscopic diffusive properties of porous media or more genera lly of random heterogeneous media with high contrasts. Therefore the multi -sca le homogenizati on approach developed in the ensuing chapter to predict the macroscopic diffusion coefficients of cementitious materials 'incorporates the MCSA model. materials (e.g. [START_REF] Oh | Prediction of diffusivity of concrete based on simple analytic equations[END_REF][START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi -coated sphere assemblage model[END_REF]) attribute these significant changes to the geometrical percolation of capil lary porosity. However, this explanation may not really be in accordance with the cri tical pore diameters measured by MIP, as shown below.

As already mentioned in Part I, gel pores sizes are assumed to be situated between a few nm and 0.2 µm and capillary pores ones to range from 0.2 µm and a few 11m, even though the size range of capillary pores varies in the different classifications for the pore stnicture of HCP proposed in literature (e.g. Powers and Brownyard l 948). [START_REF] Diamond | Mercury porosimetry-an inappropriate method for measurement of pore size distributions in cement-based materials[END_REF] evidenced that tbe critical pore diameter, defined as 1he pore width corresponding to the highest rate of mercury intrusion per change in pressure, allows for assessing the largest scale at which pores percolate. The values obtained from diverse MTP tests reported in the literature (see Table 1.5) are much smaller than the capillary porosity size range (> 0.2 µm). Th is means that the capillary pores, that are mostly inaccessible to mercury injected at low pressure, shou ld not percolate through the paste. Conversely, it is possible to find percolating paths of gel pores regardless of the volume fraction of total porosity in HCP, as already asserted previously.

Oo the basis of these indications given by MIP results, the present chapter proposes an approach focusing on the percolating gel pores located in the CSH. The strategy adopted for building a mu lti-scale homogenization model capable of reproducing strong variations of the diffusivities of HCP is thus the following: Bary (2007) computed witi the hydrntion morlel of Je1111i11gs ( 2000) aud Tennis and J en11i11gs (2000) am/ tire experime11tlll 11tells 111•em e11ts of capillary pores (lg11raslti et al. 2004).

Two additional parameters, namely the densities f "'' and/"•" of composite spheres with le porosity in the MCSA representing the CSH;,,, and CSH""' respectively, still need to detennined. [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi -coated sphere assemblage model[END_REF] . 5 ,,,., that vary from 0~1 1 ..,,( wl c = 0.30) = 7. 17 x IO • 11 m 2 /s D~'S 11 ._,, (w/ c = 0.65) = J. 04x I0-' 0 m'/s depend ing on its gel porosity. Jt is noteworthy that t' high-density CSI I;.,, that are much less diffusive than CSUcxi could be approximated as a no diffusive phase without strongly in fluencing the fina l results. However, it seems mo consistent to model this porous phase as diffus ive, s ince the CSl-1 1 "' contain a signi fica (Ric/ref et al. 1997) a1td predicted respectively by the m11/ti-scale model with f= 0.546 and by the simplified model in Stora et al. (2006b), as <1 fun ction of their total porosity.

For the sake of comparison, the results obtained with the two-step representation proposed in are modeled as a MCSA (Stora et al. 2006b).

The presenl multi-scale representation of the pore structure is sti ll very simplified in comparison with the real pore network. Nevertheless, it reproduces some of its important characteristics. First, the proposed description represents in a simple manner the percolating gel pores, wh ich is cons istent with the critical pore diameters measured by M IP. Furthermore, the gel porosity ofCS ll 0 " in the present model augments with the w/c ratio, while/is kept constant. Under these conditi ons, it can be seen from Eq. (111.56) tbat the quantity of percolating gel pores also increases in the MCSA. This is in good agreement wi th the micrographies of Bcjaoui ct al. (2006), in which the gel porosity of CSH"' appears to be of greater size and to be more connected in high w/c pastes tha11 in low w/c ones (Fig 1 .2).

The present homogenization model is quite similar to the one proposed by Bary and Bcjaoui (2006). The mai n difference lies in the modeling of the external coating, since in the present representation gel pores percolate instead of capi llary ones. It should be emphasized that most of the homogenization methods ex isting in literature focus on the perco lation of capi llary porosity and require tbe knowledge of its percolation threshold. But as outlined by Bcjaoui and Bary (2007), no consensus exists concerning its actual val ue because the defin ition of capillary porosity and the method fo r computing its percolation threshold vary from one author to another. The perco lation thresholds of capillary pores in HC P are often estimated with 1he help of advanced cement simulation tools (Garboczi and Bentz 200 1;[START_REF] Ye | Percolation of capillary pores in hardening cement pastes[END_REF].

Nevertheless, the diverse values proposed by these simul alion tools, globall y ra nging from 0.03 to 0.2, vary significanlly from one author to the other (Garboczi and Bentz 200 I;Y c 2005) and some of these estimations appear to be in contradictio n with the cri1ical pore thresho lds measured by MIP. The present model proposes an alternative approach rather focusing on the percolating gel pores located in CSll. !Ji the present descrip1ion, the capillary pores only percolate through a network of small er pores and the sudden varia1ions of HCP diffusivi ty are caused by the change in connectivity of gel porosity.

Eva/11utio11 of the macroscopic diffusion coef ficiems of leached HCP

The model is now applied to 1he case of a hydralcd CEM I paste subjected to leaching by pure water. The degradation scenario retained here is exactly the same as in [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi -coated sphere assemblage model[END_REF]. T he reader should refer to 1his paper for more details. The decalci ficatio n is supposed to be, in term of the volume of solid phase replaced by porosity, linearly related lo the decreasing calcium concen1ra1ion in 1he pore solution Cn: CH torn ll y dissolves before any other hydra1ion producls between c. = 21.54 and c. = 20.3 1 mol/m 3 , then half of AF dissolve bc1ween c. = 20.3 1 and c. = 3.08 mol/m 3 and the CS I I are progress ively degraded between c. ~ 20.3 I and c. = 1.09 mol/m 3 • More precisely, on ly 5 % of the CSH arc di ssolved and replaced by add i1i onal capillary pores in the mosl decalcified slate bul their internal porosity sign ifica111ly increases during degradation (Galle et al. 2004;lleukamp 2002). Their gel porosity is quan1i ficd by subtracting the sum of the initial capillary pores and the additional o nes, issued from the dissolution of hydration producls, from the measure of total porosi1y by M rP. Th is method is applied to a CEM I paste with w/c = 0.45 and the values of porosity comprised in the CSH before and after leaching indicate 1hat a to tal volume frac1ion of supplemeniary percolating gel porosity of 0.04 appears in the CSll during degradation. The effeclive diffusivi1ies of CSH;,,, and CSHc", wh ich arc updated from Eq. (llI.63) and from 1hese higher values of gel poros i1y, sign ificantly increase when decalci ficalion occurs.

An im portant issue for the application o f the present model 10 the case of degraded cement pastes is the possible format ion of percolating paths composed of capillary pores due to the disso lution of hydra1ion products. Neve1thclcss, lhc critical pore diameler measured by MJP for leached pastes (Galle ct al. 2004) remains in ferio r to the typical sizes o f dissolved hydrati on producls (CH, AF) situated at 1he micromc1er sca le, though ii increases by an order of magni tude (Table 1.4). Th is result suggests 1ha1 the add iti onal capillary pores fonned during 1he leach ing process do not percolate or only through sma ller pores. ii is consequently Look ing at these curves from the rig ht to the left, the macroscopic diffusivities of the pastes appear to first increase sudden ly. This augmentation is caused by the dissolution of all the po1tland ite, whi le the other hyd ration products (AF, CSIJ) are still intact. These solid phases then progressively dissolve as the calcium concentration decreases and the effective diffusivities continue to augment but less rapid ly. At the end or the leaching process, the increase of the effective diffusion coefficient is about o ne order of magnitude for the totali> leached cement pastes with low w/c ratio (0.30 -0.40) and a factor of 6 to 9 for the ones witt The present predictions are also close to the results of Bary and Bejaou i (2006) obtained wit!• a comparable assemblage, where the same degradation scenario was adopted. But th~ advan tage of the present model compared to the previous one is that the MCSA allows fo1 representing in a more simple and conven ient manner the percolat ing porosity than the SC scheme, that can be real ized only by a complex microgeometry (M ilton 1985).

Estimations of the macroscopic diffusive properties of mortars

Many models already exist to deal with such applications (e.g. [START_REF] Care | Application ofa n-phase model to the diffusion cocflicient of ch loride in mortar[END_REF][START_REF] Garboczi | Analytical fonuu las for interfacial transition zone propert ies[END_REF]. It is worth saying that the effective diffusivity of a mortar is approximatively of the same order as the diffusivity of its cement matrix. Some authors [START_REF] Nguyen | Couplage degradation chimique -comportement en compression du heton[END_REF] argue that it is due to the fact that the presence of aggregates in a hydrate cement paste matrix probably has two opposite effects on the transport properties that compensate each other. On the one hand, it increases the tortuosity of the matrix and reduces the total space available to diffusion and, on the other hand, the presence of porous ITZ enhances the diffusion process.

The homogenization model presented below is based on a few assumptions. First, the macroscopic diffusion coefficient of the cement matrix of mortar will be supposed equal to the one of the bulk cement paste. In addition, the ITZ is supposed to be a homogeneous intcrphase, even though it would be more realistic to model it as an inhomogeneous one (Lutz and Zimmennan 2005). A doubly coated-spheres model is adopted to represent mortars (Fig. llL20). Precisely, the core, the intermediary layer and the external one are respectively occupied by sand particles, ITZ and HCP, which diffusive properties result from the previous step. The sand aggregates are non-diffusive and, as in linear elasticity, the diffusivity of ITZ is computed by means of a simplified model [START_REF] Care | Application ofa n-phase model to the diffusion cocflicient of ch loride in mortar[END_REF]). The macroscopic diffusivity of mortar can then be estimated by the GSCS or modified DIM (Eq . (111.25)). This homogenization approach for computing the diffusivity of mortars will be applied and va lidated in Part V, which proposes simulations of the chemical degradation of mortar samples.

Sand particle

CH or aluminates

Capillary pores

(> 0.2 µm)
Part III: Prediction of the 1naeroscopic diffusiviti..:s of ccmentitious n101tc.rials

I .-~---~•~ .. , •. \ I \ ' • I : I CSHUI A • • ' \ • • ...... ..

CONCLUSJONS OF P ART Ill

The contributions of this Part are both theoretical and practical. The estimations of the macroscopic transport properties of porous med ia such as concrete materials represent a challengi ng issue for homogenization theories because of che difficulty of reproducing tJ1e highly non linear changes of these properties observed experimencally (e.g. [START_REF] Oh | Prediction of diffusivity of concrete based on simple analytic equations[END_REF]. The brief review performed reveals that only a small group of common ly used EMTs, like GEM or Kirkpatrick models, exh ibits che capacity co model the sudden variation of che effective diffusion coefficient of porous med ia d ue to a critical microstructure change. lt is in particular po inted out that commonly used matrix-inclusion type EMTs are to be excluded for predicti ng such rapid changes of diffusiv ity, since they are inadequate for modelling percolating pores.

A novel estimate associated to a MCSA apt to account for pore connectivity has thus been developed. By construction, this MCSA scheme automatically respects the HS rigorous bounds. Another salient feanire of this estimate is its capacity to model connected phases formi ng tortuous paths in a geometrically simple and clear way, which is essential for the prediction of effective diffusion properties of porous media. In this sense, as an estimation model, it provides a real improvement on HS bounds and other EMTs and it is expected to be applicable for a large class of porous materials, such as cementitious ones, granular media and rocks.

From a practical standpoint, a three-step homogen ization model, wbere the two types of CSH containing connected gel porosity are modelled as a MCSA, is specifically developed for tbe prediction of the macroscopic diffusivities of HCP and validated by confronting with experimental measurements performed on CEM I pastes. But the success of models developed for engineering applications also depends on their simplicity of use and on the number of parameters it involves. The present model well complies with these criteria. Apart from the usual intrinsic parameters, such as phase volume fractions and diffusivities, it depends only on two geometri cal parameters that allow for characterizing the connectivity of gel pores i11 the CSH'"' and in the CSH"''. The model distinguishes itself from already existing approaches in the sense that it does not require the knowledge of the percolation threshold of capi llary porosity. The multi-scale homogenization model is furthermore applied to cement pastes submitted to leach ing. The significant increase of cement paste diffusivities predicted highlights the highly detrimental influence of the dissolution of hydration products.

Ap pendix Ill.A: Derivation of the inhomogeneous Eshelby tensor for a sp herical isotropic inclusion

We are presently interested in solving the boundary value problem depicted below (Fig.

III.

A. I) so as to derive an analytic expression for the inhomogeneous Eshelby tensor of a spherical isotropic inclusion. All phases are supposed isotropic so that it is possible fonnulate the reasoning with scalars instead of second-order tensors.

Figure Ill.A.I: Schematic of a splterica/ isotropic i11c/11sio11 witlz a ltomoge11eo11s interpltase enclosed i11 an infinite matrix.

The genera l so lution for this problem can be expressed in the following way:

c, = B,r+~. r / 28
.1

J, = -Dcil B, -~-g ~ B Cc1 = 83r +~, r r' J l ei = -De{ B, - 2 ~•} I CM= Bsr+.!1, ,.1 l J M =-D~{ 85-2 ,. ~} (I ll.Al) (lll.A2) (ll l.A3)
where B; (i e (1 ,6]) arc constants tliat can be detcnnined from the interfaces and boundary conditions. It is straightforward that 8,_ = 8 5 = 0. It furthennorc comes from the definition of the inhomogeneous Eshelby tensor in Eq. (111.13) that 8 1 1 = .A 1 .g•1 (Duan et al. 2006). T he interfaces conditions at r = r; and r = r, respectively read: (Ul.A4) (III.AS) from the interface conditions written in Eq. (lll.A4), it comes chat 8>=(-•l;"-j)g' and B, = 8 ;;' (IJl.A6)

The unknown constant 8 6 is lhen gained from Eqs. (111.A5-A6) :

B, =[( J, -i)r,' +jr,' ]g•. (lll.A7)
By injecting the expressions of Eqs. (lll.A6-A7) into (111.AS), the following expression is finally obtained for ii;":

A:' =(1+ 2p: 1 p').

(Ill.AS) Appendb: 111.B: Derivation of Eq. (J il.61) This Appendix details the calcu lations required to obtain Eq. (lll.61). To begin with, Eq. (111.54) is rewritten as:

fl ' -{J' • fl' -n 2 (1 -) • - 2C2 t • -1-'1 -C2 • (Il l.BJ)
Using Eq. (11 1.55), /J! can also be expressed as a function of c; : fl ' _ /J' c, -Jc;

• -2 .

1-/ (ITI.82)

By recall ing that the connected volume fraction er"' of phase 2 vcri fies er'' =Jc;, Eqs.

(TII.81 -B2) become:

o.t I Owl'\ p: =fl; c, -c, ' fl.' = fl.2ll -1-J. 1-f f (Ill.83)
Substituting the expressions (Jll.B3) of /J} and /J. 2 in Eq. (ITI.56) yields Eq. (Il l.61 ). D2 Dz (III.Cl)

We next cons ider the limit of (III.Cl) when 0 2 ~ oo, in order to make the formation factor F, appear. Noticing tl1at /J~ tends to one when D 2 ~co, Eq. (111.CI) simpl ifies into:

-

1 [(1+2/-c,)- 1 +2(l -/-c 2 ))=0. (lll.C2) F, F 2
Two cases are to be treated. If F 2 ~oo (i.e. D 0 / D 2 ~O), no relation can be established between f and F 2 . This case corresponds to the situation in which the phase 2 does not percolate. Consequently the lower HS bound, also obtained by taking f = 0 in the MCSA estimate, should be applied. If F 2 is finite, Eq. (JI l.C2) yields:

f =c (2F 2 +11_ I 2F -2J 2 • (llI.C3)
Analogously, it is possible to express fas a function of F, by inverting the indexes 1 and 2 and substituting/by (1 -/):

f =l -c (2F

MODELLING OF DAMAGE OF CEMENTITTOUS MATERIALS

The damage of heterogeneous materials such as concrete is a complex nonlinear dissipative phenomenon, which has been the subject of extens ive researches (e.g. Krajcinovic 1996).

Damage in cement-based materials is generall y qual ified as quasi-brittle and is caused by tbe nucleat ion, growth and coalescence of microcracks. These cracks grow preferentially in certain directions depending on the loads app lied thus induci ng an an isotropy of the damaged material. Many models, which are more or less phenomenological, have been proposed for treating damage in concrete (e.g. [START_REF] Mazars | Application de la mecanique de l'endommagement au comportement non lioeaire ct a la rupture du beton de structure[END_REF]1985). But some phenomena such as the unilatera l effects of the opening and closing of microcracks represen t challenging issues for researchers (e.g. Connery and Wclemanc 2002). Aim ing at proposing a damage model that can be iJJcorporated into a FE code, the present Pait adopts a hybrid approach using homogenization techniques to determine the number and nature of damage variables and t11e free energy for a given damage state but also revisit the mai n ideas underlying the most popu lar models in concrete (e.g. [START_REF] Mazars | Application de la mecanique de l'endommagement au comportement non lioeaire ct a la rupture du beton de structure[END_REF]1985).

The framework of continuum damage mechanics presently employed 10 develop a physically sound model is usually associated with the thermodynamics of irreversible processes with internal variables (e.g. [START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF]. As noticed by He and Curnicr (I 995), a damage model within this framework is generally composed of three parts. First, the damage variab les are adequate ly chosen to characterize as accurately as possible the damaged state of the concerned material. Second, the material behaviour is formulated for a given damage state.

For this purpose, homogen ization techniques are employed to remedy the lack of uniformity and rigor in formulating the free energy function. The evolution laws of the damage variables for a loading history are then established. The damage model presently developed docs not clai11J to tackle complicated issues of damage, such as unilateral effects or complex load ing.

In lhe manuscript, the model is only app lied in tension but could be extended to the case of compression w ith the aid of the work of Dcudc ct al. (2002).

T he present work first assesses the capacity of prominen t homogenizat ion schemes to predict the mechanical effecti ve properties for a given damage state. Then different evolution laws for the damage variables arc discussed and a strain-based criterion is established. The damage model developed is then applied in uniaxial and biaxial tension tests. It is outl ined that we aim at bui lding a relatively simple model based on the ensu ing s implifying assumptions. The undamaged material is considered to be isotropic. The assumptions of small strains and of isotherma l conditions arc adopted. Damage in cement-based materials being quasi-brittle, plasticity may be neglected for simplicity at least in tension. In addition, the unilateral effects due to the opening or closure of microcracks and the frictional effects are neglected.

DEFINITION OF STATE VARIABLES

The state variables are defined as the group of variables which current values characterize the physical state of a system at equilibrium. Only mechanical state variables are presently considered. The state variables are constituted of the macroscopic strain t associated with the Many anisotropic damage models use a second-order damage tensor (e.g. Ladeveze 1983), although such choice may lead to strong shortcomings (Krajcinovic 1996). Among them, Connery and Welemane (2002) evidenced that the spectral decomposi tions of these secondorder damage tensors may lead to certain inconsistencies, for example the non uniqueness of the thennodynamic potential. Micromechanical approaches (e.g. [START_REF] Kachanov | Effective elastic properties of cracked solids: critical review of some basic concepts[END_REF]) are more recent than phenomenological models but may reveal very difficult to implement in a FE code.

The basic idea used by the micromechanical models for dealing with damage consists in representing the microcracks as degenerated ellipsoidal voids. By degenerateo it is meant that the lengths of one or two of its axis are reduced to zero. In the present part, only pennyshaped spheroids are considered. Damage variables that characterize material mierodefects are the crack density parameters d 1 characterizing a family i of cracks having the same nonnal vector e, associated with the driving forces Ydi defined in the ensuing. More precisely, these crack densi ty parameters introduced by Budiansky and O 'Connel (1976) are defined as d,~N,a/, where N 1 is the volume density of the family i of microcracks and where a 1 is the maximum axis of the degenerated spheroid represent ing the microcracks of type i. The damaged material is then viewed as a homogeneous matrix of cement-based material in which the microcracks arc distributed. The idea of modelling these cracks as penny-shaped spheroids Permits to account in an intuitive manner for the anisotropy caused by loads. In the ensuing, the presently defined crack density parameters d 1 are chosen as damage parameters. The total number n, of families of cracks necessary to correctly model the damaged material is discussed later. where the effective damaged stiffiiess tensor c• depends on the homogenization scheme adopted. The scope of the next part consists in finding a relevant one providing accurate estimations for the effective damaged stiffi1css tensor c•. Different micromcehanical schemes arc presently reviewed to find which ones provide the best estimations. However, the present review is only focused on explicit schemes, that can easi ly be implemented in a FE code and do not require any numerical resolution to obtain the effective damaged stiffness tensor c•.

.J Gener al presentation

A brief review of the micromechanica l schemes, that cannot pretend in any case as being exhaustive, is presented, starting obviously from the n1dimcntary dilute model (e.g. [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF]) to more recent ones. To simplify the ensuing calcu lations, the following two tensors, wh ich are both linked to the Eshelby tensor r.:' of the penny-shaped inclusion representing the type i fami ly of cracks embedded in the sound material, are introduced:

1f, = lim 'I (I -:i :::' )-1

, lFI~'' ='!._tr d, T, : s.,,

11->0 3 (IV .2)
where r 1 is the aspect ratio of the penny-shaped inclusion representing the type i fami ly of cracks and lFI~' defined in Eq. (11.34) is called the dilute compliance increment tensor of type i microcracks. It is recal led that I and S., designate the fourth order identity tensor and the compliance tensor of the undamaged material behaving as a matrix entrapping the pennyshaped crack, respectively. The tensor llll;" has got a simple expression, since the only non vanishing tenns are [START_REF] Mura | Micromechanics of defects in solids[END_REF]):

H,~:

16(1-v 2 ) d • H '., 3£ ,, i/lj " 8(1 -v 2 ) d . . 3(2 -v)E"' ' (.1 ;t 1 ) ' (IV .3)
where £., and v respectively designate the Young modulus and the Poisson ratio of the undamaged material. Using the tensorial operators, the tensor IHI;"' can also be written as:

llll",' =" 16(1 -v ?) d [t®L\ + L\ ® 1-vL\ . ® L\ .] I ~ 3/!i/ (2-v) I -, 1 - I I ~ {IV.4)
where 11, = e. ® e, and where it is recalled that 1 is the second order identity tensor. It is noteworthy that, if all the cracks are aligned, the tensor Jl[~' is then singular.

The estimation for the effective stiffness tensor in Eq. (IV.I) by the dilute model computed with homogeneous displacements applied at infinity (e.g. Eshelby 1957) varies linearly with the crack density parameters:

4 C~, = C,,:(I-p, T,), with p 1 =37rd,.

(IV.5)

Jt is however well-lmown that the dilute model does not account for the interactions between microcracks. The MT model capable of accounting for interactions in a certain manner between the cracks and the matrix, gives tbe following expression of the damaged effective stiffness tensor (e.g. Bcnveniste 1987):

. ( where n~, = c., : (1 -.r.::,) denotes the eigenstiffness tensor of the double-inclusions surrounding the family i of cracks. These double-inclusions practically characterize the spatial distribution of cracks. Recalling that spherical double-inclusions are generally adopted for isotropically distributed cracks, the non-overlapping condition is respected provided that crack densities d, remain interior to 3/47r (sec Appendix 11.B). For higher crack densities , the full -range !DD proposed by Zheng and Du (200 I) employs spheroidal double-inclusions to fulfill this non-overlapping condition but leads to complex computations. It is noteworthy that, even with spherical double-inclusions, a simple analytical expression for lll'" 0 such as the one in Eq. (IV.4) for the di lute scheme is difficult to reach for materials with at least two families of cracks having different orientations. The matrix representation of lll' 00 then cornprises extra-diag(lnal terms and as a result the expression of c; 0 ,, may be tedious for elastic bodies weakened by many families of cracks. A slightly simplified version of the IDD

A general expression/or explicit schemes

By taking benefits of similarities between the compliance increment tensors of MT and IDD schemes written in Eqs. (IV.4) and (fV.9), a general expression denoted Ill for the latter tensor bas been found to cast IDD and MT schemes in an identical framework:

Ill = £~, [aL\©L\+b(t~L\+L\~1 -21.\©L\ ) ]. ( ) _, . d d with a = -1-- d~"', de (IV.IO)
where Q DG is the deviatoriC part Of the eigenstiffneSS tensor Q~ Of the double-inCIUSiOn characterizing the distribution of cracks, de denotes a critical crack density parameter and where d;' corresponds to the critical crack density for which the dilute estimation of the Young modulus goes to zero. Tbe IDD and MT estimates may be retrieved for particular values of de and n,.oc;. For n oc;=E., (7-5v)t[t s(t -v 2 )] and dc= d:J'°= 45/128 , the previous equation coincides with the expression of the compliance increment tensor in Eq.

(IV.9) provided by the IDD scheme. Observe that d:;' 0 corresponds to the crack dens ity for which the 100 estimation of the Young modulus goes to zero. Simi larly, Eq. (IV.10) coincides witl1 the MT esti mate in Eq. ( IV.4) for D. 00 =E" t[2(1 + v)] ~nd d e going to infinity.

It is furthermore pointed out that simple general expressions can be obtained from the G,"" ( d

' J_ , ( 16( 1-v)d)_, G,11 = , 1 + ( l +v)(2-v)dt' = I+ 3(2-v) '
and if de= d~', they coincide with the dilute estimation:

G'"'

and -'-= I c ...

16{1 -v)d ( 1 ) ( ) +O d , 3 2 -v Page 146 (IV.13)
where d~' appears clearly to be U1e critical crack density for which the di lute estimation of the Young modulus goes to zero. Finally, the JOO predictions are also retrieved for Q oc ""E., (7-5v)t[t5(1-vl )] and de =d~oo E,' 00

=( 1 _ l28d ) [i-128d(I 45 J]-' = I -60p( l -v' ) (JV. l 4 ) E,, \ 45 45 12Rdt ' 15JT+4p(7 -I Sv 2 ) ' .::L--I+ 2-v Gmo [ d ( G"' - ( l +v)d~., ( ) 15(1-v' )d J'-' ] -'= I (7-5 v)d~' 60p(l -v) 15JT(2 -v)+8p(4-5v)'
w ith p = 4JTd/3. The effective Young and shear moduli of the damaged material along c 1 axis predicted by the simplified IDO can be written as:

E:' . 00 =[I+ ~' l ' I+ ~;0 )d ,]_, E.., de de (IY.1 5)
ln the ensu ing, these estimations of the elastic moduli obta ined with tl1e dilute, !DD, MT and simplified IDD homogenization models are compared to each other and confronted to numerical results.

1.2 Evolution of the e.rp/icit estimates with crack density

For the sake of comparison, the respective pred ictions of the previously mentioned schemes -- Clear:c11t differences may be observed on Fig. IV .1 between the curves respectively estimated by the dilute, !DD and MT schemes. The elastic properties predicted by MT estimate evolve very slowly when the crack density is high, while the elastic moduli predicted by the dilute model vanish rapidly. By the way, that is why the dilute scheme is considered as a lower bound by K.rajcinovic ( 1996). The curves computed with the IDD method are situated between the dilute and MT ones and go to zero for certain values of the crack dens ity parameter. It is recalled that the interaction effects between cracks are neglected in the dilute estimate and that they are only integrated in an indirect way io JOO and MT schemes, since they can be interpreted as dilute models where the stress applied at infinity is modified to The different results obtained in the case of al igned cracks for the evolution of the Young modulus with crack density are gathered in Fig. IV.2 and compared with the predictions given by micromechanical schemes. lt is pointed out that numerical resu lts (Feng ct al. 2003a; Shen and Li 2004} are systematically situated between MT and IDD or PCW estimates. These simulations arc performed with the Kachanov's method [START_REF] Kachanov | Elastic solids with many cracks: A simple method of analysis[END_REF] to account for the interactions between cracks and consider an isotropic distribution of cracks. It is complicated to determine wh ich model between MT and lDD ones is best suited for estimating the damaged effective properties. The IDD scheme seems to be in better agreement with the numerical results for low and moderate crack densities (p < I), whereas the MT predictions lay closer to the simulations for very high density of microcracks.

•~ 0.4 0.2 . \ \ \ \ ---- ........... < . - > 0.8 0.6 0.4 0.2 0.5 I 1.5 2 2.5 Crack density parameter p ---MT • -• -• Oilule • • • • • • • Full-
It would consequently be va luable to find a model capab le of scanning all the values comprised between TDD and MT predictions. The full-range lDD defined in Part II , which predictions arc situated between the ones of IDD and MT schemes, is thus clearly better than IDD, which suffers from the fact that the effective Young modulus becomes zero for a moderate value of p on Fig. JV.2. However, even though the full-range IDD is explicit, it is quite difficult to use it in practice, since the expression of the cigensti ffness tensors n.;;, of the double-inclusions characteri zing the distribution of cracks changes for different values of the crack density parameters (see Appendix 11.B}. The simplified version of IDD proposed in Eq. (lV.8} yields surprisingly good results. It then represents a good compromise, since it is both simple of use and accurate. It is however worth noting that the differential method (l lashin 1988) also provides accurate predictions, according to Feng and Yu (2003a;b) and [START_REF] Shen | A numerical simulation for effective elastic moduli of plates with various distributions and sizes of cracks[END_REF]. But a decisive argument in favour of the different IDD schemes presented is thei r si mplicity compared to impl icit models such as tlic SC, GSCS a11d the differential one, wh ile providing results close to the differential one. The differences between IDD and MT models lie on the respective distribution of inhomogeneities considered. While the distribution of cracks is necessari ly flat in the MT model (Stora et al. 2006a), this distribution in the PCW or lDD model depends on the aspect ratio of tbe double-inclusion that cou ld thus constitute an interesting parameter. The choice of a spheroidal double-inclusion may somehow accoljnt f9r the anisotropy induced in the distribution by the mechanical loads. It may thus introduce a further anisotropy aspect due to the distribution of cracks, i11 addition to the classical anisotropic behaviour of the damaged material, but it leads to complicated calculations. This is the reason for which spherical double-inclusions are adopted in the ensuing for the computations of these models.

li.2.2 Comparison for an isotropic damaged material

The analytical expressions for an isotropic damaged material can easi ly be deduced from the relations derived for the case of al igned cracks by performing the following average operation on all the possible orientations denoted by curly brackets (e.g. Benven iste 1987):

I 1 ' I )
Q= {T}= -T J+-lT --T DC.

3 """ 5 ,,.. 3 """

The following relations are thus.obtained for the concerned schemes: .

J/.2.3 Comparison for a11 ortl1otropic damaged material

The case of a material containing three families of cracks, which nonnal vectors are respectively oriented following e., e2 and e3, is now considered. The resulting material is thus orthotropic:

H"" = L P1 T. :S.,,

I (lV.21)
where i denotes the family of microcracks which normal is oriented following e, (i = l ,2,3). In the present case, the tensor Ill'"' has got a simple expression and is not singular if di. d 1 and d 3

are not equal to zero, on the contrary of the case where all the microcracks are parallel:

,. 1

16(1 -v' )d• H '"' = 8(1 -v' ) (d +d)• Hail = 8(l -v 2 ) (d +d )• H ;;;, 3£., " " " 3(2-v)E., 1 l ' mi 3(2-v)E., 2 ' ' H"" -8(1 -v' ) (d +d) 1212 -3 ( 2 -v)E., 1 2 ,
or written in tensorial form:

111• 111 = 16 (1-v' ) [~ d ( t ®i.\ +i.\ © t -vA ®11.)]• 3£M(2 -V) ~ ' . f I • I ' (IV.22)
The compliance increment tensor JHI' 0

" and H' 00 predicted by the original and simplified 11.3 Link between t he macroscopic behavior and t he nu mber of microcracks

The number of fam ili es of cracks chosen to represent the microdetects of a damaged material is an aspect of primary importance for the use of homogenizat ion procedures detailed before.

A compromise needs to be found so that the model developed accurately represents the actual roicrodefects and is at the same time tractable analytically. To answer to this problem, the directional dependence of the Young modulus predicted by the IDD model for different numbers of fami lies of cracks is therefore investigated.

Let a damaged material be represented by an elastic body weakened by nc families of cracks.

The estimation of the effective compliance tensor of the latter material by the IDD model is denoted as s; 00 . The predicted effective Young modulus of the damaged composite along an arbitrary direction of the space can be deduced from this compliance tensor with the help of the subsequent relation: The rcccDt application of micromechanics for modelling damage is a promising way to relate the damaged materia l behaviour to its microstructure. However, relatively few micromechanically-based models have been incorporated in FE codes (e.g. [START_REF] Bary | A coupled therrno-hygro-mechanical model for concrete subjected to moderate heating[END_REF][START_REF] Zhu | Micromechanical modelling of anisotropic damage in brittle rocks and application[END_REF]), because they are usually more difficult to handle than phenomenological models especially in anisotropic cases.

In order to buil d a micromechanically-bascd damage model i11corporable in FE codes, different explicit homogenization schemes have been reviewed. Their predictions of the effective mechanical properties of an elastic body weakened by aligned microcracks have been confronted with 20 numerical simulations accounting for interactions between microcracks. The rather good agreement of the IDD scheme, providing identical predictions as the PCW and DI type models, with these numerical results constitutes an argument in its favour. A simplifi ed version of this estimate that seems better su ited for dea ling with materials weakened by diverse families of cracks has also been proposed (see Eq. (lV.22))

and is employed in the ensuing to formulate the damaged material behavior.

The crucial q uestion of the number of fam ilies of cracks necessary to represent the an isotropy of the damaged material is also addressed. In plane stress conditions, four families seem to be adequate for such representation. Nevertheless, si11ce the damage model developed will only be applied to quite simple cases of loadings, only two families of perpendicu lar cracks are adopted in the ensu ing for simpl icity reasons. The model sho uld however not be applied for anisotropic cases that are more complex than orthotropy. The next chapter aims at developing evolutio n laws for the crack dens ity of these two families. The present chapter aims at bui lding suitable evolution laws for these damage variables while taking care that they respect the second principle of thermodynamics, i.e. the Claus ius-Du hem inequality wdtten just above. 12.J C riteria based on the generalized standa rd materials (G SM) formalism

The evol utio n law of the damage variables are established adopting the GSM formalism . detai led below. The framework of GSM constitutes a convenient method to automatica lly guarantee 1he fulfillment of the second principle of thermodynamics. The grounding paper of this approach was written by [START_REF] Halphen | Sur lcs matcriaux standards generalises[END_REF] and the underlying mathematics, the theory of convex analysis, can be found in detai ls in severa l books, for instance [START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF] .

12.J.J Prese11tatio11 of the GSM formalism

I Basic definition

The basic definition proposed by [START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF] for the GSM fo1malism (e.g. [START_REF] Halphen | Sur lcs matcriaux standards generalises[END_REF][START_REF] Marigo | Modelling of brittle and fatigue damage for elastic material by growth ofmicrovoids[END_REF][START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF] stipulates that the constitutive laws of a given material arc generalized standard if they are associated with a free energy and a dissipation potential having the following characteristics: they depend on the current va lues of the state variables and the dissipation potential is convex with respect to the .flux. According to the GSM model , the constituti ve relations arc thus fully defined through the choice of a specific internal energy functional <I> and of a specific dissipation functional '1 1 , w hich depend on the local values of the state variables. A simple illustrative example is now given to show the philosophy of the GSM formal ism, the mathematical basis of this theory being presented later.

Simple illustrative example

To automatically guarantee the fu lfillment of the Clausius-Duhem inequality (IV.24), the basic idea ca1Tied o ut by the GSM models consists in finding a convex dissipation potentia l 'fl that is minimal in zero and depends on ly on the internal variab les so that: r _ a•.pcJ,)

.,, -()(J • , ( JV.25) if 'I' is differentiable, the case for which Y' is not differentiable being treated later. For instance, if the following convex dissipation potential is employed:

(IV.26)
where Y 0 is the damage thresho ld in terms of elastic energy density. The positiveness .of ihe intrinsic dissipation is then immediate: Dis= Yod/ (?:. 0) . However, the driving forces thus obtained depend on the rate of damage:

y di = 2 r.,ci, .
(lV.27) (IV.28) Usually, t11e evolution laws proposed in literature for brittle or quasi-brillle materials arc rateindependent (e.g. [START_REF] Marigo | Modelling of brittle and fatigue damage for elastic material by growth ofmicrovoids[END_REF]. The utili sation of the GSM theory for the case of rateindcpendent materials treated subsequently requires some more efforts, even though the basic ideas remain the same.

12.J.2

Application of the GSM formalism for quasi-brillle materials

Jn the case of rate-independent materials such as brittle materials considered here, the dissipation potential 'fl is not necessarily differentiable. Therefore the notion of sub-gradient of 'fl is introduced to apply the GSM fom1al ism to non differentiable dissipation potentials •

• °"'[ • • ] . {0 if Y"' E K '1 1 (d 1 ,fJ,) =Lt >;.di + .7~. (d,) with 'JK (Y d,)= . , , ~ if~eK (fV.32)
where the indicator function .7~, (d) enforces the positivity of the damage rates and where K is a convex domain. By setting d, = 0 in Eq. (rY.29), the intrins ic dissipation appears clearly to be non-negative:

(IV.33)

In addition, simple.damage threshold functions are derived from the dissipation potentia l, the details of the demonstration being provided in Appendix TV.A:

.Ii= Y,1; -Yo.

(IV.34)

The latter fracture-type criterion may be too simple to faithfully represent the evo lution of damage. Indeed, if Y,. does not depend on the damage variables, the damage surface does not evolve during the damage process. More versati le evolution laws adopting the GSM fonnalism are proposed in the next part.

12./.3 Development of evolution laws adopting the GSM formalism

In order to develop more advanced criteria than the one given in the previous equation, tlle following form is used for the Helmhotz free energy: and characterize the evolution of the damage s urface threshold . They are associated with driving forces denoted X 1 • By analogy with the concept of hardening in plasticity, the function <p corresponds to the free energy stored by damage so f!ening. Practicall y, it represents the effect of the damage state on the further development of damage. It is noteworthy that, 1he damage so f!ening parameter /J; being con sidered as internal variables • the intrinsic dissipation reads as follows:

<J> = _!_ t: C'(d,): t + q:i(, 8
Dis = -il<ILJ, _ il!JL fJ, = Y, 11 d 1 + X 1 (J;(' 2 0) .
'dd, o/J;

(fV.36) [START_REF] Lorentz | Analysis of non-local models through energetic formu lations[END_REF] showed that by adopting the following expressions for the dissipa1ion potential and the free energy: where the function !n 1 represents the evo lution of the damage threshold surface. The convex ity of the proposed diss ipation potential taking its min imal value for /J, = d, = 0, with i e [I, 2], guarantees the ful lllment of the second principle of thennod ynamics. Furthermore, the i11dicator function ioi (.lJ, -d,) en forces that the interna l damage variables d 1 and /}; remain equa l to each other.

'P(d,,iJ,) = L:[ Y 0 d 1 + ~• (d,)+ 1<oi (jJ, -d,)] , ' (
Concerning the damage threshold functions, the forces .!' ; 11 are obtained by homogenization, wh ile the expressions of !n, arc unknown and may be identified in different manners (e.g.

Mari go l 985) so as to find an adequate evolution law. In particular, it is sufficient 10 adopt the dissipation potential in Eq. (JV.37) and to rewrite the Hclmhotz free potential as: <P = .!. ( IV.41)

The lauer evolution law is further investigated in a simple case.

12. 1.4 Application in the case of multi-axial tension tests

The Mari go criterion ( 1985) iJltroduced in the previous subsection is presentl y employed to define the evolution laws of a mortar subjected to uniaxial and biaxial tension tests. The purpose of this part consists in testiL1g if the Marigo criterion provides an adequate damage evolution law so that the stress-strain curve of a mortar subjected 10 uniaxial tension predicted by the micromechanically-based damage model shows gond accordance with the experimental curve measured by Le Bellego {2001). The case of biaxial tension tests is also treated to observe how the crack densities of the different fami lies of cracks evolve during non uniaxial tests. The mortar is initially assumed to be linear isotrop ic and is supposed to be io a plane stress state:

;:;:., = O:n = a-n = 0 . The consistency condition imposes that : In the case of a uniaxial tension following the e 1 axis, the tension applied merely affects the fami ly of cracks with normal vectors oriented along the e 1 axis so that we can consider that d2 = 0 and 4 2 = 0. The set of eq uations just above then simplifies into: --------MT ------~-----~----- In more genera l cases, the system of differential equations in Eq. (IV.45) can be written in a matrix form io order to find the damage amplitudes A. 1 and Ai:

-- -• -• -• -• --JOO .... ~-• -• -• 2 3 x 10°" 0.4 .-
i)ydl ---(J) i)d, i)Yd2
ad, (IV.48)

When the simplified TDD model is used to estimate the free energy, the forces Y,1; do not depend on dj U"' I) so that no coupling exists between the two criteria. The system can then be solved using for instance the basic expl icit Runge-Kutta algorithms incorporated in MATLAB, since the matrix above is diagonal and the system reduces into two decoupled equations. It shou ld be emphasized that the complex ity of computations of the evolution laws further increases if many families of cracks are involved. The main difficulty concerning the use of the TDD estimate arises from the fact that the effective stiffness tensor does not vary linearly with the crack density parameters. A hint used by Zhu ct al. (2007) .... 0.3 "13 s 0.2 "13-0.1 of damage variables are adopted for the three models

••• -•-•-•-•-•-. . -•-•-•--•-•
.~;.-...,:.;;;;.._~ The application of'Marigo evolution law to the micromechanical damage model appears to be rather complicated to implement in a FE code w ithout making simplifying assumptions. T he next subchapter is devoted to strai n-based criteria that are frequent ly emp loyed in concrete in order to find a simpler method to establish the evolution laws.

• -• -• Simp. !DD •--•-•-•-•-• --IDD ---MT
12.z Strain-based criter ia /1.

2.I Review ofstrain-based criteria

The damage loading function/can be expressed using an equivalent strain denoted c :

l = &-e 0 ,
(fV.49)

where e 0 designates the in itial damage threshold. One of the great interests of having a strainbased criterion is that it accounts for the sensitivity of the material to extension. Two forms proposed for the equivalent strain respectively by Ma7.ars (1984) and De Vree et al. ( l 99S) present valuable features for dealing with damage in concrete. In particular, they account for the differences of behavior of cement-based materials in traction and in compression. It is therefore of interest to thorough ly assess and compare these two theories of equivalent strains.

The following strai n invariants are introduced:

1 1 1 '> = tr£, 1~•l = ~[ ( trn ) 2 -trt 2 ] . Iy> = dett , l j•> =~tr(&-~( trt)l J, (IV.SO)
where tr denotes the trace of a tensor and det its determinant. Identical invariants, obtained by substituting r. by a in the previous equation, may be used for the stress tensor. The criteria proposed by [START_REF] Mazars | Application de la mecanique de l'endommagement au comportement non lioeaire ct a la rupture du beton de structure[END_REF] --,(k~~J---• ---• -• . . . : . . 12.2.2

Review of the Mazars model

The Mazars model is without contest the most popular dall'lagc model for concrete. Its success seems mainly due to its simplicity of use in a FE code and to its capacity of accounting in a simple and relatively cfficienr 111anner for the differences of behavior of concrc1e in tension and in co111pression. W here different expressions were proposed for K(tr:D) by [START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi -brittle materials[END_REF] to provide a &ood fitting with the experimental stress-strain curves in tensio11 and in compression. The damage multiplier is then determined from the consistency cond ition and the damage evolution law takes the general form:

(IV.61)

Jn the next subsection, a similar reasoning is carried out to establish an adequate evolution law for the crack densities parameters that are very different from the macroscopic damage variables employed by Ma7.ars ( 1984) and Oesmorat et al. (2007).

12.2.3 Deji11itio11 of a strain-based evolution la,., for the 111icro111ecl1anic11fly-base<I damage model

On the view of the previous considerations, the ensuing type of strain-based cri terion 1s

proposed as a starting point

f =6 0 , -"( trd ), (JV.62)
where d is a second-order micromcchanical damage tensor that reads in the (e 1 , e 2 , e 3 ) orthononnal basis: ' and c=-g t ov .

c c c

These damage loading functions can be extended to the case of two families by using the trace of the damage tensor d : where Y, is the driving force associated with the micromechanical damage tensor d and R is a projection operator that projects the principa l strain tensor on the (ei, ei, e3) reference basis.

I= K _, (
The derivation of the dissipation potential yields the damage evolution law:

. ()'!' ' (

)2 d = .it i)Y = 4 R.c,,,.

• (IV.69)

An essential difference of the present approach with phenomenological models, such as the one of [START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi -brittle materials[END_REF], is that the principal directions of the damage tensor are now fixed. The damage multiplier is then determined from the consistency condition and the damage evolution law takes the general form:

ci = d~-• ( iov ) 2 (Rt,)~ .
(JV.70) dl:ov tr R .c,, + It was observed that, with the Marigo theory, two criteria are required in the case of two families of perpendicular cracks and that the two damage multipliers have to be solved numerically. In the present case, on ly one simple criterion given in Eq. (JV.68) provides explicitly the evolutions of the diverse crack dens ity parameters. This characteristic is of high interest in view of an implementation in a FE code. However, on the contrary to the Marigo criterion that systematically respects the Clausius-Duhem inequality, the positivity of the intrinsic dissipation with the present strain-based criterion is not guaranteed and has to be carefully verified. The next subchapter is therefore devoted to the assessment of the thermodynamic admissibi lity of the damage evolution obta ined by means of strain -based criteria.

12.2.4

Positivity of the intrinsic 1/issipati<m For the strain-based criterion presented before, the fu lfilment of the Clausius-Ouhem inequality is not automatically guaranteed, since the GSM formalism was not employed. 1t is consequently necessary to verify that the second principle of thermodynamics is respected. It is recal led that in the presently developed damage model the intrinsic dissipation reads: The three derivatives above all adm it three non-vanishing eigenva lues designated by ( , (ie (1 ,3]) and obtained with the help of the symbolic toolbox ofMATLAB:

rtD(J _ ~. - E.,d;' (1-2v + 3v 2 ) ;rr _ .,, £., dt' (2-v) [ ] 2' ( , - ( , [
]2 , ,dt(2-v) 2 (d~1 (2-v)+2Qocd)

(l +v)( l-2v)dr.' +(1 -v)d (1 +v)(2-v)dc"' +d 81 (I-v 2 ) E.,dt 1 (1-2v +3v 2 ) [9(1 -v)( l-2 v }(I + v)dc"' + ( 6(1 -v )(1-2 v ){I+ v)n DG -\3v 2 + 16v -7)d r' rwtJ _ rlDD _ E.,d;' (2-v) ?2 -'> l - 2, [( 1+v)(2-v)dt' + (1 -n DG (l + v))d]
JD _I'

--------'------~-:-

? -l;i 3 - 2 • [ n ,,ad' +d~;/ (2-v)d +(! + v)(2-v ) 2 J P•ge 169 (IV.74)
ft is straightforward that all these eigenvalues are positive. The eigenvalues of the opposite of the derivatives of the effective stiffness tensor being all positive or null, the them1odynamic forces are consequently not negative. The fonn of the damage evolution law written in Eq .

(JV.70) furthcnnore ensures that d, :<: 0 (ie (1,2]) , so that the second principle of thermodynamics is respected for the three homogenization schemes considered.

Application ill the case of 11111/ti-axial te11sio11 tests

The strain-based criterion introduced just above is presently employed to define the evolution Jaws of a mortar subjected to uniaxial and biaxial tension tests. As done previously with the Mari go criterion, the pred ictive capacity of the micromechan ically-based damage model with the strain-based criterion is tested in the case of a mortar subjected to uniaxial tension (Le Bellego 200 I). The case of biaxial tension tests is also treated to observe how the crack densities of the different fami lies of cracks evolve during non uniax ial tests. For conciness, the different micromechanically-based approaches will be referred to as MT, !DD or simplified IDD damage models depending on the homogenization scheme chosen to compute the elastic properties of the damaged material. The mortar is initially assumed to be linear isotropic and is supposed to be in a plane stress state.

The consistency condition applied to the criterion defined in Eq. (IV.67) takes the form: part V

of . < ;r-•tv,, ~ = t,,,

SIMULATIONS OF THE CHEMO-MECHANICAL DEGRADATIONS :SY LEACHING OF CEMENT-BASED MATERIALS

Tlte service life of cement-based materials, wlticlt can serve as engineering barriers for the disposal of long-term nuclear wastes, is affected by the deterioration factors due to chemical attacks and mechanica l damage, but also by the detrimental interactions between these phenomena. Precisely, the concrete underground structures can be both altered by: (i) dissolution -precipitation reactions caused by ionic migration, governed by the material transpo1t properties, between the interstitial solution and grow1d water; (ii) damage due to external mechanical loadings and possibly precipitation of secondary solid phases generating internal pressures (e.g. sulfate attack).

Various experimental studies evidence significant negative interactions between these deterioration factors. For example, some authors [START_REF] Tognazzi | Couplage fissuration-degradation chimique dans les matcriaux cimentaires: caraeterisation ct modelisation[END_REF] Some approaches to model chemo-mcchanica l degradations of cement-based materials have been developed by the CEA/LECBA [START_REF] Bary | Simplified coupled chemo-mechanical modeling of cement pastes behavior subjected to combined leaching and external sulfate attack[END_REF]) but also elsewhere (e.g. [START_REF] Gerard | Contribution des couplagcs mccanique-chimie-tronsfert dans la tcnue long tcrme des o uvrnges de stockage de dechets radioactif[END_REF]Heukamp 2002;Kuhl et al. 2004a;b;[START_REF] Bellego | Coupled mechanical and chemical damage in calcium leached cementitious structures[END_REF][START_REF] Nguyen | Couplage degradation chimique -comportement en compression du heton[END_REF]. They are however mostly based on simplifying assumptions, since it is generally supposed that only calcium ions governs the chemical equ il ibrium and/or that the influence of damage on diffusion is not accounted for.

The French Atomic Energy Commission (CEA) has developed io collaboration with ANDRA and EDF a numerical integration platfonn ALLIANCES capable of simulating problems that couple chemistry, transport and mechanics (Montamal et al. 2006;2007). This platform may allow for a more thorough modelling of chemo-rnecbanical degradations compared to existing approaches (e.g. [START_REF] Gerard | Contribution des couplagcs mccanique-chimie-tronsfert dans la tcnue long tcrme des o uvrnges de stockage de dechets radioactif[END_REF]Kuhl et al. 2004a;[START_REF] Bary | Simplified coupled chemo-mechanical modeling of cement pastes behavior subjected to combined leaching and external sulfate attack[END_REF]) by gathering within rhc same simulation environment a code resolving chemica l equ ilibrium and a finite volume or element software dealing with transport and mechanical problems. The aim of the present work is to implement a comprehensive multi-physical model in this platform that:

• allows for predicting the effects of coupled chemo-mcchanical deteriora tions on cerncntitious materials;

• may be applied to any cement-based material , provided its initial composition and the physical properties of its elementary phases arc known. (2006; 2007). The ALLIANCES platform allows for combining and connecting together diverse numerical codes dealing with different physical problems. This platform is developed in collaboration between CEA, ANDRA and EDF for the simulation of ouclear waste disposal. This tool programmed in Python is presently employed to couple chemistry, with the code CHESS (van der Lee and de Windt 2002), transport and mechanics, with the finite clements software CAST3M (CcA), so as to perform numerical simu lations of the mu lti-physica l degradation of cement-based materials. In the present context of long-tcnn deterioration of concrete underground structures in nominal phase of waste storage, the following simpli fying assumptions are made:

• the material is in saturated and isothcnnal conditions;

• electro-difTusivc phenomena are neglected, ionic transfers being only due lo molecular difTusion;

• the effects of creep are djsregarded;

• the same diffusion coefficient is affected to all aqueous species;

• the chemical reactions arc instantaneous compared to the diffusion process.

1. I Coupled chemical-transport problem solved by ALLIA NCES

The chemical degradation of cement-based materia ls is a chemical-transport coupled problem.

For example, in the case of the concrete underground structures in contact with ground waler, the existence of concentrations gradients of ions between the interstitial solution and ground water provokes ionic migration inside the interstitial solution, governed by the material transpon propenies. This movement of ions disturbs the chemical equilibrium state between the solid skeleton and the pore solution of the material and causes dissolution-precipitation reactions in the material. Therefore, the evolutions of the diffusion coefficients of ions in the material are imponant data for predicting its chemical alteration, as already mentioned in Part Ill.

Different types of chemical-transport coupled simu lations have been performed on concrete materials. Some authors (e.g. [START_REF] Adenot | Durabilite du Seton: Caraccerisation et Modelisation des Processus Physiques ct Chimiques de Degradation du Ciment[END_REF][START_REF] Gerard | Contribution des couplagcs mccanique-chimie-tronsfert dans la tcnue long tcrme des o uvrnges de stockage de dechets radioactif[END_REF] proposed simplified approaches, where only ca lcium ions are supposed to govern the main chemical reactions. More complete simula1ions arc recently avai lable (e.g. [START_REF] Moranville | Physicochemical equilibria of cement• based materials in aggressive environments--experiment and modeling[END_REF][START_REF] Plane | Les Effets Couples de la Precipitation d'Especes Secondaires sur le Comportement Mecanique et la Degradation Chimiquc des Betons[END_REF], where the current Inside this medium, the species that may move arc the fluid lilling the porosity and the chemical species concentrated in the flu id. The fin ite volume method is used for the computatio ns. The medium is thus represented by a number w of subvolumes V 1 each comprising Mi mineral phases supposed homogeneous. The intersti tial solution in each subvolume contains N aqueous species, most of them being ions. For simplici ty, the cinematic dispersion is neglected. Under such conditio11s, the conservation law imposes that the divergence of the sum of the average diff11sive flux vector of ions j denoted J; through a given subvolume Vi and of the average velocity vector denoted U is equal to the variation of concentration designated by C 1 of ionsj in V,:

-

- ()C -div(J 1 +C 1 U)= cp 0 /, (V.I)
where Cp is the average porosity inside the subvolumc Vi. Jn the present case, the velocity vector U of the fluid is null because it is assumed that no pressure gradient provokes water transport by pemieation and the previous equation s impl ifies into:

ac

-div(J 1 ) = c,-,!-. Of (V.2)
T he ions flux vectors depend on f,>Tadients of ions concentration and/or on electric gradients.

However, the possible influence of the electric charges of ions on diffusion being presently neglected, the ions nux vectors are expressed by F ick's law.

At the scale of the heterogeneities of the materia l, Fick's law, stipulating that the m icroscopic nux of ions j is proportional to the microscopic g radient of concentration, reads:

(V .3)
where g r ad C, (x) is the g radient of concentration of ions j and tlie operator '.' indicates a simple contraction. The ensuing relations are obtained by averaging in each subvolu me V,:

J / : -o; .g rad C 1 (x).

(V.4)

The diverse homogenizati on techniques (e.g. Torquato 200 1) can provide estimations for the effective second order diffusion tensors o; of the d iverse ionic species through the material.

In the ALLIAN CES plat fonn, all these tensors are supposed to be identical for a ll the species, c" aa~' =div( o' .grad C, ),V'ie{ l, ... ,N}, (V.5)

where N refers to lbe total number of aqueous species. In addition to the transport process, the effects of chemical reactions in lhe case of concrete have to be accounted for. The ccmentbased material is initially taken in a stable chem ical stale. The possible leak of aqueous species such as Ca 2 ' precised by the transport equations may disturb this equilibrium state.

The new equilibrium state of the perturbed system can be computed by means of the chemical equilibrium code CHESS (van der Lee and de Wind! 2002), assuming that lhe chemical reactions are instantaneous in comparison with the transport process.

Chemical reactions computed by CHESS

The dissolution reaction of a mineral specie denoted em in contact wilh an aggressive ionic specie, designated by ,.,

where K~ is the equilibrium constant of the dissolution reaction and the brackets surrounding a given specie designates its activity. The activity of a component is generally related to its concentration by: {V.8)

where y, is the activity coefficient that can be considered equal to one for dilute solutions and p, designates the concentration in mol per liter of solution of the compone_ nt or specie i. For higher concentrations, diverse melhods accounting for example for ion interactions exist to relate the concentrations and activities of the species. The CHESS software proposes several models for computing the activity coefficient, and in the following we retain the modified Davies melhod. It must be borne in mind that the dissolution of a mineral, such as portlandite in the case ofleaching (see Eq (1.6)), can also occur by itself without requiring a reaction With aggressive ionic specie.

The precipitation reaction of a minera l specie e .. from aqueous species over the saturation limit inside the interstitial solution reads:

(V.9)

with the following action mass law: •' "' l lF,.' ''' -F'""l l <Pr ~F"".' II ,

with F" " = LL Xi~P .. (le {1, ... , l,. 0 ,1), (V. 13)

J ~I
where Pr designates the precision, I is the iteration index and I ,.,,, is the maximal number of iterations desired. ALLJANCES then adapts the time steps so that convergence is obtained within this number of steps. It is oflen necessary to impose a precision less than I o• 6 resulting in small time steps, practically a few hundreds of seconds, to ensure convergence of the computations. Step Ill: Estimations of the material effective elastic propcnies n . .

Chemistry

<.,?

Step

l np111 t/11111

The input data necessary for the chemistry • transport • mechanics (CTM) simu lations are listed in Table V. With this system, the following chemical species mainly appear in the interstitial solution:

[Ca 2 ', so; , Al'', Si0 2 (aq)]. Alkalis (Na+, K ') and ions such as Fc 3 ' or Mg 2 + are not taken into account, even though they arc usually present in the interstitial sol ut ion.

Chemistry Mechanics

• Concentrations of the initial mineral species:

Pm,me [C H, AFm,CSH, .. ,, AFt or Hydrogarnet];

• pH and composition of 1he external solution .

• Elast ic properties of mineral species: • Diffusivities of all the mineral species: diasporc, gypsum, hydrogarnc1]; £,,.,v., , with me ICH, AF1, AFm, C 1 . 6 sSH, C, ,,SH,

• Parameters of the damage model; 4-~H. Si0 2 (am), diaspore, gypsum, hydrogarnetj;

• Initial and boundary conditions. • DifTusivi1ies of gel and capillary pores: Dw., Dcp; it is intended that the sample has been so severely attacked that it reaches a unifonn degraded state. It is supposed tbat the effective diffusivities and Young moduli of the two heterogeneous CSH evolve as a function of their C/S ratios comprised between 0 and 1.65 in x to"' Bu i I 1992) on a similar paste that is about 0.15 mm I days 0 5 (see Table V.3).

•••---•--• •• •• •• -. .. -•-•• • .... -
The results obtained with Tognazzi's law and with our model are far better than those provided with a proport ional law. This shows the importance of using a realistic diffusion law. The multi -scale homogenization model presents the benefit to be more general than the empirical Tognazzi 's law that has been specifically developed for CEM I pastes on the basis of diffusion experiments on several pastes with different porosity [START_REF] Tognazzi | Couplage fissuration-degradation chimique dans les matcriaux cimentaires: caraeterisation ct modelisation[END_REF]. This law that was proved to be reliable for the standard pastes chosen for our simulations is not necessari ly applicable for other types of cement pastes,. such as CEM V. The multi-scale homogeni7.ation model is therefore adopted for the subsequent simulations.

8 __ ---------------------------•-•-•sco,l'• •.a ._. ('Cu ---ea2• :::: ~ ~ ~ ,-_ -• --• -• -• -• -• -• -• -• -• -• -• -• -• -• -• -• -• -• -• -• -• -• __ _ Si{0 2 )(aq) -• ~ 0 c -10• 5 L . , .... ,.. ••.' - . 0 •- ••• ------------------------------------ <->.~ ... . 0 0.2 G.4 0.6 G.8 I 1.2 1.4 1.6
1.8

Depth (mm)

------------------------------------- -- -----------------------------------

• , .....:: ---•-•--•-•-•-•--•-•-•-•--•-•-•-•-•-•-•--•--•-•-•-•-.. / ""' ..... " ' ••.:-: .. --. .. ------------------------------- ' l'•. . . 0.5 1.5
Depth (mm) ALLIANCES is that it is prescotly not possible to affect different diffusion coefficients tc each aqueous specic.s, which might lead to unreal istic resu lts. For example, we observe 011

Influence of tlte composition of cement pastes on tlte cltemica/ degradation

In order to perform a rigorous validation of the simulations and investigate the influence of the composition of the cement pastes on their chemical degradations, two CEM I pastes, that are well characterized (Adenot and Bu i! 1992;[START_REF] Baroghel-Bouny | Caracterisation des pates de ciment ct des betons. Methodes, ana lyse et interpretations[END_REF]Galle 200 1) and which mechanical behavior has also been studied [START_REF] Carde | Caractcrisation ct modclisation de !'alterat ion des propric1es mccaniques due;\ la lixi viation des matcriaux cimentaires[END_REF]Le Bellego 2001;Galle et al. 2004;Moranvi lle et al. 2004) are employed: a CEM l 42.5 of Origny with w/c = 0.40 [START_REF] Tognazzi | Couplage fissuration-degradation chimique dans les matcriaux cimentaires: caraeterisation ct modelisation[END_REF]; Le Bellego 200 1) and a CEM 1 52.5 HTS from Lafarge [START_REF] Galle | Effect of drying on cement-based materials pore structure as identified by mercury intrusion porosimetry: A comparative study between oven-, vacuum-, and freezedrying[END_REF]Galic ct al. 2004;Moranvi lle et al. 2004). T he comparison between the two cement pastes provides furthermore some insights about the effect of the hydration rate and of the w/c ratio on the chemical degradation. The CEM I 52.5 IJTS paste with w/c = 0.45 (Galle et al. 2004) has a qu ite di fferent composi tion from the other one with five phases supposed present init ially (Bejaoui and Bary 2007): UC, C1.6sSH, CH, AFm and hydrogarnet (Table V.2). T his ini tial composition is obtained by means of the Jennings model (Bejaoui and Bary 2007). The simulations of chem ical degradations of the CEM 1 HTS paste are confronted w ith experi mental results from pure water leaching tests [START_REF] Galic | Effect of an accelerated chemical degradation on water permeability and pore structure of cement-based materials[END_REF] The dissolution front of portlandite propagates faster in a CEM I HTS paste than in a CEM l 42.5, since the CEM I 52.5 HTS is much more porous and has a higher diffi.tsivity (see Tables V.2 andV.3). An additional possible reasou why the CEM I 42.5 paste resists better to leach ing than the CEM I 52.5 HTS one is that it contains less portlanditc and more alumiuatcs showing a better resistance to pure water. The calcium fluxes are funhennore underestimated in both simulations (see Table V .3) and appear to be much smaller than the values measured experimentally by [START_REF] Adcnot | Modelling of the corrosion of the cement paste by deionized water[END_REF] and

lnfl11ence of the aggressive sof11tio11 on the cltemicaf ilegradatio11

Although the degradation of the small cement samples in the previous subsection is significant, the effect of leaching by water at the scale of a rea l concrete structure are only observable for very long times, since it roughly takes 300 years to leach 4 cm of a concrete structure by extrapolating results from leaching experiments [START_REF] Adenot | Durabilite du Seton: Caraccerisation et Modelisation des Processus Physiques ct Chimiques de Degradation du Ciment[END_REF]). More agressivc solutions, such as ammonium nitrate (NH 4 N0 3 ), arc consequently common ly used by experimentalists to enhance the degradation process (Cal'de 1996). ln particular, a very concencratcd solution containing 6 mol/L (6M) of H.N0 3 is generally employed for the (Galle et al. 2004). Instead of suppos ing that, in the uniformly leached state, 5 % of the CSH are d issolved and replaced by capil lary pores [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi -coated sphere assemblage model[END_REF], it is presently assumed that they are substituted by additional percolating gel pores. As a consequence, an add itiona l volume fraction f!.c~ = 0.065 of gel porosity is supposed to percolate through the uniformly leached paste. By summing these additional percolating gel pores and capillary ones replacing the dissolved hydration products (CH, A Ft), the total quantity of additional porosity measured by mercury porosimetry (Galle et al. 2004) is retrieved. As may be seen in Fig. V. 12, the diffusivity obtained with this new hypothesis augments by a foctor three in the most leached zone and the depth of the Cl I dissolution front is increased by about 0.1 mm for a 50 days pure water leaching simulation (see F ig. V.13) thus leading to a better agreement with the experimental decalcification front propagation.

In addition, the calcium flux computed that is about These results high light the importance of the amount of the percolating porosity in the MCSA model (Stora et al. 2006b). ln the ensu ing chapter devoted to the chemo-mechanical couplings, the chemical deteriorations are exclusively simulated adopting this increased quantity of percolati ng gel pores for the homogenization computations.

Conclusions of chapter 13

The multi-scale homogenization model developed in Part Ill has been• implemented within the ALLIANCES platform to compute the evolutions of the diffusive properties of concrete materials. S imu lations of both pure water and accelerated leaching of these materials have been run and confronted to both experiments (e.g. [START_REF] Adenot | Durabilite du Seton: Caraccerisation et Modelisation des Processus Physiques ct Chimiques de Degradation du Ciment[END_REF]) and recent numerical results [START_REF] Moranville | Physicochemical equilibria of cement• based materials in aggressive environments--experiment and modeling[END_REF]). The concentration profi les of the aqueous and mineral species inside average size of a mesh element shou ld be at least superior to 2 mm in the case of mortars. In a first attempt, a coarse mesh composed of 200 elements was used to ensure reasonable calcu lations times (about one hour) but a 20 mesh with I 600 elements (more than two hours) was finally preferred to examine more precisely the di ssolution front propagation. Th is propagation was observed to be indepeudeut of tl1e mesh adopted. T he area of a mesh element is 8 mm 2 for the mesh with 1600 elements and its dimensions are 2 111111x4 mm . This choice is a compromise between the R VE size condition (> 2 mm) and the computation precision for damage, for wh ich it is generally recommended to avoid too big clements. the interface between sand aggregates and paste and through the sand aggregates themselves.

Displacemen t imposed

The micromechan ical damage model consequently assumes that the cracks are embedded in an effective medium having the same properties as undamaged mortars, accord ing to the fact that the cracks may grow anywhere. Another argument for representing the cracks inside the already homogenized undamaged mortars is that they may be very extended and have a correlation length that is much longer than the particulate phase such as sand or clinker grains.

In the case of concrete, cracks are also likely to grow inside and span over the coarse aggregates.

The chemical-transpo1t input data are collected in Appendix V.B. It is necessary to pay attention to the fact that the mineral composition of the cement paste inside mortar is different from the plain HCP. Indeed, the total water porosity inside the mo1tar presentl y considered with 50 % of sand grains is measured to be equal to 0. 18, whereas the total water porosity of the plain HCP is 0.27 (Le Bellego 2001). Assuming that the sand aggregates are impermeable, the total water pllrosity of the cement paste inside mortar that is more porous than the plain HCP is then about 0.36.

The mechanical input data for the damage model (sec Table V

Results and comments

Before testing the residual mechan ica l resisl• nee of the ml.ll"t'1r he111n, it is irul ispens~hle to first correctly reproduce the chemical deterioration of the leached material. T he computation time for the numerical test of the accelerated leaching is about three hours o n a standard PC-L inux machine depending on the precision defined in Eq. (V. 13), presentl y I 0-8. The CH dissolution front reaches about 17 mm (Fig. V. 19) on mortar beams after 114 days of accelerated leaching. This depth is in quite good agreement with the one obtained experimentally by Le Bellego despite the fact that the ion activities evaluated with the Davis modified method should be computed more accurately in tl1e case of Nl LiN03 attack.

II appears on F ig. V.20 that the liquid ca lcium concentration in contact with tl1c 6M Nll 4 N03 solution goes up to about 2 mol/ L near the portland itc dissolution front. This value is not far from the one considered by [START_REF] Tognazzi | Couplage fissuration-degradation chimique dans les matcriaux cimentaires: caraeterisation ct modelisation[END_REF] and [START_REF] Hcukamp | Cberuomechanics of calcium leaching of cement-based ma1erials a1 different scales: the role of CH dissolution and CSH degradation on strength and durability performance of macrials and structure[END_REF], who assert that the CH dissolution front in presence of a 6M NH.N0 3 solution occurs for a liquid calcium concentration inferior 10 2. ln a first attempt, simulations of flex ion tests on the leached mortar have been run using exactly the same values for tbe damage parameters as for the sound specimen (see Table V .6).

The results obtained appear on Fig. V.22 (right) to be unsatisfactory thus confinn ing the fact that the damage threshold is modified by the leaching process. The empirical method described in previously has therefore been employed to have the strain damage threshold co evolved. The pre-peak part of the numerical curve then agrees relatively well with the experimenta l curve on Fig. V.22 (right) but the forces predicted by the simulations tend to be overestimated. There are diverse reasons to explain this difference. The first explanation cou ld be that the Young modulus in the most leached zone, which is estimated to be equal to about 4.9 GPa by the multi-scale homogenization model, is overestimated, even though th is va lue already seems to be qu ite low. For instance, [START_REF] Hcukamp | Cberuomechanics of calcium leaching of cement-based ma1erials a1 different scales: the role of CH dissolution and CSH degradation on strength and durability performance of macrials and structure[END_REF] The possible benefits of the present work are multiple. First, the homogenization schemes proposed, such as the MCSA estimate, appear well suited for cemcntitious materials but arc also of potential interest for other kinds of comPQsites (e.g. porous media, granular materials).

Second. the analytical multi-scale approach can be employed for computing the elastic and diffusive properties of any cement paste or mortar, which composition is accurately known, and their evolution with time. It furthermore nqu ires very Few parameters and is therefore useful for engineering applications. Finally, the chemo-mechanical simu lations perfonned with the aid of this deductive model can be easily applied to other cement pastes and mortars.

Moreover, the numerical results are instructive and help for better understanding the mechanisms of degradation. For instance, they clearly show the detrimental impact of Portlandite dissolution and of the progressive CSH decalcification on the elastic and diffusive properties of HCP.

Proposals for future research

The models and simulations proposed in the manuscript may obviously be further improved.

Many interesting paths for future ameliorations can be envisaged. Double-inclusion type models have been shown to be of interest for their simplicity and versati lity but they could be This numerical work should then he extended to chemical attacks generating intemal pressures and possibly microcracks, such as sulphatic attacks. lt is also possible to extend tl1e simulations to concrete and eventually to real underground structures.

Perspectives

The present manuscript provides results that are particularly interesting in the context of nuclear waste disposal. But the present work is more generally justified by the increasi11g necess ity of adopting coupled multi-physical approaches for the assessment of the lifetime of existi11g structures. I.n the perspective of concrete durabil ity, more and more investigations are dedicated to coupled multi-physical problems, like the present one or tberrno-bydromechanical ones (e.g. [START_REF] Gawin | Numerical analysis of hygrolhermal behaviour and damage of concre1e at high temperature[END_REF]). This study put together witl1 other approaches (e.g. Bary ct al. 2007) could lead at term to the development of powerfu l prediction tools for the durabil ity of concrete facilities. JI constitutes a huge task because of the necessity of solving thorny issues, such as unsaturated conditions. These tools would then find very wide industrial applications, from nuclear reactors to C0 2 storage.

  SM macroscopic compli ance tensor of the matrix material s• macroscopic compl iance tensor of the effective material the inclusion r Q~1 cigcnstiffoess tensor of the double-inclusion embedded in the effective material E~ Eshelby tensor of a homogeneous inclusion I embedded in an infinite matrix t;1 Eshelby tensor of an inhomogeneous inclusion I embedded in the matrix

figure 1 . 1 :Figure 1 . 2 :Figure 1 . 3 :Figure 1 1 Figure 1 . 7 :Figure 11 . 7 :Figure 118 :

 1112131117117118 figure 1.1: Images of a CEM I 52.5 cement paste with a water to cement ratio (w/c) equal to 0.43 obtained by SEM: N representative area of the sample, B/ detailed image of an Outer Product Zone (taken from Bejaoui ct al. 2006) ......................................................................... 6 Figure 1.2: Images of the microstrncture of CSH;,,, and CSll'"'' in CEM I 52.5 cement pastes with diverse w/c ratios obtained by Transmission electron microscopy (TEM) (taken from Bejaou i et al. 2006) ................................................................................................................... 8 Figure 1.3: Views of cement particles generated from the spherical hannonic expansion (taken from (Garboczi and Bullard 2004) .......................................................................................... I 0 Figure 1.5 (semi-logarithmic scale): Evolution of the macroscopic diffusion coefficients measured experimentally for both sound CEM land CEM V pastes with their w/c ratio, taken from diverse sources in the literature (adapted from Richet ct al. 1997) ................................ 19 Figure I.6: Photo of a CEM I 42.5 paste sample w ith w/c = 0.40 leached during 3 months by pure water (taken from Adenot 1992). .................... .. ........................................................ 2 1 Figure 1.7: Degradation process of the main hydrated phases in tenns of calcium concentration in solid phase Sc, as a function of calcium concentration in solution Cea (l ine = resu lts obtained with the chemical equilibrium code Cl IESS; dotted line = simpli fied curve) (adapted from Bary and Bejaoui 2006) ................................................................................... 22 Figure II. I: Schematic illustrating the replacement of the heterogeneous R VE by an equivalent homogeneous medium . ........................................................................................... 28 Figure 11.2: Schematic of a spherical inclusion w ith a homogeneous intcrphase enclosed in an infinite matrix ........................................................................................................................... 32 Figure IJ.3: Schematic picture of the equ ivalent homogeneous inclusion ............................... 33 Figure 11.4: Two-dimensiona l representation illustrating the three-phase model on which is based the GSCS ........................................................................................................................ 37 Figure 11.5: Derivation of the MT (left) and IDD (right) models for a multi-phase composite from the dilute configuration by modifying the far-field stresses ............................................ 39 Figure 11.6: Schematic of the original three-phase configuration a) used for the derivation of DlM and representation of the different steps b), c), d) and e) employed for its clerivation .... 41 Figure 11.7: Schematic illustrating the link between the I Jori and Ncmat-Nasser DIM and other micromcchanical schemes (I lu and Weng 2000) ............................................................ 44 Figure 118: Schematic illustrating the discrepancies of the !Jori and Nemat-Nasser DIM ..... 45

Figure IV. 7 :

 7 Figure 11.10: Schematic illustrating the main steps employed to derive the modified DIM .... 5 I Figure 11.11: Evolutions of the effective Young modu li estimated by different micromechanica l schemes and by FE simulations (Wriggcrs and Moftah 2006) for two types of concretes as functions of the volume fractions of aggregates a) type I and b) type 11. ........ 54 Figure 11.12: Evo lutions of the effective elastic bu lk (left) and shear (right) modul i estimated by different EM Ts and by FE simulations (Garboczi and Be1Tyman 200 I) for both sound and

  .......................................................................... 192 Tab le V.5: Comparison of the decalcification front propagations obtained in the new simulations performed with ALLlANCES of pure water leaching with experimental values . ...................................................................................................... ................. : ....................... 196

  submitted to a uniaxia l traction and of the mortar beam tested in flcxion (Le Bellcgo 2001) .................................................................. 200 J GENERAL I NTRODUCTI ON

  Parts II and Ill devoted to homogen ization techniques . a multi -scale mode ling approach based on a realistic representation of cementitious materials is developed to estimate the evolution with time of their elastic and diffusive macroscopic properties, respectively. Jn Part JV, a micromcchanical damage model based on some simplifying assumptions is developed lo predict the quasi-brittle behavior of concrete and integrated in the FE code CAST3M. Part V is fi nally dedicated to the modelling and the numerical simulations of chem ica l and of coupled chemo-mcchanical degradations of leached cement-based materia ls. These simulations are perfonncd by incorporating the models proposed in the previous Parts to predict the macroscopic properties of cement-based materials into the ALLIANCES platfonn.Part I PRESENTATION OF CEMENT-BASED MATERIALSCement-based materials are massively employed in the domain of construction. A country like Ch ina consumes more than one hundred millions of tons of concrete per year. Three types of cementitious materials are studied in the present manuscript: hardened or hydrated cement paste (HCP), mortar and concrete. A particular emphasis is put on the cement pastes that constitute the matrix in mortars and concretes and that are very sensitive to degradations.

  YSICAL PROPERTIES OF CEMENT-BASED MATERIALS 2.1 Presentation of the multi-scale microstructure of cement-based materials 2.1.1 Microstructure of cement pastes 2. 1. 1.I Hydration of cement pastes Cement pastes are complex, multiphase, porous materials, which microstructurc evolves with time. The material primarily consists in anhydrous cement grains mixed with water. These grains call ed clinker are fabricated from a mix of limestone an~ clay transfom1cd by heating them at about I 450°C and by crush ing. Their diameter can vary from I to I 00 µm. A basic Portland clinker is generally composed of four dom inant compounds: tricalcium si licate (C 3 S), dicalcium silicate (~S), tricalcium aluminate (C3A) and calcium aluminoferrite (C 4 AF). For s implicity, the following cement chem ists ' notations are used: C = CaO; S = Si~; A = Al 2 0 3; F = Fe 2 0 , ; H = HiO). Some additives, such as Gypsum, fly ash or slag, may be also adjoined to this clinker. base of the standard Portland cement, to modify the paste microsirucnire and thus its macroscopic properties.

  It is commonly accepted that two different types of porous CSH arc associated with each layer. lligh-density CSH are present inside che inner layer and lowdensity CSH fonn inside che outer layer (e.g. Richardson 2000; Tennis and Jennings 2000). Tennis and Jennings (2000) classified them as high-density and low-density, whereas Richardson preferred to call them inner and outer products. IJ1 the ensuing, these two types of CSH are respectively called as internal and external products, denoted CSHin• and CSHc.c. More precisely, in the inner layer, the CS!lin• behave as a matrix phase embedding non• diffusive inclusions of C H and AF. Sim ilarly, the external layer has matrix-inclusion type morphology, where Cll, AF and some pores play the role of inclusions enclosed in the CSH""'. It is notewonhy that many mineral phases (AF, CH, CSl-li••, CSH' "", ... ) are involved in the microstrucrure of HCP and a closer look is given in the next subsection to each one separately.

  Fig ure f.2:Images oftlt e 111icrostruct11re ofCSH 1 "' and CSH'".' i11 CEM T 52.5 cem ent pastes tvitlt diverse w/c ratios obtained by Tramm1issio11 electron microscopy (TEM) (take11 fmm[START_REF] Bejaoui | Experimental and modeling studies of the link between microstructure and effective diffusivity of cement pastes[END_REF]).

Figure /. 3 :

 3 Figure /.3: Views of cem e11t particles generated from tlte splteric:ul harmonic expansion (taken from Garboczi u11d Bullard 2004).

Figure 1

 1 Figure 1.4: Differential intruded P"re volume versus pore diameter curve m eas ured by MI P (takenjr(lm Galle 2001).

  Figure J.6: Photo "fa CEM J 42.5 paste sample with wlc = 0.40 leached during 3 months by pure water (taken from A1/enot 1992).

Figure I. 7 :

 7 Figure I. 7: Degradation process of the main hydrated phases in terms of calcium concentration in solid phase Sc. as a f 1111ctio11 of calcium conce11tratio11 in solution Cc. (line= re.mlt1• obtained with the chemical equilibrium code CHESS; dotted line = simplified curve) (adapted from[START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi -coated sphere assemblage model[END_REF]).

  measured a decrease from 23 GPa to 7 GPa (a 7 1 % reduction) on the dynamic elastic modulus of a cylindrical sample with a 40 mm diameter of a CEM I 52.5 J ITS w/c = 0.45 paste subjected to a complete degradation during 30 weeks. Constantinides and Ulm (2004) further measured that the stiffness modulus of an asymptotically leached OPC type I paste decreased to about 3

  or displacement (Le Bellego 200 I) imposed at the center of the upper face and immcrged in an ammonium nitrate solution. These tests rresent the advantage of sollicitating in traction the most chemical ly degraded part of the beam and generate in that zone open cracks that are susceptible to enhance the leaching process.

Fig

  Figure //. I: Sc!tematic illustrating the replace111e11t of the heterogeneous RVE by an equivalent lto111oge11eo11s m edium.

  Eshe/by tensor to spherical inhomogeneities ~uan ct al. (2006) recently derived a generalization of the Eshelby tensor for a spherical lllhornogeneity with a graded shell embedded in an infinite matrix. In particular, they gave !he lbllowing closed-form solution for the volume average of the Eshelby tensor denoted as t:' of a spherica l inclusion with a homogeneous interphase enclosed in an infiJJite matrix, a5 depicted in Fig. ll.

Fig

  Fig ure ll.2: Schematic of a spherical inclusion with a hom oge11eo11s interp hase enclosed in an infinite matrix .

Figure 11 . 3 :

 113 Figure 11.3: Schematic picture oftl1e equivalent ltomoge11eo11s i11cl11sion.

1 :

 1 11.5) with unbounded matrix material subjected to a modified effective stress a';,,, defined as: <T:O = [I -~ c,H;" : n~,: <T~.

(Figure / 1

 1 Figure /1.5: Derivatio11 of the MT (left) and /DD (right) models/or a multi-plrase composite from the dilute configuration by modifying tire far-field stresses.

  phase representation considered byJlori and Nemat-Nasser ( 1993) are arbitrary and arc respecti vely designated by C,, C ci and C 0 (see Fig. 11.6). Two parameters are important in this model: -the double-inclusion geometry that should be representative of the distribution of the inclusions inside the composite; -the choice of the reference medium that should be representative of the immediate surroundings of the particulate phases. Pan II: Prediction of the el .. stic propertii;:,s of cemC1lt-ba.sed msteriH1sHowever, the choice of the reference medium is in reality quite limited, since the three-phase configuration should have a physical interpretation. The most usual choices consist in taking the matrix of the material with stiffness tensor C,, or the yet unkown effective material With stiffness tensor c• as the reference medium[START_REF] Hu | The connec1ions between the double•inclusion model and the Ponte Castaneda-Wi llis, Mori-Tanaka, and Kus1cr-Toksoz models[END_REF]. The main steps necessary for deriving this model are recapitulated below. The detailed demonstration can be found inHori and Nemat-Nasser ( 1993). These authors practically apply the concept of the equivalent homogeneous inclusion method (Fig.11.3) to the case of a double-inclusion. Two eigenstrains respectively denoted f; and t~, are then necessary, as represented inFig. ll.6. As in Eq .

  and En' respectively denote the strain perturbation tensors of the inclusion and of its coating. The next step consists in computing respectively the volume averages of the strain tensor tt and eci over the inclusion and its coating. For this purpose, the problem of the homogeneous fictive double-inclusion corresponding to the situation b) depicted in Fig.11.6 is decomposed in three subproblems referred to as c), d) and e)(Shodja and Sarvcstani 2001).

  ). The first term of the right-hand side of the expression above corresponds to the situation c) represented in Fig. II.6, the second one to the situation d) and the third one is issued from the resolution of the situation e) (see Fig. n.6).

1 :

 1 Co1 -c,(11.59) which can be re-ammged by using the fact thatI:~ = n -S 0 :Q~ and (C, -c 0 r' :C 0 = -s, : ll!,-• into: where Part II: Prediction of the elasric properties of cement-based materials F / _, . -• IF'-' . A ---= -[ r.: + (Ci -c 0 r' C 0 J' : S 0 = [Ill!/+ n~T' , with) = (1 ,Cl) and k = (1 ,Dl ] . (JI.62) By solving this system of two equations, the averages of the two eigenstrains can be re lated to the far field-stress by: . -£,, = U ci : .,-and 6.f = 1I.J 1 : .,-, with (fl.63) Uc, =IFf,, :[n-IF:-• :[I F:,, :IF:-'+ c, . F:;f :(n~-n~,)r (F~,-IFi:)].(11.64)

Figure 11 . 7 :

 117 Figure 11.7: Schematic illustrating the li11k betwee11 the Hori a11d Nemat-Nasser VIM a11d other micromecha11ical schemes (Hu amt Weng 2000).

For

  figure, DIM is constituted of four regions: the ellipsoidal inclusion J w ith a volume fraction c, representing a particulate phase with a stiffness tensor denoted as IC, is enclosed in a confocal ellipsoidal coating with a volume fraction cn occupied by a phase having its stiffness tensor designated by ICn. The double-inclusion thus obtained is in turn embedded in another confoc!l l el!ipsoidal coating with a vol ume fraction c., = l-c 1 -cc 1 fi lled by the matrix phase lia~ing for sti ffness tensor IC., immersed in an infinitely large matrix medi um with stiffness tensor IC., .

  last three equations, it comes that: a=[ I-n~, : (at:+ llll1'')] : a* and i: = [ s.., -r.~, : (1!1~: + JHI1") J: a• .(Jl.81) By eliminating a" in the equations just above and using the fact that n~, = n:', the following expression can be obta ined for the effective increment compl iance tensor:(11.82) which coincides w ith the MT estimate written in Eq. (ll.38) for three-phase composites. To our knowledge, this connection with the MT scheme has not yet been noticed. It is however unexpected that, in the present case, DIM leads to the same results as the class ical MT model for tliree-phase composites, the microgeometry considered for the derivation of the DIM being very different from the simple o ne employed by MT (seeFig. 11.5). It would mean that whether one inclusion phase is enclosed in the other one or on the contrary d ispersed far away one from another does not change the elastic properties. Th is result appears contradictory by considering the following example. A stiff inclusion is supposed to be completely trapped in water constituting a coating layer due for instance to a partia l dissolution. Tn such case, the stiff inclusion should logically not be able to reinforce the material, whereas according to the DIM estimate it does.

  II.84) where&:. can be expressed by means ofE4. (II.73) if the interactions between the particulate phases are neglected.

  , + ~Ill, .IC.,. !:DI . §., J .Dl! .C.,, (ll.85) which Jeads to the following expression for the compliance increment tensor after a ff= C.,-C..,: 1+111 :C., :Em :IH! :CM -,,, =[c., :(l+lll"' :C,, : l:~,) :S.,-!H!di/ :c.,J 1 (1+l!:l' 111 :C,, : E~,) :S.., -S..,, (ll.86) =[ll-H'"' :(-C,, :r.~+c..,)J' :(n+lHl' 111 :C.., :r.~, ):S,,-S..,,

  the next subsection, a new estimate based on the re. cent works of Duan et al. (2006) is Pfoposcd to remedy the shortcomings of DIM for composites with coated inclusions. Part 11: Prediction oflhc elastic properties ofcenlcnt•base<l ITT:;llcrials 4.3.4 Development of a modified DIM fo r composites with coated spherical iliclusionsThe scope of thi s subsection is to develop a double-inclusion type estimate that remedies some of the shortcomings of DIM noticed previous ly. Several attempts(Shodja and Roullli 2005;[START_REF] Shodja | Elastic fields in double inhomogeneity by the equivalent inclusion method[END_REF] to improve the scheme of Hori andNcmat-Nasser (1993) exist in the literature but most of these models requires numerical computations. For instance ' Shodja and Sarvestani (2001) applied polynomial eigenstrains instead of uniform ones to the Hori and Nemat-Nasser DIM. However, closcd-fonn solutions are difficult to reach with their theory. It is now proposed to derive differently DIM by employing the inhomogeneous Eshelby tensor in Eq . (11.21) derived by Duan et al. (2006). The configuration considered for the present derivation consists in a composite pattern embedded in an infinite matrix mediurn and subjected to a far-field stress o' .. . This composite pattern comprises various double. inclusions denoted as D,, with re [1,P] , enclosed in a matrix region, each of them being constituted of a spherical inclusion with stiffness tensor C, surrounded by a spherical coating with elasticity tensor Cc, . DIM was proved to lead to unexpected results for this type of microgcometry that is commonly encountered in cement-based materials, as wi ll be shown in chapter 6. The present derivation comprises two main steps. The effective stiffness tensors of the different double-inclusions are first estimated with the aid of the work of Duan ct al. (2006}. The particular configuration described on Fig. 11. I Oa) is thus approximated by a simpler one described on Fig. II.I Ob).

Figure

  Figure JI.JO: Sche111<11ic ilf11strati11g the main steps employed to derive the modified f)lM.

  where W~~ designates the di lute strain local ization tensor, also called the Wu tensor[START_REF] Wu | The effect of inclusion shape on the elastic moduli of a two-phase material[END_REF] of the double-inclusion D,, with re(1,P], and 2:~, denotes its Eshelby tensor. The Wu tensor practically links the volume average of the strain tensor inside an ellipsoidal inclusion to the far-field strain. The consistency condition written in an average sense: C,. : t, =Cc, : ("ii,. -t; ),(11.90) provides the following relation for the average eigenstrain £,: in the inclusion:

1 :

 1 Where !:~, is the inhomogeneous Eshelby tensor of the double-inclusion D, derived by Duan et al.(2006) and expressed in Eq.(11.21). It is emphasized that even though the inclusion and double-i nclusion both have a spherical shape, their inhomogeneous Eshclby tensors are not identical, since their expressions given in Eq. (n.21) also depend on their relative size characterized by the parameter p. The mean strain and stress tensors inside the doubleinctusion are further expressed with the help of Eqs. (11.53 -54):F.n,= [ 1-:;, i:;;_ : [ ( C, -Cc,J': Cc, + E:' T' l W~~: S.,: o-. (11.93) 0 0 , = Cc, : [ w~: + :;, S ,, : Q~, :[(c, -Cc, f' :Cc.• + i:~' r•: w:;, } s,, :a-,(ll.94) where Q~, =CM : ( n -i:~, ) is called the inhomogeneous eigenstiffness of the double. inclusion D,. By eliminating a• in the last two relations, the following expression can be obtained for the effective stiffness tensor of this double-inclusion : Jee,:[ 1 + :;, s,, :n~, :[(c, -Cc,f' :Cc, +i:' .' J' J Co, = l[ I-:;, E~,: [ ( C, -Cc.) Cc, + E~' exactly the same reasoning as in Eqs. (11.83-85) (see subsection 4.3.3.1), an expl icit estimation can then be ach ieved for the effective stiffness tensor of the composite with coated inclusions: c• =[ 1-( s,, + ~111~~ :c., : E~D/ :s,,) ':( ~111~~ )Jc.,, (ll.96) where E~DI is the Eshelby tensor of the equivalent composite pauem depicted on Fig. Il.IOc)

  Figure 11 . .11: Evolutions of the effective Yo1111g moduli estimated by different micromechanical schemes and by FE simulations (Wriggers and Mo/fall 2006) for two types of concretes as fun ctions of the volllme fractions of aggregates a) type I a11d b) type II.

Porosity

  Figure 11.12 : Evolutions oft/ie ef f ective elastic bulk (left) and shear (right) moduli estimated by differe11iEM1'.\• and by FE simulations (Garboczi an d Berryman 2001) for both sound and degraded HCP us a fun ction of e = Ern I E 11 cr •

and

  scheme and versatility rigorous non-spherical coated bounds inclusions inclusions Dilute ** * * *

Part

  II: Prediction of the elastic properties of cen1ent•bascd 1naterials P~gc 59 r of spheroids presents the benefit of being analytically calculable (sec Eq. (11.20)).

1 (

 1 Fig. lT.14 schematically represents the randomly oriented spheroids corresponding to an identical particulate phase r and their surrounding double-inclusions inside the RVE. For simplicity these sphero ids are assumed to have the same size. The number of spheroids per unit volume is denoted by N, . The volume fraction c, of phase r can thus be writteo as: ! N .fr a 3 r 1 for oblate spheroids 3 ' er = N,.Vs. 1 ,1~""" = 3 4 b N, 7r -2 for prolate ones 3 r 11.99)

Fig. 11

 11 Fig. 11.14 (bottom right).

  HCP are far from being crack-like. Two examples on ord inary Portland cement pastes (CEM I) are considered to verify the efficiency of the shape parameter<;. These CEM I pastes wi th w/c = 0.25, 0.40 and 0.50 used for this study are defined in Tables 11.2 and 1.3 in terms of volume fractions and mechanical properties of their constituent phases .. The follow ing representation of the microstructure is adopted to compute the HCP effective elastic properties: the CSH gel acts as a matrix in which the main particulate phases presented in Part I, namely AFt, A Fm, Cl-I, CP and UC, are embedded and play the role of inclusion (Fig. 11.15). Th is representation is close to the one of the Constantinidcs and Ulm model (2004), except that a homogenized CSH gel is used instead of the two CSl-1 considered in their descript ion. , constant and b depending on c; b) b constant and c, depending on c;

Figure 11 .Figure 11

 1111 Figure 11.14: schematic representation oft/le e1•olution of the dimensions of prolate spheroids corresponding to an i11clusio11 phase r with the shape parameler ~: a) c, cunstant; b) b co11stant(Stora et al. 2006a).

  tensor n;~ applied to this representation of HCP microstnicture with spheroidal inclusions (Fig. ll.15) then results in: 111 100 [ • {ud•f . r.CSll } ]-I. {udll } UCJ) :::: .a.llll 11cP . ~ "'/),• • Ill ncP ' (i i.I 00) with Part 11: Prediction of the elastic properties of ccment•basecl 1 naterials Pago 63 (II.101)where the curly brackets denote the average over all possible orientations, where tr denotes we trace operator and where r = CH, CP, UC, APt or AFm. The tensors n<:, 1 and Ill~;,, are defined in Eqs. (11.31) and (II.33). The macroscopic Young modul i of the HCP can be deduced from the latter expression by using I.lqs. (U.6-7) and (11. 12-13).

Figure 1 /

 1 Figure 1/.16: Evolutions of the 11ormalhed effective Young modulus for sound CEM 1 pastes with capillary pores as afunctio11 of the shape parameter <;for different volume fractions. Solid lines: wlc = 0.50 (cCP = 23. 7%), dashed lines: w/c = 0.40 (ccp = 18.3%).

  ~l~achcd HCP is negligible as shown on Fig.11.18. On tile contrary, the morphology of hllhal capillary pores clearly plays a s ign ificant role due to the infinite contrast between lllatri)( and porous inclusions, as already high lighted in the preceding subsection. The relative sign ificance of this porous phase depends on its volume fraction (cep = 11 %, 18.3 % and 23.7 % for w/c = 0.25, 0.40 and 0.50 respectively). The shape ofunhydrated clinker may also have a non negligible effect, since an increase of 25 % can be obtained as <; goes to 0 (see Fig.

  ll.1 9, tl1e decrease of the effective Young modulus is much more pronounced for oblate spheroids than for prolate ones. Consequently, the e lastic properties are more sensitive to the fl attened morphology of CI I than 10 the in itial capillary pores one, represented not on ly by obl ate spheroids but also by prolate ones (see Fig. II .1 9).

  discussions 011 the morphology of the phase composing HCP microstnicture i1; Part I have highl ighted the asphericity of all the hydrated phases and unhydrated clinker. Hence, it is natura l to pose the foll owi11g questions: what is the error induced by the spherical shape approximation of the inclusions of HCP w ith respect to the homogenized isotropic elastic Properties? Is it possible to improve the micromechanical predicti ons by adopting non-S[>hcrical inclusions? 5.3.J Assessment of the spherical particle assumptio11 All the particulate phases are approximated as spherical inclusioos for simpl icity, as depicted on the right-hand side of Fig. ll.15. By combini ng Eqs. (Il.6), (11.12), (U. l8) and (11.43 -44), the estimations of the effective bulk and shear moduli by the IDD model thus take the following form: I + :~:::c, (a, -I) and Gt:[ = GCSJ,I+'°' c (.2.' -/ 3. -1 ) . I 03) I+ Lc,(/J, -t) r rwhere re {AFt,AFm,CP,CH ,UC), where a, and fl,. are given in Eq. (IT.1 9),K, and G, respectively denote the bulk and shear moduli of inclusion phaser and where Kc.,, and GcsH respectively designate the bulk and shear modu li of CSH matrix. To check the validity range of the spherical shape approx imation , the micromechanical estimations given by the previous equation are compared with the simulations of[START_REF] Kamali | Comportement et simulation des materiaux cimentaires en environnements agrcssifs: lixiviation et temperature[END_REF] applied to a representative panel of sound and degraded HCP. Two NIST models are used for the FE simulations:CEMHYD3D and ELAS3D[START_REF] Bentz | Three-Dimensional Computer simulation of Portland Cement Hydration and Microstructure Development[END_REF][START_REF] Bohn | User manual for Ii •t 1 progmms• a parallel version of NIST IR 6269[END_REF] Garboczi 1998). The first one is a three-dimensional cement hydration and microstrucn1re development modeling package. The second one is a linear elastic FE program developed for computing the li near elastic properties of random materials whose microstrncture has been stored in a 2D or 30 digital image. CEMHYD3D is probably the most well-known and efficient hydration model. A unit cell of JOO µ m in size was used for all the simulations. The limitat ions of these simulations come from the size of the voxels being on ly of l µm 3 • For further details, the reader should refer to the m1icle of ll aecker et al. (2005). Three CEM I and three blended Po11land cement pastes with 7.7 % silica fome content (CEM lllA) with various w/c ratios (0.25, 0.40, 0.50) are considered. The estimations provided by Eq. (II. l 03) arc plotted on Fig.

Fig. 11 .

 11 Fig. 11.21. These rather empirical dimensions provide a morphologica l parameter <;A~1 = 0.69, having an insignificant influence according to Fig.II.18. portlandite is further assumed to inheri1 from a sim ilar shape inside the HCP as in non-confined environments, where it crys1all izes in massive hexagonal plates[START_REF] Richardson | The nature of the hydration products in hardened cement pastes[END_REF]. Considering such a plate, whose thickness is four times smaller 1han the maximum length of the regular hexagonal base, c,c 11 = 0.70 is obtained (seeFig. fl.21). This value leads to a relative variation of Young modu lus con111riscd between 0.75 and 0.90 for the three HCP, which is not negligible.

  scheme to study early-age cement pastes and predict the solid phase percolat ion during hydration process. They also obtained resu lts consistent with experimental values.Constantinides and Ulm (2004) distinguish two microstructural levels respecting the scales separation condition: the first one denoted as l and ranging from I 0 nanometers to 0.1 micrometer is characteristic of the two types of CSH, while the second designated by JI and varying from I micrometer to 100 micrometers corresponds to the scale where CSH can be regarded as a homogeneous matrix w ith Cl I crystals, capillary pores, anhydrous residuals and aluminates as inclusions. The first step consists in calcu lating CSH matrix effective properties from CSU'"' and CSHcx' intrinsic values and the second one is to determine HCP effect ive properties wi th the aid of the previous result for CSH matrix.

Fig

  Fig. JI.22 represents the hydrated cement grains. During the hydration of cement particles,two layers presently defined as inner and outer layers form successively from cement grains surface. The inner layer is less porous than the outer one, since the first one results from higher confinement conditions and from poorer water accessibility during hydration process.Generally, an anhydrous part of the cement particles remains after hydration has stopped and constitutes a rigid core surrounded by these two heterogeneous layers. The GSCS and modified DIM derived in the present manuscript are the two most suited effective medium schemes for estimating the elastic properties of composite with layered inclusions, as evidenced in subsection 4.4.2.2, and are thus employed to compute such a model of microstrucrure. The present description is based on the assumption that the cement grains supposed initially spherical hydrate forming doubly-coated spheres that fill tbe HCP microstrncture. The spherical particle approximation was shown to be reasonable for modelling unhydrated clinker in the previous chapter.

Fig

  Fig. ll.23 displays the effective Young moduli estimated for sound pastes and a unifom1ly leached one and shows that they are in good agreement with the experimental data from Constantillides and Ulm (2004) and from Galic et al. (2004). It is worth noting that the :tiniations of the mu lti-scale model computed respectively with modified DlM and GSCS at ~cl ll (Fig. 11.23) are presently almost confonded due to the scarce volume fractions of ehnker • residuals. Experimental data on elastic properties of uniformly degraded pastes are

Figure

  Figure Jl.23: Comparison between the effective Young m oduli estim ated by the m ulti-scalemodel with experim ental m easurem ents (Consta11tiflides and Ulm 2004;Galle et al. 2004) on both sound and degraded H CP.

  addition, the relative volume fractions of these two types of CSH are computed by means of Eq. (1.2) with a hydration rate assumed equal to 0.60. The predictions of the multi-scale model for the macroscopic Young modulus of undecalcified pastes are plotted in Fig. ll.24 and are very close to the numerical results. Two different schemes, namely modified DIM and GSCS, have been employed to compute the elastic properties of level 11 of the model. The GSCS provides the closest estimations to the FE results. T he mean deviation between the results obtained with the modified DIM and with the GSCS and the numerical ones are about 3 % and 7 %, respectively.

  degraded CSH'"' and CsH•''. It has been observed that the relative amount of both CS H types seems not to change when leaching occurs(Constantinides and Ulm 2004; Thomas et al. 2004). It is furthennore assumed that the CSH elastic moduli are not affected by dccalcifi cation as in the simulations of Kama li (2003). This approximation may be j ustified by the fact that the progressive evolution of their elastic properties during the leaching process . difficult to characterize experimentally.Velez ct al. (2001) showed by means of IS panoindcntation tests that the Young moduli of the partially decalcified CSH do not significantly change especially if their C/S ratio is superior to I.

•Figure 11

 11 Figure 11.24: Evolutions of the effective Young moduli estimated by the multi-scale model and by FE simulations (Kamali 2003) for both a) sound and b) partially degraded HCP as a fun ction of capillary porosity volume fraction.

Figure

  Figure Jl.25: Evo/11tio11s of the effective Yo1111g mot/11/i estimated by the Co11.1•ta11ti11itles anti U/m motlel, by tlte present 11111/ti-scale one and by FE si11111/atio11s (Kamali 2003) for botlt a) so1111d mu/ b) partial(v tlegrmletl HCP a., . a function of capillary porosity vo/11111e fraction.

  experimentally and also has to be estimated. The predicti ons for the volume fraction of ITZ fluctuate from one author to another depending for instance on the method employed and on the thickness retained for ITZ. As an illustration, the values estimated can vary from about 8 o/o[START_REF] Sun | Modeling the elastic properties of concrete composites: Experiment, differential effective medium theory, and numerical simulation[END_REF]) to 30 %(Heukamp 2002). The use of statistical models (e.g.[START_REF] Lu | Nearest-surface distribution functions for polydispersed particle systems[END_REF] taking into account the overlapping between these transition zones is generally required to compute accurately this volume fraction. The multi-scale model developed below furthermore presents the salient feature to be computable without having to look after the volume fraction of rrz.The use of GSCS represented in Fig.11.4 is w idespread to predict the elastic properties of mortars (e.g.Ramesh ct al. 1996; Hashin and Monteiro 2002). In the present work, mortars are represented as a more detailed multi-scale microgeometry depicted on Fig.11.26. Diverse coated spheres having different sizes are nested inside a matrix region filled by the bulk cement paste. The coated-spheres are defined as follows: the core and the outer layer are respectively occupied by sand particles and lTZ. A few representative grains of different sizes are chosen to discretize the wide size distribution of sand particles with the aid of experimental data on the mortar grad ing. The grading curve of the mortar characterizing the granulometry of its sand particles is generally provided by the producer. As an example, the mo11ar with fine Nevada sand particles employed by Heukamp (2002) is chosen for the present study. Three representative sizes of sand particles, w ith radii R, = 0.085 mm, R2= 0.20 mm and R3 = 0.315 mm, are adopted to represent the sand aggregates of this mortar, on the basis of the grading data ofHeukamp (2002) collected in

  Figure ll. 26: Schematic of the representation used/or the estimation of the effective elastic properties of mortars.

1 •

 1 Figure /J.B. I: Schematic illustrating tire 11011-01•erlapping co11dition and the variatio11 of tltt! do11hle-i11c/11sion geometry with tire volume fraction of a particulate phase represe11tedrespectively by oblate u11d prolate spheroids.

  more adequate for predicting these non-linear effects arc discussed. A novel estimate for the transport properties of random heterogeneous media, such as conductivity and diffusivity, is then proposed by exp loiting the main ideas underlying a ce lebrated exactly solvable assemblage of I .lash in and Shtrikman( 1962). For random heterogeneous media consisting of high contrast phases, the derived estimate is apt to capture the sudden variations of the effective diffusivity of HCP generally observed experimentally (Fig.1.5). The third chapter presents the multi-scale homogenization approach incorporating the latter estimate developed for predicting tbe macroscopic diffusion coefficients of HCP and mortars. The model developed will then be implemented into the ALLIANCES platform(Montarnal et al. 2006; z001) so as to perform simulations of chemical degradations of cement-based materials presented in Part V.

  Fig. JI. I, composed of a matrix phase, indexed by M, and of P particulate phases, wh ich are assumed to be individually homogeneous (Fig.11.i). Their diffi.1sion tensors are respectively denoted as 0 , (r = I, .. ., /'). The same notations are used as in the previous Part.

  where the operator r • (y -x) is related to the Green function G _. (y -x) by: r. ( yx )=G. (y -~) . ( 111. 9) Pan Ill: Prcdiciion of ihe macroseopic diffusivities of' ce1nc-nli1ious niaterials J>agc 93

  of the Eshelby tensor to spherical isotropic inhomogeneities On the basis of the work of Duan et al. (2006) in linear elasticity, the Eshelby tensor in diffusion for a spherical isotropic inhomogeneity with a homogeneous sl)elr (see Fig ll.2) embedded in an infinite matrix is derived in Appendix III.A. By introducing the quantity Pf called "polarizibility" by analogy wi th dipoles in electricity defined as (e.g. Torquato 200 I):

  P == r, Ir,, where the radii 'i and r, have been defined in Part II (see Fig. 11.2). The Polarizibility Pf practically Jinks the perturbation of the concentration gradient due to the insertion of a fictive concentration gradient. lts use will be very pract ical to lighten the diverse EMTs presented in the ensuing. The latter formula is of great utility, since it allows for Pan Ill: Prcdic1ion of the rnacroscopic diffusivi1ies or cementitious mat erilll ~ determining the average concentration gradient inside the domain occupied by the spherical inhomogeneity and its homogeneous interphase in response to a prescribed uniform fictive concentration gradient in the inclusion (Fig. II.2):

1 •

 1 2 ) and substituting its expression in (llI.l 7 1 ) the following solution is obtained for ii, : g, =[1+A,.o,,, '.(o,-o.,)r' .g 0 • (lll.l 8) By carrying out an analogous reasoning as in Eqs. (II.28-30), the vo lume average J , of the microscopic flux can be expressed as: HI r . J ~' with 0 : 1 = 0 .,.(I -A I ) and HI = 0 ,-• -0 " _, ' {111.19)

H

  '" = H~'.[1 -~0;". H;"'] I (11 1.23) However, as pointed o ut in the previous Part, the latter equation leads exactly to the sarne predictions as Eq. (IJJ.22) derived with the dilute model and yields the following expression for the effective diffusivity of a macroscopically isotropic material: [ 1+2.l::c,.P:' l o;"=D., l -~c,/3:' . (JlT.24) The GSCS has been applied previously to estimate with a good accuracy the linear effective elastic prope11ies of HCP and mortars modeled as composites w ith coated spherical inclusions such as the one depicted in Fig. III. I. On this figure, a spherical inclusion I with a volume fraction c, and with a diffusion tensor denoted as O, is enclosed in a coating with a volume fraction cCt occupied by a phase having its diffusion tensor designated by Der • The doubleinclusion thus obtained is in tum embedded in another coating with a volume fraction c.., =l-c 1 -cc 1 fi lled by the matri x phase hav ing for d iffusion tensor D., immersed in an infinitely large effective medium w ith diffusion tensor o•.

Figure

  Figure II/.J: Two-dimensional represe11tatio11 illustrating the application of the GSCS for computing the tlij]i1sive properties of a composite with coated spherical inclusions.

  schemes are respectively:

  in linear elasticity, the last formula leads to the same estimations as the MT scheme (Eq. lll.23) for a composite with dispersed inclusions, as illustrated in Fig.11.8. To avoid thisUJicxpcct~d result, a modified version of DIM has been proposed in Part JI. Jts deri vation in ~ case of diffusion is proposed below by employing the inhomogeneous Eshclby tensor in • (111. 13) derived in Appendix Ill.A. T he con figuration considered for the present derivation consists in a composite pattem embedded in an infinite matrix medium and subjected to a far-field flux J-. This composite pattern comprises various double-inclusions denoted as D, (re {I, P]) enclosed in a matrix region, each of them being constituted of a spherical inclusion with diffusivity tensor D, surrounded by a spherical coating with diffusivity tensor De,. The effective diffusion tensors of the different double-inclusions are first estimated with the aid of the work of Duan ct al. (2006). TI1e particular configuration described on Fig. l 11.2a) is thus approx imated by a simpler one described on Fig. lll.2b). The effective diffusive

Figure Jll. 2 :

 2 Figure Jll.2: Schematic i/111strating tlte main steps employed to derive the modified DIM.

  of the concentration gradient inside the inclusion coat ing ic: is obtained as: gc,'= -c~ (A~, -A~).[(o, -De,.) ' .De, + A:']_, .[ D,, + A 0 , .(D, -o., )T' .J-, is the inhomogeneous Eshelby tensor of the double-inclusion D,. expressed in Eq. (111.13). The mean concentration gradients and flux inside the double-inclusion are further expressed with the help ofEqs. (111.35-36): g 0 ,=[1 ~ :,:, •1\:::_.[ (o, -De, ) '.De, + .&.: 1 J' J.[o., + A,,,.(D,. -o., )T' .J-, (Ill.38) Jo,=Dc,• I + : ;, (t-A~, ).[(o, -De, r' .De, +A~ J' l[ D., + Ao,.(D, -D,, )r ' .J -, (111.39) By eliminating J~ in the last two relations, tbe following expression can finally be obtained for the effective diffusion tensor of this double-i nclusio n: De,•[ • +S...( I -A~,) .[(o, -De, r' .De, + A~']"' -]-I J I I -c;, A~,. : (D,-Dc,) . Dc, + A ~ o,,, = (111.40) By carrying out exactly the same reasoning as in Eqs. (Jl.83 -85) (see subsection 4.3.3.1 ), an expl icit estimation can then be achieved for the effective diffusion tensor of the composite "'ith coated inclusioos:where A~0 1 is the Eshelby tensor of the equivalent composite pattern (seeFig. III.2c) and

7. 3 .

 3 J Importance of the percolation oftlte diffusive phase ln cases of infinite contrasts such as D, I D 1 ~ 0 , many authors (e.g. Kirkpatrick 197 1; Torquato 200 I) assert by means of numerical simulations that the effective diffusivity or conductivity of the medium tends to obey the scaling law: r ~~ ~ 0, for c, $ c;h, l r>' --(c -c 1 • )'f. or c >c''' e; is called percolation threshold and 1 varies between 1 and 2 (e.g.[START_REF] Torquato | Random Heterogeneous Media: Microstructure and Macroscopic Properlies[END_REF]).

  o;,cp(wlc = 0.45) = 5.21 x10-12 m 2 /s(Ricbet et al . 1997), whereas its effective Young modulus and its Poisson ratio are taken equal to 23 GPa and 0.25(Ga lle et al. 2004), respectively. T he total volume fraction of these runnels spanning over the whole paste is assumed to vary between 0 and 0.25 and we are interested in estimating their effects on the material elastic and diffusive properties by means of the full -range !DD scheme(Zheng and Du 200 1 ). On Fig.

Figure

  Figure 111.3: l!:l'ofutio11 of the 11omw/izetl ej]e£•tive efa.l'tic propertie. ~ a) flllli oftl1e tl(f/11.l'ivity b) of the pe1for111ed ce111e111 pa.l'te wit ft the volume ji•actio11 of porom cylimlricaf t 111111 e I.<.

  Geometrical prope1ties of the constituent phases, such as their tortuosity (e.g.[START_REF] Bear | Introduction to modeling of transport phenomena in porous media[END_REF], are closely related to these formation factors. Given the fact that Pan 01: Prediction of the macroscopic diffusivities of ccmcntitious materials rage 103 f)'(D, ,D, )I D 2 = D" (D, I D 2 , !) (Milton 2002), a comparison of the two previous equations gives the relation -i: c 2 = II F 2 for such materials. The latter indicates that the formation factor of a given phase incorporates information on both the tortuosity and the volume fraction of diis phase. 4 Review of matrix-inclusion type EMTs from the viewp oint of connectivity and 1. percolation A brief review of the capacity of the main existing analytical homogenization schemes to accurately model actual percolating paths of pores and capture strong variations due to the creation or annihilation of percolati ng paths in random heterogeneous media is proposed.

1 Figure 111

 1111 Figure 111.4 (logaritltmic scale): Evol11tio11 of tl1e effecti11e diffusion coefjicie111 of a macroscopically isotropic two-phase 11wterial with D 2 I D, = I 0 3 respectil'ely estimated by the MT (Mori and Tanaka 1973) a11d tlte /DD (Zheng anti Du 2001) scheme.5 as afr111ctio11 of Ille volume fraction of the diffusive phase. The latter phase is model et/ 11s proltzte spheroids with diverse aspect ratios r.

1 )

 1 11s-' +c, -a1 ' D,,s-D2 l+c, -/3,2 'I C2 JJ2 I c, I(111.46) with /3/ defined in Eq . (In. 12). The lat1er formula was derived by means of variational principles[START_REF] Hashin | A variational approach to 1he theory of the elastic behaviour of polycrys1als[END_REF]) that arc explained in many monographs (e.g. Mi lton 2002; Torquato 2001) and are thus not presently recalled. The applicat ion of the HS bounds to composite materials w ith high contrasts is problematic, since the bounds arc not tight enough to bi nd the effective transport coefficients, as may be seen on Fig.111.4. rt is possible to improve the HS bounds if more infonnation , in the form of con-elat ion ftinctions(Beran 1965), formatio n factors[START_REF] Berryman | Bounds and estimates for transport coefficients of random and porous media with high contrasts[END_REF] or experimental measurements[START_REF] Bergman | Calculation of bounds for some average bulk properties of composite materials[END_REF], is avai lable for the composite. On the basis of these pieces of information, a good number of tighter bounds (e.g.Beran 1965;[START_REF] Bergman | Calculation of bounds for some average bulk properties of composite materials[END_REF] Torquato 200 1;[START_REF] Milton | The theory of composites[END_REF]) have been derived from the variational principles of Hash in and Sh1rikman ( 1962).

D 1

 1 +2y D 2 +2y(111.47) This function depends monotonically on y , which has the dimension of diffusivity and is referred to as a transfonnation parameter.[START_REF] Berryman | Bounds and estimates for transport coefficients of random and porous media with high contrasts[END_REF] evidenced that most of the kJlown bounds, such as Reuss and Voigt, HS or Beran bounds, can be gained from the latter equation for simple values of y. rr the parameter y is equa l to Di or D 2 , the lower and upper HS bounds arc respectively retrieved. If the parameter y is equal to ( ,D, + ( ,D, or ( ( 1 1 D 1 +( 2 10 2 ) ' , the lower and upper Beran bounds are respectively retrieved (Berryman 2005). The microstructura.l parameters (, a1;d ( ; = l -( 1 are known as three-point parameters and are related to the three-point spatial correlation functions (e.g. Torquato 2001) of the compos ite microstrucn1re. Let us assume that an experimental value for the material effective diffusivity, denoted as D' , is available at a given volume fraction of porosity. Bergman (l 976) then derived improved bounds incorporating this piece of information and taking the fo llowing form: Do-= A(yo_) and Do+ = A(y 11 +) , Di--D2-Yo+ = ( vD 1 + (1v )o 2 i with v = (A-'(D') )-Di . D1 -D2 (11149) These Bergman bounds have been recentl y argued as being adequate to deal with 1he cases of high con trasts (Berryman 2005). To give an example of the curves pred icted by Bergman 1><>unds, it is presently supposed that D'I D 2 == 0.1. The HS, Beran and Bergman bounds for the effective diffusivity of random heterogeneous media with phase diffusivities such as D I D ~ 0 and D2 I Di = I 0 3 are plotted in Figs. Hl.5 and IIl.6, respectively. The bounds I 2 plotted on Fig. UI.5 appear to have simi lar shape as the curves on Fig. 111.4, which may not be suited for capturing percolation effects.

Figure 111

 111 Figure 111.6: Evo/11tio11s of HS, Beran and Bergmtm bo1111ds for a 11111croscopically isotropic two-phase material with D, I D,~ 0 as afi111ctio11 of the vo/11111efractio11 of the high diffusive phase.

dFigure

  Figure Ill. 7 (semi-logarithmic scale): Evollltions of the effective diffusion coefficient of a macroscopically isotropic hvo-phase material with D 2 I D, = I Ol estimated by SC type schemes as a function of the volllme fra ction of the high diffusive phase.

Figure 111 . 8 :

 1118 Figure 111.8: Evollltions of the effective diffusion coefficient of a macroscopically isotropic two-phase material with D, I D 1 ~0 estimated by SC type schemes as a fun ction of the volume fra ction of the tliffusive phase.

  c~ ::2/z . The latter equation has only one positive root corresponding to the Kirkpatrick estimate:

  ~cal exponent. The Mclachlan model is quite similar to the Kirkpatrick one but involves Para111eters instead of one. It is noteworthy that for p ~ I, GEM coincides with the Kirkpatrick est imate. Consequently, the McLachlan scheme can also violate HS bounds for certain val ues of the parameter p .

Figure

  Figure Ill.JO: Two-di111ensio11a/ represe11tatio11 of the CSA developed by Has/tin and

  1 = 0.56, c 2 ~ 0.44, D 2 I D, = I 0 and D I D, = 20 , where D, and D, are recalled to designate the diffusivities of the solid and porous phases, respectively. The lower and upper HS bounds for this maccrial are plotted o n Fig. 111. 11 .

Figure 111

 111 Figure111.11 (semi-logarithmic scale): Evolution of HS bounds as a/unction of the porous phase volume fraction and graphical represe11tatio11 of the geometric parameter f of MCSA(Stora et al. 2006b).

  c,)p; )-D, [1+2(/ +c, )fl;], c-2D,D, [cf -2c 2 )/J; -I] . (Ill.57) The solution of this equation provides only one positive root that constitutes an estimate for the effective diffusion coefficient of the material: o• = -b+ .Jb 2 -4oc . 2a (111.58) According as/= 0 or I, the previous equation yields the HS. or HS+ bound. Furthermore, D'

Figure

  Fig11re J/l.13: Evo/111io11s of the effective diff11sio11 coefficient of u macroscopically isotropic two-phase material with D 2 ID, = I O' estimated by the MCSA scheme as fi111ctio11s of the higli-d(f/usive phase volume fractio11 for diverse values of the parameter j:

Figure

  Figure JJI.15: Evolution of the volume fra ctions of the high-diffusive phase respectively dispersed in the shells of Cl spheres (a) and in the cores o/C2 spheres (b) as functions of the total volume fraction of the high-diffusive phase for diverse contrasts behveen phase diffusivities, the geometric parameter f being equal to 0.5.

9

  CONSTRUCTION OF A MULTI-SCALE HOMO GENIZATION MODEL FOR EST IMATING THE EFFECTIVE DIFFUSIVITIES OF CEMENT-BASED MATERIALS As already argued previously, cement-based malerials have complex microstructure with a multi-scale porosity. It is consequently quite natural to adopt mu lti-scale approaches to deal with such media. In the present chapter, a multi-scale homogenization model is first developed for HCP that constitutes the matrix of concrete materials, and is then extended to mortars. The main difficulty for building a su ited homogenization model for HCP consists in correctly representing their multifaceted porosity and in reproducing the highly non linear changes of their diffusive properties measured experimentally (Fig. l.5). Several homogenization models developed for predict ing the transport properties of cement-based

Figure 111 . 16 :

 11116 Fig. IT.22 represents the hydrated cement grains. During the hydration of cement particles,two diffusive layers presently defined as inner and outer layers fonn successively from cement grains surface. The inner layer is less porous than the outer one, since tbe first one results from higher confinement conditions and from poorer water accessibi lity during hydration process. Generally, an anhydrous part of the cement particles remains after hydration has stopped and constitutes an impcnneable core surrounded by these two heterogeneous layers. As in linear elasticity, the GSCS is employed to model such a microstructure at the micrometer scale. The level 11 also depicted in Fig.Il.22 corresponds to the scale of the heterogeneities of the two layers. It is common ly accepted that two different types of porous CSH are associated with each layer. In the inner layer, the CSI-1"" behave as a matrix phase embedding nondiffusive inclusions of CH and AF. Similarly, the external layer has matrix-inclusion type morphology, where CH, AF and capillary pores play the role of inclusions enclosed in the CSH"'. Funhennore, according to the critical pore diameters measured by M IP (TableI.5),

  assumed that only a very small volume fraction ofO.O percolates through the CSH;.'. Using this value, Eq. (Jll.56) yields /"'= 0.805. The effecti diffusi vity of CSIJ;., is then estimated from Eq. (HJ.63): D~. S""•• = 8.30xl0 13 m 2 / s . T geometric parameter of the MCSA representing csu•XI is identified from an experimcn value of the macroscopic diffusion coefficien t of tritiated water in CEM I paste with w/c 0.50 (Richel ct al. 1997): D;,cr iwl c = 0.50) = 8.9 l x Io-" m'ls . The value thus obtained f the density .f of low-porosity composite spheres is .f ""' ' ~ 0.546. Eq. (111.63) provid esti mations of o;

  Fig. ITI.17 and arc confronted to experimental resu lts issued fromRichel et al. ( 1997). lt is recalled that a HCP becomes more porous as its w/c ratio increases. They are in quite good agreement with the values measured, since the mean relative error is about 16 %. It is outl ined that the results can be notably improved by taking/as a linear function instead ofa constant.Furthcm1ore, using the identica l three-step homogenization model, the predictions obtained by employing MT with the gel pores embeddi ng sol id spherical inclusions instead of the MCSA for the effective di ffusivities of the CSHcxo are provided in Fig.111.1 7 and highly overestimate the experimental results thus confirmi ng the inadeq uacy of this matrix-inclusion type scheme in the particular mentioned configuration for predicting their effective diffusivities.

  Figure 111.18: Two-dimensional representation of HCP u~•i11g two scale.~: at the first scale, 11on-tliffusive a11d little diffusive phases are modeled as S/Jlterical i11cl11sio11s embedded in a solid diffusive phase; at tlte second scale, the homogenized solid phase and tlte porous 011e

Part

  Ill: Predictioo of 1he 1nt1croscoJ)ie diffusivitics of ce1ncntitious 1n:1tcrials sound to use a matrix-inclusion type scheme for estimati ng the effective properties of the layers of the doubly coated spheres during leaching. The effective diffusion coefficients thus obtained versus calcium concentration for CBM I pastes with different w/ c ratios are shoWn on F igure 111. 19. Two principal domains are observed on the curves displayed on this figure.

Figure 111

 111 Figure111.19: Evolutions of the ef f ective tfijf11sio11 coejjicie11ts t>fCEM I pastes with different w/c ratios as af1111ctitm of calcium co11ce11tratio11 in pore sol11tio11(Storti et tll. 2006b).

Figure 111

 111 Figure 111.20: Schem(ltic of the represe11tatio11 used/or the estimation of the effecti11e diffusive am/ elastic properties of m ortars.

  stress o and of parameters chosen to characterize the damaged state of the material. Two main categories of damage models based on different definitions of the damage parameters may be distinguished: the phenomenological models (e.g. L1deveze 1983) focusing on the damage effects at the macroscopic level and micromechanical approaches (e.g. Oonnieux et al. 2006; Pensee and Kondo 2003) linking the macro-response of the material to microscopic parameters characterizing its microdefects (Budiansky and O'Connell 1976).

11

  Ml C ROMECH ANICAL ESTI MATIONS OF T HE DAMAGED MATEIUAL BEHAVIOR In the case of a stra in formulation, the Helmhotz free energy practically corresponding to the energy stocked by elastic strains is written as : (TY. I)

  ')-' c.,,. = s,, +llll (JV.6) where IHl"'' defined in Eq. (11.33) is the dilute compliance increment tensor of the damaged body. As already mentioned in Part TI, different double-inclusion schemes providing quite simi lar predictions have been developed to account for the cracks spatial distribution (e.g. KOstcr-Toksoz 1974; Hori and Nemat-Nasscr 1993; Ponte-Castaneda Willis 1995; Shen and Yi 200 I; Zheng and Du 200 I). In the case of an isotropic distribution of cracks, identical estimations can thus be obtained from the IDD model (Zheng and Du 2001 ), PCW one (Ponte-Castaneda Willis 1995) or DIM (e.g. Hori and Nemat-Nasser 1993):

  previous equation for the effective axial Young modulus £,', the longitudinal Poisson ratio v;, = v;2 and the longitudinal shear modu lus o; = 0:2 = o;, of the damaged material allowing for retrieving IDD, MT and di lute homogenization schemes for particular values of the critical crack density parameter de:(IV. I I) If Qoc=E.,t[2(1+v)] and de tends to infinity, the expressions of the Young and shear moduli given in the previous equation coincide with the estimation of the MT model written below: E,'" rt d J

  for the effective axial Young modulus, the longitudinal Poisson ratio and shear modulus of a damaged material with aligned cracks are plotted on Fig. IV. I as functions of the crack density parameter p. This crack density parameter used by Ponte-Castaneda and Willis ( 1995) has a very clear geometrical signification. It corresponds to the volume fraction occupied by spheres circumscribing each penny-shaped crack.

  Figure I V.I: Evolutions oflfte effective axial J'o1111g modullls, tfte 1011gifudi11al l'oiss<m ratio and the /011git11dinal shear modulus of an elastic body with v = 0.25 weakened by aligned penny-shaped cracks 011 the (e2, eJ) plane estimated by diverse homogenization scheme.I' us functions of the crack density parameter p = 4n:d/3 (3D).

  account for the presence of other microcracks, as is shown in Part I I (Fig. U.5). Most of the Papers dedicated to the calculation of cracks interactions are numerical, since it is a tremendously difficult task to calculate the exact stress intensity factors of a given rnicrocrack, which interacts with all the other microcracks. Thorough numerical s imulations taking into account interactions between cracks have been pcrfom1ed by Shen and Yi (200 1), Feng and Yu (2003a,b) and Shen and Li (2004) in 20 to predict the damaged mechanical properties. Tt is consequentl y interesting to compare the present estimates w ith these advanced simulations to gauge which one is the most accurate.

Figure

  Figure JV.2: Evolutions of the effective axial Young modulus of an elastic body with v = 0.25 weakened by aligned penny-shaped cracks on t!te (e 1 , eJ) plane e.<timated by diverse homogenization ~•cl1emes and by numerical simulations as functions of the crack density parameter p = 1t d (2D).

Figure

  Figure JV.3: Evolutions oftlte effective Young m odu/11s (up le/I), the Poisson ratio (up rig/ti), the bulk (down leji) a11d shear moduli (down right) of an elastic body with v = 0.25 weakened by ra11do111/y oriented p e1111y-slwped cracks predicted by >arious ho111oge11izatio11 schemes as fi111ctio11s of the crack density parameter p (J D).

Figure

  Figure IV.4: Evolutions of the effective Young moduli and shear moduli a/011g the three principal axis e 1 = c 2 = e 3 of an elastic body with v = 0.25 weakened by three families of pen11y-shaped cracks orie11ted according to the principal directio11s with ide11tical crack densities Pi = p 2 = P:• = p I 3 (up) and with crack de11sities p i = p 2 = p I 2 a11d fJ3 = 0 (dow11) as jimctions of the crack density p arameter p.

FigureFigure

  Figure JV.5: Rosette represe11ti11g the relative decrease of the effective Young modulus predicted by the !DD scheme for hvo families of cracks alig11e1/ with the e1 a11d e2 directions which cr11ck densities are d i = </, d 2 = 0 with d = 0.1 (dash-dotted), d 1 = d, d2 = d/2 (dashed) a11d d 1 = d2 = d (plain), respectively.

  J2 ESTABLIS HMENT OF T HE EVOLUTION LA \.VSThe establishmen t of the evolution laws is generally performed m the framework of the thcm1odynam ics of irreversible processes. According to the second principle of thermodynam ics, the intrinsic dissipation Dis has to be non-negative (e.g. Ngu yen 2000):. , 11 are the driving forces associated w ith the micromcchanica l damage variab les d,.

  (e.g.[START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF]:(IV.29) where iJtp denotes the sub-gradient of 'fl. This concept generalizes the definition of derivative and can still apply even if 'P is not differentiable. The fol lowing equivalency is useful to relate the dissipation potential 'P to damage threshold function/; (e.g. Nguyen 2000):Y ., e o'f'(d 1 ) ~ Y ., realises max(Y.,d,). damage threshold functions have been determined, the consistency conditions and the normal associated flow rules, classical ly used in plasticity, can be expressed by means of Kuhn-Tucker theorem:.Id = ,t of, ~i; realises max (Y"'d,) ~ ' oY"'where ~ designate the damage multipliers, that may be expressed from the consistency condition. The following convex diss ipation potential may for instance be employed (e.g.Lorentz and Andrieux 2003):

  ,), (IV.35) 2 where f3, are called damage softening variables and the function q:i characterizes this damage softening. These sotlening damage variables are introduced as additional internal variables Part IV: Modelling of da1nage of ce1nentitious materials Pagc 1sg

  ,,/l,) = 2 e:C'(d,) :e + Z:: f(!Yl, (s)-Y 0 )ds , I 0 (IV.38) the following thresho ld functions expressed with respect to the thermodynamic forces associated to the internal variables d 1 and /); are derived:/, = Y.,,-!H,(/J, ),(IV.39)

  e :C' (d,): & +.!. r 0 w (Z::A ~ -L:JJ, ).

  is a damage parameter, to retrieve the ensuing Mari go criterion ( 1985): J.= Y,.-Y,,(l+{J))d, .

  ensuing, the macroscopic stresses or strains are denoted without any superscript. The stress tensor G estimated by the present damage model is expressed as: ( d<P ) . G = -ac = C (d 1 ):e. (IV.43) Jo the case of two famil ies of perpendicular cracks, two damage threshold functions for the two crack density parameters are involved and their derivatives read: Jf, . = ard, .. + arJ, d + ar., (J _ax, R I ac . C ad, I ad2 2 ap, 1-'Il j; = at;11 • £+ ard, (J + ar.2 (J _ax, P

  iJr., . . iJY, " d. iJY" d. ax, a -o --. &+ --I +--, --,.,, -. criterion, tbe derivative of the driving force X 1 with respect to •/3, is simply equa l to (J) (sec Eq. (IV.41 )). Moreover, g iven the fact that d 2 = 0, the latter equation yields: d . -(iJY d, _ )-' il.l'.,, . • ( IV.47) I -dd (JJ d& . f. . I The evolution of the intemal variable d 1 and the stress-strain curve predicted by different homogenization schemes, namely MT, IDD and simplified IDD, with the Marigo criterion are plotted in Fig. JV.7. The Young modulus and Poisson ratio of the sound mortar are respectively equal to EM = 44 GPa and v = 0.25 (Le Bellego 200 I). By default, the initial damage threshold is taken as Yo= 8000 J ni'' and the value of the parameter (J) in the Mari go criterion is set to zero by default. The influence of the parameter (J) on the stress-strain curve predicted by JDD is illustrated in Fig. IV.8 for information. On Fig. !V.7, the results obtained with the three micromechanically-bascd models are confronted with the experimental measurements from Le Bellego (200 I) on mortar samples. The stress-strain curve predicted by the IDD or PCW model appears to reproduce quite well the post-peak part of the strcss-strain curve but goes to zero quite rapidly. Visible differences between the curves obtained with MT, IDD and simplified !DD schemes may be observed on Figs. !V.7, even though the evolutions of damage variables are computed with Marigo criterion using the same parameters ( Y 0 = 8000 J m•' and (J) = 0) for the three models. The MT and simplified !DD models are seen to be far from Le Bellego's results for this set of parameters but it is difficult to improve their predictions even by changing the values of Y 0 and {JJ. To obtain a better agreement with the experimental resu lts, it would be preferable to adopt a stress formulation for the Helmhotz free energy in cq (IV.I)[START_REF] Pensee | Contribution de la micromecanique a la modclisation tridimensionnelle de l 'endommagcment par mesofissuration[END_REF].

2

  Fig ure IV. 7: Compariso11 of the stress-strai11 curves pre<ficted of a mortar subjectetl to 1111iaxia/ te11sio11 along the e 1 axis by differe11t micromecha11ically based damage models with Marir:o criterion with the one experim e11tally recorded by Le Bel/ego (2001) (up); evolutio11 for the cmck density parameter d 1 predicted by the latter models (dow11).

Figure

  Figure IV.8: Comparison of the stress-strai11 curvel• predicted of a mortar subjected to u11iaxial tension along the e 1 axis by the 1DD damage model with Marigo criterio11 with the one experime11tally recorded by Le Bel/ego (2001) (up); evolutio11 for the crack de11sity parameter d 1 predicted by the latter model (down).

  to avoid this Problem is to consider a Taylor development at the first order of the driving forces, so that the extra diagonal terms in the previous matrix are annihilated. However, such an assumption tnight not be fully satisfactory, since the accuracy of the homogenization scheme is lost in thi s Taylor development at the first order. The stress-strain curves predicted by the tnicromechanically-based damage model with the Marigo criterion for mortar samples subjected to biaxial tensions along the two axis e 1 and e2 with & 11 =11 22 is plotted on Fig. IV.9. The plotted curves clearly differ from the uniaxial case (Fig. TV.[START_REF] Lo | Solutions for effective shear properties in 3 phase sphere and cylinder models[END_REF]. This is due to the fact that the stiffness tensors predicted by the MT, 100 and simplified schemes do not vary linearly with damage variables and thus the driving forces Ya; depend from tlie two i. nvolved crack damage variables. This dependence practically results in different stress-strain curves in uniaxial and in biaxial tests. It is fu11henn ore pointed out that the extra diagonal tenns of the square matrix in the previous equation do not vanish and the system of ordinary differential equations has to be solved numerically, using preferenti ally an implicit Runge-Kuna algorithm. As in the un iax ial case, strong differences between the stress-strain curves obtained with MT, IDD and simplified TDD schemes arc visible on Fig. IV.9, when the same parameters the evolutions (Y 0 =8000Jm•l and cv=O).

Figure

  Figure IV.9: Stress-strain curves along the two principal a..:is of a mortar subjected to biaxial tension along the two axis e 1 and e2 with c,, = c 22 predicted by different homogenization models with Marigo criterion (up); evolution for the crack density parameters d 1 and d 2 predicted by the latter model (down).

I

  and De Vree et al. ( l99S) are respectively recalled below:e,., = J(e ,)' . , with (e,). = e, :ie ,I, 2k l <<lEvv = 2k(l -2v) 1 +-2k l-{'--l--2 -' v"7) ' 1 + {l +v) 2 2 ' (IV.52)where e 1 (i = J,ll,111) are the principal strains, e 0 is the strain threshold corresponding to the initiation of damage in traction and k denotes the ratio of the compressive strength over the tensile one. These two equations are very different at first sight, since the first one is expressed in terms of principal strains and tbe second uses the first two invariants of the strain tensor.The surfaces representing these two threshold functions arc plotted in Figs. JV.IO and IV.I I for v = 0.2S and Eo = I 0 .. . A sound criterion should respect some important conditions: (i) to be systematically non negative; (ii) to be closed co11vcx. The Mazars criterion fulfills the first condition but not the second one (Fig. IV. I I). The De Vrce criterion respects both conditions (De Vree ct al. I 99S). In addition, the strong connections ex isting between the De Vree criterion and the one recently proposed by Cbristensc11 (200S) are evidenced in Append ix IV.B. The damage threshold surface evolves during the damage process so that a suited law is required to account for this evolution.

Figure

  Figure IV.JO: Damage yield surfaces respectively representing the Mazars criterion and theDe Vree one for k = I and k = 3 in the space of principal stresses.

  -~-'-~~~~~~~ ..... ~~-'-~--JL-4--'

Figure

  Figure IV. I I: Damage yield surfaces respectively representing the Maza rs criterion and the De Vree one for k = I and k = 3 in the space of principal strains.

  It has been extended to fatigue(Papa I 993) and recently to anisotropic damage[START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi -brittle materials[END_REF]. A non-local version of the Mazars model was furthennore proposed byPijaudier-Cabot and Bazant ( 1987) in order to improve !he numerical results and limit the mesh dependency. Basically, theMazars model (1984) consists in defining the following criterion: where '.D is a scalar macroscopic damage variable expressed as: function K(.V) rules the evolution of the damage surface. This function ic(.V) is chosen so as to obtain the following evolution law:'.D = I •o(~-A) Aexp(-8(&,.. -r. 0 )), c ~"' (!V.55)where £.,. is the Mazars equivalent strain de fined in Eq. (IV.52) and where A and 8 are parameters that are identified with different values in traction and in compression. The stressstrain behavior is then given by: (JV.56)[START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi -brittle materials[END_REF] proposed a convenient extension of the Mazars evolution to anisotropic cases by adopting a non-associated evolution law deriving from the latter dissipation potential:'fl. = Y :(r.,, ):, (IV.'.>7)where Y is the dri ving force associated with the second order damage tensor '.D and where tp is the tensor of the principal strains. This tensor admits the following representation:(TV.58)where ® is recalled to stand for the tensor product and e, (i = 1,2,3) are the orthonormal vectors defining the principal strain directions. The derivation of the dissipation potential thus yields the following evolution law:(IV.59)In other words, the damage tensor rate is proportional to the square of the positive part of the principal strain tensor. The Mazars criterion is reused by substituting the previous isotropic damage variable '.D by the trace of the second order damage tensor :D presently considered: f= c.., -K(tr'.D),(IV .60)

  vectors of tl1e two families of perpendicular cracks retained in the micromcchanical model arc respectively oriented along e 1 and e 2 • The next stage consists in identifyi ng a suitable function "(trd ). To perform such task, on ly one fami ly of aligned cracks with no rmal vectors parallel to e 1 is first considered for simplicity. It is pointed o ut from Eqs. (IV.52-53) that the elastic stiffness deterioration along the direc. tion e 1 with the Mazars model follows the evolution: computations, the parameter A is taken equal to I . It is furthermore recalled that MT, IDD and simpl ified IDD estimations of the effective Young modu lus £~ of the damaged material along the nonnal direction e 1 to ali gned cracks are expressed analytica ll y as functions of the crack density parameters in Eqs. (IV. I I), (IV. 14) and (IV.15), respectively.By combining these equations with Eq. ( !V.64) where the Mazars equivalent strain is substituted by the one of De[START_REF] De Vree | Comparison of nonlocal approaches in continuum damage mechanics[END_REF], the following damage load ing functions f can be establ ished for MT, lDD and simplified IDD damage models, respectively:d~"'g(fm, ) _d !,,, " g E:ov = -exp -E:ov -E:o an 2a a = 1g(l,,,. ) b= l -g(l0 v)

  e 0 , )-trd , (IV.67) where the expression of K depends on the homogenization scheme employed to compute the effective mechanical properties of the damaged material. By analogy with Desmorat et al. (2007), a simple expression is proposed for the dissipation potential: tp' = Y,: (Rt,,):, (IV.68)

  To show the thermodynamic admissibility of the difforent micromechanically-based damage model with the stra in-based criteria defined in Eq. (IV.66), the respective positivities of the thennodynamic forces YJ 1 and of the damage rates d, are verified. To prove that Ydt ;,: 0 , it is sufficient to show that the derivatives of c• with respect to the crack density parameters are semi-definite positive, for example by computing its detenninant or by looking at the signs of its eigenvalues. For this purpose, it is convenient to rewrite the stiffness tensor predicted by the different homogenization schemes in a matricial form . Using the fact that:_ac• = C••()~;° c• with i e (l,2), the derivati ves of the stiffness tensor estimated by MT, roo and simpl ified IDD with respect to the crack density parameter d, (or similarly d, ) read respectively: ac:,,. -c• •(4 . s ) . c• -(}d, -;rr • 3 . 11-' f. . ,, . ;rr, -ac;OD = c• . dRl!/00 . C' ac mr, -c • . . alill . c" -~ -I/JO . (}d, • ttJtJ> (IV.73)

'

  Part lV: Modelling of damage ofcemcntitious rnaterials 9£.,d~" (1-2 v +3v 2 )[9(1 -v)d;!" +(12(1 -v)U 00 +4)d] (.'DD: (l-v)[ (6(1-v)Q 00 + 2)d 2 +9d-;' (I +v)d + 9(1-2v}(l +v)J' t 0 -100 E.

Fig

  Fig ure IV.12: Comparison of the stress-strt1in curves predicted of a 111ortt1r subjected to uniaxiill ten sion by the da111age 111odel with "strain-based criterion with the one experi111e11tt1//y recorded by Le Bel/ego (200 I) (11p); evo/11tio11 for the crack density pt1rt1111eter.~ d 1 and di predicted by the latter model (down).

FigllreFigure

  Figllre IV.I 3: Stress-strain curves along tlie hvo principal axis of a mortar Sllhjected to biaxial tension along tlie two axis e 1 and ei with e,, = c 22 predicted by tlie damage model with Marigo criterion (up); evolutions fo r the crack density parameters d 1 and d 2 predicted by tire latter model (down).

12. 3

 3 Conclusions of chapter 12The evolution law has been established within the framework of thennodynamics Of irreversible processes. Two theories, Marigo's one based on the GSM formalism and the other one using the concept of equivalent strain, have been assessed and applied to the micromechanically-hased damage model. Their tbennodynamical admissibi lity has also been investigated, showing the coherence of both theories. For our particular ortbotropic model comprising two internal damage variables, the strain-based evolution law that seems easier to implement in a FE code is adopted. The criterion proposed by Marigo is attractive but requires some simplifications and perhaps more efforts(Zhu ct al. 2007) for its use in numerical computations. CONCLUSIONS OF PART I V An orthotropic damage model based on both micromcchanics and thennodynamics of irreversible processes has been developed. The model comprises two damage variables, representing the density of cracks in two perpendicular directions, which is suited to deal with 01thotropic damage under plane stress cond itions. Different homogenization schemes have been tested for estimating the Helmhotz free energy. The IDD or PCW methods are shown to be of interest provided the damage variables do not trespass a certain critical crack density value. A simplified version of the !DD model has been developed to circumvent this difficulty. However, the original lDD or PCW scbeme that is more standard is chosen for computing the damaged material behavior for a given damage state in the ensuing FE simu lations. A strain-based evolution law is established for the two micromechanical damage variables and the them1odynamical admissibility of the developed damage model has been verified. An application to the case of mortar samples subjected to uniaxial and biaxial tension tests bas been proposed. Jn the subsequent Part, the !DD damage model is implemented in the FE code CAST3M and more complicated mechanical problems, such as flex ion tests, arc addressed.

  showed the existence of highly degraded areas around artificial cracks simi lar to those originated by mechanical damage. Hence, the nucleation and growth of cracks in the concrete may accelerate the transport phenomena and then enhance the chem ical degradation process, which in tum creates an additional porosity that affects its overall mechanical behaviour. Jn addition, numerous investigations on the influence of calcium leaching on the residual mechanical behaviour of cement-based materia ls (e.g. Carde l 996; Gall e et al. 2004; Heukamp ct al. 2005; Le Bellego et al. 2003; Nguyen 2005) have concluded on the significant reduction of their elastic moduli and of their residual strength. They generally consist in performing mechanical tests such as uniaxial compression or traction tests (e.g. Carde 1996; Le Bellego 2001; Nguyen 2005; Heukamp et al. 2005) or 3 points flexion tests (e.g.Le Bellego 2001). on samples previously subm itted to an accelerated leaching. Unfortunately, these experiments, designated as « residual resistance» tests, do not account for the effects of damage on the chemical degradation process. Nevertheless, some fully coupled experiments, designated as "life-time", have been performed among others by LeBellego (2001) and[START_REF] Schneider | Deterioration of high-performance concrete subjected to attack by the combination of ammonium nitrate solution and flexure stress[END_REF], where the material is submitted to a mechanical sollicitation and undergoes simultaneously a chemical attack by an aggressive solution of ammonium nitrate. These experiments may lead to tbe complete rupture after a few months.

For

  this purpose, the multi-scale homogenization model developed in Parts II and 111 and the micromechanical damage model proposed in Part IV are implemented into the numerical platform to estimate tbe material mechanical behavior as well as its diffitsive properties. One of the interests of these ho mogenization theories is to propose a deductive approach that may apply to diverse cementitious materials, provided the elastic and diffi.1sivc properties of their elementary phases are known. T he integration in the ALLIANCES platfonn of a comprehensive homogenization model could thus penn it to develop an advanced prediction tool of the degradation of real concrete structures. Furthe1more, these upscaling methods help for a better comprehension of the influence of the 111icrostn1cture and its evolution with time on the chemical and mechanical behaviors of these materials. The scope of the present Part is two-fold. The first chapter is specifically devoted to the simulat ions of chemical degradations. After a brief presentation of how to model chemical degradations with the ALLIANCES platfonn, one-dimensional simulations of pure water and accelerated leaching of cement-based materials a.re thus proposed and validated. In the second chapter dedicated to the in teractions between damage and chemical deteriorations, coupled chemo-mcchanical simulations of simple structures subjected to dccalcification and sol icitated in flex ion are performed; confrontation of the numerical resu lts with available experimental data (Le Bcllego 2001) is carried out and analyzed. 13 MODELLI NG AND SIMU LATIONS OF TH E C H EMICAL DEGRA DATIO N W ITH ALLI ANCES PLATFOR M 13. 1 Presenta tion of ALLIANCES The scope of this part is to detai l the main steps for 1nodelli1Jg chemo-mechanica l degradations of cement-based materials with the ALLJANCES platfonn that is briefly Part V: Simulations of1hechcmo•mcchanical degrad•tions by leaching ofcement,.bascd tnarcrials Page 177 presented below. For more details concerning the laner platform, the reader should refer to t;tontamal et al.

  chemical equilibrium state is obtained by means of a chemical equilibrium code. Plane! (2002) thus modelled the sulfate ~ttack of HCP with HYTECH (van der Lee et al. 2003), based on both geochemistry and transport codes. Moranvi lle ct al. (2004) ran with the help of the CEMHYD3D model of NIST (Bentz et al. 2002) simulations of pure water degradations that are quite comparable to the ones presently realized. It is pointed out that the ALLIANCES platfonn is capable of treating mu lti-species reactive transport problems on the COnirary to many simplified coupled approaches developed for cement-based materials (e.g. Pan V: Si1nulntions of the chc1»0•mechanical degr.-idations by leaching of ccn1ent•based nHuerials Page 178 Gerard 1996). This is important in the case of dccalcilicatio n for modelling for instance the migration of alkalis or simulating accelerated lcachings by NH.N03 because of the penetration ofN03-. J 3.1.1.J Transport equations solved by CASTJM To present in a simple manner the transport equations solved by CAST3M, transport is first supposed to occur in a non reactive porous medium representing the cement-based material .

Part V :

 : Simulations ofthcchcmo-mcchanical degradations by leaching of cement-based materials Page 179so that only one effective tensor o• is involved. Furthennore, tbe cementitious materials are coDsidered to be macroscopically isotropic so that this effective second order diffusion tensor o• simpli fies into D't , where D' is a scalar. By combining Eqs. (V.2) and (V.4), tbe 1ransport equations solved by CAST3M are written in the following way:

NK~=

  Tif e .. I[ e;' r•-. chemical-transporr equations solved by ALLIANCES The ALLIANCES platform solves the chemical-transport problem by coupling the chemical equ ilibrium code (CHESS) with transport (finite volume, CAST3M) in the following manner: = J+L:v .. Pm ,'Vj e {l, ... ,N},'v'me {l, ... , Mi} , .. .. where c., , c,. respectively denote the volume fractions of the mineral phase m and of porosity, v;. designates the molar volume (Umol) of the minera l m, Mi is the tota l number of mineral species and the concentration C 1 of ions j is expressed in 11101 per liter of solution.The operators Tr and Ch, respectively computed by CAST3M and CHESS, are such as:Tr(c 1 .,cm,C 1 ) = div(D 0 (c,.,cm).gn1d( C)) , , K~ and Zim have been defined previously. The effective diffusion coefficient o• depends of the porosity and of the volume fractions of minerals. Eq. (V. l J) presents how these quantities arc computed with ALLIANCES. It is noteworthy that diverse empirica l diffusion laws, such as the ones of Winsauer and Tognazzi , are already implemented in ALLJANCES[START_REF] Nozourtier-Mazauric | Prise en compte des variations de porosite dans le couplage transport-chimie mis en reuvre dans ALLIANCES[END_REF]. In order to show the importance of the evolution law chosen for the macroscopic diffus ion coefficient, difTcrcnt types of laws, such as Tognazzi's one ( 1998), are tested in the first simul ations of leaching on cement pastes.

Prut V :

 : Fig. V.I. The first step consists in implementing a homogenization model in order to estimate the material effective diffusive properties from its composition. By communicating with a chemic<•I eq uilibrium code (CHESS), ALLIANCES in a second step updates at eacb time step 1he mineral composition and the porosity of the cement and c. omputcs the evolution of the diffusivity inside the material from its current composition. This practically corresponds to the chem ical-transport loop depicted on the !en-hand s ide of Fig. V . I. The coupling algorithm is sequent ial iterative(Yeh and Triparhi 1989) and has been validated on different tests(Monlarnal et al. 2007). The fixed point algorithm used to so lve the transport e<1uations is implicit in time (Nozourticr-Mazauric 2004), but the scheme employed to solve the chem ical parl is explicit. The precis ion concerning the chemical resolution is fixed by the user and is defined at a given time step t + I from the total concentrations of the constituting species as follows:

  Figure V.l: Scltematic of the 111otlelli11g of che1110-111ecl1t111ical degradatio11s of ce111e11fbased 1110/erials 111itl1 ALLIANCES.

PFigure V. 2 :

 2 Figure V.2: Schematic of the 011e-dimensio11al system employed.for the s i11111/atio11s of pure waler leaching of H CP.

  I. In order to obtain reasonable computations times, non altered HC P is assumed to be composed only of four or five main phases: [UC, CH, AFm, C1 .6sSll, AFt or hydrogarnet]. During the leachi ng simulations, the chemical code CHESS searches for the new mineral phases that precipitate using its huge database. To fi1rthcr decrease the time compu tation, the database is reduced to the following system for altered llCP: [C H, AFt, A Fm, C1.~sSH , Cu;SH, Co.\lOS H, Si02 (am). diaspore, gypsum, hydrogarnctj, where Si02 (am) refers to the amorphous silicon gel. The chemi ca l formu lae and the equi librium constants of Parl V: Siinuta1ions of the che1 no-mcchanical dcgrada1ions by leaching oi cc1 ncnt-bascd m;iteri;i.ls P~ge 183 these mineral phases are g iven in Appendix V .A. On the contrary to portlandite that can totally dissolve during the degradation of a cement paste, the decalcification of CSH rathe r implies a progressive decrease of their C/S ratio. As a consequence, theiJ diffusive and elastic properties are gradually diminished. To model this degradation in a simplified way, three types of CSH with diverse C/ S ratios (C/S = 1.65, 1.25, 0.90) corresponding to more or less decalcified states, are introduced in CHESS. Silicon gel Si02(am) is fmthermore introduced in the chemical system to represent the most decalcified state of CS H. The anhydrous residuals are supposed to be unreactive, since they only represent a few percent in terms o f volume fractions and modelling their hydration would require very long computation times.

  Transport/:, ",.., v,,,, \Vi th me (CH, A Ft, AFm, C, 6 ,SH, C, ,,SI I, C 0 . 90 SH, Si0 2 (am),

  MCSA parameters:/ '"' :ind /"' (Pa11 Jll)Coupling Chcmistry -, transportCoupling CT M Parameters of the coupling algorithm.Frequency of the computations of the dama~e state with CAST3M.

The

  CSll "'' a nd CS H'" arc not distinguished in the c hemical computat ions, since their relative proportions inside the paste are not affected by leaching (Constantinides and Ulm 2004). It is however necessary to differentiate them in the homogeni zation model, because they have different elastic and diffusive properties, as shown previously in the manuscript. Their relati ve volume fractions a re g iven by[START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland c. ement pastes[END_REF]. Constantin ides and[START_REF] Constantinidcs | The effect of two types ofC-S-H on the elasticity of cement-based materials: Results from nanoindentation and micromechanieal modeling[END_REF] have measured the Yo ung modul i of CSH 1 "' and CS H"' both for sound and asymptotically leached stales, their C/S ratio decreasing to 0.8(1-leuka mp 2002). By asymplo tical ly leached, Part V: Simulations of the chemo-.n1echanicnl degradations by leaching of cement.based materials J>age 184

  13.2.3 Influence of the dif/11sio11 law on the chemical degradationSimulations with three different diffusion laws enlisted below are performed on the CEM I 42.5 paste to put in evidence the influence of the evolution of diffusivity on the leaching process. The first simulation is thus made employing a simple proportional law designated by D~Qf': (V. 15) the second one adopts Tognazzi's law denoted as D;.,, . for predicting the evol ution of the diffusion coefficients: D;.W> = D~1 exp(9.95{c. -c~'J), (V.16) whi le the third one uses the multi-scale homogenization model defined in Part fl for estimating the diffusivities. In the equations above, o;. 1 refers to the eOective diffusivity of the non altered paste, Cp a11d c;' ' denote the current and initial total porosities of the paste, respectively. The mineral compositions of the CEM I 42.5 paste with w/c = 0.40 degraded by pure water after 50 days, computed at each center of the subvolumes (see Fig. V.3) with ALLIANCES using respectively the three types of laws, as a function of the depth of the simulated samples are plotted on Fig. V.3. As a complement, Fig. V.4 shows the evol ut ions of the diffusion coefficient inside the CEM I paste for the three simulations. Pan V: Simulations of the cherno-1ncchanical dcgraditio n~ by leaching Of (:l!mcnt•based materials .,,....,_. ....... ..., ........ -+j-AF•

Figure V. 3 :

 3 Figure V. 3: Mineral compositions in terms of volume fractions insi<le a sample ofCEM I 42.5 paste with wlc = 0.40 subjected to a 50 days attack by pure water simulated with ALLIANCES using") a proportional law, b) Togn"zzi's law and c) the multi-scale homogenization motfel, respectively.

Figure V. 5 :

 5 Figure V.5: Evolutions of the pH of the interstitial solution and of the calcium concentrations i11 solid phase and i11 solution inside a CEM 142.5 paste with wlc = 0.40 subjected to a 50 days auack by pure water simulated with ALLIANCES 11.~ing a proportional law (red), Tognazzi's law (blue) and the multi-scale ho111oge11izatio11 model (dashed), respectively.

  ~t~r-~--..~~-.-.~---,.-~--.~~-.--.~--.~~-...~~-r;====:c:==::::;i .9c; ....... AJ • • g

Figure V. 6 :

 6 Figure V.6: Evolution of the co11ce11trati<ms i11 solution of the main aqueous species compared with the i11itial ones as af11nctio11 of the depth of the s"mple ofCE1 W 142.5wit!t w/c = 0.40 f'"Ste after 50 d"ys of pure w"ter degrad"tion simulated with ALLIANCES 11s i11g a proporti()lwl fow (up), Togn"u,i's law (middle) a11d the m1tfti-sc"fe /l()moge11hation model (down), respectively.

Fig

  Fig. V.6 that Si0 2 (aq) move inside the interstitial solution towards the exterior of the paste, whi le they should hardly move in theory because of the big size of the Si02(aq) molecu les. Since there is a large number of factors that affect the chemical degradation of cement, it is ol practical sign ifi cance to identify factors that strongly affect the deterioration . For this purpose. the respective effects of the composition of the cement paste, of the aggressive solution, of the inclusion shape of Cl-1 and of the CS H percolating porosity on the leach ing of I ICP are investigated below. The last two factors are parameters of the mu lti -scale homogeni7.ation model that significantly modify the macroscopic diffusive and mechanical prope11ies ol cement pastes (Stora ct al. 2006a; b).

(

  )art V: Sinlula1ions of the che1 no•mechanical degradations by leaching of cement-based 1naterials Page 191

  but also with other results from recent simulat ions (Moranvi lle ct al. 2004).

Fig ure V. 7 :

 7 Fig ure V. 7: Mineral co111positio11s i11 terms of1•0/11111efraction. ~ inside " . mmple o/CEM I 52 . .5 HTS paste with w/c = 0.45 subjected to a 50 day.1 • attack by pure water si11111/ated witlt ALLIANCES. The figure above presents the mineral profile in a CEM I IJTS paste wi th w/c ~ 0.45 leached wi th pure water during 50 days. The comp lete dissolu tion of phases, such as Cl I, hydrogarnet and A Fm. may be observed on the left-hand side of the top of Fig. V.7, corresponding to the alterated zone. It is noteworthy that there is a reprecipitation of /\Fm in the zone where

  Fig. V .8 eonfinns that the degradation propagates faster in the CEM I HTS paste. The decrease of the pl I and of the concentration of calcium in solid phase is much more significant than inside the CEM I 42.5 paste due to the dccalcification of the C12sSH.

  Bourdette ( 1994) on a CEM I 42.5 paste with w/c = 0.40. It is consequently necessary to adjust the MCSA parameter of the mu lti-scale homogen i7.ation model in order to obtain a better agreement with the experimental results. 111• 2 d• 05 (Adenot 1992) 1.75 11101111" ct•O.> experimentaly I. I mol m• 2 d-0.5 (Bourdcttc 1994) (Galle el al. 2002) Calcium Flux (ofter 50 days of 0.55 mol m• 2 d•<l.> 0.82 mol 111• 2 d.o.> sin1ulatc<I pure waler leaching)

Figure

  Figure V.8: E1,o/11tions oftlte /1H oftlte interstitial solution and oftlte concentrations of cafci11111 ifms in solid pf1t1se and in solution inside 42.5 (w/c = 0.40) and 52.5 HTS (w/c = 0.45) CEM f pastes subjected to a 50 days a/lack by pure water.

  coupled chemo-mechanical degradation experiments (sec e.g. Le Bcllcgo 200 I). II is therefore necessary to be able to simulate accelerated leaching by this 6M NII 4 N0 3 solution to reproduce the available chemo-mechan ical degradati on experiments. The rectangle with 3 mm of length used for pure water leacb ing simulations (see Fig. V.2) is not adapted for the 6M NH 4 N03 attack due to tbe very fast propagation of the dissolution front. A bigger rectangle with I cm of length and I mm of height and a bigger mesh composed of 20 rectangular subvolumes are consequently employed to obtain reasonable computation times. The mineral profiles obtained for a CEM 1 42.5 paste with w/c = 0.40 respectively auacked by pure water and by a 6M NH 4 N0 3 solution during I 0 days is plotted on Fig. V .9. The dissolution front is at least one order of magn itude deeper for the accelerated attack. The decalcification front propagations obtained for different cement pastes are presented in TableV.4 and appear to be at least one hundred times faster than for pure water Jeacbing. However, the simulated dissolution front propagations are clearly underest imated compared to those 1J1easured experimentally. This shortcom ing may be caused by a too low diffusion coefficient in the degraded zone. It is furthermore pointed out that the calculations of ion activities by the Davis mod ified arc inaccurate and probably overestimated, wh ich shou ld rather enhance the simulated propagation of the dissolution front. According to Heukamp(2002), the Davis modified law used by default in ALLJANCES does not suffice for computing the right activities and equilibrium concentrations in the case of a 6M NH4N01 solution and other laws such as Pitzer equations should be used. Indeed, the equilibri um concentrations of the hydration products are strongly modified in the presence of a 6M NH4N01 solution.

•\

  Figure V. 9: Mineral co111positio11s i11 terms of volume fractions im•ide a sample of CEM I 42.5 paste with a w/c = 0.40(le1Jellego 2001) s ubjected to a JO days a/lack 011 its left side by pure water (left) and by a 6M NH 4 N 0 3 solu1io11 (right).

Figure

  Figure V.10: Evolutions of the macroscopic diffusivity i11side" sample ofCEM I 42.5 pas te (.vie = 0.40) subjected to a 50 days affack by pure water simulated with the multi-scale homoge11izatio11 model with CH modelled as a sphere (blue) am/ flS obfilte spheroids (dashed), respectively.

Figure2

  Figure V.12: Evol11tio11s of the macroscopic diffusivity inside a sample <JfCEM I 42.5 fJllSte (w/c "' 0.40) s ubjected to a 50 days attack by pure water simulate1l with the multi-scale homogenization model by co11sideri11g diff erent quantities of additional percolllting gel pores ( f!.c~) = 0.04 and f!.c~~ = 0.065 ).

  the paste predicted by the simulations are qu ite close from what is generally observed experi mentally. Tbc resu lts obtained with the latter model are better than those ob1ained with empirical laws (e.g.[START_REF] Tognazzi | Couplage fissuration-degradation chimique dans les matcriaux cimentaires: caraeterisation ct modelisation[END_REF]). However, the dissolution front propagation and the calcium flux tend to be underestimated in comparison with the experimental measu res but the results have been improved by increasing the quantity of percolating gel pores in the mu ltiscale representation of llCP. After this presentation of the coupled chemical-transport problem solved by ALLIANCES, the next chapter is specifical ly devoted to the couplings between chem istry and mechanics. Pan V: Si1nul:uions of 1hc ehcmo-mcchanical dcg:raclation.s by leach in~ of ccinenl•bascd maierials p 3 g.e 201 Ulm (2004) funhcr measured that the stiffness modulus of an asymptorically leached paste decreased to about 3 GPa. The homogenized Young modulus of the most dccalcified subvolumc on the lefi-hand side of Fig. V.15 is of the same order.

Fi1:11re V. 15 :(

 15 Evnl11tif111 of tlte Young 11wtf11f11s insitle a s11111pl e of CEM I 42.5 paste witlt wlc • 0.40 (Le Bellego 2001) s11bj ec1etf '"" 50 tfflys pure w11ter allflck. 14.2 S imulations of th e mechanical residual resistance t~sts on cement-based m aterials 14.2. I Tensile tests 011 1111iformly leacltetf specimens I lcukamp (2002) performed interesting types of experiments testing m traction or compression both sound and asymptotically leached samples of cement or monar. By asymptotically leached, it is intended that the sample has been so severely attacked that it reaches a unifonn degraded state with much lower elastic propenies and strength. To simulate such deterioration, the prev ious one-dimensiona l system in Fig. Y.2 is reused except that a 6M NI r.N03 solution is now emp loyed instead of pure water. The initial mineral composition of the OPC type I cement paste with w/c , 0.50 is computed by means of the Tenn is and Jennings model and displayed in Appendix V.B. After 50 days, the sample is already entirely leached and a uniaxi al tensile test along the e 1 axis is simulated on tbis specimen.The eventual cracks inside the paste solicitated in traction along the venical axis on the beam should mainly appear oriented along the hori7.0ntal axis. For simplicity, the normal vectors of all the cracks in the present damage model arc thus assumed to be parallel to the horiwntal axis e 1 • The subscripts can thus be disregarded, since only one family of parallel cracks is involved. The undamaged material being isotropic, the effective damaged one is consequently transversely isotropic, according to our model. Pa11 V: Sinn1lation.s of !he che1110-1 nech.;lnical deg.rad:nions by teaching of ccrncnl•bascd nlulcrials Page 202 T he damage parameters such as tensile strength in the sound and uniformly leached specimens are obtained from Heukamp et al. (2005) and enlisted on Table V.6. A fter simulating a 100 days attack by a 6M NH 4 N03 solution, the sample is uni formly leached. The stress-strain curves of these two samples obtained by FE arc plotted in Fig. V .I 6, illustrating the stro ng degradation of the mechanical properties of chemically deteriorated cements. The relative decrease obtained for the Young modul i are qu ite close to the ones measured by I lcukamp et al. (2005). Unfortunately these tests did not record the post-peak behavior. Therefore, the developed tool, and in particular the micromcehan ically based damage model, needs to be further val idated by comparing w ith the more detailed experiments from Hcukamp ct al. 2005)Table V. 6: Meclt1111ic11l input data adopted for tlte micromecl11111ically-b11setl tl1111111ge model in the cases of a HCP sample (He11kamp et al. 2005) s11hmittetl to " 1111i11Xi11l traction and of the mortar beam tested i11 jlexion (le Bellego 200.1).

Figure

  Figure V.16: Stress-str11i11 curves of a so1111d and a11 asymptotically leached cement paste with 11 w/c mtio equal to 0.50 (He11kt1111p et 11/. 2005) submitted to a 1111iaxial tn1ctio11 followi11g tlte e, flxis.

  For testing the mechanical resid ual resistance of cement-based materia ls, the case of a beam in flex ion is now s imulated so as to reproduce existing experiments (e.g. Le Bcllcgo 200 l ;[START_REF] Schneider | Deterioration of high-performance concrete subjected to attack by the combination of ammonium nitrate solution and flexure stress[END_REF]. Le Bcllego (200 I) in particu lar cond11ctc. d a large series of three. point bending tests of partially leached mortars and recorded the whole curve linking the displacement imposed and the reaction force. It is emphasized that their results are more detai led than the ones of[START_REF] Schneider | Deterioration of high-performance concrete subjected to attack by the combination of ammonium nitrate solution and flexure stress[END_REF], who perlo nned the same kind of tests but have on ly recorded the ultimate strength. In the present simu lations, the smallest beam tested by LeBellego (2001) is used. Its total length and height are 320 mm and 80 mm, respective ly. T he lateral surfaces and the 60 mm long sections at the corners of the beam are insulated by an epoxy coating (see Fig. Y.17). T herefore on ly the central pa11 of the lower face, which wi ll be subsequently subjected to tensile stresses, is in contact w ith a 6M NH 4 N03 so lution . T he 20 mesh depicted on Fig. Y. l 7 is employed for the chemical and mechanical computations on the beam. The chemical attack of the beam by this aggressive solution lasts 114 days. Le Bcllcgo measured with phenolphthalein that the degradation depth reached 18.2 mm from the bottom of the beam. After this attack, the deteriorated beam is fu lthcr subjected to a three-point flex ion test schematicall y represented in Fig. Y.17. The following mechanical boundary cond iti ons arc imposed:u,,(x = 0,y = 0.08.t = I l 4d) = 11 0 ; u ,(x = O,y,t) = O; u,.(x = 0.12,y = 0,t )= 0. (V.17) After I 14 days of leaching, an increasing displacement 11 0 is thus imposed o n the center of the upper face (rig. Y. 17). The non-l inear mechanical behaviour of the leached sample is then evaluated by FE using the micromechanically based damage model detail ed in Section 2. The beam is assumed to be in plane stress conditions. In the present system, eventual cracks inside the chem ica lly degraded zone presently solicitated in traction along the horizontal axis because of the flexural disp lacement imposed on the beam shou ld main ly appear oriented along the vertica l ax is.A rather coarse 20 mesh comprising 1600 elements is employed for the chemical and mechanical computations on the 111011ar beam, since the smallest element of the mesh shou ld be bigger than the typical size of the RYE. It is recalled that this size is about I 0-100 ~lm for cement pastes and about I• l 0 mm for mortars. According to the work of Pensee and He (2007) accou nting for the effects of the size of the RYE, the laller should be sign ificantly bigger than the typical si:les of the particu late phases. Lutz et a l. ( 1997) for exampl e considered an average equ ivalent diameter of 700 ~1 111 for sand aggregates. Consequently, the Part V: Simul:uions of the cheJtlO•mechanical dcgr.1da1ions by leaching of cc1 ncn1-bnscd ma1crials Page 204

Figure

  Figure V.J 7: Schematic of the two-dime11sio11al m esh employed for the sim11/atio11s of m echa11ical resista11ce tests (up); meshes of200 elem ents (dow11 lefl) tmd of1600 elements (down right) used/or the computatio11s.

Figure

  Figure V.18: Identification of the stmin damage threshold ofa leached OPC type I paste (Heukamp el al. 2005) from the stress-strain curve of the damaged u11leaclted one.

  7 mol/L. In addition, !he diffusivity in the leached zone of the beam (Fig. V.20) augments by more than two o rders of magnitude compared with the sound part. Pan V; Simulation,; of the chcmo•nlCchanical degn1d1uions by lc:.ching of ecmcn1-ba.c;ed rn.

Figure V. 19 :

 19 FigureV.19: Profile oftlte portla11dite co11ce11tratio11 (moVL) i11 11 mortar be11111 after a 114 days attack by a NH,N01 .m/11tio11 (11p); crack de11sity partmieter d 1 predicted by tlte /DD damage model in tlte leaclted beam s11bmitte1l 111 a j lex11ra/ displaceme11t of 39 11111 (dow11).

Figure V. 20 :Figure

 20 Figure V.20: Profiles of the dijful•io11 coefficients (down) and oftlte liquid calcium co11ce11tratio11 (11101/L) (up) ill a mortar beam after a 114 days attack by a 6M NH 4 N0 1 solutio11.

  measured that the effective Young modu lus on a unifonn ly leached mortar is equal to 4.3 GPa. Another reason may be that the dissolution front propagation, tliat affects the material residua l mechanical behavior, is slightly underestimated in comparison with the one gained experimentally, as more sophisticated model surmounting the lack of knowledge on CSH diffusivities is proposed for predicting the macroscopic diffusivities of cement pastes and mortars. In this multi-sca le model, the two types of CSH containing connected gel porosity are estimated with the help of the MCSA scheme presently developed. An orthotropic damage model has been developed for concrete materials by combining the Mazars model (1984) and micromechan ical concepts. Jt comprises two damage variables, representing the density of cracks in two perpendicular directions, which is suited to deal with orthotropic damage under plane stress conditions. It is less involved than other simi lar approaches in the literature (e.g. Pensee and Kondo 2003) but presents the advantage to be relatively simple to implement numerically. Simulations of chemical and of coupled chemo-mechanical degradations of leached cemcntbased materials have been performed by integrating into ALLIANCES the different models developed in the rest of the thesis to pred ict the evo lutions of the diffusivity and of the mechanical behavior of cement-based materials. The numerical results obtained have been confronted with experimental tests and are encouraging.

  further developed as proved by the proposed derivation of a mod ified DIM. Re. cent progress in analytical techniques of homogenization (e.g.[START_REF] Berryman | Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media[END_REF] Duan et al. 2006) may for instance be very promising for predicting with better accuracy the effective properties of cement-based materials.The prediction of the non-linear mechanical behavior of cement-based materials needs to be improved. For this purpose, it would be desirable to develop and implement a more advanced damage model, including phenomena such as plasticity or creep. The integration of a nonlocal fonnulation is also central to get better estimations of tbe post-peak behavior of tile damaged material.The chemo-mechanical simulations presented in the thesis are promising. Still, some more work should be done. Important aspects of the chemical degradation process, such as the progressive CSH decalcification, could be described more thorough ly. The retroactive effects of cracks on the cbemical degradation have also to be taken into account in order to reproduce fully coupled chemo-mcchanica l experiments(Le Bellcgo 2003;[START_REF] Schneider | Deterioration of high-performance concrete subjected to attack by the combination of ammonium nitrate solution and flexure stress[END_REF].

Table I .

 I I: Mineral composition in terms of volume fractions of standard CEM I hydrated pastes obtained by the model of[START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland c. ement pastes[END_REF] 2000) .......................... 7

Table 1 .

 1 2: Critical pore diameter values measured by MIP tests for various types of cements pastes in sound or leached states issued from diverse sources . ............................................... 13 Table I.3: Elastic properties of the main phases in a sound state present in HCP microstructure, taken from diverse sources in the literature . .................................................. 16 Table 1.4: Young moduli measured experimentally of standard cement pastes, taken from diverse sources in the literature ............................................................................................... 17 Table 1.5: Table summarizing some important experimental results of accelerated leaching tests of HCP, taken from diverse sources in the literature . ..................................................... 22 Table IJ. I: Table confronting the merits of the different EMTs presently reviewed ............... 58

Table II

 II

.2: Input volume fractions of principal hydrates, unhydrated Cl inker and capillary porosity for six different HCP (Kamali 2003) ......................................................................... 63 Table Jl.3: Input volume fractions of principal hydrates, unhydrated Clinker and capillary porosity for two different HCP ................................................................................................. 79 Table 11.4: Grading of the fine Nevada sand used for the mortars (Heukamp 2002) ............... 83 Table 11.5: Comparison of the micromechanical estimations with experimental data on cement pastes and mortars (Heukamp 2002). .. ................................................................................... 85 Table llJ. I: Comparative table aiming at testing homogenization models on three simple criteria .................................................................................................................................... 112 Table 111.2: Composition in tem1s of volume fractions of hydrated CEM l pastes obtained by combining the mineral compositions given in

Table V .

 V 3: Comparison of the decalcification front propagations obtained in the simulations performed with ALLIANCES of pure water leaching with experimental values .................. 190 Table V.4: Comparison of the decalcification front propagations obtained in the different s imulations perfonned with ALLIANCES of accelerated leaching by a 6M NH.N03 solution with experimental values

and the experimental measurements of capillary pores (lgarashi ct al. 2004): ....... : ................................................. 128 Table V. I : Table summarizing the main input data for performing CTM coupled simulations using the homogenization model with ALLIANCES ............................................................ 181 Table V .2: Initial composition of the two CEM 1 pastes used for the simulations ................ 183

  1.2. The relative volume fractions of the two types of CSH arc computed in this work by means of the Tennis and Jennings model (2000):

	M • = 3.0 I 7(w/cg-J .347~ + 0.538,
	c:CSJ/1111 =	M, -M,M, PcsHrnr	,	(1.2)
		M,M,

PcsHm

where cCSH

1 ,,,, CcsHm, PcsH'"', Pcs 11 ..,,

are respectively the volume fractions and densities of CSil 1 "' and CSHex', M, is the total mass of CSH, M • is the ratio of the mass of CSH'"' to the total mass of CSH, c; the hydration rate.

Table 1 .

 1 2: Critical pore diameter values meas ured by MIP tests/or various types of cements pastes i11 sound or leached states issued from diverse sources.

	Type of cement paste CEM I	w/c 0.45 -	Sa mple preparation Sou nd Critical p or e diameter Leached 0.2 11m Oven-dryed 20 run (30 weeks with at60°C NH,N03)	Sources (Galle et al. 2004)
	ASTM Type I cement	0.45	Vacuum-dried	40nm		(Delagrave et al. 1998)
	CEM I	0.30	Vacuum-dried	30 nm		(Richel et al. 1997)
	CEM I	0.40	Vacuum -35 -40 dried nm		(Richel et al. 1997)
	CEM I	0.60	Vacuum-70 -80 dried nm		(Richet et al. 1997)
	CEM I	0.40	Vacuum-dried	60 nm	.	(Galle 200 I)
	CEM I	0.40	Oven-dryed at 105°C	170 nm	-	(Galle 200 I)

Table 1 .

 1

		SOUND	LEACHED
	Phases	HC P			HCP	References
		E (GPa)			v
	CH	42.3	0.324 0	0	Monteiro and Chang
						( 1995)
		38	0.3 1			Constantin ides	and
						Ulm (2004)
	Unhydrated	11 7.6	0.3 14 117.6	0.314	Boumiz et al. ( 1997)
	clinker
	Capillary	0	0	0	0
	Porosity
	A Ft,	22.4	0.25 0	0	Assumed by Karna Ii
	Hydrogamet,					(2003)
	C. AH n	25	0.25			Haecker et al. (2005)
	A Fm	42.3	0.324 0	0	Assumed by Kama Ii
						(2003)
	CSII gel (CSHex' 22.4	0.25	22.4	0.25	Damidot ct al. (2003)
	+ CSH;"')
	csu<' "	2 1.7	0.24 3	0.24	Constantinidcs	and
						Ulm (2004)
	CS H'"'	29.4	0.24 3	0.24	Constantinides	and
						Ulm (2004)

v E (GPa) * T he n umerical values used for the calculations are in bold. 3: Elastic properties of the mai11 phases i11" sound state present in HCP microstr11ct11re, tllke11 from diverse sources in the literllture.

Table

Table 1 .

 1

	4: Young moduli measured experi111e11tally of sta11dard ce111e11t pastes, take11 from
	diverse source. ~ i11 the literat11re.

P11rt 1110ltr leochinJl rates are;,, parentlre.s;s to give a cn1nparison

		(2004)	Galle et al. (2004)	Cardc ( 1996)	Heukamp (2002)
	Estimated hydration rate	0 76	0.88	0.8	0.8
	Initial measured by water porosity	0.39	0.326	0.405	0.397
	Leaching tests	19 days with 6M	16 weeks with	I day wich 5.46M
		ofNH, NQ3	6M of NH,NO,	ofNH, NO,
	Porosity measured by water on the	.	0.6 12	0.5098	0.632
	leached soccimen
	Accelerated leaching rate•	1.56 mm d•"•' (0.1 40 mm d• 0 " )	1.74 mm d"'" (0.190 nun d' 0 " )	I .3 I mm d' 0 " (0.125 nun d"'")	2.25 mm d" 05
	Table 1.5: Table summarizi11g some important experi111e11tal results of accelerated leaching
	tests a/ HCP, take11 from diverse sources in the literature.

*

  1 = a.l)>l>J + Ps,mlK,

			(II.18)
	with
	a, , , , , , , -(3K., + 4G.,) and Ps"'	5(3KM + 4G.,)' 6(KM + 2G.,)	(fl.19)

  in her simulations.Ire plotted on Pig. 11.l 6 for the three CEM I pastes. It is worth noting that for this case of

		Volu me fractions in %
		CEM I paste	C EM 11/A paste
	w/c =	0.5	0.4	0.25	0.5	0.4	0.25
	CH	16.4	16.9	15.7	10.5	10.8	8.5
	CSH	47.3	47.4	42. 1 49.4	49.7	45.I
	UC	10.04 14.46	28	13	18.4	31.4
	A Ft	2.4	2.8	2.9	2.7	2.3	2.9
	A Fm	0.16	0. 14	0.3	0.4	0.4	I
	Porosity(%) 23.7	18.3	11.0	24.0	18.4	11. l
	Table 11.2: Input volume fractions of principal hydrates, 1mhydrllted clinker a11d
		capillary porosity for six different H CP (Kamali 2003).
	The first example consists in approx imating the initial capillary porosity respectively as
	prolate and oblate spheroids with varying shapes, while the other pa11icu late phases are
	modeled as spherical inclusions. The different volume fractions of these initial capi llary pores
	•re Ccp = 11 %, 18.3 % and 23.7 % in the CEM I pastes with w/c = 0.25, 0.40 and 0.50,
	ltspcctively. The effects of both types of porous spheroids on the effective Young modulus

~rous inclusions the spherical shape optimizes the Young modulus. This can be understood llltuitively considering the fact that for a given area the spherical surface maximizes the inner tl>lun1c (e p 1 .. . g. o ya and Szego 1951 ). The curves o n F 1g. 11.16 show 1hat prolate and oblate Porous Spheroids with an identical value for the morphologica l parameter do not have the llllle effects on Young modulus except for c; close to I. The weakening effects of prolate porous spheroids compared to the spherical case are quite negligible, whereas oblate 0 nci

  . .

	t.2
	.:: LI S • .ll-~ ' • ttl I.I	'	' ' '	'	" .............. Oblate spheroids •• ....
						"• ........ . ......
			Prolate spheroids	" • " •
	I 0			0.2	0.4	0.6	0.8
					Morphological Parameter<;

Fig11re JJ.

1 7: Evo/11tio11s

of normalized ef f ective Young mod11/11s for sound CEM I pastes with 1111hydrated ,•/inker respeclively represented by oblate and prolate spheroids, as a fun ction oftltc shape parameter ~• Solid lines: ><Ve -0.50; dashed lines: wlc = 0.40;

  .2.

	From the microstrnctural description detailed in Part I, it appears that needle-like ettringite
	•hould rather be cbaracterized by prolate shapes and disk-like portlandite by oblate
	hlor(lhologies. Concerning unhydrated clinker, AFm and initial capillary pores, no evidence
	indicates whether to preferentially use oblate or prolate spheroids for characterizing thei r
	llbapc, Thus, they are represented by 50 % of both.oblate and prolate spheroids in this work.

l'he influence of the shape of aluminates and portlandite on effective elastic properties of

  " ... ,..... ..

			------............. .	1n er
	Ponlandite (up) ..., 0.8 Aluminates (down) . ~ kl	"	,.•	. "	....	....
	. ....... 0.6 kl					Initial Capillary
	0.4						Pores
	0.2
	0 0	0.2	0.4			0.6	0.8
		Morphological Parameter<;

Figure 11.18: Evolutions oftlte normalized effective Young 11101/ulusfor sound CEM I p11stes, 11s a fun ction of tlte morphological parameter <; . Solid lines: wlc = 0.50; daslted lines: wlc = 0.40; dotted lines: wlc = 0.25

Table JJ

 JJ

		Volume fractions in %
		CE M I 52.5	OPC type I p aste
		(Galle et al. 2004)	(Constantinides and Ulm 2004)
	w/c =	0.45	0.50
	Hydration rate	0.87	0.77
	CH	15. l	11.3
	CS Hin•	21.6	15.6
	CSH""'	28.8	42.8
	UC	4.3	5.0
	Aluminates	14.2	6.3
	CP	3.6	6.0
	Porosity(%)	3 1.9	38.5
	Water porosity(%)	32.6	39.7

.3: /11p11t volum e fractiom; of principal hydrates, 1111/iydrated clinker a11d capillary porosity for hvo differe111 H CP.

Table 11 . 4 :

 114 Table 11.4. The total volu me fraction occupied by sand is about 50 % (Heukamp 2002). Grading of the fine Nevada sand used for mortars by Heukamp (2002).

	Size of sand	Relative
	particles	proportion
	0-0.17 mm	30%
	0.1 7 -0.23 mm	30%
	0.23 -0.50 mm	40 %

Table 1 /

 1 Table 11.5 and compared with experimental data (Heukamp 2002).

	Ph ases	Yo ung mod ulus	Young modulus
		predicted (GPa)	predicted (GPa)	Experimental
		w ith 3 sizes of	with 1 size of	Measures (GPa)
		a!!gre!!atcs	av<>r c!!ates
		Sound	Leached	Sound	Leached	Sound	Leached
	ITZ	4.6	2.8	4.6	2.8	-	-
	HCP	23. 1	3.0	23.1	3.0	22.8	3.5
	Mortar	25. I	3.7	27.4	5.3	25.1	4.8

.5: Comparison of the microm echan ical estimations with experimental data 0 11 cement pastes a11d m ortars (Heukamp 2002).

  = 4 violates them for the ranges of volume fractions c 2 < 0.25. On the contrary, lhe classical SC estimate, corresponding to a coordination number of z = 6[START_REF] Kirkpatrick | Classical transport in disordered media: scaling and effective-medium theories[END_REF]

	1973), always satisfies I IS bounds.
	A model called the general effective media (GEM), used for instance by Oh and Jang (2004)
	lo predict the diffilsion coefficients of!ICP, has been proposed by McLach lan (1987):

are based seems quite far from the actual continuous microstnictures of disordered materials. In addition, though the Kirkpatrick estimate may provid~ nrore realistic thresholds than the original SC model, they violate HS bounds for non negligible ranges of volume fractions, as may be seen in Fig. lll.7. More precisely, it is seen on this figure that the Kirkpatrick estimate obtained for z

  Nevertheless, the demonstration detailed in Appendix III.C of the previous equation is valid only if both fonnation factors are finite. This fact clearly defines the situations for which MCSA are of interest: they are useful provided that both formation factors are finite, i.e. the two phases percolate. When one of the fonnation factors is infinite, MCSA degenerates into a simple CSA.

	I 2F -2 2	/	1'1 . 2 2F. -2 ; /	(111.62)

Table 111

 111

	.2.

Table 11

 11

/.2: Composition in terms of volume fractions of hy rlraterl CEM I pastes o/Jtaine by combining tire mineral compositions given in Bejtw11i and

Table V .

 V

	1: T<1ble s1111111111rizillg the main i11p11t data for perfor111i11t: CTM co11p/etf
	si11111/atio11s 11si11g the ho111oge11izatio11 111odel with ALLIANCES.

  •where the superscripts S and UL mean sound and unifonnly leached, respectively. The values of the Young moduli Ef'SHm•«m and E~J 111 .,,.,, .. , are taken from Constantin ides andUlm (2004) The CEM I 42.5 paste is presently assumed to be initially composed only of five phases (Le Bellego 2001): UC, C1. 65 SH, CH, Afm and Aft. The initial composition of the material (in mol/L), computed by LeBellego (2001) by means of the simplified model from[START_REF] Adenot | Durabilite du Seton: Caraccerisation et Modelisation des Processus Physiques ct Chimiques de Degradation du Ciment[END_REF], is given in TableV.2. The hydration rate is estimated in the calculations of phase volume fractions so as to retrieve the same value for total porosity as the water porosity measured experimentally (LeBellego 2001). The composition of the paste is a little particular because of its high concentration in A Fm.

	the fol lowing manner: rifC / S>08, lifC / S S 0.8, Ji f CI S > 0.8, l if C / SS0.8, Phases Estimated hydration F. •'(..'SJl"'r0tl!n --(I CIS-0.8)Eut C / S-0.8 ,.s . I .GS-0.8 C$Jlin1(11 '•"' + I . 6 S-(),g c.,CSllb1t0te.xt' D -(I-CIS-0.8)ou1 C / S -0.8 0 $ . CS/II"'""" -1.65 -0.8 C$Hi"' 0 ""' + 1.65-0.8 l'$1fw.,n" (V.14) D DUI. Lafarge CEM I 52.S paste (w/c = 0.40) HTS paste (w/c = 0.45) (Le Bell ego 200 I; Adenot I 992) (Galle et al. 2004) rate 0.80 0.88 CH (moVL) 3.54 4.56 Aft (mol/L} 0.0873 0.00 AFm(moVL) 0.45 0.199 Hydrogarnct (mol/L) 0.00 0.478 CSH (mol/L) 3.86 4.44 <.S.Jli11t@!l v:1 = CSIJ111HJ# ,;;1 Origny CEM I 42.S UC 0.074 0.043
	Warcr nnrositv	0.27	0.319
	• D 1 , 1 (m / s) 2	2.80 to• 12 (Tognazzi 1998)	5.21 10• 12 (Richer ct al 1997)

and the diffusivitics Dt 511 ..,,., .,,, and D:!;""'•"" are estimated by means of the MCSA model developed in Part Ill. The Poisson ratio is assumed to remain constant during leaching

(Stora et al. 2006a)

.

T he reference cement paste used for the simulat ions is the Origny CEM I 42.5 paste with w/c = 0.40, since it corresponds to the one used by Le Bellcgo (200 I) in her experiments. l'urthennore, this type of cement paste is well characterized (e.g.

[START_REF] Adenot | Durabilite du Seton: Caraccerisation et Modelisation des Processus Physiques ct Chimiques de Degradation du Ciment[END_REF][START_REF] Baroghel-Bouny | Caracterisation des pates de ciment ct des betons. Methodes, ana lyse et interpretations[END_REF]

and its mechanical behavior has also been studied

[START_REF] Carde | Caractcrisation ct modclisation de !'alterat ion des propric1es mccaniques due;\ la lixi viation des matcriaux cimentaires[END_REF][START_REF] Bellego | Coup I ages ch imic-mecanique dans lcs structures en beton attaquees par l'eau: Etudc experimentale et analyse numerique[END_REF]

.

Table V .

 V 2: Initial composition of the hvo CEM I pastes used for the simulations.

  • • • • •

	1.5	2

Table V .

 V 3: Comparison of the decalcijicatio11 front propagations obtained i11 tire simulations performed tVith ALLIANCES of1mre water leaching with experimental values.

Table V

 V

	.4: Comparison of tire decalcijicatio11/ront propagatio11s o btai11ed i11 tire
	dijf erell/ simulation s perfo rmed with A LLIANCES of a ccelerated leaching by a 6M
	N H 4 N03 solution with exp erime11tal values

  • • •

	Dissolution of
	1>orllandite
	Oecalcification
	of the CSH
	1	1 2	1 •	1 6	1.s	2
	Dcplh (mm)

oo ~~~~~~,--~~_..~_..~_._~~~~~~ o 2 O.A o.s o.a

ACKNOWLEDGEMENTS I first of all would like to acknowledge DSNl/Reactcurs Betons for its financial support. I wish to thank B. Bary and Professor Q.-C. He, who directed in a comple1J1entary way my PhD thesis and taught me a lot of things. l thank also all the people at tbe LaM at the university of Paris-Est Mame-la-Vallee and at the CEA/DPC/SCCME/LECBA in Saclay. I thank Professors D. Kondo and G. Pijaudicr-Cabot, for having reviewed my dissertation in a few weeks only, and Professor van der Lee who examined my work. A special think also to L. Sanguigno and Professors A. Vautrin aad G. Mensiticri, who incited me to do a PhD.

Appendix A: Input data for the comparison between EMTs and numerical simulations 'fhe basic input data of the simulations ofGarboczi and [START_REF] Garboczi | Elastic moduli of a material contain ing composite inclusions: effective medium theory and finite element computations[END_REF] [START_REF] Garboczi | Elastic moduli of a material contain ing composite inclusions: effective medium theory and finite element computations[END_REF].

Appendix Jl .B: Non-overla pping condUion and dete r mination of t he double.inclusion geometry for highly conce nt rated spheroidal inclusions

If an inclusion overlaps its double-inclusion, the homogeni7.ation calculations fail to Worf<.

The inclusion volume fraction c1 is represented by the ratio of inclusion vol ume on the double-inclusion one. When r, '# I, the condition on volume fractions which has to be respected in order to avoid overlapping of inclusions reads (Ponce-Castaneda and Willis For concentrations highe r than c 1 , , lhc double-inclusion can no longer be iakcn as spherical.

To avoid any risk of overlap, it is sufficient to adupt its geometry to the inclusion one (see Fig. 11.B.I). Its aspect ra1io rm is suitably chosen so that the !DD remains valid for any concentration as illustrated on Fig. In other words, the determination of the double-inclusion geometry is now linked to the choice of inclusion shape. 111c IDD estimale taken with such a shape for the double-inclusion works for any concentration and is called the full-range IDD [START_REF] Zheng | An explicit and un iversally appl icable estimate for the effective properties of multiphase composites which accounts for inclusion distribution[END_REF].

Appendix Jll.C: Link between f and the formation factors This Appendix aims at establishing the links between the geometric paramcter/ofMCSA and the fo1mation factors defined in Eq. (111.45). For this purpose, the quadratic equation (Ill.57) is first divided by Di : ,;. )

As the MT and original JOO model, this simplified IDD scheme can be cons idered as an improved dilute model (see Fig. 11.5), where each family of cracks modeled as a single pennyshaped inclusion is immersed in the unbounded matrix material subjected to a modified farfield stress.

It is worth noting that imp licit schemes such as the SC (Budiansky and O'Connell 1976), the GSCS [START_REF] Huang | Several variations of 1he generali zed sclfconsistcnt method for hybrid composites[END_REF]) and the differentia l scheme [START_REF] Hashin | The differential scheme and its application to cracked materials[END_REF]) may also be applied but the expressions of these estimates can become very complex even if only one fami ly of cracks is involved. However, with the explicit schemes presented above, the analytical expressions of the effective damaged stiffness tenso; may also be quite tedious when many fam ilies of cracks are involved. Therefore, three particular cases are treated in the next subsection to draw a comparison between the different explicit schemes: al igned, isotropic and orthotropic.

11.2 Comparison of the homogenization schemes in simple cases

11.2.1

Comparison for a material with aligned cracks

All tbe cracks are assumed to be aligned with their nonnal vectors parallel to the same ax is e 1 •

The subscripts can thus be disregarded, since only one family of parallel cracks is involved.

Tbe effective damaged material is transversely isotropic, since the undamaged one is assumed to be isotropic. In such case, simple analytica l expressions are available for most of the studied schemes (Nemat-Nasser and Hori 1993). A general expression including lDD and MT schemes as particu lar cases is proposed below.

Appendix IV.A: Link between the dissipation potential and the damage thresh old functions

As an example, we look after the dissipation potential providing the classical fracture-type criterion f written in Eq. (IV.34). For simplicity on ly one damage variable is considered.

Using the definition of Legendre-Fenchel transform, the following equivalence can be established (e.g. [START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF]:

where 'P' is the Legendre-Fenchel transform of 'f/. for the particular case of a dissipation potential 'fl positive homogeneous of degree one, i.e. \;;/d, \;;/k ?. 0, 'l'(kd

where K is a convex domain defined by a threshold function f . It appears by etllploying the properties of the Legendre-Fenchel transform and the threshold function/given in Eq. (IV.34) that:

(IV.A3)

More precisely, the following dissipation potential homogeneous of degree one is deduced [START_REF] Lorentz | Analysis of non-local models through energetic formu lations[END_REF]:

where Yo is the damage threshold in tenns of elastic energy density and where the indicator fonction .7oc.' (d) enforces the positivity of the damage rates.

Appendix I V.B: Links between t he De Vr ce and the Christensen criteria

In order to draw a comparison between the Christensen and the De Vree criteria, the latter is written in terms of stress invariants:

(IV.Bl) using those relations:

It is easy to demonstrate that, when the De Vrcc criterion is verified fDv = 0, the Christensen one is then automatically verified. The previous equat ion is first elevated to power square:

and is then expanded yielding the following relation:

By factorizing the terms with the first invariant of the stress tensor, we notice that:

which simplifies into the fom1 proposed by Christensen (2005):

Conversely, supposing the latter equation verified, it can be recast in the following form:

The left-hand member is then expanded: 1.3) then serve as input parameters for the estimations of the elastic properties.

, ' , , , . .. \ ' \

• '. :.<: ..

, • with the same strain damage thresho ld as for the sound material •• with a different strain damage threshold in the sound and leached cases

CONCLUSIONS OF PART V

The present Part is devoted to the development of a tool incorporated in the ALLIANCES numerica l platform that allows for predicting the long-term evolutions of the mineral composition of leached cement iti ous materials and of their mechanical a nd diffusive properties. This numerical instn11nent is built up by incorporating the multi-scale homogen i7..ation approach proposed in Parts II and Ill to predict respectively tl1e evolutions of the diffusivity and of the elastic behavior of cement-based materials into ALLIANCES. The micromechanical damage model developed in Part JV has also been integrated to estimate the damage propagation in these materia ls.

Pnr1 V: Simuht1ions of1hc chemo•nlcChitnic11J degrada1ions hy lc3ching ofccJncnl•bascd rnatc1ials P:.gc 211

Owing lo the numerical device developed, s imu lations of the chemical behavior ofdecalcified cement-based materials have been carried out. The good agreement of the simulated dissolution front propagation and stiffoess reduction with experimental data constitute a further validation of the multi-scale homogenization approach proposed in the manuscript.

Residual resistance experiments illustrating the couplings between damage and leaching have also been s imulated. The numerical resu lts confronted with experimental tests arc encouraging but may require some further refinement for the pred ictions of' damage inside the materia l. Nevertheless, they give some interesting in terpretations summarized below for the measures ga ined from these tests that are particularly useful for assessing the durabil ity of cement-based materials. The simulated behaviour of sound and leached mortar beams in flexion has underl ined the necessity to have the damage threshold presently expressed in deformat ions evolved w ith decalcification. An empirical method that assim ilates the decrease of the Young modulus d ue to leaching to the one caused by damage is then employed to predict more accurately the maximum load that the deteriorated material can sustain.

According to this method, the fonnation of cracks is then influenced by dccalcification, the leached zones being less damaged than the chemically sound ones.

Appendix V.A : C hemical formul ae and eq uilib r ium constan ts of t he m iner al phases in the reduced datab ase.

IS G ENERAL CONCLUSI ONS AND P ERSP ECT IVES

The present chapter recapitulates the main findings and co 1 • nc us1ons of the thesis. Some discussions and proposals for future research arc also fomrnlated.

J Main conclusio ns

The findi ngs or achievements of the thesis concern different domains that can be separated in the subsequent categories:

(i) homogenization theories;

(i i) modelling and prediction of the macroscopic properties of cement-based materials;

(iii) numerica l simulations of coupled chemo-transport-mechanical degradations of cement-based materials.

In the first Parts of the manuscript, diverse homogeni zation techniques have been assessed

using different criteria and tested on particular cases that have been solved numerically.

Among the reviewed schemes, double-inclusion type models and GSCS exhibit the most suited attributes for esti mating the linear elastic propert ies of cement-based materials. By revisiting the D IM of Hori and Nemat-Nasscr ( 1993), a modified version of this model overcoming some of its origina l shortcomings in the case of coated inclusions has been derived.

T he merits of homogen ization theories for predicting the effective diffusive properties of porous media presenting high contrast between phase diffusivities have also been examined .

Many models such as the matrix-inclusion type schemes are pointed out to be inappropriate, since they are inade<Jtiate for modelling percolating or connected pores. A novel estimate associated to a space-fill ing assemblage mixing two types of composite spheres pertinent to account for pore connectivity has thus been developed. By construction, this MCSA scheme automatically respects HS rigorous bounds and depends only on one geometric parameter which vari ation allows the realization of any diffusivi ty ranging within these bounds.

By taking benefit of EMTs reta ined from the preceding review, a two-step homogenization model is developed for the predicti on of the macroscopic linear elastic properties of HCP and mortars and validated by comparison with experimental measurements. T he developed model is also va lidated in the case of cement pastes and mo rtars subm iucd to severe leach ing. T he in fluence of inclusion shape on the micro mechanical estimations of the effective elastic properties of HCP has also been investigated. The spherical particle phase approximati on that is genera lly adopted for simplicity is shown to be valid in most cases, except for very porous leached pastes. Using the same framework as the two-scale approach developed in elasti city. a