F Lavergne
email: francis.lavergne@cerema.fr

A Ben Fraj

I Bayane

J F Barthélémy

Estimating the mechanical properties of hydrating blended cementitious materials: an investigation based on micromechanics

Keywords: homogenization, cement, silica fume, hydration, compressive strength

 [1] has been extended to blended cements and coupled to a micromechanical scheme similar to that of Pichler & Hellmich (2011) [2] to estimate the Young modulus and the compressive strength of cementitious materials as a function of time. A finite aspect ratio of 7 is introduced to describe the shape of the hydrates and improve the estimate of the early age strength by the micromechanical scheme. Furthermore, accounting for the stress fluctuations in the cement paste partly explains the fact that the compressive strength of a concrete can be lower than that of its cement paste. Finally, the estimated physical properties are compared to numerous experimental measurements from the literature and new experimental measurements on blended cement pastes featuring significant weight fractions of limestone filler, fly ash or silica fume. It is shown that the present model slightly overestimates the dilution effect.

Introduction

Deriving the mechanical properties of cementitious materials from their compositions is a matter of interest for the design of optimized formulations and accurate estimates of stresses and strains in structures. Standards models often rely on the measurement of the compressive strength at 28 days f c,28 as the sole indicator of the mechanical performance of the concrete. An empirical formula is therefore applied to estimate the secant Young modulus of the concrete E 28 . For instance, the one of the Eurocode 2 [START_REF] Afnor | Eurocode 2, calcul des structures en béton, section 3[END_REF] for concrete containing quartzite aggregates is E 28 = 22 (f c,28 /10) 0. 3 , where E 28 is in GPa if f c,28 is provided in MPa. The Young modulus can be 20% higher for basalt aggregates or 10% to 30% lower for limestone and sandstone aggregates. While the influence of the aggregate type is accounted for, this formula can not be applied to lightweight aggregates or recycled aggregates. Indeed, a corrective factor η E = (ρ/2200) 2 , where ρ is the dry density of the concrete in kg/m 3 , is introduced to handle the case of lightweight aggregates (ρ < 2200kg/m 3 ). The standard itself considers this correction as inaccurate and requires an effective measurement of the Young modulus whenever this property becomes critical. The ACI 318 standards [START_REF]Building code requirements for structural concrete[END_REF] specifies a similar equation, except that the exponents are very different: [START_REF] Dilnesa | Synthesis and characterization of hydrogarnet ca3(alxfe1 -x)2(sio4)y(oh)4(3 -y)[END_REF] 20

E 28 ≈ 21 ρ 2300 1.5 f c,
0.5 (1)
The ACI 318 standards also points out that the modulus for concrete is sensitive to the modulus of the aggregates and that the measured values range typically from 120 to 80 percent of the estimated value [START_REF]Building code requirements for structural concrete[END_REF]. Moreover, it is well-known that this equation often overestimates the Young moduli of concretes featuring compressive strengths above 41 MPa [START_REF]ACI Committee 363[END_REF]. Lastly, these empirical formulae require a prior knowledge of the compressive strength of the concrete. Consequently, they become inadequate to design optimized formulations. Models have been proposed in order to improve the accuracy of the estimates of the mechanical properties by relying on material sciences. For instance, Freyssinet proposed in 1933 that the strength and stiffness of cementitious materials are related to what is nowadays called the gel-space ratio [START_REF] Freyssinet | Idées et voies nouvelles[END_REF]. Following the same reasoning, Powers later assumed that the strength of cement gel is the sole source of strength of the porous cement paste and uncovered a practical relationship between the gel-space ratio and the compressive strength of cement pastes [START_REF] Powers | Structure and physical properties of hardened portland cement paste[END_REF]. Such models can address the heterogeneous nature of cementitious materials, the changes of their microstructures due to the hydration reactions, the stiffness of the components involved at microscopic scale and the failure mode if the compressive strength is estimated. At first, different homogenization schemes were introduced to estimate the properties of concrete starting from those of the cement paste [START_REF] Hirsch | Modulus of elasticity iof concrete affected by elastic moduli of cement paste matrix and aggregate[END_REF][START_REF] Counto | The effect of the elastic modulus of the aggregate on the elastic modulus, creep and creep recovery of concrete[END_REF][START_REF] Nilsen | Concrete: A three phase material[END_REF][START_REF] De Larrard | Structures granulaires et formulation des bétons -Concrete Mixture -Proportionning -A scientific approach[END_REF]. Then, coupling the existing knowledge about the chemical reactions occurring in concrete [START_REF] Powers | Studies of the physical properties of hardened portland cement paste[END_REF][START_REF] Jennings | Model for the developing microstructure in portland cement pastes[END_REF][START_REF] Berliner | Quasielastic neutron scattering study of the effect of water-to-cement ratio on the hydration kinetics of tricalcium silicate[END_REF] and measurements of the stiffness of individual phases by nanoindentation [START_REF] Constantinides | The effect of two types of c-s-h on the elasticity of cement-based materials: Results from nanoindentation and micromechanical modeling[END_REF] allowed the development of multiscale homogenization procedures dedicated to cementitious materials [START_REF] Bernard | A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials[END_REF][START_REF] Constantinides | The effect of two types of c-s-h on the elasticity of cement-based materials: Results from nanoindentation and micromechanical modeling[END_REF][START_REF] Pichler | A multiscale micromechanics model for the autogenous-shrinkage deformation of early-age cement-based materials[END_REF][START_REF] Sanahuja | Modelling elasticity of a hydrating cement paste[END_REF][START_REF] Ghabezloo | Association of macroscopic laboratory testing and micromechanics modelling for the evaluation of the poroelastic parameters of a hardened cement paste[END_REF][START_REF] Pichler | Upscaling quasi-brittle strength of cement paste and mortar: A multi-scale engineering mechanics model[END_REF][START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF][START_REF] Termkhajornkit | Dependence of compressive strength on phase assemblage in cement pastes: Beyond gel-space ratio -experimental evidence and micromechanical modeling[END_REF], in which the cement paste itself is considered as an heterogeneous material. Furthermore, numerical simulations of the microstructure of the cement paste have been achieved [START_REF] Van Breugel | Simulation of hydration and formation of structure in hardening cement-based materials[END_REF][START_REF] Bentz | A three-dimensional cement hydration and microstructure program. I hydration rate, heat of hydration, and chemical shrinkage[END_REF][START_REF] Bishnoi | µic: A new platform for modelling the hydration of cements[END_REF]. Such numerical simulations require some data about the particle size distribution of the cement. Then, the generated artificial microstructures may be used as inputs of 3D numerical simulations so as to estimate the Young modulus or the strength of the material [START_REF] Šmilauer | Microstructure-based micromechanical prediction of elastic properties in hydrating cement paste[END_REF][START_REF] Qian | 3d simulation of micromechanical behavior of cement paste[END_REF]. Nevertheless, these computations can prove time-consuming: optimizing a formulation, validating the model against experimental results or using it for every element of a finite-element model of a structure still are intricate challenges. Consequently, the mean-field homogenization models based on a scalar description of cement hydration remain valuable for their ease of use and their low computational demand [START_REF] Buffo-Lacarrière | Multiphasic finite element modeling of concrete hydration[END_REF]. Indeed, extending these models to account for new evidences is often straightforward.

Recent advances on our understanding of cement hydration are to be incorporated in the model that is introduced in the present article. First, the products of the hydration of C 4 AF, a phase of cement, have to be updated according to new evidences [START_REF] Dilnesa | Synthesis and characterization of hydrogarnet ca3(alxfe1 -x)2(sio4)y(oh)4(3 -y)[END_REF][START_REF] Vespa | Identification of the thermodynamically stable fecontaining phase in aged cement pastes[END_REF] and the conditions for the formation of calcium monocarboaluminates have been defined by a thermodynamical framework [START_REF] Lothenbach | Thermodynamic modelling of the effect of temperature on the hydration and porosity of portland cement[END_REF]. Second, the effect of pozzolanic additions such as silica fumes and fly ashes can be modeled by accounting for the pozzolanic reactions [START_REF] Waller | Relations entre composition des bétons, exothermie en cours de prise et résistance en compression[END_REF]. Indeed, most of the existing multiscale homogenization models are focused on plain cements, while modern formulations often includes these additions to improve the durability or the mechanical performances, reduce the costs or the environmental footprint. Third, the effect of the fineness of cement and fillers, the relative humidity and temperature on the kinetic of the chemical reactions are modeled so as to provide an estimate of the mechanical properties as a function of time, not as a function of the hydration degree. Finally, the present model is validated against measurements published in the literature. The accuracy of the hydration model can be verified by XRD-Rietveld analysis, thermogravimetry and calorimetry. The estimates of compressive strength and Young modulus are also compared to the corresponding measurements. In addition, new measurements have been performed on blended cement pastes so as to extend the database of useful experimental results. The formulations and experimental procedures are detailed in order to help further validations of homogenization models.

Evolution of the volume fractions of constitutive phases

At early age, the evolutions of the mechanical properties of concrete are driven by the progressive hydration of the cement paste. In the present section, the volume fraction of the constitutive phases of concrete are estimated as functions of the time. First, an isothermal hydration model is introduced in order to rapidly produce an estimate of the volume of each phase of a cement paste. Then, the effect of temperature and relative humidity are accounted for. Lastly, the pozzolanic reactions are introduced.

The hydration of the cement

The considered hydration model is similar to that of Thiery et. al. [START_REF] Thiery | Un modèle d'hydratation pour le calcul de la teneur en portlandite des materiaux cimentaires comme donnée d'entrée des modèles de carbonatation[END_REF] as modified by Nguyen [START_REF] Nguyen | Modelisation des couplages entre hydratation et dessication des matériaux cimentaires à l'issue du décoffrage. etude de la dégradation des propriétés de transfert[END_REF]. It is combined to the empirical expressions of the hydration rate of clinker phases described by Parrot & Killoh [START_REF] Parrot | Prediction of cement hydration[END_REF][START_REF] Lothenbach | Thermodynamic modelling of the effect of temperature on the hydration and porosity of portland cement[END_REF][START_REF] Gawin | Modelling and experimental study of hydration for ordinary Portland cement[END_REF].By describing the reaction rates as a function of the remaining amounth of reactant and other environmental parameters, the present hydration model is fully compliant with the concept of chemical affinity [START_REF] Bernard | A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials[END_REF]. New evidences regarding the products of cement hydration [START_REF] Lothenbach | Thermodynamic modelling of the effect of temperature on the hydration and porosity of portland cement[END_REF][START_REF] Dilnesa | Synthesis and characterization of hydrogarnet ca3(alxfe1 -x)2(sio4)y(oh)4(3 -y)[END_REF][START_REF] Vespa | Identification of the thermodynamically stable fecontaining phase in aged cement pastes[END_REF] have been incorporated as updates.

The constitutive phases are the unhydrated reactants of cement, water, the products of the chemical reactions and filler. The different phases are listed in table 1.1, along with some of their physical properties. The hydration reactions are then listed in table 1.2 along with their volume changes per mole of solid reactant ∆V r and balances of capillary voids per mole of solid reactant ∆V r,cap . The volume change ∆V r is related to the chemical shrinkage : products need less space than the reactants if water is included. Yet, since solid products need more space than the solid reactants, hydration reactions progress at the expense of the capillary porosity. Table 1.2: The considered set of chemical reactions is similar to those of Thiery et. al [START_REF] Thiery | Un modèle d'hydratation pour le calcul de la teneur en portlandite des materiaux cimentaires comme donnée d'entrée des modèles de carbonatation[END_REF]. Since the formula C 1.7 SH 4 corresponds to saturated C-S-H, the balance of volume of solid phases reflects the change of capillary porosity. The values of heat released by hydration are coming from [START_REF] Odler | 6 -hydration, setting and hardening of portland cement[END_REF], on page 271.

The kinetics of the reactions at the constant temperature of 20°C is to be described. The model of hydration of Thiery et. al. [START_REF] Thiery | Un modèle d'hydratation pour le calcul de la teneur en portlandite des materiaux cimentaires comme donnée d'entrée des modèles de carbonatation[END_REF] makes use of the power laws adjusted by Papadakis et. al. [START_REF] Papadakis | Hydration and carbonation of pozzolanic cements[END_REF] to describe the rate of hydration of clinker compounds C 3 S, C 2 S, C 3 A and C 4 AF. The kinetics introduced by Ngugen [START_REF] Nguyen | Modelisation des couplages entre hydratation et dessication des matériaux cimentaires à l'issue du décoffrage. etude de la dégradation des propriétés de transfert[END_REF] is more complex so as to account for the induction period, a period of nucleation and growth of hydrates and a period during which the kinetics of hydration is controlled by the diffusion of ions. The empirical hydration rates of Parrot and Killoh [START_REF] Parrot | Prediction of cement hydration[END_REF], used by existing hydration models [START_REF] Lothenbach | Thermodynamic modelling of the effect of temperature on the hydration and porosity of portland cement[END_REF][START_REF] Gawin | Modelling and experimental study of hydration for ordinary Portland cement[END_REF], also distinguish physically based limiting factors for the hydration rate, but it ensures that the hydration rate is finite and continuous. Hence, the latter will be considered in the present study.

The hydration degree 0 ≤ α i (t) ≤ 1 traces the evolution of the molar quantity N i (t) of solid reactant i ∈ {C 3 S, C 2 S, C 3 A, C 4 AF} consumed by the reaction:

α i (t) = 1 - N i (t) N i (0) (2)
where N i (0) is the initial quantity of solid reactant i in 1 cm 3 of fresh cement paste. In the model of Parrot and Killoh, three phenomena limit the rate of hydration of the solid reactant i. Consequently, the actual rate of hydration αi is the lower of three rates corresponding the phenomena: αi = min( αi,1 , αi,2 , αi,3 )

The first rate αi,1 arises from a nucleation-growth mechanism. Hence, it is modeled by an Avrami equation [START_REF] Avrami | Kinetics of phase change. I general theory[END_REF][START_REF] Avrami | Kinetics of phase change. II transformation-time relations for random distribution of nuclei[END_REF]:

αi,1 = k i,1 n i,1 (1 -α i ) (-ln(1 -α i )) 1-n i,1 (4)
i [START_REF] Parrot | Prediction of cement hydration[END_REF] reported in [START_REF] Lothenbach | Thermodynamic modelling of the effect of temperature on the hydration and porosity of portland cement[END_REF] are also used in the present study to model the isothermal hydration of the clinker compounds at 20°C.

C 3 S C 2 S C 3 A C 4 AF k i,1 , day
where the parameters k i,1 and n i,1 are parameters identified by Parrot and Killoh [START_REF] Parrot | Prediction of cement hydration[END_REF] (Tab. 1.3). The second rate αi,2 describes the diffusion of the solid reactant through the layers of hydrates that were previously formed. Hence, it is modeled according to a Jander equation [START_REF] Jander | Reaktionen im festen zustande bei höheren temperaturen. reaktionsgeschwindigkeiten endotherm verlaufender umsetzungen[END_REF]:

αi,2 = k i,2 (1 -α i ) 2 3 1 -(1 -α i ) 1 3 (5)
The last rate αi,3 changes according to a power law describing the formation of a thick shell around unreacted cement particles [START_REF] Gawin | Modelling and experimental study of hydration for ordinary Portland cement[END_REF]:

αi,3 = k i,3 (1 -α i ) n i,3 (6)
where k i,3 and the order of the reaction n i,3 were adjusted by Parrot and Killoh [START_REF] Parrot | Prediction of cement hydration[END_REF] (Tab. 1.3). These adjustments are valid for a cement of Blaine fineness B 0 =385m 2 /kg. To account for the effect of the Blaine fineness on the rate of cement hydration, the rate of hydration is multiplied by the ratio B/B 0 , where B is the Blaine fineness of the cement [START_REF] Parrot | Prediction of cement hydration[END_REF][START_REF] Gawin | Modelling and experimental study of hydration for ordinary Portland cement[END_REF]. Indeed, hydration mechanisms such as the Avrami rate law regarding growth and the Jander rate law are related to the size of the cement particles [START_REF] Khawam | Solid-state kinetic models: Basics and mathematical fundamentals[END_REF].

The model of Parrot and Killoh considered in the present study suffers from many drawbacks. It must be mentioned that using the values of parameters of Parrot and Killoh [START_REF] Parrot | Prediction of cement hydration[END_REF], there is no nucleation-growth step for C 2 S, and no diffusion-controlled step for C 3 S and C 2 S. Indeed, the hydration of C 2 S is in fact restricted to the power law rate and the hydration of C 3 S directly jumps from the nucleation growth step to the power-law rate. On the contrary, modern numerical models of cement hydration relying of an 3D description of the microstructure do not need to introduce the third stage. Lastly, these models accurately account for the effect of the particle size distribution of the cement, while the Blaine fineness of the cement is often the only parameter of this distribution provided to the model of Parrot and Killoh. This limitation is of particular importance for modern cements : the span of their particle size distribution is smaller than those of old cements due to improvements of grinding process. As a result, there are fewer particles larger than 65µm in modern cements and these cements hydrate more rapidly despite comparable Blaine fineness. But the kinetic model of Parrot and Killoh is not able to model this effect. Nevertheless, this model proved both sound and practical for decades in spite of its drawbacks.

The previous set of equations describes the consumption of solid reactant i as a function of time. But C 3 A and C 4 AF can take part in many reactions (see table 1.1 for formulae and table 1.2 for reactions). To choose which reaction consumes the solid reactant, it is assumed that C 3 A first reacts with gypsum (C SH 2 ) to form the ettringite (C 6 A S3 H 32 ). The reaction consuming ettringite starts as the gypsum is depleted. Following thermodynamical evidences [START_REF] Lothenbach | Thermodynamic modelling of the effect of temperature on the hydration and porosity of portland cement[END_REF], monocarboaluminate (C 4 A CH 11 ) is formed except if the calcite is depleted or if the temperature exceeds 48°C. Otherwise, the ettringite reacts with the C 3 A to produce monosulfoaluminate (C 4 A SH 12 ). Lastly, the reaction forming hydrogarnets (C 3 AH 6 ) is activated if and only if both the gypsum and the ettringite are depleted. Compared to existing models [START_REF] Thiery | Un modèle d'hydratation pour le calcul de la teneur en portlandite des materiaux cimentaires comme donnée d'entrée des modèles de carbonatation[END_REF][START_REF] Nguyen | Modelisation des couplages entre hydratation et dessication des matériaux cimentaires à l'issue du décoffrage. etude de la dégradation des propriétés de transfert[END_REF], the reactions of C 4 AF have to be modified to account for new evidences. Indeed, the formation of FH 3 is hard to detect in real cement paste since it is amorphous [START_REF] Brouwers | The work of Powers and Brownyard revisited: Part 2[END_REF] and iron has been detected in silicious hydrogarnet [START_REF] Taylor | An electron microprobe study of a mature cement paste[END_REF][START_REF] Rodger | Electron microscopy study of ordinary portland cement and ordinary portland cement-pulverized fuel ash blended pastes[END_REF][START_REF] Dilnesa | Fe-containing phases in hydrated cements[END_REF]. Moreover, chemical investigations have shown that Fe-containing silicious hydrogarnet are more stable than Fe-ettringite and Fe-containing alumina, ferric oxide, monosulfate (AFm) phases [START_REF] Dilnesa | Synthesis and characterization of hydrogarnet ca3(alxfe1 -x)2(sio4)y(oh)4(3 -y)[END_REF]: the formation of this phase is thermodynamically favored in aged paste [START_REF] Vespa | Identification of the thermodynamically stable fecontaining phase in aged cement pastes[END_REF]. Consequently, the reactions accounted for in the present model are similar to those of Brouwers [START_REF] Brouwers | The work of Powers and Brownyard revisited: Part 2[END_REF] until all sources of silica are depleted. The product of the reactions is a mix between the hydrogarnets C 3 FS 1.34 H 3.32 and C 3 AS 0.84 H 4.32 investigated in [START_REF] Dilnesa | Synthesis and characterization of hydrogarnet ca3(alxfe1 -x)2(sio4)y(oh)4(3 -y)[END_REF]: C 6 AFS 2.18 H 19 . The additional water improves consistency with the formula introduced in [START_REF] Brouwers | The work of Powers and Brownyard revisited: Part 2[END_REF].

In addition, an estimate of the mass of chemically bounded water m wb in a cm 3 of cement paste can be computed as justified by Brouwers [START_REF] Brouwers | The work of Powers and Brownyard revisited: Part 2[END_REF]:

m wb M H = 1.2(N CSH + N CSHp ) + N CH + 9N C 6 A S3 H 32 + 10N C 4 A SH 12 + 11N C 4 A CH 11 + 7.64N C 6 AF S 2 H 19 + +2N C SH 2 + 6N C 3 AH 6 + 3N F H 3 (7)
Finally, the hydration model provides an estimate of the chemical shrinkage δV ch (t) and capillary porosity Φ cap (t). Indeed, the chemical shrinkage writes:

δV ch (t) = j v j (N j (t) -N j (0)) (8)
where v j is the molar volume of the chemical species j in cm 3 /mol (Tab. 1.1). As the hydrates are formed in their saturated states, the volume of the water v H N H (t) per 1cm 3 cement paste represents the water-filled capillary porosity. Hence, the capillary porosity, that is the volume fraction of capillary pores in the cement paste, writes:

Φ cap (t) = (-δV ch (t) + v H N H (t))/1cm 3 (9)
As-is, hydration stops when either the capillary water or the anhydrous particles are totally consumed. The effect of the exhaustion of capillary water on the hydration kinetics is introduced in the next section.

Influence of the relative humidity on hydration kinetics

It is known that either the water content or the relative humidity h can affect the pace of hydration reaction. For instance, the reduction coefficient β h introduced by Parrot et. al. ([44], cited by [START_REF] Van Breugel | Simulation of hydration and formation of structure in hardening cement-based materials[END_REF]) is applied:

β h = h-0.55 0.45 4 if h > 0.80 β h = 0 if h ≤ 0.80 (10)
Indeed, since the relative humidity is related to the chemical potential of water vapor, hydration reactions may become impossible at relative humidity lower than 75%-80% for thermodynamic reasons [START_REF] Jensen | Thermodynamic limitation of self-desiccation[END_REF][START_REF] Flatt | Why alite stops hydrating below 80% relative humidity[END_REF]. For instance, it has been observed that the relative humidity in hydrating cement pastes remains above 80%, even for a 7-days-old cement paste with a water to cement ratio w/c of 0.3 [START_REF] Chen | Prediction of self-desiccation in low water-to-cement ratio pastes based on pore structure evolution[END_REF]. This thermodynamic stopping of hydration introduces a discontinuity of the reaction rate. Nevertheless, the coefficient β h at 80% relative humidity is 0.095, implying that the hydration rate has already been divided by 10 compared to the 99% hydration rate. The Kelvin-Laplace equation can be applied to estimate the capillary pressure p cap and the diameter d w of the largest pore full of liquid water if the relative humidity is high [START_REF] Lura | Autogenous shrinkage in high-performance cement paste: An evaluation of basic mechanisms[END_REF][START_REF] Lin | Modeling shrinkage of portland cement paste[END_REF]:

p cap = - RT ln(h) v H = 4 γ w d w (11)
where R = 8.314 J/mol/K is the universal gas constant, v H = 18.015 cm 3 /mol is the molar volume of water and γ w = 0.07197 N/m is the surface tension of water/air interface. The largest diameter d w is estimated according to the hydration model of HYMOSTRUCT [START_REF] Van Breugel | Simulation of hydration and formation of structure in hardening cement-based materials[END_REF]. Nevertheless, the adsorbed water is not accounted for in the present study. Hence, the capillary pore size distribution of the cement is assumed to be of the form:

Φ <d = a. ln d d 0 ( 12
)
where Φ <d is the volume fraction of capillary pores of diameter smaller than d in the cement paste and d 0 = 4 nm is the diameter of the smallest capillary pore. The coefficient a can be tuned to fit the span of the pore size distribution. Indeed, the diameter of the largest pore is d max (t) = d 0 exp(Φ(t)/a). Moreover, a numerical model to estimate a has been designed [START_REF] Koenders | Simulation of volume changes in hardening cement-based materials[END_REF] : it is likely that a depends on the temperature history, the hydration degree or the water to cement ratio. Following the results reported in [START_REF] Koenders | Simulation of volume changes in hardening cement-based materials[END_REF], the following trend is considered:

a = 0.1 if w/c > 0.4 0.1 + 0.4 (0.4 -w/c) if w/c < 0.4 (13) i E i (kJ/mol) E i /R (K) C 3 A 54 6500 C 3 S 42 5000 C 4 AF 32 3800 C 2 S 21 2500
Table 1.4: The activation energies of the mineral species considered in the present study are taken from [START_REF] Maekawa | Modeling of concrete performance : hydration, microstructure formation and mass transport[END_REF][START_REF] Poole | Modeling temperature sensitivity and heat evolution of concrete[END_REF].

This trend ensures a rapid decrease of d max (t) for low w/c ratios. The volume of water in the capillary pores is then estimated as:

Φ w = Φ cap + δV ch /1cm 3 = v H N H /1cm 3 (14)
Finally, the diameter of the largest pore full of water d w is such that:

Φ w = a ln d w d 0 (15)
Once d w is estimated, equations ( 11) and [START_REF] Nilsen | Concrete: A three phase material[END_REF] are successively applied to compute h and β h . Lastly, if h is lower than 80%, the coefficient β h is set to zero so as to stop the hydration. This feature is of paramount importance to properly estimate the hydration degree of cement paste featuring a low w/c ratio. The influence of the temperature is introduced in the next section.

By performing NRM relaxometry [START_REF] Muller | Densification of c-s-h measured by 1h nmr relaxometry[END_REF][START_REF] Muller | Characterization of porosity and C-S-H in cement pastes by 1H NMR[END_REF], it has been shown that there is no capillary water at 28 days in a sealed sample of cement paste featuring a water to cement ratio of 0.4. Nevertheless, the relative humidity of the sealed sample is still above 91%: pores of 23nm in diameter are still full of water according to the Kelvin equation. As a consequence, the statement that the hydration continues to take place in the absence of remaining capillary water clearly refers to the capillary water identified by NRM, which strongly differs from the capillary water considered by Powers. As a result, the proposed model considering the effect of the relative humidity does not contradict recent evidence from NRM relaxometry.

Influence of temperature on hydration kinetics

The temperature and the rates of the hydration reactions are tightly coupled. On the one hand, raising the temperature increases the rates of the hydration reactions. On the other hand these chemical reactions release heat which triggers a raise of temperature in adiabatic conditions. Hence, semi-adiabatic calorimetry tests are performed to estimate the evolution of the hydration degree with time. In the present section, the hydration model is adapted to account for the effect of temperature on hydration kinetics and a semi-adiabatic calorimetry test is modeled.

Acceleration of hydration reactions with temperature

It is well-known that increasing the temperature speeds up the chemical reactions of hydration. The thermo-activation of the reactions is modeled by the Arrhenius law which links the actual time t to an equivalent time t eq at the reference temperature T 0 [START_REF] Hansen | Maturity computer for controlled curing and hardening of concrete[END_REF] :

t eq (t) = t 0 e -Ea R 1 T (t) -1 T 0 dt ( 16
)
where Ea is the activation energy in J/mol and R = 8.314 J/mol/K is the gas constant. A default uniform value of the activation energy is often assumed to perform structural analysis [START_REF] Granger | Comportement différé du béton dans les enceintes de centrales nucléaires : analyse et modélisation[END_REF][START_REF]itech, Cesar-lcpc v5 analyse du béton au jeune âge[END_REF][START_REF] Briffaut | Numerical simulations of the qab and langavant semi-adiabatic tests : Analysis and comparison with an experimental measurement campaign[END_REF][START_REF] Briffaut | Analysis of semi-adiabiatic tests for the prediction of early-age behavior of massive concrete structures[END_REF]]. Yet, the apparent activation energy of cementitious materials depends on the temperature, the hydration degree [START_REF] Van Breugel | Simulation of hydration and formation of structure in hardening cement-based materials[END_REF][START_REF] Xiong | Isothermal calorimetry study of blended cements and its application in numerical simulations[END_REF], the cement type [START_REF] Pang | Effects of curing temperature and pressure on the chemical, physical, and mechanical properties of Portland cement[END_REF], the cement composition [START_REF] Van Breugel | Simulation of hydration and formation of structure in hardening cement-based materials[END_REF][START_REF] Maekawa | Modeling of concrete performance : hydration, microstructure formation and mass transport[END_REF][START_REF] D'aloia | Determining the "apparent" activation energy of concrete: Ea-numerical simulations of the heat of hydration of cement[END_REF][START_REF] Schindler | Effect of temperature on hydration of cementitious materials[END_REF][START_REF] Lin | Modeling of hydration kinetics and shrinkage of Portland cement paste[END_REF][START_REF] Lothenbach | Thermodynamic modelling of the effect of temperature on the hydration and porosity of portland cement[END_REF], the Blaine fineness [START_REF] Schindler | Effect of temperature on hydration of cementitious materials[END_REF][START_REF] Lin | Modeling of hydration kinetics and shrinkage of Portland cement paste[END_REF] and the water to cement ratio [START_REF] Poole | Modeling temperature sensitivity and heat evolution of concrete[END_REF]. In the hydration model of Maekawa et. al. [START_REF] Maekawa | Modeling of concrete performance : hydration, microstructure formation and mass transport[END_REF], the activation energy E i /R depends on the mineral i ∈(C 3 S, C 2 S, C 3 A, C 4 AF) and the stage of hydration. Indeed, the apparent activation energy decreases towards 21 kJ/mol during the diffusion stage of hydration [START_REF] Suzuki | Quantification of hydration-heat generation process of cement in concrete[END_REF]. It must be noticed that this activation energy is comparable to the activation energy for ion diffusion in water (18 kJ/mol) [START_REF] Talekar | Temperature dependence of activation energies for self-diffusion of water and of alkali ions in aqueous electrolyte solutions. a model for ion selective behavior of biological cells[END_REF].Consequently, the activation energies are lower for the minerals that hydrate later :

E C 3 A > E C 3 S > E C 4 AF > E C 2 S (Tab. 1.4)[60, 64].
As-is, isothermal calorimetry tests at different temperatures can be modeled. Nevertheless, a very common test on cement hydration is the semi-adiabatic calorimetry test, where the heat released by the exothermic chemical reactions partly induces an increase of the temperature of the mortar. Therefore, a minimal thermodynamic framework is presented in the next section to investigate this coupling.

Modeling semi-adiabatic calorimetry test

The change of enthalpy of a system dH balances the released heat δQ as long as quasistatic isobaric processes are considered. Moreover, the enthalpy of the system is a function of its temperature T , its pressure P and the molar content N i of each chemical species i. Consequently, the following equation can be written for quasi-static isobaric processes:

-δQ = dH = C p (P, T, N i )dT + i ∆ f H i (P, T )dN i ( 17
)
where C p (P, T, N i ) is the total heat capacity of the system and ∆ f H i (P, T ) is the molar enthalpy of formation of the chemical species. The heat capacity reads:

C p (P, T, N i ) = i C p,i (P, T )N i ( 18
)
where C p,i (P, T ) is the molar heat capacity of the chemical species i. It is assumed that the heat capacity is independent from the temperature, which is reasonable given the narrow range of temperature that is considered for cement hydration (10°C-70°C). This hypothesis is no longer valid if the solubilities of the various chemical species and their phase diagrams were to be studied. Lastly, the dependence on pressure is beyond the scope of the present study : it is assumed that the hydration occurs at a constant pressure equal to the standard pressure of P = 0.1MPa (1bar). The molar enthalpy of formation ∆ f H i (P, T ) of the chemical species i depends logically on temperature through the corresponding molar heat capacity:

∆ f H i (P, T ) = ∆ f H i (P, T 0 ) + T T 0 C p,i (P, θ)dθ (19)
where ∆ f H i (P, T 0 ) is the standard enthalpy of formation at the reference temperature of 25°C and P = 0.1MPa as given in the tables of chemical thermodynamic properties [START_REF] Wagman | The nbs tables of chemical thermodynamic properties. selected values for inorganic and c1 and c2 organic substances in si units[END_REF]. The changes of the molar contents is related to the chemical reactions:

-δQ = dH = ( i C p,i .N i ) dT + i (∆ f H i (P, T 0 ) + (T -T 0 ).C p,i ) dN i (20)
The change of enthalpy related the consumption of reactants and formation of products can be grouped by reaction:

-δQ = dH = ( i C p,i .N i ) dT + r (∆ r H(P, T 0 ) + (T -T 0 ).C p,r ) dα r N 0,r (21)
where ∆ r H(P, T 0 ) is the heat released by an isothermal reaction of one mole of the reference reactant (one of C 3 S, C 2 S, C 3 A, C 4 AF) of the reaction r, N 0,r is the initial quantity of the reference reactant and C p,r is the difference of heat capacity of the reactants and the products, for one mole of the reference reactant:

∆ r H(P, T 0 ) = i∈r ν r,i ∆ f H i (P, T 0 ) C p,r = i∈r ν r,i C p,i (22)
where ν r,i is the signed stochiometric coefficient of reactant i in reaction r, the stochiometric coefficient of the reference reactant being ν r,ir = -1. The effect of temperature on the enthalpy of formation is limited in the considered range of temperature due to the orders of magnitude of the molar changes of heat capacity (≈ 100 J/mol/K) and molar enthalpy of reactions (≈ 100 kJ/mol).

The heat released Q depends on the calorimeter and the temperature outside of it T out as in [START_REF] Afnor | Méthodes d'essais des ciments partie 9 : Chaleur d'hydratation -méthode semi-adiabatique[END_REF][START_REF] Boulay | Quasi-adiabatic calorimetry for concretes: Influential factors[END_REF][START_REF] Briffaut | Numerical simulations of the qab and langavant semi-adiabatic tests : Analysis and comparison with an experimental measurement campaign[END_REF][START_REF] Briffaut | Analysis of semi-adiabiatic tests for the prediction of early-age behavior of massive concrete structures[END_REF].

δQ = (a(T -T out ) + b(T -T out ) 2 )dt (23)
where the coefficients a and b result from the calibration of the calorimeter according to the standard NF EN 196-9 [START_REF] Afnor | Méthodes d'essais des ciments partie 9 : Chaleur d'hydratation -méthode semi-adiabatique[END_REF].

The enthalpies of formation of the chemical species come from [START_REF] Wagman | The nbs tables of chemical thermodynamic properties. selected values for inorganic and c1 and c2 organic substances in si units[END_REF][START_REF] Odler | 6 -hydration, setting and hardening of portland cement[END_REF][START_REF] Taylor | 5 hydration of the calcium silicate phases and 6 hydrated aluminate, ferrite and sulphate phases[END_REF]. The enthalpy of formation of C 1.7 SH 4 is set so as to comply with the heat of hydration of C 3 S and C 2 S. The value of ∆ f H = -3287 kJ/mol for C 1.7 SH 4 is comparable to the value of -2890 kJ/mol estimated by Fujii and Kondo [START_REF] Fujii | Communications of the american ceramic society estimation of thermochemical data for calcium silicate hydrate (c-s-h)[END_REF] for C 1.7 SH 2.617 if the enthalpy of formation of the additional water (4-2.161)H is similar to that of liquid water ∆ f H = -285.83 kJ/mol. The heat capacity of C 1.7 SH 4 is estimated at C p = 297 J/mol/k starting from the heat capacity C p = 190.10 J/mol/K of C 1.6 SH 2.58 obtained by Blanc et. al. [START_REF] Blanc | Chemical model for cement-based materials: Temperature dependence of thermodynamic functions for nanocrystalline and crystalline c-s-h phases[END_REF][START_REF] Andra | Cementitous phases[END_REF] if the heat capacity of the additional water is similar to that of liquid water C p = 75.291 J/mol/K. Heat capacities measured by Ederová and Šatava [START_REF] Ederová | Heat capacities of c3ah6, c4assh12 and c6ass3h32[END_REF][START_REF] Matschei | Thermodynamics of cement hydration[END_REF] are colored in blue in table 1.1. Thermochemical values of Fe(OH) 3 are available in the NIST-JANAF thermochemical tables [START_REF] Chase | Nist standard reference database 13 nist janaf thermochemical tables 1985 v1.0, Tech. Rep. V1.0[END_REF]. Finally, the molar heat capacity and molar enthalpy of formation of C 6 AFS 2.18 H 19 are the sums of the ones of C 3 FS 1.34 H 3.32 and C 3 AS 0.84 H 4.32 as reported in [START_REF] Dilnesa | Synthesis and characterization of hydrogarnet ca3(alxfe1 -x)2(sio4)y(oh)4(3 -y)[END_REF] plus 11.36 times that of liquid water. Derived values of the heat of hydration ∆ r H(P, T 0 ) and changes of heat capacities C p,r of each reaction r are reported in table 1.2.

Effect of additional silicate

As in [START_REF] Lu | Phase composition of hydrated dsp cement pastes[END_REF][START_REF] Bentz | Modelling drying shrinkage in reconstructed porous materials: application to porous vycor glass[END_REF], a pozzolanic reaction consumes the silicate S and Portlandite to produce pozzolanic C 1.1 SH 3.9 featuring a C/S ratio lower than that of the C 1.7 SH 4 produced by the hydration of C 3 S. The molar volume is assumed to be 101.8 cm 3 /mol, to be compliant with the measured chemical shrinkage of 0.2 cm 3 per gram of hydrated silicate [START_REF] Lu | Phase composition of hydrated dsp cement pastes[END_REF][START_REF] Bentz | Modelling drying shrinkage in reconstructed porous materials: application to porous vycor glass[END_REF]. Similarly, the enthalpy of formation of C 1.1 SH 3.9 is set to -2841 kJ/mol so that the heat released by the pozzolanic reaction is 870 kJ/kg of S [START_REF] De Larrard | Structures granulaires et formulation des bétons -Concrete Mixture -Proportionning -A scientific approach[END_REF]. It must be noticed that this specific heat is often higher than that of the cement. Fly ashes may contain aluminate as part of their oxyde composition and a reaction system similar to that of C 3 A is considered [START_REF] Waller | Relations entre composition des bétons, exothermie en cours de prise et résistance en compression[END_REF](Table 1.2). Hence, the fact that some of the aluminate can be incorporated into the C-S-H phase is not accounted for in the present model. The pozzolanic reactions are more sensitive to temperature than the hydraulic reactions of cement and the influence of temperature on the rate of pozzolanic reactions is modeled by an activation energy of E S,A = 83, 14 kJ/mol (E S,A /R=100 K).

The reactivity of silica fume is due to its high specific surface, which is measured at about 20000m 2 /kg by B.E.T. technics [START_REF] Waller | Relations entre composition des bétons, exothermie en cours de prise et résistance en compression[END_REF]. It could be compared to the Blaine surface of cements (350-400 m 2 /kg) or fly ashes (375-450 m 2 /kg). Nevertheless, it must be mentioned that the measurement procedures refer to different physical phenomena (adsorption and air permeability) which may not depend on the same surface. B.E.T. and Mercury Intrusion Porosimetry suggest that the specific surface of silica fume could be considered as 20 times higher than that of a typical cement [START_REF] Olek | Determination of surface area of portland cement and silica fume by mercury intrusion porosimetry[END_REF]. Finally, since our hydration model makes use of the Blaine fineness, a specific surface of 2000m 2 /kg is considered as default for silica fume to keep the same kinetics for both silica fumes and fly ashes and provide a reasonable estimate of the degree of hydration of cement blended with silica fumes.

The kinetics of the pozzolanic reactions is set according to the results of Waller [START_REF] Waller | Relations entre composition des bétons, exothermie en cours de prise et résistance en compression[END_REF]. The degree of pozzolanic reactions α S of a fly ash of Blaine fineness 400m 2 /kg at 20°C writes:

α S = 1 1 + τ t n ( 24
)
where n = 0.7 and τ = 80days are coefficients adjusted to the results of analysis of Scanning Electron Microscopy (SEM) images reported in [START_REF] Weerdt | Hydration mechanisms of ternary portland cements containing limestone powder and fly ash[END_REF], assuming a fly ash of pozzolanic activity of 60%. The corresponding reaction rate writes [START_REF] Waller | Relations entre composition des bétons, exothermie en cours de prise et résistance en compression[END_REF]:

αS = n τ (α S ) 1-1 n (1 -α S ) 1+ 1 n (25)
Finally, this equation can be tuned to account for the effect of the Blaine fineness, the relative humidity and temperature as shown in the previous sections. The updated hydration model produces estimates of the volume fractions of the constitutive phases of blended cement pastes as a function of the hydration degree and time (Fig. 1.1). These volume fractions are provided as inputs to a micromechanical scheme in the next section.

Using homogenization schemes to estimate the mechanical properties

Estimating the overall behavior of concrete

In the present section, concrete is depicted as a multiscale composite material. A brief overview of up-to-date mean-field homogenization models dedicated to cementitious materials is to be performed. At each scale, depending on the microstructural morphology, either the Mori-Tanaka scheme [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] or the self-consistent scheme [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF] are applied to estimate the timedependent mechanical properties of the composite material starting from the properties of the constitutive phases and their geometrical shapes. The expressions of the overall stiffnesses derived by using these schemes are recalled in Appendix A.

Multiscale models designed to upscale the mechanical properties of cementitious material can significantly differ. For instance, in the micromechanical model of Constantinides and Ulm [START_REF] Constantinides | The effect of two types of c-s-h on the elasticity of cement-based materials: Results from nanoindentation and micromechanical modeling[END_REF], the low density C-S-H (LD-C-S-H) and the high density C-S-H (HD-C-S-H) [ [START_REF] Jennings | Model for the developing microstructure in portland cement pastes[END_REF][START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of portland cement pastes[END_REF] are merged by using the Mori-Tanaka scheme, the low density C-S-H being the matrix. This mix defines the matrix of the cement paste, in which the capillary porosity, unhydrated particles and CH are embedded. Hence, the Mori-Tanaka scheme is used to estimate the behavior of the cement paste. Finally, at the scale of the mortar, the aggregates and the cement paste are respectively the inclusions and the matrix in the Mori-Tanaka scheme applied to estimate the overall elastic properties. Modeling of the setting of concrete is achieved by introducing the self-consistent scheme at the scale of the cement paste [START_REF] Bernard | A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials[END_REF].

To model the autogenous shrinkage, Pichler et. al. [START_REF] Pichler | A multiscale micromechanics model for the autogenous-shrinkage deformation of early-age cement-based materials[END_REF] have introduced an additional scale, that of porous C-S-H, where the C-S-H and the capillary porosity are merged using the Mori-Tanaka scheme to define the matrix of the cement paste. Similarly, Sanahuja et. al. [START_REF] Sanahuja | Modelling elasticity of a hydrating cement paste[END_REF] handled HD-C-S-H as a coating of unhydrated particles at the scale of the cement paste.

The properties of the porous LD-C-S-H are defined by a self-consistent scheme and an aspect ratio of LD-C-S-H of 0.033 is introduced to recover an acceptable estimate of the hydration degree at setting. On the contrary, Ghabezloo recovered a reasonable estimate of the overall stiffness by using the self-consistent scheme at the scale of the cement paste, but all phases (LD-C-S-H, HD-C-S-H, big capillary pores...) featured a spherical morphology [START_REF] Ghabezloo | Association of macroscopic laboratory testing and micromechanics modelling for the evaluation of the poroelastic parameters of a hardened cement paste[END_REF]. In the model of Pichler and Hellmich [START_REF] Pichler | Upscaling quasi-brittle strength of cement paste and mortar: A multi-scale engineering mechanics model[END_REF][START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF], which relies on the hydration model of Powers and Brownyard [START_REF] Powers | Studies of the physical properties of hardened portland cement paste[END_REF], the self-consistent scheme is applied to estimate the elastic properties and the compressive strength of the porous hydrates, the hydrates being considered as infinitely long cylinders mixed with the capillary porosity. Yet, by performing experiments on C 3 S and C 2 S pastes, Termkhajornkit et. al. [START_REF] Termkhajornkit | Dependence of compressive strength on phase assemblage in cement pastes: Beyond gel-space ratio -experimental evidence and micromechanical modeling[END_REF] have questioned the fact that the gel-space ratio, related to the capillary porosity, is a good indicator of the strength of a mortar [START_REF] Powers | Structure and physical properties of hardened portland cement paste[END_REF]. Their micromechanical model relies on a porous C-S-H matrix defined as a mix between solid C-S-H and the total porosity, including gel porosity, so as to ensure a better agreement with their experimental data on compressive strength. The molar volume of solid C 1.7 SH 1.8 is set to v sC-S-H =70 cm 3 /mol according to NMR Relaxometry [START_REF] Muller | Densification of c-s-h measured by 1h nmr relaxometry[END_REF] and the Young modulus and aspect ratio of the solid C-S-H respectively are 57 GPa and 15. The Portlandite and the unhydrated clinker are then introduced as inclusions in the porous C-S-H. But this homogenization model significantly underestimates the overall elastic stiffness of the cement paste.

Finally, introducing infinite, large or small aspect ratios in a porous microstructure suggests buckling as a potential failure mode, as introduced in [START_REF] De Larrard | Structures granulaires et formulation des bétons -Concrete Mixture -Proportionning -A scientific approach[END_REF].

Indeed, a cylinder of aspect ratio 35, stiffness 29GPa, fixed at both ends will reach the Euler critical strength and experience buckling as soon as the axial compressive stress exceeds 120MPa, which corresponds to the critical Von Mises value considered in Pichler and Hellmich for infinite aspect ratio [START_REF] Pichler | Upscaling quasi-brittle strength of cement paste and mortar: A multi-scale engineering mechanics model[END_REF][START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF]. Since the only reason for introducing elongated particles in the self-consistent scheme is to produce an acceptable estimate of the setting and early strength development, accounting for the hydration degree at setting is required. First, the hydration degree at setting is in the range 2.7-4.3% for cement pastes of water to cement ratio in the range 0.25-0.4 [START_REF] Zhang | Early hydration and setting of oil well cement[END_REF]. These figures correspond to a gel-space ratio of about 7% below which the compressive strength is null. It has been observed on mortars of w/c in the range 0.3-0.5 that the onset of the shear moduli measured by ultrasonic tests corresponds to a minimum heat release of at least 100J/ml of water [START_REF] Lootens | On the relation of setting and early-age strength development to porosity and hydration in cement-based materials[END_REF]. Retaining an heat of hydration of 500kJ/kg for the cement, it corresponds to gel-space ratios of about 13%. Though setting at gel-space ratios of 7% and 13% would be properly described by choosing aspect ratios of 35 or 21 in the self-consistent scheme, an aspect ratio of 7 is chosen so as to ensure a better agreement with experimental measurements of the compressive strength on cement pastes and mortars [START_REF] Taplin | A method for following the hydration reaction in portland cement paste[END_REF][START_REF] Lerch | Long-time study of cement performance in concrete[END_REF]. Hence, the multiscale homogenization scheme introduced in the present article (Fig. 2.1) is very similar to the one introduced by Pichler and Hellmich [START_REF] Pichler | Upscaling quasi-brittle strength of cement paste and mortar: A multi-scale engineering mechanics model[END_REF][START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF][START_REF] Hassan | Downscaling based identification of nonaging power-law creep of cement hydrates[END_REF] as it proved able to estimate the elastic stiffness, the compressive strength and the short term non-aging creep strain of cement pastes:

1. At the smallest scale, all the hydrates are merged by using the self consistent scheme.

The Young modulus of the C 1.7 SH 4 and C 1.1 SH 3.9 of respective density 2.08g/cm 3 and 1.9g/cm 3 is set to 25 GPa and the Poisson ratio to 0.24. This value is a median of the stiffness of HD-C-S-H and LD-C-S-H identified by Constantinides and Ulm [START_REF] Constantinides | The effect of two types of c-s-h on the elasticity of cement-based materials: Results from nanoindentation and micromechanical modeling[END_REF] on C-S-H foams. To compute Eshelby tensors, all phases are considered as spherical at this scale. 2. At the scale of the hydrate foam, the hydrates are merged with the capillary porosity using the self-consistent scheme. As underlined in [START_REF] Bernard | A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials[END_REF], the bulk modulus of the pores strongly affects the estimated overall Poisson's ratio. Consequently, the bulk modulus of the pores is considered to be null for static measurements of the Young modulus, but the bulk modulus of water (2. Young modulus and dynamic Poisson's ratio. In this case, a bulk modulus K cap is set for the water-filled fraction of the capillary porosity, which turn the hydrate foam into a three-phase material. An aspect ratio of 7 is chosen for the fibrillar, directional morphology of the hydrates, which has been observed at the scale of 100-500nm on Transmission Electron Microscopy (TEM) images [START_REF] Richardson | Tobermorite/jennite-and tobermorite/calcium hydroxide-based models for the structure of c-s-h: applicability to hardened pastes of tricalcium silicate, -dicalcium silicate, portland cement, and blends of portland cement with blast-furnace slag, metakaolin, or silica fume[END_REF]. Moreover, this aspect ratio, combined to the self-consistent scheme, ensures a reasonable estimate of the strength at early age. 3. At the scale of cement paste (1 µm-65 µm), all reactants of chemical reactions and the fillers are considered as inclusions embedded in the hydrate foam. Hence, the Mori-Tanaka scheme is applied to estimate the elastic stiffness and the strength of the cement paste. Mechanical stiffness are reported in table 2.1. To estimate the brittle failure of the cementitious material, a Von Mises criteria is set for the hydrate inclusions as in [START_REF] Pichler | Upscaling quasi-brittle strength of cement paste and mortar: A multi-scale engineering mechanics model[END_REF][START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF]. The quadratic mean of the Von Mises stress is estimated by using Kreher's lemma [START_REF] Kreher | Residual stresses and stored elastic energy of composites and polycrystals[END_REF] (see appendix Appendix A) on a collection of 20 orientations for hydrate inclusions introduced at the scale of the hydrate foam. For each orientation, this quadratic mean is compared to the Von Mises stress at failure set to σ V M,hyd = 120 MPa. It is perfectly consistent with the 70MPa criteria introduced by Pichler and Hellmich on the quadratic average of the deviatoric part of the stress field, which must be multiplied by √ 3 if the Von Mises stress is considered. The cement paste fails as soon as a single orientation reaches the criterion, thus defining the compressive strength of the cement paste f c,p . 4. The scale of concrete (> 65 µm): the sand and the entrapped air are considered as inclusions embedded in the cement paste. Again, the elastic stiffness is estimated according to the Mori-Tanaka scheme, the cement paste being the matrix. The Interfacial Transition Zone that could exists at the interface between the aggregates and the cement paste is not accounted for in the present model and a perfect bonding is assumed between the aggregates and the cement paste, which is considered as homogeneous at this scale. Nevertheless, estimating the compressive strength of the mortar starting from that of the cement paste requires a particular attention. The new procedure and the underlining assumptions are described in the following section.

Estimating the compressive strength of mortar

Even if the quadratic mean of the equivalent stress in the hydrate is estimated according to Kreher's equation [START_REF] Kreher | Residual stresses and stored elastic energy of composites and polycrystals[END_REF], the stress concentrations in the cementitious matrix due to the aggregates are not properly accounted for. Indeed, in the article of Pichler et. al. [START_REF] Pichler | Upscaling quasi-brittle strength of cement paste and mortar: A multi-scale engineering mechanics model[END_REF], the estimated compressive strength of a mortar featuring a water to cement ratio of 0.5 is estimated to be similar to that of a cement paste of water to cement ratio 0.42. Hence, the homogenization method of Pichler et. al. suggests that a mortar is significantly stronger (up to about 20%) than a cement paste featuring the same water to cement ratio and the same hydration degree. It contradicts experimental evidences : the strengths of mortar and concrete are expected to be lower that those of their cementitious matrices (down to about -10% for mortar and about -30% for concrete), even if the tested prismatic samples all feature the same aspect ratio of 3:1 to reduce the confining effect of end restraint [START_REF] Cong | Role of silica fume in compressive strength of cement paste, mortar, and concrete[END_REF]. On the one hand, this contradiction could be resolved by introducing an heterogeneous cementitious matrix featuring an Interfacial Transition Zone as the weakest link. On the other hand, a better way to estimate the stress fluctuations in the cementitious matrix can be introduced.

In the present article, the stress fluctuations in the cementitious matrix are accounted for as performed in the range of polycrystals [START_REF] Brenner | Elastic anisotropy and yield surface estimates of polycrystals[END_REF]. But the theoretical analysis need to be adapted to matrix-inclusion composite materials and checked. To this end, the local values of the stress are assumed to be multivariate normally distributed, the covariance matrix being isotropic. Hence, the deviatoric part of the stress in the matrix s is 5-variate normally distributed using σ 11 , σ 22 , √ 2σ 23 , √ 2σ 13 , √ 2σ 12 as scalar variables. Indeed, the density function of the deviatoric stress is:

f (s) = 1 (2π) 5 2 V 5 2 exp - 1 2V (s -s 0 ) : (s -s 0 ) ( 26
)
where s 0 is the mean deviatoric stress in the matrix computed according to the Mori-Tanaka scheme and the standard deviation V drives the fluctuation of the stress field. Consequently, the scaled double dot product (s : s)/V is distributed according to the noncentral chi-squared distribution of 5 degrees of freedom and noncentrality parameter λ = (s 0 : s 0 )/V :

s : s ∼ V χ 2 5 s 0 : s 0 V (27)
Hence, the average < s : s > writes:

< s : s >= 5V + s 0 : s 0 (28)
As < s : s > is estimated according to Kreher's lemma [START_REF] Kreher | Residual stresses and stored elastic energy of composites and polycrystals[END_REF] and s 0 : s 0 is estimated according to the localization tensor (Eq. A.5), the previous equation allows access to the variability V of the deviatoric part of the stress field. In particular, if the considered material is homogeneous, the local stress is uniform and < s : s >= s 0 : s 0 . Hence V is null, and the noncentral chisquared distribution becomes a degenerate distribution. The consistency with the fact that the local stress is uniform is to be noticed. Finally, the cementitious matrix is expected to fail if and only if a volume fraction c f ail of it has reached a Von Mises stress higher than the compressive strength of the cement paste f c,p . The criteria to enforce is:

F V χ 2 5 (λ) 2 3 f 2 c,p ≥ 1 -c f ail (29)
where

F V χ 2 5 (λ)
is the repartition function of the double dot product s : s. To estimate the compressive strength of the mortar, it must be noticed that the noncentrality parameter λ is independent from the magnitude of the overall stress: it is evaluated once for all by setting an a priori value of the overall uniaxial stress Σ. Then, the critical value V c is such that:

V c = 2 3 f 2 c,p F -1 χ 2
The case of an infinite β corresponds to an homogeneous stress in the cementitious matrix. In this particular case, the quadratic average of the Von Mises stress on the matrix is considered so as to define the failure of the mortar.

A value of c f ail = 1% is set so as to comply with the observed decrease of compressive strength of mortars compared to the one of cement pastes (10-15%) [START_REF] Cong | Role of silica fume in compressive strength of cement paste, mortar, and concrete[END_REF]. The compressive strength is almost unchanged for c f ail = 10% and the estimated compressive strength of mortars would become 10% higher than the strength of the paste if c f ail were set to 25%.

The most critical step of this estimate is the assumption regarding the distribution of the Von Mises stress in the matrix. To investigate the effect of these assumptions, the estimated distribution of the Von Mises stress is compared to that estimated by performing 3D numerical simulation on a cubic representative elementary volume, as performed for polycrystals [START_REF] Lavergne | Effects of grain size distribution and stress heterogeneity on yield stress of polycrystals: A numerical approach[END_REF]. The considered periodic unit cell features 597 spherical inclusions packed to a volume fraction of 61% thanks to the Lubachevsky-Stillinger algorithm (Fig. 2.2) [START_REF] Lubachevsky | Geometric properties of random disk packings[END_REF][START_REF] Lavergne | Investigation of the effect of aggregates' morphology on concrete creep properties by numerical simulations[END_REF]. Periodic boundary conditions are applied to the unit cell, the macroscopic loading being an uniaxial stress Σ XX . The inclusions are stiffer than the matrix, its Young modulus being 65GPa compared to 15GPa for the matrix. The Poisson ratio of both phases is 0.2. Then, the FFT algorithm [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF][START_REF] Eyre | A fast numerical scheme for computing the response of composites using grid refinement[END_REF] is applied to compute the elastic response of the unit cell, that is the values of the stress field for each voxel on a regular 160 × 160 × 160 grid. The behavior of "grey voxel", overlapping on both the matrix and the inclusions, is set according to the Reuss bound, using 64 sensing points in each voxel to estimate the local volume fraction as in [START_REF] Lavergne | Investigation of the effect of aggregates' morphology on concrete creep properties by numerical simulations[END_REF]. The estimated overall Young modulus is about 34.4GPa, comparable to 33.5GPa for a Mori-Tanaka scheme, which corresponds to the Hashin-Strikman lower bound in this case. In addition, full-field and mean-field estimates of the arithmetic and quadratic averages of the Von Mises stress in the matrix are similar. Indeed, the arithmetic averages divided by Σ XX respectively are 0.71 and 0.72, while the scaled quadratic averages respectively are 0.74 and 0.81. The proposed mean-field model of stress fluctuation is interesting as the span of the distribution of the Von Mises stress is comparable to that obtained by full-field simulations (Fig. 2.2). As a result, 1% volume of the matrix features Von Mises stress above 1.40Σ XX according to full-field simulations, compared to 1.16Σ XX according to the proposed meanfield estimate. Regarding the numerical simulations, modifying the inclusion size distribution by requiring that half the volume of inclusions must consist in inclusions featuring volumes 8 times larger than inclusions from the other half does not significantly change the numerical values listed above and the estimated distributions. As a consequence, the last difference can be attributed to the fact that the distribution of the components of the stress in the matrix is not multivariate normally distributed. In addition, the distribution of the Von Mises stress in the matrix is similar to that of the component σ xx (x) of the stress as other components are smaller. Moreover, this component is high in particular areas of the matrix, where the load is transferred from one inclusion to the closest ones in the direction of loading (Fig. 2.2). As a conclusion, the proposed model of stress fluctuations provide a practical estimate of the Von Mises stress distribution in the matrix, though the accuracy of 3D numerical simulations is clearly not achieved. Nevertheless, a single 3D elastic computation takes more than a minute: since hundreds of cases are to be considered, the proposed mean field estimate is preferred in the sequel.

Comparisons to existing experimental results

Measures regarding the hydration process from the literature

The hydration model is now to be validated against evidences from the literature, cement compositions being reported in table 3.1. Various experimental procedures have been designed to assess the accuracy of such models. For instance, the kinetics of hydration reactions described by the model of Parrott and Killoh [START_REF] Parrot | Prediction of cement hydration[END_REF] were adjusted to measurements obtained by quantitative analysis of X-ray diffraction patterns. The results are reproduced in [START_REF] Van Breugel | Simulation of hydration and formation of structure in hardening cement-based materials[END_REF]. In addition, the degree of hydration of each phase estimated by the model of Parrott and Killoh can be compared to novel results of XRD/Rietveld analysis [START_REF] Escalante-García | Effect of temperature on the hydration of the main clinker phases in portland cements: part i, neat cements[END_REF][START_REF] Kocaka | Development and evaluation of methods to follow microstructural development of cementitious systems including slags[END_REF][START_REF] Schepper | A hydration study by xrd/rietveld analysis of cement regenerated from completely recyclable concrete[END_REF] (Fig. 3.1, 3.2, 3.3). Although the model features correct characteristic times for the hydration rate of C 3 S, C 2 S and C 3 A, the error on the estimate of the hydration degree can be significant. Moreover, the measurements of the hydration degree of C 4 AF seem uncertain and the corresponding estimates can prove inaccurate. Additional sources of uncertainty includes the Bogue composition, which is different if computed according to oxide composition measured by XRD/Rietveld [START_REF] Kocaka | Development and evaluation of methods to follow microstructural development of cementitious systems including slags[END_REF], and the Blaine fineness. Nevertheless, the influence of temperature on the hydration degree of each phase of cement is correctly modeled as shown by comparison to the measurements of Escalante García & Sharp [ Results of isothermal [START_REF] Lerch | Long-time study of cement performance in concrete[END_REF][START_REF] Waller | Relations entre composition des bétons, exothermie en cours de prise et résistance en compression[END_REF][START_REF] Pane | Investigation of blended cement hydration by isothermal calorimetry and thermal analysis[END_REF] or semi-adiabatic calorimetry tests [START_REF] Waller | Relations entre composition des bétons, exothermie en cours de prise et résistance en compression[END_REF] can be compared to estimates produced by the hydration model (Fig. 3.3,3.4,3.5). The heat of hydration of cement pastes of water to cement ratio equal to 0.4 is satisfyingly modeled for different ASTM cement types and different temperature tested by Lerch and Ford [START_REF] Lerch | Long-time study of cement performance in concrete[END_REF]. However, for a given temperature and a given cement type, the estimate of the heat of hydration depends on both the Bogue composition and the Blaine fineness. Since the Bogue composition is computed according to the oxide composition of the cement, a significant uncertainty on the estimated heat of hydration is introduced. The adiabatic calorimetry tests of Waller [START_REF] Waller | Relations entre composition des bétons, exothermie en cours de prise et résistance en compression[END_REF] were performed to measure the effect of silica fumes and fly ashes on the heat of hydration. On the one hand, the present hydration model overestimates the final temperature for low water to cement ratios. On the other hand, the final temperature is accurately estimated for high water to cement ratios. A potential explanation is that concrete featuring a low water to cement ratio can reach adiabatic temperature higher than 70°C if the initial temperature of the mix is 20°C. Hence, the validity range of the hydration model must be limited to temperatures Samples were cured at 20°C for one day and then stored at different temperatures. lower than 70°C. Pane and Hansen [START_REF] Pane | Investigation of blended cement hydration by isothermal calorimetry and thermal analysis[END_REF] have also investigated the hydration of cement by isothermal calorimetry and thermogravimetry. The measured heat of hydration, mass of bounded water and mass of Portlandite are successfully compared to their corresponding estimates. In particular, displaying the mass of Portlandite as a function of the mass of bounded water for different temperatures reveals the fact that pozzolanic reactions are more sensitive to temperature than hydraulic reactions. . The mean percent error and the mean absolute percent error are computed for the compressive strengths and the Young Moduli estimated by the present model. They are compared the those of the Young Moduli estimated by the ACI209 and Eurocode.

Y exp i -Y model i Y exp i MAPE= 1 n n 1 Y exp i -Y model i Y exp i n compressive strength,

Measures of Young modulus and compressive strength from the literature

The micromechanical model is to be validated against various experimental results regarding the mechanical properties of mortar and concrete [START_REF] Lerch | Long-time study of cement performance in concrete[END_REF][START_REF] Taplin | A method for following the hydration reaction in portland cement paste[END_REF]104,[START_REF] Haecker | Modeling the linear elastic properties of portland cement paste[END_REF][START_REF] Termkhajornkit | Dependence of compressive strength on phase assemblage in cement pastes: Beyond gel-space ratio -experimental evidence and micromechanical modeling[END_REF]. The investigations of Termkhajornkit et. al. [START_REF] Termkhajornkit | Dependence of compressive strength on phase assemblage in cement pastes: Beyond gel-space ratio -experimental evidence and micromechanical modeling[END_REF] on the dependence of compressive strength on phase assemblage in cement paste reveals the specific role of C-S-H. The present model is not able to reproduce these results as all hydrates, including the Portlandite and the C-S-H behave as binders at the scale of the hydrate foam. Nevertheless, for the considered pure C 3 S/C 2 S pastes, the gel-space ratio at setting and the ultimate value of the compressive strength are reasonable (Fig. 3.6). Lerch and Ford have performed compressive strength tests on mortars of water to cement ratio 0.53 and sand to cement ratio 2.77 at different ages. The effect of the cement type on the development of mortar strength is satisfyingly described during the first month (Fig. 3.7). Yet, the long-term strength of the mortars are overestimated. If the samples were stored in air, it is likely that drying could have slowed down or even stopped hydration, while the present model considers hydration without exchange of moisture with the environment. If the samples were stored in water, calcium leaching could have occurred. Taplin [START_REF] Taplin | A method for following the hydration reaction in portland cement paste[END_REF] investigated the effect of the water to cement ratio on the compressive strength of cement pastes: his experimental measurements have also been used to validate the micromechanical model of Pichler and Hellmich [START_REF] Pichler | Upscaling quasi-brittle strength of cement paste and mortar: A multi-scale engineering mechanics model[END_REF][START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF]. Unsurprisingly, the present model is also able to account for this effect (Fig. 3.8): the use of a finite aspect ratio for the hydrates even improves the estimate of the setting time. Moreover, Haecker et. al. have measured the growth of the Young moduli of cement pastes featuring different water to cement ratios in order to appraise the quality of their 3D elastic numerical simulations [START_REF] Haecker | Modeling the linear elastic properties of portland cement paste[END_REF]. The present hydration model does not perfectly estimates the hydration degree, which triggers some errors on the estimated Young moduli (Fig. 3.9). It also expects lower moduli for the cement paste of cement H than for those of cement D because the cement H contains more inert fillers than cement D according to the chemical phase composition. However, the experimental measurements do not feature such a discrepancy. Lastly, the results of tests on more than 3000 concretes have been gathered by Tomosawa and Noguchi [104] along with some data about their formulation. Despite the lack of information about the Bogue compositions of the cements, this database can prove useful to challenge the present hydration model. Nevertheless, concretes featuring a water to cement ratio lower than 0.35 are discarded from the comparison if the curing conditions did not correspond to a sealing of the sample. Similarly, cases where the sand or the gravel saturated density is lower than 2.4g/cm 3 were discarded because the Young moduli of the corresponding light aggregates are not known. Moreover, steam-cured samples are assumed to experience a temperature of 65°C. Yet, the major source of uncertainty is the moduli of the aggregates. Indeed, the aggregates considered in the database can feature different stiffness depending on their type and density. Indeed, some concretes are made of stiff Bauxite aggregates of density 3.16g/cm 3 . Unsurprisingly, the Eurocode2 standards does not seems to be adequate to estimate the Young moduli of such concretes starting from their compressive strengths. On the other hand, the ACI 318 standard accounts for the density of the concrete. But it does not prove more reliable than the Eurocode2 because most of the concretes of the database are made of hard sandstone, which can be a rather soft material despite its density. Assumptions regarding the stiffnesses of aggregates are reported in table 2.2 and the estimates of the compressive strengths and Young Moduli are compared to reported measurements for more than 600 samples (Fig. 3.10). On average, the present model slightly overestimates the strengths of the concretes as shown by computing the mean percent error (Tab. 3.2). But the estimated Young modulus is slightly less biased than that estimated by the ACI 318 code and the Eurocode2. Regarding the Eurocode 2 estimate, the 0.7 reduction factor applied to the case of sandstone aggregates could be responsible for the bias. Results of Lam et. al [START_REF] Lam | Degree of hydration and gel/space ratio of high-volume fly ash/cement systems[END_REF] on cubes of side 70.7mm and Pichler et. al [START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF] on cylinders of diameter 29mm and height 58mm are added. One of the major difficulty in assessing the accuracy of micromechanical models of cementitious materials is the lack of complete database on formulation, properties of raw materials and corresponding measurements of moduli and strength. The present section aims at providing a comprehensive set of data about the formulation of different cement pastes containing different additions. Results of semi-adiabatic calorimetry and compressive strength tests are gathered and compared to the corresponding estimates produced by the hydration model.

Blends are defined based on weight fractions. For instance, the blend denoted by C85L15 contains 15% weight of limestone filler, 5% weight of silica fume are used in the blend C95SF05 and C50FA50 contains 50% weight of class F fly ash. Two water to binder ratios are considered: 0.32 and 0.5. The cement that has been used is a CEM I 52.5 N CE CP2 NF from the plant of Beffes (Tab. 4.1). The considered silica fume is made of 95% SiO 2 . The Blaine fineness of the limestone-based filler is 462 m 2 /kg. To ensure sealed hydration conditions, the three 4cm×4cm×16cm samples are taken out of their steel mold at 24 h and stored at 20°C in sealed watertight plastic bags until the flexural and compressive strength tests are performed. As a result, the mass of the samples at the time of the test are close to those measured when the samples are unmolded (relative differences of weights are less than 1%). Compressive strength tests of cement pastes were performed at 1, 3, 7, 14, 28 and 90 days according to . Three flexural tests are performed and the six resulting halves are loaded and crushed during compression tests. Measurements of the compressive strength are reported in table 4.2.

Semi-adiabatic calorimetry tests are performed according to the standard NF EN 196-9 [START_REF] Afnor | Méthodes d'essais des ciments partie 9 : Chaleur d'hydratation -méthode semi-adiabatique[END_REF]. To avoid large temperatures, about 1000g of sand was added to the cement paste (Tab. 4.3). The heat capacity of the calorimeter is about 380 kJ/K and its released heat can be estimated by using a = 75. J/h/K, b = 0.260 J/h/K 2 in equation [START_REF] Bentz | A three-dimensional cement hydration and microstructure program. I hydration rate, heat of hydration, and chemical shrinkage[END_REF].

Experimental results

On the one hand, fillers may accelerate the hydration reactions due to their specific surface. On the other hand, the heat released per unit weight of the binder is lower if a filler is substituted to the cement since the filler does not react (except for the formation of monocarboaluminate). In the second case, the reached temperature are lower during semiadiabatic tests, a phenomenon which is largely visible on the experimental results (Fig. 4.1). The compressive strength can be underestimated by the model, in particular at early age and for high weight fractions of limestone fillers. Comparisons to measurements performed on cement pastes containing pozzolanic additions lead to similar conclusions (Fig. 4.2). Indeed, the heat of hydration at the end of semi-adiabatic calorimetry tests is correctly estimated for both water to cement ratios. But the model overestimates the reduction of compressive strength due to substitution of the cement by the fly ash. Indeed, for 50% weight substitution, the long term compressive strength is similar to that of the pure cement paste while the model expects a significant decrease.

Limitations of the present model

Unsurprisingly, the duration of the induction period is not accurately estimated by the present hydration model. Indeed, studying this period would require a precise modeling of the dissolution kinetic of species accounting for the concentration of ions in the solution and the effect of additives or organic acids.

The micromechanical model introduced above relies on the average of the Von Mises stress on the hydrate needles to define the failure of the material as in [START_REF] Pichler | Upscaling quasi-brittle strength of cement paste and mortar: A multi-scale engineering mechanics model[END_REF][START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF]. The most serious problem of this approach is that the failure criteria is symmetric in traction and compression. Hence, the failure at the local scale is not properly described since the traction strength of concrete can be more than ten times smaller than its compressive strength. Recent investigations on the failure of C-S-H and multiscale modeling were able to account for discrepancies between compression and traction [START_REF] Hlobil | Micromechanical multiscale fracture model for compressive strength of blended cement pastes[END_REF]. Nevertheless, the micromechanical model of Pichler and Hellmich correctly depicts the failure of the cement paste in case of biaxial compressive loadings [START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF], where confinement improves the strength. Even if the proposed model is able to estimate the strength of the cement paste in case of biaxial compressive loadings, the effect of confinement is not accounted for in the statistical procedure introduced herein. As a consequence, the proposed upscaling of strength must indeed be limited to the predicition of uniaxial compressive strengths for mortars. In this particular case, the stress in the cementitious matrix remains mostly uniaxial as shown on figure 2.2 and relying on an estimate of the uniaxial compressive strength of the cementitious matrix is sufficient. Otherwise, the proposed statistical procedure can be extended to multiaxial loadings, though the resulting estimate must be validated. Indeed, local stresses in the cement paste can be generated according to the proposed estimate of its distribution and compared to the multiaxial criterion as computed in [START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF]. Then, the strength of the mortar can be defined by introducing the volume fraction c f ail . However, it must be recalled that concrete is rarely linear elastic up to failure, thus limiting the ultimate scope of the proposed model.

The present homogenization scheme does not describe the nanometric scale of the C-S-H, which is a mix of the gel porosity and globules. Indeed, in the present model, the density of the C-S-H is assumed to be uniform with respect to time. On the contrary, by combining Rietveld analysis and thermogravimetry analysis, it has been shown that curing at high temperature significantly reduces the gel porosity of the C-S-H [108]. As a result, the density of the C-S-H increases from 1.91 at 5°C to 2.27 at 60 °C [108]. On the one hand, increasing the density raises the stiffness of the C-S-H. On the other hand, the capillary porosity is also increased, thus leading to a significant decrease of the strength [START_REF] Burg | The influence of casting and curing temperature on the properties of fresh and hardened concrete[END_REF]. Consequently, as the present model has been validated against various tests of samples stored at temperatures around 20°C, it would be reasonable to limit its use to concrete experiencing temperatures in the range 10°C to 30°C. In addition, NMR relaxometry [START_REF] Muller | Densification of c-s-h measured by 1h nmr relaxometry[END_REF] points out that the global density of the C-S-H gel is driven by the available precipitation space, thus implying a variable gel porosity [START_REF] Königsberger | Densification of c-s-h is mainly driven by available precipitation space, as quantified through an analytical cement hydration model based on {NMR} data[END_REF]. Indeed, the density of the saturated B-class C-S-H introduced in [START_REF] Königsberger | Densification of c-s-h is mainly driven by available precipitation space, as quantified through an analytical cement hydration model based on {NMR} data[END_REF] can be as low as 1.34g/cm 3 or higher than 1.89g/cm 3 , corresponding to porosities in the range [79%, 44%]. According to nanoindentation results on cement paste featuring w/c=0.5, the packing density triggering a percolation of the gel is about 50% [111]. It would correspond to an available precipitation space of about 51% in the model of Königsberger et al. [START_REF] Königsberger | Densification of c-s-h is mainly driven by available precipitation space, as quantified through an analytical cement hydration model based on {NMR} data[END_REF]. However, such an available precipitation space is not reached at 28 days according to the present hydration model, for a w/c of 0.5 at 20°C. As result, the authors acknowledge that combining the knowledge stemming from NMR relaxometry to a predictive micromechanical scheme accounting for evidences brought by nanoindentation measurements is not achieved by the proposed model, likely because it requires a deeper understanding of the gel structure. Different classes of C-S-H of prescribed densities and mechanical properties can also be distinguished by performing nano-indentation measurements [START_REF] Vandamme | Nanogranular packing of c-s-h at substochiometric conditions[END_REF]. But the volume fractions of each C-S-H class depends on the water to cement ratio, hydration degree and temperature history. Combining these data into a single consistent model might lead to improved estimates of the cement mechanical properties.

Lastly, the Young modulus of cement pastes at early age can be significantly underestimated and the dilution effect for large weight fraction of fly ash or limestone filler seems overestimated. Indeed, this feature can be explained by the fact that some of fillers, mineral additions and unreacted particles are fully incorporated in the solid skeleton of the cement paste, thus influencing its setting and the rapid development of the mechanical properties [START_REF] Bentz | Limestone and silica powder replacements for cement: Early-age performance[END_REF]. Numerical models [START_REF] Van Breugel | Simulation of hydration and formation of structure in hardening cement-based materials[END_REF][START_REF] Bentz | A three-dimensional cement hydration and microstructure program. I hydration rate, heat of hydration, and chemical shrinkage[END_REF][START_REF] Haecker | Modeling the linear elastic properties of portland cement paste[END_REF][START_REF] Bishnoi | µic: A new platform for modelling the hydration of cements[END_REF] naturally account for this phenomenon while most of existing micromechanical models do not consider these phases at the scale where the selfconsistent model is applied.

Conclusion

Despite its simplicity, the updated hydration model is able to provide a reasonable estimate of the hydration degree and mass of each phase of the cement paste as a function of time. Indeed, providing a couple of inputs on the binder composition, the formulation and the environmental conditions is sufficient to recover valuable estimates of the heat of hydration and masses of the various co-existing chemical species of the cement paste. Introducing molar volumes, mechanical properties and a simplified description of the morphology of the cement paste allows the use of mean-field schemes producing estimates of the compressive strength. The model of Pichler & Hellmich [START_REF] Pichler | Upscaling quasi-brittle strength of cement paste and mortar: A multi-scale engineering mechanics model[END_REF] was adopted and modified. First, introducing a finite aspect ratio to describe the hydrates at the scale of the hydrate foam improves the the estimated compressive strength at early age. Second, refining the estimate of the stress fluctuation in the cement paste at the scale of the concrete partially explains the fact that a mortar and a concrete can be weaker than its cementitious matrix. Nevertheless, the estimated compressive strength is underestimated at early age and for high weight substitution of the cement by limestone filler of pozzolanic additions.

Knowing the secant Young modulus as defined in the Eurocode and ACI 318 standards [START_REF] Afnor | Eurocode 2, calcul des structures en béton, section 3[END_REF][START_REF]Building code requirements for structural concrete[END_REF] is also required to estimate creep strains, to anticipate strains and bending deflection and to address buckling issues. Indeed, the present estimate of the elastic stiffness as a function of time and composition can prove useful to estimate the basic creep of concretes as long as loading occurs after a few days. For instance, in the solidification model [114] and in the model of Granger [START_REF] Granger | Comportement différé du béton dans les enceintes de centrales nucléaires : analyse et modélisation[END_REF], the changes of the volume fractions of phases at the scale of the cement paste is held responsible for both the increase of the elastic stiffness and the aging property of basic creep. Indeed, the model of Granger directly uses the Young modulus at the time of loading as a parameter while the solidification model refers to a function describing the volume fraction of the load-bearing solidified matter, which affects both the static Young modulus and the basic creep compliance. Hence, these parameters could be tuned depending on the composition prior to a simulation at structural scale. Nevertheless, given the limited accuracy of the present model, performing systematic experiments to measure the Young modulus is clearly recommended whenever strains become critical.

The most serious drawback of the present model is that it is limited to simple curing conditions. Indeed, a significant improvement of the understanding of effects of curing conditions could be performed if the present model were tightly coupled to a structural analysis.

Appendix A. Multiscale homogenization by mean field schemes in brief

In the present appendix, the overall stiffness of a composite material is estimated by using mean field schemes [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF][START_REF] Hill | A self-consistent mechanics of composite materials[END_REF].

Let C m being the elastic stiffness of the matrix C i , c i , a i , φ i being the stiffness, the volume fraction, the aspect ratios and orientation of the class i of inclusions. For each class i, the Eshelby's tensor S i (C m , a i , φ i ) is introduced [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF]. Closed-form expressions of the Eshelby's tensor for ellipsoidal inclusions are detailed in [START_REF] Mura | Micromechanics of Defects in Solids, Mechanics of Elastic and Inelastic Solids[END_REF]. As computed by Eshelby, the relation between the strain in the inclusion ε i and the strain ε m in the remote matrix defines the influence tensor A i (C m , C i , a i , φ i ) such that ε i = A i : ε m , where:

A i (C m , C i , a i , φ i ) = 1 + S i (C m , a i , φ i ) : C -1 m : (C i -C m ) -1 (A.1)
where 1 is the identity tensor of order 4. In the Mori-Tanaka scheme [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF], all inclusions are considered to be embedded in the matrix, experiencing the same strain ε m in a distance. Hence, the average of the strain E in the composite writes:

E = i c i A i (C m , C i , a i , φ i ) : ε m + (1 - i c i ) ε m (A.2)
Similarly, the overall stress Σ writes:

Σ = i c i C i : A i (C m , C i , a i , φ i ) : ε m + (1 - i c i ) C m : ε m (A.3)
Hence, the overall stiffness estimated according to the Mori-Tanaka scheme C M T is:

C M T = i c i C i : A i (C m , C i , a i , φ i ) + (1 - i c i ) C m : i c i A i (C m , C i , a i , φ i ) + (1 - i c i ) 1 -1 (A.

4) 31

If an overall stress Σ is applied to the composite material, the average stress in the matrix σ m writes:

σ m = C m : i c i A i (C m , C i , a i , φ i ) + (1 - i c i ) 1 -1 : C -1
M T : Σ (A.5)

An alternative to the Mori-Tanaka scheme is the self-consistent scheme [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF], in which all phases are assumed to be embedded in a matrix as stiff as the overall composite itself. In this case, the overall stiffness C SC must satisfy the following implicit equation:

C SC = i c i C i : A i (C SC , C i , a i , φ i ) : i c i A i (C SC , C i , a i , φ i ) -1 (A.6)
These schemes can be combined to design multiscale homogenization procedures, thus delivering estimates of the overall stiffness C o as well as estimates of the mean stress in each microscopic phases. Lastly, the quadratic mean of the Von Mises stress in the phase p can be estimated according to Kreher's lemma [START_REF] Kreher | Residual stresses and stored elastic energy of composites and polycrystals[END_REF] as performed by Pichler and Hellmich [START_REF] Pichler | Upscaling quasi-brittle strength of cement paste and mortar: A multi-scale engineering mechanics model[END_REF] as: 3 δ ij δ kl + 1 2 δ ik δ jl + 1 2 δ il δ jk )e i ⊗ e j ⊗ e k ⊗ e l is the deviatoric fourth order unit tensor, G p is the shear modulus of the phase p and c p it's volume fraction.

< σ 2 eq > p = 3

Figure 1 . 1 :

 11 Figure 1.1: Evolutions of the volume fractions of the main constitutive phases of sealed cement pastes as functions of the hydration degree of the cement α. Left: a plain cement paste w/c = 0.5, (Tab. 4.1), right: a cement blended with 10% silica fume, w/(c + sf ) = 0.35 (Tab. 4.1).

Figure 2 . 1 :

 21 Figure 2.1: The different scales considered for the homogenization of cementitious materials. The phases and the homognization schemes are detailled at each scale.

  mean-field scheme Von Mises, 3D numerical simulation Components, 3D numerical simulation

Figure 2 . 2 :Figure 3 . 1 :

 2231 Figure 2.2: Left: local values of the Von Mises stress divided by Σ XX within a periodic unit cell as predicted by 3D numerical simulations. Right: Comparisons between the distribution of the Von Mises stress in the matrix estimated by 3D numerical simulations and the proposed mean field method. Distributions of the local components of the stress estimated by 3D numerical simulations are also reported.

Figure 3 . 2 :

 32 Figure 3.2: Results of XRD/Rietveld analysis [100] are compared to corresponding estimates. Samples were cured at 20°C for one day and then stored at different temperatures.

Figure 3 . 3 :Figure 3 . 3 :

 3333 Figure 3.3: Results of XRD/Rietveld analysis [102] are compared to corresponding estimates.

45 Figure 3 . 4 : 55 Figure 3 . 5 :

 45345535 Figure 3.4: Results of isothermal calorimetry and thermogravimetry [103] are compared to corresponding estimates.

  Termkhajornkit et. al, C3S Termkhajornkit et. al, C2S Pichler et al. 2012, OCP Lam et al. 2000, OCP model

Figure 3 . 6 :

 36 Figure 3.6: Results of compressive strength tests of Termkhajornkit et. al. [21] on pure C3S/C2S paste are compared to corresponding estimates. Water to cement ratios are respectively 0.45 and 0.43. Tests are performed on cylinders of diameter 7.4mm and height 22.7mm.Results of Lam et. al[START_REF] Lam | Degree of hydration and gel/space ratio of high-volume fly ash/cement systems[END_REF] on cubes of side 70.7mm and Pichler et. al[START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF] on cylinders of diameter 29mm and height 58mm are added.

Figure 3 . 7 :

 37 Figure 3.7: Results of compressive strength tests of Lerch and Ford [88] are compared to corresponding estimates. w/c = 0.53, a/c = 2.77. Tests are performed on 2-inch cubes according to ASTM C109-40T.

8 Figure 3 . 8 :Figure 3 . 9 :

 83839 Figure 3.8: Results of compressive strength tests of Taplin[START_REF] Taplin | A method for following the hydration reaction in portland cement paste[END_REF] on cement pastes featuring different w/c ratios are compared to corresponding estimates. The results of Taplin had to be scaled because he published the measured ultimate force instead of the compressive strength. The loaded area of the rectangular prismatic sample reported in[START_REF] Pichler | Effect of gel-space ratio and microstructure on strength of hydrating cementitious materials: An engineering micromechanics approach[END_REF] is 0.373×0.5 square inch, the height being 0.5 inch.

  The Young modulus as a function of the compres-Comparison between the estimated Young Moduli and the measured one. Comparison between the Young moduli estimated by the ACI209 standards and the measured ones.

  Comparison between the Young moduli estimated by the Eurocode standards and the measured ones.

Figure

  Figure 3.10: Comparisons related to the Japanese database [104]. Lines correspond to equality between estimates and measured results and 15% relative errors.

32 Figure 4 . 1 :

 3241 w/c = 0.Effect of limestone fillers on the hydration and hardening of the cement paste. For calorimetric tests, the measured temperature and released energy (thick lines, crosses) are compared to the estimated one (solid lines). The same thing is performed for the compressive strength on the right.

Figure 4 . 2 :

 42 Figure 4.2: Effect of silica fume and fly ash on the hydration and hardening of the cement paste. For calorimetric tests, the measured temperature and released energy (thick lines, crosses) are compared to the estimated one (solid lines). The same thing is performed for the compressive strength on the right.

2 <

 2 σ : K : σ > p = where K = (-1

Table 1 .

 1 1: The constitutive phases of the cement paste. Cement notation are used except for calcite : C=CaO; S=SiO 2 ; A=Al 2 O 3 ; F=Fe 2 O 3 ; S=SO 3 ; C=CO 2 . Molar masses are computed according to the python module pyisocalc, based on the data gathered by IUPAC[118].

	name	formula	molar mass density molar volume enthalpy of formation	C p
			g/mol	g/cm 3	cm 3 /mol	kJ/mol, @ 25°C	J/mol/K, @ 25°C
	Cement Bogue composition
	Alite	C 3 S	228.335	3.15	72.5	(-2929.2)	171.88
	Belite	C 2 S	172.258	3.27	52.7	(-2307.5, β)	(128.78, β)
	Tricalcium Aluminate	C 3 A	270.194	3.03	89.2	-3587.8	209.87
	Tetracalcium aluminoferrite	C 4 AF	485.96	3.71	131.0	-5090.3	396
	gypsum	C SH 2	172.172	2.31	74.5	-2022.6	186.02
	C-S-H	C 1.7 SH 4	227.496	2.1	108.3	(-3283)	(297)
	Portlandite	CH	74.093	2.26	32.8	-986.1	87.49
	Ettringite	C 6 A SH 32	1255.108	1.78	705.1	-17539	2174
	MonosulphoAluminate	C 4 A SH 12	622.519	2.02	308	-8778	942
	iron(III) oxide-hydroxide	FH 3	213.734	2.2	97.2	(2×-832.616)	(2×101.671)
	Fe-Si hydrogarnet	C 6 AFS 2.18 H 19	1071.43	2.18	491.5	-14088	1682
	hydrogarnet	C 3 AH 6	378.285	2.52	150.1	-5548	459
	water	H	18.015	1.0	18.0	-285.83	75.291
	amorphous silicate	S	60.103	2.22	27.1	-903.49	44.4
	C-S-Hp	C 1.1 SH 3.9	192.048	1.9	101.1	(-2841)	(297)
	calcite	C C	100.087	2.71	36.9	-1128.79	81.88
	calcium monocarboaluminate	C 4 A CH 11	568.449	2.17	262.0	-8250	881
	stochiometry of reaction	volume change balance capillary void enthalpy of reaction ∆rH @ 25°C change of heat capacity Cp,r
			cm 3 /mol(C3S/C2S/C3A/C4AF)		kJ/mol	J/mol/K
	C3S+ 5.3H → C1.7SH4 + 1.3CH	-17	-78		-121	-160
	C2S+ 4.3H → C1.7SH4 + 0.3CH	-12	-65		-42	-129
	C3A+3C SH2+26H → C6A S3H32	-76	-392		-452	-551
	C3A+0.5C6A S3H32+2H → 1.5C4A SH12	-15	-21		-238	-34
	C3A+6H → C3AH6		-47	-61		-245	-203
	C4AF+3C SH2+30H → C6A S3H32+CH+FH3	-60	-480		-438	-748
	C4AF+0.5C6A S3H32+6H → 1.5C4A SH12+CH+FH3	1	-109		-224	-231
	C4AF+10H → C3AH6+CH+FH3	-31	-149		-232	-399
	C4AF+2.18 C3S +23.54H → C6AFS2.18H19+4.54 CH	-73	-351		-360	-464
	C4AF+2.18 C2S +21.36H → C6AFS2.18H19+2.36 CH	-62	-323		-189	-396
	1.1CH+S+2.8H → C1.1SH3.9	-13	-38		-52	-54
	C C+C3A+11H→C4A CH11	-62	-136		-389	-239
	A+5CH+25H+C SH2→ C6A S3H32	-9	-441		-1777	-411
	A+3CH+0.5C6A S3H32→1.5C4A SH12+H	4	14		-62	60
	A+3CH+3H→C3AH6		-28	-26		-69	-108
	A+3CH+C C+8H→C4A CH11	-43	-101		-213	-145

Table 1 . 3

 13

		-1 1.5	0.5	1.0	0.37
	n i,1	0.7	1.0	0.85	0.7
	k i,2 , day -1 0.05 0.006 0.04 0.015
	k i,3 , day -1 1.1	0.2	1.0	0.4
	n i,3	3.3	5.0	3.2	3.7

: Parameters of Parrot and Killoh

Table 2 .

 2 2 GPa) is accounted for to estimate the dynamic 1: The stiffnesses of the different phases of the C-S-H foam and the cement paste. The Young modulus E and the Poisson's ratio ν of each phase are given.

	i	E, GPa	ν	ref.
	HD-C-S-H	29.4	0.24	[15]
	LD-C-S-H	21.7	0.24	[15]
	C 3 A,C 3 S,C 4 AF,C 2 S	130.	0.3	[25]
	C SH 2	45.7	0.33 [119, 120]
	S,A	72.8	0.167	[105]
	C C	83.8	0.31	[121]
	CH	42.3	0.324	[122]
	C 6 A SH 32	22.4	0.25	[105]
	C 3 AH 6	22.4	0.25	[105]
	C 6 AFS 2.18 H 19	22.4	0.25	[105]
	FH 3	22.4	0.25	[105]
	C 4 A CH 11	42.3	0.324	[105]
	C 4 A SH 12	42.3	0.324	[105]
	i	E, GPa	ν
	fillers		65	0.2
	bauxite		150	0.2
	andesite, basalt	80	0.25
	sand		70	0.07
	quartzite		70	0.1
	limestone		60	0.15
	sandstone		45	0.15
	other		65	0.15

Table 2 .

 2

2:

The default stiffnesses of the different inert phases involved at the different scales. The Young modulus E and the Poisson's ratio ν of each phase are given.

Table 3 .

 3 100]. 1: Parameters of cements considered in the present study.

	Name	C 3 S C 2 S C 3 A C 4 AF gypsum	CaCO 3 Blaine source
	A	68 25.3 3.6		3.9			470	[101]
	B	49.1 25.5 5.8	10.7 4.3			310	XRD-Rietveld
	C	64.5 16.2 6.7	8.7	3.3			200	p21, Blaine from
	OPCN OPCS	71.6 10.9 3.7 65.6 16.4 8	10.7 3.1 7.4 2.6			d50 p106 using [123] 376.5 [100] 359.8
	OPC CRC	52.5 18.2 67.6 13.1 2.7 5	10.9 0.4+ 2.2C S 8.9 2.4C SH 0.5 +	2.5 0.9	380 350	[102]
						1.2C	S
	OPC I	53.7 18	9.1	7.6	5.07C SH 0.5		429
	Fly ash		S=57.4, A=18.2, C S=1.5, activity=0.4	380	[103]
	SF			S=94.5, activity=0.9		(2000)
	ca	58	15	7.7	8.5	4.2C S		397	[31]
	cv a		S=49.1, A=26.5, C S=0.9, activity=0.6	(400)
	cv b		S=54.4, A=25, C S=0.2, activity=0.6	(400)
	cv c		S=55.1, A=29.7, C S=0.7, activity=0.6	(400)
	cv d		S=49.2, A=35.6, C S=0.9, activity=0.6	(400)
	cv e		S=45.9, A=23.5, C S=0.9, activity=0.6	(400)
	fs a→d			S=90.9→92, activity=0.9		(2000)
	c11	50	22	12.1	7.2	2.7C S		343.6
	c11T	51	21	12.1	7.1	2.6C S		364
	c12	45	28	12.6	7.3	2.7C S		327.6
	c12T	46	27	12.5	7.1	2.7C S		315.2
	c13	50	26	10.1	6.5	2.8C S		342.8
	c14	42.5 32	8.2	9.2	2.9C S		342.4
	c15	64.5 10	12.1	7.5	3.2C S		322.9
	c16	53.5 21	7.5	10.7 2.9C S		326.1
	c16T	52.5 22	7.9	10.4 3.1C S		311.4
	c17	52	23	10.4	9.3	2.9C S		398.5	[88], Bogue
	c18	44.5 28	13.2	6.8	3.1C S		326.8	estimated from
	c18T	44 28.5 13.2	6.7	2.9C S		375.6	chemical
	c21	40	41	6.4	9.7	2.1C S		289.1	composition
	c21T	38	40	6.6	9.7	2.2C S		298.3
	c22	41.5 33.5 6.6	11.7 2.4C S		306.5
	c23	51	24	3.7	16.6 2.6C S		310.9
	c24	41	29	5.4	14.8 3C S			369.7
	c25	34	39	4.7	14.9 3.2C S		328.7
	c31	56	17	10.8	6.4	3.8C S		579.5
	c33	60	13	10.4	7.7	3.9C S		527.2
	c33T	57	16	10.4	7.3	3.7C S		516.9
	c34	64 10.5 5.7	10.1 2.9C S		496.9
	c41	20	51	4.5	15.2 3.4C S		367.9
	c42	27	55	3.5	8.2	2.6C S		350.1
	c43	25	48	6.2	13.8 3.6C S		384.6
	c43A	29	52	5.3	9.3	3.2C S		363.3
	c51	41	39	3.7	10	2.4C S		348.3
		41.4 34	9.8	7.5	5			312	[87], reported by
									[63]
	H	64.4 8.9	4.8	9.1	1.8+2.1C SH 0.5 + 1.7C S	360	[105]
	D	71.2 9.5	8.4	1.3	1.4+3.4C S	4.5	400

Table 3 .

 3 2: Comparisons related to the Japanese database[104]

	present model	-14%	19%	712
	Young modulus, present model	2%	12%	675
	Young modulus, ACI209	-6%	12%	634
	Young modulus, Eurocode	9%	18%	634

Table 4 .

 4 3.10: Comparisons related to the Japanese database[104]. Lines correspond to equality between estimates and measured results and 15% relative errors. 1: The Bogue composition of the clinker and the composition of the cement.

	clinker	96%
	C 3 S	65
	C 2 S	11
	C 3 A	11
	C 4 AF	8
	CaCO 3	3.5%
		100%
	gypsum	4.6
	Blaine fineness 380 m 2 /kg
	density	3.12 g/cm 3
	4. Additional experiments on blended cement pastes
	4.1. Raw materials, formulations and procedures

Table 4 .

 4 2: Measurements of the compressive strength f c , MPa. Sample standard deviations

	s =	1 N -1	(f Name	binder(g) dry sand(g) water(g)
			w/b=0.5
			C100	371	1113	196
			C95L05	503	947	257
			C90L10	517	924	252
			C85L15*	516	905	258
			C70L30	566	870	291
			C95SF05	510	971	263
			C85SF15	536	932	277
			C75FA525	546	873	280
			C50FA50	580	773	297
			w/b=0.32
			C100	566	1131	191
			C95L05	616	1173	208
			C90L10	629	1143	211
			C85L15	645	1121	216
			C70L30	686	1055	229
			C95SF05	567	1080	191
			C85SF15	593	1030	199
			C75FA25	601	963	202

c -fc ) 2 are reported in parenthesis. *: 2×2 compressive tests only.

Table 4 .

 4

3: Mix proportions for semi adiabatic calorimetry. The water to binder ratio accounts for the 0.9% absorption of the dry sand. *: The water to binder ratio is 0.48.

(λ) (1 -c f ail ) (30)The function stats.ncx2.ppf of scipy[START_REF] Oliphant | scipy.stats.ncx2[END_REF] is applied to evaluate the inverse of this repartition function. Again, it must be noticed that the value of √ V varies directly with the magnitude of the overall uniaxial stress Σ, the ratio β = Σ/ √ V being evaluated at the same time as λ. Finally, the compressive strength of mortar f c,m writes:f c,m = β V c(31)