0 / (1 21 0 3 1 4 1 1 0 (

Thanks to their oxide layer, aluminium alloys are remarkable for their ability to resist corrosion. However, in welding, this protective layer acts as a barrier which must be broken in order to succeed in the thermomechanical joining of aluminium.

The chosen alloy (6082-T6 or AlSi1MgMn) has been subjected to various deformation path. The first of them consists in the channel-die (plane strain) compression of two cuboids, one above the other. Considering the configuration of the test, the surface size between the two samples rises, so that the fragmentation of the oxide layer creates welding bonds. However, the friction effects in the channel lead to a heterogeneous deformation, so that the contact surface undergoes different behaviors: a microscopic study then shows that the welds appear in areas with significant shear.

Channel-die and uniaxial compressions of beveled samples confirm that more significantly than the global deformation, the shear strain is the most active phenomenon for achieving an effective thermomechanical joining.

Another approach is the cumulative deformation as a result of a cyclic load: a tube is cut through its section and undergoes both a compression and cyclic torsion load. The contact surface between the two semi-tubes is under a shear behavior and the combination between plastic deformation and local heating leads to a fragmentation of the oxide layer: all this factors allow the thermomechanical joining of aluminium alloys.

Weight reduction and longer working life of structures are major challenges in the competition between light alloys and composites for new generations of aircrafts or cars. To this end, new grades of aluminum alloys are under development, either to reduce their density with equivalent mechanical properties or to increase the mechanical strength, allowing to reduce thickness.

Alongside the selection of materials for specific characteristics, another approach is mass gains via new assembly techniques. Indeed, compared to conventional methods (rivets, bolts, etc.), some welding methods do not require any filling metal (friction stir welding, pressure welding, ultrasonic welding, etc.) and thus reduce weight and manufacturing costs of the structure up to 25%.

During plastic deformation of metallic materials, the hydrostatic pressure has an essential impact on the local and thermomechanical joining in general and on the mechanism of healing of metallic materials in particular (closure of cavities by local welding). These processes of metal junction will be studied from two approaches, both mechanical (the deviatory and spherical load allows the inner walls to come closer in a global and local scale (surface roughness)) and chemical (properties of oxide layers and their evolution during the local plastic deformation). The underlying notion is the fragmentation of the oxide layer during deformation, implying the creation of "fresh surface": this non-oxide material is physically unstable and therefore naturally bonds with its closest neighbor.

The chosen material is the aluminium alloy EN AW-6082-T6, also referred as AlSi1MgMn, whose chemical composition is described in Tab. 1. The addition of manganese controls the grain structure which in turn results in a stronger alloy, thus replacing 6061 in many structural applications (Tab.2). Fatigue properties and analysis could be seen thanks to Borrego [START_REF] Borrego | Analysis of low cycle fatigue in AlMgSi aluminium alloys[END_REF] Table 1 -Chemical composition of AW-6082-T6 aluminium alloy (wt%) The first deformation path investigated is the plane-strain compression, using a channel-die device which is the most relevant for simulation of rolling of metallic alloys. Since the width w 0 remains constant during the deformation, the main singularity of the test with respect to other compression tests is that only two components (out of five, assuming we are dealing with incompressible materials) of the strain rate tensor are not prescribed by the test: the shear strain rates 12 ε and 13 ε (where x 1 , x 2 and x 3 denote respectively the Rolling, Transverse and Normal Directions, Fig. 1) [START_REF] Ch | Propriétés générales de l'aluminium et de ses alliages[END_REF].

Fig. 1 -Schematic representations of the specimens used in channel-die compression

Here, the objective of the tests is the study of thermomechanical joining of metallic metals. Thus, the compression is made on two cuboids, set one above the other (Fig. 1). Considering the configuration of the test, the surface size between the two samples rises, so that the fragmentation of the oxide layer leads up to the emergence of fresh surface, creating welding bonds.

Various parameters have been studied in order to decide on their influence on the welding. Of course, the first parameter is the set strain and then the objective was to determine, for a given set of parameters, the stain necessary to obtain a junction of the two samples. Here the equivalent von Mises strain was up to 1 for a sustainable assembly. The influence of temperature (room temperature, ≈20 ° C or hot working, 350 ° C), the strain rate (at hot temperature only) and the surface roughness of the samples at the interface (from raw to mirror finish) were investigated. As one might have expected, at hot temperature, the diffusion and the reactivity of the material make easier the junction and the joinability threshold falls significantly from 0.92 to 0.67. In contrast, the strain rate does not appear to be an influential parameter, even if the compression should be fast enough to avoid excessive oxygen diffusion at the interface, which could lead to an oxidation of the appearing fresh surface. Finally, the roughness can also reduce the strain required to achieve a thermomechanical joining, since a mirror finished surface allows the local walls to come closer.

The first macroscopic observation highlighted that the local crystal orientations led to local heterogeneous deformation, so that the initial surface roughness (mirror finish) is not retained. However, similar tests on single crystals (Al0.3%Mn Cube and Al0.1%Mn Goss) didn't lead to sustainable joining. See also [START_REF] Vanderesse | Channel-die compression at high temperature[END_REF] about the channel-die compression of crystals.

The second macroscopic observation deals with the final deformation of the sample: as in uniaxial compression, the friction effects stands in to the upper and lower surfaces of the channel, but here the friction also extends to the lateral faces of the sample, leading to a heterogeneous deformation (Fig. 2). After deformation, when the assembly is sustainable, a microscopic study has been carried out. Coupled with a FEM simulation on Abaqus ® , it then showed that the welds appear in zones with significant shear values and gradient (Fig. 2). Using an EBSD analysis, it was also showed a static recristallisation in welded zones. In order to confirm this result, the compression of beveled samples (angle of 10°) has been carried out. It showed that more significantly than the global deformation, it's the local behavior, especially the shear strain, which is the most active phenomenon for achieving an effective thermomechanical joining.

To conclude, the thermomechanical joining of aluminium in plane strain compression succeed and showed that, if the global deformation remains the first factor for achieving a joining, especially in terms of surface growth, the local behavior (crystallographic orientations and shear) seems to be the most active phenomenon for achieving a sustainable joining. To confirm the predominance of the local behavior on the global deformation, a second method has been set up: the cumulative shear deformation as a result of a cyclic load. The test specimen, made of the same aluminium alloy (6082-T6), is a tube which is cut through its section (Fig. 3). The testing machine used is a biaxial servo-hydraulic of traction-compression and alternating torsion. This equipment is endowed by an axis alignment system and fluid hydrostatic bearings. Here, while the upper part remains static, the lower part is free in axial displacement (compression or traction) and/or in rotation. The height of the specimen is 150 mm and the effective area, where the theory of circular tube is applied, is 30 mm high and 3 mm thick, the outside diameter being 30 mm. Given the configuration of the test, the interface is under a pure shear behavior during the rotation.

For an tube of outer and inner radius R e and R i and length L >>R e under a compression force F z and a torque T z , the strain and stress fields describing the test are given by the equation:

2 2 2 2 4 4 2 ( ) 3 2 ( ) z zz e i ext VM z zz ext e z z e i F R R R T R R θ θ σ π σ σ σ σ π  =  -  = +   =  -  (1) 1 2 e z R L θ α ε = (2)
Note that in our case, a realistic value for the length L -and so ε θz -is hard to define since the non-effective area must also be considered and since the sliding at the interface may cause only a partial drive of the upper part by lower one.

To find a window of joinability, many parameters have to be set: the compression force and its type of solicitation (monotonic or cyclic, with a variable mean value or not), the torsion angle amplitude and its period (thus defining the speed of rotation), the number of cycles, the surface polishing at the interface. In this n-dimension space, the test being rather complex, it has been necessary to decide on configuration which can lead to a sustainable joining.

Here, both the compression force and the defined angle are set so that the test specimen remains near the yield strength σ Y at each cycle: instead of a large deformation as in the first method, the underlying idea is here to consider very small plastic deformation cumulated on a lot of cycles. The Section τ Z θ R lower part is firstly loaded under compression and retained during the entire test in under for the walls of the interface to get closer. Secondly, the lower part is under a cyclic torsion load.

In our configuration, the compression stress σ zz remains elastic while the force F z is below 66 kN and the torsion strain ε θz has an elastic behavior as long as the angle α doesn't exceed 1,214° (to obtain this value, the sliding has been neglected and the non-effective area has been considered up to 1/3, leading to L=57 mm). It also important to note that the test specimen isn't heated and is made at room temperature: the only variation of temperature is the consequence of the local heating which occurs as a result of friction between the two parts.

Firstly, the tests have been conducted with a controlled surface finishing (26µm) in order to be sure of the repeatability of the test. The compression force was up to 50 or 60 kN (which is close to 66kN, value for the entry in the elastic-plastic domain). The variation of angle were chosen linear and with an amplitude from 1.4 to 7°, but for angle of 5° or higher, the interface acted as in a dry machining, with production of metallic chips. To prevent from this inconvenient, both the angle and the compressive force have been decreased. Hence the input signal of the torsion angle α has been set sinusoidal, with α 0 the amplitude and T the period:

0 ( ) sin(2 / ) t t T α α π = ⋅ (3)
Eventually, the parameters to achieve a sustainable joining are: F z =30 kN, α 0 =1°, T=0.6 s and, in order to improve the reactivity of the surfaces at the interface, a mirror finish (1 µm) has been set: thanks to this surface finish, the width of the oxide layer is perfectly controlled and regular, thus allowing a better closure of the inner walls. As regards the duration, the cyclic load lasted a little more than 7h, i.e. 42000 cycles, but despite of many measures during the test, nothing indicates a precise moment when the joining, even partial, has been effective. A characterization of the joining is in progress, both with nondestructive testing (tomography X), destructive testing (classical traction or compression tests on samples taken on either side of the original specimen, compared with full sample) and microscope analysis (optical, SEM, EBSD). The goal of those results is to determine whether the joining is total or partial and then if the junction is strong. Moreover, some other tests are in progress too in order to measure the displacements of the two samples thanks to stereo correlation device.

Nevertheless, the combination of shear, friction and local heating at the interface leads to a fragmentation of the oxide layer, thus allowing a thermomechanical joining.

The purpose of this paper was to study the effects of the shear on the thermomechanical joining of aluminium alloys. The first method -plane strain compression -lead to the conclusion that, in order to achieve a sustainable joining, an important deformation rate is needed, but the predominant phenomenon is the shear: a comparison between the observation on a optical microscope and a FEM analysis lead to the conclusion that the joining were effective in the zones with important shear stress. In order to valid this interpretation, a second protocol has been conveyed: a torsion test specimen has been cut through its section and undergoes a compression followed by a cyclic torsion with small angles during a great number of cycles. In this condition, the interface is under a pure shear behavior and a sustainable thermomechanical joining happened. Finally, this paper showed the importance of shear for breaking up the oxide layer, leading to a thermomechanical joining of aluminium alloys.

Fig. 2 -

 2 Fig. 2 -Deformed shape, shear stress σ 13 distribution and welded zones.

Fig. 3 -

 3 Fig. 3 -Description of the compression-torsion test.

Fig. 4 -

 4 Fig. 4 -Real-time data acquisition -Von Mises equivalent stress (zoom on a short period)

Table 2 -

 2 Mechanical and physical properties of AW-6082-T6 aluminium alloy

This work is supported by the "MACODEV" research cluster of the Région Rhône-Alpes, France, through a collaboration between the PECM laboratory of École des Mines de Saint-Étienne and the 3S-R laboratory of Grenoble-INP.


Frédéric Péralès

Adrien Socié

Yann Monerie

Frédéric Perales

Effects of the microstructural uncertainties on the poroelastic and the diffusive properties of mortar

Keywords: microporomechanics, homogenization, global sensitivity analysis, interfacial transition zone, mortar, Sobol variance decomposition

Introduction

The behavior of concrete is a topic of great concern in the context of the durability of civil engineering structures. The solicitations induced by the environment lead to the mineralogical evolution and delayed deformation of the concrete. The kinetic and the amplitude of the degradation depend on the properties and mineralogy of the mature cementitious material. For example, several works bring out the effect of the initial water-cement ratio on the concrete delayed deformation and chemical reactions, such as for sulfate attack [START_REF] El-Hachem | Multi-criteria analysis of the mechanism of degradation of Portland cement based mortars exposed to external sulphate attack[END_REF][START_REF] Planel | Long-term performance of cement paste during combined calcium leachingśsulfate attack: kinetics and size effect[END_REF][START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF], calcium leaching [START_REF] Heukamp | Chemomechanics of Calcium Leaching of Cement-Based Materials at Different Scales: The Role of CH-Dissolution and C-S-H Degradation on Strength and Durability Performance of Materials and Structures[END_REF][START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF] or drying shrinkage [START_REF] Tognevi | Modélisation multi-échelle et simulation du comportement thermo-hydromécanique du béton avec représentation explicite de la fissuration[END_REF]. The water-cement ratio affects the cement paste composition and its porosity and thus modifies the overall poroelastic and diffusive properties of the material.

To consider the effect of the morphological multi-scale properties on the mechanical and chemical responses of concrete under different loading, analytical and numerical models have been developed [START_REF] Bary | Simplified coupled chemo-mechanical modeling of cement pastes behavior subjected to combined leaching and external sulfate attack[END_REF][START_REF] Bernard | Predicting the evolution of mechanical and diffusivity properties of cement pastes and mortars for various hydration degrees ś A numerical simulation investigation[END_REF][START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF]). The material properties of mature concrete based on multiscale models are performed in two steps: first, the microstructure properties, such as volume fraction of inclusions or phases' assemblage, are estimated, then, a micromechanical model is employed [START_REF] Bary | Simplified coupled chemo-mechanical modeling of cement pastes behavior subjected to combined leaching and external sulfate attack[END_REF][START_REF] Bernard | Predicting the evolution of mechanical and diffusivity properties of cement pastes and mortars for various hydration degrees ś A numerical simulation investigation[END_REF][START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF][START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF][START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF][START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF]. At cement scale, the volume fraction of hydrates and the capillary porosity are estimated by a hydration model. [START_REF] Bary | Simplified coupled chemo-mechanical modeling of cement pastes behavior subjected to combined leaching and external sulfate attack[END_REF][START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF][START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF][START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF][START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF] and at mortar or concrete scale, an additional function is considered to estimate the volume fraction of the so-called Interfacial Transition Zone (ITZ) between the matrix and the inclusions [START_REF] Garboczi | Analytical formulas for interfacial transition zone properties[END_REF][START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF]. Each model depends on a wide variety of parameters (size of the inclusions, clinker composition...) that affect the estimated volume fraction and thus the macroscopic property [START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF][START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF][START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF]. Furthermore, multi-scale models are function of the material properties of each considered phase. These properties are dependent on experimental measurement [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] Haecker | Modeling the linear elastic properties of Portland cement paste[END_REF], on post-processing used (i.e. inverse analysis) [START_REF] Hashin | An inverse method to determine the elastic properties of the interphase between the aggregate and the cement paste[END_REF][START_REF] Seigneur | Transport properties evolution of cement model system under degradation -Incorporation of a pore-scale approach into reactive transport modelling[END_REF] or molecular simulations results [START_REF] Jelea | On the Laplace-Young equation applied to spherical fluid inclusions in solid matrices[END_REF][START_REF] Hajilar | Nano-scale investigation of elastic properties of hydrated cement paste constituents using molecular dynamics simulations[END_REF]Honorio et al. 2020b).

Therefore, a large number of parameters are required and their inherent uncertainty influences the overall predicted properties. Uncertainty and sensitivity analysis have already been carried out to quantify the stochastic variation due to hydration models and material properties from nanoscale to macroscale (concrete) [START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF]Honorio et al. 2020a;[START_REF] Sudret | Sensitivity analysis of nested multiphysics models using polynomial chaos expansions[END_REF][START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF]. [START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF] reveal that the uncertainty due to the input parameters is magnified during the upscaling processes and uncertainties of inclusion properties take a major role in the overall elastic variation. [START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF] and [START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF] determine the overall uncertainties of the hydration processes through the effects of the parameters. [START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF] show the effect of the hydration model uncertainties increases with the number of input parameters. [START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF] and [START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF] highlight that the elastic parameters have the most significant impact on the variability of the homogenized poroelastic properties. These works mostly focus on the elastic properties except [START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF] who have studied the poroelastic properties but only at the cement scale. Honorio et al. (2020a) have studied the variability of the response obtained with an analytical homogenization model in order to obtain the concrete's electrical properties. The multiscale model gives the electrical value from the C-S-H to the concrete scale. The electrical conductivity homogenization scheme is similar to the one of the diffusion and following diffusion assumption, the electrical conductivity only occurs in the pore solution. The propagation of the Monte-Carlo simulation applied to a micromechanical model led to the study of the impact of the uncertainties across each scale induced by the pore conductivity variability.

The works previously mentioned are dedicated to the uncertainties of the analytical homogenization model applied to the estimation of the poroelastic or conductivity properties. The model developed in this study allows to estimate the poroelastic and diffusive properties of mortar used to simulate the chemo-poromechanical behavior of concrete submitted to chemical attacks [START_REF] Socié | Chemo-poro-mechanical modeling of cementitious materials (diffusion-precipitation-cracking)[END_REF][START_REF] Socié | Multibody approach for reactive transport modeling in discontinuous-heterogeneous porous media[END_REF]. The work mainly focuses on two aspects: the effect of the ITZ material properties and volume fraction, and the effect of the uncertainties relative to the clinker composition. Note that, to our knowledge, theses aspects have never been studied in the literature before. The first point has been driven by the lack of information relative to the interfacial zone even though multiscale models consider its effects [START_REF] Patel | Diffusivity of saturated ordinary Portland cement-based materials: A critical review of experimental and analytical modelling approaches[END_REF]). Indeed, as described in (Honorio et al. 2020a), even though the ITZ plays a major role in the efficient mortar properties, and multiscale approaches should be taken into account [START_REF] Hashin | An inverse method to determine the elastic properties of the interphase between the aggregate and the cement paste[END_REF][START_REF] Heukamp | Chemomechanics of Calcium Leaching of Cement-Based Materials at Different Scales: The Role of CH-Dissolution and C-S-H Degradation on Strength and Durability Performance of Materials and Structures[END_REF][START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF][START_REF] Sun | Modeling the elastic properties of concrete composites: Experiment, differential effective medium theory, and numerical simulation[END_REF]), the challenge rests on knowing its volume fraction and its composition. The second point is based on the study of [START_REF] Stutzman | Uncertainty in Bogue-calculated phase composition of hydraulic cements[END_REF] that shows that the Bogue constant [START_REF] Bogue | Calculation of the Compounds in Portland Cement[END_REF][START_REF] Taylor | Modification of the Bogue calculation[END_REF]) introduces uncertainties that can have a significant effect on the cement paste microstructure estimated by hydration models. A part of the study focuses on the impact of these uncertainties instead of the uncertainties relative to the hydration model such as developed in [START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF]Honorio et al. 2020a;[START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF]). The study presented in this article has been also extended to the main uncertainties relative to the hydrate's properties. Finally, the identification of the main contributor to the output variation and their interactions has been carried out by a sensitivity analysis.

This paper begins with a description of the microporomechanics model (Section 2). Section 3 is dedicated to its validation. The impact of the main variable in the homogenized properties is underlined. The uncertainties of the material properties and microstructure parameters are studied in Section 4. At first, each uncertainty is studied separately to exhibit the most predominant parameters. The study highlights the main effect of the cement paste matrix, C-S-H, and ITZ to respectively the cement paste and mortar variation. Secondly, the global uncertainties are presented for three mortars distinguished by their water-cement ratios: 0.3, 0.4 and 0.5. The main results show that the uncertainty increases with the upscaling and the predominant impacts of the ITZ characteristic on the chemo-mechanical mortar properties. A Sobol variance decomposition is studied in Section 5 to highlight the impact of the main input variable and their interactions.

Microporomechanical model

The multi-scale model permits the estimation of the poroelastic parameters and the diffusion coefficient of the chemo-poromechanical model developed in [START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF][START_REF] Socié | Chemo-poro-mechanical modeling of cementitious materials (diffusion-precipitation-cracking)[END_REF]. The model is dedicated to the study of geomaterial expansion due to the strong precipitation of a solid inside the porous medium. The chemical software describes the mineralogy evolution and the poromechanical model estimates the swelling by the volume fraction of the main precipitated solid, denoted 𝜑 ms . The porous medium is described with an isotropic elastic poromechanical model, where the pressure depends on the volume fraction of the main precipitated solid and the strain tensor [START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF][START_REF] Socié | Chemo-poro-mechanical modeling of cementitious materials (diffusion-precipitation-cracking)[END_REF]:

𝝈 = C : 𝜺 -𝑏𝑃 𝑰 𝑃 = 𝑁 ⟨𝜑 ms -⟨𝜑 0 ms + 𝑏 tr 𝜺⟩ + ⟩ + (1)
where 𝝈 is the Cauchy stress tensor [Pa], C is the fourth order stiffness tensor [Pa], 𝜺 is the linearized strain tensor, 𝑃 is the pore pressure [Pa], 𝑰 is the identity second order tensor, 𝑏 is the Biot coefficient assuming overall isotropy [-], 𝑁 is the Biot skeleton modulus [Pa], ⟨𝑥⟩ + = (𝑥 + |𝑥 |)/2 are the Macaulay brackets, and 𝜑 0 ms represents the quantity of initial pores to be filled by the solid to induce swelling [-].

The microporomechanical model aims to estimate the following mortar variables from the microstructure:

• the Biot coefficient 𝑏 m and Biot skeleton modulus 𝑁 m , • the bulk and shear moduli, 𝑘 m and 𝑔 m , and thus the Young modulus 𝐸 m and the Poisson ratio 𝜈 m , • the diffusion coefficient 𝐷 m .

The subscript 𝑚 defines the phase of the mortar material. The Biot parameters are obtained considering that the minerals inducing the pore pressure precipitate in the capillary pores and the pores of the mortar's Interfacial Transition Zone. Section 2.2 describes the homogenization function based on a multi-scale representativity detailed in Section 2.1.

Description of the mortar microstructure

We consider a two-scale modeling illustrated in Figure 1: Level I: cement paste The Portland cement paste (CEM I) is a material with an isotropic matrix, called C-S-H, containing inclusions assumed spherical and randomly distributed in space. The inclusions correspond to the capillary porosity, the unhydrated cement and the main solid hydrated phases: portlandite, ettringite, and an AFm phase which can be katoite, hydrogarnet or monosulfoaluminate. Depending on the hydration model, the unhydrated cement is taken into account. Level II: mortar scale Mortar is a three-coated sphere assemblage with an isotropic matrix (cement paste) containing inclusions randomly distributed in space and in orientation (sand) and a percolation phase called the Interfacial Transition Zone (ITZ). The C-S-H phase can be described as a porous medium with two types of C-S-H characterized by their densities. The C-S-H density affects the mechanical and diffusive properties [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[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] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF][START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes[END_REF][START_REF] Ulm | Is concrete a poromechanics material?-A multiscale investigation of poroelastic properties[END_REF]) and models have different cement paste representations or consider a dedicated representation for the C-S-H scale [START_REF] Bary | Coupled chemo-transport-mechanical modelling and numerical simulation of external sulfate attack in mortar[END_REF][START_REF] Béjaoui | Experimental and modeling studies of the link between microstructure and effective diffusivity of cement pastes[END_REF][START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF][START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF][START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF][START_REF] Ulm | Is concrete a poromechanics material?-A multiscale investigation of poroelastic properties[END_REF]. Furthermore, some studies [START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF][START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF]) consider a representation of the ITZ phase whereas [START_REF] Hashin | An inverse method to determine the elastic properties of the interphase between the aggregate and the cement paste[END_REF]; [START_REF] Heukamp | Chemomechanics of Calcium Leaching of Cement-Based Materials at Different Scales: The Role of CH-Dissolution and C-S-H Degradation on Strength and Durability Performance of Materials and Structures[END_REF] estimate its properties from the cement paste value. Since the microstructure and the associated coefficients of C-S-H and ITZ are difficult to obtain, we choose to not describe their microstructures. Their coefficients are obtained with a probabilistic distribution function.

Estimation of the chemo-poromechanical properties

The homogenization schemes follow the recommendations of the literature for a mature material and are similar for mechanical and diffusive properties: Cement paste Mori-Tanaka scheme [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] used in [START_REF] Bary | Simplified coupled chemo-mechanical modeling of cement pastes behavior subjected to combined leaching and external sulfate attack[END_REF][START_REF] Heukamp | Chemomechanics of Calcium Leaching of Cement-Based Materials at Different Scales: The Role of CH-Dissolution and C-S-H Degradation on Strength and Durability Performance of Materials and Structures[END_REF][START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF][START_REF] Ulm | Is concrete a poromechanics material?-A multiscale investigation of poroelastic properties[END_REF][START_REF] Le | Modélisation multi-échelle des matériaux viscoélastiques hétérogènes : application à l'identification et à l'estimation du fluage propre d'enceintes de centrales nucléaires[END_REF]), Mortar Generalized Self-Consistent scheme [START_REF] Hashin | An inverse method to determine the elastic properties of the interphase between the aggregate and the cement paste[END_REF] used in [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF][START_REF] Tognevi | Modélisation multi-échelle et simulation du comportement thermo-hydromécanique du béton avec représentation explicite de la fissuration[END_REF][START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF]). Note that the Mori-Tanaka scheme is not suitable for early-age concrete when there is not enough C-S-H to consider this phase as a matrix. Our model is applied to mature material and cement paste admitting a high hydration rate.

Cement paste

The cement paste is assumed to be composed of 𝑁 ph phases: 𝑁 sol solid phases and capillarity pores, contained in a matrix of C-S-H. We consider that all inclusions are spherical. The bulk 𝑘 cp and shear 𝑔 cp moduli depend on the volume fraction 𝜑 i , shear modulus 𝑔 𝑖 and bulk modulus 𝑘 𝑖 of phase 𝑖 such that

𝑘 cp = 𝑘 csh + 4𝑔 csh 3 𝑁 ph 𝑖=1 𝜑 i 𝑘 𝑖 -𝑘 csh 𝑘 𝑖 + 4/3𝑔 csh 1 - 𝑁 ph 𝑖=1 𝜑 i 𝑘 𝑖 -𝑘 csh 𝑘 𝑖 + 4/3𝑔 csh , 𝑔 cp = 𝑔 csh + 𝐻 csh 𝑁 ph 𝑖=1 𝜑 i 𝑔 𝑖 -𝑔 csh 𝑔 𝑖 + 𝐻 csh 1 - 𝑁 ph 𝑖=1 𝜑 i 𝑔 𝑖 -𝑔 csh 𝑔 𝑖 + 𝐻 csh , (2)
and 𝐻 csh = 𝑔 csh (3/2𝑘 csh + 4/3𝑔 csh )/(𝑘 csh + 2𝑔 csh ) where the subscripts łcshž and łcpž denote C-S-H and cement paste respectively. We assume that the solid, called here łmain precipitated solid 𝜑 ms ž, see Equation (1), inducing the macroscopic swelling precipitates in the capillarity porosity and ITZ porosity [START_REF] Socié | Multibody approach for reactive transport modeling in discontinuous-heterogeneous porous media[END_REF][START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF]. Accordingly, we do not consider the C-S-H poroelastic properties. In that way, the Biot tensor 𝑏 cp 𝑰 and Biot skeleton modulus 𝑁 cp of the cement paste only depend on the hydrostatic part of the localization tensor 𝐴 ℎ 𝑖 [START_REF] Ulm | Is concrete a poromechanics material?-A multiscale investigation of poroelastic properties[END_REF]:

𝑏 cp 𝑰 = 1 - 𝑁 sol 𝑖=1 𝜑 i 𝐴 ℎ 𝑖 𝑰 , 1 𝑁 cp = 𝑁 sol 𝑖=1 𝜑 i (1 -𝐴 ℎ 𝑖 ) 𝑘 𝑖 (3)
with

𝐴 ℎ 𝑖 = 1 + 3𝑘 csh 3𝑘 csh + 4𝑔 csh 𝑘 𝑖 𝑘 csh -1 -1 𝑁 ph 𝑖 𝜑 i 1 + 3𝑘 csh 3𝑘 csh + 4𝑔 csh 𝑘 𝑖 𝑘 csh -1 -1 -1 . (4)
The effective diffusion coefficient 𝐷 cp of the cement paste depends on the diffusion of the phases [START_REF] Dormieux | Microporomechanics[END_REF]:

𝐷 cp = 𝐷 csh 1 + 2 𝑖 𝜑 i 𝐷 𝑖 -𝐷 csh 𝐷 𝑖 + 2𝐷 csh 1 - 𝑖 𝜑 i 𝐷 𝑖 -𝐷 csh 𝐷 𝑖 + 2𝐷 csh . (5)
Moreover, we assume that the capillary porosity and the C-S-H are the only diffusive phases [START_REF] Bary | Coupled chemo-transport-mechanical modelling and numerical simulation of external sulfate attack in mortar[END_REF][START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF][START_REF] Bogdan | Morphological multisscale modeling of cementitious materials -Application to effective diffusive properties prediction[END_REF][START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF]).

Mortar

The mortar is mainly represented as a three-coated sphere assemblage where the Generalized Self-Consistent scheme is used in order to take into account the ITZ phase as an interphase coating of the aggregate particle. The mortar properties depend on the ITZ, the sand and the cement paste properties. In this section, the subscripts łitzž and łsž define the ITZ and sand, respectively. The bulk modulus is defined by [START_REF] Hashin | An inverse method to determine the elastic properties of the interphase between the aggregate and the cement paste[END_REF][START_REF] Nguyen | A composite sphere assemblage model for porous oolitic rocks[END_REF][START_REF] Tognevi | Modélisation multi-échelle et simulation du comportement thermo-hydromécanique du béton avec représentation explicite de la fissuration[END_REF])

𝑘 m = 𝑘 cp + 𝜑 s + 𝜑 itz 1 𝑘 * -𝑘 cp + 3𝜑 cp 3𝑘 itz + 4𝑔 itz (6) with 𝑘 * = 𝑘 itz + (𝜑 s /(𝜑 s +𝜑 itz ))/(1/(𝑘 𝑠 -𝑘 itz ) + 3𝜑 itz /(𝜑 s +𝜑 itz )/(3𝑘 itz + 4𝑔 itz ))
. The shear modulus is deduced from an implicit system depicted in [START_REF] Hashin | An inverse method to determine the elastic properties of the interphase between the aggregate and the cement paste[END_REF][START_REF] Nguyen | Apport de la modélisation mésoscopique dans la prédiction des écoulements dans les ouvrages en béton fissuré en conditions d'accident grave[END_REF][START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF]. For a sake of compactness, we refer the reader to these papers for details. The Biot coefficient and skeleton modulus depend on poroelastic phases values [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF][START_REF] Nguyen | A composite sphere assemblage model for porous oolitic rocks[END_REF]:

𝑏 m =𝑏 cp + (3𝑘 cp +4𝑔 cp ) (1-𝜑 cp ) [(𝑏 itz -𝑏 cp ) (4𝑔 itz (1-𝜑 cp ) +3𝑘 * 1 ) -𝜑 s (𝑏 itz -1) (3𝑘 itz +4𝑔 itz )] 9𝑘 𝑠 𝑘 itz 𝜑 cp (1-𝜑 cp ) +12𝑔 itz 𝑘 * 2 𝜑 cp + (4𝑔 cp +3𝑘 cp (1-𝜑 cp )) (4𝑔 itz (1-𝜑 cp ) +3𝑘 * 1 ) (7)
with

𝑘 * 1 = 𝑘 itz 𝜑 s + 𝑘 𝑠 𝜑 itz and 𝑘 * 2 = 𝑘 𝑠 𝜑 s + 𝑘 itz 𝜑 itz , and 1 𝑁 m = 𝜑 s + 𝜑 itz 𝑁 * + 𝜑 cp 𝑁 cp + 3(𝑏 * -𝑏 cp ) 2 𝜑 s (𝜑 s + 𝜑 itz ) 3(𝜑 s + 𝜑 itz ) (𝑘 cp -𝑘 * ) + 3𝑘 * + 4𝑔 cp 1 𝑁 * = 𝜑 s /(𝜑 s + 𝜑 itz ) 𝑁 𝑠 + 1 -𝜑 s /(𝜑 s + 𝜑 itz ) 𝑁 itz + 3(𝑏 itz -𝑏 𝑠 ) 2 (1 -𝜑 s /(𝜑 s + 𝜑 itz ))𝜑 s /(𝜑 s + 𝜑 itz ) 3𝜑 s (𝑘 itz -𝑘 𝑠 )/(𝜑 s + 𝜑 itz ) + 3𝑘 𝑠 + 4𝑔 itz 𝑏 * = 𝑏 itz + (3𝑘 itz + 4𝑔 itz ) (𝑏 m -𝑏 itz )𝜑 s /(𝜑 s + 𝜑 itz ) 4𝑔 itz + 3𝑘 𝑠 (1 -𝜑 s /(𝜑 s + 𝜑 itz ) + 3𝑘 itz 𝜑 s /(𝜑 s + 𝜑 itz )) . ( 8
)
The sand is considered nonreactive with a negligible diffusion coefficient. Based on [START_REF] Bogdan | Morphological multisscale modeling of cementitious materials -Application to effective diffusive properties prediction[END_REF][START_REF] Heukamp | Chemomechanics of Calcium Leaching of Cement-Based Materials at Different Scales: The Role of CH-Dissolution and C-S-H Degradation on Strength and Durability Performance of Materials and Structures[END_REF][START_REF] Hashin | An inverse method to determine the elastic properties of the interphase between the aggregate and the cement paste[END_REF][START_REF] Salah | Identification of a cohesive zone model for cement paste-aggregate interface in a shear test[END_REF], elastic and diffusive properties of ITZ are deduced from the cement paste coefficients. As a first approach, due to the high porosity of the ITZ phase and because the phase is mostly composed of matrix solid phases [START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF], the poroelastic properties of the ITZ are estimated from a bimaterial representation [START_REF] Dormieux | Microporomechanics[END_REF]:

𝑏 itz = 1 - 𝑘 itz 𝑘 csh ; 1 𝑁 itz = 𝑏 itz -𝜙 itz 𝑘 csh . (9)
The diffusive coefficient of the mortar is estimated with the so-called three-coated sphere model [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF][START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF]:

𝐷 m = 𝐷 cp + (1 -𝜑 cp ) [𝐷 itz -𝐷 cp + 𝐷 * 1 (𝐷 * 2 ) -1 ] -1 + 𝜑 cp 3𝐷 cp -1 (10) with 𝐷 * 1 = 𝜑 s /(𝜑 s + 𝜑 itz ) and 𝐷 * 2 = (𝐷 𝑠 -𝐷 itz ) -1 + (1 -𝐷 * 1 ) (3𝐷 itz ) -1 .
Adrien Socié et al.

Effects of the microstructural uncertainties on the poroelastic and the diffusive properties of mortar

Validation

This section aims to validate this simple and closed-form model by comparison with a more sophisticated model [START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF]) and the model's availability to capture the microstructure variation.

Impact of the microstructure evolution

The ability of the model to describe the microstructure evolution is depicted with the effect of the water-cement mass ratio on the overall properties. The cement paste microstructure is estimated by [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF]; [START_REF] Tognevi | Modélisation multi-échelle et simulation du comportement thermo-hydromécanique du béton avec représentation explicite de la fissuration[END_REF] [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF][START_REF] Tognevi | Modélisation multi-échelle et simulation du comportement thermo-hydromécanique du béton avec représentation explicite de la fissuration[END_REF]). We consider one C-S-H phase and its volume fraction is equal to the sum of the inner and the outer C-S-H volume fractions given in [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF][START_REF] Tognevi | Modélisation multi-échelle et simulation du comportement thermo-hydromécanique du béton avec représentation explicite de la fissuration[END_REF]). In the validation part, the AFm is modeled by the monosulfoaluminate. In the applications part, Sections 4 and 5, the AFm phase is represented by the katoite to be consistent with the chemo-poromechanical model [START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF].

we consider one C-S-H phase and its volume fraction is estimated equal to the sum of the inner and the outer C-S-H volume fraction of [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF][START_REF] Tognevi | Modélisation multi-échelle et simulation du comportement thermo-hydromécanique du béton avec représentation explicite de la fissuration[END_REF]). The AFm is modeled by the monosulfoaluminate. The material properties are described in Table 2. The mechanical results agree well with the experimental measurements of [START_REF] Haecker | Modeling the linear elastic properties of Portland cement paste[END_REF] shown in Figure 2(a). [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF]). The considered C-S-H diffusion coefficient is equal to 8 × 10 -12 m 2 s -1 to fit the experimental result. The diffusion model can catch the effects of the microstructure and fits well the experimental results. Nevertheless, for a low water-cement ratio, the model overestimates the diffusion value due to the simplified representation of the C-S-H phases. As described in [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF], the model needs to consider the influence of the water-cement ratio in the volume fraction of the different C-S-H phases (inner and outer).

𝐷 [m 2 s -1 ] 𝐸 [GPa] 𝜈 References C-S

Validation of the Micromechanical representation

In Table 3, the model is compared to the reference model of [START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF] which has a microstructure representation of the C-S-H and the ITZ phases. The microstructure and the hydrated properties are taken from [START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF]). In our model, the properties of the C-S-H and the ITZ phases are estimated by a stochastic model. The range of values are estimated using a Monte-Carlo experiments on 10000 samples to take into account the uncertainties associated 2005) and (b) normalized diffusion properties compared to the experimental results of [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF]. [START_REF] Bellégo | Coupled mechanical and chemical damage in calcium leached cementitious structures[END_REF][START_REF] Gallé | Effect of an accelerated chemical degradation on water permeability and pore structure of cement-based materials[END_REF]) and analytical homogenization model [START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF]. Young modulus in [GPa] and diffusion coefficient in [10 -12 m 2 s -1 ]. * Experimental value from [START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF]) and measured by [START_REF] Bourdette | Durabilité du mortier, prise en compte des auréoles de transition dans la caractérisation et la modélisation des processus physiques et chimiques d'altération[END_REF].

with the C-S-H and ITZ properties. The parameters are described by a log-normal distribution where the mean value and standard deviation parameters of the normal law distribution are respectively summarized in Tables 2 and4. The diffusion and elastic properties fit well with the model and experimental results, except for the material of [START_REF] Gallé | Effect of an accelerated chemical degradation on water permeability and pore structure of cement-based materials[END_REF] where the model estimates a lower Young's modulus value.

Main results

The model can both predict the poroelastic and diffusive properties from an estimated microstructure and capture the effects of the cement mineralogy in the overall behavior.

Uncertainty propagation

The propagation of uncertainty of the poroelastic and diffusive mortar properties are explored through a Monte-Carlo method. At first, each uncertainty is studied separately and divided into two domains: material properties, reported in Section 4.1, and microstructural characteristics, in Section 4.2. We consider a mortar composed of a CEM I cement paste, detailed in Table 1, with a water-cement ratio of 0.4, an ITZ volume fraction of 0.185 and a volume fraction of sand of 0.55. Each parameter is investigated independently, the others set to their mean value, see Tables 1 and2. Secondly, we study the impact of the global uncertainties on three water-cement ratios: 0.3, 0.4 and 0.5. The study has been developed to simulate a sulfate attack where the aluminate phase is modeled by the katoite [START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF]. In that way, we assume that the AFm phase is represented by the katoite phase. Our results can be applied to a system where the aluminate is represented by the monosulfoaluminate. Indeed, recent molecular simulations (Honorio et al. 2020b) highlight that the elastic properties of monosulfoaluminate are similar to the ettringite and thus the katoite.

A Monte-Carlo study with 10 000 simulations by parameter is performed. In the sequel, the input parameters of the multi-scale model are described by log-normal distributions to ensure the strict positivity:

𝑝 (𝑥) = 1 𝑥𝑉 √ 2𝜋 exp - ln 𝑥 -𝑒 𝑉 √ 2 2 , (11)
where 𝑒 and 𝑉 are the mean and the standard deviation of the normal law distribution, respectively. For each study, the output mean and standard deviations are estimated. Appendix A is dedicated to the estimation of the parameters uncertainties and summarizes the various coefficients used in the literature. (2013), we study the impact of the uncertainties of the Young Modulus of the two hydrates. For the diffusion part, the study is focused on the C-S-H phase. Indeed, in the analytical model benchmark led by [START_REF] Patel | Diffusivity of saturated ordinary Portland cement-based materials: A critical review of experimental and analytical modelling approaches[END_REF], the authors highlight that every tested multi-scale and empirical model has to fit the diffusion associated with the matrix phases that induce a wide variability of values (see Appendix B). This wide variability is explained by the difference in the microstructure representation and the fact that the calibration value depends on the experimental techniques used to obtain the cement paste diffusivity [START_REF] Patel | Diffusivity of saturated ordinary Portland cement-based materials: A critical review of experimental and analytical modelling approaches[END_REF]. Furthermore, the C-S-H's diffusion coefficient is difficult to identify because of the sorption reactions between the cation and anion in solution and the solid phases [START_REF] Seigneur | Transport properties evolution of cement model system under degradation -Incorporation of a pore-scale approach into reactive transport modelling[END_REF].

ITZ properties At mortar scale, the ITZ value mainly affects the poroelastic and diffusion properties [START_REF] Heukamp | Chemomechanics of Calcium Leaching of Cement-Based Materials at Different Scales: The Role of CH-Dissolution and C-S-H Degradation on Strength and Durability Performance of Materials and Structures[END_REF][START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF]). Tables C.5 and C.6 show the large variability in the literature. The values are generally estimated by inverse analysis [START_REF] Heukamp | Chemomechanics of Calcium Leaching of Cement-Based Materials at Different Scales: The Role of CH-Dissolution and C-S-H Degradation on Strength and Durability Performance of Materials and Structures[END_REF][START_REF] Hashin | An inverse method to determine the elastic properties of the interphase between the aggregate and the cement paste[END_REF][START_REF] Garboczi | Stress, displacements, and expansive cracking around a single spherical aggregate under different expansive conditions[END_REF][START_REF] Patel | Diffusivity of saturated ordinary Portland cement-based materials: A critical review of experimental and analytical modelling approaches[END_REF]) which depends both on the analytical homogenization/numerical scheme used and the experiment uncertainties [START_REF] Aït-Mokhtar | Experimental investigation of the variability of concrete durability properties[END_REF][START_REF] Patel | Diffusivity of saturated ordinary Portland cement-based materials: A critical review of experimental and analytical modelling approaches[END_REF].

The input parameters are summarized in Table 4. The mean value and standard deviation of portlandite's Young modulus are identified from [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] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF]. The C-S-H value is estimated for water-cement ratios of 0.3 and 0.5 by analytical homogenization (Mori-Tanaka scheme) from the measurement of the uncertainties [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 high and low-density C-S-H volume fractions are estimated from [START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes[END_REF]. The mean value and standard deviation of the C-S-H diffusion and ITZ properties are estimated from the literature and presented in Appendices B and C.

𝐷 csh [m 2 s -1 ] 𝐸 csh [GPa] 𝐸 ch [GPa] 𝐷 itz [m 2 s -1 ] 𝐸 itz V 2.84 × 10 -12 2 5 6.93 × 𝐷 cp 0.175 × 𝐸 cp

Results

The results are reported in Table 5 and plotted in Figure 3. At cement scale, the C-S-H has the main uncertainties impact. In contrast to the results of [START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF] and [START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF], the portlandite and C-S-H elastic properties do not have a strong impact on effective mortar properties. The mortar poroelastic properties are mainly affected by the uncertainties associated with the ITZ rigidity (strong input dispersion, see Table C.6) and the poroelastic properties of the transition zone [START_REF] Nguyen | A composite sphere assemblage model for porous oolitic rocks[END_REF]. The uncertainties propagation from cement paste to the mortar admit a quite similar variance.

The diffusion coefficient has a larger variance than the elastic properties. First of all, the C-S-H uncertainties have a significant effect on the mortar and cement paste properties. The uncertainty propagation is stable and the mortar standard deviation is larger than cement paste one. This effect is undergone by the ITZ diffusion coefficient that is a function of the cement paste properties. At mortar level, the ITZ still plays a major role in the uncertainty but, instead of the poroelastic properties, there is a competition between the C-S-H and ITZ uncertainties on the mortar diffusion variation. The C-S-H and ITZ have the main uncertainties impact respectively at cement paste and mortar scales. Only for the diffusion coefficient, the C-S-H uncertainties play a major role in the global mortar variance.

Uncertainties 𝐸 cp [GPa] 𝐸 m [GPa] 𝑏 m 𝑁 m [GPa] 𝐷 cp /𝐷 0 [×10 -3 ] 𝐷 m /𝐷 0 [×10 -

Uncertainty from the microstructure 4.2.1 Input parameters

We consider two main impacts of the microstructure in each level: the hydrate volume fraction and the ITZ volume fraction.

Hydrate volume fraction Most of the hydration models [START_REF] Buffo-Lacarrière | Multiphasic finite element modeling of concrete hydration[END_REF][START_REF] Papadakis | Physical and chemical characteristic affecting the durability of concrete[END_REF][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 based on the alite, belite, aluminate and ferrite mass fractions estimated from the chemical clinker composition by Bogue's formula [START_REF] Bogue | Calculation of the Compounds in Portland Cement[END_REF][START_REF] Taylor | Modification of the Bogue calculation[END_REF]). [START_REF] Stutzman | Uncertainty in Bogue-calculated phase composition of hydraulic cements[END_REF] highlighted that the Bogue constant introduces a significant uncertainty that mainly impacts the alite and belite mass fractions (around 10 %) and thus the portlandite and C-S-H volume fractions. Furthermore, we notice a strong variability of C-S-H molar volume in the literature: for a calcium silica ratio of 1.65, the molar volume is equal to 0.084 L mol -1 [START_REF] Planel | Les effets couplés de la précipitation d'espèces secondaires sur le comportement mécanique et la dégradation chimique des bétons[END_REF] or 0.078 L mol -1 [START_REF] Lothenbach | Cemdata18: A chemical thermodynamic database for hydrated Portland cements and alkali-activated materials[END_REF]. This variability impacts the overall properties estimated by the hydration-homogenization model [START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF]. In order to capture both the impact of Bogue's constant and molar volume uncertainties in the microstructure properties, a Monte-Carlo study is carried out using the hydration model of [START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF], see Appendix A. The associated standard deviations are applied to the studied microstructures, as reported in Table 6. The variation of the solid volume fraction

ITZ volume fraction

The ITZ volume fraction depends on the sand granulometry [START_REF] Garboczi | Analytical formulas for interfacial transition zone properties[END_REF], the sand mean size particle [START_REF] Garboczi | Analytical formulas for interfacial transition zone properties[END_REF], the hydration rate [START_REF] Sun | Modeling the elastic properties of concrete composites: Experiment, differential effective medium theory, and numerical simulation[END_REF]), the inclusion properties [START_REF] Keinde | Etude du béton à l'échelle mésoscopique : simulation numérique et tests de micro-indentation[END_REF]) and the water cement ratio [START_REF] Sun | Modeling the elastic properties of concrete composites: Experiment, differential effective medium theory, and numerical simulation[END_REF][START_REF] Tognevi | Modélisation multi-échelle et simulation du comportement thermo-hydromécanique du béton avec représentation explicite de la fissuration[END_REF]). The ITZ volume fraction is usually estimated from the particle size distribution of sand and considering a constant ITZ thickness by Garboczi's formula [START_REF] Garboczi | Analytical formulas for interfacial transition zone properties[END_REF], such as [START_REF] Heukamp | Chemomechanics of Calcium Leaching of Cement-Based Materials at Different Scales: The Role of CH-Dissolution and C-S-H Degradation on Strength and Durability Performance of Materials and Structures[END_REF][START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF][START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF][START_REF] Tognevi | Modélisation multi-échelle et simulation du comportement thermo-hydromécanique du béton avec représentation explicite de la fissuration[END_REF]. The great uncertainty of ITZ thickness influences the estimation of the volume fraction [START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF]Honorio et al. 2020a). The mean value and the standard deviation are deduced from the volume fractions used in the literature and summarized in Table C.7. The variation of the ITZ volume fraction impacts the cement paste volume fraction and satisfies

𝜑 cp = 1 -𝜑 itz -𝜑 s .
Finally, to study the influence of the ITZ porosity on the poroelastic properties ( 9), an uncertainty analysis is carried out with the mean value of 𝜙 itz /𝜙 equals to 1.75 and a standard deviation of 0.25, based on the global values used in the literature, see Table C.8.

Results

The results are presented in Table 7 and plotted in Figure 4. At cement paste scale, the variations of poroelastic properties and diffusion coefficient increase with the volume fraction uncertainties. This effect is due to two aspects. First of all, the Young modulus of each hydrate remains similar and we do not consider their diffusivity, see Table 2. Secondly, the porosity is linearly dependent on the volume fraction (12). Because the katoite is the hydrate inclusion (excluding C-S-H) that admits the larger volume fraction uncertainty, the efficient properties admit a larger disparity for this phase. Note that the variation of the solid fraction of the aluminate phases can also impact the chemo-mechanical response, such as sulfate attack. The matrix (C-S-H) remains the phase whose uncertainties have the most impact on the estimated properties.

Uncertainties 𝐸 cp [GPa] 𝐸 m [GPa] 𝑏 m 𝑁 m [GPa] 𝐷 cp /𝐷 0 × 10 3 𝐷 m /𝐷 0 ×
At mortar scale, the mortar variation is, first of all, induced by the ITZ volume fraction uncertainty. [START_REF] Patel | Diffusivity of saturated ordinary Portland cement-based materials: A critical review of experimental and analytical modelling approaches[END_REF] observed a low impact of the ITZ phase on the diffusion properties, which may call into question the main influence of the mortar homogenization scheme obtained. As reported by [START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF], the uncertainty of the cement paste parameters is magnified during the upscaling processes but, for the elastic and diffusion coefficients, their influences are Adrien Socié et al.

Effects of the microstructural uncertainties on the poroelastic and the diffusive properties of mortar negligible compared to the ITZ uncertainty. The Biot properties are impacted by the porosity and the volume fraction of ITZ. Note that the ITZ porosity has been little studied in the literature that could have a strong impact on the overall properties, see

Main results

The variations of poroelastic properties and diffusion coefficient increase with the volume fraction uncertainties of the inclusion and their impacts are magnified during the upscaling processes. The main sources of uncertainty at the cement paste and mortar scales remain respectively the C-S-H and ITZ properties.

Impact of all uncertainties

The impacts of material and microstructure uncertainties on the analytical estimated results are investigated. Based on the remarks of [START_REF] Stutzman | Uncertainty in Bogue-calculated phase composition of hydraulic cements[END_REF] regarding the uncertainties associated with the Bogue formula rendering łsupposedly distinct classes of cement practically indistinguishablež, the homogenization propagation uncertainties are studied for three watercement ratios.

The evolution of the apparent material properties is shown in Table 8 and plotted in Figure 5. The mean parameters follow the tendencies specified in the literature (experimental, numerical and analytical results): the diffusion increases and the Young Modulus decreases with the water-cement ratio [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF][START_REF] Béjaoui | Experimental and modeling studies of the link between microstructure and effective diffusivity of cement pastes[END_REF][START_REF] Bernard | Predicting the evolution of mechanical and diffusivity properties of cement pastes and mortars for various hydration degrees ś A numerical simulation investigation[END_REF][START_REF] Patel | Diffusivity of saturated ordinary Portland cement-based materials: A critical review of experimental and analytical modelling approaches[END_REF][START_REF] Tognevi | Modélisation multi-échelle et simulation du comportement thermo-hydromécanique du béton avec représentation explicite de la fissuration[END_REF]. As previous results, the uncertainties increase with the mortar parameters, in particular the diffusion coefficients. Furthermore, the Biot coefficient evolution with the water-cement ratio is not clear. The standard deviation increases with the upscaling due to the propagation of the uncertainties through the ITZ properties and volume fraction. The standard deviation of the diffusive properties decreases with the water-cement ratio that is in accordance with the results of Honorio et al. (2020a) for the conductivity properties of concrete material. We note an inverse tendency for the young modulus. These effects are relatively low and are more pronounced in (Honorio et al. 2020a). In our case, the volume fraction of the main inclusion (except the portlandite) and matrix remain quite similar but the capillary porosity increases due to the dissolution of the unreacted cement, see Table 1. The variation of the standard deviation with 𝑤/𝑐 can be due to the effect of the microstructural variability on the porosity which increases with the capillary porosity.

𝑤/𝑐 𝐸 cp [GPa] 𝐸 m [GPa] 𝑏 m 𝑁 m [GPa] 𝐷 cp /𝐷 0 × 10 3 𝐷 m /𝐷 0 ×
𝑤/𝑐 𝐷 𝑖 /𝐷 0 [×10 -3 ] 𝐷 m /𝐷 0 𝐷 cp /𝐷 0
The coefficient variation 𝑉 /𝑒 is mainly impacted by the water-cement ratio and so for a low water-cement ratio, the predicted values admit a small error.

Sensitivity analysis

Section 4 highlights the impact of the morphologic and the material properties parameters in the global response. To identify the main contributor to the output variation, a sensitivity analysis study is carried out.

The sensitivity analysis is based on the variance-based method. The influences of each input are measured through the Sobol indices (between 0 and 1). The first order sensitivity Sobol index, denoted 𝑆 𝑖 , is used to measure the sensitivity of each input parameter:

𝑆 𝑖 = 𝑉 x 𝑖 (𝑒 (y|x 𝑖 )) 𝑉 (y) , (13)
where x 𝑖 is the vector of input parameters, y is the output values given by the model, and 𝑖 defines the input index. For more details, see [START_REF] Gilquin | Iterative estimation of Sobol' indices based on replicated designs[END_REF][START_REF] Saltelli | Making best use of model evaluations to compute sensitivity indices[END_REF].

The system is computed using the Sobol-Saltelli method [START_REF] Gilquin | Iterative estimation of Sobol' indices based on replicated designs[END_REF][START_REF] Saltelli | Making best use of model evaluations to compute sensitivity indices[END_REF][START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF][START_REF] Sudret | Sensitivity analysis of nested multiphysics models using polynomial chaos expansions[END_REF] where the Sobol indices are estimated through a Monte-Carlo experiments. Let us consider 𝑿 𝑎 ∈ R 𝑁 𝑖 ×𝑁 𝑠 and 𝑿 𝑏 ∈ R 𝑁 𝑖 ×𝑁 𝑠 two independent input sample matrices, where 𝑁 𝑖 and 𝑁 𝑠 are respectively the number of input values for each model and the number of Monte-Carlo samples. To compute the first order Sobol indices, a matrix, denoted 𝑿 𝑐,𝑖 , where all factors except the row 𝑗 is equal to 𝑿 𝑏 , is defined for each variable. The row 𝑖 corresponds to the row of the matrix 𝑿 𝑎 . The estimation of 𝑆 𝑖 is obtained by the products of the vector of output values estimated from 𝑿 𝑎 and each matrix 𝑿 𝑐,𝑖 , ∀𝑖 ∈ 𝑁 𝑖 , denoted respectively y 𝑎 and y 𝑐,𝑖 [START_REF] Gilquin | Iterative estimation of Sobol' indices based on replicated designs[END_REF][START_REF] Saltelli | Making best use of model evaluations to compute sensitivity indices[END_REF]:

𝑆 ′ 𝑖 = 1 𝑛 𝑛 𝑗=1 (𝑦 𝑎 𝑗 × 𝑦 𝑐,𝑖 𝑗 ) - 1 𝑛 𝑛 𝑗=1 𝑦 𝑎 𝑗 2 1 𝑛 𝑛 𝑗=1 (𝑦 𝑎 𝑗 ) 2 - 1 𝑛 𝑛 𝑗=1
𝑦 𝑎

𝑗

The first-order sensitivity indices measure only the direct effect of one parameter on the response of the micromechanical model. The total Sobol indices representing the interaction effect between parameters are estimated in the previous studies (Section 4.1 and Section 4.2). We focus here on the first order to distinguish the Sobol indices at the cement paste and at the mortar scales to highlight the highest Sobol index value. Moreover, the study allows us to consider the influence of the interaction effects of input properties on homogenized values. Only diffusive and elastic properties are considered because the Biot properties lead to high variation.

Analysis and results

Young Modulus Figure 6 shows the first order Sobol indices for the Young modulus of the cement paste and the mortar.

𝐸 csh 58 6.9 For the cement paste, the total sum of the indices is almost equals to 1, which means there are negligible interaction effects between the morphologic and the material properties. This result confirms the conclusion of the uncertainties studies. The C-S-H phases play a major role. The Young modulus must be estimated properly to simulate the elastic cement paste properties accurately. Furthermore, the viscoelastic properties of the cement paste, not studied here, are impacted by the C-S-H properties and volume fraction [START_REF] Le | Modélisation multi-échelle des matériaux viscoélastiques hétérogènes : application à l'identification et à l'estimation du fluage propre d'enceintes de centrales nucléaires[END_REF][START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF][START_REF] Sanahuja | Mean-field homogenization of time-evolving microstructures with viscoelastic phases: application to a simplified micromechanical model of hydrating cement paste[END_REF]. The aluminate phase (here the katoite) has a greater effect than the portlandite elastic properties and other morphological properties. Note that the uncertainties could be more important for the monosulfoaluminate phase because its Young Modulus is closed to that of Portlandite phase and it is greater than that of the katoite phase [START_REF] Haecker | Modeling the linear elastic properties of Portland cement paste[END_REF]. Nevertheless, the compressibility modulus value of the phase is still in debate, recent studies using the molecular simulations show the monosulfoaluminate properties are closed to that of the ettringite phase and thus the katoite (Honorio et al. 2020b). The volume fraction uncertainties are deduced from Bogue's equation [START_REF] Stutzman | Uncertainty in Bogue-calculated phase composition of hydraulic cements[END_REF]. Hydration properties based on the total concentration of aluminates, instead of the mass fraction of C 3 A [START_REF] Papadakis | Physical and chemical characteristic affecting the durability of concrete[END_REF][START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes[END_REF], could reduce the global uncertainties. The effect of other hydrates is negligible.

𝐸
The mortar is impacted by the uncertainties associated with the ITZ phase. The total sum of the first order indices is 0.95 which highlights an impact negligible of the interaction between the variables. The main interaction is due to the ITZ strength, see Tables 5 and7. The key parameter is the ITZ Young Modulus which is hard to deduce. From an experimental point of view, the identification of the microstructural and mechanical properties of the interphase remains a challenge [START_REF] Jebli | Experimental characterization of mechanical properties of the cement-aggregate interface in concrete[END_REF][START_REF] Salah | Identification of a cohesive zone model for cement paste-aggregate interface in a shear test[END_REF][START_REF] Sun | Modeling the elastic properties of concrete composites: Experiment, differential effective medium theory, and numerical simulation[END_REF]. Numerically, the representation of the ITZ phase [START_REF] Honorio | Multiscale estimation of ageing viscoelastic properties of cement-based materials: A combined analytical and numerical approach to estimate the behaviour at early age[END_REF][START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF]) can reduce the uncertainties, but additional variations can appear due to the material representativity. Such as the hydration, adding model parameters can increase the variation [START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF]. Furthermore, the ITZ strength properties depend on the aggregate and sand which impact the homogenized response [START_REF] Keinde | Etude du béton à l'échelle mésoscopique : simulation numérique et tests de micro-indentation[END_REF]. Our approach seems to be a convenient choice to consider the impact of the variation of ITZ elastic properties independently of the inclusion characteristics. At cement paste scale, the interaction between the microstructure and the local diffusivity is negligible. The diffusivity depends essentially on the C-S-H diffusion. This result agrees with [START_REF] Patel | Diffusivity of saturated ordinary Portland cement-based materials: A critical review of experimental and analytical modelling approaches[END_REF]. Furthermore, distinguishing the C-S-H phases and identifying the diffusion coefficient for each phase remain an experimental challenge [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] Korb | Microstructure and texture of hydrated cement-based materials: A proton field cycling relaxometry approach[END_REF][START_REF] Seigneur | Transport properties evolution of cement model system under degradation -Incorporation of a pore-scale approach into reactive transport modelling[END_REF].

At mortar scale, the sum of the first Sobel indices highlights the main impact of the interaction term. The interaction is due to the upscaling impact of the cement paste volume fraction variation and the relationship between the ITZ and the cement paste diffusion properties. In the first order, the cement paste properties have the main effect on the homogenized coefficients. The effects of interaction and the C-S-H properties could be one of the major roles in the multi-scale model. The mortar is also mainly impacted by the uncertainties associated with the ITZ phase (volume fraction and diffusion coefficient). The three-coated homogenization scheme models the percolation of the transitional phase and thus the volume fraction is also a key parameter to estimate.

Main results

This study highlighted the main sources of uncertainty of the model and the interaction between the model parameters. The interaction between the microstructure and material properties remains negligible except for the diffusion coefficient of the mortar. The main conclusion highlighted with the uncertainty study is confirmed in this study: the ITZ and C-S-H are the main parameters affecting the efficient properties. The effect of the C-S-H Young modulus is quite negligible at the mortar scale for the Young modulus compared to the ITZ parameters.

Conclusions

To study the chemo-mechanics behavior of concrete, a multi-scale microporomechanics model was proposed in [START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF][START_REF] Socié | Chemo-poro-mechanical modeling of cementitious materials (diffusion-precipitation-cracking)[END_REF][START_REF] Socié | Multibody approach for reactive transport modeling in discontinuous-heterogeneous porous media[END_REF]. The model allows to estimate the diffusive and poroelastic properties of mortar and to highlight the effect of the microstructural characteristic. The microporomechanical model is based on a representation at cement paste and mortar scales. The homogenization scheme follows the recommendations of the literature for a mature material.

The model was validated for various cement pastes and mortars by comparison with analytical homogenization models and experimental results. A probabilistic approach was applied to the multi-scale model to investigate the uncertainty propagation of the microstructural and material properties. The results highlighted the main impact of the cement matrix (C-S-H) uncertainties at the cement paste scale and the Interfacial Transition Zone at the mortar scale for the diffusive and the poroelastic properties. Based on [START_REF] Stutzman | Uncertainty in Bogue-calculated phase composition of hydraulic cements[END_REF], the influence of Bogue's uncertainties through the initial volume of each hydrated was investigated. Despite the C-S-H phases, the aluminate hydrate has a major impact on the poroelastic properties. The propagation of the global uncertainties for three water-cement ratios showed the ability of the model to capture the effect of the material evolution on the diffusive and elastic properties. However, the uncertainties of the Biot parameters are too important to obtain quantitative estimation.

The last part of the paper was dedicated to a sensitivity analysis based on the first Sobol indices. This study highlighted the negligible interaction between the microstructure and the local diffusivity for the diffusive and elastic properties of the cement paste and for the mortar Young modulus. For a solid phase, the model was mainly dependent on the material properties rather than the volume fraction. For the Young modulus, the main conclusion of the uncertainty analysis was similar. The C-S-H was the main uncertainties input parameter at the cement paste scale and the ITZ, at the mortar scale. The cement paste was also impacted by the volume fraction of the aluminate phases and the effect of the portlandite phase remained negligible. For the diffusivity at the cement paste scale, the C-S-H was the main factor, in agreement with [START_REF] Patel | Diffusivity of saturated ordinary Portland cement-based materials: A critical review of experimental and analytical modelling approaches[END_REF]. For the mortar diffusivity term, the interaction between the input parameters and the C-S-H effect was predominant. The study underlined the predominant impact of the ITZ volume fraction and properties to estimate mortar properties.

Finally, the study highlighted the large variety of microstructures, due to the uncertainties associated to the Bogue formulation, that mainly impact the C-S-H and aluminate phases, which are the limiting reactant of the Delayed Ettringite Formation and External Sulphate Attack [START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF]. The results could be applied to a chemo-mechanical Finite Element simulation to consider the impact of the overall uncertainties on the chemical and mechanical degradation kinetics.

A Uncertainties associated to the hydration model

Bogue's formula [START_REF] Bogue | Calculation of the Compounds in Portland Cement[END_REF][START_REF] Taylor | Modification of the Bogue calculation[END_REF][START_REF] Stutzman | Uncertainty in Bogue-calculated phase composition of hydraulic cements[END_REF]) is used to estimate the main cement phases from the clinker chemical properties. [START_REF] Stutzman | Uncertainty in Bogue-calculated phase composition of hydraulic cements[END_REF] studied the impact of the uncertainties associated with the formula. According to the authors, the uncertainties associated with the Bogue formula have a great impact on the microstructure and material properties estimated by the hydration model. Indeed, the main cement phases C 2 S, C 3 S, C 3 A and C 4 AF are commonly used in the hydration model to predict the initial concrete properties [START_REF] Bary | Coupled chemo-transport-mechanical modelling and numerical simulation of external sulfate attack in mortar[END_REF][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]. To study the impact of the Bogue model uncertainties on the poroelastic and diffusive properties, a Monte-Carlo method is carried out on the hydration model developed in [START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF]). The hydration model differs from the one used by [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF]; [START_REF] Tognevi | Modélisation multi-échelle et simulation du comportement thermo-hydromécanique du béton avec représentation explicite de la fissuration[END_REF], which is the model of [START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes[END_REF], because it is applied for a mature concrete and a totally hydrated reaction is assumed. The cement paste microstructure is estimated by a thermodynamical chemical model developed in [START_REF] Socié | Multibody approach for reactive transport modeling in discontinuous-heterogeneous porous media[END_REF]). This methodology is mainly used and validated in reactive transport [START_REF] De Windt | Modeling the degradation of Portland cement pastes by biogenic organic acids[END_REF][START_REF] Planel | Les effets couplés de la précipitation d'espèces secondaires sur le comportement mécanique et la dégradation chimique des bétons[END_REF][START_REF] Seigneur | Predicting the atmospheric carbonation of cementitious materials using fully coupled two-phase reactive transport modelling[END_REF]). The article focuses on the role of the microstructure uncertainties and the material properties on the analytical homogenization results. Our study rests on the microstructure from [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF]. The hydration model of [START_REF] Socié | Modélisation chimio-mécanique de la fissuration de matériaux cimentaires : vieillissement et tenue des enceintes de confinement des centrales nucléaires[END_REF] is considered to estimate the uncertainties propagation reported in [START_REF] Stutzman | Uncertainty in Bogue-calculated phase composition of hydraulic cements[END_REF]. The comparison of different hydration models achieved in [START_REF] Göbel | Uncertainty analysis in multiscale modeling of concrete based on continuum micromechanics[END_REF]Honorio et al. 2020a;[START_REF] Venkovic | Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age[END_REF] The thermodynamic chemical solver [START_REF] Socié | Multibody approach for reactive transport modeling in discontinuous-heterogeneous porous media[END_REF] gives the concentration of each main phase of the cement phases and the molar fractions are simply deduced by multiplying the concentration by the molar volume. A Monte-Carlo scheme is carried out on the results of [START_REF] Stutzman | Uncertainty in Bogue-calculated phase composition of hydraulic cements[END_REF], see Table A.1, using a log-normal distribution for each phase. Furthermore,

C 3 S C 2 S C 3 A C 4 AF Gypsum
Mass fraction (%) 58.4 ± 9.72 14.5 ± 9.68 9.9 ± 2.6 7.1 ± 1.56 6.1

Table A.1 Mass fraction used, based on the cement paste used in [START_REF] Planel | Long-term performance of cement paste during combined calcium leachingśsulfate attack: kinetics and size effect[END_REF], with a water cement ratio of 0.4. The standard deviation is based on [START_REF] Stutzman | Uncertainty in Bogue-calculated phase composition of hydraulic cements[END_REF].

the C-S-H molar volume is hard to measure because it depends on the type of C-S-H [START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes[END_REF] as well as the calcium silica ratio of the phase [START_REF] Stora | Modelling and simulations of the chemośmechanical behaviour of leached cement-based materials: Leaching process and induced loss of stiffness[END_REF]). We study two values of C-S-H and the other molar volumes are summarized in

B C-S-H diffusion coefficient

The C-S-H diffusion coefficient used in the literature are listed in Table B.4. In case of two C-S-H phases, the C-S-H homogenized value is estimated using the Maxwell scheme. Note that we do not include the following values 19.4 × 10 -12 m 2 s -1 (Bogdan 2015) and 25 × 10 -12 m 2 s -1 [START_REF] Seigneur | Transport properties evolution of cement model system under degradation -Incorporation of a pore-scale approach into reactive transport modelling[END_REF]) because theses values are well beyond the mean value found in the literature. We find the mean value and standard deviation 4.64 ± 2.84 × 10 -12 .

C Mortar uncertainties

The section lists the ITZ property values used or measured in the literature.

Figure 1

 1 Figure1Multiscale heterogeneous microstructure of mortar, inclusions are assumed spherical.

Figure 2

 2 Figure 2(b) compares the diffusion coefficient to the experimental results reported in[START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF]. The considered C-S-H diffusion coefficient is equal to 8 × 10 -12 m 2 s -1 to fit the experimental result. The diffusion model can catch the effects of the microstructure and fits well the experimental results. Nevertheless, for a low water-cement ratio, the model overestimates the diffusion value due to the simplified representation of the C-S-H phases. As described in[START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF], the model needs to consider the influence of the water-cement ratio in the volume fraction of the different C-S-H phases (inner and outer).

Figure 2

 2 Figure 2 Effect of the water-cement ratio 𝑤/𝑐 on the material properties of cement paste: (a) elastic properties compared to the experimental results of Haecker et al. (2005) and (b) normalized diffusion properties compared to the experimental results of[START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF].

4. 1

 1 Uncertainty from the material properties 4.1.1 Input parameters Solid uncertainties at cement paste scale At cement paste scale, the porous medium is affected by the uncertainties associated with the solid phases (hydrates). The global sensitivity analysis led by Venkovic et al. (2013) have shown that the poroelastic properties are essentially impacted by the properties of C-S-H and portlandite. Based on the work of Venkovic et al.

Figure 3

 3 Figure 3 Uncertainty propagation from the material properties. Mean value and standard deviation obtained through Monte-Carlo simulation are respectively represented by data symbols and error bars: (a) poroelastic properties and (b) normalized diffusion properties. Legend: 𝐸 cp , 𝐸 m , 𝑏 m 𝑁 m , 𝐷 cp /𝐷 0 , and 𝐷 m /𝐷 0 .

Figure 4

 4 Figure 4 Uncertainty propagation from the microstructural properties (volume fractions and porosities). Mean value and standard deviation obtained through Monte-Carlo simulation are respectively represented by data symbols and error bars: (a) poroelastic properties and (b) normalized diffusion properties. Legend: 𝐸 cp , 𝐸 m , 𝑏 m 𝑁 m , 𝐷 cp /𝐷 0 , and 𝐷 m /𝐷 0 .

AdrienFigure 5

 5 Figure 5 Impact of material and microstructural uncertainties on the overall effective coefficient of cement paste and mortar for three water-cement ratios: (a) Young Modulus, (b) Biot coefficients, and (c) diffusion coefficients.

Figure 6

 6 Figure 6 Sensitivity analysis of Young Modulus: first order Sobol indices of (a) cement paste and (b) mortar.

Adrien

  Socié et al. Effects of the microstructural uncertainties on the poroelastic and the diffusive properties of mortar Diffusion coefficient Figure7shows the first order Sobol indices for the diffusion coefficient of the cement paste and the mortar.

Figure 7

 7 Figure 7 Sensitivity analysis of diffusion coefficient: first order Sobol indices of (a) cement paste and (b) mortar.

  Adrien Socié et al.Effects of the microstructural uncertainties on the poroelastic and the diffusive properties of mortar

Table 1

 1 and summarized in Table1. As previously depicted, Cement paste composition

	𝑤/𝑐	Ettringite	AFm	Portlandite	Unreacted cement	C-S-H
	0.3	0.074	0.072	0.152	0.137	0.494
	0.35	0.075	0.072	0.156	0.094	0.507
	0.4	0.072	0.073	0.155	0.064	0.509
	0.45	0.069	0.073	0.151	0.043	0.504
	0.5	0.065	0.075	0.144	0.03	0.494
	0.6	0.058	0.068	0.131	0.014	0.48

Table 2

 2 Material properties of mortar phases.

	-H Portlandite	4.64 × 10 -12 0	23.8 38	0.24 0.324	Constantinides and Ulm 2004; Venkovic et al. 2013, Table B.4 Haecker et al. 2005; Venkovic et al. 2013
	Ettringite, Katoite		0	22.4	0.25	Haecker et al. 2005
	Monosulfoaluminate	0	38	0.324	Haecker et al. 2005
	Anhydrous cement		0	117.6	0.314	Haecker et al. 2005
	Sand		0	62.4	0.21	Heukamp 2003
	Capillary pore ITZ	0 10.3 × 𝐷 cp 0.175 × 𝐸 cp 𝜈 cp 0 𝐷 0 = 2 × 10 -9	Bogdan 2015 Tables C.5 and C.6

Table 4

 4 Standard deviation of the normal law distribution associated to the material properties uncertainties. The mean values are specified in Table2.

Table 5

 5 Effect of the material properties uncertainties on the homogenized properties.

Table 6

 6 Adrien Socié et al.Effects of the microstructural uncertainties on the poroelastic and the diffusive properties of mortar impacts the overall properties through the capillary porosity 𝜙 = 1 -Standard deviation of the normal law distribution associated to the microstructural properties uncertainties. The mean values are specified in Table1for 𝑤/𝑐 = 0.4 and ITZ volume fraction of 0.185. The subscripts 'kat' and 'ett' respectively define the values associated with the katoite and ettringite.

	𝑁 sol
	𝜑 i .
	𝑖

Table 7

 7 Impact of the microstructural properties uncertainties on the homogenized properties.

  Table C.8.

	Poroelastic properties [GPa]	30 40 (a)
		20	𝜑 csh	𝜑 ch	𝜑 ett	𝜑 kat	𝜑 itz 𝜙 itz /𝜙 cp	𝜑 csh	𝜑 ch	𝜑 ett	𝜑 kat	𝜑 itz

Table 8

 8 Input parameters, properties uncertainties study.

  could form an extension of the present study. The total concentration of each main component is estimated on the mass fraction of each component of the clinker, denoted 𝑓 𝑚 𝑖 for the solid 𝑖, and the water-cement mass ratio 𝑤/𝑐:Ca 2+ = 3C C 3 S + 2C C 2 S + 3C C 3 A + C CaSO 4 + 4C C 4 AF (C Al(OH) - 4 = 2C C 3 A + 2C C 4 AF , 𝑀 𝑖 𝑓 𝑚 𝑖 𝑚 c𝑖 𝑉 total , ∀𝑖 ∈ [1, 4] (A.2)where 𝑖 is the species with corresponding molar mass 𝑀 𝑖 [kg mol -1 ], cement mass 𝑚 c𝑖[kg], cement density 𝜌 c𝑖 [kg m -3 ]; also 𝜌 w is the water density [kg m -3 ] and 𝑉 total is the total volume [m 3 ].

	(C aq tot ) aq tot ) H 2 SiO 2-4 = C C 3 S + C C 2 S (C aq tot ) SO 2-4 = C CaSO 4 (C aq tot ) (A.1)
	with (Planel 2002):
	𝑚 c𝑖 =	𝑉 total 𝜌 c𝑖 𝑤 𝑐 𝜌 c𝑖 𝜌 w + 1	; (C

aq tot ) 𝑖 =

  Table A.2. The results are Chemical equations log 10 (𝐾 sol ) Molar volume [m 3 mol -1 ] C-S-H(1.65) ⇔ 1.65Ca 2+ + 3.3OH -+ SiO 2 -2H 2 O Table A.2 Hydrate properties considered in the hydration model: chemical reactions, dissolution equilibrium constants log 10 (𝐾 sol ), and molar volumes (Planel 2002; Lothenbach et al. 2019).summarized in TableA.3. The solid aluminate phase, here the katoite, and the C-S-H area are impacted by the variation. Table A.3 Results obtained via the hydration computation.

	Portlandite Katoite Ettringite	-17.64 -5.19 -20.5 4 + 4OH -+ 26H 2 O -44.9 4 + 3Ca 2+ + 4OH -⇔ Ca 2+ + 2OH -⇔ 2Al(OH) -⇔ 2Al(OH) -4 + 6Ca 2+ + 3SO 2-	0.084, 0.078 0.033 0.707 0.15
	Phases		C-S-H	Portlandite	Ettringite	Katoite
	Volume fraction	0.374 ± 0.018	0.184 ± 0.005	0.144 ± 0.007	0.07 ± 0.011

  TableB.4 C-S-H diffusion coefficient 𝐷 csh used in the literature. In case of articles using two C-S-H phases, the homogenized value is estimated using Maxwell scheme.

	𝐷 csh [10 -12 m 2 s -1 ]	References
	2.2	Bernard and Kamali-Bernard (2012)
	3.2	Bary (2008)
	5.097	Stora et al. (2009)
	5.97, 1.23, 4, 2.66, 2.9	Patel et al. (2016)
	8	Socié (2019)
	11	Korb et al. (2007)

Journal of Theoretical, Computational and Applied Mechanics June 2022 jtcam.episciences.org

Acknowledgements Authors would like to thank Jean Baccou for his help and advice for the sensitivity analysis.

Ethics approval and consent to participate Not applicable.

Consent for publication Not applicable.

Competing interests

The authors declare that they have no competing interests.

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Effects of the microstructural uncertainties on the poroelastic and the diffusive properties of mortar