Yushan Gu

Patrick Dangla
email: patrick.dangla@ifsttar.fr

Othman Renaud-Pierre Martin
email: renaud-pierre.martin@univ-eiffel.fr

Omikrine Metalssi
email: othman.omikrine-metalssi@univ-eiffel.fr

Teddy Fen Chong

Othman Omikrine Metalssi

Teddy Fen-Chong
email: teddy.fen-chong@univ-eiffel.fr

Modeling the sulfate attack induced expansion of cementitious materials based on interface-controlled crystal growth mechanisms

Keywords: External Sulfate Attack, Delayed Ettringite Formation, Poromechanics, Correns equation, Ostwald-Freundlich equation, Crystal growth

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Sulfate attack is a well-known phenomenon of the degradation of concrete, which is associated with an expansion. The most common mechanism used to explain the expansion is the formation of ettringite in pores: capillary pores for External Sulfate Attack (ESA) [START_REF] Yu | Mechanism of expansion of mortars immersed in sodium sulfate solutions[END_REF], and gel pores for Delayed Ettringite Formation (DEF) [START_REF] Famy | Expansion of heat-cured mortars[END_REF][START_REF] Johansen | Simultaneous presence of alkali-silica gel and ettringite in concrete[END_REF]. Many pieces of research have been conducted on ESA and DEF separately due to their different sources of sulfate ions. ESA is described as a degradation inducing both an expansion of the material and a decalcification of C-S-H with the sulfate ions coming from external solutions. The formation of secondary sulfate-bearing phases further leads to the loss of mechanical performance. In DEF, the sulfates coming from the material react with Al-bearing phases to form the primary ettringite during the curing period. This primary ettringite is decomposed when the material is exposed to a heat treatment above approximately 65 • C [START_REF] Taylor | Delayed ettringite formation[END_REF], leading to the release of sulfate ions in the pore solution, which may be trapped on C-S-H [START_REF] Odler | Interaction between gypsum and the CSH phase formed in C 3 S hydration[END_REF][START_REF] Fu | Delayed ettringite formation in Portland cement products[END_REF]. In the subsequent cooling storage, the temperature drop provokes the formation of delayed ettringite, leading to a homogeneous expansion in the paste [START_REF] Johansen | Simultaneous presence of alkali-silica gel and ettringite in concrete[END_REF]. Despite the different sources of sulfate ions concerning ESA and DEF, their expansions are induced by ettringite formation. Even though some scholars attribute the damage to the formation of gypsum [START_REF] Tian | Does gypsum formation during sulfate attack on concrete lead to expansion?[END_REF], it happens in a condition where the concentration of sulfates is high [START_REF] Gollop | Microstructural and microanalytical studies of sulfate attack III. sulfate-resisting portland cement: Reactions with sodium and magnesium sulfate solutions[END_REF][START_REF] Schmidt | Physical and microstructural aspects of sulfate attack on ordinary and limestone blended portland cements[END_REF][START_REF] Marchand | Theoretical analysis of the effect of weak sodium sulfate solutions on the durability of concrete[END_REF] or the pH of the surrounding environment is low [START_REF] Ragoug | Durability of cement pastes exposed to external sulfate attack and leaching: Physical and chemical aspects[END_REF]. The formation of gypsum is even observed after the occurrence of cracks [START_REF] Santhanam | Mechanism of sulfate attack: A fresh look: Part 1: Summary of experimental results[END_REF], which means that the precipitation of gypsum is not the (main) reason for expansion. Therefore, we assume that ettringite formation is the only reason inducing damage in cement-based materials exposed to sulfate attacks. The precipitation of ettringite in the pores of the material leads to a tensile pressure, and cracks appear when this pressure is higher than the tensile strength of the samples [START_REF] Santhanam | Mechanism of sulfate attack: A fresh look: Part 1: Summary of experimental results[END_REF][START_REF] Flatt | Thermodynamics of crystallization stresses in DEF[END_REF]. The microstructure of materials, before and after sulfate attacks, has been analyzed in our previous work [START_REF] Gu | Pore size analyses of cement paste exposed to external sulfate attack and delayed ettringite formation[END_REF][START_REF] Gu | Locating ettringite due to DEF at the pore scale of cement paste by heat-based dissolution tests[END_REF] where we assumed that capillary pores range between 10 nm and 1 µm and gel pores have a diameter less than 10 nm based on the pore size classification in [START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF]. A common expansion mechanism for both ESA and DEF has been proposed, indicating that ettringite forms in the biggest capillary pores firstly without generating an obvious expansion and then penetrates smaller capillary pores and gel pores from which large expansion can be observed.

The crystal growth of ettringite is believed to be controlled by interface attachment kinetics rather than diffusion. Otherwise, the crystal/liquid interface would be at the solubility limit, and no pressure could be tolerated at the interface [START_REF] Scherer | Supersaturation in porous media[END_REF]. The chemical potential difference between the crystal and the liquid [START_REF] Adamson | Physical chemistry of surfaces[END_REF] drives the crystal growth, and the crystal stops growing toward the pore wall due to a large repulsive force between the crystal and the pore wall. At the interface of the crystal and the pore wall, a thin film of solution remains [START_REF] Correns | Growth and dissolution of crystals under linear pressure[END_REF], where the concentration is assumed to be the same as that of the remaining pore solution (fast diffusion). This concentration may exceed the equilibrium value of the tip of the crystal that is free to grow into the pore space. The crystal will keep on growing until its tip surface reaches a curvature corresponding to an equilibrium with the solution. In this way, the chemical potential is converted to mechanical work, the driving forces being the chemical activities of the reactants in the pore solution.

A considerable number of models have been proposed in the literature to simulate ESA. The volumetric expansion models have been proposed by linking the expansion to the net solid volume and the original capillary porosity [START_REF] Clifton | Sulfate attack of cementitious materials: volumetric relations and expansions[END_REF][START_REF] Ikumi | Simplified methodology to evaluate the external sulfate attack in concrete structures[END_REF]. A more precise model applied to calculate the volume of ettringite by considering the molar volumes of different components of cement paste, degree of hydration, and capillary porosity has also been proposed [START_REF] Tixier | Modeling of damage in cement-based materials subjected to external sulfate attack. i: Formulation[END_REF]. In this model, the ettringite, excluding part of it forming in capillary pores, has been considered as the cause of expansion. A continuum damage mechanics approach was adopted to evaluate the structural damage. A chemical system controlled by the sulfate and calcium species and a poroelasticity approach to describe concrete microcracking have been proposed in [START_REF] Bary | Simplified coupled chemo-mechanical modeling of cement pastes behavior subjected to combined leaching and external sulfate attack[END_REF]. Continuing this work, a fully coupled chemo-transfermechanical model considering the decalcification of hydrated phases such as leaching and the migration of sulfate ions has been proposed in [START_REF] Bary | Coupled chemo-transport-mechanical modelling and numerical simulation of external sulfate attack in mortar[END_REF]. The effective mechanical and diffusive properties have also been estimated in this paper. However, the elastic theory adopted in this model should be modified as the material is no longer in the elastic range when cracks occur. For the simulations concerning DEF, the models focus more on empirical studies. Some parameters characterizing the expansion behavior of materials have been proposed [START_REF] Brunetaud | Étude de l'influence de différents paramètres et de leurs interactions sur la cinétique de l'amplitude de la réaction sulfatique interne au béton[END_REF] to simulate the expansion curve that has a sigmoid shape. A model considering the early-age thermal history to predict the expansion has been established in [START_REF] Baghdadi | Effect of early age thermal history on the expansion due to delayed ettringite formation: experimental study and model calibration[END_REF], and no expansion is believed to be observed if the temperature remains below a threshold value. The thermodynamic equilibrium of ettringite, monosulphoaluminate, and portlandite, the sorption of ions into C-S-H under different environmental conditions of temper-atures, as well as moisture and leaching have been added in a physicochemical model to predict the behavior of concrete [START_REF] Salgues | DEF modelling based on thermodynamic equilibria and ionic transfers for structural analysis[END_REF]. The supersaturation that may result in ettringite formation and the crystallization pressures concerning DEF has been evaluated in [START_REF] Flatt | Thermodynamics of crystallization stresses in DEF[END_REF]. The crystallization pressure was found several times higher than the tensile strength of the samples, which leads to the damage of the material. However, the evolution of the expansion of specimens was not simulated by this approach. All these models have been proposed to investigate the most important factors concerning sulfate attacks, while the porous microstructure of materials has not yet been considered in them.

This paper proposes a poromechanical model founded on the mechanism of surface-controlled crystal growth and other physicochemical properties both for ESA and DEF in Portland cement-based materials, despite their different sulfate sources. The crystallization pressure of ettringite is believed to be the driving force responsible for expanding cementitious materials. The ettringite formation in pores will be explained and simulated in this paper, considering a pore invasion process and a deformation process. In the former process, the crystal of ettringite forms by invading the pore network from the largest to the smallest pores, while in the second process, it forms at the pore wall by deforming the pore. In the latter process, the damage of the material can occur so that the deformation includes the formation of new cracks that accommodate ettringite. Two kinetic constants A e and A p will be proposed to represent these two processes.

Damage poromechanical modeling

Crystals nucleate heterogeneously and randomly in the pores. Small crystals tend to dissolve before significant stress develops and transfer their solute to the larger crystals until crystals only exist in the largest accessible pores where stress can then eventually develop. The idea that stress can develop within small pores and small nuclei before the coarsening process has extended, has been investigated in [START_REF] Scherer | Factors affecting crystallization pressure[END_REF]. This idea is not retained here since we think that it could only occur upon high rate of desorption and thus upon high rate of supersaturation. In view of the time scales involved in the experimental observations and the fact that only saturated situations are considered in this model, we exclude such paths. The expansion of cement-based materials exposed to sulfate attacks is assumed to originate from the crystallization pressure exerted at the crystal surface in contact with the solid, after further growth of the ettringite is resisted by the pore wall. A thin film of solution of nanometric size is acknowledged to remain between the crystal and the solid wall [START_REF] Scherer | Crystallization in pores[END_REF], where the transport of ions can occur to feed the crystal. The film is assumed to remain in equilibrium with the free solution so that its composition is the same as that of the free pore solution. As a general rule the crystallization pressure is the pressure needed to stop the growth of crystals in equilibrium with a supersaturated solution. However in the current state of non equilibrium, the crystallization pressure could not be able to stop the growth of the crystal. To quantify the departure from equilibrium, we define the saturation index of the supersaturated solution, β p , that would be in equilibrium with the current crystallization pressure, P C . β p indicates the ratio of the ion activity product to the solubility product constant. This equilibrium results from the thermodynamical equilibrium between the solid crystal submitted to a normal pressure P C and the dissolved crystal ions in a liquid solution of pressure P L . This thermodynamical equation results in the well known Correns' equation [START_REF] Correns | Growth and dissolution of crystals under linear pressure[END_REF] under the form:

P C -P L = RT V C ln(β p ) ( 1
)
where R is the ideal gas constant, T is the temperature, V C is the molar volume of crystals, P L is the pressure of the liquid solution where the crystal can dissolve. So the notation β p is no more than the saturation index associated, through equilibrium, to the current crystallization pressure, P C , exerted on the crystal at the crystal-solid interface. If the current saturation index of the crystal in the surrounding solution, β, is greater (resp. lower) than β p , the crystal is not in equilibrium.

It will grow (resp. dissolve) and increase (resp. decrease) its crystallization pressure. The growth will stop until β = β p . The following section will give the kinetics of the crystal growth caused by the difference between these two saturation indexes.

In this research, a poromechanical approach is used to model the constitutive behavior of cement paste submitted to the pore pressures P C and P L . The porous space of material is assumed to be saturated by the liquid and the solid crystal only, i.e., no air occupies the porous space, which means that unsaturated situations are not considered in this model.

It is assumed that the mechanical behavior of the material is governed by an effective stress tensor, σ ij , responsible for the strain ij . The stress-strain relationship can be modeled in the framework of the plasticity or damage theories. Even though both of them could have been chosen, in this paper we will focus on the isotropic damage modeling, and the stress-strain relationship is given by [START_REF] Kondo | Basic concepts and models in continuum damage mechanics[END_REF][START_REF] Pijaudier-Cabot | Damage models for concrete[END_REF]:

E d ij = (1 + ν)σ ij -νσ kk δ ij (2)
where E d = (1-d)E, E is the Young's modulus of undamaged material and d is the damage variable;

σ kk represents the trace of the tensor. The loading function of damage is f

( ij , κ) = ˜ ( ij ) -κ,
where ˜ is a scalar measure of the strain level called equivalent strain. Usually it is defined as

˜ ( ij ) = (< 1 > 2 + < 2 > 2 + < 3 > 2 ) 1/2
, where i are the principal values of the strain tensor and < • > are the McAuley brackets denoting the positive part. κ is the current elastic limit expressed in terms of equivalent strain, and its initial value κ 0 is related to the tensile strength f t of the material in uniaxial tension, f t = Eκ 0 , as illustrated in Fig. 1. Damage can grow only if the current state reaches the boundary of the elastic domain, f = 0. The evolution of the damage and threshold variables can be expressed through the so-called flow laws as:

d dt (d) = λ ; d dt (κ) = λh κ ( ij , κ) (3)
and subjected to the classical Kuhn-Tucker conditions:

f ≤ 0 ; λ ≥ 0 ; f λ = 0 (4)
An integrated form of Eq (3) which defines implicitly the function h κ [START_REF] Jirásek | Consistent tangent stiffness for nonlocal damage models[END_REF], is:

d = 0 if κ ≤ κ 0 (5) d = 1 - κ 0 κ exp(- κ -κ 0 κ f -κ 0 ) if κ > κ 0 (6)
where κ f is a parameter affecting the ductility of the response (see Fig. 1) and is related to the local dissipation density g t .

g t can be expressed numerically as the area under the uniaxial stress-strain curve (g t = f t (κ fκ 0 /2)), and it has the meaning of energy dissipated per unit volume of the totally damaged material under uniaxial tension. It can be calculated through the fracture energy G f and the width of crack band w via: G f = g t w [START_REF] De Borst | Numerical aspects of cohesive-zone models[END_REF]. Typical values for cement-based materials are given in Tab. 1. Fracture energy (N/m) G f 15 [START_REF] Petersson | Crack growth and development of fracture zones in plain concrete and similar materials[END_REF] 75 [START_REF] Petersson | Crack growth and development of fracture zones in plain concrete and similar materials[END_REF] Crack width (µm) w 2 [START_REF] Attiogbe | Submicroscopic cracking of cement paste and mortar in compression[END_REF] 3 [START_REF] Petersson | Crack growth and development of fracture zones in plain concrete and similar materials[END_REF] Local dissipation density (kPa) gt 7.5 -25.0 -

We will adopt the following expression for the effective stress tensor:

σ ij = σ ij + b d (S L P L + S C P C )δ ij (7)
where σ ij is the stress tensor of the material; S L and S C are saturation degrees concerning the liquid and crystal, i.e. the volume fraction of liquid and crystal to the initial pore volume. The parameter b d is the Biot's coefficient of the damaged material. In accordance with the homogenization theory linking the Biot's coefficient and the bulk modulus of the solid matrix, the damaged biot's coefficient is assumed to be given by a simple law b d = b + d(1 -b) where b is the Biot's coefficient of the undamaged material.

Modeling the formation of crystals in pores and cracks

Crystal forms in the existing pores of the material and in the cracks generated during the damage process. We define the volume fraction (or the partial porosity) of crystals φ C such that φ C dV 0 is the porous volume that the crystal occupies in the current deformed volume element dV t of the porous material, the volume of which is dV 0 in its initial undeformed state. If the material were rigid, the volume φ C dV 0 = φ 0 S C dV 0 would only result from the invasion of the porous volume or the receding process from it under the action of the surface tension γ CL between the crystal and the solution. In a deformable porous material, the volume φ C dV 0 now results from a twofold process:

the possible invasion of the porous volume or the receding process from it; and the deformation of the porous volume that the crystal currently occupies under the action of the crystallization pressure resulting in the partial porosity change ϕ C dV 0 . So accordingly, the partial volume fraction of crystals can be split into two parts [START_REF] Coussy | Revisiting the constitutive equations of unsaturated porous solids using a Lagrangian saturation concept[END_REF]:

φ C = φ 0 S C + ϕ C (8)
The first term on the right hand side φ 0 S C is the volume fraction filled by crystals before any deformation, thus, related to the invasion process. The second term ϕ C is the volume fraction filled by crystals due to the deformation of pores, including the cracks generated during the damage process. The rate of the latter term is assumed to be a fraction of the volumetric strain rate:

d dt (ϕ C ) = χ C d dt ( kk ) (9)
where we assume here that χ C = S C for the sake of simplicity.

The invasion process corresponds to the displacement of the crystal surface in contact with the liquid through the pore entrance, which depends on the pore size distribution (PSD) of the material. The crystal-liquid interface, at the tip of the crystal, is located at the pore entrance and is submitted to the liquid pressure P L . Its mean radius of curvature is denoted with r and is closely linked to the radius of the entry pore. Consistently with what was done for the crystal-solid interface, in the current state, the crystal tip could not be in equilibrium with the solution. To quantify the departure from equilibrium, we define the saturation index of the supersaturated solution, β e , that would be in equilibrium with the current radius of curvature r. Their relationship is derived from the well-known Ostwald-Freundlich equation:

RT V C ln(β e ) = 2γ CL r ( 10
)
where γ CL is the surface energy of the crystal. So the notation β e is no more than the saturation index associated, through equilibrium, to the current radius of curvature, r, of the crystal-liquid interface at the tip of the crystal. Eq. 10 expresses a thermodynamic equilibrium and tells us that the solubility of the crystal increases with the inverse of its size so that a small crystal is less stable than a larger one. The consequence is that the precipitation of crystals will first occur in the biggest pores and will penetrate ever smaller ones. We must emphasize that this process does not take into account the nucleation stage, which is a complex process occurring randomly on the surface of the pore network. The modeling of this nucleation process is out of the scope of this paper. The random nucleation process is expected to be followed by the growth of nuclei. After some time, the biggest nuclei located in the biggest pores will swallow the other nuclei through the well-known Ostwald ripening process. So some time after the nucleation process has started, it turns out that the crystal precipitates first in the biggest pores and then penetrates ever-smaller pores consistently with the Ostwald-Freundlich equation. Even if this nucleation stage may take some time, it is assumed that this stage is not critical for modeling the ESA/DEF-induced expansion processes. The variation of PSD of cement paste before and after sulfate attacks performed in [START_REF] Gu | Pore size analyses of cement paste exposed to external sulfate attack and delayed ettringite formation[END_REF] is consistent with this analysis.

Therefore the crystal saturation S C should be related to the volume of pores larger than a given size.

The PSD of the porous material can be characterized by a function S(r) representing the volume fraction occupied by pores having a pore entry radius lower than r. S(r) is usually characterized by Dynamic Vapor Sorption (DVS) techniques. As stated previously, whether ESA or DEF is considered, we assume that the crystal phase forms in the largest pores, i.e., those which have a pore radius larger than the radius of curvature of the crystal-liquid interface r:

S C (r) = 1 -S(r) (11)
Here, for the sake of simplicity and without loss of generality, we assume that the curvature radius of the crystal at the crystal-solid interface equals to the pore radius. S(r) can be approximated from Van Genuchten model [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF] as:

S(r) = (1 + (r 0 /r) 1/(1-m) ) -m (12)
Typical values for r 0 and m are given in Tab. 2.

Crystallization-induced expansion kinetics

The mechanism of the expansion kinetics in a porous material whose pores are filled with a supersaturated solution is now envisioned as follows, and an illustration of the crystal growth in pores is shown in Fig. 2. The crystal is separated from the pore wall by a film of solution of a nanometric size where ions can transport and make the crystal grow in the direction of the pore wall [START_REF] Correns | Growth and dissolution of crystals under linear pressure[END_REF][START_REF] Coussy | Deformation and stress from in-pore drying-induced crystallization of salt[END_REF]. This interface between the solid and the solution film is named the crystal-solid interface to be distinguished from the other interface that has direct contact with the pore solution. At the crystal-solid interface the crystal will reach equilibrium rapidly (i.e. β = β p ) by exerting a stress on the wall according to Eq. 1 and stop growing, see Fig. 2(a). The crystal-liquid interface undergoes a pore invasion process depending on the PSD, which describes crystals penetrating from larger pores into smaller ones. At a given time, the current saturation index of the surrounding solution β is greater (resp. lower) than β e , indicating that the crystal is not in equilibrium. It will grow (resp. dissolve), and the radius of curvature of the crystal-liquid interface will then decrease (resp.

increase) until β e = β, see Fig. 2(b). The pore invasion process leads to increasing effective stress according to Eq. 7. When the material cannot resist the stress, cracks occur, and a deformation process induced by the occurrence of cracks starts (Fig. 2(c)) according to the damage mechanism described in section 3. The crystal growth, in the deformation process, consumes the solute in the adjacent liquid, which may lead to the dissolution of the crystal at a small entry radius, as mentioned in [START_REF] Scherer | Factors affecting crystallization pressure[END_REF]. At the crystal-solid interface, crystallization pressure decreases, allowing the growth of crystal towards the pore wall, until the crystal at the crystal-solid interface reaches equilibrium again, as illustrated in Fig. 2(d).

Let us now model the growth rate at the crystal-solid and crystal-liquid interfaces. It is assumed that the growth rate is governed by an interface-controlled mechanism. The growth rate of the interface Y is given by [START_REF] Kirkpatrick | Crystal growth from the melt: A review[END_REF]:

Y = Y 0 (1 -exp(- G r RT )) (13)
where the constant Y 0 can be evaluated from Y 0 = f k B T 3πλ 2 η with f : the fraction of sites on the crystal surface available for attachment, k B : the Boltzmann constant, λ: the thickness per molecular layer, and η: the viscosity of the solution. Typical value for Y 0 is 0.64 m/s (λ = 1 nm, η = 0.001 Pa.s, T = 293 K). In Eq. 13, G r is the reaction Gibbs energy of the crystal dissolution reaction, i.e. G r = i ν i µ i -µ C where µ i is the chemical potential of ions in solution, ν i is the stoichiometric coefficients and µ C is the chemical potential of the solid crystal at the curved interface. The latter term can be related to that of a flat interface through the Gibbs-Thomson equation,

µ C = µ ∞ C + 2γ CL V C /r at the crystal-liquid interface or µ C = µ ∞ C + (P C -P L )V C at the crystal-solid interface.
It turns out that G r can be expressed as G r = RT ln(β/β e ) at the crystal-liquid interface and G r = RT ln(β/β p ) at the crystal-solid interface. Two kinetic laws, in the same form as Eq. 13, can be formulated to describe the rate of precipitated crystals (in mol/L) at the two interfaces under the form:

d dt (n e ) = a e (S C )(1 - β e β ) at the crystal -liquid interface; (14)
d dt (n p ) = a p (S C )(1 - β p β ) at the crystal -solid interface. ( 15
)
where n e is the crystal mole content per unit volume of the material formed during the pore invasion process by the accretion of new crystals at the crystal-liquid interface (entry pore), and n p is the crystal mole content per unit volume of the material formed during the pore deformation process by the accretion of new crystals at the crystal-solid interface (pore wall). It is worth to mention that the crystal growth is assumed to be slower than the diffusion of reactant ions, which means that the crystal growth is not affected by the diffusion process. Moreover, the nucleation process before crystals exert pressure on the pore wall has not been taken into account, and no impingement among crystals has been considered in the model. Assuming a constant molar volume of the crystal, we can infer from Eq. 8 the relations

V C n e = φ 0 S C and V C n p = ϕ C .
Here it is clearly assumed that the deformation process of pores ϕ C is mainly caused by cracking. Since the growth rate linearly depends on the surface area of interfaces, the two functions a e (S C ) and a p (S C ) are expected to model the evolution of these surface areas with the volume fraction of the crystal S C . Due to the lack of knowledge, in the following we will assume that (i) the surface area of the crystal-liquid interface is constant, so that V C a e (S C ) = φ 0 A e and (ii) the surface area of the crystal-solid interface, is proportional to the crystal saturation so that V C a p (S C ) = A p S C . From the kinetic laws (Eqs.

14 and 15) and the above assumptions:

d dt (S C ) = A e (1 - β e β ) (16)
d dt (ϕ C ) = A p S C (1 - β p β ) ( 17
)
where A e and A p are kinetic constants in the pore invasion and pore deformation processes, respectively.

According to Eq. 17 the crystallization-induced pore deformation stops when the crystal at the crystal-solid interface is in equilibrium (β = β p ). This equilibrium state is reached rapidly in the elastic range because the crystallization pressure rises very quickly upon crystal growth in such a confined situation. However, this equilibrium breaks when cracks occur. Without the constraint of the pore wall, when cracks occur, the saturation index of crystals at the crystal-solid interface drops quickly, according to Eq. 1. Two subsequent, but highly important, situations will occur:

• If the source of reactant ions is unlimited (e.g., in the case of ESA), the saturation index of crystals in the pore solution (β) is kept high and constant, crystals at the crystal-solid interface will always be out of equilibrium (e.g., in Fig. 2(c)) and the crystal will keep on growing;

• If the source of reactant ions is limited (e.g., in the case of DEF), it means that the saturation index of crystals in the pore solution (β) will decrease during crystallization.

Crystals at the crystal-solid interface may obtain equilibrium again (e.g., in Fig. 2(d)), from which moment crystals stop growing and the swelling reaches a plateau.

Effect of heat treatment on cement composition

Given the composition of oxide components of Portland cement (like SiO 2 , CaO, Al 2 O 3 , SO 3 ), the mole balance equations for oxides are employed to get the content of hydrates in cement under the forms:

n SiO2 = n C-S-H (18)
n CaO = n CH + 1.7 n C-S-H + 3 n C3A + 6 n AFt + 4 n AFm (19)
n Al2O3 = n C3A + n AFt + n AFm (20)
n SO3 = 3 n AFt + n AFm + n ads SO3 + n rel SO 2- 4 (21)
where n i stands for the mole content of i per unit volume of the material, n rel SO 2- 4 and n ads SO3 stand for the mole content per unit volume of the material of the released sulfate ions from heat treatment and the ones adsorbed on C-S-H. In the above equations we have neglected the free Si, Ca and Al dissolved in the pore solution. After the normal hydration of Portland cement and before any further heat treatment, the hydrated cement is assumed to be composed of CH, C-S-H, C 3 A, AFt only, i.e. with no AFm (ettringite is more stable in a cement system than AFm at ambient temperature [START_REF] Damidot | Thermodynamic investigation of the CaO-Al 2 O 3 -CaSO 4 -H 2 O system at 25 • C and the influence of Na 2 O[END_REF]) and no released and adsorbed sulfates: n AFm = n rel SO 2- 4 = n ads SO3 = 0. Considering this state as the initial state of cement (at t = 0), we get from the above mole balance equations:

n C-S-H (0) = n SiO2 ( 22
)
n CH (0) = n CaO -1.7 n SiO2 -3 n Al2O3 -n SO3 (23)
n C3A (0) = n Al2O3 - 1 3 n SO3 (24)
n AFt (0

) = 1 3 n SO3 (25)
n AFm (0) = 0 (26)

n rel SO 2- 4 (0) = 0 ( 27
)
n ads SO3 (0) = 0 ( 28
)
When the material is exposed to a heat treatment with a temperature above the critical value, the primary ettringite is transformed into AFm. We assume that this transformation happens to all the primary ettringite, due to a lack of experimental measurements. The decomposition of ettringite releases sulfate ions, so that the cement paste after the curing process (taking place after

n C-S-H (t c ) = n SiO2 (29)
n CH (t c ) = n CaO -1.7 n SiO2 -3 n Al2O3 - 1 3 n SO3 (30)
n C3A (t c ) = n Al2O3 - 1 3 n SO3 (31)
n AFt (t c ) = 0 (32)

n AFm (t c ) = 1 3 n SO3 (33)
n ads SO3 (t c ) + n rel SO 2- 4 (t c ) = 2 3 n SO3 (34)
We can infer from these equations that the content of released sulfates and the ones adsorbed on C-S-H equals two times of the content of AFm, namely,

n ads SO3 (t c ) + n rel SO 2- 4 (t c ) = 2 n AFm (t c ).
Following [START_REF] Barbarulo | Chemical equilibria between C-S-H and ettringite, at 20 and 85 • C[END_REF], sulfates are bound on C-S-H so that the quantity n ads SO3 previously introduced can be written in the form:

n ads SO3 = n C-S-H B SO3 (35)
where B SO3 is the fraction of bound sulfates or the S/Si ratio i.e. the total amount of adsorbed sulfates onto C-S-H per unit of mole of C-S-H. The S/Si ratio can be related to the sulfate concentration in the pore solution, ρ SO 2- 4 (see Fig. 4 of [START_REF] Barbarulo | Chemical equilibria between C-S-H and ettringite, at 20 and 85 • C[END_REF]), and the adsorption isotherm of SO 2- 4 ions on C-S-H is governed by a Langmuir law [START_REF] Divet | Delayed ettringite formation: the effect of temperature and basicity on the interaction of sulphate and C-S-H phase[END_REF]. A relationship similar to that established for chloride bindings [START_REF] Nguyen | Physicochemical modelling of chloride ions ingress in cementitious materials[END_REF], the binding isotherm of which also follows the Langmuir law, can be proposed under the form:

B SO3 = B 0 ρ SO 2- 4 ρ 0 + ρ SO 2- 4 (36)
Typical values are: B 0 = 2.55 and ρ 0 = 0.87 mol/L (for pH=13) [START_REF] Divet | Delayed ettringite formation: the effect of temperature and basicity on the interaction of sulphate and C-S-H phase[END_REF]. This equilibrium depends on the temperatures, e.g. the amount of bound sulfates at 85 • C is 10 times more than the one at 20 • C. It means that putting a system equilibrated at 85 • C to a decreased temperature of 20 • C breaks its equilibrium and leads to supersaturated ettringite in solution (see Fig. 5 in [START_REF] Barbarulo | Chemical equilibria between C-S-H and ettringite, at 20 and 85 • C[END_REF]).

Determination of the saturation index of ettringite and its evolution with time

Ettringite forms from the reaction of sulfate, calcium and aluminium ions. In this model, aluminium is provided by C 3 A-rich hydration products in case of ESA or AFm in case of DEF as commonly acknowledged in the literature [START_REF] Tixier | Modeling of damage in cement-based materials subjected to external sulfate attack. i: Formulation[END_REF][START_REF] Tixier | Modeling of damage in cement-based materials subjected to external sulfate attack. ii: Comparison with experiments[END_REF]. Examples of such reactions are:

C 3 AH 6 + 3 Ca 2+ + 3 SO 2- 4 + 26 H 2 O → C 3 A(C S) 3 H 32 (37)
C 3 A(C S)H 12 + 2 Ca 2+ + 2 SO 2- 4 + 20 H 2 O → C 3 A(C S) 3 H 32 (38)
However in seldom cases some already present ettringite evenly distributed within the microstructure re-precipitates in large pores without the need of C 3 A-rich products. This kind of ettringite is neglected in this model. It is assumed that C 3 A in Portland cement is present in the form of C 3 AH 6

(the most stable form of hydrogarnets [START_REF] Damidot | Thermodynamic investigation of the CaO-Al 2 O 3 -CaSO 4 -H 2 O system at 25 • C and the influence of Na 2 O[END_REF]). We must note however that probably a wider range of hydrogarnets can be formed during different conditions of hydration (e.g. autoclaved cements).

Denoting with β j the saturation index of j and with K j the solubility product constant of j, the saturation indexes are defined as:

β AFt = (a Ca 2+ ) 6 (a Al 3+ ) 2 (a SO 2- 4 ) 3 (a OH -) 12 (a H2O ) 26 /K AFt ( 39
)
β AFm = (a Ca 2+ ) 4 (a Al 3+ ) 2 (a SO 2- 4 )(a OH -) 12 (a H2O ) 6 /K AFm ( 40
)
β C3AH6 = (a Ca 2+ ) 3 (a Al 3+ ) 2 (a OH -) 12 /K C3AH6 (41)
β CH = (a Ca 2+ ) (a OH -) 2 /K CH ( 42
)
where a j stands for the activity of j in the pore solution and approximated here to its concentration, ρ j in mol/L. The water activity a H2O is assumed to be equal to unity. Additionally the ion concentrations must satisfy the electroneutrality of the solution:

2 ρ Ca 2+ + 3 ρ Al 3+ + ρ Na + = 2 ρ SO 2- 4 + ρ OH - (43)
where the concentration of sodium ion ρ Na + in the pore solution is considered to be known.

The calcium needed to form ettringite is provided by portlandite, and the portlandite is assumed to be at the solubility limit once it is not depleted, i.e. β CH = 1. When all portlandite is consumed, its saturation index is lower than 1. Therefore we add some Kuhn-Tucker conditions as follows:

n CH ≥ 0 ; β CH ≤ 1 ; n CH (β CH -1) = 0 (44)
We then end up with the above 6 equations (Eqs. 39 -44) and 8 unknowns: 4 concentrations

(ρ Ca 2+ , ρ Al 3+ , ρ SO 2-

4

, ρ OH -) and 4 saturation indexes (β AFt , β AFm , β C3AH6 , β CH ). 2 more equations are required to determine the saturation index of ettringite β AFt , which is the input for the poromechanical modeling. The nature of these two additional equations depends on the case that is dealt with, either ESA or DEF.

ESA case

It is assumed here that the aluminium for ettringite formation is provided by hydrogarnets C 3 AH 6 (so it is the only form of C 3 A and the reaction to form ettringite is shown in Eq. 37). C 3 AH 6 is assumed to be at the solubility limit, i.e. β C3AH6 = 1 when it is not depleted and otherwise β C3AH6 < 1. Kuhn-Tucker conditions can be written as follows:

n C3A ≥ 0 ; β C3AH6 ≤ 1 ; n C3A (β C3AH6 -1) = 0 (45)
The sulfates are provided by the external solution, and thus the sulfate concentration ρ SO = n ads SO3 = 0) that C 3 A and CH contents are not depleted:

n C-S-H (t) = n SiO2 (46) n CH (t) = n CaO -1.7 n SiO2 -3 n Al2O3 -3 n AFt (t) (47)
n C3A (t) = n Al2O3 -n AFt (t) (48)
n AFm (t) = 0 (49)

n rel SO 2- 4 = 0 (50)
n ads SO3 (t) = 0 (51)
The hypotheses that β CH = β C3AH6 = 1 hold as long as n CH (t) > 0 and n C3A (t) > 0 respectively, otherwise Eqs. 44 and 45 must be used.

DEF case

It is assumed here that the aluminium forming ettringite is provided by AFm (see Eq. 38), which is decomposed from the primary ettringite during heat treatment. AFm is at the solubility limit, i.e. β AFm = 1 when AFm is not depleted. Again, Kuhn-Tucker conditions associated to AFm can be written as follows:

n AFm ≥ 0 ; β AFm ≤ 1 ; n AFm (β AFm -1) = 0 (52)
One additional equation is given by adsorption law linking the content of adsorbed sulfates with the C-S-H content and the concentration of free sulfates in the pore solution under the form:

n ads SO3 (t) = n C-S-H (t) B SO3 (t) (53)
Table 3: Set of mechanical constitutive equations and chemical equations with inputs marked in red and outputs marked in blue. For the sake of simplicity, we assumed that the liquid pressure P L = 0. The output β (the saturation index of ettringite) from the chemical set of equations serves as an input for the mechanical part. In the case of DEF, the output φ C (the volume fraction of ettringite crystal) serves as input for the chemical part. After discretization in time steps, the algorithm switches from one set to the other until convergence.

Mechanics Chemistry

Inputs:

β = βAFt, PL = 0, σij or ij Inputs: ρ SO 2- 4 (ESA) or φC (DEF)
Equations Equations

E d ij = (1 + ν)σ ij -νσ kk δij βAFt = (ρ Ca 2+ ) 6 (ρ Al 3+ ) 2 (ρ SO 2- 4 ) 3 (ρ OH -) 12 /KAFt σ ij = σij + b d (SLPL + SC PC )δij βAFm = (ρ Ca 2+ ) 4 (ρ Al 3+ ) 2 (ρ SO 2- 4 )(ρ OH -) 12 /KAFm E d = (1 -d)E βC 3 AH 6 = (ρ Ca 2+ ) 3 (ρ Al 3+ ) 2 (ρ OH -) 12 /KC 3 AH 6 b d = b + d(1 -b) βCH = (ρ Ca 2+ )(ρ OH -) 2 /KCH f ( ij , κ) = ˜ ( ij ) -κ ≤ 0 2 ρ Ca 2+ + 3 ρ Al 3+ + ρ Na + = 2 ρ SO 2- 4 + ρ OH - d = 1 -κ 0 κ exp(-κ-κ 0 κ f -κ 0 ) nCH ≥ 0 ; βCH ≤ 1 ; nCH(βCH -1) = 0 RT V C ln(βp) = PC -PL --------------------- RT V C ln(βe) = 2γ CL r Additional equations for ESA d dt (SC ) = Ae(1 -βe β ) nC 3 A ≥ 0 ; βC 3 AH 6 ≤ 1 ; nC 3 A(βC 3 AH 6 -1) = 0 d dt (ϕC ) = ApSC (1 - βp β ) --------------------- SL(r) = (1 + (r0/r) 1/(1-m) ) -m
Additional equations for DEF

SC (r) = 1 -SL(r) nAFm ≥ 0 ; βAFm ≤ 1 ; nAFm(βAFm -1) = 0 φC = φ0SL + ϕC n ads SO 3 = nC-S-HB0ρ SO 2- 4 /(ρ0 + ρ SO 2- 4 ) d dt (ϕC ) = SC d dt ( kk ) n ads SO 3 + ρ SO 2- 4 φ = 2/3 nSO 3 -2 nAFt nAFt = VC φC Outputs: φC , ij or σij Output: β = βAFt

Simulations of experiments in literature

In this section, the expansive behavior of Portland cement-based materials exposed to ESA and DEF in restrained and free conditions is simulated. The mechanical properties and the porosity of materials are adopted from Tab. 1. The parameters, which represent the pore structure of materials and determine their saturation degree of the crystal, are referred to Tab. 2.

Simulation of ESA in restrained conditions

We simulate hollow cement paste cylinders exposed to ESA in restrained conditions as studied in experiments conducted by Ma et al. [START_REF] Ma | Experimental and numerical study on cement paste degradation under external sulfate attack[END_REF]. In these experiments, three types of longitudinal restraints were applied on the hollow cement paste cylinders (CEM I 42.5 N) employing a spring and steel bars (middle diameter 3 mm for low-restraint and 7 mm for high-restraint) through the specimens to facilitate non-, low-and high-restraint conditions. PVC molds with stainless steel rods in the center were made to obtain the thin-walled hollow cement paste cylinders. The thin-walled hollow cylinders were firstly immersed in saturated limewater at 20 • C for 90 days, then together with three types of restraints immersed in sodium sulfate solutions that had the concentrations of 1.5 g/L and 30 g/L, respectively. Strain gauges were glued on the middle part of the stainless steel bars to measure the local restrained expansion in the axial direction.

In these experiments, the uni-axial compression is assumed to result from poroelasticity (no damage) so that the stress-strain equation can be expressed as:

σ zz -ν(σ rr + σ tt ) = E zz ( 60
)
Although there is a fretting effect near the end faces of the specimen, its middle section is considered in an uni-axial stress state for the simplicity:

σ zz = σ zz + bS C P C , σ rr = bS C P C , σ tt = bS C P C .
The previous equation turns into:

σ zz = E zz -(1 -2ν)bS C P C (61)
Both the axial strain and the compressive axial stress were measured by Ma et al. and are plotted in Fig. 3(a-b). The linear relationship between the compressive axial stress and the axial strain actually stems from the linear elasticity of the steel bar so that the tensile force sustained by the steel bar is the product of the cross section area of the specimen and -σ zz . Therefore σ zz = -k zz where the slope k is around 9.257 GPa from Fig. 3(a) and so: The generated stress under low restraint and the corresponding simulated results based on Eq. 62 are presented in Fig. 3(b). In this figure, the compressive axial stress is presented as positive. A kinetic constant A e = 4.0 × 10 -8 s -1 is calibrated from the generated stress of specimen 3mm-30g/L, in which case the diffusion process is faster and is assumed completed. Then this constant

440 σ zz = -(1 -2ν)bS C P C /(1 + E/k) (62

445

A e is used to predict the behavior of another specimen in the case of 3mm-1.5g/L. It shows that the predicted axial stress for 3mm-1.5g/L is lower than that for 3mm-30g/L as observed from the experimental measurements after 18 days.

Simulation of DEF in confined conditions

We simulate here the confining stress which is needed to stop DEF as studied by Burgher et al.

[51] through the use of Hoek cells, where a continual supply of water was set to allow for water to circulate to and from the ends of cylinders. The concrete cylinders (8.57 cm in diameter by 17.78 cm in length) were fabricated and exposed to DEF with a maximum temperature of 96 • C for 12h, and then they were cured in limewater for long-term storage at 23 • C. Two cylinders (MC1 and MC2) were fabricated and cured in the same way. They were removed from limewater at approximately 110 days with expansions: 0.61% for MC1 and 0.51% for MC2, indicating that they were damaged before being placed under mechanical stress. The cylinders were placed inside Hoek cells, initial axial and confining pressures were applied on them to ensure that the cylinders were tightly confined by the cells. The evolution of axial and confining pressures was recorded during the test with the formation of delayed ettringite. The confining pressure of the hydraulic oil increased until it reached a level of confining pressure at which the internal stress was balanced by the external pressure (see Fig. 4(a)). However, a lesser degree of confining stress was exerted on specimen MC1, which the authors suspected due to a hydraulic oil leak.

The specimens were situated in a confined condition with axial stress σ zz , radial stress σ rr , and hoop stress σ tt (σ rr = σ tt = confining stress, see Fig. 4(a)). Again the specimens are assumed to have a poroelastic response. No axial strain zz was measured in the experiment, one possible explanation could be that Hoek cells have a high stiffness and the axial strain can be neglected.

Then, we assume in the axial direction:

σ zz -ν(σ rr + σ tt ) = E zz = 0 (63)
which gives:

σ zz = 2νσ rr -(1 -2ν)bS C P C (64)
First, a kinetic constant A e = 6.2 × 10 -8 s -1 is calibrated from the axial stress of specimen MC1 according to Eq. 64 considering that the measured confining stress σ rr is prescribed by the experimental setup and given in Fig. 4(a). Then, the same value of A e is used to predict the axial stress for MC2, see Fig. (b) Simulated axial stress of cylinders [START_REF] Burgher | Investigation of the internal stresses caused by delayed ettringite formation in concrete[END_REF]. axial stress for MC2 is a bit larger than that for MC1. This is also observed in the measurements but to a bit larger extent.

ρ SO 2- 4 (tc) (mol/L) n ads SO 3 (tc) (mol/L) Ae (s -1 ) MC1 0.075 0.283 6.2 × 10 -8 MC2 0.075 0.283 6.2 × 10 -8

Simulation of ESA in free swellings

We simulate the free swelling of mortar due to ESA as conducted in [START_REF] Ferraris | Mechanics of degradation of portland cementbased systems by sulfate attack[END_REF]. The sulfate concentration

ρ SO 2- 4
is known based on the experimental conditions. It is clearly observed from the experimental 480 swelling that the curve includes two parts: one slowly increasing part and one accelerated part (see, e.g., Fig. 5). It turns out from our modeling that the former part is governed by the pore invasion process when the material is in elastic state, and the latter part is mainly governed by the pore deformation process when cracks are appearing. The connection point of these two parts corresponds to the threshold of damage. Therefore, the strain corresponding to this point should 485 be fixed from the swelling curves first. Then, two kinetic constants A e and A p could be calibrated from the increase rate of expansion which, in a free stress condition, is given by:

zz = b d 3K d S C P C (65)
Ferraris et al. studied ESA by exposing mortar specimens to sodium sulfate solutions with different pH conditions and different geometries [START_REF] Ferraris | Mechanics of degradation of portland cementbased systems by sulfate attack[END_REF]. Some of their experimental results are selected and presented in this study. The mortar prisms (25 × 25 × 276 mm 3 ) were stored in pH uncontrolled and pH = 7 conditions with an initial concentration of sodium sulfate solution of 5% by weight.

Then the initial concentration of sulfate ions and alkalis are:

ρ SO 2- 4 = 5% ρ liquid /M Na2SO4 and ρ Na + = 2 ρ SO 2- 4
, where M Na2SO4 is the molar mass of Na 2 SO 4 (142 g/mol), and ρ liquid is the density of the liquid (assumed to be 1 g/mL). In this way, ρ SO 2- 4 is 0.35 mol/L and ρ Na + is 0.70 mol/L.

The solution that was not pH controlled, was renewed each time when an expansion measurement was made. The pH controlled condition was maintained by adding sulfuric acid. This method was proved to be able to fix the hydroxyl ion concentration as well as the sulfate ion concentration in solution [START_REF] Brown | An evaluation of the sulfate resistance of cements in a controlled environment[END_REF]. Therefore, ρ SO 2- 4 and ρ Na + are kept constant in the simulation for both conditions. s -1 and A p = 1.6 × 10 -9 s -1 . The evolution of the content of CH, C SH 2 , C-S-H, AFm, C 3 AH 6 , and ettringite are shown in Fig. 6(a) when pH is uncontrolled. Then, these two parameters are used to predict the swelling curve of the specimen, which is in a pH controlled condition at 7 (see Fig. 5). However, no expansion is predicted. A condition of pH=7 contributes to a rapid dissolution of portlandite and precipitation of gypsum as shown in Fig. 6(b). No ettringite is formed because it is not stable at pH=7. In this model, the formation of ettringite is supposed to be the only cause of expansion. In other words, this model cannot capture the expansion induced by the formation of gypsum. However, the precipitation of gypsum at pH=7 from the simulation and the observed expansion in experiments support the idea that gypsum is responsible for expansion in ESA [START_REF] Tian | Does gypsum formation during sulfate attack on concrete lead to expansion?[END_REF][START_REF] Wang | Sulfate attack on hardened cement paste[END_REF],

especially at a low pH condition [START_REF] Ragoug | Durability of cement pastes exposed to external sulfate attack and leaching: Physical and chemical aspects[END_REF]. Fig. 7 shows the expansion of mortar cylinders with three different diameters: 25 mm, 50 mm, and 75 mm. Specimens start to expand macroscopically when a similar fraction of the specimens are affected by the penetrated sulfate ions [START_REF] Yu | Mechanism of expansion of mortars immersed in sodium sulfate solutions[END_REF]. Small specimens start to expand earlier and faster than large specimens due to their weak ability to overcome the expansive forces. The transport of sulfate ions depends on the diffusion process, which is mandatory when studying the effect of dimensions.

Simulation of DEF in free swellings

We simulate the free swelling of concrete due to DEF as studied in [START_REF] Kchakech | Experimental study of the influence of the temperature and duration of heat treatments at an early age on the risk of 805 concrete expansion associated with delayed ettringite formation[END_REF], where sigmoid expansion curves have been observed (see, e.g., Fig. 10). The sigmoid expansion curve includes a latent 525 period, an accelerated increasing phase, and a plateau part (or a non-linear slowly increasing part).

Our model can explain these three periods of behavior as follows. The material transforms from the elastic to damage range at the connection point between the latent and accelerated period.

This threshold point of damage is determined by the physical and mechanical properties of the material. Then the strain increases rapidly and reaches a plateau when at least one of the reactants is depleted.

Simulating the plateau is a challenge in this work. After heat treatment, the content of sulfates s -1 . Fig. 8 shows the precipitation of AFt by consuming AFm and sulfates (free sulfates in the pore solution and the ones adsorbed on C-S-H), and the AFt content plateau when reactants are depleted. Fig. 9 illustrates the evolution of the saturation degree of crystals S c and the strain. We can observe that a relative higher A e leads to a higher S C , namely a higher volume fraction of AFt formed in the pore invasion process, and therefore a lesser fraction in the deformation process and further a lesser expansion degree. Therefore, the distribution of ettringite between the pore invasion process and the pore deformation process depends on kinetic constants A e and A p . To sum up, the initial content of sulphur n SO3 and A e determine the kinetics of the elastic range, which together with A p affect the kinetics of damage range and the value of plateau. In the following simulations, the point connecting the latent and the accelerated period will be calibrated from the swelling curve first. Then, two kinetic constants will be calibrated from the increase rate of expansion based on the inputs of ρ Experimental investigations were performed to study the effect of thermal history on DEF of con-550 crete prismatic specimens (11 × 11 × 22 cm 3 ) in [START_REF] Kchakech | Experimental study of the influence of the temperature and duration of heat treatments at an early age on the risk of 805 concrete expansion associated with delayed ettringite formation[END_REF]. Different maximum heating temperatures (61, 71, and 81 • C) with different heating durations were adopted in their study. However, concrete specimens preheated with 61 • C did not show expansions while the others did. Thus a temperature above 65 • C was considered to be the threshold temperature for DEF occurrence. Moreover, a pessimum effect was found in the effective energy in [START_REF] Kchakech | Experimental study of the influence of the temperature and duration of heat treatments at an early age on the risk of 805 concrete expansion associated with delayed ettringite formation[END_REF], which is a combination of temperature above 65 • C and heating duration. It means that as the heating duration increases with a temperature above 65 • C, the expansion may decrease after the effective energy is beyond a threshold value. In this section, the expansion of specimens preheated with a maximum temperature of 71

• C and different heating durations are simulated. The specimen, which was heated at 71 • C for 14 days, showed a maximum expansion, in which case it is assumed that all primary ettringite was decomposed during the preheating treatment. The content of AFm and sulfates after the heat treatment are: n AFm (t c ) = 0.15 mol/L, n SO 2- 4 (t c ) = 0.075 mol/L, and n ads SO3 (t c ) = 0.283 mol/L. The specimen preheated with a short duration, 71-7days, for example, showed a smaller expansion probably due to a lack of sufficient energy to decompose all the primary ettringite. The specimen preheated for 28 days showed a smaller expansion as well, which may be related to the definitive aluminum substitution of silica in C-S-H and resulted in a decrease in the quantity of precipitated ettringite [START_REF] Salgues | Modélisation des effets structuraux des réactions sulfatiques internes: application aux barrages en béton[END_REF]. In these cases, the total ettringite that can be formed when the expansion of the specimen reaches the plateau phase is less than the one in specimen 71-14days. Therefore, the values of ρ SO 2- 4 (t c ), n ads SO3 (t c ) and n AFm (t c ) are decreased so that the final amount of ettringite that could be precipitated is reduced. First, A e and A p are calibrated from the expansion curve of specimen 71-14days. Then, these two parameters are kept constant to calibrate ρ SO 2- 4 (t c ), n ads SO3 (t c ) and n AFm (t c ) according to the plateau value of expansion. Fig. 10 illustrates the predicted swelling curves for specimens with A e = 4.9 × 10 -8 s -1 and A p = 2.8 × 10 -9 s -1 , which are calibrated from the expansion curve of 71-14days specimen. For specimens 71-7days and 71-12days, the simulations show a shorter latent period compared to the actual one, while a slower kinetics of latent period is predicted for specimen 71-28days. This could be explained by the fact that during the short-term (7 and 12 days)/ long-term (28 days) preheating, the microstructure of the material is modified, which leads to a higher/ lower capacity to accommodate ettringite and results in a longer/ shorter elastic range. Our model cannot illustrate this factor. For specimen 71-2days, the model fails to capture the accelerated phase even though the latent and plateau phases are well predicted. With a limited heating duration, its microstructure may be quite different from the one concerning 71-14days. Therefore the kinetic constant A p is not applicable for specimen 71-2days.

Conclusions and discussions

A poromechanical damage model based on the crystallization pressure and the surface-controlled In this model, two crystal growth mechanisms are considered: one is associated with the pore invasion process and the other with the pore deformation process. Both mechanisms are surfacecontrolled: the growth rate is controlled by the supersaturation of ettringite in the pore solution.

The growth rates involve two kinetic parameters, denoted with A e and A p in this paper, that could be theoretically related to the increase rate of the elastic and damage strains of the material.

So these parameters were calibrated to fit the observed strain from some experiments and then used to blind-test the model through the simulations of other experiments. The overall result is satisfactory, and a summary of the calibrated parameters is presented in the following Tab. 5.

Despite the assumption of homogeneous state, the range of variations of the two calibrated kinetic parameters were not so wide since they vary by a factor of 2.8 only over all the considered cases.

In addition, the ratio between the two kinetic parameters turned out to be rather constant over all the considered cases (ranging between 18 and 19). We believe that these rather similar factors reflect the physics of the surface-controlled growth mechanism. From a physical point of view, whatever ESA or DEF is considered, the major conclusions that can be drawn from the simulations are:

• The latent period during which minimal deformation occurs is an elastic response of the material where ettringite crystals first fill the biggest pores and generate small crystallization pressure and tensile effective stresses.

• The accelerated period starts when the amount of crystal and the crystallization pressure are such that the effective stress -strain relationship results from the damage process and the post-peak tensile behavior of the material.

• The deformations originate from the same mechanisms except for what concerns the source of sulfate ions.

The proposed poromechanical model can be adapted to any class of constitutive equations, including elasticity, plasticity, and damage. The latter has been chosen in this paper for convenience. So it could be worth including a plastic component in the current model, especially when loading cycles are to be considered. The mechanical response is coupled to the chemistry of the pore solution and, consequently, to the pH of the solution in the case of ESA. The calibrated parameters on experiments performed at high pH (actually when the pH was not controlled) were used to blind-test the model in simulations of experiments at controlled pH=7. However, no expansion was observed. This was due to the precipitation of gypsum instead of AFt. The formation of ettringite is considered in the current version of the model as the only reason for expansion, and that is why no expansion was predicted. Considering gypsum as another source of crystallization pressure could be a promising way of improving the modeling. Taking into account the diffusion is another perspective. The diffusion of ions in specimens with various dimensions has been well simulated by simply imposing sulfate ions and alkali boundary conditions with a linear increase rate.

In the case of DEF, the potential amount of delayed ettringite that could eventually precipitate after the reaction has completed, seems to depend, non-trivially, on the heat history that the material has endured (temperature and duration of the treatment). In the model, the final precipitated delayed ettringite could be related to the initial content of sulfur. So, in addition to the kinetic parameters, we also calibrated the initial content of AFm, released sulfates, and the ones adsorbed on C-S-H to fit the plateau of the sigmoid expansion curve. In the end, these parameters were adopted to predict the expansion curve. The damage behavior was well predicted, while the kinetic of the latent period was not satisfying and could be improved. This is because the effect of the heating process on the microstructure was neglected, which is an important factor for the elastic behavior. Some assumptions have been considered in the model, e.g., all the primary ettringite in cement was assumed to be decomposed after the heat treatment. It will be interesting to quantify the transformation of ettringite to AFm via chemical analyses methods and determine this process as a function of the heating temperature and duration.

The proposed model suggests a method to explain the crystallization of ettringite and the expansion due to sulfate attacks in Portland cement-based materials. However, this model is applicable for other types of cement by modifying the main hydrates, for example. Considering the experimental conditions will be other aspects to improve the model, e.g., the mortar and concrete mix composition, preheating histories, curing conditions, among others. It will provide better estimations of the saturation index of ettringite.

Acknowledgement

Thanks to Cerema and Université Gustave Eiffel for allowing the creation of the joint research team entitled " Équipe de Recherche Commune sur les Matériaux pour une Construction Durable (ERC MCD)", within which these research works were carried out.

Figure 1 :

 1 Figure 1: Typical stress-strain behavior of cement-based materials.

( a )

 a Pore invasion process -Crystals at the crystal-solid interface reach equilibrium βp = β by exerting a crystallization pressure according to Eq. 1, namely P C = RT V C ln(βp); (b) Pore invasion process -Crystals in equilibrium at the crystal-solid interface βp = β and the crystal-liquid interface βe = β; (c) Pore deformation process -Crystals not in equilibrium at the crystal-solid interface but in equilibrium at the crystal-liquid interface βe = β; (d) Pore deformation process -Crystals in equilibrium at both the crystal-solid interface βp = β and the crystal-liquid interface βe = β.

Figure 2 :

 2 Figure 2: Illustration of the crystal growth in the pore invasion and pore deformation processes.

a 4 )φ = n rel SO 2 - 4 .

 424 curing time t = t c ) is composed of CH, C-S-H, C 3 A, AFm, free sulfates in the pore solution, and some sulfates adsorbed on C-S-H. The concentration of the released sulfates in the pore solution depends on the porosity of materials, and it can be expressed as: (ρ SO 2- So after the heat treatment we can infer from the mole balance Eqs. 18 -21 and from the assumption n AFt (0) = n AFm (t c ) that:

  known. This provides the two equations needed to determine the saturation index of ettringite. Once the saturation index of ettringite β AFt has been determined and the ettringite content n AFt (t) has been derived from the model, we can check from the mole balance equations (assuming also n AFm = n rel SO2- 4

  Stress-strain relationship in the axial direction under low-restraint as measured in[START_REF] Ma | Experimental and numerical study on cement paste degradation under external sulfate attack[END_REF];

Figure 3 :

 3 Figure 3: Simulation of axial stresses in experiments [50].

  4(b), where the compressive stress is presented as positive. The predicted

Figure 4 :

 4 Figure 4: Simulations of axial stresses in experiments [51].

Figure 5 :

 5 Figure5: Simulations and calibration of parameters for ESA[START_REF] Ferraris | Mechanics of degradation of portland cementbased systems by sulfate attack[END_REF]

  Evolution of the content of hydrates when pH is not controlled; Evolution of the content of hydrates when pH=7.

Figure 6 :

 6 Figure 6: Evolution of the content of hydrates when (a) pH is not controlled and (b) pH=7.

Figure 7 : 4 = 0 .

 740 Figure7: Simulations and calibration of parameters for ESA[START_REF] Ferraris | Mechanics of degradation of portland cementbased systems by sulfate attack[END_REF]

(ρ SO 2- 4 ( 4 (t c )φ = 2 n 4 (

 4424 t c ) and n ads SO3 (t c )) and the content of AFm (n AFm (t c )), which are predicted from the amount of sulfur in cement and have a relation of n ads SO3 (t c ) + ρ SO 2- AFm (t c ) (see section 5), determine the total amount of ettringite formed in the end. To better understand these parameters, three cases are simulated where n AFm (t c ) = 0.15 mol/L, ρ SO 2- t c ) = 0.075 mol/L, n ads SO3 (t c ) = 0.283 mol/L, A p = 2.8 × 10 -9 s -1 , and the only variable is A e ranging from 4.4 × 10 -8 s -1 to 5.4 × 10 -8

SO 2- 4 (

 4 t c ), n ads SO3 (t c ) and n AFm (t c ). Evolution of the content of phases when Ae = 4.9 × 10 -8 s -1 .

Figure 8 :

 8 Figure 8: Transformation of AFm to AFt with the variable Ae.

Figure 9 :

 9 Figure 9: Evolution of saturation degree of crystals and strain with the variable Ae.

Figure 10 :

 10 Figure10: Simulations and calibration of parameters for DEF[START_REF] Kchakech | Experimental study of the influence of the temperature and duration of heat treatments at an early age on the risk of 805 concrete expansion associated with delayed ettringite formation[END_REF]

585

  ettringite growth mechanism is proposed for Portland cement-based materials submitted to sulfate attacks. External Sulfate Attack (ESA) and Delayed Ettringite Formation (DEF) are intended to be addressed by the same mechanisms, except for their different source of sulfates. The concentrations of sulfate ions and alkalis concerning ESA are determined by the external solution, whereas, for DEF, they are determined by cement materials. This model is applied in a homogeneous state, i.e., without accounting for the diffusion of ions, and is compared with experiments conducted on Portland cement-based materials in different configurations. The results show that the model rather satisfactorily describes the swelling and the stress induced by both attacks.

Table 1 :

 1 Typical model parameters.

	Parameter	Symbol Cement paste	Mortar/Concrete
	Young's modulus (GPa)	E	21.7	[34] 30.1	[34]
	Poisson's ratio	ν	0.24	[34] 0.26	[34]
	Porosity	φ	0.34	[34] 0.23	[34]
	Biot's coefficient	b	0.69	[34] 0.54	[34]
	Tensile strength (MPa)	ft	2.0	[35] 3.1	[36]

Table 2 :

 2 Typical values of r 0 and m for cement paste, mortar and concrete[START_REF] Gu | Experimental pore scale analysis and mechanical modeling of cement-based materials submitted to delayed ettringite formation and external sulfate attacks[END_REF].

	Parameter Cement paste Mortar Concrete
	r0 (nm)	16	156	128
	m (-)	0.44	0.25	0.38

Table 4 :

 4 Physico-chemical parameters.

	Physico-chemical parameters
	Molar volume of ettringite	VC	710 [48]	cm 3 /mol
	Surface energy of ettringite	γCL	0.1 [49]	J/m 2
	Kinetic coefficient 1	Ae	-	1/s
	Kinetic coefficient 2	Ap	-	1/s
	Activity product constant of AFt at 25 • C	log(KAFt)	-44.55 [48] -
	Activity product constant of AFm at 25 • C	log(KAFm)	-29.43 [48] -
	Activity product constant of C3AH6 at 25 • C	log(KC 3 AH 6 ) -20.49 [48] -
	Activity product constant of CH at 25 • C	log(KCH)	-5.14 [48]	-
	Mole content of C-S-H/ unit volume of the material	nC-S-H(0)	1.393	mol/L
	Mole content of CH/ unit volume of the material	nCH(0)	1.530	mol/L
	Mole content of C3A/ unit volume of the material	nC 3 A(0)	0.114	mol/L
	Mole content of AFt/ unit volume of the material	nAFt(0)	0.150	mol/L
	Binding fraction coefficient	B0	2.55	-
	Reference concentration	ρ0	0.87	mol/L

Table 5 :

 5 Calibrated kinetic constants Ae and Ap from investigated studies.

	Parameters ESA restrained	ESA free 1	ESA free 2	DEF constrained	DEF free
	(s -1 )	(section 8.1)	(section 8.3) (section 8.3)	(section 8.2)	(section 8.4)
	Ae	4.0 × 10 -8	3.0 × 10 -8	8.4 × 10 -8	6.2 × 10 -8	4.9 × 10 -8
	Ap	-	1.6 × 10 -9	4.4 × 10 -9	-	2.8 × 10 -9

where n ads SO3 (t) and n C-S-H (t) are determined from the mole balance Eqs. 18 -21. Combining Eq. 36, this equation can be expressed under the form:

where

Then, after an exposure time t to DEF, the content of each phases are under the forms:

where it is assumed that C 3 A does not contribute to the source of aluminium, so n C3A is assumed to be constant and equal to its initial value. As long as AFm and CH are not depleted, i.e. n AFm (t) > 0 and n CH (t) > 0, the hypotheses β CH = β AFm = 1 hold.

Summary of constitutive laws

The set of equations is summarized in Tab. 3 and typical parameters that will be used in the following simulations are listed in Tab. 4.