MODELING CONCRETE AT EARLY AGE USING PERCOLATION Lavinia Stefan1,2 Farid Benboudjema1, Jean Michel Torrenti 3, Benoît Bissonette2 1 LMT, ENS Cachan, 61 Avenue du président Wilson, 94230 CACHAN, France 2 Département de Génie Civil, Pavillon Adrien-Pouliot , local 2928B, Université Laval, Québec, Canada, G1V 0A6 3 LCPC, 58 boulevard Lefebvre, 75732 Paris cedex 15, France. ABSTRACT The prediction of early age behavior of cementitious materials is of particular importance when it comes to the prediction of the crack occurring risks. Amongst the most important parameters that define the hydrating material are its elastic properties and the changes in volume that arise due to the very reaction of hydration. On a discrete generated microstructure, a percolation – type approach is applied. A forest fire algorithm allows taking into account the binding role played by the hydrates, and it reveals a threshold of hydration below which the rigidity of the concrete is negligible. The evolution of elastic characteristics is obtained by using a homogenization method applied to the percolated microstructure. Autogenous shrinkage is assumed to be due to the rise of a capillary pressure, the latter itself being a consequence of the hydration reaction. The capillary pressure is obtained from a model for desorption isotherm and is applied to the deformable skeleton corresponding to the percolated microstructure. Using this approach and Biot's theory, it is possible to compute the autogenous shrinkage and its evolution around the threshold of percolation. 1 INTRODUCTION Concrete is a material likely to be subjected to shrinkage and therefore to cracking risks. At early age, the mechanical characteristics of the material follow a fast evolution. Its Young’s modulus rap- 2 idly evolves from 0, when the material is in a liquid state, to values close to the service value, after several days. An intrinsic property of the hydration reaction is that it leads to a volume reduction of the reaction products. Moreover, due to the hydration phenomenon, capillary pressure arises, pressure that leads to autogenous shrinkage after the set. In the case that deformations are restrained, the material is stiff enough to oppose to the volume reduction induced by the hydration reaction, so cracks may occur if its strength is not yet fully developed. Therefore, in order to have a predictive model, the evolution of the mechanical properties, as well as the evolution of the autogenous shrinkage, are needed. The prediction of early age behaviour of concrete is subject of an important number of studies in literature and several models were developed in order to predict the risk of cracking [1]. However, progress is still possible, especially at very early age when concrete evolves from a state close to that of a liquid to a solid state. This transition can be modelled by an mechanical percolation approach [2, 3, 4], given a discrete generated hydrating microstructure, used to model the evolution of the material from the very instant of water – cement contact, up to its hardened state. The advantage of such an approach is that it permits to predict the evolution of the mechanical properties and of the autogenous shrinkage of concrete at early age, function of its hydration degree, as it shall be shown further on. In order to apply the percolation algorithm, we first need to have a virtual microstructure. This requires the use of a hydration model that can predict at any step the volumetric fractions of each phase involved in the hydration reaction. Several models that are able to ensure a correct computation of the above mentioned phases can be found in the literature ([5] to [10]). In the present study the computations are based on different hydration models proposed by Powers [4] and Jennings and Tennis [6]. Using these models one can estimate the amounts of the phases in hydrating cement pastes. Powers model considers only three phases (anhydrous cement, hydrated cement and porosity). Jennings and Tennis model allows explicit computation of the evolution of anhydrous phases (C3S, C2S, C3A and C4AF) and of the hydrated phases (C-S-H, portlandite and Afm). Therefore, it is straight forward to generate an evolving random microstructure that includes the 3 mentioned phases. More physical hydration models ([8] to [10]) exist. The proposed approach can be applied to these models, given slight development of the method. Once the virtual microstructure is generated, a percolation algorithm is then applied [4] in order to capture the solidified part, which contributes to the support of mechanical loadings, for each value of hydration degree. A self consistent scheme on the percolated cluster is used to obtain evolution of the elastic properties. Such an approach is fundamental to predict a correct evolution of mechanical properties at early-age. Otherwise, decrease of mechanical properties and/or non-zero Young modulus before hydration can be predicted [4]. The autogenous shrinkage is then predicted after the computation of evolution of desorption isotherms with respect to hydration degree and using Biot's theory. 2 HYDRATION MODELS 2.1 Powers model [4] This model has been used by several researchers (see for example [19, 14]). The cement paste is composed of three phases: the anhydrous cement grains, the hydrates and the porosity. It is assumed that the hydrates occupy a volume 2.31 times larger than that of the reactants, leading to the following relationships: V (1 − α ) Vanh (α ) = (1) w 1 + 3.2 c V hyd (α ) = V 1 + 1 .31α − V anh (α ) w 1 + 3 .2 c V pore (α ) = V − V hyd (α ) − V anh (α ) (2) (3) 4 where Vanh, Vhyd, Vpore and V represent respectively the volume of anhydrous cement, the volume of hydration products, the volume of pores and the total volume of the system. w/c is the water to cement ratio and α is the hydration degree. It is straightforward to calculate from this simple model the volume of each phase of a cement paste for a given water to cement ratio. 2.2 Jennings and Tennis model [6] In this model one consider the evolution of contents of anhydrous cement phases (C3S, C2S, C3A and C4AF), of hydrated phases (C-S-H, portlandite and Afm) and of capillary porosity with respect to time. The composition of the cement is obtained from the modified form of the Bogue calculation proposed by [10]. Independent hydration of C3S, C2S, C3A and C4AF is assumed. Aft is not considered. A set of Avrami-type equations is used for the approximation of the degree of hydration of each compound. The general form of the equation is: αi = 1 − e c − ai ( t −bi ) i (4) where αi = degree of hydration of reactant i (i = C3S, C2S, C3A or C4AF) and t is the age (in days). Note that we use this set of equations in isothermal conditions (if the temperature is not constant thermo-activation must be taken into account). The constants ai, bi and ci are those determined empirically by [11]. They are given in Table I: C3S C2S C3A C4AF TABLE I: CONSTANTS USED IN EQ. 4 [11] ai bi ci 0.25 0.9 0.7 0.46 0 0.12 0.28 0.9 0.77 0.26 0.9 0.55 5 The overall degree of hydration α is given by a weighted average Wi of the degrees of hydration of the individual compounds i: α = i = 4 ∑ αW (5) i i i =1 Using data (density and molecular weight) compiled from various authors, the following equations for the prediction of the volume (in cm3) Vj of each phase j were obtained by [6] (anhydrous, portlandite, Afm and C-S-H, respectively, for 1 g of cement paste): Vanh = 1−α 1 (6) w ρ c 1+ c 1 VCH = ( 0.189α C S WC S + 0.058α C S WC S ) 3 3 2 2 (7) ( 0.849α C AWC A + 0.472α C AF WC AF ) 3 3 4 4 (8) w 1+ c V Afm = 1 1+ w c VC − S − H = 1 1+ w ( 0.278α C S WC S + 0.369α C S WC S ) 3 3 2 2 (9) c where w/c = water to cement ratio and ρc = density of cement (3.15 g.cm-3). C-S-H may also be dissociated into LD (low density) and HD (high density) C-S-H [7]. The capillary porosity Vcp is obtained, knowing the volume change due to the difference in solid volume between the products and the reactants ∆i (c.f. Table II, [6]). 6 Vcp = 1− 1 1+ w 4    1 + ∑ αiWi ∆i   i =1  (10) c Table II. DIFFERENCE IN SOLID VOLUME ∆i [6] ∆i [cm3/g] C3S C2S C3A C4AF 0.437 0.503 0.397 0.136 2.3 comparison of the two models The two models give the volumetric fractions of hydrated and anhydrous phases. Jennings and Tennis model has a richer description. So it could take into account differences between cements. Note also that for the same degree of hydration it gives a smaller quantity of hydrates (so it will give also a smaller Young's modulus). In the following applications we will use mainly Jennings and Tennis model. Powers model will be used for some examples. 2.4 generation of the microstructure The volumetric fractions of initial anhydrous cement phases and water for α = 0 are computed and then randomly disposed within a 90 × 90 elements two-dimensional representative elementary volume (REV). The size of the REV was chosen after several simulations that were made for microstructures of different sizes (starting from 50 × 50 elements REV up to 300 × 300 elements REV). Beginning with a 90 × 90 REV, the threshold of the evolutions of the mechanical characteristics, such as the Young modulus, does not vary with the increase of the number of the elements in the microstructure [4]. The initial microstructure evolves with the degree of hydration, following a set of rules: for each increment of hydration degree, the volumetric fractions of anhydrous cement phases that reacted and the hydrates formed are computed (Eq. 3 – 7). The anhydrous grains 7 that are surrounded by the greatest number of pixels corresponding to water (pore phase) are replaced by the hydrates. Then, the remaining hydrates shall replace the pores that are found in the vicinity of anhydrous cement grains. Figure 1B corresponds to a degree of hydration α = 0.60 and it represents an example of the evolution of the initial microstructure represented in Figure 1A, for a water to cement ratio of 0.45. A B Figure 1. A. Initial generated microstructure; B. Hydrating microstructure α = 0.6. w/c = 0.45. black = water, white = anhydrous, grey = hydrates. Powers model. Such a description of the microstructure is enough for the calculation of the volumetric phases and the evaluation of the poromechanics properties. Of course it will not be the same if one consider the non linear behaviour of a cementitious material. 3 PERCOLATION Once the virtual microstructure is generated, we can apply the percolation algorithm. The need of such a calculus arises from the particularity of cementitious materials. Indeed, for very low water to cement ratios, anhydrous cement particles are in contact from the very beginning. Still, the material has no stiffness or strength while it founds itself in the viscous state. It is only after setting that it starts to develop its mechanical properties. In order to capture the effect of “glue” played by hydrated phases and therefore the development of cohesion in cement-based materials, a “burning” type algorithm is 8 used [2]. It is adapted to match the specificity of cement – based materials. As hydrates are formed, connections between solid fractions are made throughout the matrix, connections that propagate with the advancement of the hydration, ensuring the development of the solid skeleton. Once such paths will cross from one end of our virtual microstructure to the other, we can consider that a “set – like” phenomenon took place. The percolated cluster is identified and isolated, so we can then proceed to mechanical computations. The different steps of the “burning algorithm” are: 1. Fire is set on one of the sides of the microstructure, and only solid parts can participate to the percolated path (hydrates or anhydrous elements next to at least one hydrate); 2. Depending on the nature of the lighted pixel, fire propagates only under certain imposed conditions: - If the lighted element is an anhydrous grain, fire propagates to other hydrated grains found in the neighbourhood, grains which have never been previously lighted. - If the lighted element is a hydrate, fire propagates to both hydrated and unhydrated neighbours, which have never been previously lighted. Thus, 2 anhydrous grains of cement that are in contact with each other cannot ensure the cohesion of the system, so they shall not be considered as part of the percolation cluster, but an anhydrous grain that is found between 2 hydrates will be included in the percolation cluster. This set of rules ensures that for very low water to cement ratios there is not percolation for a degree of hydration equal to zero, knowing that the grains are in contact (there is percolation if you consider the contacts between grains like in the case of a sand; in the present context we need a percolation that simulates the cohesion in the medium). 3. Step 2 is repeated until fire stops to propagate. Mechanical percolation occurs, only if pixels on the opposite side have been lighted. The percolated microstructure corresponds to the assembly of pixels which have been lighted. 4. Percolation is tested from left to right, right to left, up to down and down to up. Only the common percolated pixels in all of 9 the four tests will be kept. This allows the removing of « dead end » parts. 5. Finally, mechanical calculations are performed on the final percolated cluster. A typical percolation test for a microstructure of w/c = 0.45 is shown in Fig. 2. The first percolating cluster appears for a hydration degree of 0.26. It corresponds to about 6 hours after the water to cement contact. α = 0.22 α = 0.25 α = 0.26 α = 0.29 Figure 2. Percolation algorithm applied to the hydrating microstructure 4 EVOLUTION OF PORO-MECHANICAL PROPERTIES Elastic properties of each phases of cement paste are now well known, due to the nano-indentation techniques developed in the last years ([12] to [16]). Mean values are presented in Table III. 10 TABLE III: ELASTIC PROPERTIES OF PHASES Phase Ei [GPa] references νi 135 0.3 [12] C3S 130 0.3 [12] C2S 145 0.3 [12] C3A 125 0.3 [12] C4AF 42 0.31 [15] CH 22.5 0.24 [14] C-S-H 22.4 0.25 [16] Afm Knowing the volume fractions of each phase of cement paste, the computation of elastic properties is usually achieved by using one of the following 2 methods. The first method is to make use of the finite element computations ([17], [18]). However, there are several disadvantages as it can be quite time-consuming and significant numerical problems (i.e. mesh dependencies) can occur due to stress concentrations between different phases. The second method is to use homogenization schemes that are able to compute the elastic properties of composite materials, provided that the elastic properties of each of the phases are known ([18] to [21]). Results depend on the morphology of the heterogeneous material and the shape of the phases. The time of calculus is dramatically reduced, and all problems that are linked to the type of finite element used, or the mesh density are no longer encountered. Due to its simplicity and good results that it provides, in the present paper a homogenization method was applied to the percolated cluster. The self-consistent scheme (with a spherical morphology) is adopted among various other methods, since it is the one that fits the best to a hydrating matrix. A Mori – Tanaka scheme will not provide good results as it is difficult to isolate at early-age a matrix surrounded by inclusions. The equations for the prediction of elastic mechanical properties are as presented by [19]: kh   ki i =n = ∑ Vi k i 1 + α m  k i =1  h  −1  ki   i = n  − 1    ∑ Vi 1 + α m  k   h   i =1    − 1      −1 (11) 11   gi i =n g h = ∑ Vi g i 1 + β m  g i =1  h  −1  gi   i = n  − 1    ∑ Vi 1 + β m  g   h   i =1    − 1      −1 (12) where kh and gh = homogenized bulk and shear moduli, ki and gi = bulk and shear moduli of phase i, Vi is the volumetric fraction of phase i, αm and βm are functions of the mechanical properties of phases [19]. The mechanical characteristics are thus estimated by using the self-consistent scheme that can be applied to the percolation cluster only. In this manner, the volume of solids that is not found in the percolation cluster shall be evaluated as a void in the total volume. In Figure 3A we can see the evolution of total solid volume function of the degree of hydration, for different water to cement ratios. When the simulation takes into account only the percolated cluster, there is a threshold below which the solid fraction is not taken into account (continuous line). Indeed, before set, even if connections are made between solids, they provide nor strength or stiffness to the viscous material. Computations made on the entire microstructure will take into account the total amount of solid phase found in the REV (dotted line). In this case, when mechanical computations are performed, and especially for low water to cement ratios, a Young modulus for a zero degree of hydration is predicted (Figure 3B). This represents the essential difference between the present approach and the classical application of the self-consistent scheme [19], [20]. The latter gives excellent results, provided that the computations are not performed in the neighbourhood of the percolation threshold. For more important degrees of hydration, when almost all of the solids belong to the percolation cluster, both methods can be successfully applied. 12 1 w/c = 0.25 A Solid fraction 0.8 w/c = 0.65 0.6 0.4 0.2 0 0 Young's modulus (GPa) 35 0.2 B 0.4 Hydration degree 0.6 0.8 w/c = 0.25 30 25 20 15 w/c = 0.65 10 5 0 0 0.2 0.4 0.6 0.8 Hydration degree Figure 3. A. Evolution of solid fraction; B. Evolution of Young’s modulus with the degree of hydration. Entire microstructure (dotted line) and percolated cluster (continuous line) The Biot’s coefficient bi is computed using the following relation [10]: bi = 1 − kh ks (13) where ks is the bulk modulus of the solid skeleton only. In order to compute it, it is necessary to « extract » the porosity and the unpercolated part of the microstructure and thus to consider a virtual microstructure that contains only the anhydrous and hydrated phases. 13 Biot's coefficient By making these assumptions, the solid bulk modulus can be calculated by applying the homogenisation scheme on the solid part of the percolated microstructure (equations (11) and (12)). We shall use as input in our computations the volumetric fractions of anhydrous grains and hydrates phases respectively (f’anh and f’hyd) which are found in the total solid volume of the percolated cluster, represented by the sum fhyd + fanh : f hyd f anh f ' anh = & f ' hyd = (14) f anh + f hyd f anh + f hyd 1 w/c =0.60 0.9 w/c =0.45 w/c = 0.30 0.8 0.7 0.6 0.5 0.4 0.1 0.2 0.3 0.4 0.5 0.6 Hydration degree Figure 4. Evolution of Biot’s coefficient for different w/c ratios. 5 AUTOGENOUS SHRINKAGE With the advancement of the hydration reaction, the anhydrous cement grains and water are consumed and, provided the water to cement ratio is sufficiently low (inferior to 0.6), the material undergoes a desaturation. The relative humidity is below 100%, so according to Kelvin’s law, the water that is found in the partially saturated pores is subjected to a capillary depression pc. This capillary 14 depression will be the cause of the autogenous shrinkage (see [22] for example). By combining Kelvin’s law with the relation between the saturation degree S and the relative humidity given by Van Genuchten’s relation, one obtains: pc ( S ) = a (S − b − 1) 1−1 / b (15) where pc is the capillary depression, and a and b are constants. The a and b coefficients can be obtained by performing a desorption test. This type of test can be performed on hydrated materials. However, for hydrating materials, this is no more possible since this type of test needs about 1 year to be performed. These parameters can be obtained by an inverse analysis from the pore size distribution (determined from low temperature calorimetry [23] or mercury intrusion [24]). We use here another approach, since all these measurements induce a modification of the pore microstructure (cracking) and give a wrong image of the porosity (due for instance to the “ink bottle” effect). As shown by [25] on mature cement paste for various w/c ratios, we assume that the mass water content w50 at 50% RH is proportional to the C-S-H amount V’C-S-H (cm3 per g of dry hcp): w50 = kV 'C −S − H (16) From Eq. (16), the corresponding saturation degree S50 at 50 % RH is calculated. By assuming that b coefficient in Eq. (15) remains constant and equal to 0.5 during hydration, one can predict the evolution of desorption isotherm (after the calculation of the capillary pressure with the Kelvin law). For k = 0.1 and w/c = 0.25, one obtains the results given in Figure 5. 15 1 0,9 α=0,27 Saturation degree 0,8 α=0,36 0,7 0,6 α=0,46 0,5 0,4 0,3 0,2 0,1 0 0 0,2 0,4 0,6 0,8 1 Relative humidity Figure 5. Evolution of desorption isotherm for w/c = 0.25 with respect to hydration degree. Assuming an isodeformation of the porosity subjected to a given capillary pressure in the case of an unsaturated isothermal poroelastic approach, [26] and [27] showed that we have the following relation: dσ = k h dε − bi Sdpc (17) where kh is the bulk modulus, bi is the Biot coefficient (in our case it is directly obtained by applying the self-consistent scheme), σ and ε are respectively the mean spherical stress and the mean spherical strain. In the case of free shrinkage, the average stress is null and one thus has: dε = 1 bi Sdpc kh (18) In this relation, kh, bi, S and pc depend on the degree of hydration. From our results (equation 18), one can finally deduce the evolution 16 -3 Autogenous shrinkage (10 ) of the autogenous shrinkage. Figure 6 presents this evolution for the w/c = 0.25. These calculations lead to a qualitatively correct evolution of the autogenous shrinkage for this water to cement ratio. It is found almost proportional to the hydration degree, as measured experimentally by several authors. However, we have to improve our model for larger water to cement ratio and to take into account the viscous behaviour of the material at early age in order to be more predictive. Indeed, Eq. (17) assumed only elastic strain. Due to creep under the capillary pressure, predicted autogenous shrinkage amplitude will be higher. Besides, due to hydration, saturation degree is decreasing, which slows the hydration rate. This feature needs also to be addressed in the future. 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 Hydration degree Figure 6. Evolution of autogenous shrinkage for w/c = 0.25. CONCLUSIONS The percolation approach of early age of cementitious materials behavior makes it possible to describe the evolutions of the mechanical characteristics of these materials: Young modulus, Poisson ratio and Biot’s coefficient. Assuming an isotherm of desorption independent of the degree of hydration, an isodeformation of porosity and an ag- 17 ing elastic behavior we can then deduce the evolution of the autogenous shrinkage due to the capillary depression. Calculations lead to a good order of magnitude for the predicted autogenous shrinkage. However, this approach needs to be supplemented by taking account the viscous behavior at early age. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] Acker P., Torrenti J.M., Ulm F.J., “Comportement du béton au jeune âge”, Paris, Hermès, 2004. Torrenti J.M., Benboudjema F, “Mechanical threshold of concrete at year early age”, Materials and Structures, vol. 38, No 277, 2005, p.299-304. Smilauer V., Bittnar Z., “Microstructure-based micromechanical prediction of elastic properties in hydrating cement paste”, Cement and Concrete Research 36 (2006). Stefan L., Benboudjema F., Torrenti J.M., Bissonnette B., “Prediction of elastic modulus of cement pastes at early ages”, proposed for publication in Cement and Concrete Research, 2009. T.C. Powers, T.L. Brownyard, “Studies of the physical properties of hardened portland cement paste (nine parts)”, Journal of the American Concrete Institution 43 (Oct. 1946 to April 1947). Jennings H.M. & Tennis P.D. 1994. “Model for the developing microstructure in Portland cement pastes.” J Am Ceram Soc 77(12): 3161-3172. Tennis, P.D. & Jennings H.M. 2000, “A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes.” Cement and Concrete Research 30: 855-863. Bentz D.P., “CEMHYD3D: three-dimensional cement hydration and microstructure a development modelling package, version 2.0.”, NISTIR 6485, U.S. Department of Commerce, (2000), http://ciks.cbt.nist.gov/monograph. van Breugel K., “Simulation of hydration and formation of structure in hardening cement-based materials.” PhD Thesis, Technical University Delft, Netherlands, (1997). 18 [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] Bishnoi S., Scrivener K, “Microstructural Modelling of Cementitious Materials using Vector Approach.”, Proceedings of the 12th International Congress on the Chemistry of Cement, Montreal, Canada, July 2007 Taylor, H. F. W. 1989, “Modification of the Bogue Calculation.”, Adv Cem . Res 2(6): 73-77. Velez, K., Maximilien, S., Damidot, D., Fantozzi, G. & Sorrentino, F. 2001, “Determination of elastic modulus and hardness of pure constituents of Portland cement clinker.” Cem. Conc. Res. 31(4): 555-561. Damidot D., Velez K., Sorrentino F. 2003, “Characterisation of interstital transition zone (ITZ) of high performance cement by nanoindentation technique”, 11th International Congress on chemistry of cement; Proc. intern. symp., Durban, South Africa, 11-16 May 2003. CD-ROM. Constantinides, G. & Ulm, F.-J. 2004. “The effect of two types of C-S-H on the elasticity of cement-based materials: results from nanoindentation and micromechanical modeling.” Cem. Concre. Res. 34(1): 67-80. Monteiro, P.J.M. & Chang, C.T. 1995. “The elastic moduli of calcium hydroxide.”, Cement and Concrete Research 25(8): 1605– 1609. Kamali, S., Moranville, M., Garboczi, E.G., Prene, S. & Gerard, B., “Hydrate Dissolution Influence on the Young’s Modulus of Cement Paste.” In Li et al. (ed.), Fracture Mechanics of Concrete Structures (FraMCoS-V); Proc. intern. symp.,, Vail, 2004. Haecker, C.-J., Garboczi, E.J., Bullard, J.W., Bohn, R.B., Sun, Z., Shah, S.P., Voigt, T. 2005. “Modeling the linear elastic properties of Portland cement paste.” Cement and Concrete Research 35: 1948–1960 Smilauer, V.& Bittnar Z. 2006, “Microstructure-based micromechanical prediction of elastic properties in hydrating cement paste” Cement and Concrete Research 36: 1708–1718 Bernard, O., Ulm, F.J. & Lemarchand, E. 2003, “A multiscale micromechanics-hydration model for the early-age elas- 19 [20] [21] [22] [23] [24] [25] [26] [27] tic properties of cement-based materials.” Cem. Conc. Res. 33(9): 1293-1309. Sanahuja, J., Dormieux, L. & Chanvillard, G. 2007. “Modelling elasticity of a hydrating cement paste.”, Cement and Concrete Research 37: 1427–1439. Sun, Z., Garboczi, E.J. & Shah, S.P. 2007. “Modeling the elastic properties of concrete composites:Experiment, differential effective medium theory and numerical simulation.”, Cement & Concrete Composites 29: 22–38. Hua C, Acker P., Ehrlacher A., “Analyses and models of the autogenous shrinkage of hardening cement paste: I Modelling at macroscopic scale”, Cem. Concr. Research. 25 (7) (1995) 1457 –1468. Michaud, P.M., 2006. Vers une approche chimio-poroviscoelastique du comportement au jeune age des betons. PhD thesis, INSA de Lyon, Lyon. (in french) Haouas, A., 2007. Comportement au jeune âge des matériaux cimentaires – caractérisation et modélisation chimio-hydromécanique du retrait. PhD thesis, Ecole Normale Supérieure de Cachan, Cachan. (in french) Baroghel-Bouny, V., 2007. “Water vapour sorption experiments on hardened cementitious materials Part I: Essential tool for analysis of hygral behavior and its relation to pore structure”. Cement and Concrete Research 37: 414–437 Chateau X, Dormieux L, “Micromechanics of saturated and unsaturated porous media”, Int. J Numer. Anal. Meth. Geomech., 2002. Coussy O., “Revisiting the constitutive equations of unsaturated porous solids using a Lagrangian saturation concept”, International Journal for Numerical and Analytical Methods in Geomechanics, V31, n°15, 2007, Pages: 1675-1694