Isabelle Guilloteau

Etienne Arnoult

Bernard Peseux

Jean-Marc Berthillier

M Berthillier

Aircraft Engine Blades-casing Contact Study

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Industrial Context

During the qualification stage of a civil aircraft engine, a series of tests are carried out. One of these tests is a blade off test that demonstrates the engine capability of containing damage without catching fire and without failure of its mounting attachments when operated for at least 15 seconds, unless the resulting engine damage induces a self shutdown. The engineers need to predict the real loads applied to the engine structure in order to optimize the design stage. For this, a lot of numerical simulations are carried out but current software does not give adequate results, and in particular the finite element simulations involving contact. The aim of our work is to propose simplified models providing accurate results. The first part of this paper deals with the numerical simulation of contact, while the second part is an experimental study.

Algorithmic Aspects

In order to build a simplified model of the dynamic contact phenomenon, classical contact-impact methods and time integration operators are studied.

CONTACT METHODS

Two parameters must be defined to characterize the contact conditions : the gap function g, which represents the distance between the bodies in potential contact, and the contact pressure A. Thus, the unilateral contact law is expressed as an impenetrability condition (g~O), a compressive condition ('A~O), and a complementary condition (g'A=O).

The discretized contact problem consists in finding the displacement vector U(t), so that MU+Fint(U,U)-Fext+Fcontact=O and G(t)=G The contact methods derive from the optimization study with constraints. The first one is the penalty method, adopted by commercial software of dynamic mechanics like LS-DYNA3D, PAMCRASH or RADIOSS. It consists in introducing artificial springs between the contact nodes. This method is easy to implement in a finite element code but admits body penetrations and induces high frequency oscillations. The problem becomes : find U so that

MU+ pint (U, U) + [ Q Ta.Q]u _ pext + Q Ta.Go =O (I)
where the contact force vector is A = aG and a is the penalty parameter matrix.

The second method is the Lagrange multiplier method, which has the advantage of enforcing the exact geometric contact condition but introduces additional variables, namely the contact forces. The problem is then to find the couple (U,A) so that MU+ pint(U, U)-pext +QT A=O [START_REF] Billet | Caracterisation des modes de vibration d'une coque de revolution a l'aide d'un vibrometre laser a faisceau tournant[END_REF] with the condition G(t)=G 0 +QU = 0. Another kind of method is a mix of the penalty functions and the Lagrange multipliers : for example, the augmented Lagrangian or the perturbed Lagrangian methods. We will not deal with these methods in the present paper.

TIME INTEGRATION SCHEMES

The scheme generally chosen for high velocity impact problems is an explicit one : the second-order accurate central difference method Un+l = un +~TiJn+l/2, Un+l/2 = Un-1/2 +~TUn [START_REF] Carpenter | Lagrange constraints for transient finite element surface contact[END_REF] with ~T = tn+l-t 0 = tn+l/2-tn-112 • This scheme is conditionally stable, i.e. the time step must be lower than a limit value, depending on the discretization of the structure : ~ T ~ 2/wmax, where Wmax is the highest frequency component in the discretized system.

In the case of low velocities, an implicit scheme can be used. The most common one is the Newmark time integration scheme u n+t = u n +~TUn+ ~T2 [<Yz- ~run+ ~Dn+t] (4) [START_REF] Staples | Excitation of travelling wave response in axi-symmetric structures[END_REF] where ~ and y are the Newmark parameters. Integration is globally first-order accurate for y=l/2 and a choice of 2P~y~l/2 leads to an unconditional stability. If P=O and y=l/2, the scheme corresponds to the central difference method.

FORWARD INCREMENT LAGRANGE MULTIPLIER

If the discretized contact problem is resolved using the central difference method for time integration and the Lagrange multiplier for contact, there appears a singularity : the contact forces An+l have no influence on the displacements Un+l ! Carpenter et al. [3] propose an alternative formulation compatible with the explicit scheme :

•• int • ext T MU 0 +F (U 0 ,U 0 )-F 0 +Qn+l A 0 =0, Gn+t=O (andnotGn=O). * c
The formulation leads to a predictor-corrector process Un+l = Un+l + Un+l, where U~+ 1 is the vector of nodal displacements calculated without contact. The vector U~+l = -~T 2 M-1 Q~+tAn is a displacement correction due to contact and the force that permits us to impose the contact constraint is

[ 2 -1 T ]-l ( * 0) An= ~T Qn+IM Qn+l Qn+lUn+l +G • (6)

Numerical Results

The different methods of dynamic contact are tested on simple examples. The first one is an impact of two identical elastic rods, initially separated, as described in [START_REF] Carpenter | Lagrange constraints for transient finite element surface contact[END_REF]. The methods used to resolve this problem are the penalty and the forward Lagrange multiplier methods, with an explicit time integration scheme. Figure 1 shows that the Lagrange multipliers give very accurate results compared to the penalty method.

If the penalty parameter is increased for a better precision of the results, the nodal penetration decreases but a lot of numerical oscillations appear in the velocity and contact force curves. Theoretically an infinite value must be used for the penalty parameter, but in practice, especially with an explicit time scheme, the contact stiffness should be of the same magnitude as the structure stiffness, otherwise the time step must be widely reduced. That is the reason why the results are not always satisfactory.

With the Lagrange multiplier method, the results are accurate, and the explicit stable time step does not need decreasing [START_REF] Belytschko | Contact-impact by the pinball algorithm with penalty and Lagrangian methods[END_REF]. Thus, in order to model the following examples, the Lagrange multiplier method is retained.

The second example deals with the impact of an elastic rod against a rigid wall, resolved using the Lagrange multiplier method with implicit and explicit time integration schemes and LS-DYNA3D, with the default contact values. The Newmark scheme with (~=0.25 ; y=0.55) does not give satisfactory results ; there appears numerous oscillations in particular in the contact force curve [START_REF] Chaudhary | A solution method for static and dynamic analysis of threedimensional contact problems with friction[END_REF]. The (~=0.50 ; y=0.55) implicit and the explicit schemes, coupled with the Lagrange multiplier method, give similar results, while the explicit-penalty method (LS-DYNA3D) induces a slight nodal penetration. The last numerical test is the rebound, without friction, of a 2D rotational elastic beam against a rigid circle (cf. Figure 2). The steel beam describes an eccentric circle with the contact one and is subjected to a torque at its extremity on the hinge . The experimental part of the study deals with vibratory interaction between the engine and the fan bladed disk. When an aircraft engine bears a blade loss, high unbalanced loads are generated. The bladed disk and the casing come into contact, shocking and rubbing each other. In the meantime the two structures are loaded over a huge frequency spectrum and numerous natural modes of vibration are excited. Some particular conditions are required to make the whole system come into resonance and be submitted to severe stresses. The dynamic instability can lead to the ruin of the engine. Special attention is therefore to be paid on the exact conditions which give rise to the instability.

CONTACT EXTREMITY DISPlACEMENT

Both the casing and the bladed disk are supposed to be axisymmetric structures. As a consequence, their natural modal shapes can be represented by the number of their nodal diameters n. If fc denotes the casing natural frequency for n nodal diameters and fb the bladed disk natural frequency for the same n diameters mode, if f denotes the rotation frequency of the engine, then a condition for dynamic instability is given by [START_REF] Staples | Excitation of travelling wave response in axi-symmetric structures[END_REF]:

fc + fb = n.f (7)
This equation means that waves in opposing directions are travelling both in the casing and the blades.

In order to avoid such a coincidence, engine designers are to respect important frequency constraints which usually lead to increasing the casing stiffness, weighting the engine down and making it clumsy. Other solutions have to be found.

EXPERIMENTAL TEST RIG

The purpose of the experimental approach is to examine more precisely the behaviour of the blades when rubbing on the casing, and validate numerical simulations. In a second part, the test rig will be used to determine which parameters are involved in the apparition of interaction, in addition to equation [START_REF] Taylor | On a finite element method for dynamic contact/impact problems[END_REF].

The test rig used is deliberately simplified (see Figure 3). It consists in a part of a casing of an aircraft engine bolted on a steel ring which is fixed on the table of a vertical lathe, and in a blade clamped in the turret. In this configuration, the rotating part is the casing, and no aerodynamic effect due to the rotation of blades has to be taken into account. The use of a single blade instead of a whole set is convenient to get rid of the problem of the direction of the travelling waves through the blades. The contact between the blade and the casing, as well as the lathe rotation, are controlled by the experimenter. The instrumentation is made of classical devices: accelerometers (B&K), amplifiers (B&K), spectral analyzer (HP), and is completed with a Laser Doppler Velocimeter (Polytec ), useful for non-contact measurements on the rotating casing.

A first set of measurements was taken to characterize the test rig, especially to determine the natural frequencies and modal shapes of the casing. A finite element model including the casing and the steel ring was established and modified with respect to the experimental results. For a chosen rotation frequency of the lathe, interaction will be induce with a selected mode of the casing and a corresponding frequency of the blade.

FINITE ELEMENT MODEL-MODAL ANALYSIS

As a first approximation neglecting geometric and material defaults, the casing which looks like a frustrum of a cone is considered as an axisymmetric structure. The natural modes of this kind of structures have the particularity to be double ones: to each natural frequency correspond two different modal shapes. When the axisymmetry is perfect, and if e denotes an angular parameter of the structure, the modal shapes in the radial direction are cos(n9) and sin(n9), where n stands for the number of nodal diameters. No special orientation is noticeable. In case of imperfections, the symmetry is broken and the system is said to be mistuned. Each natural frequency splits into two distinct ones, and the modal shapes are modified according to the importance of the unbalancement. Particular orientations appears [START_REF] Tobias | The influence of dynamical iraperfections on the vibrations of rotating disks[END_REF].

The frequency response of the casing was recorded while the lathe was stopped, with a method using a small rotating mirror making the laser spot describe a circle on the inner side of the casing [START_REF] Billet | Caracterisation des modes de vibration d'une coque de revolution a l'aide d'un vibrometre laser a faisceau tournant[END_REF]. Once the identification of natural frequencies was complete (see Figure 4), time acquisitions were made to extract the modal shapes (see Figure 5). It appeared that the structure was mistuned both in frequency and time response, the biggest frequency split being obtained for the 4 nodal diameters mode (this is due to the way the steel ring was fixed on the lathe). The lowest modes were the most perturbed ones.

A first finite element model was made with SAMCEF supposing the perfect axisymmetry of the casing. A modal analysis carried out on the model clearly indicated that such an assumption was unacceptable. A fine geometrical study of the casing revealed that for a given altitude, the radius was not constant, the small variation, which never exceeds 1% being significant enough to disturb the modal behaviour. Another finite element model was then made including the geometric alterations.

The new results are compared with the experimental results in Figure 6. The biggest differences are attributed to the imperfections of the casing embedding.

CREATING INTERACTION

The test rig has been used in its current configuration to make first attempts on the interaction phenomenon. The choice of a mode with regular modal shape is more advisable. Moreover, to avoid the high modal density at the beginning of the spectrum, and because of its weak damping, the 10 nodal diameters mode was finally selected. Using the highest rotation frequency of the lathe (5Hz), and adapting equation [START_REF] Taylor | On a finite element method for dynamic contact/impact problems[END_REF] to the situation in which a single blade is implied, the first natural frequency of the blade was set to satisfy the mathematical condition of interaction. The contact was then created on the rotating casing.

The major issue which has to be faced is the lack of symmetry of the casing. The contact is unfortunately not permanent, and the structure is not excited by a constant force. As a result, no significative amplitude in the frequency response is noticeable neither on the casing nor on the blade. However, it is interesting to remark that for lower rotation frequency, the 10 nodal diameters mode's answer is higher than the others (20 dB). At these rotation speeds, the contact is quite similar to a harmonic excitation. When rotating at 5 Hz, the whole structure experiments impulsions whose effect is the excitation of every mode which is confirmed by the frequency response.

In order to suppress the effect of the geometrical default it is considered to fix an abrasive material on the inner side of the casing and to machine it so as to force the blade/casing contact to be permanent.

Conclusion and Future Prospects

The interaction phenomena, which have motivated the present work, are of major concern to aircraft designers. The present study is composed of a numerical part dealing with dynamic contact simulations on classical examples, in particular with the Lagrange multiplier method, and an experimental part which consists in identifying the parameters involving modal interaction. The next stage of our work is the simulation and the experimental study of the blade/casing dynamic contact.

Figure 1 .

 1 Figure 1. Impact of two identical elastic rods.

EXTREMITY S 1 .Figure 2 .

 12 Figure 2. Beam rebound on a circular contact zone.

•

  The next stage of our numerical study consists in modelling contact with a flexible axisymmetric surface.

  urr

Figure 3 .

 3 Figure 3. Experimental test rig.

Figure 4 .

 4 Figure 4. Casing frequency response.Figure 5. Example of modal shape.

Figure 5 .

 5 Figure 4. Casing frequency response.Figure 5. Example of modal shape.

Figure 6 .

 6 Figure 6. Comparison experimental results -finite element model.


Ve ´ronique Baroghel-Bouny
email: veronique.baroghel-bouny@lcpc.fr

Pierre Mounanga

Abdelhafid Khelidj

Ahmed Loukili

Noureddine Rafar

Autogenous deformations of cement pastes Part II. W/C effects, micro-macro correlations, and threshold values

Keywords: Cement paste, Shrinkage, Hydration, Ca(OH) 2, Microstructure

A broad experimental study has been performed, from the end of mixing up to 2 years, on a set of plain cement pastes prepared with the same type I ordinary portland cement (OPC) and various water-to-cement ratios (W/C), and cured at various constant temperatures. Several parameters have been measured on the hydrating materials, such as chemical shrinkage, volumetric and one-dimensional autogenous deformations, degree of hydration of the cement, Ca(OH) 2 content and Vicat setting times. Drying shrinkage has also been measured on the mature materials. In this part II of the paper, the effects of W/C within the range 0.25-0.60 have in particular been analysed in relation to the microstructural characteristics of the materials. This micro-macro analysis has highlighted a W/C threshold value (located around 0.40) both at the macro-level (on autogenous, but also on drying deformations and durability-related properties) and at the micro-level (characteristics of the hydration products, MIP porosity and pore size distribution, etc.).

In addition, volumetric and one-dimensional autogenous shrinkage deformations have been compared in the case of W/C=0.25 and T=20 8C. Finally, a critical twofold (chemical and structural) effect of calcium hydroxide has been found. When significant structural effects, generated by the formation and the growth of large-size Ca(OH) 2 crystals, take place, swelling can become prominent, as observed for onedimensional autogenous deformations in the case of medium and high W/C, and deviations are recorded on linear relationships.

Introduction

Nowadays, engineers have to take into account the required service lifetime, when selecting a concrete mixture and designing a structure. Hence, in recent years, the durability of reinforced or prestressed concrete structures has becoming a main concern. However, the most reliable, scientific-based, and accurate methodology used in the design stage to insure the required service lifetime for the structure will remain quite inefficient, if the early hours of the material life are not treated with the same accuracy in terms of execution quality and curing. The long-term properties of the material and even the whole life of the structure itself will indeed be determined by the early-age behaviour of concrete. For example, autogenous shrinkage as well as thermal dilation, which occur at early age, may yield, if restrained, the development of prejudicial cracking, particularly in highperformance concrete structures. In addition to mechanical problems, species aggressive to concrete or to reinforcement can more easily be carried from the surface through these (micro)cracks and then reduce the durability of the structure.

Various physical and chemical mechanisms are at the origin of autogenous volume changes and cracking of cementitious materials. The precise mechanisms are not very well understood, in particular because complex moisture and temperature combined effects take place. This makes a reliable predictive approach to the behaviour of concrete structures impossible.

As regards the test methods, autogenous deformations of cement pastes can be measured in two different ways: measurement of volumetric deformations (see for example Refs. [START_REF] Paulini | A weighing method for cement hydration[END_REF][START_REF] Sellevold | Highperformance concrete: early volume change and cracking tendency[END_REF][START_REF] Justnes | Total and external chemical shrinkage of low W/C ratio cement pastes[END_REF][START_REF] Garcia-Boivin | Retrait au jeune a ˆge du be ´ton-De ´veloppement d'une me ´thode expe ´rimentale et contribution a `l'analyse physique du retrait endoge[END_REF][START_REF] Mitani | Autogenous shrinkage of different types of cement and different types of admixtures at early ages[END_REF]), and measurement of one-dimensional deformations (e.g. Refs. [START_REF] Tazawa | Autogenous shrinkage caused by selfdesiccation in cementitious material[END_REF][START_REF] Jensen | A dilatometer for measuring autogenous deformation in hardening Portland cement paste[END_REF][START_REF] Baroghel-Bouny | Effect of mix-parameters on autogenous deformations of cement pastes-microstructural interpretations[END_REF]). The first type of measurement is particularly suitable to obtain shrinkage data at (very) early age, whereas the other is more often performed for longterm investigations.

Within the framework of the study of cementitious materials shrinkage, in addition to the fact that a lot of research work has already been done regarding the longterm period, it seems particularly relevant from both the scientific and practical points of view, as previously mentioned, to focus on the early age, in particular the first 24 h. The question of the selection of a relevant initial (reference) time t 0 (b24 h) for the deformation measurements has, for example, not been solved at the present time. Besides, along with the parameters related to the hydration process, it is well known that mix parameters, and in particular the water-to-cement ratio (W/C), influence autogenous deformations (see for example [START_REF] Tazawa | Autogenous shrinkage caused by selfdesiccation in cementitious material[END_REF][START_REF] Baroghel-Bouny | Effect of mix-parameters on autogenous deformations of cement pastes-microstructural interpretations[END_REF][START_REF] Jensen | Autogenous deformation and change of the relative humidity in silica-fume-modified cement paste[END_REF][START_REF] Schiessl | Influence of binders and admixtures on autogenous shrinkage of high performance concrete[END_REF][START_REF] Mitani | Variations volumiques des matrices cimentaires aux tre `s jeunes a ˆges: approche expe ´rimentale des aspects physiques et microstructuraux[END_REF]). As far as chemical shrinkage is concerned, Justnes et al. [START_REF] Justnes | Total and external chemical shrinkage of low W/C ratio cement pastes[END_REF] did not find any significant effect of W/C (ranging from 0.30 to 0.50) on the final chemical shrinkage value of cement pastes at 20 8C, from 0 to 48 h after mixing. Boivin et al. [START_REF] Boivin | Experimental assessment of chemical shrinkage of hydrating cement pastes[END_REF] confirmed these results and pointed out a purely kinetic influence of W/C on chemical shrinkage. Nevertheless, a broader range of investigations is still required, in order to accurately characterize the influence of W/C. The purpose of this paper is to improve the knowledge of cementitious materials and to contribute to the understanding of the mechanisms related to autogenous deformations. This step is of primary importance in order to enable the prediction of both early-age (autogenous) and delayed (drying, carbonation, etc.) deformations. Within this framework, the monitoring of the hydration process (through degree of hydration of the cement, Ca(OH) 2 content, Vicat setting times, etc.) and the characterisation of the cement paste microstructure have been carried out here as tools to help understand and interpret deformation results. The paper is separated in two parts. Part I was devoted to the analysis of curing temperature effects on early-age (V24 h) volume changes and to the investigation of the relevance of the maturity concept. In addition, micro-macro relationships have been pointed out in part I between volume changes and chemical and physical parameters. In this part II, the effects of W/C within the range 0.25-0.60 will be investigated on various properties (chemical shrinkage, autogenous deformations, drying shrinkage, etc.), in relation to the micro-structural characteristics of the materials. Moreover, a comparison between volumetric and one-dimensional autogenous shrinkage will be carried out in the case of W/C=0.25 and T=20 8C.

Experimental program, materials and test methods

Experimental program and materials

A broad experimental study has been performed from the end of mixing up to 2 years on a set of plain cement pastes prepared with the same type I ordinary portland cement (OPC) (CEM I-52.5, according to the EN 197-1 European standard). W/C ranged from 0.25 up to 0.60 and the mixtures were submitted to an isothermal curing process at 10, 20, 30, 40, or 50 8C (F0.1 8C). The cement contained 70.2% C 3 S, 7.8% C 2 S, 3.8% C 3 A, and 6.0% C 4 AF (Bogue calculation). Its loss on ignition was 2.3% and its Blaine specific surface area was 3320 cm 2 g À1 . The gypsum content was 5.2%. The chemical composition and the particle size distribution of the cement are given in Ref. [START_REF] Mounanga | Predicting Ca(OH) 2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF]. The cement was mixed in a 5-L laboratory mixer for 1 min, before addition of distilled water and a further mixing period of 3 min, in order to ensure a good homogeneity. A specific procedure has been carried out for the preparation of the samples intended for length change measurements. This procedure will be detailed further in the paper. As far as the curing temperature is concerned, the mixing water was heated or cooled on a case-by-case basis. This ensured a rapid equilibrium of the mixture at the required temperature.

In this study, chemical shrinkage, volumetric and onedimensional autogenous deformations, and drying shrinkage have been measured, as well as other parameters characterizing the hydration process or the microstructure of the material (degree of hydration of the cement, Ca(OH) 2 content, size of Ca(OH) 2 crystals, C/S ratio and thickness of the C-S-H rims, MIP porosity and pore size distribution, etc.).

Determination of degree of hydration and Ca(OH) 2 content

The evolution of both the non-evaporable water content and the calcium hydroxide content of the cement pastes has been monitored as a function of time (age) by means of thermogravimetric analysis (TGA) conducted between 20 and 1100 8C, in an inert N 2 atmosphere (1 bar). The tests have been performed at 3, 6, 9, 12, 15, 18, 21, and 24 h after mixing (curing temperatures from 10 to 50 8C), and also at the ages of 28 days, 6 months, and 2 years (curing at T=20 8C). The specimens were not submitted to crushing or drying prior to TGA measurement, in order to reduce their perturbation. The testing and calculation procedures applied are described in Ref. [START_REF] Mounanga | Predicting Ca(OH) 2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF].

The non-evaporable water amount is defined here as the mass loss recorded between 145 and 1050 8C, minus the mass loss due to CO 2 release caused by calcite decomposition between 600 and 800 8C [START_REF] Taylor | Cement chemistry[END_REF][START_REF] Loukili | Etude du retrait et du fluage de Be ´tons a `Ultra-Hautes Performances[END_REF]. This nonevaporable water amount has been used to calculate the degree of hydration of the cement (a). The Ca(OH) 2 content of the cement pastes has been determined by considering that dehydroxylation took place between 400 and 600 8C [START_REF] El-Jazairi | A simultaneous semi-isothermal method of thermogravimetry and derivative thermogravimetry, and its application to cement pastes[END_REF].

Chemical shrinkage measurement

(Total) chemical shrinkage has been measured on the various cement pastes by using a weighing method [START_REF] Paulini | A weighing method for cement hydration[END_REF][START_REF] Boivin | Experimental assessment of chemical shrinkage of hydrating cement pastes[END_REF]. The testing and calculation procedures applied are detailed in Ref. [START_REF] Mounanga | Predicting Ca(OH) 2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF]. The contact surface with the water bath is 20 cm 2 and the specimens are thin enough (7 mm), in order to permit absorption of water from the bath at the rate that the water is being consumed by the chemical reactions. The whole system, including the cement paste sample in its flask, is hung from a balance and is immersed in distilled water at the required temperature, 10, 20, 30, 40, or 50 8C (F0.1 8C). The measurement started 10 min after watercement contact and last for 24 h.

Autogenous deformation measurements

Early-age volume change measurement

The test method used here for early-age volume change (also called external chemical shrinkage) measurement consists of measuring, by hydrostatic weighing, the buoyancy variations of a sample constituted by a latex membrane filled with cement paste and immersed in a thermo-regulated bath. The test method is described in part I of the paper. The sample is placed on a nacelle hung from a balance, and immersed in the bath at the required temperature, 10, 20, 30, 40, or 50 8C (F0.1 8C). The accuracy of the balance is 0.0001 g. Both the bath temperature and the mass of the system are continuously recorded from 30 min after water-cement contact up to 24 h. In order to guarantee the absence of bleeding (see part I), volume changes have been measured here only on the cement pastes with W/C=0.25.

Length change measurement

Unrestrained one-dimensional autogenous deformations have been measured on F 20Â160-mm cement paste samples (W/C ranging from 0.25 up to 0.60) by using a simple technique, allowing length change measurement by means of dial gauge (with an accuracy of 10 Am/m), from initial Vicat setting time up to 1 year, at T=20 8C (F1 8C).

The samples were cast in polyethylene moulds ensuring tightness and low restraint, and also avoiding trapping air bubbles during the pouring of the fresh paste. The moulds have a longitudinal opening system allowing an initial (reference) length measurement inside the mould, and also an easy demoulding of the paste samples at the age of 24 h in order to carry out the further measurements on demoulded and sealed samples. The set-up and the test procedure are detailed in Ref. [START_REF] Baroghel-Bouny | Effect of mix-parameters on autogenous deformations of cement pastes-microstructural interpretations[END_REF].

The fresh paste was poured from the bottom of the mixing bowl into the moulds without stopping the mixer. This ensured a good homogeneity of the paste [START_REF] Roy | De ´formations instantane ´es et diffe ´re ´es des be ´tons a `hautes performances[END_REF][START_REF] Kheirbek | Influence des parame `tres de formulation sur les retraits endoge `ne et de dessiccation de la pa ˆte de ciment[END_REF]. Two stainless steel plugs remained gripped to the paste by means of embedded screw rods (see Fig. 1) and constituted the ends of the mould as well as of the sample. Each end had a recessed stud placed at the centre for length change measurement. From casting, the samples in their moulds were kept in rotation until at least the initial Vicat setting time, in order to minimize segregation and bleeding expected in particular with medium and high W/C. Just after demoulding, the samples were sealed by wrapping them in three self-adhesive aluminium foil sheets, in order to avoid moisture exchange with the surroundings (see Fig. 1).

The specimens were weighed at each length change measurement time. A 0.005% relative weight loss was measured for the mix with W/C=0.25 (vs. 0.018% with W/ C=0.60) within 1 year, thereby confirming weak water losses during this long-term experiment. The observed deformation values were not corrected by the thermal dilation coefficient of the materials, as the temperature rise was low inside the small samples tested here and disappeared very early compared to the ages of length change measurements.

Drying shrinkage measurement on mature materials

After 1-year length-change measurements in autogenous conditions, the self-adhesive aluminium foil sheets of some of the F20Â160-mm sealed samples were removed and the hardened cement pastes (hcp) were exposed in sealed chambers to various relative humidities (controlled by saturated salt solutions), in order to measure bpureQ drying shrinkage at T=20 8C (F1 8C). The same measurement device (dial gauge) as in autogenous conditions was used.

Other chemical, physical and microstructural investigations

The initial and final setting times of the cement pastes have been assessed by means of the Vicat (needle) test. The microstructure of the hcp has been investigated at various ages by means of secondary or backscatter mode scanning electron microscopy (SEM) after vacuum-drying, along with energy dispersive X-ray (EDX) semi-quantitative analysis. In particular, the size of the Ca(OH) 2 crystals has been estimated by secondary mode SEM on newly broken surfaces, whereas the C/S ratio and the thickness of the C-S-H rims have been quantified by BSE-SEM on polished sections in the neighbourhood of the unreacted cement cores [START_REF] Chaussadent | Influence du rapport E/C sur l'hydratation, la microstructure et les de ´formations endoge `nes de pa ˆtes de ciment durcies[END_REF]. Moreover, porosity and pore size distribution have been determined by mercury intrusion porosimetry (MIP), after freeze-drying of the specimens, with an apparatus ( P max =200 MPa) allowing the investigation of pore radii ranging from 3.7 nm up to 60 Am.

W/C effects: experimental results and discussion

Calcium hydroxide content

The calcium hydroxide content of the cement pastes with W/C ranging from 0.25 up to 0.60 and cured at T=20 8C is plotted in Fig. 2a vs. the degree of hydration of the cement (from 3 h up to 2 years). The degree of hydration of the cement vs. the age is displayed in Fig. 3 (see Ref. [START_REF] Mounanga | Predicting Ca(OH) 2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF] for results obtained at other temperatures).

The results show that for 0.25VW/CV0.40, both at early age and in the long term, the amount of Ca(OH) 2 produced at a given a is not influenced by W/C (note that a given a is obtained at different ages for different W/C). A linear relationship is exhibited between Ca(OH) 2 content and degree of hydration within the whole range of degrees of hydration, from a=7% (threshold corresponding to the precipitation of Ca(OH) 2 , as detected by TGA, see part I) up to ac80%.

Conversely, when W/CN0.40, the Ca(OH) 2 content seems to increase as a function of W/C, for the same degree of hydration (ranging up nearly 100%) (see Fig. 2a). The latter observation is confirmed by the results obtained as regards the C/S ratio of the C-S-H, which is the major other solid phase including calcium. The investigations do point out that for the same degree of hydration, the C/S ratio decreases when W/C increases (see Fig. 2b).

Chemical shrinkage

The chemical shrinkage of the cement pastes with various W/C is plotted vs. the age in Fig. 4, from about 10 min after water-cement contact up to 24 h, for each temperature investigated [START_REF] Schiessl | Influence of binders and admixtures on autogenous shrinkage of high performance concrete[END_REF][START_REF] Justnes | Correlating the deviation point between external and total chemical shrinkage with the setting time and other characteristics of hydrating cement paste[END_REF][START_REF] Barcelo | Early age shrinkage of concrete: back to physical mechanisms[END_REF][START_REF] Beltzung | Dissolution of cement and early chemical shrinkage of cement paste[END_REF], or 50 8C). Each plot is obtained from the mean value of two measurements. As seen on Figs. 3 and4, for a given temperature, very slight and unsystematic variations are recorded as a function of W/C on the bultimateQ (at 24 h) chemical shrinkage value. These results are in agreement with previously published works [START_REF] Justnes | Total and external chemical shrinkage of low W/C ratio cement pastes[END_REF][START_REF] Boivin | Experimental assessment of chemical shrinkage of hydrating cement pastes[END_REF][START_REF] Justnes | Correlating the deviation point between external and total chemical shrinkage with the setting time and other characteristics of hydrating cement paste[END_REF]. Furthermore, they extend the range of validity of such a statement within the W/C range 0.25-0.60 (as displayed in Fig. 4b in the case of T=20 8C) and within the temperature range 10-50 8C. In addition, Fig. 5a shows that, whatever the water-to-cement ratio, for Tz30 8C, the temperature has no more influence on the bultimateQ chemical shrinkage value (see also Ref. [START_REF] Mounanga | Predicting Ca(OH) 2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF]).

Besides, for a given temperature, the initial slope value of the chemical shrinkage vs. age plot corresponding to W/ C=0.25 seems slightly detached, whereas the values obtained with the other W/C are merged. This effect is depicted in Fig. 5b, where the initial slope values are plotted vs. the temperature, for the various W/C tested. This points out an acceleration effect on the rate of chemical shrinkage, induced by low W/C. This acceleration phenomenon is indeed significant between W/C=0.25 and W/C=0.30, but is no more noticeable when W/C ranges from 0.30 up to 0.60. The possible explanation of this finding combines chemical and physical processes:

-by decreasing W/C, the number of contacts between solid particles is increased. As contact zones are preferential sites for heterogeneous nucleation of C-S-H (or even act as nuclei), the precipitation starts earlier and reactions are accelerated [START_REF] Nonat | Interactions between chemical evolution (hydration) and physical evolution (setting) in the case of tricalcium silicate[END_REF][START_REF] Nonat | Calcium hydroxide concentration in solution: parameter determining the kinetics of the early hydration of tricalcium silicate and the characteristics of the products[END_REF], and so the chemical shrinkage, -in low W/C cement pastes, the dense packing of the cement particles, rapidly consolidated by C-S-H layers, hinders diffusional ion exchanges. This means that a greater ionic concentration and a rapid supersaturation take place in the interstitial liquid phase. This induces a faster precipitation of hydrates (e.g. Ca(OH 2 )) and therefore an accelerated chemical shrinkage.

Volumetric autogenous shrinkage

The volumetric autogenous shrinkage of the cement pastes with W/C=0.25 and cured at various temperatures [START_REF] Schiessl | Influence of binders and admixtures on autogenous shrinkage of high performance concrete[END_REF][START_REF] Justnes | Correlating the deviation point between external and total chemical shrinkage with the setting time and other characteristics of hydrating cement paste[END_REF][START_REF] Barcelo | Early age shrinkage of concrete: back to physical mechanisms[END_REF][START_REF] Beltzung | Dissolution of cement and early chemical shrinkage of cement paste[END_REF], or 50 8C) is plotted vs. the age in Fig. 6, from initial Vicat setting time (this moment is taken as the zero point on the time scale of the curves) up to 24 h. Each plot is obtained from the mean value of three samples. The knee-point observed on each plot (see also part I) appears later than the final Vicat setting time, for any temperature within the range investigated and for W/C=0.25 (see Fig. 7). These results are in accordance with those reported by Justnes et al. [START_REF] Justnes | Correlating the deviation point between external and total chemical shrinkage with the setting time and other characteristics of hydrating cement paste[END_REF] and complement them within the temperature range 10-50 8C. These authors found that the knee-point of the volumetric autogenous shrinkage vs. age plot matched the final Vicat setting time for W/C=0.40, but occurred later for W/C=0.30 and earlier for W/C=0.50, at T=23 8C. Note that from tests carried out on cement pastes cured at 20 8C and with W/C ranging from 0.30 to 0.40, Garcia was able to rather situate the knee-point between initial and final Vicat setting times [START_REF] Garcia-Boivin | Retrait au jeune a ˆge du be ´ton-De ´veloppement d'une me ´thode expe ´rimentale et contribution a `l'analyse physique du retrait endoge[END_REF]. It can be deduced from the whole of these studies that, at early age, a threshold takes place at about W/C=0.40 (where a coincidence between autogenous shrinkage knee-point and setting time is observed) for the cements used. This bpercolationQ threshold seems linked to the degree of connectivity of the pore (and solid) network, as it will be detailed further in this paper.

One-dimensional autogenous deformations

The unrestrained one-dimensional autogenous deformations (length changes) measured on the cement pastes with various W/C at T=20 8C are plotted vs. the age in Fig. 8, from initial Vicat setting time (this moment is taken as the zero point on the time scale of the curves) up to 1 year. The results are the mean values obtained from at least six samples. Note that a good brepeatabilityQ was recorded, since the standard deviation was of the order of magnitude of the accuracy of the device. The experimental results point out that, at a given age, the magnitude of autogenous shrinkage increases linearly as W/C decreases from 0.60 down to 0.25 (an increase of 1025 Am/m has been recorded on the bultimateQ, i.e. 1-year, value, see Fig. 9). In the case of mixtures with low W/C, a great proportion of the long-term autogenous shrinkage value occurs before 24 h (e.g. 540 Am/m has been recorded at 24 h, from initial Vicat setting time, with W/C=0.25, which accounts for 42% of the bultimateQ, i.e. 1-year, value), illustrating thereby the need of an initial measurement far earlier than 24 h for this type of materials. In the case of mixtures with medium or high W/C, swelling is exhibited from a few hours up to several days (À230 Am/m has been recorded at 21 days with W/C=0.60, see Fig. 8).

Various physical and chemical processes linked to cement hydration are involved in the unrestrained (isothermal) autogenous deformations measured on cement paste samples [START_REF] Sellevold | Highperformance concrete: early volume change and cracking tendency[END_REF][START_REF] Jensen | Autogenous deformation and change of the relative humidity in silica-fume-modified cement paste[END_REF][START_REF] Baroghel-Bouny | Effect of mix-parameters on autogenous deformations of cement pastes-microstructural interpretations[END_REF][START_REF] Mitani | Variations volumiques des matrices cimentaires aux tre `s jeunes a ˆges: approche expe ´rimentale des aspects physiques et microstructuraux[END_REF]. These deformations include in the general case three components: (total) chemical shrinkage, structural (chemical) swelling due to crystal growth, and self-desiccation shrinkage. The component which is prominent depends on the age at which the system is observed and on the mix-composition of the cement paste [START_REF] Baroghel-Bouny | Effect of mix-parameters on autogenous deformations of cement pastes-microstructural interpretations[END_REF]. In the long term, another component may be added: the matrix creep under hygral stress [START_REF] Ulm | Creep and shrinkage coupling: new review of some evidence[END_REF]. The starting time of the period where self-desiccation shrinkage is prominent, the other components becoming negligible, can be assumed for each mixture to be the time when the relationship between autogenous shrinkage and internal relative humidity (RH) becomes linear, as experimentally exhibited for ordinary and high-performance cement pastes and concretes in Ref. [START_REF] Baroghel-Bouny | Experimental investigation of self-desiccation in high-performance materials-Comparison with drying behaviour[END_REF]. Within this period, the correlation between W/ C and autogenous deformations can thus be mainly explained by the self-desiccation process. Assuming that self-desiccation shrinkage is based on the capillary tension mechanism, a finer initial pore structure will generate lower equilibrium internal RH according to Kelvin-Laplace law and will create higher tension in the pore water. Therefore, the lower the initial pore volume and the finer the pore network, the higher the self-desiccation shrinkage. As a matter of fact, when W/C decreases, the porosity and mean pore radius, measured by MIP on the hcp at a given age, also decrease (see Fig. 10), which trend is consistent with the increasing shrinkage value measured at the same age (see Fig. 9). This interpretation is also in agreement with the work of Hua et al. [START_REF] Hua | Analyses and models of the autogenous shrinkage of hardening cement paste: I. Modelling at macroscopic scale[END_REF], who modelled autogenous shrinkage, assumed as self-desiccation shrinkage, on the basis of MIP data.

The internal RH of the cement paste samples has been measured by RH-sensors on duplicates at the age of 1 year (360 days). Autogenous deformations are plotted vs. internal RH, at 360 days, in Fig. 11. The linear correlation observed confirms that the deformations recorded at this time are mainly the result of the self-desiccation process.

Drying shrinkage

It can also be deduced from Fig. 11 that the internal RH decreases as low as about 75% for the cement paste with W/ C=0.25, after 1-year sealed curing. Therefore, the first RH step selected for the subsequent drying process at T=20 8C after 1-year curing in autogenous conditions was RH=71.5% (obtained by means of a saturated salt solution based on SrCl 2 ). Once the equilibrium state was reached at this RH, the hcp samples were submitted to RH=63% (NH 4 NO 3 salt solution). The length changes registered when the hcp samples were submitted to RH=63%, from equilibrium state at RH=71.5%, are plotted vs. drying time Fig. 9. Correlation between one-dimensional autogenous deformations measured at T=20 8C and W/C, at the ages of 28 and 360 days. in Fig. 12a. It can be seen that, contrary to autogenous deformations, the bultimateQ drying shrinkage value increases when W/C increases from 0.25 up to 0.60. More precisely, the relationship between the bultimateQ drying shrinkage values at RH=63% and W/C seems to be linear above and below a threshold value located at W/C=0.40 (see Fig. 12b).

Interpretation of the swelling observed on length changes with medium and high W/C

Autogenous swelling within the first days has also been reported in the literature by other authors, for cementitious materials with medium or high W/C (e.g. Refs. [START_REF] Tazawa | Autogenous shrinkage caused by selfdesiccation in cementitious material[END_REF][START_REF] Schiessl | Influence of binders and admixtures on autogenous shrinkage of high performance concrete[END_REF]). Nevertheless, until now any deep insight into its associated mechanism has not really been performed.

The swelling observed on the experimental results presented here cannot be attributed to thermal effects due to the exothermic nature of chemical reactions. These effects disappeared indeed quickly as a result of the small size of the test specimens, unlike the swelling increase, which last several days (about 2 weeks, in the case of W/C=0.60).

Among the various components previously mentioned, the swelling, which appears here after setting may rather be attributed to the prominence of structural swelling, more precisely to the formation and the growth, through the hydration reactions of the cement, of large-size Ca(OH) 2 crystals (and also AFt needles, AFm crystals, etc.). During this period, swelling is prominent, although all the chemical reactions forming these products lead to a decrease in solid volume(s) and contribute to chemical shrinkage. Note that the manufacturing of expansive cements is based upon this principle (formation and growth of AFt needles). As the growth rate of platelet-like crystals is the slowest (compared to cubic or needle-like crystals), and according to the amount and the size of the products, the Ca(OH) 2 crystal growth is probably the main contributor to the blateQ deformations registered here. Another argument supporting the major influence of Ca(OH) 2 , as compared to that of AFt, is the low aluminate content of the cement tested here. The crystal growth generates a crystallization pressure on the pore walls, related to the supersaturation rate according to Riecke's law. The pressure can induce expansion of the microstructure (with creation of porosity) [START_REF] Vernet | Monitoring of the chemical and mechanical changes in high performance concretes during the first days[END_REF][START_REF] Scherer | Crystallization in pores[END_REF], as long as a water film is present between the crystal and the pore wall (allowing an easy circulation of ions). Expansion is favoured at high W/C probably as a result of the greater compliance of those samples (lower stiffness), which allows them to expand more under a given crystallization pressure (see further). In mixtures with high W/C, large-size Ca(OH) 2 crystals (and AFt needles) do have enough time, water amount and space for growing up, before the strength of the solid matrix and the self-desiccation process become strong enough to balance the crystallization pressure, and before self-desiccation shrinkage becomes prominent (when a gas phase appears in the capillary pores). In porous areas, calcium hydroxide crystals can indeed become bigger than the initial clinker particles. Microstructural investigations of the hcp tested here provide arguments to support this explanation (see Fig. 13a). The amount and the mean size of Ca(OH) 2 crystals, measured by TGA and SEM-EDX analysis, respectively, and therefore the volume occupied by the calcium hydroxide crystals, increase both with hydration time (even after 28 days) and W/C. For example, at 28 days, the Ca(OH) 2 content was 18.2%, 24.6%, 30.6%, Fig. 11. Correlation between one-dimensional autogenous deformations and internal RH, for cement pastes with W/C ranging from 0.25 up to 0.60 and cured at T=20 8C, at the age of 360 days. Fig. 10. Results obtained by MIP on hcp with various W/C, after curing at T=20 8C and freeze-drying. and 32.7%, and the mean size of the crystals was 4-10, 10-20, 10-30, and 80-100 Am, when W/C was respectively 0.25, 0.35, 0.45, and 0.60. Likewise, at 2 years, the Ca(OH) 2 content was 21.3%, 29.5%, 34.6%, and 38.6%, and the mean size of the crystals was 8-15, 10-30, 20-40, and 100-120 Am, when W/C was respectively 0.25, 0.35, 0.45, and 0.60. Therefore, even if on the one hand the increase in the Ca(OH) 2 amount is not very large when W/C increases from 0.45 up to 0.60, on the other hand the crystal size increases drastically. With low W/C, SEM observations showed that calcium hydroxide appeared mainly as small crystals spread out among the C-S-H phase (see Fig. 13b). Likewise, AFt could not be easily detected in the microstructure of hcp with W/C=0.25, whereas it appeared as well-crystallized needles in the case of W/C=0.45 and W/C=0.60. This last point indicates that the contribution of AFt to early-age swelling, before its dissolution, is probably more important in high W/C mixtures.

The fact that expansion, attributed here mainly to the growth of large-size Ca(OH) 2 crystals, proceeds for several days is also in agreement with the Ca(OH) 2 data reported by Bentz and Stutzman [START_REF] Bentz | Evolution of porosity and calcium hydroxide in laboratory concretes containing silica fume[END_REF] and obtained by means of quantitative image analysis. The authors showed that the area occupied by blargeQ Ca(OH) 2 crystals (areaN16 Am 2 ) increased with the age for at least 7 days, in a material with W/solid=0.45, as the hydration process goes on and as blargeQ crystals continue to grow up at the expense of smaller ones.

Furthermore, note that small Ca(OH) 2 crystals, invisible by SEM, could also be responsible for expansion.

Thus, the basic role of Ca(OH) 2 , with respect to autogenous deformations, is pointed out again (see part I), but here as a (micro)structural effect.

Another mechanism [START_REF] Van Breugel | Simulation of hydration and formation of structure in hardening cement-based materials[END_REF][START_REF] Barcelo | Early age shrinkage of concrete: back to physical mechanisms[END_REF] could also be suggested, in order to explain the swelling observed at medium and high W/C. This mechanism is based upon the formation of inner C-S-H rims (surrounding the residual unreacted cement cores), whose volume is higher than that of the portions of anhydrous grains they replace. These products do require water from outside the borderline of the cement grains for their formation (by topochemical reaction [START_REF] Baroghel-Bouny | Characterization of cement pastes and concretes-Methods, analysis[END_REF]). They are thus developing with a (local) increase in bsolidQ volume (contrary to outer C-S-H). Here, when W/C increases from 0.25 up to 0.60, the thickness of the C-S-H rims increases from 1 to 10 Am in the 28-day-old hcps. In particular, the thickness increases greatly above W/C=0.40. The volume of pseudomorphic (inner) C-S-H is therefore greater with high W/C cement pastes, and so would be the contribution of this Fig. 13. SEM images (G=2000Â) on newly broken surfaces of 28-day hardened cement pastes cured at T=20 8C. 1: Ca(OH) 2 crystals. Fig. 12. Drying shrinkage (length changes) measured at T=20 8C on F 20Â160-mm hcp samples submitted to RH=63%, after 1-year curing in autogenous conditions and equilibrium at RH=71.5%. mechanism to swelling deformations. This is in accordance with the experimental deformation results, supporting thereby this mechanism along with the existence of a threshold located at W/C=0.40 with the cement used here.

For both mechanisms and according to the literature, significant stresses will be raised in the case of confinement, which occurs when the pore space is restricted (i.e. in small pores, in Hadley grains, or in the case of low W/C), if solid contacts are required (e.g. between the C-S-H rims). But that condition is not necessary if the stresses can be transmitted through the water phase, which always saturates the pore space of high W/C materials at least at early age. As regards the mechanism based on the crystal growth (crystallization pressure), significant stresses will be raised in the case of high supersaturation, which rather takes place in low W/C materials. Nevertheless, at early age, the low crystallization pressures that prevail in high W/C materials (as a result of low supersaturation) are able to generate a significant expansion of the microstructure and thus swelling deformations, owing to the low stiffness of the materials.

Finally, it can be concluded that the swelling observed is probably linked to the amount and the size of the various hydration products (Ca(OH) 2 , inner C-S-H, AFt, etc.) likely to develop at early age in a concomitant way, in the complex cementitious system. Further investigations would be needed, in order to evaluate the actual contribution of each product. For example, tests performed on alite paste would allow to beliminateQ the contribution of aluminate phases (e.g. AFt). On the basis of the results obtained with the cement tested here, Ca(OH) 2 seems to be the main contributor.

Micro-macro analysis: existence of a W/C threshold value

In part I of the paper, it has been pointed out that the earlyage (V24 h) volumetric autogenous shrinkage of cement pastes with W/C=0.25 was linearly related to the degree of hydration (a) of the cement or to the Ca(OH) 2 content, from a=7% and whatever the temperature. In the present part II, it is possible to analyse further the role of these two parameters, in particular in the long-term range, and as a function of W/C. Fig. 14a andb show that one-dimensional autogenous deformations at T=20 8C, at a given age, are linearly related to the degree of hydration or to the Ca(OH) 2 content, respectively, but below and above a threshold, whose value is found again around W/C=0.40 (a deviation is observed at this point on the linear relationships).

In addition, these results confirm the existence of a W/C threshold located at 0.40 for the cement tested here, at the macroscopic scale, with respect to one-dimensional autogenous deformations at T=20 8C. It has indeed previously seen that, when the reference of the length-change measurements was taken at initial Vicat setting time, for W/Cz0.40 swelling was exhibited at early age and continued for several days. Conversely, for W/Cb0.40, shrinkage was recorded, whatever the age (see Fig. 8).

Likewise and as previously mentioned, the same threshold can be pointed out from the results presented in this paper and from those reported in the literature regarding the relative location of the knee-point of the volumetric autogenous shrinkage vs. age plot and the Vicat setting times [START_REF] Justnes | Correlating the deviation point between external and total chemical shrinkage with the setting time and other characteristics of hydrating cement paste[END_REF].

Moreover, the same threshold is found (at W/C=0.40) with respect to drying shrinkage: a deviation is recorded on the linear relationship between bultimateQ drying shrinkage values (at RH=63%) at T=20 8C and W/C, as illustrated in Fig. 12b.

Furthermore, the same threshold has been reported as regards the various durability-related properties or processes (leaching, etc.). For example, a carbonation front has been observed after an accelerated carbonation test on the cement pastes tested here for W/Cz0.45, whereas no carbonation took place for W/CV0.35 in the same experimental conditions [START_REF] Rafar | Comparaison et validite ´des me ´thodes de mesure de la carbonatation[END_REF]. Likewise, W/C=0.40 is widely recognized as a critical threshold, regarding the frost behaviour.

The same threshold value is obtained at the microscopic scale, confirming these findings and interpretation. For example, equivalent pore volumes are obtained by MIP, below and above 0.1 Am (two-mode pore size distribution), in the case of W/C=0.45 and W/C=0.60 at 28 days, whereas the major contribution to the MIP pore volume corresponds to pores with radius below 0.1 Am, in the case of W/C=0.25 and W/C=0.35 (single-mode distribution) (see Fig. 10a). Furthermore, the MIP porosity seems to increase to a greater extent from W/C=0.40 (see Fig. 10b).

Likewise, the amount, morphology, size, and distribution of the hydration products are very different above and below W/C=0.40, as described for Ca(OH) 2 , AFt, and inner C-S-H in the previous section. In addition, the C/S mole ratio of the C-S-H rims, measured by BSE-SEM, becomes very low above W/C=0.40. As a matter of fact, when W/C is 0.25, 0.35, 0.45 and 0.60, the C/S mole ratio at 28 days is respectively 1.93, 1.86, 1.51 and 1.61, and the C/S mole ratio at 2 years is respectively 2.00, 1.75, 1.43 and 1.50 (see Fig. 2b). More precisely, the C/S mean value is 1.7 and C/S values are below 1.7 for W/C=0.45 and W/C=0.60, and above 1.7 for W/C=0.35 and W/C=0.25. According to the literature, this change in stoichiometry can be associated with a change in the silicate chain length [START_REF] Nonat | Interactions between chemical evolution (hydration) and physical evolution (setting) in the case of tricalcium silicate[END_REF][START_REF] Nonat | Calcium hydroxide concentration in solution: parameter determining the kinetics of the early hydration of tricalcium silicate and the characteristics of the products[END_REF][START_REF] Lu | Phase composition of hydrated DSP cement pastes[END_REF][START_REF] Richardson | The nature of C-S-H in hardened cements[END_REF]. A review of the literature shows that the 1.7 value found here is a quite usual mean value measured on C 3 S pastes. Higher C/S values have been experimentally obtained here on pastes with low W/C. A low W/C means that hydration takes place in water-deficient conditions, that internal RH is lowered as a result of self-desiccation, and that dense C-S-H are produced. As a matter of fact, for W/Cb0.40 and at the age of 2 years, the degree of hydration of the cement is below 80% (see Fig. 2a) and the internal RH is below 80% (see Fig. 11 and Ref. [START_REF] Baroghel-Bouny | Experimental investigation of self-desiccation in high-performance materials-Comparison with drying behaviour[END_REF]). The higher C/S values measured in these conditions are in accordance with the literature. For example, somewhat higher C/S values than 1.7 have been measured on cement pastes [START_REF] Lu | Phase composition of hydrated DSP cement pastes[END_REF][START_REF] Richardson | The nature of C-S-H in hardened cements[END_REF]. More precisely, Richardson [START_REF] Richardson | The nature of C-S-H in hardened cements[END_REF] reported that inner C-S-H generally had a higher mean C/S ratio than outer products. Likewise, Nonat found that high C/S (high lime concentration) corresponded to dense C-S-H (in very dilute solutions) [START_REF] Nonat | Interactions between chemical evolution (hydration) and physical evolution (setting) in the case of tricalcium silicate[END_REF][START_REF] Nonat | Calcium hydroxide concentration in solution: parameter determining the kinetics of the early hydration of tricalcium silicate and the characteristics of the products[END_REF]. Further, Jensen et al. [START_REF] Jensen | Clinker mineral hydration at reduced relative humidities[END_REF] measured high C/S values (e.g. C/S=3, after 3 months at RH=83% and at T=40 8C) when studying hydration of C 3 S and C 2 S phases exposed to water vapour at RHb100%. Lower C/S values than 1.7 have been measured here in the case of high W/C, but such low values have seldom been reported until now in the literature. Nevertheless, these experimental results may be related to the observation made by Nonat (in very dilute solutions) [START_REF] Nonat | Interactions between chemical evolution (hydration) and physical evolution (setting) in the case of tricalcium silicate[END_REF][START_REF] Nonat | Calcium hydroxide concentration in solution: parameter determining the kinetics of the early hydration of tricalcium silicate and the characteristics of the products[END_REF]. The author showed that a discontinuity appears between the ranges [1.2À1.5] and [1.8À2], when the C/S of the C-S-H obtained by complete C 3 S hydration is plotted vs. the lime concentration in solution. This discontinuity was attributed by the author to a change in the mode and kinetics of growth of the C-S-H, associated with a change in their structure and morphology. Similar changes may occur in cement pastes when W/C varies to a great extent.

In addition, the threshold pointed out in this paper at about W/C=0.40 is quite consistent with the C-S-H model proposed by Jennings, where 50% of the products are lowdensity C-S-H and 50% are high-density C-S-H, for mature materials with W/C=0.40 [START_REF] Jennings | A model for the microstructure of calcium silicate hydrate in cement paste[END_REF].

Finally, the threshold value found in this paper at W/ C=0.40 can be attributed to chemical and physical phenomena linked to the hydration process, and can be related to the initial amount of capillary water. This W/C value probably corresponds to the borderline between well-connected and weakly connected pore networks (i.e. percolation threshold of the capillary phase, according to the definition given by Bentz and Garboczi [START_REF] Bentz | Percolation of phases in a three-dimensional cement paste microstructural model[END_REF]). Below this threshold, the capillary phase connectivity drops drastically and selfdesiccation appears in moisture-insulated systems at the ages considered. In these conditions, the hydration process is hindered by the lack of water or space, and the reaction rate is drastically reduced by the slow diffusion of the reactants through the C-S-H layers. Therefore, hydration cannot completely be achieved, even over the long term. This is in agreement with the literature (see for example Ref. [START_REF] Lu | Phase composition of hydrated DSP cement pastes[END_REF]) and with the previously mentioned experimental results with respect to degree of hydration and internal RH.

Comparison between volumetric and one-dimensional autogenous shrinkage for W/C=0.25

A comparison between volumetric and one-dimensional autogenous shrinkage measurements has been performed at T=20 8C for the cement paste with W/C=0.25, after converting the volumetric deformations into corresponding 1-D deformations (assumed as 1/3 of the volumetric ones) in Am/m. The results are plotted in Fig. 15a when the reference is taken at about 2.4 h, which approximately corresponds to the initial Vicat setting time. A factor of about 4 is found between the two types of measurement at 24 h. The discrepancy is further increased when the samples used for one-dimensional measurement are not kept in rotation until setting (see Fig. 15a). When the reference (of the deformations) is taken at the knee-point observed on the volumetric autogenous shrinkage vs. age plot, the difference is drastically reduced (see Fig. 15b), suggesting a better agreement between both types of measurement after the bknee-point periodQ (see part I) and a fortiori after the setting period (see Fig. 7). This illustrates once again that the choice of the reference (t 0 ) for expressing deformation data is critical for a relevant comparison of experimental results. Nevertheless, this does not provide sufficient explanations to the discrepancy observed.

Such discrepancies have already been reported, despite the very few comparisons available in the literature. For example, Barcelo et al. [START_REF] Barcelo | Linear vs. volumetric autogenous shrinkage measurement: material behaviour or experimental artefact[END_REF] observed a large difference between the results obtained from volumetric measurements on paste samples continuously rotated, from horizontal (enabling measurement from casting), and from vertical one-dimensional tests (see also Ref. [START_REF] Charron | Early age deformations of hydrating cement systems: comparison of linear and volumetric shrinkage measurements[END_REF], where mortars with W/C=0.35 and W/C=0.45 were tested). In addition to artefacts such as bleeding water re-absorption and internal underpressure inside the latex membrane (this effect is enhanced if air bubbles are present at the sample surface), the authors suggested the possibility of an anisotropic behaviour of the material at early age. Note that Beltzung and Wittmann [START_REF] Beltzung | Dissolution of cement and early chemical shrinkage of cement paste[END_REF] observed different vertical and horizontal shrinkage magnitudes before the age of 3 h and similar ones thereafter, on cubic HPC samples with W/C=0.32. Furthermore, on the one hand, an influence of the immersion depth of the sample or of a weak water penetration through the latex membrane in volumetric measurements, and on the other hand, possible artefacts occurring between initial and final Vicat setting times in one-dimensional measurements, could also be suggested, as well as a possible contribution of creep in the measured deformations.

All of these results illustrate that autogenous deformations are often difficult to measure and that the results appear to depend on the apparatus/procedure used. This is particularly true for (very) early volume changes occurring through the setting process.

Nevertheless, the results presented in this paper provide useful indications as regards the selection of a relevant initial time (t 0 ), valuable for both volume-and lengthchange measurements. Of course t 0 b24 h, unlike the conventional measurements, in order to take into account the significant proportion of autogenous shrinkage (and the associate cracking risk), which occurs before 24 h with some materials (low W/C). In addition, t 0 Nt(K), where K is the volumetric autogenous shrinkage knee-point, since before t(K), volumetric and 1-D measurements seem to quantify different processes.

Summary and concluding remarks

Experimental tools for assessing key-parameters have been presented in this paper. These tools can be used in view of designing and optimising mixtures, within the framework of the prediction of early-age deformations and of the prevention of early-age cracking. Nevertheless, as illustrated for example by the comparison between volumetric and one-dimensional autogenous shrinkage measurements, further investigations are still required with respect to both the techniques and the test procedures, before the two methods could be efficiently used in a complementary way.

Numerous experimental results have been obtained by means of these tools on a set of plain cement pastes prepared with the same type I OPC, with W/C ranging from 0.25 up to 0.60, and isothermally cured at various temperatures ranging from 10 up to 50 8C. These results constitute a data base useful for further researches and for developing engineering models, capable of predicting both early-age and delayed deformations of concrete, from the mix-parameters and/or from the microstructural characteristics of the material. This database could be completed in the near future by results obtained at higher temperatures and on concrete mixtures with the one-dimensional system.

In the following sections are summarized, first, the effects of W/C, and secondly, the determining role of calcium hydroxide.

W/C effects

n Both at early age and in the long term, the amount of Ca(OH) 2 produced at a given degree of hydration is not influenced by W/C within the range 0.25-40.

Conversely, within the range 0.45-0.60, at a given degree of hydration, the Ca(OH) 2 amount seems to slightly increase as a function of W/C. In addition, the crystal size, and therefore the contribution to the swelling observed, greatly increases when W/C increases. n W/C does not influence the rate and the magnitude of chemical shrinkage in a significant manner within the W/ C range 0.30-0.60 and for curing temperature ranging from 10 up to 50 8C. However, W/C has a kinetic effect within the range 0.25-0.30. Moreover, whatever the W/

Fig. 1 .

 1 Fig. 1. Scheme of the sealed cement paste sample, wrapped in self-adhesive aluminium foil sheets, used for length change measurements in autogenous conditions.

Fig. 2 .

 2 Fig. 2. Correlation between Ca(OH) 2 content (a) or C/S ratio of the C-S-H (b) and degree of hydration, for cement pastes with W/C ranging from 0.25 up to 0.60 and cured at T=20 8C, from 3 h up to 2 years.

Fig. 3 .

 3 Fig. 3. Degree of hydration of the cement vs. age for cement pastes with various W/C and cured at 20 8C.

Fig. 4 .

 4 Fig. 4. Chemical shrinkage vs. age for cement pastes with various W/C and cured at 10 8C (a), 20 8C (b), 30 8C (c), 40 8C (d), or 50 8C (e).

Fig. 5 .

 5 Fig. 5. bUltimateQ (at 24 h) chemical shrinkage value (a) and initial slope of the chemical shrinkage vs. age plots (b), as a function of temperature, for cement pastes with various W/C.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Volumetric autogenous shrinkage vs. age for cement pastes with W/C=0.25 and cured at 10, 20, 30, 40, or 50 8C (zero point of the time scale: t 0 =initial Vicat setting time).

Fig. 8 .

 8 Fig.8. One-dimensional autogenous deformations vs. age, measured from initial Vicat setting time up to 1 year on F20Â160-mm sealed samples of cement pastes with various W/C, at T=20 8C (zero point of the time scale: t 0 =initial Vicat setting time).

Fig. 14 .

 14 Fig. 14. Correlation between degree of hydration of the cement (a) or Ca(OH) 2 content (b) and one-dimensional autogenous deformations at a given age, for cement pastes with W/C ranging from 0.25 to 0.60 and cured at T=20 8C.

Fig. 15 .

 15 Fig. 15. Comparison between volumetric and one-dimensional autogenous shrinkage for the cement paste with W/C=0.25 and cured at T=20 8C: (a) reference (t 0 ) taken at the initial Vicat setting time, (b) reference (t 0 ) taken at the knee-point (K) indicated in (a).

Acknowledgments

The authors are grateful to Roger Coue ´, from IUT de Saint-Nazaire, for his assistance in setting up the chemical shrinkage device and to Ali Kheirbek, from LCPC, for his contribution as regards the length-change measurements.

C, for Tz30 8C, the temperature has no more influence on the magnitude. n At W/C=0.25, and for temperatures ranging from 10 up to 50 8C, the final Vicat setting time is sooner than the knee-point of the volumetric autogenous shrinkage vs. age plot. n At a given age, the magnitude of one-dimensional autogenous shrinkage increases linearly as W/C decreases from 0.60 down to 0.25, as a result of the self-desiccation process and of the structural swelling (and in agreement with the microstructural characteristics of the materials). Conversely, the magnitude of bpureQ drying shrinkage decreases when W/C decreases from 0.60 down to 0.25, n A W/C threshold has been pointed out (located around 0.40, for the type of cement tested here: CEM I 52.5) at early age and in the long term, at the macro-level with respect to volumetric and onedimensional autogenous deformations, as well as to drying shrinkage and durability-related properties. The same threshold has been found at the micro-level, with respect to the amount and characteristics of the hydration products assessed by means of SEM-EDX and TGA, as well as to the MIP porosity and pore size distribution.

Chemical and (micro)structural effects of Ca(OH) 2

All the findings presented in this paper show the critical twofold effect of calcium hydroxide:

-chemical effect: the Ca(OH) 2 content of the cement paste plays the role of a chemical indicator of the hydration (and of the setting) process, and hence of autogenous deformations, similar to that of the degree of hydration of the cement, -(micro)structural effect: this effect is generated by the formation and the growth of large-size Ca(OH) 2 crystals.

When the chemical effect (amount) is prominent, it has been found that autogenous shrinkage was linearly linked to the Ca(OH) 2 amount, as seen in part I on early-age volumetric autogenous shrinkage of cement pastes with W/C=0.25. When significant structural effects take place, swelling can become prominent, as observed in this part II on one-dimensional autogenous deformations of cement pastes with medium and high W/C, and deviations are registered on the linear relationships.