Abstract

An experimental non destructive device based on the propagation of acoustic waves (compressional waves) at low frequency (100Hz-1kHz) is used to characterize continuously the setting of hydraulic concrete. The results are presented for different formulations. From the recorded signals, the wave velocity, the damping coeficient and the ratio of pressure levels in two orthogonal directions are presented. They enable to follow up material setting and hardening phases and also to quantify by an inverse analysis, the evolution of the complex viscoelastic modulus from the fluid to the solid phase. The rheological behavior of setting heterogeneous concretes is studied from a theoretical approach by means of the homogenisation technique of periodic medium. Different assumptions are proposed. This material is first considered as a viscoelastic medium, then as a suspensions of gas bubbles at finite concentration in a viscoelastic matrix with low compressibility. The macroscopic equivalent medium behaves a compressible viscoelastic medium the mechanical parameters of which are presented as a function of time throughout the setting.

KEYWORDS: rheology, wave propagation, hydraulic concrete, inverse analysis, homogenisation technique, two phases modelling, oedometric modulus, shear modulus.

1 Introduction

This work deals with an experimental and theoretical approach of the rheological behaviour of materials such as mortars, concretes, throughout setting. This phenomenon, which plays a significant part in the industrial process, arises from binder hydration where the initial fluid mixture is changed into a solid. This progressive evolution modifies significantly the character and the behaviour of the initial "fluid" mixture.
We consider in this paper the case of hydraulic concrete composed of solid aggregates, cement, water and air bubbles (trapped during the mixing process, the volumic concentration of the gas phase is about 1-5 %). Other materials have been tested [1], [2], [8], [9]. At the beginning, the mixture is a viscous suspension of solid particles. With the hydration reactions, solid components without regular structure (hydrates) are developing, thus increasing solid concentration. The process going on, the percolation threshold is reached. This constitutes the transition between the fluid and solid states. Progressively, the solid skeleton, plunged into the surrounding fluid, is reinforcing. Then the material is hardening. This process lasts around 6 hours.
Structure evolution and the presence of heterogeneities, the size of which ranges from a millimetre to a centimetre, are the subject of a complex rheological analysis. In addition, conventional tests are ineffective to monitor the setting continuously. First, the measurement ranges for rheometrical tests are too limited. Second, the presence of gas bubbles leading to low initial wave velocity (C » 100m/s), ultrasounds are scattered onthe large heterogeneities and strongly attenuated [14], [13]. So, large frequency spectrumis required [10]and it is observed that the frequency span of the received signals is modifiedthroughout setting; or ultrasounds are applied on fresh desaerated cement pastes [6].

In the first part, we present the experimental device enabling to monitor the rheological evolution of different formulations of hydraulic concrete. Its principle is based on the propagation of compressionnal waves through the setting material. Another configuration of the device enables to study the propagation of shear waves in the material [3]. In the second part, the analysis of the experimental data is achieved with different assumptions. The first one consists in assuming the bulk material as a viscoelastic medium. Then a more precise analysis takes into account the presence of gas bubbles inside the evolutionary paste. The homogenisation technique is developed to characterize the behaviour of a suspension of gas bubbles in a viscoelastic matrix [4]. The behaviour law obtained characterized a viscoelastic compressible medium the mechanical parameters of which are given as a function of the component parameters, is then used to analyse the experimental measurements. For each assumption, experimental data are analysed and the evolutions of the rheological parameters according to time are presented.

2 Experimental device and measurements

2.1 The device
The laboratory device called "vibroscope"( [8], [1]), enables to study the propagation of compressionnal waves in the material. The principle of the device is based on the large variations of the macroscopic compressibility of concretes while they evolved from a fluid state to a solid one. Due to the presence of gas bubbles, the inverse compressibility of which is high as compared to those of the fluid and the solid phases, the wave velocity is very low (C_p » 100m/s). Large wavelengths (l ³ 12; 5cm) as compared to the heterogeneity size (h » 1cm) are then achieved when using very low frequencies (f £ 1kHz). Applied stresses and recorded signals show a narrow frequency spectrum around 800Hz which ensures a good scale separation between wavelength and heterogeneity size.

Fig 1: Diagram of the experimental device.

By means of a At regular time intervals, transient plane waves are generated by a vibrator attached to a plate. Waves are sensed by means of three pressure transducers positioned inside the paste and removed at the end of each experiment (see Fig. 1).

Two tranducers (P_1// and P_2//) are positionned in the direction of the wave propagation. From the signals, the wave velocity C_p and and the ratio of the pressure levels || P_2// || / || P_1// || = D_p are deduced. Note that D_p is inversely proportional to the damping coeficient.
A third one (P₃) senses the waves in a normal direction to the wave propagation. This pressure transducer is positioned just beside the sensor P_1// and thus, the ratio of the levels of pressure (P₃/P_1//) is to be connected to the Poisson's effect which develops in concrete that is setting. This measure is very interesting because this ratio takes value 1 in a fluidand a value lower than 1 for a solid.
Fresh cement pastes are very sensitive to the thermodynamical environment. Due to the lagged mould, the vibroscope does not disturb the the kinetics of the chemical reactions.

In this test, very small solicitations are applied to the material to avoid damaging of the microstructure which is building up during setting. We present in the next section, some signals recorded and we verified the linearity of the behaviour.

2.2 Experimental measurements
Results are presented for various formulations of hydraulic concrete.

2.2.1 Formulations
We present here results achieved for different classical formulations of hydraulic concrete (granulates 1767kg/m ³, cement 420kg/m ³ , water 177kg/m ³ , air bubbles between 1 to 3% per volume). Two different parameters were modified : the ratio water / cement (0, 4 and 0, 6) and the temperature of casing (10, 20 and 30°C). So six different experiments were carried out. The hydration of cement particles and complex chemical reactions lead to built a solid skeleton and so, to the setting and hardening of concrete. This evolution lasts between five to eight hours depending on the formulations.

2.2.2 Behaviour linearity
For the tests, in order to avoid damaging the microstructure which is buiding up during setting, very small levels of load (acceleration about 5m/s ² ) are applied to the material. We recorded the following maximum levels :

• displacement » 10mm,
• strain » 10^-5
• rate of strain» 4.10^-3s^-1
• and pressure levels ranging from 10 (early aged concrete) to 300Pa (hardened concrete).

Moreover, we verified experimentally at different moments during the setting, that the behaviour remains linear in a wide range around these values. For diferent level of load (acceleration A ranging from 0, 25 to 12m/s ² ), the pressure signals have been recorded forall the transducers. Then we normalised each signal by the maximum value of the pressure. In the one hand, we compared the normalised pressure signals by superimposing them for each tranducers, and in the other hand, the normalisation factors have been compared. We present here two different examples achieved on early aged concrete (30 min after pourring) and another one for hardened concrete (3 hours after the pouring). We ploted the normalized signals obtained for different level of load (30 min after pourring, see Fig. 2, 3 hours after the pouring, see Fig. 3) We observe a very good superposition of the curves even for low levels of acceleration.

The normalisation factor are then compared for all the signals emitted, the ratio between the normalisation factors P_i/P₀ (subscript _i -resp. ₀- stands for one unspecified load-resp. reference load-) are ploted as a function of the ratio of normalisation load factorA_i/A₀.
We observe on the one hand, the very good superposition of the curves normalized in the broad range around the usual value (A » 5m/s ² ), and on the other hand, that the ratios of the values of normalisations gather around the first bisectrix (curve x = y).
We can conclude that in the range of load tested, this test is non destructive and that the behaviour remains linear.

Fig 2a:	Fig 2b:	Fig 3a:	Fig 3b:
Fig 2: Linearity of the behaviour, superposition of normelized recorded signals, example 1: concrete aged 30 minutes.		Fig 3: Linearity of the behaviour, superposition of normelized recorded signals, example 2 : concrete aged 3 hours.

Fig 4: Linearity of the behaviour, comparison of the normalization coeficients.

2.2.3 Velocity and damping coeficients
In case of compressional waves, for the different formulations investigated, the evolutions of the wave velocity C_p and and the ratio of pressure levels D_p = || P_2// || / || P_1// || are plotted versus time [11](Fig. 5and 5). From the very simple measurement of C_p(50 £ C_p £ 2500m/s), we observe clearly the evolution of the material in a wide range, from its fluid state to its solid state.

Time changes are detected by the device. For all the experiments on hydraulic concrete, two stages appear (Fig. 5 and 6). During the first one, the mechanical evolution is weak but not null, during the second stage, this evolution is significant. During the tests, wave velocity is multiplied by a factor 15 and the report/ratio increases of a factor 5 what means that damping decreases. The same two stages are also observed on the last parameter measured R_p. At the beginning, the values are between 0, 6 and 0, 8 depending on the experiment (except the first measurements for one case). Let us note that this ratio would take value 1 in a perfect fluid. Then we observe that the ratio is decreasing during a second stage. In all cases, we observe a final value of about 0, 1.

The transition can be linked to the threshold percolation. In all cases investigated, we may observe with these measurements, a significant evolution of the materials and we note that these evolutions are very consistent on both velocity and pressure ratios evolutions. The device allows to monitor the evolution and it is sensible to the mechanical changes related to the formulations tested. Classical in uence of the different parameters investigated are observed, for example an increase of the setting process when increasing the pouring temperature or decreasing the water/cement ratio.

Fig 5: Evolution of Cp and Dp coeficients versus time.

Fig 6: Evolution of the ratio Rp of pressure levels in two normal directions.

3 Analysis

The measurements achieved are used to calculated rheological parameters for the material. The inverse analysis consists in assuming a behaviour law for the material and then to study the propagation of compressional waves in such a medium.

Different assumptions are proposed. The first basic analysis is to consider that the fresh concrete is viscoelastic. Then we are abble to calculate from the experimental data, the macroscopical oedometric modulus. Then a more detailled analysis is presented. The method uses the homogenisation technique of periodic medium to determine the behaviour of the macroscopically equivalent medium assuming that the fresh concrete is a suspension of gas bubbles in a viscoelastic matrix.

3.1 Viscoelastic medium
A first analysis can be achieved in terms of viscoelasticity : C_p and D_p are related to the complex oedometric modulus
E = | E | e ⁱ. The evolutions of | E | versus time are presented (Fig. 7), where E = K + (a + 2b)M^* in the next analysis). This gives an instantaneous rheological measurement on a wide range (factor 1000) allowing to monitor the setting process. Experiments clearly highlight, first the significant rheological variations for a given material during the setting, second the setting kinetics for each material.

3.2 Bi phasic modelling
A more precise analysis is achieved by means of a theoretical approach. The macroscopic behaviour of such materials is obtained by means of the homogenisation method. The macroscopic constitutive law of these heterogeneous media is deduced from a physical analysis at the microscopic scale using the homogenisation theory [12]. This method is well adapted when the medium presents a elementary representative volume W (ERV) and when the scale separation between the ERV size l and macroscopic length L related to the wavelength l, L = l=2p, is marked. L and l introduce the small parameter Î = l/L. For the experiments achieved [11], the value of Î is about 10^-3. The medium is considered as a periodic arrangement of identical cells W.

The behaviour law:We consider in this study that the medium is composed by gas bubbles (assumed to be purely compressible) in suspension in a viscoelastic medium. The interbubble mixture, called hereafter matrix, is constituted by solid elastic particles and viscous uid. For simplicity, we assume that the matrix is a viscoelastic medium which evolves progressively with binder hydration. This chemical process leads to material setting; it is suficiently slow (about 6 hours) to allow with short duration tests (about 25ms), the identification of the instantaneous rheological behaviour at any time. We apply the homogenisation method to a suspension of air bubbles (c is the finite gas concentration and 1/K_g its compressibility) in a viscoelastic matrix characterised by its complex viscoelastic shear modulus M_m and its compressibility 1/K_m.We do not detail the homogenisationprocess. It is presented in [4].

The macroscopic behaviour of the equivalent medium is characterised by the following set of equations eq.1; eq.3, eq.2:
is the macroscopic strain tensor. The form tensors a (resp. b) of rank 2 (resp. 4) have real values and become scalar numbers in case of isotropic macroscopic behaviour. eq.2, eq.3, eq.1 define the macroscopic behaviour law.
At the macroscopic scale we obtain the behaviour of a viscoelastic compressible medium. The shear and volume moduli are of the same order of magnitude and depends on the properties of the matrix and the VER geometry by the form factors a and b. The matrix only appears by its shear modulus M_m and the macroscopic moduli (shear and volume) are proportional to M_m by form tensors a and b. The contribution of the gas appears in the effective compressibility K, the value of which is generally complex. It can degenerate into real values (adiabatic or isothermic) according to frequency and bubble size. The inertial effects are related to the matrix.
It is important to note that even under macroscopic purely isotropic strain, it is the shear properties of the interbubble matrix which acts. The behaviour law ( eq.2, eq.1 ) corresponds to the interesting case where the viscoelastic modulus M_m is of the same order of magnitude than the gas compressibility 1/Kg. Modifying the order of magnitude of M_m and following the same approach, one can prove that :

• when
• when
• when M_m = O(K_m), the matrix compressibility must be taken into account and we obtain the classical behaviour of a porous medium with viscoelastic skeleton.

As the shear behaviour of setting material strongly evolves, these various descriptions can be observed as the material is passing from a uid state to a solid state.

Evolution of rheological parameters: We assumed that the material tested is an isotopic medium. So, the behaviour law becomes :

(4)

The scalars (a and b) only depend on the local geometry of the elementary representative volume. The gas phase remains in quasi-adiabatic conditions because for the frequency used ( » 800Hz), the thickness of the thermal layer ( » 0; 1mm). is small as compared to the bubble size (1mm or more). So, where g = 1, 4 is the ratio of specific heat coeficients , the pressure of the gas phase is P ^e _g = 10 ⁵ Pa and c, the gas phase concentration, is measured precisely (about 1 to 3 % in case of hydraulic concrete).

The interest of this analysis is to obtain rheological measurements directly linked the evolving matrix properties. It is important to note that under compressional wave propagation, the macroscopic shear modulus (bM^*) is proportional to the shear modulus of the interbubble mixture M^*. The dometric modulus K+(a+2b)M^*, (a+2b)M^* and the phase f of M^* can be deduced from the measurement of C_p and D_p. They are presented for all the experiments (Fig. 7and 8).

Fig 7: Rheological modulus : presentation of the complex bulk oedometric modulus, modulus and phase.

Fig 8: Rheological evolutions of the complex volume modulus (modulus and phase).

For the hydraulic concrete, rheological quantities are consistent with the following mechanical interpretation. At the beginning, the interbubble matrix is characterized by a viscous behavior (suspension in water of solid particles and crystals generated by binder hydration). At this stage, K and (a + 2b)M^* are of the same order of magnitude. A transition is reached with the percolation threshold of the hydrates. When they become connected, a rigid elastic skeleton is being built. The development of the crystallization increases both the elastic and viscous properties of the material, as water is trapped inside the weak skeleton. The dominant term in E is thus related to (a+2b)M^* and not anymore to K. This transition is considered as the beginning of the mechanical setting. This effect goes on with a continuous increase of | (a + 2b)M^* | corresponding to the hardening of the matrix (Fig. 8).

4 Conclusions

These experimental and theoretical studies enable to monitor the rheological evolution of fresh concrete that set from the study of compressional wave propagation. The device based on non destructive tests allows to determine the wave velocity, damping coeficient and the ratio of pressure in two orthogonal directions as function of time. From these measurements, we can distinguish different stages in the mechanical evolution of the material. During the first stage, a slow evolution is observed. Then an intermediate phase occurs where velocity is increasing, damping is decreasing and the ratio of pressure levels on two normal directions is decreasing. This transition can be interpreted like the threshold of percolation of the hydrates in formation. Moreover, the parametric study on the formulation and the initial conditions of manufacture of the concrete make it possible to find the usual in uences of the parameters (initial temperature, ratio W/C) from a qualitative point of view but also to determine the quantitative in uence of it more precisely. For example, it is possible to indicate for each formulation, the mechanical in uence of a variation of the initial temperature.
Then, these data are used to calculate mechanical parameters attached to the evolving material. An inverse analysis is necessary as the tests are non homogeneous.

Different levels of analysis are presented. They allow to follow up throughout the setting of hydraulic concrete the evolutions of some rheological parameters of the mixture:

• to follow up the mechanical evolution of the material in a qualitative way,
• to calculate rheological parameter of the macroscopic medium,
• to evaluate rheological parameter for the interbubble matrix that sets.

From the homogenisation technique, the macroscopic behaviour law has been defined for different cases : the material is assumed to be successively a viscoelastic medium and a suspension of purely compressible bubbles in a viscoelastic medium. These different assumptions lead to give a more detailled analysis and the rheological parameters versus time are characterizing successively the viscoelasticity of the bulk material and the viscelasticity of the interbubble matrix. A theoretical work concerning tri phases mixtures has been developed [4].

Complementary adaptations of the device allow to obtain complementary values for the behaviour law in particular from the generation of shear waves. Works are developed to study at the same time the propagation of both compressional and shear waves.

This approach is adjustable for the study of many evolving materials. It can be adjusted to test other time-dependant heterogeneous materials during their mechanical evolutions in civil engineering and also in others fields, involving different evolutive process.

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