Fresh concrete is analyzed as a granular mixture (considering the entire population of particles, from cement to silica fume to gravel) in a water suspension. In this analysis, the content of entrapped air will be ignored. The minimum volume of water is that which corresponds to the porosity of the dry system. A concrete with zero workability is therefore, by definition, a packing in which the porosity is just saturated with water (Figure 11).

An increase in the water content, beyond the minimum to fill the pores makes possible a water-filled spacing between the particles in the mixture, and, consequently, sliding between particles can be initiated (see Figure 12). If the shearing of the system is confined, a deformation will appear if this applied shear stress is sufficient to counteract the friction forces between the solid particles. Thus, the yield stress will be governed not by the liquid phase, the only role of which in the material is to define the average distance between particles, but by the number and nature of the contacts between particles. Hence, for a mixture with a single size class of particles, there will be a relationship between yield stress and packing of the following form:

(6)

where is the volumetric fraction of solid material (with respect to a total volume of one), * is the maximum value of for close packing (or packing density of the dry mixture) and f is an increasing function. The ratio /* is an expression of the relative concentration of solids, compared to the maximum packing. The function f increases with /*, because the yield stress increases with increasing values of /*.

To investigate the microstructure of the flowing material, we will assume that the speed of each particle is equal to the macroscopic speed of the homogeneized fluid, i.e., fresh concrete. Then, if the fluid remains in the laminar regime while flowing between the solid particles, its contribution to the shear resistance will remain proportional to the overall strain gradient.

Thus, in the classic form of the Bingham model:

=  0 + µ

(7)

the term  ₀ is the contribution of the skeleton and the term µ is the contribution of the suspending liquid. Based on the preceding analysis, a general form for the plastic viscosity, µ , can be deduced:

(8)

where µ ₀ is the plastic viscosity of the suspending fluid and g is an increasing function of /*. The function g is increasing because the plastic viscosity increases with the increase of /*. A further explanation of this statement will be given in section 4.3.2.

In the case of particles of several sizes, the value of the function f should depend on all the contributions of the various size classes of particles. Therefore the yield stress is as follows:

(9)

where _i is the volume fraction of granular size class i and _i* is its maximum value for close packing, all of the other _j (j   i) being constant.

When the size and surface roughness of the particles change, the number of contacts between particles and the roughness of the particles change. It is therefore to be expected that the contribution of each size class to the yield stress includes size and roughness parameters relative to the particle fraction.

As to the plastic viscosity, it is already known, in the case of suspensions of single-size spheres, that the plastic viscosity does not depend on the size of the particles [25]. For binary systems of spheres, it appears that the parameter /* continues to control the apparent viscosity, the influence of the size distribution being contained in the packing density term * [26]. Clearly, it would be desirable if this assertion were to remain valid for particle systems with a large range of sizes in which the particles are not spherical. Thus, as a first step, an attempt was made to verify this assumption for the mortars and concretes that were tested.

If we refer now to the Herschel-Bulkley model, which was shown in 3.1.4 to be more applicable than the Bingham model for describing the rheology of fresh concrete, there is the question: "What is the physical origin for its mathematical form?" It can be inferred that the exponent of the relationship only reflects the increase (or reduction) in the plastic viscosity as the shear rate increases. The basic assumption that permits us to explain the Bingham model physically, i.e., the plastic viscosity does not depend on the shear strain rate, is probably true only for a narrow range of shear strain rates. Flocculation of particles, or the appearance of local turbulence, probably modify the flow conditions of the liquid phase in the interstices of the granular phase. Moreover, it is in the mixtures without HRWRA that, on the average, the strongest non-linearities have been found (i.e. the greatest values of the coefficient b in the Herschel-Bulkley model). It is also in these mixtures that the forces between fine particles are the most important, because the cement particles are not dispersed by the HRWRAs.

It follows from the previous considerations that the calculation (or measurement) of the packing density of the dry mixtures, defined as the maximum concentration (by volume) that the "dried out" suspension could attain, constitutes an essential preliminary step to modeling the rheological properties. A recent model developed for mathematically describing compaction of powders is described in the following section.

LCPC has been conducting studies for several years to develop models to predict the packing density of granular mixtures. A linear model for the packing density of particles mixtures was first developed [27], followed more recently by the Solid Suspension Model [28 ,29], incorporated in the RENÉ-LCPC software [13]. The essential innovation in this second model consisted in distinguishing the actual packing density of a mixture, which was attained by using a given placement and compaction procedure, from the virtual packing density, the maximum packing density that could be attained only by putting the particles in place one by one. The virtual packing density was calculated as follows:

=  Min (  i )
	(Min = minimum value of i )	(10)
1  i   n,   i   0

(11)

in which y_j is the proportion by volume of size class i, i.e., the ratio of the volume of size class i to the total solid volume; i is the virtual packing density of the i-class compacted alone; _i is the virtual packing density of the mixture when class i is dominant (i.e., when it is responsible for the "blockage" of the mixture); and the parameters a_ij and b_ji are interaction coefficients describing the "loosening" of the particles and wall effects, respectively. The loosening and wall effect can be defined as follows. In the vicinity of an isolated coarse grain, the packing of a small grain is perturbated. This effect is called the wall effect. Conversely, when an isolated fine grain is introduced in an interstice of a coarse grain packing, the coarse grain arrangement is disrupted. This second effect is called loosening effect [33].

In this model, using the _i parameters, a reference plastic viscosity could be calculated. This plastic viscosity depends on the processing method and the mixture composition. By analyzing a large number of test data (original tests or measurements taken from the literature [30, 31]), it was possible to tabulate reference viscosities against the placement method. Finally, the model provides an equation, which when solved numerically gives the theoretical value for the packing density. At the time the "Solid Suspension Model" was constructed, an empirical determination was made of the coefficients a and b (by smoothing the experimental values, which are presumed to be a function of the ratio between particle sizes). By calculating the _i parameters for the packing density measurements related to a narrow range of the constituents, it was possible to calculate the theoretical packing density of each combination of these components. During validation studies, it was shown that an average error of less than one percent resulted when the model was used for predicting the packing density of the dry mixtures. It was also anticipated that the model would be equally suitable for predicting the plastic viscosity of the concentrated suspensions.

Subsequent evaluation of the model disclosed that:

• The fitting that leads to the interaction equations could be improved [32];
• The predictions of the plastic viscosity of binary sphere suspensions compared with experimental values taken from the literature [26] proved to be disappointing, and
• An error appeared in the calculation of the plastic viscosity with regard to a mixture of several granular size classes.

Consideration of these findings lead to the definition of a new model, called the Compressible Packing Model [33] . In this model, the hypothesis that led to the definitions of the virtual packing density values are preserved. However, the empirical expressions that make it possible to predict the values of the interaction coefficients have been modified. The new expressions provide for better fitting of the experimental data and satisfy certain continuity conditions that were not taken into consideration before. These coefficients are as follows:

(12)

where d_i and d_j are the diameters of the granular classes i and j as defined by sieve sizes.

The concept of reference viscosity, which is difficult to justify physically, was replaced with the concept of a compaction index K. Like the reference viscosity in the previous model, this factor is assumed to be characteristic of the placement of the mixture. Thus, the emphasis is placed on the assumption that a granular packing is not a frozen structure, but may be considered as a compressible object, the state of compaction being described by the parameter K. Using considerations of the additive nature of particle contributions and of self-consistency, the following equation is proposed to define the compaction index:

(13)

where * is the packing density of the granular mixture. The partial compaction indices K_i represent the contribution of each class to the overall index K. The values _i* are the maximum values of the partial volume fraction _i when packing is carried out (i.e., the maximum volume of particles i that can be placed in the mixture, the other values _j, with j  i, remaining constant).

Compared with the solid suspension model, the improvement in predicting the experimental values of packing density is marginal [33]. Nevertheless, the compressible packing model appears to be better justified physically. The compaction index could thus be directly linked to the energy provided to pack the system. Typical values of the compaction index for various placement procedures for dry particles are given in Table 3. For the water-demand tests performed on mixtures of binders [34], a value 6.7 for the index K is found for this type of system (compaction of the fine particles in a very concentrated suspension under the effect of mixing and capillary forces) [35].

Until now, attempts at direct validation of packing density models have only concerned granular materials of a smaller range of particle size than that of concrete [29]. It was, therefore, hoped that the applicability of the new model to cement-and-aggregate mixtures could be verified. We first measured the dry packing density of the cement, using the same procedure as used for the aggregates (see section 2.1.1) (the value found was 0.553). Given the compacting process and the compaction index associated with it, the value of the virtual packing density is less than that of the cement in the presence of water (0.61 instead of 0.64). This shows that the forces of interaction between particles and the friction forces they generate are more significant in air than in water, despite the formation of hydrates, which occurs from the first contact between the water and the anhydrous cement. In other words, if the fresh concrete were dried out, e.g. by using a drained vibro-compaction procedure to squeeze out the maximum amount of the interstitial water, one would expect a higher packing density than for the dry mixture.

The dry mixtures were then prepared using the same composition of dry ingredients (sand, coarse aggregates and cement) as selected mixtures, i.e., BO2, BO5, BO7 and BO11(Chapter 3) and their dry packing densities were measured. The agreement with the predictions of the compressible packing model was excellent (Figure 13), and confirms that the model is applicable throughout the range of particle sizes used in this study.

Figure 13. Comparison of the experimental values of packing density with the predictions of the Compressible Packing Model for dry cement-sand-gravel mixtures

The effect of the presence of water on the virtual packing density of the cement shows that the parameters _i needs to be calculated from measurements made on the binders in the presence of water. Recall that _i is the virtual packing density of a single-size fraction [13]. Also, the experimental program includes intermediate mixtures with dosages of HRWRA between 0 and the maximum value (the saturation dosage). Therefore, the variation in the demand for water by the cement, as a function of the percentage of HRWRA, "SP%", has been systematically measured and the change in the packing density of the cement, c, (corresponding to a K value of 6.7) has been determined (Figure 14). This maximum packing density of the cement is effectively expressed by an empirical equation of the parabolic type:

(14)

Using these estimated values of packing density, the values of the virtual packing density, i, for the cement with various doses of HRWRA were deduced (see Table 4). It was then possible to calculate the packing density, *, of each mixture group for a compaction index of 9. Based on our work with dry mixtures, a compaction index of 9 is characteristic of a highly compacted random packing. The values of * are given in Appendix VII.

Table 4. Values of the virtual packing density of the cement for different dosages of HRWRA

Hu examined the literature concerning rheological models linking mixture composition and viscosity of suspensions [7]. He found that most authors [7, 36, 37, 38] analyzed fresh concrete as a paste/aggregate composite and tried to deduce the plastic viscosity of the concrete from the plastic viscosity of the paste by multiplying it by a function that took into account the volume and nature of the granular phase. Some authors even extended this analysis to the cement paste, using the Farris approach. In order to calculate the plastic viscosity of the multi-modal suspensions they performed an iterative calculation, the whole being made up of the suspending fluid and the finest classes being dealt with homogeneously at the scale of a given class [25]. As elegant as they might be, these models suffer from not taking into consideration the inter-particle interactions. In fact, most concrete mixtures have a more or less continuous size distribution, so that the division into a number of discrete classes is arbitrary. Even the distinction between cement paste and aggregate, which is pertinent in the case of hardened concrete, is difficult to justify for fresh concrete. The large particles of cement are of comparable size comparable to the finest sand particles and their respective contributions to the rheology of the whole are not of a different nature (at least as long as the hydration of the cement remains negligible). One way of attempting to link the rheology of the neat cement paste with that of concrete was to introduce another factor, i.e., the gap existing between the aggregates [39]. However, this approach requires measuring the rheological behavior of the paste through independent means, which was not done in the present study.

The model described herein is based on the work of Chang and Powell, in which the relative concentrations of the suspensions were treated as controlling their plastic viscosity [26]. When the experimental plastic viscosity µ' measured in this study is plotted as a function of the ratio between solid volume fraction and packing density, the plot in Figure 15 is obtained. The solid volume fraction that is used that of the "de-aired" mixture, which means that the quantity of air has been disregarded (see Figure 16). This assumption is not entirely valid. In fact, it is known that mixtures consolidated by vibration always contain a certain volume of entrapped air, which varies from 1% for fluid concretes to 4% and more for mortars with HRWRA. Moreover, the plastic viscosity is measured for the sheared (and thus unconsolidated) mixtures, the air content of which is probably greater than that of unsheared or consolidated mixtures. However, this volume of air is difficult to measure (tests deal with consolidated concrete, while concrete under shear exhibits dilatancy) and in any case is governed by the rheology of the system, i.e., a given mixture increases its dilatancy with a decrease of water content. In the absence of reliable data on this subject, the volume of aggregate that would have been obtained after total consolidation was used.

The relative concentration of solids has significant effect on the plastic viscosity. It is quite remarkable that the experimental points in Figure 15 are grouped about a single curve, regardless of the nature of the mixture (mortars or concretes, with or without HRWRA, and with or without silica fume). An empirical equation for the best fit curve is:

(15)

and provides an evaluation of the experimental plastic viscosity with an average relative error of 27%. The uncertainty concerning the packing density parameter * (on the order of +1% in terms of absolute value explains part of the dispersion of the experimental points around the calculated curve. The volume fraction of solids is also subject to a certain error since the sheared mixture is less compacted than the consolidated mixture. Finally the plastic viscosity µ', calculated from the Herschel-Bulkley parameters, valid for a limited range of the shear strain rate, also has an error, which is about 10% for the cohesive mixtures [7] and probably more for mixtures mixtures exhibiting bleeding and segregation. This is one reason that the present model does not appear to be easily improved by taking into account secondary parameters other than the fundamental parameter of relative solid volume concentration of solids.

According to this model, the effect of the HRWRA on the plastic viscosity is through deflocculation of the cement, which is expressed by a lower water demand and by a greater packing density, *, of the mixtures.

When the yield stress is plotted as a function of the relative concentration of solids (Figure 17), the same trend as observed for the plastic viscosity is not obtained. It appears that it is necessary to consider the contributions of the various granular fractions, according to the nature of the materials.

Returning to the formulation of the compaction index (equation 8), it is found that the form of this expression is similar to the equation type needed. The terms K_i are functions of the relative volume concentration of size class i (the terms _i/*_i) and this type of expression is the only one that permits the terms to be additive and self-consistent [33]. It is thus tempting to develop models from linear combinations of the partial compaction indices K_i, since one can then sum the terms K_i relating to different fractions having a similar size composition (for example, the different fractions of sand, a material with a large grading span compared to the other components). However, in calculating the terms K_i, it is necessary to take into account the solid volume in the de-aired concrete (or in the mortar) and not in the corresponding dry mixture. In order to avoid this confusion, the compaction indices related to de-aired concrete will be called K'_i. They are calculated with a value for the parameter equal to 1 minus the volume of free water (the difference between the total water in the mixture and the water absorbed by the aggregates). The following two equations have been obtained by a fit to mimimize the absolute difference between the measured and calculated values. For mixtures without HRWRA we obtained:

' 0 =  exp ( 2.537 + 0.540 K'g +0.854  K's +1.134 K'c  )

(16)

and, for the mixtures with 1% HRWRA (without silica fume), we obtained:

' 0 = exp  (2.537 + 0.540 K'g +0.854  K's +0.224K'c )

(17)

In these equations, '₀ is the yield stress obtained by fitting of the rheometer results in accordance with the Herschel-Bulkley model. The indices g, s and c relate to gravel, sand and cement, respectively.

The average error of these models is 163 Pa (see Figure 18). The error is reduced to 109 Pa if we exclude from the set the mixtures BO4C, BHP4A and BHP4B. For these mixtures the model underestimates the yield stress, because their high gravel content and low cement content, promote segregation and aggregates interlocking.

By comparison, the error due to the uncertainty regarding the friction of the rheometer joint is on the order of 60 Pa [8], to which must be added the repeatability error due to the sampling of the material, the value of which is on the order of 10% of the yield stress. Therefore, the precision of the model is felt to be satisfactory considering the experimental uncertainties. As to the number of adjustable parameters, it may appear high (five multiplier coefficients for the K's factors), but it remains small compared with the number of experimental points used to determine the coefficients (49).

In the model given by equation 16, it is found that the multiplier coefficients of the partial compaction indices are ordered logically according to the particle size: they increase as the size decreases. However, this increase is not proportional to the specific surface area since the elementary friction due to contact between the particles probably decreases when the number of contacts per unit volume increases. This may be why all of the approaches based on taking the specific area of the aggregates as a fundamental parameter for the rheology have always failed or required empirical correction [40].

In the second model represented by equation 17, the contribution of the aggregates is unchanged while the cement contribution is strongly reduced. It is thus seen that the introduction of the HRWRA into the mixture has two effects. In the first, the HRWRA increased the deflocculation of the cement, which permits the fine particles to pack more efficiently in the interstices between the large particles, reducing the water demand. In the second, the organic molecules are adsorbed onto the solid surfaces and lubricate the contacts between particles. This lubrication reduces the shear stresses at the contact zones, explaining the reduction of the coefficient of K'_c by a factor of 5. In mixtures with high dosages of HRWRAs, it is the granular skeleton that is mostly responsible for the yield stress. This is why the yield stress is directly linked to the relative concentration of aggregates (Figure 19).

As described in section 2.3.4, four silica-fume concretes were generated from the central mixture with HRWRA. Increasing percentages of silica fume were added while the solid-volume proportion of cement was kept constant, and the water was adjusted to remain in the region of measurable flow. The dosage of HRWRA was increased to maintain a condition of maximum deflocculation of the fine particles. The reduction in the water dosage did not compensate for the addition of silica fume, so that the volume of paste continuously increased. The yield stress does not continuously decrease with silica fume dosages varying between 0 and 30% of the mass of cement. A minimum is reached at around 15% silica-fume (at the limit of significance), while the yield stress clearly increases for greater dosages of silica fume (see Figure 20). Thus, it appears that the silica fume has a specific effect on the yield stress, even in the presence of HRWRA. In order to evaluate the multiplier coefficient of the term K'_sf added to equation 17, we first searched for a value that would produce a theoretical curve (yield stress, dosage of silica fume) that was approximately parallel to the experimental curve. This condition is satisfied for an fitted value of 1.5 (Figure 20). If a simple statistical approach to minimize the deviation between the model estimates and the experimental values on all four silica-fume concretes is pursued, a value on the order of 0.806 is obtained for the K'_sf coefficient. Additional studies are needed to choose between these two options.

Nevertheless, it appears that the contribution of the finer material (diameter less than 80 µm) to the yield stress in the presence of HRWRA is not a function of their size alone. The affinity of the surfaces to the polymer appears to play a role. In addition, it is known that silica fume adsorbs the naphthalene sulfonates because of its silanol groups [41]. However, the quantity adsorbed per unit of surface area is probably less than that of the portland cement, and hence the effectiveness of the HRWRAs in reducing the shear stress would probably be less.

The results obtained in this study were also used for our third goal, i.e, to develop a simple and cheap field test. We will describe the test and use the same data as in the rheology study to validate the use of the new device, which is a modification of the standard slump test.