RESULTS AND ANALYSIS OF THE RHEOLOGICAL DATA

The test results are given in Appendix IV for ordinary concretes, Appendix V for concretes with admixtures, and Appendix VI for concretes with silica fume and with intermediate dosages of HRWRAs. For each mixture, the data collected from the measurements of the modified slump tests, the density, and the Bingham parameters are given. Also the parameters calculated using the Herschel-Bulkley function are listed (see section 3.1.4). Visual observations reported bleeding and segregation, the qualitative ranking (0 to 3) is included in the tables (0 = good and 3 = worst case).

Slumps were obtained throughout the entire range from 0 mm to 290 mm (for self-leveling mortar). For the most fluid mixtures, the slump flow was also measured (i.e., the average diameter of the spread after stabilization, measured parallel to the sides of the rectangular metal plate on which the tests were performed). Some concretes exhibited segregation, manifested by the accumulation of coarse aggregates surrounded by a "lake" of mortar. Within each group corresponding to the same dry mixture, the slump increased with the dosage of water, except for the BHP1 and BHP4 groups. Overall, it was found that the sensitivity of the slump test to the amount of water varied from one series to the other.

The rheometer test was performed on all of the mixtures that were made. However, for some mixtures, because the slumps were too low or the size distributions were too far from the central mixture, it was not possible to obtain usable measurements, i.e., the shearing was not uniform. The mixtures with bold lettering in the tables of Appendix IV and Appendix V are those for which the rheometer measurements were valid. Some concretes, such as those of the BHP8 series, which had HRWRAs and high cement contents, exhibited the peculiarity of being self-leveling (very low yield stress), along with high plastic viscosity, which was visually apparent by their extreme slowness in reaching equilibrium position during the slump test. The torque necessary to shear them at the maximum velocity (0.8 rps) sometimes exceeded the capabilities of the rheometer. Hence, after a nonproductive test, the test was repeated while limiting the range of the shear rate, called "reduced shear rate" tests (Appendix V). Only one test was conducted for each mixture, therefore no calculation of uncertainty can be made. From previous tests [7], the error on the torque and the rotation speed can be estimated to be less than 1%.

The intensity of bleeding varied a great deal from mixture to mixture. In the series of concretes without HRWRA, all the concretes with low cement contents (groups BO2 and BO3 in Appendix IV) exhibited appreciable bleeding during the course of the rheometer test. A majority of the concretes with HRWRA also displayed the same phenomenon, particularly when the slump was increased by adding water. The "intermediate" concretes behaved similarly, while the silica fume concretes remained homogeneous during the tests, confirming the well-known stabilizing of ultra-fine particles.

The negative effect of the HRWRA on the cohesiveness of the concretes was also observed as the segregation increased when the amount of gravel was increased. This consequence of the dilatant property of the mixtures [7] was, as might have been expected, manifested particularly for the mixtures that had HRWRA and high coarse aggregate content.

For each rheometer test, a linear regression analysis was performed on the values of the net torque (i.e., the torque after subtracting the contribution of the seal) as a function of the rotation velocity in order to deduce the Bingham parameters, the yield stress and plastic viscosity, in accordance with the procedure given in ref. [7]. The results are given in tables in Appendices IV to VI.

The plastic viscosity values were distributed over a wide range. Generally, the mixtures with HRWRA have higher plastic viscosities than those without HRWRA. This is attributed to the HRWRAs having a greater effect on the shear yield stress than on the plastic viscosity, while water reduction causes a notable increase in both parameters [1 ,7]. In the series of intermediate concretes, while the slumps remain in the same range, it is striking to observe how the plastic viscosity increases as the central mixture with HRWRA is approached (see Appendix IV). On the contrary, incorporation of silica fume makes it possible to negate the effect that lowering the water dosage has on plastic viscosity, in accordance with the known filler effect. When the central mixture with HRWRA (BHP1C) is compared with the mixture richest in silica fume (FS30), reducing the water/binder ratio from 0.38 to 0.24 decreases the plastic viscosity from 530 to 400 Pa· s. This study confirms the beneficial effect of silica fume on the plastic viscosity of high-performance concretes [20] because a lower plastic viscosity makes the placement easier.

Among concretes having slumps equal to at least 100 mm, the yield stresses that are obtained remain lower than 2000 Pa (Appendix IV-VI), which was expected on the basis of past experience [7]. On the other hand, negative yield stresses are obtained with seven mixes, sometimes with values below (-300) Pa, which are not explainable by the uncertainty associated with measurement (on the order of 100 Pa). A negative yield stress also implies that the data have a high degree of non-linearity and the Bingham model cannot, therefore, be used to extrapolate the yield stress. Moreover, examination of the curves of the torque as a function of the angular velocity of the rheometer blades shows a varying degree of nonlinearity (see Figure 6).

Figure 6. Measurement of torque as a function of the angular velocity for three groups of concrete, and fit with a model of the Herschel-Bulkley type. In each group, the curves are ranked as expected as a function of the water content of the mixtures (i.e., the torque decreases as the water content increases).

Based on the above anaylsis using the Bingham model, it was necessary to develop a non-linear model to represent the flow behavior. Prompted by an approach borrowed to describe the "rheology of mud" [21], a granular material that has a number of similarities to fresh concrete, we applied the Herschel-Bulkley model. This provides a relationship of shear stress to shear strain rate based on a power law [22]:

= ' 0 +ab

(2)

where is the shear stress, is the shear strain rate imposed on the sample, and '₀, a, and b are the new characteristic parameters describing the rheological behavior of the concrete. The relationship between the torque and the angular velocity of the rheometer is similar to the previous equation, and is calculated by integration of the function relating the velocity field and the torsional motion imposed by the geometry of the test. It is in the following form:

= 0 + A  N b

(3)

where is the torque, ₀ is the minimum torque necessary to shear the sample, N is the angular velocity in revolutions per second and A and b are the parameters that depend on the concrete and the dimensions of the apparatus. By following the same calculation steps as in the analysis of the test using the Bingham model [7], one obtains the following formulas for the characteristic parameters relating to the concrete:

(4)

where R₁ and R₂ are the interior and exterior radii of the concrete sample in the rheometer (equal to 20 mm and 120 mm, respctively), and h is the height of that sample (equal to 100 mm). The coefficient 0.9 in the equation for the coefficient a takes into consideration the resistance effect of the friction of the side walls on the flow of the concrete in the apparatus.

This Hershel-Bulkley model was fitted to the experimental curves of torque versus angular velocity. The Herschel-Bulkley model fits the experimental curves quite well. Examples of these curves are given in Figure 6, and the values of the rheological parameters for the various mixtures, appear in the tables in Appendices IV to VI. One finds that all of the new yield stress ('₀) values calculated using equation (4) are positive. As to the values of the parameter b, they are generally greater than one (1.53 on the average for the concretes without HRWRA and 1.36 for the concretes with HRWRA), which clearly shows that the actual behavior of the fresh concrete generally differs from the linear behavior described by the Bingham model.

The connection between the Hershcel-Bulkley yield stress and the slump has been shown both theoretically and experimentally [7,23]. The present tests confirm this relationship (Figure 7). If all the results obtained, including the concrete without HRWRA for which the slump was less than 100 mm are used, Figure 7 shows that the data can be approximated by a single curve. This suggests that the range of validity of the rheometer tests, in regard to the "new" yield stress (Herschel-Bulkley), extends beyond concretes having a very plastic consistency, i.e., high slump or low yield stress. We were also interested in the final slump flow, which has been considered to provide a better measurement (than the standard slump) of the consistency of self-leveling concretes [24]. Self-leveling concretes have a slump value of 260 mm or higher. Excluding the mixtures whose slump tests exhibited segregation by accumulation of coarse aggregates in the middle of the concrete, it is found that the slump flow is related to the yield stress (see Figure 8). Therefore, the measurements obtained with the slump test support use of the Herschel-Bulkley model for analysis of rheological tests.

Figure 7. Relationship between yield stress from the Herschel-Bulkley (HB) model and slump for all of the mixtures tested.

Figure 8. Relationship between yield stress from the Herschel-Bulkley (HB) model and slump flow (spread) for the self-leveling concretes.

3.2.1. Non-linear behavior

In order to minimize segregation during rheometer testing, we adopted (see section 2.2.2) a procedure that consisted of measuring five points in the selected range of angular velocity. The drawback of this choice appeared afterwards when we realized that it would be necessary to determine by regression, three parameters, instead of two, based on these five experimental points. Because three parameters must now be determined using only five points, the number of degrees of freedom is very low. Therefore, the uncertainty in the fitted parameters is greater and it is more difficult to conclude with certainty that the parameter b differs from 1.0. In order to judge the accuracy obtained on the fitted parameters, we used one of the concrete mixtures tested in accordance with the procedure previously defined (test B08C) and repeated the test using an identical procedure, except that the range of the angular velocity was extended in both directions (i.e., one point was added at a higher velocity and one at a slower velocity). The measurements were still done by decreasing velocity. The result of this additional test (B08Cbis) is shown in Figure 9.

A considerable difference is found between the torque values obtained in the two tests (regular procedure, Bo8C, and the extended range, B08Cbis, of shear strain rates), which is shown by the marked differences in the yield stress in Table 2. Shearing the concrete at higher velocities has a tendency to promote segregation [5]. Therefore, this further justifies the precautions taken in the procedure that was adopted. On the other hand, when the results of the second test, B08Cbis, are considered in isolation, it turns out that the Herschel-Bulkley parameters are not very sensitive to the number of points considered, whether it consists of a total of seven points or is reduced to the five central points, as in the standard test. Thus, if all of the test had been performed in the same range of shear strain rate, but with a greater number of measurement points, the rheological parameters obtained would probably not have been very different from those found.

Another question concerns whether the centrifugal force during the tests could explain the nonlinear behavior. The contribution of the outer layers to the resistant torque of the sample (from which the distance to the axis is greater than half of the radius of the apparatus) is predominant, and the outer layers could have an over-concentration of aggregate. We believe that the non-linearity is not due to the effect of the centrifugal force for the following reasons:

• The acceleration due to the centrifugal force is at most 3 m/s² (for an angular velocity of 0.8 rps and a radius of 0.12 m), i.e., less than a third of the acceleration due to gravity. Thus, any radial segregation due to the circular movement should be less than that which inevitably results from gravity.
• Mixture BHP8C exhibits a strongly nonlinear behavior (b=1.88). However, the maximum rotation velocity during the rheometer test was only 0.34 rps (corresponding to a maximum accelaration of 0.5 m/s²). In addition, given the extreme plastic viscosity of this material, it is hard to see how the gravel could have segregated during the test.
• Finally, a comparison of the behavior of mortars (groups B06-BO10-BO11 and BHP6-BHP10-BHP11) with that of concretes shows that the average value of the Herschel-Bulkley coefficient b for the mortars is 1.45. This is of similar magnitude to that of the concretes, whereas the mortars are generally much less sensitive to segregation because of the limited size of the coarse aggregate and the increased importance of the surface forces with respect to the volume forces in the relative movements of the particles.

Therefore, we conclude that the rheological behavior of the mixtures that were studied is nonlinear, with some exceptions.

3.2.2. Suitability of the Bingham model for fresh concrete

An unexpected result of the present study consists in the need to provide at least three parameters to describe the rheological behavior of the fresh concrete: a significant complicating factor due to the fact that only one parameter, the slump, is currently used by engineers. The concrete community is reluctant to adopt the Bingham approach, itself more complex than the usual approach, according to which a single subjective property called "workability" is used to describe the flow behavior of fresh concrete. To what extent is the additional sophistication inherent in the Herschel-Bulkley model necessary? If one wishes to characterize the behavior of the concrete in a shear strain rate range from 0 to 6 s^-1, which encompasses the range of the rheometer, one may consider that the Bingham model is still sufficient, even if the method of calculating the yield stress and plastic viscosity parameters has to be changed. In fact, it is sufficient for this purpose to use a linear approximation of the Herschel-Bulkley curve in the shear strain rate domain in question. As to the plastic viscosity, the following equation can be established by minimizing the deviation between the two models (using the least-squares method):

(5)

where µ' is the slope of this straight line (Figure 10) and _max is the maximum shear strain rate achieved in the test. The yield stress,  ₀, is equal to the yield stress, ₀', as calculated by the Herschel-Bulkley equation. The Bingham model with  ₀ and µ as values of the yield stress and of the plastic viscosity makes it possible then to maintain a relatively simple approach to the behavior, without having the disadvantage of a negative yield stress being calculated by Tattersall's original approach. The values of µ' were calculated for the concretes that were tested in the present project (Appendix IV to VI). The parameters µ and µ' are of the same order of magnitude, the latter being a little smaller when the value of the exponent b is greater than 1 (which is true in most of the cases).

This approach, using '₀ and µ' instead of  ₀ and µ, might not be suitable to represent the material rheological behavior in the case of shear strain rates larger than the range of measurement performed with the rheometer because the linear regression leads to larger errors in the range of shear rates higher than the region tested (Figure 10). Or otherwise state, because the modified Bingham model would not be able to represent the non-linear behavior at the large strain rates.For instance, in the case of pumping concrete, it may be necessary to use the original equation with all three parameters of the Herschel-Bulkley model.

Analysis of the collected rheological data, was used in the establishement of models linking rheological behavior with mixture composition. The results are given in the next chapter.