Fitting the Experimental Curves with the Herschel-Bulkley Model

It seems that a better model for representing the experimental points is the following

                                                                       (2)

where is the measured torque (bulk value minus the contribution of the empty rheometer [10] ), N the speed of rotation (in rev/s), ₀ , A and b numerical parameters determined by the least square difference method. This type of model has been referred to as the Herschel-Bulkley (HB) model. HB behavior is the same as the Bingham behavior when the exponent is equal to 1 [11]:

= ' ₀ + a ^b                                                                                   (3)

Following this model, the description of the rheological behavior requires three parameters: ' ₀ (the HB yield stress), a and b.

Integrating the contribution of each surface element to the torque, the following equations are obtained:

where R₁ and R₂ are the inner and outer radii, and h the height of the sheared concrete specimen. In BTRHEOM, R₁ = 20 mm, R₂ = 120 mm and h = 100 mm.

These equations can be inverted to deduce the material parameters from the fitting of the bulk curve. A 10 % correction is applied to the a parameter to account for the skirt friction effect, as is normally done in determining the plastic viscosity [6]. Therefore, we have for the HB parameters:

The fitting of the experimental points appears in Figure 2, and the parameters found are given in Table 2. The experimental relative errors are about 0.7 % and 1 % for the torque and for the rotation speed, respectively [5]. Here, the coefficients of variation provided by the regressions have been found close to 1% Thus, it appears that the HB model provides an excellent approximation to the measurements. The HB yield stress is always positive, and, within each series, it decreases when the water content increases. Also, the exponent b differs significantly from 1, which shows the limits of the Bingham model.

In favor of the Bingham model, it could be argued that the Bingham yield stress values provide a better ranking of the mixtures than the HB yield stress, with respect to the slump values (see Table 2 ). However, it is thought that the ability of the Self-Compacting Concretes to produce high spread in the slump test is due not only to the low yield stress, but also to the high amount of cement paste (here, about 40 %), which lubricates the aggregates up to the end of the test. The paste volume of the high-performance concretes is less (32 %), then a part of the paste is squeezed from the skeleton during the slumping of the sample, until friction between aggregates stops the flow.