#### 4.1. Rheological Parameters

The rheological parameters of yield stress, viscosity, and thixotropy were calculated as described in

Section 3. The calculated parameters are presented in

Table 2. All parameters are the average from three individual measurements. The results describe an expected decrease of the calculated Bingham yield stress

${\tau}_{0,B}$ with an increase of the mini slump flow value, which was calculated with the mini slump flow yield stress

${\tau}_{0,sF}$ as well, and a decrease in calculated plastic viscosity

$\mu $. Also, with increasing slump flow the flow length generally increases, which leads to decreased calculated L-box yield stress

${\tau}_{0,L-Box}$. Moreover, with increasing PCE amount, which decreases attractive particle interactions, the thixotropy

${A}_{thix}$ of the tested samples decreased. For a better comprehension and description of measured values and their rheological correlations,

Figure 5 and

Figure 6 are provided. Roussel et al. assume the correlation between dynamic yield stress and the mini slump flow value [

10]. The correlation between the two parameters calculated with Equation (3) is shown in

Figure 5 for all testing series. The general assumption that yield stress decreases with increasing slump flow is valid for all series and serves as base for the subsequent comparison of yield stress values in correlation with workability (L-box flow).

#### 4.2. Flow Analysis

Step 1: Applying yield stress values

In

Figure 6, the L-box flow length is compared to the dynamic yield stress from the three aforementioned yield stress calculation approaches: the rheometric investigation

${\tau}_{0,B}$, calculated yield stress according to the mini slump flow test

${\tau}_{0,SF}$, and calculated yield stress according to the L-box test

${\tau}_{0,LBox}.$ Generally, for low flow lengths the calculated yield stress from L-box flow is always the lowest, followed by the yield stress calculated from slump flow value. The measured rheometric yield stress is the highest. With increasing flow length, the gap between calculated and measured yield stresses decreases. For three of the four testing series (0.45_p, 0.45_m and 0.52_p), the yield stresses at the end of flow are very similar. For the last testing series, 0.52_m, yield stress calculated from L-box flow is generally the highest. Subsequently, there is no all-in-all correlation for only yield stress and L-box flow. In particular, the slope and the gap between calculated and measured yield stresses vary depending on the mixture composition. Thus other rheological parameters like viscosity and structural buildup have to be taken into account.

For the analysis and interpretation, three aspects should be considered:

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The rheometric dynamic yield stress measurement took place after the static yield stress measurement. The high shear of 30 seconds before the dynamic shear profile served for a homogenization and deagglomeration after the time of rest. Still, the agglomerate network within the colloidal paste changed over time and with rest, with a major impact on rheometric values (cf. [

38])

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The L-box flow was assumed as free flow. Still, within a short time of rest a low hydrodynamic pressure from the top to the bottom throughout the vertical column built up, which pushed the flow as soon as the gate was lifted.

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The rheometric measurements were conducted with a vane-in-cup system using the Reiner–Riwlin equation for the calculation of shear stress from rotational torque. With increasing non-Bingham-like material behavior, the regression fits less.

The mentioned aspects, whose content leads to poorer comparability of the values, present difficulties in rheometric investigations. For a better understanding of the correlation of different rheological parameters depending on mixture composition and form-filling behavior of cementitious materials, not only yield stress but also viscosity was taken into account in a second step.

Step 2: Applying velocity values

Cement paste and concrete possess shear-dependent viscosities. Plastic viscosity during steady shear might suffice for relative comparisons or the estimation of flow velocity during steady flow. Indeed, during form-filling the flow velocity decreases until rest. An estimation of shear rate and thus the correlation of a shear rate-dependent viscosity to each part of flow is difficult and needs numerical simulation. Thus, for a simple demonstration, the velocity depending on flow length was calculated for each paste (

Figure 7a) and mortar (

Figure 7b). The data points are calculated data points for each second of flow during a flow time from two seconds after flow start until the end of flow. The dashed lines show the potential decreasing trend functions.

After the gate is lifted, flow occurs with maximum velocity. The maximum flow velocity is furthermore increased with increasing the slump flow values of the mixtures. Therefore the velocity calculation shown in

Figure 7a for all pastes and

Figure 7b for all mortars invariably presents flow velocity ongoing from two seconds after gate-lifting. Analyzing the velocity for each mixture, some conclusions can be drawn:

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Within the first two seconds, the flow length is different for each slump flow value. The higher the slump flow value, the higher the flow velocity and thus the initial flow length after two seconds

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The velocity decreases tremendously but with a reduced rate during the experiment

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The change of velocity decrease is more pronounced for highly flowable mixtures and for mortars than for pastes

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Very slow flow takes place for the last few centimeters of flow

Step 3: Applying thixotropy at rest

With a change of velocity during flow, rheological parameters change. For shear-thinning mixtures, viscosity increases with decreasing velocity and thus shear rate. Therefore, not only a constant viscosity parameter should be taken into account for the calculation of concrete flow but either a flow-dependent viscosity or just the velocity. Moreover, with decreasing velocity and thus shear rate, structural buildup increases. Therefore, thixotropic structural buildup should be taken into account for flow analysis during form-filling, which was previously shown by the author in [

55]. In

Figure 8, a general correlation of thixotropy values (while containing the same initial slump flow values and thus dynamic yield stress) and flow length shows the decrease of flow length with increase in thixotropy. In the diagram, all mixtures are shown (0.45_p, 0.45_m, 0.52_p, and 0.52_m). For all samples the slump flow was adjusted through different amounts of superplasticizer; therefore it is not distinguished between different samples in the diagram. The depiction between thixotropy and flow length shall be shown solely exemplarily depending on the slump flow value. It is once more shown that taking only initial dynamic yield stress into account is not sufficient to predict form-filling ability.

The testing series 0.45_p and 0.45_m contain a w/c ratio of 0.4, wherefore thixotropic structural buildup due to colloidal forces is thus relatively low for these testing series. The flow length decreases without the impact of thixotropy, which is nearly the same value for all mixtures in the testing series 0.45_p and 0.45_m. The decrease in flow length is pronounced and nearly linear, compare

Figure 9. In comparison, the testing series 0.52_p and 0.52_m contain a w/c ratio of 0.3 and thus pronounced thixotropic structural buildup due to lower particle distances and therefore higher interparticle forces. The decrease in flow length correlates to increasing structural buildup; still, the decrease in flow length is less pronounced and shows a nearly potential decrease of flow length (especially

Figure 10 a). Obviously, the effect of thixotropy on flow changes.

Step 4: Combination of rheological parameters

As shown in steps 1–3, the flow length cannot be depicted by the investigation of only one rheological parameter. Indeed, the flow length is not only dependent on yield stress, viscosity, and structural buildup but on the interaction between these parameters as well. Increasing viscosity implies slower and thus longer flow, which might lead to an increase in thixotropic structural buildup as soon as the velocity gets too low to provide shear rates for full structural breakdown. At the same time, slow flow with subsequent residual structural buildup leads to an increase in static yield stress

${\tau}_{0,s}$ and therefore even faster stoppage of the flow. The interdependence between these parameters is not trivial and hardly predictable, wherefore a general correlation to common aforementioned equations does not seem appropriate. For a comprehensive depiction of the correlation between the flow time, flow length and thixotropy,

Figure 10 is shown. For all mixtures, both the total time of flow until the end of flow and thixotropy are shown in correlation with L-box flow length.

A general idea of form-filling behavior is not possible. Indeed, different cases of concrete flow can be determined. If no thixotropy occurs, flow length is dependent on yield stress. In

Figure 10a the thixotropy of the sample 0.45_p is quite low (0.08–0.11 Pa/s); moreover, the values change only slightly for different flow lengths. It thus can be assumed that the total time of flow and the flow length correlate directly depending on the suspension’s yield stress, as given in Equation (4). The flow predictability is possible without the knowledge of more parameters, and the inertia effects are negligible. The time of flow or the flow velocity do not affect the final flow length. The flow velocity indeed affects the viscosity, which is not constant but dependent on the shear rate: with changing shear rate the viscosity can be shear-thinning or shear-thickening, depending on the relative solid volume fraction. Still, shear-dependent viscosity does not affect the flow length directly as long as no relevant thixotropy occurs. As soon as thixotropy has to be taken into account, the prediction becomes more complicated: the thixotropy of the mortar in

Figure 10b is higher and has a wider range (0.25–0.56 Pa/s). The time of flow is not only dependent on yield stress but on thixotropic structural buildup as well. Flow stoppage occurs as soon as thixotropic structural buildup leads a static yield stress which surpasses the suspension’s required yield stress to stop flow. These effects are even more pronounced for the samples in

Figure 10c,d. Due to the suspension’s actual solid volume fraction of

${\mathsf{\Phi}}_{act}$ = 0.52, the viscosity generally is already higher than for pastes with

${\mathsf{\Phi}}_{act}$ = 0.45. Due to lower particle distances, higher thixotropy occurs. Slow flow due to higher viscosity and higher residual thixotropy values thus lead to faster flow stoppage, even if the yield stress values (adjusted through the mini slump flow values) theoretically are the same as in the aforementioned series. The calculation of flow length thus is dependent on thixotropy and therefore also on the shear-dependent viscosity. The viscosity determines (1) the speed of flow and thus the shear-dependent effective thixotropic structural build, and (2) the total duration of flow which itself determines the residual time for structural buildup. Therefore, a general flow prediction without knowledge of the interdependencies between the rheological parameters and workability is not easily possible.

Simulations of concrete flow, e.g., using methods like CFD, can help to investigate the actual shear rate during flow. With the knowledge of the residual thixotropy value per shear rate, an implementation of thixotropy or shear-dependent residual structural buildup will help to predict the actual flow length not only depending on yield stress, but also depending on time of flow and thixotropy.