Abstract
The accessibility to rheological parameters for concrete is becoming more and more relevant. This is mainly related to the constantly emerging challenges, such as not only the development of highstrength concretes is progressing very fast but also the simulation of the flow behaviour is of high importance. The main problem, however, is that the rheological characterisation of fresh concrete is not possible via commercial rheometers. The socalled concrete rheometers provide valuable relative values for comparing different concretes, but they cannot measure absolute values. Therefore, we developed an adaptive coaxial concrete rheometer (ACCR) that allows the measurement of fresh concrete with particles up to
1 Introduction
In the last decades, the development of new concretes gained more and more importance, which is mainly related to emerging challenges such as highperformance concretes, 3D printable concretes, or predictive flow simulations. To achieve this and superior concrete performance, the profound understanding of the rheology of fresh concrete is an essential requirement and therefore of high significance [1,2,3,4].
However, determining the exact rheological properties of fresh concrete is still very difficult today. This is mainly due to the fact that the measuring gap of commercial rheometers is too small for big aggregates. Therefore, the socalled “concrete rheometers” were developed, which are mostly similar to a coaxial system [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], but there are also some that work on the principle of a parallelplate system [21,22,23,24] or characterise the concrete via spreading tests [25]. In many cases, these rheometers are research projects and thus not commercially available. However, there are some commercial “concrete rheometers” offered by Schleibinger with the Viskomat XL, Viskomat NT, and eBTV models. For determining the pumpability of concrete, Schleibinger also offers a very practiceoriented measuring device called SLIPER, which, however, cannot be compared with a rheometer purely in terms of principle.
All these devices are usually referred to as “concrete rheometers,” which, with a few exceptions, is due to the fact that the characteristic measured values are rotational speed and torque (analogous to common rheometers). In most cases, the Reiner–Riwlin equation [26], which is based on the Bingham model, is used to determine the flow properties, regardless of the actual flow behaviour. Therefore, even if the designation “rheometer” implies that absolute rheological parameters can be determined, this is only possible with difficulty or not at all possible with almost all “concrete rheometers” developed in the past [18]. This is due to the measuring geometries, which do not provide a mandatory analytical describable shear field (e.g. vanelike geometries, paddles, or spheres). This inevitably also means that different concrete rheometers are not comparable.
To counteract this problem at least for selected concrete rheometers, roundrobin tests were carried out in 2,000 at LCPC in France and in 2003 at Master Builders Solutions in the USA [14,27]. The aim was to develop correlations between the rheometers by measuring identical concrete mixes, so that in future it would be possible to compare different concrete mixes measured in different concrete rheometers. The rheometers used were the BTRheom [21,22], the IBB [11], the Twopoint rheometer [6], the ConTec BML [8,9], and the CEMAGREFIMG [12]. Although different rheological properties, such as yield stress and plastic viscosity, were analysed, it was not possible to find a general solution for correlating these devices. Therefore, different suggestions for future investigations were raised, e.g. the establishment of standardised measurement methods for concrete and the use of suitable reference materials [14,27]. One of the goals pursued with these suggestions was the development of better rheometers, with which absolute values of concrete can be determined and used in simulations [27]. Since such rheometers have not been developed in the recent years, we developed an innovative concrete rheometer, the adaptive coaxial concrete rheometer (ACCR), to tackle the aforementioned challenges, which were met with great success.
2 Theory
2.1 Rheometry of concentric coaxial cylinder systems
The measuring gap of concentric coaxial cylinder systems is formed by two cylinders with the same symmetry axis, the inner cylinder (bob, radius
For Newtonian fluids, the shear rate can directly be expressed by equation (2), where
2.2 Correction of unknown flow behaviour
Since equations (1)–(3) are only valid for Newtonian fluids, different correction methods have been developed for unknown flow behaviour.
A rather simple method is the Schümmer correction, which uses the representative shear rate
However, this correction method is mainly accurate for fluids with certain flow behaviour, so that a more general and accurate correction method developed by Krieger and Elrod should be applied, if possible [28,31].
The Krieger–Elrod correction is based on the general formulation of the shear rate, which was developed in a power series, equation (6). The parameter
The accuracy of this correction depends on the factor
2.3 Flow behaviour of fresh concrete
The most established model for describing the flow behaviour of cementbased suspensions is the Bingham model [14,27,33,34,35]. Bingham fluids are characterised by a yield stress with subsequent Newtonian flow behaviour and can be described mathematically by equation (7) [34].
The slope of the straight line in the
However, nonlinear flow behaviour of especially selfcompacting concretes has been observed in numerous investigations [36,37,38,39,40] and the application of the Bingham model led to physically impossible negative yield stress values [36,37,41]. Therefore, the socalled Herschel–Bulkley model was increasingly used to describe the rheological properties, equation (8).
This model replaces the original linear Newtonian relationship between
Even though the Herschel–Bulkley model initially appears to be suitable for characterizing nonlinear behaviour, some further investigations have concluded that the yield stress values determined are strongly dependent on the flow exponent. The comparison of this model with other models has consistently shown a yield stress that is too low for shearthinning materials [42] and too high for shearthickening materials [43,44]. To overcome this problem, the Bingham model was extended by the quadratic term
When applying the modified Bingham model, it has been observed that the yield stress determined via this model always lies between the yield stress determined via Herschel–Bulkley and Bingham and therefore, seems to be a more accurate estimation [42]. In addition, according to Feys et al., the yield stress is determined independent of the degree of shear thickening, which can further support the accuracy of this model [40].
3 Development and construction of the ACCR
In principle, rheometry offers a wide range of rheometer types, but the choice of a suitable measuring system for determining absolute flow properties as accurately as possible is limited. The most common rheometers are parallelplate systems, coaxial systems, and capillary rheometers. However, for a highly concentrated, coarsegrained suspension such as concrete, which, depending on its composition, tends to sediment, parallelplate rheometers and capillary rheometers are not suitable. For a parallelplate system, even a slight sedimentation would cause the upper plate to apparently slip and the sheer dimension of the measuring gap, due to the particle size, would most probably lead to a flowout of the concrete during or even before the measurement itself. In a capillary rheometer, there would be no possibility to prevent wall slip or to perform time dependent or yield stress measurements. Thus, only a coaxial rheometer is suitable. Figure 1 shows the schematic structure of the ACCR.
3.1 Basic dimensions
When dimensioning a coaxial system for suspensions, the maximum particle size is the first thing to be determined. The maximum measurable particle size
If this relationship is neglected, measurement errors due to friction and the influence of the particles on the flow field cannot be excluded [29,45].
Second, according to ISO 3219, the radius ratio
To maintain this ratio is mandatory to achieve an uniform shear rate and stress distribution among the gap [29,46]. For manufacturing and handling reasons, a maximum measurable particle size of
With regard to the rheometer dimensions, the ISO 3219 further specifies that the following relationship must apply to the ratio of the inner cylinder length
However, since there is no rheological justification to exact this ratio in ISO 3219 and a corresponding length of
3.2 Rotating bottom plate
In all coaxial systems, front face influences occur on the upper and lower front faces of the inner cylinder, which, without correction or other measures, lead to erroneous results. The upper front face influence can be eliminated by filling the measuring gap to the upper edge of the inner cylinder only. Since the upper face and other components of the rheometer above it are only in contact with air, the influence is in general negligibly small.
The situation is different on the lower front face of the inner cylinder. An additional parallelplate system with a theoretical measuring gap
In the case of the ACCR, a design with this specification is not expedient because of handling issues and a significantly higher, impractical sample volume. Instead, the effect was inhibited with a rotating bottom plate below the inner cylinder moving at the same speed, Figure 2. Thus, the material between the rotating bottom plate and the inner cylinder is not sheared at all and therefore the torque is not increased. In order to ensure that the rotational speed of the inner cylinder and the bottom plate match as closely as possible, both components are driven by identical motors and gears.
Furthermore, the whole bottom part of the rheometer is, unlike commercial rheometers, detachable to empty and clean the rheometer after measurement, Figure 2.
3.3 Adaptive measuring profiles with integrated cooling
When measuring suspensions, apparent wall slip is a common problem, so that wall adhesion can no longer be guaranteed and the use of roughened or profiled measuring surfaces is usually practiced [47,48,49,50,51]. To allow a slip inhibition for the ACCR, an adaptive approach with several, exchangeable measuring profiles was implemented. The measuring surface of the coaxial system therefore consists of six individual measuring profiles at the inner and outer cylinders, which, when lined up together, form a circular ring. The standard measuring profiles for materials prone to wall slip are made of stainless steel with a length of
In addition, the standard profiles (stainless steel) for the outer cylinder cool the measured fluid in the gap. A copper coil is integrated in each profile to be fed with a temperaturecontrolled cooling fluid, Figure 3. The cooling fluid is fed in parallel through the profiles, ensuring a homogeneous temperature. Alternatively, a cooling coil wrapped around the outer cylinder can be used for cooling, which is sufficient for nonexothermally reactive materials and was used in the context of comparative measurements together with the smooth aluminium profiles, see Section 4, and partly during the first series of tests, see Section 5.1. It is important to mention that a cooling device is always required because of the selfheating caused by the shearing. In commercial rheometers, cooling can be realized with little effort, since the sample volume is very small. However, in the ACCR with a much larger sample volume, heat is generated not only by the shear process, but also by the reaction of the fresh concrete itself. Therefore, the cooling of the concrete is generally a very difficult challenge, but one that has been mastered very well with the cooling profiles when the temperature values achieved are considered, see Section 5.
3.4 Overall design and key parameters
Figure 4 shows the setup of the whole rheometer. The inner cylinder and the bottom plate are each driven by an AC asynchronous motor from Lenze (type MXXMA), which has a nominal rotational speed of
3.5 Error analysis
To assess the validity of the results presented in this study, it is necessary to estimate the measurement accuracy of the ACCR. Therefore, all possible sources of error were identified and evaluated. The relevant errors for the rheological parameters shear rate and shear stress were characterised in more detail in a Gaussian error propagation, which lead to relative inaccuracies of
4 Comparative measurements
For evaluation purposes, the ACCR was tested first with simple, controllable fluids and compared to a commercial Anton Paar MCR 501 using a CC27 standard system (
For the sugar solution and the cream bath, a positive shear rate ramp (PSR) starting with
Fluid  MS  Temperature (°C)  Correction  Fit 

Deviation (%) 

Sugar solution 


Schümmer 



Cream bath 


Krieger–Elrod 





Suspension 


Krieger–Elrod 



Mayonnaise 


Schümmer  — 


The fitnumbers represent the used polynomial degree, except for the mayonnaise, where the HerschelBulkleymodel was used.
4.1 Newtonian fluid–sugar solution
The aqueous sugar solution contained a sugar mass fraction of
4.2 Shear thinning fluid–cream bath
For the cream bath measurements, an additional preshearing step for
4.3 Shear thinning suspension–cream bath with polyamide particles
To evaluate the ACCR regarding suspensions, it had to be first decided whether the exact same suspension or a similar suspension with the same particle diameter to gap ratio should be used, while both the variants promise different advantages. Since preliminary measurements showed that there is nearly no tendency for wall slip (plateplate and coaxial systems), it was decided to use the exact same suspension (particle volume fraction
4.4 Yield stress fluid–mayonnaise
For mayonnaise, a decrease in the measured values was observed with the increase in the individual measurements for both the rheometers, so that the individual measurements (X.1–X.3) are presented instead of a single mean curve. The reason for this behaviour might be rising air bubbles during the measurements that had been entrapped in the filling process, structural changes in the mayonnaise due to drying processes, or even an influence of the different loading processes. The effect is stronger for the ACCR, which corresponds with the mentioned assumptions. Wall slip can be excluded as a reason since preliminary investigations showed no tendency. Due to the high yield stress, a correction via Krieger–Elrod is not possible, so that the Schümmercorrection was applied. However, the Herschel–Bulkley model could be fitted to the values for calculating the deviations and analysing the yield stresses, Figure 5(d). The results show that the values of the ACCR are below the values of the MCR. There are deviations in the range between
4.5 Conclusion
The first three comparisons clearly show that the ACCR delivers consistent absolute rheological values within a close error margin to commercial measuring systems. On a closer look, a constant offset of about
5 Rheological characterisation of fresh concrete
One of the most important reasons why a generally valid rheological classification of fresh concrete is much more difficult than for other materials is the fact that the term “concrete” does not refer to a clearly defined substance, but rather to an entire class of materials. Concrete consists of four components (cement, water, aggregates, and various additives) in different ratios. Even the mixing time and the time to the measurement are relevant, since it is a reactive material. Therefore, it is necessary to define a mixture as well as an accurate mixing and measuring procedure to allow for comparable measurements. The exact composition and handling are described in the supplementary material S2. All measurement results are approximated by representative values according to Schümmer. From this point, the following wordings are used to distinguish between different sets of measurements:
Individual measurements
Measurement series (consisting of multiple individual measurements)
Investigation series (consisting of multiple measurement series)
Series of tests (consisting of multiple investigation series)
5.1 First series of tests
During the first series of tests, the self compacting concrete (SCC) was investigated in two investigation series, which are presented and discussed below. The first investigation series consisted of measuring the SCC using originally smooth aluminium profiles, which were fitted with brass flat bars (
5.1.1 Standard measuring procedure – aluminium profiles
In the first investigation series, the SCC was measured in three measurement series, with one measurement series consisting of five individual measurements without changing the specimen. The results of all
It can be seen that the shear stress increases significantly with the increase in the individual measurement number. This can be attributed to the reaction of the concrete and the structural buildup over time. This behaviour is particularly important in the context of the reproducibility of future measurements, as it highlights the relevance of adhering to a strict schedule when measuring concrete.
Furthermore, the flow behaviour of the concrete can be characterised in more detail. For low shear rates of approx.
Furthermore, the shear thinning behaviour at low shear rates must be considered in more detail. During some measurements, a completely unsheared area was formed, probably due to a drop below the yield stress, resulting in a reduction in the measuring gap starting from the outer cylinder, Figure 7. In such a state, the shear rate is effectively higher than assumed and the influence of coarsegrained particles in the reduced gap is bigger, so that the measured shear stresses lose validity. However, this phenomenon occurred primarily in the later individual measurements, so that at least the first individual measurements are valid for classical rheological investigation. A correction of this block building phenomenon is almost impossible due to the large number of influencing parameters. Not only the yield stress of the fresh concrete, which depends on the time and the load history, is relevant, but also the parameters already mentioned, such as the particletogap ratio, the particle shape, the orientation of the particles during shear, and the particle migration. It is also unclear to what extent the local concrete composition changes with this reduction in the measuring gap, since, at least apparently, an increased water content could be observed in the reduced gap. Furthermore, the occurrence of the gap reduction can only be detected by observing the measuring gap and not from the measurement values themselves.
A possible solution to this problem would be the increase in the geometry parameter
In the following, the applicability of the common flow models (Bingham (B), HerschelBulkley (HB), modified Bingham model (MB)) for concrete, see Section 2.3, will be used for further characterisation. Since the values in the high shear rate range (
1st individual measurement  2nd individual measurement  3rd individual measurement  

Bingham 











Herschel–Bulkley 















Mod. Bingham 















Taking a closer look at Table 2, it can be seen that the yield stress increases significantly for all models with the increase in the individual measurement and is about twice as high for the third individual measurement. A reason for this might be the structural buildup. However, the yield stresses are of the same order of magnitude for all models with very low deviations.
The parameters describing the viscosity (
To this extent, it is not possible to clearly identify which of the models is best suited to describe the flow behaviour of the SCC based on the model parameters listed in Table 2 and Figure 8. If, due to the gap reduction, only the first individual measurement is considered, the Herschel–Bulkley model seems to be the best in terms of both error sum of squares and visual matching with the measured values. However, since the differences are minute and this evaluation is based on only two measurements, further investigation is necessary, see Section 5.2.
The reaction of the aluminium was assumed to be negligible in this investigation because the contact area of the aluminium to the concrete had been very small compared to the total sample material. Nevertheless, during this investigation series, it was found that the gas bubbles had risen. For this reason, all subsequent investigations have been carried out with stainless steel profiles.
5.1.2 Hysteresis measurements
In the context of this work, the flow behaviour of SCC is generally characterised by conducting an NSR starting at high shear rates and then decreasing the shear rate step by step. Nevertheless, investigation via hysteresis measurements can provide a deeper insight into the rheology of SCC. Therefore, a PSR starting at low shear rates and then increasing the shear rate was added to the standard measurement procedure, which is performed directly after the negative ramp. Figure 9 presents the results of the hysteresis measurements where two measurement series were performed with two individual measurements each. Due to the longer investigation period, only two measurement cycles were possible before stronger hardening became relevant. The mean temperature for the first measurement series was
Therefore, the dilatant behaviour at the beginning of the NSR might occur due to insufficient preshearing since this behaviour could not be observed in the PSR. These results clearly emphasize how important a sufficient preshearing and the use of NSR is.
5.2 Second series of tests
The first series of tests have provided valuable results, especially with regard to the rheological properties of fresh concrete and the test procedure. However, many sources of error were also identified, which must be considered for a more reliable rheological characterisation of fresh concrete.
As the temperature was a major problem in both the measurement series of the previous tests with a deviation of
The results presented below are divided into three sections: reproducibility, rheological characterisation, and additional measurements. The additional measurements focus on sedimentation. Since the concrete already showed a high degree of consolidation in the individual measurements 4 and 5 in the previous series of tests, only the first two individual measurements were investigated from here on.
5.2.1 Reproducibility
All
In principle, the evaluation of the coefficient of variation depends on the considered area, but in general, coefficients below
5.2.2 Rheological characterisation of fresh concrete
Since the previous obtained values showed a very good reproducibility, they were averaged for the individual measurements 1 and 2 to obtain only one curve for each individual measurement, Figure 14(a). The mean temperatures are
While the course of the first individual measurements is almost linear for shear rates
Since the course of the first individual measurements is now also almost linear for
Comparing Figure 14(a) and (b), it can be clearly seen that the onset of the apparent shear thickening shifts to higher shear rates, which again is an indication for insufficient preshearing, especially for the second individual measurements. The hysteresis measurements, see Section 5.1.2, also showed no increase in this shear rate range for the PSR, while showed an increase for the NSR. Therefore, the described effect can clearly be attributed to insufficient preshearing and not to dilatant flow behaviour. This also becomes obvious when looking at the course of the torque, see supplementary material S3, since the torque for the first measurement value has not yet reached a pure steady state value in the second individual measurement, but has already done so in the first individual measurement. Therefore, the measured value at the highest shear rate of the second individual measurements is not included in the further evaluation. Again, the effect of gap reduction occurred at low shear rates, as discussed in detail in Section 5.1.1. The start of the gap reduction can only be determined to a limited extent via the measured values, which makes it a critical problem for the measurement. It is only possible to identify a kind of “step course” in both the individual measurements from a shear rate of approx.
Looking at the first individual measurements, the values show a linear and thus Newtonian behaviour. Although the first point (
First individual measurement  Second individual measurement  

Bingham 








Herschel–Bulkley 











Mod. Bingham 











The situation is different for the second individual measurements. As already described in the consideration of the entire shear rate range, a slight tendency to shear thinning up to
Usually, the yield stress is related to exactly such phenomena, which means that an increased structure formation must also result in an increase in the yield stress. Due to the NSR, it could be argued that an increasing destruction of the structures takes place over the measurement period and that both the individual measurements have the same initial state and thus the same yield stress at the end of the measurement (
In future investigations, the characterisation of fresh concrete should be extended to shear rates
5.2.3 Investigation of sedimentation tendency
In order to assess the sedimentation and segregation tendency, additional measurements were carried out during the adjustment of the concrete mix and during the second test series. These consisted of a socalled cylinder sedimentation test [55] and representative sample taking after individual measurements.
The execution of the cylinder sedimentation test is shown schematically in Figure 15. For this purpose, three stacked cylinders (
For the concrete investigated in this work with largest aggregates
Cylinder  Mass of aggregates

Deviation to mean value



Mesh size

Upper 


Middle 



Lower 






Mesh size

Upper 


Middle 



Lower 





Therefore, a lance was constructed to take samples directly from the rheometer gap at different heights. The lance has four sealable cups (A1 (top) to A4 (bottom)) with a volume of
After the individual samples were washed over a sieve with a mesh size of
6 Conclusion
The ACCR presented in this study was developed for the purpose of determining the rheological properties of suspensions, primarily fresh concrete, with a maximum particle size of
After the difficulties of concrete measurements encountered in the preliminary tests had been solved, the fresh concrete could be rheologically characterised with extremely high reproducibility. The applicability of the common flow models for concrete (Bingham, Herschel–Bulkley, and modified Bingham model) could be satisfactorily analysed. It was shown that especially the Bingham and modified Bingham model are suitable for the description of the concrete used in this work, while the Herschel–Bulkley model seems to underestimate the yield stress and represents shearthinning behaviour. Furthermore, only the values of the first individual measurements are valid for further evaluation of rheological properties, since the influence of structural buildup in the second individual measurements is too high to be neglected and can lead to a false interpretation of the rheological behaviour.
However, a limited shear rate range has resulted due to the yield stress and the structural buildup of the concrete, which in particular have led to a reduced measurement gap. This mainly affected the low shear rate range. Within the scope of this work, this phenomenon could not be avoided and can also only be reliably fixed in the future by redesigning the rheometer with a

Funding information: We would like to thank the German Research Foundation (DFG) for supporting the research within the priority program SPP 2005 “Opus Fluidum Futurum.”

Author contributions: Conceptualisation: S.J. and S.J.; data curation: S.J.; formal analysis: S.J. and S.J.; funding acquisition: H.J.S. and S.J.; investigation: S.J.; methodology: S.J. and S.J.; project administration: S.J. and H.J.S.; supervision: H.J.S.; validation: S.J. and S.J.; visualisation: S.J.; writing – original draft: S.J.; writing – review and editing: S.J. and H.J.S.

Conflict of interest: The authors declare no conflict of interest.

Ethical approval: The conducted research is not related to either human or animal use.

Data availability statement: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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