# Analysing the Influence of Fibers on Fresh Concrete Rheometry by the Use of Numerical Simulation

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## Abstract

**:**

## 1. Introduction

#### 1.1. Experimental Investigations of Fiber Suspensions

#### 1.2. Numerical Investigations of Fiber Suspensions

#### 1.3. Current Research Question

## 2. Materials and Methods

#### 2.1. Governing Equations and Constitutive Modeling of Fiber Reinforced Concrete

#### 2.2. Considerations for Estimating the Relative Cell Size

#### 2.3. Interim Conclusion

## 3. Results

#### 3.1. Numerical Setup

#### 3.2. Inertia of the Matrix Fluid

#### 3.3. Investigations at Constant Rotational Speed

## 4. Conclusions and Outlook

- The experimental investigations indicated that the orientation process of the fibers during the measurements has no influence on the measured torque, because no hysteresis of the torque-rotational speed curves can be detected. The simulation results in this section did show a slight increase in torque during the start-up phase. However, this is smaller than the width of the scatter in the experiment and thus explains the observation that no orientation influence on the measured torque is recognizable.
- The torque at the ball probes is primarily determined by the orientation state at the surface of the sphere and only in areas with a high shear rate. Such areas are located on the outside and underside of the ball probe. At these locations, a clear preferred orientation and a steady-state of orientation is established very quickly. Far away from the ball probes, on the other hand, the orientation process is significantly slower and even lasts longer than the measurement duration selected in the experiment. The orientation tendency could be identified depending on the prevailing flow form. The same effects could also be observed in the flow in the ball probe rheometer (cf. argumentation around Figure 14), whereby the complex succession of flow states causes a less clear and, above all, spatially inhomogeneous orientation distribution in the ball probe rheometer.
- The orientation model requires an approximation of the so-called cell size, i.e., an effective interaction area around a fiber. Based on the cell size approximation for the two extreme orientation states (isotropic vs. uniaxial), the flow was simulated and a comparison of the torques with the experimental results was carried out. It was found that when the cell size approximation for uniaxial orientation states is selected, the experimentally determined torque can be reproduced well, whereas when the approximation for isotropic orientation is used, a significant overestimation of the torque is calculated. This can be explained after analyzing the flow: The cell size approximation is considered in the fiber constitutive law, but not in the orientation-evolution equation for Newtonian fluids. The term influences the solution in regions with large velocity gradients. In the ball probe rheometer flow, this applies to the vicinity of the sphere, where a uniaxial orientation state dominates particularly quickly. Far away from the ball probes, there is generally no uniaxial orientation. However, due to the low shear rates in these regions, the model error in the fiber constitutive law hardly affects the flow solution due to the uniaxial cell size approximation.
- The simulation model provides a linear increase in the effective viscosity with increasing fiber volume fraction compared to the matrix liquid, whereby the gradient corresponds to the experimental results. Furthermore, the shear rate in the flow field of the suspension was compared with a homogeneous Newtonian substitute fluid, which produces the same torque at the ball probes. No significant differences were found, so that a good approximation can be assumed not only in the integral measured variables, but also locally using a homogeneous substitute fluid.
- In order to investigate the influence of orientation on the torque in more detail and to enable an objective comparison with the previous Couette investigations, additional simulations were carried out at a constant rotational speed. The influence of the orientation process is shown by an increase in torque during the start-up process. A similar characteristic increase at constant rotational speed is also described in the literature for other concrete rheometer geometries such as the vane rheometer. The extent of the torque overshoot increases with increasing fiber volume fraction.

- The model assumes inflexible fibers. Fibers can only be considered inflexible if their deformation does not result in a significant change in the flow around them with the matrix fluid. This usually applies to steel fibers, but generally not to textile fibers, and in particular for polymer fibers this must be examined more closely in individual cases before the model is used.
- The model assumes a homogeneous fiber distribution.
- The model (without extension) does not take into account fiber-fiber interaction, which occurs at higher fiber concentrations. In this case in particular, increased fiber agglomeration can also occur, which results in segregation of the fiber suspension.
- No fiber-wall interaction is taken into account.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Examined ball probe rheometer eBT2 from Schleibinger Testing Systems Teubert u. Greim GmbH. with two ball probes.

**Figure 2.**Size of the dimensionless cell radius $h/R$ depending on the fiber volume fraction $\varphi $ and fiber orientation state in the different concentration regimes. In the case shown, ${a}_{r}=30$.

**Figure 3.**Pre-factor of the fiber stress term as a function of fiber volume fraction and orientation state for ${a}_{r}=30$.

**Figure 4.**Torque-rotational speed correlation on the ball probes MK105 (

**a**) and MK145 (

**b**) of the eBT2. The experimental data (symbols), the simulation results (lines) and the Equation (1) (homogeneous replacement fluid, dashed lines) adapted to the simulation results based on the dynamic viscosity are shown.

**Figure 5.**Development of the preferred orientation during the process (${h/R\parallel}_{\mathrm{uni}}$). The largest eigenvalue ${\lambda}_{3}$ of the orientation tensor ${\mathbf{A}}_{2}$ is shown both in the color coding and with the isocontour for ${\lambda}_{3}=0.8$. ${\mathsf{\Omega}}_{a}=-{\mathsf{\Omega}}_{\mathrm{sphere}}$.

**Figure 6.**Orientation ellipsoids and streamlines at the same rotational speed ${\mathsf{\Omega}}_{a}=0.08\mathrm{rad}/\mathrm{s}$ in the (

**a**) acceleration and (

**b**) deceleration curve (${h/R\parallel}_{\mathrm{uni}}$). The streamlines are colored with the local representative shear rate $\dot{\gamma}$.

**Figure 7.**Development of the torque calculated in the simulation of the fiber suspension ($\varphi =2.0\mathrm{vol}.-\%$) in relation to the fiber-free Newtonian matrix fluid ($\mu =111.6\mathrm{Pa}\mathrm{s}$) on both ball probes of the eBT2 (comparison of constant rotational speed (${\mathsf{\Omega}}_{a}=0.17\mathrm{rad}/\mathrm{s}$) with torque ramp as in the experiment).

**Figure 8.**Development of the torque calculated in the simulation of the fiber suspension ($\varphi =0.674\mathrm{vol}.-\%$) in relation to the fiber-free Newtonian matrix fluid ($\mu =111.6\mathrm{Pa}\mathrm{s}$ on both ball probes of the eBT2 (comparison of constant rotational speed (${\mathsf{\Omega}}_{a}=0.17\mathrm{rad}/\mathrm{s}$) with torque ramp as in the experiment).

**Figure 9.**Measurement of the shear stress of a cement mortar with and without the addition of fibers in a paddle rheometer (vane rheometer). The source contains no information about the fiber geometry and the fiber volume fraction [7].

**Figure 10.**Relative effective viscosity of the Newtonian fiber suspension over fiber volume fraction. Comparison between numerical simulation and experiment.

**Figure 11.**Field of the equivalent shear rate in comparison between the fiber suspension (

**a**) and a Newtonian substitute fluid (

**b**), which produce an identical torque on the spheres.

**Figure 13.**Evolution of the largest eigenvalue ${\lambda}_{3}$ of the orientation tensor ${\mathbf{A}}_{2}$ at selected observation points around the ball probes of eBT2. Constant rotational speed ${\mathsf{\Omega}}_{a}=0.17\mathrm{rad}/\mathrm{s}$, $\varphi =2\mathrm{vol}.-\%$, ${\mu}_{\mathrm{matrix}}=111.6\mathrm{Pa}\mathrm{s}$.

**Figure 14.**Development of the preferred orientation at constant rotational speed ${\mathsf{\Omega}}_{a}=0.17\mathrm{rad}/\mathrm{s}$, ${h/R\parallel}_{\mathrm{uni}}$ and $\varphi =2\mathrm{vol}.-\%$ in the Newtonian matrix fluid in the section through the ball probe equator. Streamlines and flow state index Z.

**Table 1.**Coordinates of the observation points in the spherical equatorial plane. The origin of the coordinates lies on the axis of rotation.

Ball Probe | No. | Description | Position [m] |
---|---|---|---|

MK105 | 1 | Oncoming flow | $(-0.105;\phantom{\rule{0.166667em}{0ex}}-0.047)$ |

2 | Wake flow | $(-0.105;\phantom{\rule{0.166667em}{0ex}}0.047)$ | |

3 | Outside | $(-0.15;\phantom{\rule{0.166667em}{0ex}}0)$ | |

4 | Inside | $(-0.06;\phantom{\rule{0.166667em}{0ex}}0)$ | |

5 | Far field wake | $(0;\phantom{\rule{0.166667em}{0ex}}0.145)$ | |

MK145 | 6 | Oncoming flow | $(0.145;\phantom{\rule{0.166667em}{0ex}}0.047)$ |

7 | Wake flow | $(0.145;\phantom{\rule{0.166667em}{0ex}}-0.047)$ | |

8 | Outside | $(0.19;\phantom{\rule{0.166667em}{0ex}}0)$ | |

9 | Inside | $(-0.1;\phantom{\rule{0.166667em}{0ex}}0)$ | |

10 | Far field wake | $(0;\phantom{\rule{0.166667em}{0ex}}-0.185)$ |

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**MDPI and ACS Style**

Gerland, F.; Vaupel, T.; Schomberg, T.; Wünsch, O.
Analysing the Influence of Fibers on Fresh Concrete Rheometry by the Use of Numerical Simulation. *Constr. Mater.* **2024**, *4*, 128-153.
https://doi.org/10.3390/constrmater4010008

**AMA Style**

Gerland F, Vaupel T, Schomberg T, Wünsch O.
Analysing the Influence of Fibers on Fresh Concrete Rheometry by the Use of Numerical Simulation. *Construction Materials*. 2024; 4(1):128-153.
https://doi.org/10.3390/constrmater4010008

**Chicago/Turabian Style**

Gerland, Florian, Tim Vaupel, Thomas Schomberg, and Olaf Wünsch.
2024. "Analysing the Influence of Fibers on Fresh Concrete Rheometry by the Use of Numerical Simulation" *Construction Materials* 4, no. 1: 128-153.
https://doi.org/10.3390/constrmater4010008