# Damage Evolution of Steel Fibre-Reinforced High-Performance Concrete in Low-Cycle Flexural Fatigue: Numerical Modeling and Experimental Validation

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Experimental Methods

#### 2.1. Concrete Mixtures and Fibres

#### 2.2. Experimental Setup and Loading

#### 2.3. Damage Indicators

#### 2.4. Numerical Model

## 3. Experimental and Numerical Results

#### 3.1. Load-CMOD Curves

#### 3.2. Mechanical Damage Indicators

#### 3.3. Acoustic Emission Measurements

#### 3.4. Simulations of Three-Point Bending Tests at Low Cycle for Reinforced HPC

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Three-point bending beam test: (

**a**) geometry for the experimental setup and boundary value problem and (

**b**) detailed photo of setup.

**Figure 2.**Sketches for the damage indicators in quasi-static tests: (

**a**) CMOD plastic (CMOD${}_{\mathrm{pl},i}$), (

**b**) initial (${s}_{0}$) and residual (${s}_{i}$) stiffness and (

**c**) damage energy (${W}_{\mathrm{pl},i}$).

**Figure 5.**Development of residual stiffness: (

**a**) zoomed view of (

**b**) plot for different fibre contents.

**Figure 6.**Plastic deformation after unloading: (

**a**) zoomed view of (

**b**) plot for different fibre contents.

**Figure 7.**Absorbed damaging energy ${W}_{\mathrm{pl},i}$: (

**a**) zoomed view of (

**b**) plot for different fibre contents.

**Figure 8.**Phases of deterioration, progression of (

**a**) CMOD, (

**b**) load and (

**c**) registered hits (total and weighted by respective energy) over time.

**Figure 9.**Three-point bending beam test at low cycle for fiber-reinforced HPC: distribution of stresses ${\sigma}_{x}$ (in GPa) in horizontal direction (x-axis), the equivalent plastic strain ${\alpha}^{\mathrm{HPC}}$ for HPC phase, the phase-field parameter q for HPC phase in (

**a**,

**c**,

**e**) at CMOD = 0.018 mm and (

**b**,

**d**,

**f**) at CMOD = 0.108 mm, respectively.

**Figure 10.**Three-point bending beam test for fiber-reinforced HPC: (

**a**) load-CMOD diagramm for experimental data and simulations using (

**b**) five preferred orientations between the angles of −10° to 10° and (

**c**) 24 preferred orientations between the angles of −90° to 90°. Comparison of experimental and simulated (

**d**) load-CMOD diagram and (

**e**) calculated and interpolated residual stiffness-CMOD diagram.

**Table 1.**Composition of the used concrete mixtures, data from [34].

Ingredients | Quantity in kg/m${}^{3}$, HPC |
---|---|

cement: CEM I 52,5R | 500 |

fine sand | 75 |

sand 0/2 | 850 |

basalt 2/5 | 350 |

basalt 5/8 | 570 |

silica fume | 570 |

superplasticizer | 5 |

stabilizer | 3 |

water | 176 |

steel fibres | 0/23/57/115 |

number of specimens | 3/3/3/3 |

Property | Mean Value | Unit |
---|---|---|

tensile strength ${f}_{t}$ | 5.7 | MPa |

compressive strength ${f}_{c}$ | 112 | MPa |

Young’s modulus E | 39.976 | GPa |

Poisson’s ratio $\mu $ | 0.192 | – |

Parameter | Value |
---|---|

duration discrimination time | 0.4 ms |

rearm time | 0.4 ms |

threshold | 30.1 dB |

filter | 95 kHz–850 kHz |

gain | 34 dB |

**Table 4.**Measured mechanical properties, see Table 2, and calibrated material parameters for HPC.

${\mathit{E}}^{\mathbf{HPC}}$ | ${\mathit{\nu}}^{\mathbf{HPC}}$ | ${\mathit{f}}_{\mathbf{t}}$ | ${\mathit{f}}_{\mathbf{c}}$ | ${\mathit{\psi}}_{\mathit{t}}^{\mathbf{c},\mathbf{HPC}}$ | ${\mathit{\psi}}_{\mathbf{c}}^{\mathbf{c},\mathbf{HPC}}$ | ${\mathit{y}}_{0}^{\mathbf{HPC}}$ | ${\mathit{\beta}}_{\mathbf{p}}$ | ${\mathit{\beta}}_{\mathbf{n}}$ | ${\mathit{h}}^{\mathbf{HPC}}$ | l | $\mathit{\zeta}$ | ${\mathit{E}}^{\mathbf{F}}$ | ${\mathbf{v}}^{\mathbf{F}}$ | ${\mathit{y}}_{0}^{\mathbf{F}}$ | ${\mathit{h}}^{\mathbf{F}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

GPa | − | MPa | MPa | MPa | MPa | − | − | − | MPa | mm | − | GPa | − | MPa | MPa | |

HPC | 39.976 | 0.192 | 5.7 | 112 | 4.2 $\times {10}^{-4}$ | 0.13 | 6.2 | 0.5 | 0.12 | 13,000 | 14 | 1 | 210 | 0.003 | 660 | 130 |

**Table 5.**Calibrated values for the interpolation of degradation functions for HPC in tension and compression.

q | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 |
---|---|---|---|---|---|---|---|---|---|---|---|

${g}^{+}\left(q\right)$ | 1 | 1 | 1 | 1 | 0.93 | 0.85 | 0.74 | 0.6 | 0.39 | 0.18 | 0.003 |

${g}^{-}\left(q\right)$ | 1 | 0.852 | 0.714 | 0.585 | 0.464 | 0.353 | 0.252 | 0.164 | 0.089 | 0.03 | 0.003 |

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**MDPI and ACS Style**

Gebuhr, G.; Pise, M.; Anders, S.; Brands, D.; Schröder, J.
Damage Evolution of Steel Fibre-Reinforced High-Performance Concrete in Low-Cycle Flexural Fatigue: Numerical Modeling and Experimental Validation. *Materials* **2022**, *15*, 1179.
https://doi.org/10.3390/ma15031179

**AMA Style**

Gebuhr G, Pise M, Anders S, Brands D, Schröder J.
Damage Evolution of Steel Fibre-Reinforced High-Performance Concrete in Low-Cycle Flexural Fatigue: Numerical Modeling and Experimental Validation. *Materials*. 2022; 15(3):1179.
https://doi.org/10.3390/ma15031179

**Chicago/Turabian Style**

Gebuhr, Gregor, Mangesh Pise, Steffen Anders, Dominik Brands, and Jörg Schröder.
2022. "Damage Evolution of Steel Fibre-Reinforced High-Performance Concrete in Low-Cycle Flexural Fatigue: Numerical Modeling and Experimental Validation" *Materials* 15, no. 3: 1179.
https://doi.org/10.3390/ma15031179