# Computational Fracture Evolution Analysis of Steel-Fiber-Reinforced Concrete Using Concrete Continuous Damage and Fiber Progressive Models

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

Proportions of concrete mixtures (kg/m^{3}) | |

Material | Mixture ID |

F1/F2 | |

cement CEM I 42.5R | 380 |

natural sand 0–2 mm | 220 |

gravel 2–8 mm | 1611 |

steel fibers | 25 |

water | 167 |

SP PE-220 % m.c. | 1 |

VMA VM-500 % m.c. | 0.2 |

#### 2.2. Mechanical Properties

#### 2.3. Experimental Beam Tests

#### 2.4. Numerical Analysis

#### Brief Description of the Methods Used in the Analysis

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

Damage initiation criteria | maximum strain |

Tensile strain limit | 0.1 |

Damage evolution law | material properties degradation |

Tensile and compressive stiffness reduction | 0.95 |

## 3. Results

#### 3.1. Experimental Results

#### 3.2. Numerical Results

## 4. Discussion and Conclusions

#### 4.1. Discussion

#### 4.2. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**The sample set on the measuring stand (

**left panel**); view of the measuring part of the sample with the extensometer attached (

**right panel**).

**Figure 8.**Collection of different experimental force–CMOD curves: raw data (different color different experiment, detailed description in text).

**Figure 12.**All numerical force–CMOD curves: raw data (different color different experiment, description in text).

**Figure 14.**All fracture energy curves: numerical results (different color different experiment, description in text).

**Figure 16.**Force–CMOD curves: comparison of statistical results between numerical and experimental analysis.

**Figure 17.**Fracture energy curves—comparison of statistical results between numerical and experimental analysis.

**Table 1.**Basic physical and chemical properties of the cement [55].

Cement Type | Setting Time Start/End | Compr. Strength | Specific Surface Area (Blaine) | Specific Gravity | ${\mathbf{SO}}_{3}$ | ${\mathbf{C}}^{-}$ | ${\mathbf{Na}}_{2}{\mathbf{O}}_{\mathbf{eq}}$ | |
---|---|---|---|---|---|---|---|---|

(min) | (min) | (MPa) | (cm^{2}/g) | (g/cm^{3}) | (%) | (%) | (%) | |

CEM I 42.5R | 176 | 231 | 57.9 | 3538 | 3.1 | 2.52 | 0.063 | 0.6 |

Test | ID of Mixture | ||
---|---|---|---|

F1 | F2 | ave. F1 and F2 | |

Flow (mm) | 360 | 350 | class F2 |

Compressive strength 28 days (MPa) | 62.46 [1.58] | 62.4 [1.25] | 62.43 [1.36] |

Compressive strength 134 days (MPa) | 51.98 [0.82] | 52.42 [1.50] | 52.2 [1.22] |

Flexural strength 134 days (MPa) | 5.58 [0.14] | 5.73 [0.08] | 5.66 [0.13] |

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

Concrete density | 2500 $\mathrm{kg}/{\mathrm{m}}^{3}$ |

Concrete Young’s modulus | 41.545 GPa |

Concrete Poisson’s ratio | 0.18 |

SF density | 7850 $\mathrm{kg}/{\mathrm{m}}^{3}$ |

SF Young’s modulus | 4 GPa |

SF Poisson’s ratio | 0.3 |

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

Uniaxial compressive strength ${R}_{c}$ | 62.4 MPa |

Uniaxial tensile strength ${R}_{t}$ | 6.25 GPa |

Biaxial compressive strength ${R}_{b}$ | 74.9 MPa |

Dilatancy angle $\psi $ | 30 deg |

Softening | exponential |

Plastic strain at uniaxial compressive strength ${\kappa}_{cm}$ | 0.002 |

Plastic strain at transition form power law to exponential softening ${\kappa}_{cu}$ | 0.0035 |

Relative stress at start of nonlinear hardening ${\Omega}_{ci}$ | 0.3 |

Residual relative stress at ${\Omega}_{cu}$ | 0.75 |

Residual compressive relative stress ${\Omega}_{cr}$ | 0.2 |

Mode 1 area specific fracture energy ${G}_{ft}$ | 100 N/m |

Residual tensile relative stress ${\Omega}_{tr}$ | 0.1 |

Plain Concrete (PC) | FRC | |||||
---|---|---|---|---|---|---|

Statistic | ${\mathbf{f}}_{\mathbf{R},\mathbf{B}}$ | CMOD | ${\mathbf{f}}_{\mathbf{R},\mathbf{0.5}}$ | ${\mathbf{f}}_{\mathbf{R},\mathbf{1.5}}$ | ${\mathbf{G}}_{\mathbf{f}}$ | ${\mathbf{E}}_{\mathbf{0}.\mathbf{5}-\mathbf{1.5}}$ |

(MPa) | (mm) | (MPa) | (MPa) | (N/mm) | (MPa) | |

mean | 6.247 | 0.027 | 2.903 | 2.988 | 4.869 | 36.794 |

std.dev. | 0.472 | 0.005 | 0.957 | 0.958 | 1.000 | 108.304 |

coeff. of var. | 7.55% | 17.34% | 32.95% | 32.08% | 20.54% | 294.35% |

skewness | 0.252 | 0.554 | −0.464 | −0.297 | 0.044 | 0.447 |

kurtosis | −0.699 | 0.085 | −0.847 | −1.468 | −1.463 | 0.368 |

median | 6.105 | 0.026 | 2.996 | 3.194 | 4.883 | 52.152 |

${\chi}^{2}$(0.95,4) | 6.000 | 3.600 | 5.600 | 6.000 | 2.000 | 6.000 |

Shapiro–Wilk | 0.966 | 0.963 | 0.963 | 0.899 | 0.941 | 0.968 |

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## Share and Cite

**MDPI and ACS Style**

Pokorska, I.; Poński, M.; Kubissa, W.; Libura, T.; Brodecki, A.; Kowalewski, Z.
Computational Fracture Evolution Analysis of Steel-Fiber-Reinforced Concrete Using Concrete Continuous Damage and Fiber Progressive Models. *Materials* **2023**, *16*, 5635.
https://doi.org/10.3390/ma16165635

**AMA Style**

Pokorska I, Poński M, Kubissa W, Libura T, Brodecki A, Kowalewski Z.
Computational Fracture Evolution Analysis of Steel-Fiber-Reinforced Concrete Using Concrete Continuous Damage and Fiber Progressive Models. *Materials*. 2023; 16(16):5635.
https://doi.org/10.3390/ma16165635

**Chicago/Turabian Style**

Pokorska, Iwona, Mariusz Poński, Wojciech Kubissa, Tomasz Libura, Adam Brodecki, and Zbigniew Kowalewski.
2023. "Computational Fracture Evolution Analysis of Steel-Fiber-Reinforced Concrete Using Concrete Continuous Damage and Fiber Progressive Models" *Materials* 16, no. 16: 5635.
https://doi.org/10.3390/ma16165635