# Numerical Simulation of the Fracture Behavior of High-Performance Fiber-Reinforced Concrete by Using a Cohesive Crack-Based Inverse Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material Production and Experimental Campaign

^{3}were also manufactured. All specimens were cured in a climatic chamber until 28 days of age. The cylindrical specimens were used to obtain the compressive strength, the modulus of elasticity, and the tensile strength following EN 12390-3 [34], EN 12390-13 [35], and EN 12390-6 [36], respectively. All tests were conducted at 28 days of age of the material. In Table 3, the results of the mentioned tests can be seen.

_{R}

_{1,}should be at least 40% of the load at the limit of proportionality (f

_{LOP}). Moreover, it should be also checked that the load at a CMOD value of 2.5 mm should be at least 20% of f

_{LOP}, usually termed f

_{R}

_{3}. In Table 4, it can be observed that all concrete formulations exceeded the requirements set by [22].

## 3. Numerical Simulations

**t**transmitted across the crack faces was parallel to the crack displacement vector

**w**(central forces model). If the magnitude of the crack opening vector |

**w**| does not decrease, the relation between both parameters can be stated as appears in (1)

**w**|) is the softening function for pure opening, which, in this manuscript, will be defined by multilinear functions, where unloading and reloading branches are aligned with the origin and the softening function is defined by a set of points.

**t**can be obtained as

**t**is constant along the crack, with h being the triangle height, A representing the area of the element, L representing the crack length,

**σ**representing the stress tensor, and

**n**representing the unit vector normal to that side and to the crack. A more detailed explanation of the implementation can be found in [41].

**E**stands for the elastic tangent tensor, ε

^{α}is the apparent strain vector obtained with the nodal displacements, and

**b**is the gradient vector corresponding to the shape function of the solitary node, as appears in Equation (4):

^{+}**t = σ n**and also using Equations (1) and (3), the following expression could be obtained.

**1**stands for the identity tensor. If an iterative algorithm is used,

**w**could be found to satisfy Equation (5).

_{ct}), the element behaves elastically. The tensile strength influences the limit of proportionality of the experimental curves, being greater as f

_{ct}increases. Nevertheless, if such value is exceeded, the direction of the maximum principal stress is found and a crack perpendicular to such direction is introduced. From that point onwards, the behaviour of the element is governed, if strains keep growing, by the softening function of the material, which relates the stress and the crack opening relation (σ-w). Apart from the tensile strength, the subroutine requires the modulus of elasticity of the material E. Although it is true that the presence of fibres in concrete changes the post-peak behaviour of the material when subjected to compressive stresses, given that for high crack widths the material behaviour is mainly under tensile stresses, no compressive damage was considered in the material implementation. Moreover, no experimental data appeared in [33] regarding the post-peak compressive strength of any of the mixes analysed.

## 4. Results and Discussion

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Fracture test results of SFRC-OL, SFRC-3D, and SFRC-5D. (

**b**) Fracture test results of H1 and H2.

**Figure 4.**(

**a**) Comparison of the average experimental results obtained in the fracture tests of SFRC-OL and the numerical simulation. (

**b**) Softening function used in the simulation shown in (

**a**).

**Figure 5.**(

**a**) Comparison of the average experimental results obtained in the fracture tests of SFRC-3D and the numerical simulation. (

**b**) Softening function used in the simulation shown in (

**a**).

**Figure 6.**(

**a**) Comparison of the average experimental results obtained in the fracture tests of SFRC-5D and the numerical simulation. (

**b**) Softening function used in the simulation shown in (

**a**).

**Figure 7.**(

**a**) Comparison of the average experimental results obtained in the fracture tests of H1 and the numerical simulation. (

**b**) Softening function used in the simulation shown in (

**a**).

**Figure 8.**(

**a**) Comparison of the average experimental results obtained in the fracture tests of H2 and the numerical simulation. (

**b**) Softening function used in the simulation shown in (

**a**).

**Figure 9.**(

**a**) Softening functions obtained in the inverse analysis of SFRC-OL, SFRC-3D, and SFRC-5D. (

**b**) Softening functions obtained in the inverse analysis of H1 and H2.

Component | Weight (kg/m^{3}) |
---|---|

Water | 199 |

Cement | 425 |

Limestone filler | 210 |

Sand | 947 |

Gravel | 292 |

Finer gravel | 194 |

Superplasticizer | 5.91 |

OL | 3D | 5D | PF | |
---|---|---|---|---|

Material | Steel | Steel | Steel | Polyolefin |

Shape | Straight | Hooked | Double-hooked | Straight |

Length (mm) | 13 | 30 | 60 | 60 |

Eq. diameter (mm) | 0.2 | 0.38 | 0.9 | 0.9 |

Tensile strength (MPa) | >2600 | >1160 | >2300 | >500 |

Modulus of elasticity (GPa) | 210 | 210 | 210 | 9 |

Fibres per kg. | 282,556 | 3183 | 3132 | 27,000 |

**Table 3.**Mechanical properties and their corresponding coefficient of variation (c.v.) of the concrete mixes [33].

Formulation | f_{cm} | E | f_{ct} | |||
---|---|---|---|---|---|---|

(MPa) | c.v. | (GPa) | c.v. | (MPa) | c.v. | |

SFRC-OL | 66.9 | 0.05 | 32.8 | 0.01 | 7.15 | 0.03 |

SFRC-3D | 63.1 | 0.04 | 32.7 | 0.02 | 7.96 | 0.04 |

SFRC-5D | 63.8 | 0.02 | 33.9 | 0.02 | 7.69 | 0.09 |

H1 | 64.7 | 0.02 | 40.7 | 0.03 | 8.05 | 0.01 |

H2 | 66.3 | 0.03 | 29.5 | 0.01 | 7.95 | 0.01 |

**Table 4.**Residual strength of the concrete formulation in relation with the requirements set by [22].

Strength (MPa) | f_{LOP} | f_{R}_{1} (0.5 mm) | % f_{LOP} | f_{R}_{3} (2.5 mm) | % f_{LOP} |
---|---|---|---|---|---|

SFRC-OL | 7.3 | 7.39 | 102% | 5.03 | 69% |

SFRC-5D | 5.5 | 8.11 | 148% | 8.40 | 153% |

SFRC-3D | 8.2 | 13.18 | 161% | 12.17 | 148% |

H1 | 7.0 | 9.94 | 142% | 10.33 | 147% |

H2 | 6.6 | 7.51 | 113% | 7.97 | 120% |

Concrete Mix | Experimental W_{f} (kN/mm) | Simulation Wf (kN/mm) | Experimental G_{f} (N/m) | Simulation G_{f} (N/m) | ΔG_{f} (%) |
---|---|---|---|---|---|

SFRC-OL | 98.8 | 97.1 | 5270.4 | 5179.7 | −1.75% |

SFRC-3D | 236.8 | 241.5 | 12,627.8 | 12,878.1 | 1.94% |

SFRC-5D | 170.5 | 172.8 | 9095.8 | 9213.9 | 1.28% |

H1 | 232.4 | 235.2 | 12,396.2 | 12,546.1 | 1.20% |

H2 | 178.9 | 181.1 | 9542.9 | 9657.4 | 1.19% |

Concrete Mix | SFRC-OL | SFRC-3D | SFRC-5D | H1 | H2 | |||||
---|---|---|---|---|---|---|---|---|---|---|

w (mm) | σ (MPa) | w (mm) | σ (MPa) | w (mm) | σ (MPa) | w (mm) | σ (MPa) | w (mm) | σ (MPa) | |

Point 1 | 0.00 | 4.69 | 0.00 | 4.69 | 0.00 | 4.69 | 0.00 | 4.69 | 0.00 | 4.69 |

Point 2 | 0.16 | 2.50 | 1.50 | 4.50 | 0.02 | 2.00 | 0.02 | 2.00 | 0.05 | 2.00 |

Point 3 | 3.00 | 0.70 | 4.00 | 2.00 | 0.23 | 3.33 | 0.20 | 4.20 | 0.50 | 2.80 |

Point 4 | 3.90 | 0.60 | 6.00 | 0.00 | 1.30 | 2.50 | 2.40 | 2.70 | 2.50 | 2.20 |

Point 5 | 5.00 | 0.00 | - | - | 8.30 | 0.00 | 10.00 | 0.00 | 10.00 | 0.00 |

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**MDPI and ACS Style**

Enfedaque, A.; Alberti, M.G.; Gálvez, J.C.; Cabanas, P.
Numerical Simulation of the Fracture Behavior of High-Performance Fiber-Reinforced Concrete by Using a Cohesive Crack-Based Inverse Analysis. *Materials* **2022**, *15*, 71.
https://doi.org/10.3390/ma15010071

**AMA Style**

Enfedaque A, Alberti MG, Gálvez JC, Cabanas P.
Numerical Simulation of the Fracture Behavior of High-Performance Fiber-Reinforced Concrete by Using a Cohesive Crack-Based Inverse Analysis. *Materials*. 2022; 15(1):71.
https://doi.org/10.3390/ma15010071

**Chicago/Turabian Style**

Enfedaque, Alejandro, Marcos G. Alberti, Jaime C. Gálvez, and Pedro Cabanas.
2022. "Numerical Simulation of the Fracture Behavior of High-Performance Fiber-Reinforced Concrete by Using a Cohesive Crack-Based Inverse Analysis" *Materials* 15, no. 1: 71.
https://doi.org/10.3390/ma15010071