# Fracture and Size Effect of PFRC Specimens Simulated by Using a Trilinear Softening Diagram: A Predictive Approach

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## Abstract

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## 1. Introduction

## 2. Experimental Benchmark

## 3. Embedded Cohesive Crack Model

## 4. Definition of the Trilinear Softening Diagrams

## 5. Results and Discussion

## 6. Study on the Influence of ${w}_{r}$ and ${w}_{f}$

#### 6.1. Influence of ${w}_{r}$

#### 6.2. Influence of ${w}_{f}$

## 7. Conclusions

- The complete fracture behaviour of PFRC specimens can be numerically simulated using a predictive trilinear cohesive crack model, which can be defined a priori by means of empirical expressions obtained with lab tests different from those simulated. This diagram is defined by four points, with coordinates that depend on PFRC mechanical characteristics, i.e., the tensile strength of the matrix, the proportion of fibres, and the orientation factor. Abscissa values ${w}_{r}$ and ${w}_{f}$ (see Figure 3) are fixed based on experimental results obtained in previous literature. It is still an unsolved challenge to obtain expressions to estimate ${w}_{r}$ and ${w}_{f}$ using the mechanical characteristics of the PFRC.
- The softening diagrams are not equal for all specimen sizes and should be adjusted for each of them. This is mainly due to a different orientation factor that varies with the size of the specimen.
- The maximum remanent loads obtained for each size present a linear trend on the load–displacement diagram, which does not agree completely with the experimental observations, although the load–displacement and load–CMOD curves properly agree with the experimental envelopes for the three studied sizes.
- Modifying ${w}_{r}$ and ${w}_{f}$ affects the maximum remanent load on the load–displacement diagram and modifies the last part of this diagram but cannot capture the nonlinear trend of the remanent load among specimen sizes.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Load–displacement diagram obtained in a three-point bending test with a PFRC specimen; (

**b**) trilinear softening diagram.

**Figure 2.**

**Left**: scheme of a three-point bending test and specimen geometry;

**right**: scheme of crack propagation from the notch tip during the test.

**Figure 3.**Scheme of a trilinear softening function. Load–unload branches follow a line towards the origin.

**Figure 4.**Potential crack paths (

**left**) and geometrical definitions of $w$, $n$, and ${b}^{+}$ (

**right**).

**Figure 8.**Numerical results compared with the experimental envelopes; each specimen size is identified by a different colour. Experimental envelopes correspond to three specimens tested.

**Figure 9.**

**Left**: trilinear diagrams used to study the influence of ${w}_{r}$ on the numerical simulations;

**right**: detail of the diagrams around point r (see Figure 3). Continuous lines represent the trilinear functions with ${w}_{r}$ = 1.65, dashed lines the functions with ${w}_{r}$ = 1.85, and dotted lines the function with ${w}_{r}$ = 2.05.

**Figure 10.**Numerical results compared with the experimental envelopes to study the influence of ${w}_{r}$ on the numerical simulations. The trilinear functions used in each case are shown in Figure 9. Experimental envelopes correspond to three specimens tested.

**Figure 11.**Left: trilinear diagrams used to study the influence of ${w}_{f}$ on the numerical simulations; upper right: detail of the diagrams around point r; lower right: detail of the diagrams around point f (see Figure 3). Continuous lines represent the trilinear functions with ${w}_{f}$ = 6.0, dashed lines, the functions with ${w}_{f}$ = 6.5, and dotted lines, the function with ${w}_{f}$ = 7.0.

**Figure 12.**Numerical results compared with the experimental envelopes to study the influence of ${w}_{f}$ on the numerical simulations. The trilinear functions used in each case are shown in Figure 11. Experimental envelopes correspond to three specimens tested.

Material | SCC10 |
---|---|

Cement (kg/m${}^{3}$) | 375 |

Limestone (kg/m${}^{3}$) | 200 |

Water (kg/m${}^{3}$) | 188 |

w/c | 0.5 |

Gravel (kg/m${}^{3}$) | 245 |

Grit (kg/m${}^{3}$) | 367 |

Sand (kg/m${}^{3}$) | 918 |

Superplasticizer (% cement) | 1.25 |

PF48 (kg/m${}^{3}$) | 10 |

Material density (g/cm${}^{3}$) | 0.910 |

Eq. diameter (mm) | 0.903 |

Tensile strength (MPa) | >500 |

Modulus of elasticity (GPa) | >9 |

Specimen | Length (mm) | Width (mm) | Height (mm) | Notch (mm) |
---|---|---|---|---|

Large | 1350 | 50 | 300 | 150 |

Medium | 675 | 50 | 150 | 75 |

Small | 340 | 50 | 75 | 37.5 |

${\mathit{f}}_{\mathit{t}}$ (MPa) | ${\mathit{G}}_{\mathit{F}}$ (N/mm) | $\mathit{\varphi}$ | ${\mathit{w}}_{\mathit{k}}$ (mm) | ${\mathit{\sigma}}_{\mathit{k}}$ (MPa) | |
---|---|---|---|---|---|

Small/Medium/Large | 3.2 | 0.13 | 1.448 | 0.07143 | 0.57715 |

$\mathit{\theta}$ | $\%\mathit{P}\mathit{u}\mathit{l}\mathit{l}\mathit{e}\mathit{d}-\mathit{O}\mathit{u}\mathit{t}$ | ${\mathit{V}}_{\mathit{f}}$ | ${\mathit{\sigma}}_{\mathit{u}}$ (MPa) | ${\mathit{\sigma}}_{\mathit{r}}$ (MPa) | |
---|---|---|---|---|---|

Small | 0.63 | 0.54 | 0.011 | 376 | 1.20 |

Medium | 0.62 | 0.54 | 0.011 | 376 | 1.18 |

Large | 0.72 | 0.54 | 0.011 | 376 | 1.37 |

Small | Medium | Large | |
---|---|---|---|

${w}_{t}$ (mm) | 0.00 | 0.00 | 0.00 |

${\sigma}_{t}$ (MPa) | 3.20 | 3.20 | 3.20 |

${w}_{k}$ (mm) | 0.07 | 0.07 | 0.07 |

${\sigma}_{k}$ (MPa) | 0.57 | 0.57 | 0.57 |

${w}_{r}$ (mm) | 1.650 | 1.650 | 1.650 |

${\sigma}_{r}$ (MPa) | 1.20 | 1.18 | 1.37 |

${w}_{f}$ (mm) | 6.00 | 6.00 | 6.00 |

${\sigma}_{f}$ (MPa) | 0.00 | 0.00 | 0.00 |

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**MDPI and ACS Style**

Suárez, F.; Gálvez, J.C.; Alberti, M.G.; Enfedaque, A. Fracture and Size Effect of PFRC Specimens Simulated by Using a Trilinear Softening Diagram: A Predictive Approach. *Materials* **2021**, *14*, 3795.
https://doi.org/10.3390/ma14143795

**AMA Style**

Suárez F, Gálvez JC, Alberti MG, Enfedaque A. Fracture and Size Effect of PFRC Specimens Simulated by Using a Trilinear Softening Diagram: A Predictive Approach. *Materials*. 2021; 14(14):3795.
https://doi.org/10.3390/ma14143795

**Chicago/Turabian Style**

Suárez, Fernando, Jaime C. Gálvez, Marcos G. Alberti, and Alejandro Enfedaque. 2021. "Fracture and Size Effect of PFRC Specimens Simulated by Using a Trilinear Softening Diagram: A Predictive Approach" *Materials* 14, no. 14: 3795.
https://doi.org/10.3390/ma14143795