# Fracture Models and Effect of Fibers on Fracture Properties of Cementitious Composites—A Review

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fracture Models

#### 2.1. ACK Model

_{f}represents the fiber diameter.

#### 2.2. FC Model

#### 2.3. CB Model

#### 2.4. PSH Model

#### 2.5. DKF Model

${K}_{I}<{K}_{Ic}^{ini}$ | Elastic Stage; No Crack |

${K}_{I}={K}_{Ic}^{ini}$ | Crack initiation |

${K}_{Ic}^{ini}<{K}_{I}<{K}_{Ic}^{un}$ | Crack stable propagation |

${K}_{I}={K}_{Ic}^{un}$ | Crack begins to destabilize and expand |

${K}_{I}>{K}_{Ic}^{un}$ | Crack instability propagation |

## 3. Effect of PVA Fiber on Fracture Properties

#### 3.1. Effect of Fiber Content on Fracture Properties

#### 3.2. Effect of Fiber Length on Fracture Properties

^{3}), and the ultimate flexural strength was 13.9 MPa (Figure 12).

#### 3.3. Effect of Fiber Surface Oiling on Bond Performance

## 4. Effect of SFs on Fracture Properties

## 5. Effects of Hybrid Fibers on Fracture Performance

_{5}> 5, I

_{10}> 10, and I

_{30}> 30 are also strain hardening composites. The load–displacement curve for the pullout of a single fiber can be used to analyze the synergistic effects of mixed fibers on the matrix, as well as the synergistic effects of other fibers. Generally, adding 1.5% SFs and 1.0% PVA fiber to the matrix can greatly improve the fracture performance of the matrix. The total dosage of PVA fiber and SF should ≤ 3%. This satisfies the modification of SFs and PVA fibers to cementitious composites.

## 6. Conclusions

- (1)
- The fracture model of the cementitious composite, the ACK model, and the PSH model are used mainly for ductile cementitious composites. ACK model can determine the optimal fiber content of matrix, PVA fiber generally no more than ${J}_{b}^{\prime}/{J}_{tip}>1$ is the basic condition for the quasi-strain hardening of cementitious composites, while a large value of ${J}_{b}^{\prime}/{J}_{tip}$ can realize the cracking of multi-saturated cracks in the matrix. The FC, BC, and DKF models are mainly applied to semi-brittle material–concrete composites. Both FC and BC models are based on tensile softening of concrete, and the value of ${G}_{f}$ can be expressed as the ability of concrete to resist crack propagation. The FC model can directly simulate the concrete with nonlinear characteristics by using the finite element method. The DKF model has the advantages of the FC and BC models, and is widely used in practical engineering. The DKF model can judge the whole process of concrete failure according to ${K}_{Ic}^{ini}$, ${K}_{Ic}^{un}$, and ${K}_{I}$ parameters.
- (2)
- The fracture energy and toughness of the cementitious composite can be increased by adding an appropriate amount of PVA fibers (1.2–2.0%) with an appropriate (16 mm) fiber length. The main objectives of PVA fiber surface oiling (approximately 1.2%) are to weaken the adhesion between the PVA fibers and the matrix, reduce the friction during fiber pullout, and prevent the fracture of PVA fibers.
- (3)
- The stiffness of SFs is higher than that of PVA fibers; thus, SFs have a better strengthening effect on the cementitious composite. SF fibers of appropriate length have a good bridging effect on macroscopic cracks in the matrix, and SFs can also absorb fracture energy. The mixing amount of SF in is cementitious composite 4%, which can greatly improve the fracture property of the matrix. In the pullout process, the SF is easily pulled out, whereas the PVA fiber is easily broken. SF and PVA fibers can be mixed in the matrix simultaneously. The pullout load displacement of the fibers is the synergistic effect of the two. Generally, adding 1.5% SFs and 1.0% PVA fiber to the matrix can greatly improve the fracture performance of the matrix. Thus, adding appropriate amounts of PVA fibers and SFs can improve the mechanical and fracture properties of the matrix.

## 7. Outlook

- (1)
- For comparison with the ACK model, the PSH, FC, BC, and DKF models are reviewed and analyzed. Although all the models have advantages, there are deficiencies for practical engineering that must be resolved.
- (2)
- However, when PVA fiber and SFs fiber are mixed, the optimal effect of SF and PVA fiber on the fracture property, mechanical property, and durability of the matrix is not discussed in depth. In the study of SFs and PVA fiber mixing, too much range of water–binder ratio was not set, and too much research on its high-temperature performance was not carried out. The pullout of a single fiber’s load–displacement curves of polypropylene fibers and SF single fibers were also not analyzed. To improve the mechanical properties of cementitious composites with low fiber contents, further research should be performed in these areas.
- (3)
- The fracture properties of fiber-reinforced cementitious composites are studied and analyzed only with regard to the basic properties and mechanisms. Therefore, it is necessary to further study the practical application of cementing composites and the freezing–thawing environment in which the cementing materials are located, as well as the wet, hot, and salt environment in saline and alkaline areas and coastal areas.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 10.**Fracture analysis with four phase (non-linear) [50]. (

**a**) Initial crack, (

**b**) stable crack propagation, (

**c**) instability, and (

**d**) instability.

**Figure 11.**Effect of the PVA fiber content on the P-COMD curve [65].

**Figure 12.**Effect of the PVA fiber length on the flexural strength vs. mid-span deflection curve of the matrix [72].

**Figure 13.**Microscopic images of cracked beams with different fiber lengths and volume fractions, including bridging fibers [73]. (

**a**) PVA length of 8 mm and fiber content of 1%, (

**b**) PVA length of 12 mm and fiber content of 1%, (

**c**) PVA length of 8 mm and fiber content of 2%, and (

**d**) PVA length of 12 mm and fiber content of 2%.

**Figure 14.**Effect of the fiber oil content on the bond performance of the matrix [43]. (

**a**) Frictional bond, (

**b**) chemical bond.

**Figure 15.**Failure mode of matrix [76]. (

**a**) Mortar, (

**b**) oil-coated PVA, and (

**c**) PVA coated without oil.

**Figure 16.**Effects of the SFs content and type on the load–deflection curve [84]. (

**a**) Micro-straight SFs (%), (

**b**) hooked-end SFs (%).

**Figure 17.**Effects of the SFs content on the load–deflection curve [86].

**Figure 18.**Typical load–deflection curves [87].

**Figure 20.**Effects of SFs and PVA fibers on the flexural strength and mid-span deflection of cementitious composites [99].

**Figure 21.**Load–displacement curve for the pullout of a single fiber [100].

Model | Crack Criterion | Mechanical Treatment of Micro-Fissure Zone | Numerical Method Used |
---|---|---|---|

ACK model [22] | ${V}_{f}\ge {V}_{c}$ ${\sigma}_{f}^{\prime}>{\sigma}_{m}^{u}$ | Stress–strain curve | Integral method |

FC model [27] | $\sigma \ge {f}_{t}$ $w={w}_{0}$ | Bilinear (or nonlinear) strain softening curve | Finite-element method |

CB model [34] | $\sigma \ge {f}_{t}$ $w={w}_{0}$ | Micro-cracks are uniformly distributed and parallel and the damage degree is expressed by the reduction of the elastic modulus | Finite-element method |

PSH model [39] | ${J}_{b}^{\prime}>{J}_{tip}$ | Energy under quasi-stress strain | Integral method |

DFK model [52] | ${K}_{I}={K}_{Ic}^{ini}$ | Load–CMOD curve | Weight functions and other numerical calculation methods |

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**MDPI and ACS Style**

Zhang, P.; Yang, Y.; Wang, J.; Jiao, M.; Ling, Y.
Fracture Models and Effect of Fibers on Fracture Properties of Cementitious Composites—A Review. *Materials* **2020**, *13*, 5495.
https://doi.org/10.3390/ma13235495

**AMA Style**

Zhang P, Yang Y, Wang J, Jiao M, Ling Y.
Fracture Models and Effect of Fibers on Fracture Properties of Cementitious Composites—A Review. *Materials*. 2020; 13(23):5495.
https://doi.org/10.3390/ma13235495

**Chicago/Turabian Style**

Zhang, Peng, Yonghui Yang, Juan Wang, Meiju Jiao, and Yifeng Ling.
2020. "Fracture Models and Effect of Fibers on Fracture Properties of Cementitious Composites—A Review" *Materials* 13, no. 23: 5495.
https://doi.org/10.3390/ma13235495