# Study on Staged Damage Behaviors of Rock-like Materials with Different Brittleness Degrees Based on Multiple Parameters

^{*}

## Abstract

**:**

## 1. Introduction

_{d}) better predicts the brittleness index. Research on determination methods for the brittleness index based on the full stress–strain curve can also enrich and improve the brittleness evaluation system [12,13,14]. Therefore, investigating the damage process of specimens with different brittleness degrees is critical.

## 2. Design of the Uniaxial Compression Test

#### 2.1. Petrographic Studies

#### 2.2. Similarity Relationship between Sandstone and Rock-like

#### 2.3. Test Scheme

## 3. Results

#### 3.1. Physical and Mechanical Parameters

#### 3.2. Division of the Fracture Evolution Stages

#### 3.3. Calculation of the Brittleness Index

#### 3.4. Basic AE Parameters

#### 3.5. RA and AF Characteristics of AE

#### 3.6. Characteristics of the AE b Value

_{dB}is the amplitude of AE in dB and N is the number of AE hits or events with an amplitude greater than A

_{dB}[62].

#### 3.7. Change in the Natural Frequency during the Uniaxial Compression Tests

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Photomicrographs of the tested specimen. Legends: Q stands for quartz, CM for clay mineral, Cc for calcite, F for feldspar.

**Figure 2.**Stress–strain curve and failure mode of the argillaceous calcareous siltstone and the rock-like material.

**Figure 3.**Specimen preparation process: (

**A**) weighing; (

**B**) dry material mixing; (

**C**) water weighing; (

**D**) gypsum retarder preparation; (

**E**) wet material mixing; (

**F**) mold preparation; (

**G**) press molding with a jack; (

**H**) grinding and standing; and (

**I**) specimen maintenance.

**Figure 4.**Testing site: 1—the AE system; 2—the AE sensor; 3—the LDV measurement point; and 4—the LDV.

**Figure 5.**Crack volumetric strain model [21].

**Figure 6.**Crack volumetric strain curves, volumetric strain curves, and stress–strain curves of specimens with different ratios of aggregates to binders: (

**a**) 1:1; (

**b**) 4:1; and (

**c**) 7:1.

**Figure 7.**Relationship between the brittleness index (${B}_{\mathrm{i}}$) and the ratios of the aggregates to binders.

**Figure 8.**Variation of the AE parameters of the specimens with different brittleness indexes (4.16, 3.36, and 2.25) during different failure stages: (

**a**,

**c**,

**e**) the ring count rate and accumulative ringing count and (

**b**,

**d**,

**f**) the energy rate and accumulative energy.

**Figure 11.**Distribution of the AE RA-AF parameters for the specimen with a brittleness index of 4.16: (

**a**) scatter diagram; and (

**b**) density nephogram.

**Figure 12.**Distribution of the AE RA-AF parameters for the specimen with a brittleness index of 3.36: (

**a**) scatter diagram; and (

**b**) density nephogram.

**Figure 13.**Distribution of the AE RA-AF parameters for the specimen with a brittleness index of 2.25: (

**a**) scatter diagram; and (

**b**) density nephogram.

**Figure 14.**Relationship between the stress, the b value, and the strain for brittleness indexes of (

**a**) 4.16, (

**b**) 3.36, and (

**c**) 2.25.

**Figure 16.**Acquisition of the natural frequency: (

**a**) time-domain illustration and (

**b**) frequency-domain illustration after a fast Fourier transform of the time-domain diagram.

**Figure 17.**Axial stress, natural frequency, and strain curves for brittleness index of (

**a**) 4.16, (

**b**) 3.36, and (

**c**) 2.25.

**Figure 18.**Law of natural frequency during the stable crack growth stage: (

**a**) linear fitting of frequency and (

**b**) linear fitting of the rise slope of the frequency and brittleness index.

**Figure 19.**Relationship between the natural frequency decline slope at the stress drop and the brittleness index.

**Figure 20.**Variation analysis of the natural frequency: (

**a**) natural frequency curves of specimens with different brittleness indexes and (

**b**) relationship between the average natural frequency during different stages and the brittleness index.

**Figure 21.**Analysis of the average natural frequency increment and energy: (

**a**) area formed by the natural frequency curve and the strain at peak strength of specimens with different brittleness indexes; and (

**b**) relationship between $\overline{\Delta {f}_{a}}$, energy and the brittleness index.

**Figure 22.**Relationship between the release of energy during failure and $\overline{\Delta {f}_{a}}$.

Group | Density (g × cm ^{−3}) | Uniaxial Compressive Strength (MPa) | Elastic Modulus (MPa) | Tensile Strength (MPa) | Internal Friction Angle (°) | Cohesion (MPa) |
---|---|---|---|---|---|---|

Argillaceous calcareous siltstone | 2.51 | 40.75 | 4274.60 | 2.03 | 47.71 | 7.57 |

Rock-like | 1.90 | 15.09 | 2429.57 | 1.01 | 47.56 | 4.64 |

${\mathit{a}}_{\mathit{\sigma}}$ | ${\mathit{a}}_{\mathit{E}}$ | ${\mathit{a}}_{\mathit{c}}$ | ${\mathit{a}}_{\mathit{\phi}}$ |
---|---|---|---|

2.70 | 1.76 | 1.63 | 1.00 |

Group | Calsite (g) | Barite Powder (g) | Bentonite (g) | Cement (g) | Plaster (g) | Ratios of Aggregates to Binders |
---|---|---|---|---|---|---|

1 | 157.5 | 135 | 67.5 | 315 | 45 | 1:1 |

2 | 157.5 | 135 | 67.5 | 45 | 45 | 4:1 |

3 | 157.5 | 135 | 67.5 | 6 | 45 | 7:1 |

Ratios of Aggregates to Binders | Density (g × cm ^{−3}) | Uniaxial Compressive Strength (MPa) | Elastic Modulus (MPa) | Poisson’s Ratio | Internal Friction Angle (°) | Cohesion (MPa) |
---|---|---|---|---|---|---|

1:1 | 1.91 | 15.30 | 2163.05 | 0.17 | 47.56 | 4.64 |

4:1 | 1.91 | 8.99 | 1321.58 | 0.21 | 46.99 | 2.95 |

7:1 | 1.90 | 5.65 | 873.53 | 0.25 | 43.27 | 1.71 |

Ratios of Aggregates to Binders | ${\mathit{\sigma}}_{\mathit{c}\mathit{c}}$ (MPa) | ${\mathit{\sigma}}_{\mathit{c}\mathit{i}}$ (MPa) | ${\mathit{\sigma}}_{\mathit{c}\mathit{d}}$ (MPa) | ${\mathit{\sigma}}_{\mathit{f}}$ (MPa) | ${\mathit{\sigma}}_{\mathit{r}}$ (MPa) |
---|---|---|---|---|---|

1:1 | 4.12 | 7.26 | 13.42 | 15.30 | 6.60 |

4:1 | 1.20 | 2.44 | 8.6 | 8.99 | 2.52 |

7:1 | 1.31 | 2.04 | 5.37 | 5.65 | 1.07 |

Ratios of Aggregates to Binders | ${\mathit{\sigma}}_{\mathit{c}\mathit{i}}$ (MPa) | ${\mathit{\epsilon}}_{\mathit{c}\mathit{i}}$ (10^{−3}) | ${\mathit{\sigma}}_{\mathit{p}}$ (MPa) | ${\mathit{\epsilon}}_{\mathit{p}}$ (10^{−3}) | ${\mathit{\sigma}}_{\mathit{r}}$ (MPa) | ${\mathit{\epsilon}}_{\mathit{r}}$ (10^{−3}) | ${\mathit{B}}_{\mathbf{i}}$ |
---|---|---|---|---|---|---|---|

1:1 | 7.26 | 3.85 | 15.30 | 9.02 | 6.60 | 10.60 | 4.16 |

4:1 | 2.44 | 1.66 | 8.99 | 9.61 | 2.52 | 12.40 | 3.36 |

7:1 | 2.04 | 1.40 | 5.65 | 8.94 | 1.07 | 13.80 | 2.25 |

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**MDPI and ACS Style**

Jiang, T.; Wan, L.; Wang, W.; Xu, C.; Liu, C.; Meng, F.; Cui, Y.; Li, L. Study on Staged Damage Behaviors of Rock-like Materials with Different Brittleness Degrees Based on Multiple Parameters. *Materials* **2023**, *16*, 2334.
https://doi.org/10.3390/ma16062334

**AMA Style**

Jiang T, Wan L, Wang W, Xu C, Liu C, Meng F, Cui Y, Li L. Study on Staged Damage Behaviors of Rock-like Materials with Different Brittleness Degrees Based on Multiple Parameters. *Materials*. 2023; 16(6):2334.
https://doi.org/10.3390/ma16062334

**Chicago/Turabian Style**

Jiang, Tong, Li Wan, Wenxue Wang, Chao Xu, Chen Liu, Fanke Meng, Yuan Cui, and Longfei Li. 2023. "Study on Staged Damage Behaviors of Rock-like Materials with Different Brittleness Degrees Based on Multiple Parameters" *Materials* 16, no. 6: 2334.
https://doi.org/10.3390/ma16062334