# Dynamic Compressive Mechanical Property Characteristics and Fractal Dimension Applications of Coal-Bearing Mudstone at Real-Time Temperatures

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## Abstract

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## 1. Introduction

## 2. Characterization and Test Methods for Coal-Bearing Mudstone

#### 2.1. Real-Time Temperature SHPB Testing Method and Process

- The tests are conducted using a high-to-low temperature gradient approach, with each rock sample pressure gradient tested at a particular temperature gradient before moving to the next one. The mudstone obtained from coal-bearing strata is placed in an external heating oven and subjected to a predetermined temperature for 10 min in a high-temperature ambient oven. The sample is then placed between the incoming and transmitting bars using a fire clamp to ensure that the temperature difference between the displayed real-time temperature and the preset temperature is within ±1 °C.
- The air pressure score should be adjusted before swiftly holding the mudstone rock sample in place with the incidence and transmission bars for impact testing. During this process, it is essential to record the impact bullet velocity and strain signals in the incidence and transmission bars.
- After the test, the incident and transmission bars are promptly extracted from the high-temperature ambient furnace. The rock samples are then retrieved using an iron spoon and transferred into an iron container. The container is subsequently placed inside an externally heated furnace with an appropriate temperature gradient to ensure that the samples remain intact and undamaged during the high-temperature ambient cooling process.
- Once the impact tests for all rock samples at a particular temperature gradient are complete, the fragments are gradually cooled to room temperature within an externally heated furnace with a temperature reduction gradient. The samples are then ready for the next round of impact tests at a different temperature gradient. For instance, if the preset temperature is 400 °C, the initial holding time is 30 min at 400 °C, followed by a temperature reduction of 50 °C at a rate of 10 °C/min, along with an additional 60-min holding time at the gradient. Finally, the temperature is further reduced until it reaches room temperature (25 °C), as demonstrated in Figure 3c.
- Upon completion of impact testing across all temperature gradients, the macroscopic damage characteristics and microscopic fracture patterns of the fractured rock samples are analyzed and quantified.

#### 2.2. Test Principle and Accuracy Verification

_{0}is the modulus of elasticity of the bar material, A

_{0}is the area of the compressional bar, C

_{0}is the wave velocity, L

_{S}is the original length of the specimen, and A

_{S}is the cross-sectional area. To ensure the accuracy of our experiment, we compare the error of the incident and reflected waves with the transmitted waves, as illustrated in Figure 4. The small, transmitted wave signal of the coal-bearing mudstone confirms the assumption of one-dimensional stress wave propagation. In contrast, the overlapping incident, reflected, and transmitted waves introduce the equilibrium index R(t) for quantitative analysis [38]. We use R(t) < 5% as the criterion for judgment and calculate it as follows:

## 3. Dynamic Compressive Mechanical Properties of Coal-Bearing Mudstone

#### 3.1. Variation of Strain Rate with Air Pressure

^{−1}to 70.712 s

^{−1}, an increase of 214.50%. At a given pressure, the strain rate first decreased and then increased with increasing temperature. However, the strain rate at high temperatures was lower than at room temperature. The difference between the maximum and minimum strain rates at different air pressure was more pronounced at high pressure than at low pressure, indicating a more significant fluctuation in the strain rate with the temperature at high pressure.

#### 3.2. The Variation Law of the Stress–Strain Curve

#### 3.3. The Change in Dynamic Compression Mechanical Properties

#### 3.4. Variation of Dynamic Mechanical Properties with Temperature

## 4. Fractal Failure Characteristics of Coal-Bearing Mudstone

#### 4.1. Macroscopic Damage Characteristics

^{−1}). Broken fragments of the coal-bearing mudstone were analyzed to determine the variation pattern of the degree of damage with temperature in this strain rate range, as illustrated in Figure 10. The results showed that the surface fragmentation degree of the large residual fragments after composite crushing initially increased and then decreased with the temperature rise. Moreover, the degree of crushed rock samples at high temperatures was greater than at room temperature. In the temperature range of 25–150 °C, the size of the large fragments of the coal-bearing mudstone increased, while the sharpness of the fragmented contour decreased, indicating a strengthening effect on the rock samples due to the expansion of the clay minerals within the coal-bearing mudstone to fill the fissures (Figure 10a–d). Conversely, in the temperature range of 200–400 °C, the average size of the fragments gradually decreased, with large fragments disappearing and the sharpness of the edges of the fragments slightly increasing. These results indicate a weakening effect on the rock samples due to the expansion of fractures caused by the over-expansion of clay minerals within the coal-bearing sandstone through temperature, which can intensify the destruction of the rock samples (Figure 10e–i). To quantify the variation of damage with temperature and strain rate, it was necessary to sieve the particle size of the fractured fragments and determine the average particle size of the damaged fragments and the fractal dimension of the fragments.

#### 4.2. Blockiness Distribution Coefficient

_{vn}of the n-th group is determined by calculating the average of the largest and smallest particle sizes in the particle size range. In the case of the >20.0 mm group, the average particle size is selected as 20.0 mm since the maximum size is unknown. W

_{sn}represents the mass fraction of the n-th group. Table 5 shows the variations of W

_{sn}and the blockiness distribution coefficient r with temperature T. By utilizing the data in the table, a variation curve of r with T is plotted, as illustrated in Figure 12.

#### 4.3. Fractal Dimension of Fragments

_{max}, this can be approximated as

_{max}is set to be the original radius of the rock specimen to maintain consistency during the calculation. Table 6 presents the variation of $\mathrm{lg}\left({M}_{R}/{M}_{\mathrm{max}}\right)$ and D with $\mathrm{lg}\left(R/{R}_{\mathrm{max}}\right)$ at different temperatures.

## 5. Discussion

#### 5.1. Different Stages of Real-Time Temperature Influence

#### 5.2. Correlation of Fractal Dimensions and Applications

## 6. Conclusions

- Peak stress and elastic modulus exhibit an initially increasing and then decreasing trend with rising temperature. They demonstrate a slight increase between 25 and 150 °C, followed by a monotonous decreasing trend between 150 and 400 °C. The blockiness distribution coefficient exhibits an initially increasing and then decreasing trend, while the fractal dimension shows an initially decreasing and then increasing trend as the temperature rises. Combined with macroscopic damage morphology, it indicates an initial decrease and subsequent increase in the extent of damage to the coal-bearing mudstone with increasing temperature.
- The influence of real-time temperature on coal-bearing mudstone can be classified into two stages. The correlation coefficients between the fractal dimension of the fragments and the peak stress, elastic modulus, and peak strain are −0.944, −0.916, and −0.460, respectively. Peak stress and elastic modulus decrease linearly with the increase in the fractal dimension.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Pressure/ Temperature | Strain Rate $\dot{\mathit{\epsilon}}$ (s^{−1}) | ||||||||
---|---|---|---|---|---|---|---|---|---|

25 °C | 50 °C | 100 °C | 150 °C | 200 °C | 250 °C | 300 °C | 350 °C | 400 °C | |

0.2 MPa | 46.021 | 42.468 | 40.455 | 22.484 | 27.065 | 27.255 | 35.896 | 37.051 | 38.498 |

0.3 MPa | 60.650 | 58.035 | 57.765 | 39.164 | 45.895 | 50.053 | 51.651 | 53.849 | 55.819 |

0.4 MPa | 70.325 | 63.654 | 60.745 | 42.065 | 54.210 | 53.516 | 57.051 | 61.251 | 67.416 |

0.5 MPa | 82.551 | 75.721 | 70.545 | 54.898 | 62.516 | 61.048 | 65.055 | 70.027 | 77.874 |

0.6 MPa | 95.874 | 87.451 | 83.151 | 65.564 | 73.846 | 70.511 | 75.842 | 83.415 | 86.135 |

0.7 MPa | 110.052 | 100.165 | 97.105 | 70.712 | 76.154 | 76.447 | 84.515 | 90.754 | 93.464 |

Pressure/ Temperature | Peak Stress ${\mathit{\sigma}}_{\mathbf{m}}$ (MPa) | ||||||||
---|---|---|---|---|---|---|---|---|---|

25 °C | 50 °C | 100 °C | 150 °C | 200 °C | 250 °C | 300 °C | 350 °C | 400 °C | |

0.2 MPa | 45.779 | 38.323 | 49.621 | 43.021 | 42.275 | 43.330 | 26.227 | 25.756 | 28.223 |

0.3 MPa | 57.709 | 60.610 | 63.945 | 64.342 | 63.279 | 50.406 | 34.617 | 38.652 | 35.731 |

0.4 MPa | 71.488 | 86.077 | 77.874 | 73.899 | 74.356 | 70.675 | 47.080 | 47.114 | 41.711 |

0.5 MPa | 89.401 | 93.601 | 89.108 | 83.001 | 84.442 | 78.231 | 54.728 | 53.970 | 43.584 |

0.6 MPa | 94.921 | 99.370 | 97.283 | 95.828 | 98.811 | 73.108 | 59.209 | 60.064 | 46.229 |

0.7 MPa | 103.437 | 106.089 | 100.876 | 98.566 | 100.469 | 82.773 | 66.970 | 67.281 | 51.809 |

Pressure/ Temperature | Elastic Modulus ${\mathit{E}}_{\mathbf{d}}$ (GPa) | ||||||||
---|---|---|---|---|---|---|---|---|---|

25 °C | 50 °C | 100 °C | 150 °C | 200 °C | 250 °C | 300 °C | 350 °C | 400 °C | |

0.2 MPa | 5.525 | 6.403 | 5.809 | 5.024 | 4.621 | 4.505 | 3.226 | 3.288 | 3.376 |

0.3 MPa | 7.343 | 6.996 | 6.776 | 6.733 | 6.192 | 5.935 | 4.041 | 4.519 | 4.418 |

0.4 MPa | 9.809 | 9.352 | 8.783 | 7.438 | 7.609 | 7.271 | 5.408 | 5.158 | 4.629 |

0.5 MPa | 10.625 | 9.671 | 10.218 | 9.570 | 9.362 | 8.845 | 7.856 | 7.764 | 6.375 |

0.6 MPa | 12.110 | 11.132 | 12.681 | 10.093 | 10.855 | 9.172 | 9.131 | 8.450 | 6.714 |

0.7 MPa | 15.185 | 13.025 | 15.623 | 13.703 | 13.240 | 12.542 | 11.178 | 11.586 | 8.679 |

Pressure/ Temperature | Peak Strain ${\mathit{\epsilon}}_{\mathbf{d}}$ (×10^{−3}) | ||||||||
---|---|---|---|---|---|---|---|---|---|

25 °C | 50 °C | 100 °C | 150 °C | 200 °C | 250 °C | 300 °C | 350 °C | 400 °C | |

0.2 MPa | 10.682 | 10.123 | 11.525 | 11.435 | 9.698 | 10.741 | 10.277 | 9.976 | 10.760 |

0.3 MPa | 10.053 | 11.386 | 9.831 | 12.791 | 12.054 | 10.632 | 10.627 | 9.962 | 10.624 |

0.4 MPa | 9.009 | 10.043 | 10.179 | 11.181 | 11.305 | 10.630 | 10.364 | 10.641 | 9.807 |

0.5 MPa | 7.779 | 11.010 | 11.407 | 10.781 | 9.808 | 10.880 | 10.594 | 9.892 | 9.588 |

0.6 MPa | 9.464 | 9.735 | 10.489 | 11.076 | 10.360 | 10.353 | 10.173 | 10.942 | 10.755 |

0.7 MPa | 9.152 | 11.736 | 10.829 | 10.181 | 9.979 | 11.271 | 9.232 | 11.550 | 8.426 |

T/°C | W_{sn} | r/m | |||||||
---|---|---|---|---|---|---|---|---|---|

<5 mm | 5–7 mm | 7–9 mm | 9–11 mm | 11–14 mm | 14–17 mm | 17–20 mm | >20 mm | ||

25 | 0.09916 | 0.10564 | 0.08423 | 0.08433 | 0.12270 | 0.11352 | 0.18766 | 0.20276 | 0.01322 |

50 | 0.09446 | 0.09393 | 0.08813 | 0.07757 | 0.10871 | 0.12454 | 0.18786 | 0.22480 | 0.01354 |

100 | 0.09583 | 0.08422 | 0.07333 | 0.07533 | 0.11506 | 0.13898 | 0.18526 | 0.23199 | 0.01374 |

150 | 0.08722 | 0.08565 | 0.08183 | 0.08275 | 0.12290 | 0.11753 | 0.18538 | 0.23674 | 0.01374 |

200 | 0.09576 | 0.08742 | 0.07908 | 0.10259 | 0.14755 | 0.12211 | 0.16103 | 0.20446 | 0.01323 |

250 | 0.09970 | 0.09489 | 0.10631 | 0.09970 | 0.13514 | 0.11652 | 0.15255 | 0.19519 | 0.01289 |

300 | 0.10293 | 0.11915 | 0.12227 | 0.09482 | 0.11978 | 0.11291 | 0.14410 | 0.18404 | 0.01249 |

350 | 0.11280 | 0.12928 | 0.12167 | 0.09379 | 0.11153 | 0.11217 | 0.13245 | 0.18631 | 0.01228 |

400 | 0.11427 | 0.14074 | 0.12137 | 0.10781 | 0.09813 | 0.10910 | 0.12395 | 0.18463 | 0.01208 |

**Table 6.**The variation of $\mathrm{lg}\left({M}_{R}/{M}_{\mathrm{max}}\right)$ and D with $\mathrm{lg}\left(R/{R}_{\mathrm{max}}\right)$ at different temperatures.

T/°C | $\mathbf{lg}\left({\mathit{M}}_{\mathit{R}}/\mathit{M}\right)$ | D | ||||||
---|---|---|---|---|---|---|---|---|

$\begin{array}{l}\mathbf{lg}\left(\mathit{R}/{\mathit{R}}_{\mathbf{max}}\right)\\ =-1.00000\end{array}$ | $\begin{array}{l}\mathbf{lg}\left(\mathit{R}/{\mathit{R}}_{\mathbf{max}}\right)\\ =-0.85387\end{array}$ | $\begin{array}{l}\mathbf{lg}\left(\mathit{R}/{\mathit{R}}_{\mathbf{max}}\right)\\ =-0.74473\end{array}$ | $\begin{array}{l}\mathbf{lg}\left(\mathit{R}/{\mathit{R}}_{\mathbf{max}}\right)\\ =-0.65758\end{array}$ | $\begin{array}{l}\mathbf{lg}\left(\mathit{R}/{\mathit{R}}_{\mathbf{max}}\right)\\ =-0.55284\end{array}$ | $\begin{array}{l}\mathbf{lg}\left(\mathit{R}/{\mathit{R}}_{\mathbf{max}}\right)\\ =-0.46852\end{array}$ | $\begin{array}{l}\mathbf{lg}\left(\mathit{R}/{\mathit{R}}_{\mathbf{max}}\right)\\ =-0.39794\end{array}$ | ||

25 | −1.00366 | −0.68867 | −0.53906 | −0.42787 | −0.30447 | −0.21497 | −0.09841 | 1.5769 |

50 | −1.02475 | −0.72494 | −0.55827 | −0.45089 | −0.33461 | −0.23111 | −0.11059 | 1.5588 |

100 | −1.01850 | −0.74461 | −0.59623 | −0.48319 | −0.35284 | −0.23452 | −0.11463 | 1.5578 |

150 | −1.05938 | −0.76228 | −0.59397 | −0.47179 | −0.33691 | −0.23816 | −0.11733 | 1.4957 |

200 | −1.01882 | −0.73712 | −0.58127 | −0.43789 | −0.29039 | −0.19756 | −0.09934 | 1.4973 |

250 | −1.00130 | −0.71088 | −0.52158 | −0.39729 | −0.27105 | −0.18558 | −0.09431 | 1.5295 |

300 | −0.98746 | −0.65349 | −0.46300 | −0.35737 | −0.25263 | −0.17272 | −0.08833 | 1.5766 |

350 | −0.94769 | −0.61604 | −0.43920 | −0.33957 | −0.24483 | −0.16670 | −0.08954 | 1.6491 |

400 | −0.94207 | −0.59344 | −0.42437 | −0.31498 | −0.23484 | −0.16026 | −0.08865 | 1.6645 |

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## Share and Cite

**MDPI and ACS Style**

Guo, S.; Zhang, L.; Pu, H.; Zheng, Y.; Li, B.; Wu, P.; Qiu, P.; Ma, C.; Feng, Y.
Dynamic Compressive Mechanical Property Characteristics and Fractal Dimension Applications of Coal-Bearing Mudstone at Real-Time Temperatures. *Fractal Fract.* **2023**, *7*, 695.
https://doi.org/10.3390/fractalfract7090695

**AMA Style**

Guo S, Zhang L, Pu H, Zheng Y, Li B, Wu P, Qiu P, Ma C, Feng Y.
Dynamic Compressive Mechanical Property Characteristics and Fractal Dimension Applications of Coal-Bearing Mudstone at Real-Time Temperatures. *Fractal and Fractional*. 2023; 7(9):695.
https://doi.org/10.3390/fractalfract7090695

**Chicago/Turabian Style**

Guo, Shiru, Lianying Zhang, Hai Pu, Yadong Zheng, Bing Li, Peng Wu, Peitao Qiu, Chao Ma, and Yiying Feng.
2023. "Dynamic Compressive Mechanical Property Characteristics and Fractal Dimension Applications of Coal-Bearing Mudstone at Real-Time Temperatures" *Fractal and Fractional* 7, no. 9: 695.
https://doi.org/10.3390/fractalfract7090695