# Quantitative Analysis of Infrared Thermal Images in Rock Fractures Based on Multi-Fractal Theory

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Process and Analysis Method

#### 2.1. Materials and Experimental System

^{3}. In order to ensure the homogeneity of the samples, the P-wave velocity of the samples was tested. The wave velocities of all samples are about 3.51 km/s and the homogeneity between samples is good.

#### 2.2. IRR Principle of Rock Fracture

#### 2.3. Multi-Fractal Theory

_{288×382}with the size of 288 × 382. The IRT matrix x

_{m×n}at the sample position (valid area) is extracted by MATLAB software, and the multi-fractal characteristics of the data in the matrix are analyzed according to the literature [38,39].

_{m×n}is divided into boxes of size $k\times k$. The probability density of the box subset (i, j) is $\left\{{P}_{ij}\left(k\right)\right\}$. The probability distribution $\left\{{P}_{ij}\left(k\right)\right\}$ is calculated for each subset:

## 3. Analysis and Discussion of Experimental Results

#### 3.1. Time Series Change in IRT

#### 3.2. Multi-Fractal Analysis of Infrared Thermal Images

_{1}-e

_{1}, a

_{2}-e

_{2}) were selected. The positions of different time points are shown in Figure 3. The multi-fractal characteristics of IRT field data at each time were calculated. Figure 4 and Figure 5a show the calculation results of sample S-1. The infrared thermal images in all the figures were redrawn by Surfer software. At the time of a

_{1}(47.03 s), the $\Delta \alpha $ is low, meaning that the degree of non-uniformity of the infrared temperature field data distribution is small, and $\Delta f$ is close to 0. With the loading process, the infrared temperature field changes as a whole. At the time of b

_{1}(190.88 s), the values of $\Delta \alpha $ and $\Delta f$ show little change. At c

_{1}(343.45 s), the loading is close to the peak stress of the specimen, and a macro fracture occurs. The temperature field of the surface is in the abnormally high-temperature region and develops continuously. At d

_{1}(345.73 s), the sample is destroyed, and there is an obvious shear fracture zone on the surface and some rock blocks fall off in the tensile fracture zone. The MaxIRT of the fracture zone is 31.16 °C and the temperature of the falling off area is the temperature of the background environment. In the process of c

_{1}-d

_{1}, the difference in infrared temperature distribution on the surface of samples gradually increases and $\Delta \alpha $ increases. The performance of high temperature data is gradually dominant, so $\Delta f$ decreases. e

_{1}(350.79 s) is the moment after the failure of the sample, at which time the temperature decreases, but the difference in the temperature distribution is still significant.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Time series change in IRT: (

**a**) S-1; (

**b**) S-2. (Different color blocks distinguish stage I, II and III).

**Figure 4.**Multi-fractal features of infrared thermal images at different time points of S-1: (

**a**) a

_{1}; (

**b**) b

_{1}; (

**c**) c

_{1}; (

**d**) d

_{1}; (

**e**) e

_{1}.

**Figure 5.**Dynamic changes of multi-fractal indicators of infrared thermal image: (

**a**) S-1; (

**b**) S-2. (Different color blocks distinguish stage I, II and III).

**Figure 6.**Multi-fractal features of infrared thermal images at different time points of S-2: (

**a**) a

_{2}; (

**b**) b

_{2}; (

**c**) c

_{2}; (

**d**) d

_{2}; (

**e**) e

_{2}.

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**MDPI and ACS Style**

Miao, B.; Wang, X.; Li, H.
Quantitative Analysis of Infrared Thermal Images in Rock Fractures Based on Multi-Fractal Theory. *Sustainability* **2022**, *14*, 6543.
https://doi.org/10.3390/su14116543

**AMA Style**

Miao B, Wang X, Li H.
Quantitative Analysis of Infrared Thermal Images in Rock Fractures Based on Multi-Fractal Theory. *Sustainability*. 2022; 14(11):6543.
https://doi.org/10.3390/su14116543

**Chicago/Turabian Style**

Miao, Bin, Xinyu Wang, and Hongru Li.
2022. "Quantitative Analysis of Infrared Thermal Images in Rock Fractures Based on Multi-Fractal Theory" *Sustainability* 14, no. 11: 6543.
https://doi.org/10.3390/su14116543