# Study on Shape Characteristics of Plastic Zone in Heterogeneous Roadway-Surrounding Rock

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## Abstract

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## 1. Introduction

^{3D}and experiments [25], while Zhang et al. studied the influence of shale bedding on the stress–strain response, failure mode, and shear strength by establishing a heterogeneous numerical simulation model [26]. Furthermore, Jonak et al. studied the rock fracture process and its influencing factors using a numerical simulation method [27,28,29].

## 2. Analytical Algorithm for Plastic Zone of Surrounding Rock in Heterogeneous Circular Roadway

#### 2.1. Elastic Mechanics of Surrounding Rock of General Circular Roadway

#### 2.2. Mechanical Model of Heterogeneous Circular Roadway Surrounding Rock

#### 2.3. Shape of Plastic Zone in Surrounding Rock of Heterogeneous Circular Roadway

- ${K}_{1}=9{(1-\lambda )}^{2}$
- ${K}_{2}=-12{(1-\lambda )}^{2}+6(1-{\lambda}^{2})\mathrm{cos}2\theta $
- ${K}_{3}=10{(1-\lambda )}^{2}{\mathrm{cos}}^{2}2\theta -4{(1-\lambda )}^{2}{\mathrm{sin}}^{2}(\phi (i,j)){\mathrm{cos}}^{2}2\theta -2{(1-\lambda )}^{2}{\mathrm{sin}}^{2}2\theta -4(1-{\lambda}^{2})\mathrm{cos}2\theta +{(1+\lambda )}^{2}$
- ${K}_{4}=-4{(1-\lambda )}^{2}\mathrm{cos}4\theta +2(1-{\lambda}^{2})\mathrm{cos}2\theta -4(1-{\lambda}^{2}){\mathrm{sin}}^{2}(\phi (i,j))\mathrm{cos}2\theta -\frac{4}{P}(1-\lambda )\mathrm{cos}2\theta \mathrm{sin}(2\phi (i,j))C(i,j)$
- ${K}_{5}={(1-\lambda )}^{2}-{\mathrm{sin}}^{2}(\phi (i,j)){\left(1+\lambda +\frac{2C(i,j)}{P}\frac{\mathrm{cos}(\phi (i,j))}{\mathrm{sin}(\phi (i,j))}\right)}^{2}$

## 3. Shape Characteristics and Evolution Law of Plastic Zone

#### 3.1. Shape Characteristics of Plastic Zone in Surrounding Rock of Single-Layer Heterogeneous Roadway

#### 3.2. Shape Characteristics of Plastic Zone in Surrounding Rock of Multi-Layer Heterogeneous Composite Roadway

#### 3.3. Directional Characteristics of Butterfly Shaped Plastic Zone

- ${{K}^{\prime}}_{1}=9{(1-\lambda )}^{2}$
- ${{K}^{\prime}}_{2}=-12{(1-\lambda )}^{2}+6(1-{\lambda}^{2})\mathrm{cos}(2(\theta 1-\alpha ))$
- $\begin{array}{l}{{K}^{\prime}}_{3}=10{(1-\lambda )}^{2}{\mathrm{cos}}^{2}(2(\theta 1-\alpha ))-4{(1-\lambda )}^{2}{\mathrm{sin}}^{2}(\phi (i,j)){\mathrm{cos}}^{2}(2(\theta 1-\alpha ))\\ -2{(1-\lambda )}^{2}{\mathrm{sin}}^{2}(2(\theta 1-\alpha ))-4(1-{\lambda}^{2})\mathrm{cos}(2(\theta 1-\alpha ))+{(1+\lambda )}^{2}\end{array}$
- $\begin{array}{l}{{K}^{\prime}}_{4}=-4{(1-\lambda )}^{2}\mathrm{cos}(4(\theta 1-\alpha ))+2(1-{\lambda}^{2})\mathrm{cos}(2(\theta 1-\alpha ))-4(1-{\lambda}^{2}){\mathrm{sin}}^{2}(\phi (i,j))\mathrm{cos}(2(\theta 1-\alpha ))\\ -\frac{4}{P}(1-\lambda )\mathrm{cos}(2(\theta 1-\alpha ))\mathrm{sin}(2\phi (i,j))C(i,j)\end{array}$
- ${{K}^{\prime}}_{5}={(1-\lambda )}^{2}-{\mathrm{sin}}^{2}(\phi (i,j)){\left(1+\lambda +\frac{2C(i,j)}{P}\frac{\mathrm{cos}(\phi (i,j))}{\mathrm{sin}(\phi (i,j))}\right)}^{2}$

#### 3.4. Shape Evolution of Plastic Zone

## 4. Error Analysis of Analytical Algorithm

#### 4.1. Error Analysis of Plastic Zone Boundary of Single Layer Heterogeneous Surrounding Rock

#### 4.1.1. Error Analysis of Plastic Zone Boundary under Isobaric Condition

#### 4.1.2. Error Analysis of Plastic Zone Boundary under Non-Isobaric Conditions

#### 4.2. Error Analysis of Plastic Zone Boundary of Multi-Layer Heterogeneous Composite Surrounding Rock

#### 4.2.1. Error Analysis of Plastic Zone Boundary under Isobaric Condition

#### 4.2.2. Error Analysis of Plastic Zone Boundary under Non-Isobaric Conditions

## 5. Discussion

#### 5.1. Shape of Plastic Zone of Roadway Surrounding Rock

#### 5.2. Range of Plastic Zone of Roadway Surrounding Rock

#### 5.3. Comparison with Existing Experimental Results

## 6. Conclusions

- (1)
- The analytical algorithm proposed by this study uses several simple and commonly known independent variables to express the important unknown quantity of the range and shape of the plastic zone of the surrounding rock of the roadway, so that the functional relationship between them is clearer, and the failure law of the roadway can be revealed by studying the properties of the function, which is incomparable with other methods such as numerical simulation.
- (2)
- The comparison results show that the shape and range characteristics of the plastic zone obtained by the analytical algorithm are highly consistent with those obtained by the numerical simulation method, and are also consistent with the existing experimental results, so the analytical algorithm is effective. Compared with other methods, this method can guide field practice more conveniently and simply.
- (3)
- There are some errors between the analytical method and the numerical simulation. In addition, the relative error of the plastic zone calculation of a multi-layer heterogeneous roadway is larger than that of a single heterogeneous roadway. According to the accuracy required by the actual project, these problems can be ignored.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Numerical calculation model and mechanical parameter assignment. (

**a**) Numerical calculation model. (

**b**) Random distribution of mechanical parameters.

**Figure 4.**Shape of plastic zone in single-layer heterogeneous surrounding rock of roadway with different lateral pressure coefficients. (

**a**) Coal. (

**b**) Mudstone. (

**c**) Sandy mudstone.

**Figure 5.**Calculation model of multi-layer heterogeneous surrounding rock. (

**a**) Analytical calculation. (

**b**) Numerical simulation.

**Figure 6.**Shape of plastic zone in multi-layer heterogeneous surrounding rock of roadway with different lateral pressure coefficients.

**Figure 7.**Direction variation characteristics of plastic zone under different rotation angles of maximum confining pressure. (

**a**) 0°. (

**b**) 30°. (

**c**) 60°. (

**d**) 90°. The blue arrow is the direction of maximum principal stress, and the orange arrow is the direction of minimum principal stress.

**Figure 10.**Calculation results of heterogeneous circular roadway-surrounding rock plastic zone boundary with lateral pressure coefficient of 1. (

**a**) Coal. (

**b**) Mudstone. (

**c**) Sandy mudstone.

**Figure 11.**Calculation results of plastic zone boundary of single-layer heterogeneous surrounding rock under different lateral pressures. (

**a**) Coal. (

**b**) Mudstone. (

**c**) Sandy Mudstone.

**Figure 12.**Calculation results of multi-layered heterogeneous circular roadway-surrounding rock plastic zone boundary with lateral pressure coefficient of 1.

**Figure 13.**Calculation results of plastic zone boundary of multi-layer heterogeneous surrounding rock under different lateral pressure coefficients.

Rock | Tensile Strength/MPa | Friction Angle/(°) | Cohesion/MPa | Elastic Modulus/GPa | Poisson’s Ratio | Compressive Strength/MPa |
---|---|---|---|---|---|---|

coal | 0.35 | [23, 28] | [2.8, 3.2] | 22.96 | 0.23 | 9.42 |

mudstone | 1.91 | [27, 33] | [5.7, 6.3] | 10.35 | 0.24 | 20.78 |

sandy mudstone | 4.39 | [32, 38] | [8.7, 9.3] | 2.69 | 0.22 | 34.57 |

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**MDPI and ACS Style**

Li, J.; Wu, Z.; Zhang, W.; Ma, N.; Guo, S.
Study on Shape Characteristics of Plastic Zone in Heterogeneous Roadway-Surrounding Rock. *Sustainability* **2022**, *14*, 9480.
https://doi.org/10.3390/su14159480

**AMA Style**

Li J, Wu Z, Zhang W, Ma N, Guo S.
Study on Shape Characteristics of Plastic Zone in Heterogeneous Roadway-Surrounding Rock. *Sustainability*. 2022; 14(15):9480.
https://doi.org/10.3390/su14159480

**Chicago/Turabian Style**

Li, Jun, Zheng Wu, Wenlong Zhang, Nianjie Ma, and Shuying Guo.
2022. "Study on Shape Characteristics of Plastic Zone in Heterogeneous Roadway-Surrounding Rock" *Sustainability* 14, no. 15: 9480.
https://doi.org/10.3390/su14159480