# Elastic-Plastic-Damaged Zones around a Deep Circular Wellbore under Non-Uniform Loading

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description and Fundamental Theory

#### 2.1. The Stress State around a Wellbore

_{0}is the radius of the wellbore. Studies have shown that the weight of the rock mass within the influence range of the wellbore can be neglected, and the error from the original problem does not exceed 5%. Therefore, the horizontal in-situ stress can be simplified to be uniform, so that the original problem becomes the plane strain problem of a circular hole, in which the vertical in-situ stress is P, the horizontal in-situ stress is λP (λ is the lateral stress coefficient), shown in Figure 2.

#### 2.2. Basic Assumptions and Constitutive Model

_{1}, which is constant along the lengthwise direction of the well.

#### 2.3. Fundamental Equations

## 3. Analytical Solution of Stress and Boundary Line Equation

#### 3.1. Elastic Zone

#### 3.2. Plastic Zone

#### 3.3. Damaged Zone

#### 3.4. Boundary Line Equation

## 4. Parameter Analysis

#### 4.1. Stress Distribution in Surrounding Rock Mass

#### 4.1.1. Effect of the Internal Friction Angle

#### 4.1.2. Effect of the Cohesion

#### 4.1.3. Effect of the Lateral Stress Coefficient

#### 4.2. Size of the Plastic Zone

#### 4.2.1. Effect of the Internal Friction Angle

#### 4.2.2. Effect of the Cohesion

#### 4.2.3. Effect of the Brittleness Coefficient

#### 4.2.4. Effect of the Lateral Stress Coefficient

## 5. Comparison of Results

#### 5.1. Comparison with the Results Using the Elastoplastic Softening Model under Uniform Stress Field

#### 5.2. Comparison with the Results Using the Ideal Elastoplastic Model under Non-Uniform Stress Field

#### 5.2.1. Effect of the Internal Friction Angle

#### 5.2.2. Effect of the Cohesion

#### 5.3. Comparison with the Results of the Perturbation Method

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Rock mass behavior models (modified from Reference [18]).

**Figure 4.**Schematic diagram of stress field decomposition: (

**a**) the in-situ stress field, (

**b**) the uniform compressive stress field, (

**c**) the two-sided compressive and two-sided tensile stress field.

**Figure 5.**The radial and hoop stress in the surrounding rock mass with different internal friction angles.

**Figure 7.**The radial and hoop stress in the surrounding rock mass under different lateral stress coefficients.

**Figure 12.**Comparison of the plastic zone and the damaged zone in surrounding rock under the non-uniform stress field (NUSF) and uniform stress field (USF).

**Figure 13.**Comparison of plastic zone shape between elastoplastic softening model and ideal elastoplastic model with different internal friction angles.

**Figure 14.**Comparison of plastic zone shape between elastoplastic softening model and ideal elastoplastic model with different cohesion.

$\mathit{P}/\mathbf{MPa}$ | $\mathit{\phi}{/}^{\circ}$ | $\mathit{\mu}$ | ${\mathit{R}}_{0}/\mathbf{m}$ | $\mathit{c}/\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{c}}^{*}/\mathbf{MPa}$ | $\mathit{\beta}$ | $\mathit{\lambda}$ | ${\mathit{P}}_{1}/\mathbf{MPa}$ |
---|---|---|---|---|---|---|---|---|

6.527 | 30 | 0.22 | 1.25 | 1.2 | 1.3 | 1 | 1.2 | 0 |

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**MDPI and ACS Style**

Shang, X.; Zhang, Z.
Elastic-Plastic-Damaged Zones around a Deep Circular Wellbore under Non-Uniform Loading. *Symmetry* **2020**, *12*, 323.
https://doi.org/10.3390/sym12020323

**AMA Style**

Shang X, Zhang Z.
Elastic-Plastic-Damaged Zones around a Deep Circular Wellbore under Non-Uniform Loading. *Symmetry*. 2020; 12(2):323.
https://doi.org/10.3390/sym12020323

**Chicago/Turabian Style**

Shang, Xiaoji, and Zhizhen Zhang.
2020. "Elastic-Plastic-Damaged Zones around a Deep Circular Wellbore under Non-Uniform Loading" *Symmetry* 12, no. 2: 323.
https://doi.org/10.3390/sym12020323