# Anisotropic Constitutive Model of Intermittent Columnar Jointed Rock Masses Based on the Cosserat Theory

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Constitutive Model of a Columnar Jointed Rock Mass

#### 2.1. Cosserat Equivalent Model of a Single Set of Intermittent Jointed Rock Masses

**a**is the joint,

**b**is the complete rock with the same thickness, ${h}_{j}$ is the joint thickness, and ${A}_{j}$ is the joint communication rate. In the analysis,

**a**and

**b**are regarded as a composite element, and ${E}^{a,b},{v}^{a,b},{G}^{a,b}$ and ${E}_{i},{v}_{i},{G}_{i}$ respectively, represent the elastic modulus, Poisson’s ratio, and shear modulus of the composite element

**ab**and the complete rock.

_{1}.

**m**is the couple stress. The sign convention in the case of plane strain is ${E}^{\prime}=E/(1-{v}^{2})$, ${v}^{\prime}=v/(1-v)$. The following relationship was obtained by a previous study [34].

_{1}- ①
- Normal stress and normal strain$$\begin{array}{l}{\stackrel{-}{\epsilon}}_{11}^{a,b}=\frac{{\stackrel{\_}{\sigma}}_{11}^{a,b}}{{E}^{\prime}{}_{1}^{a,b}}-\frac{{\nu}^{\prime}{}_{12}^{a,b}}{{E}^{\prime}{}_{1}^{a,b}}{\stackrel{\_}{\sigma}}_{22}^{a,b},\hspace{1em}{\stackrel{-}{\epsilon}}_{22}^{a,b}=\frac{{\stackrel{\_}{\sigma}}_{22}^{a,b}}{{E}^{\prime}{}_{2}^{a,b}}-\frac{{\nu}^{\prime}{}_{21}^{a,b}}{{E}^{\prime}{}_{2}^{a,b}}{\stackrel{\_}{\sigma}}_{11}^{a,b}\\ {\stackrel{-}{\epsilon}}_{11i}=\frac{{\stackrel{\_}{\sigma}}_{11i}}{{E}_{i}^{\prime}}-\frac{{\nu}_{i}^{\prime}}{{E}_{i}^{\prime}}{\stackrel{\_}{\sigma}}_{22i},\hspace{1em}{\stackrel{-}{\epsilon}}_{22i}=\frac{{\stackrel{\_}{\sigma}}_{22i}}{{E}_{i}^{\prime}}-\frac{{\nu}_{i}^{\prime}}{{E}_{i}^{\prime}}{\stackrel{\_}{\sigma}}_{11i}\end{array}\}$$
**ab**and the intact rock in 1 and 2 directions, respectively. - ②
- Shear stress and shear strain$$\begin{array}{l}{\epsilon}_{12}^{c}={D}_{33}{\sigma}_{12}^{c}+{D}_{34}{\sigma}_{21}^{c}\\ {\epsilon}_{21}^{c}={D}_{43}{\sigma}_{12}^{c}+{D}_{44}{\sigma}_{21}^{c}\end{array}\}$$
- ③
- Couple stress and curvature$$\begin{array}{l}\frac{1}{{\rho}_{}^{a,b}}=\frac{{\alpha}_{j}{m}_{1}^{a,b}}{{E}^{\prime}{}_{2}^{a,b}{I}_{}^{a,b}}\\ \frac{1}{{\rho}_{i}}=\frac{{\alpha}_{i}{m}_{1i}}{{E}_{i}^{\prime}{I}_{i}}\end{array}\}$$

_{1}to the unit body illustrated in Figure 3, the equivalent elastic constant of the composite body

**ab**can be obtained as follows [35].

_{2}to the unit body illustrated in Figure 3, the equivalent elastic constant of the composite body

**ab**can be obtained as follows.

**ab**in Equations (5) and (6) into Equations (1)–(3), the asymmetric elastic matrix element of the Cosserat intermittent jointed rock mass can be described by Equation (7):

#### 2.2. Cosserat Elastic Matrix of a Hexagonal Prismatic Jointed Rock Mass

_{3}, and the column axis is cut into the rock mass with a staggered distance,

**r**. When the selected coordinate system does not follow the main direction of the intermittent jointed rock mass, the main axis of the material coordinate system

**1-2**deviates from that of the natural coordinate system

**x-y**by the angle θ, and the relationship between the two is shown in Figure 5.

## 3. Off-Axis Elastic Matrix Equation of a Columnar Jointed Rock Mass

#### 3.1. Off-Axis Elastic Matrix Equation

**x-y-z**denotes the geodetic coordinate system,

**1-2-3**denotes the material principal axis coordinate system, α denotes tendency, and β denotes trend.

**x-y-z**can be converted into the vector $\left\{a\right\}$ in the material principal axis coordinate system

**1-2-3**by the equation ${\left\{A\right\}}_{xyz}=\left[l\right]{\left\{a\right\}}_{123}$.

**1-2-3**to the geodetic coordinate system

**x-y-z**, which is given as

**x-y-z**of the columnar joint can be obtained by the transformation matrix [L] as follows.

#### 3.2. Case Analysis

_{2}β

_{3}

^{3−2}rock layer (basalt of the upper permian emeishan formation). The diameters of the columns range from 13 to 25 cm, and the length is generally 2–3 m; the average diameter and length are 0.2 and 2 m, respectively. The staggered distance is r = 0.5 m. Additionally, according to an engineering geology report, the occurrence of the P

_{2}β

_{3}

^{3−2}rock layer is 30° N~50° E, NE ∠ 10~25°, and the conversion matrix ${\left[L\right]}_{12\times 12}\left(40\xb0,15\xb0\right)$ was obtained after taking the average value. The mechanical parameters of the rock mass and joint are listed in Table 1 [19].

_{3}is reduced by 21.2%, E

_{2}is reduced by 5.1%, and Poisson’s ratio and shear modulus also change by varying degrees. This indicates that the variation of each parameter is closely related to the occurrence of the column axis. Therefore, the influences of anisotropic parameters should be fully considered when analyzing the deformation and failure characteristics of a columnar jointed basalt diversion tunnel.

## 4. Program Implementation and Application of Constitutive Model

#### 4.1. Program Implementation of the Cosserat Constitutive Model

^{3D}, the Cosserat anisotropic constitutive model was programmed by VC++ and compiled into the dynamic link library (DLL file), and was also embedded in the software to carry out the calculation of the columnar jointed rock mass.

^{3D}numerical software are viewed as symmetric tensors, this work mainly considers the composite effect of joints and the rock mass; thus, the influence of the couple stress was neglected during programming. The calculation process of the Cosserat constitutive model is as follows. First, the basic mechanical parameters are input into the constitutive model function, and the strain increment of the current step element and the stress value of the previous step element are calculated. The steady-state strain and variable increment of the current time step of the element are calculated according to the stress value of the element. Then, based on the elastic assumption, the new stress of the element is calculated according to the elastic parameters, the elastic strain increment, and the element stress increment. The yield state of the element is judged by calling the element yield function, and the stress is adjusted according to the yield function and the plastic potential function to achieve new stress and yield states of the element. Finally, the new stress and related parameters of the element are returned to the main program of FLAC

^{3D}for the calculation of the next step and element. The final calculation results are obtained after this calculation process. According to the geological conditions of the Baihetan Hydropower Station, tunnel excavation was simulated to verify the correctness of the proposed Cosserat constitutive model and its application in underground engineering.

#### 4.2. Model Establishment

#### 4.3. Result Analysis

## 5. Conclusions

- (1)
- By considering the rock and joint surfaces as dualistic structures, a Cosserat constitutive model of a regular hexagonal columnar jointed rock mass was established by analyzing the mechanical properties of three sets of intermittent joints angled at 60°.
- (2)
- The elastic matrix expression of the columnar jointed rock mass under an off-axis condition was given by introducing the positive off-axis transformation matrix. The equivalent anisotropic parameters under the off-axis condition were investigated by an example. The results showed that the elastic modulus decreases by 21.2% in the vertical direction and 5.1% in the horizontal direction when the cylinder is deflected by 15°. Additionally, Poisson’s ratio and shear modulus were also found to change significantly and revealed obvious anisotropy.
- (3)
- With the aid of the Visual C++ program development platform, a FLAC
^{3D}interface program of the proposed Cosserat constitutive model was developed, and the deformation and failure characteristics of a columnar joint tunnel after excavation were calculated and analyzed. The anisotropic constitutive model calculation program determined the deformation and failure phenomena of the surrounding rock of the tunnel to be more complex than those calculated by the isotropic model. The main manifestation is that the plastic zone and stress distribution of the surrounding rock are asymmetric. The vertical and horizontal displacements calculated by the Cosserat constitutive model were also greater than those calculated by the isotropic model. These findings all indicate that the calculation results of the Cosserat constitutive model could better reflect the mechanical properties of the columnar jointed rock mass after excavation. The research methods and conclusions of this paper can provide guidance for further construction at the Baihetan Hydropower Station. - (4)
- The constitutive model derived in this work is based on the ideal state. Considering the more complicated geological conditions of the Baihetan Hydropower Station, the applicability of the model still requires further verification. In addition, the calculation program used in this work fails to consider other influencing factors, such as the couple stress and the joint dip angle, as well as what impact these factors will have on the tunnel excavation calculations. These issues are worthy of further research.

## Author Contributions

## Funding

## Conflicts of Interest

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Parameter | ρ (g·cm^{−3}) | E (GPa) | v | c (MPa) | ψ (°) | Kn (GPa/m) | Ks (GPa/m) |
---|---|---|---|---|---|---|---|

Rock | 2.8 | 60 | 0.2 | 12.4 | 56.3 | -- | -- |

Joint | -- | -- | -- | 0.6 | 36 | 100 | 50 |

[L]_{12×12} | E_{1} (MPa) | E_{2} (MPa) | E_{3} (MPa) | ν_{12} | ν_{13} | ν_{23} | G_{12} (GPa) | G_{21} (GPa) | G_{13} (GPa) | G_{31} (GPa) | G_{23} (GPa) | G_{32} (GPa) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

(0°, 0°) | 20.00 | 20.00 | 34.48 | 0.086 | 0.138 | 0.138 | 17.54 | 17.54 | 10.53 | 19.49 | 10.53 | 19.49 |

(40°, 15°) | 20.00 | 18.98 | 27.17 | 0.080 | 0.111 | 0.250 | 17.06 | 16.13 | 11.90 | 19.31 | 11.12 | 21.37 |

Tracking Point (Coordinate (m)) | Horizontal Displacement | Vertical Displacement | ||
---|---|---|---|---|

Isotropic Model | Cosserat Model | Isotropic Model | Cosserat Model | |

1 (0, 0, 2) | 0.000 | 0.000 | −2.042 | −2.620 |

2 (2, 0, 0) | −0.409 | −0.621 | −1.021 | −0.592 |

3 (2, 0, −2) | −0.704 | −0.642 | −0.363 | −0.188 |

4 (−4, 0, −4) | −0.036 | −0.156 | −0.463 | −0.312 |

5 (0, 0, −4) | 0.000 | 0.000 | 1.121 | 1.181 |

6 (−2, 0, −4) | 0.046 | 0.060 | 0.463 | 0.430 |

7 (−2, 0, −2) | 0.704 | 1.170 | 0.365 | 0.942 |

8 (−2, 0, 0) | 0.409 | 1.260 | 1.021 | 1.671 |

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**MDPI and ACS Style**

Lu, W.; Zhu, Z.; Que, X.; Zhang, C.; He, Y.
Anisotropic Constitutive Model of Intermittent Columnar Jointed Rock Masses Based on the Cosserat Theory. *Symmetry* **2020**, *12*, 823.
https://doi.org/10.3390/sym12050823

**AMA Style**

Lu W, Zhu Z, Que X, Zhang C, He Y.
Anisotropic Constitutive Model of Intermittent Columnar Jointed Rock Masses Based on the Cosserat Theory. *Symmetry*. 2020; 12(5):823.
https://doi.org/10.3390/sym12050823

**Chicago/Turabian Style**

Lu, Wenbin, Zhende Zhu, Xiangcheng Que, Cong Zhang, and Yanxin He.
2020. "Anisotropic Constitutive Model of Intermittent Columnar Jointed Rock Masses Based on the Cosserat Theory" *Symmetry* 12, no. 5: 823.
https://doi.org/10.3390/sym12050823