# Simulated Short- and Long-Term Deformation in Coastal Karst Caves

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## Abstract

**:**

## 1. Introduction

## 2. Contents and Methods

#### 2.1. Numerical Modeling of Short-Term and Long-Term Deformation Surrounding Caves

#### 2.2. The Mesoscopic Controlling Mechanism of Self-Balance Pressure Arches of Caves

#### 2.3. Time-Dependent Deformation under Pressure Arches

#### 2.4. The Discrete Element Analysis Method for Rheological Mechanisms under the Pressure Arches

## 3. Results and Discussion

#### 3.1. Case Study

#### 3.1.1. Overview of the Selected Case Study

#### 3.1.2. Excavation Deformation Monitoring Data

#### 3.1.3. Rock and Soil Parameters and the DEM Model

^{3}, 2000 kg/m

^{3}and 2000 kg/m

^{3}, respectively, according to Table 2. The damping coefficient in the DEM modeling process plays a role in controlling calculating stability, so we select this value as 0.5 according to a general searching of relevant studies [40,49]. The ratio of normal to shear stiffness effects the Poisson ratio of the material, which is not so important in this modeling scenario, and we select this as a general value of 1.2.

- (1)
- Select models’ modulus, friction angle, tensile and cohesion strength according to the experience.
- (2)
- Alter the modulus based on stress–strain curves or rock strength.
- (3)
- Alter the tensile and cohesion strength according to rock strength.
- (4)
- Change the friction angle according to soils’ ultimate stress in the stress–strain curves.
- (5)
- Repeat (2)~(4) to obtain a reasonable result.

#### 3.2. The Mesoscopic Mechanical Characteristics under the Self-Balanced Pressure Arch

#### 3.3. The Displacement Results Considering Rheological Deforming

#### 3.4. Discussion for Short-Term Stability under Different Overlying Loadings

#### 3.5. Discussion for Caves’ Influence on Deformation under Different Rock Strengths

#### 3.6. Discussion for the Long-term Deformation under Different Viscosity Coefficients and Elasticity Modulus

## 4. Conclusions

- This paper presented a DEM-based method to evaluate the short-term and long-term stability of coastal karst caves. The long-term settlements of sinkholes are mainly dependent on viscosity, and the short-term collapsing only appears in weak bedrocks, which causes a much larger final deformation.
- The local stress concentration surrounding caves brings potential risk to the local zones. If the bedrock has a relatively high strength, the presence of caves has little influence on the final modeling. In this study, when the value of rock strength reduced by 90%, the short-term and long-term behaviors of sinkholes were affected extensively.
- The modeling proposed in this paper can predict long-term deformation after foundation pit excavation, and the prediction error is less than 10%. The modeling process can be calibrated from geological borehole data and applied to other geotechnical engineering practices in coastal karst areas.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Schematic diagram of a numerical scheme for the simulation of rock rheology in a karst area.

**Figure 7.**Schematic diagram of a numerical scheme for the simulation of soil rheology in a karst area.

**Figure 13.**The calibration result of the stress–strain curve shows strength (

**a**) and Young’s modulus (

**b**).

**Figure 14.**The numerical time-dependent response after altering viscosity coefficients (

**a**) ${\eta}_{1}$; (

**b**) ${\eta}_{2}$.

**Figure 17.**Contact force chain characteristics with or without caves (lines in these figures reveal the force state of material; the direction and width of lines represent the force direction and magnitude, respectively).

**Figure 19.**Predicted and monitored vertical displacement of the slope using different random seeds: (

**a**) random seed 1; (

**b**) random seed 2; (

**c**) random seed 3.

**Figure 20.**The stress and mesoscopic damage in different overlying loads: (

**a**) stress–time series; (

**b**) mesoscopic damage and grains’ velocities.

**Figure 21.**Comparison on the foundation between caved or not caved condition, the settlement lines compare the long-term deformation, and the grains’ figures compare the collapsing condition and also the quantitively counted crack number; in this case, the tension strength and cohesion strength are 1.0 MPa and 0.4 MPa.

**Figure 22.**Comparison on the foundation between caved or not caved condition, the settlement lines compare the long-term deformation, and the grains’ figures compare the collapsing condition and also the quantitively counted crack number; in this case, the tension strength and cohesion strength are 2.0 MPa and 0.8 MPa.

**Figure 23.**Comparison on the foundation between caved or not caved condition, the settlement lines compare the long-term deformation, and the grains’ figures compare the collapsing condition and also the quantitively counted crack number; in this case, the tension strength and cohesion strength are 3.0 MPa and 1.2 MPa.

**Figure 24.**The time-dependent behavior of modeling using different viscosity coefficients: (

**a**) monitoring point at left side; (

**b**) monitoring point at right side.

**Figure 25.**The numerical time-dependent response after altering elastic modulus: (

**a**) ${E}_{1}$; (

**b**) ${E}_{2}$.

Type of Model | Governing Equation | Creep Equation |
---|---|---|

Maxwell model | $\sigma ={E}_{1}{\epsilon}_{k}+{\eta}_{1}\dot{{\epsilon}_{k}}$ | $\epsilon =\frac{{\sigma}_{0}}{{E}_{1}}\left(1-\mathrm{exp}\left(-\frac{{E}_{1}}{{\eta}_{1}}t\right)\right)$ |

Kelvin model | $\dot{{\epsilon}_{M}}=\frac{\dot{\sigma}}{{E}_{2}}+\frac{\sigma}{{\eta}_{2}}$ | ${\epsilon}_{M}=\frac{1}{{\eta}_{2}}{\sigma}_{0}t+\frac{{\sigma}_{0}}{{E}_{2}}$ |

Burger’s model | $\begin{array}{c}\ddot{\sigma}+\left(\frac{{E}_{2}}{{\eta}_{1}}+\frac{{E}_{2}}{{\eta}_{2}}+\frac{{E}_{1}}{{\eta}_{1}}\right)\dot{\sigma}+\\ \frac{{E}_{1}}{{\eta}_{1}}\frac{{E}_{2}}{{\eta}_{2}}\sigma ={E}_{0}\ddot{\epsilon}+\frac{{E}_{1}{E}_{2}}{{\eta}_{1}}\dot{\epsilon}\end{array}$ | $\epsilon =\frac{{\sigma}_{0}}{{E}_{2}}+\frac{\sigma}{{\eta}_{2}}t+\frac{{\sigma}_{0}}{{E}_{1}}\left(1-{e}^{-\frac{{E}_{1}}{{\eta}_{1}}t}\right)$ |

Soil Types | Unit Weight (kN/m^{3}) | Modulus (MPa) | Friction Angle (°) | Cohesion (kPa) |
---|---|---|---|---|

Filled soil | 18.5 | 4.00 | 12.0 | 4.0 |

Medium coarse sand | 20 | 4.00 | 12.0 | 4.0 |

Clay | 20 | 7.00 | 18.0 | 20.0 |

Rock Types | Uniaxial Strength (MPa) | Saturated Strength (MPa) | Modulus (MPa) | Unit Weight (kN/m^{3}) |
---|---|---|---|---|

Limestone | 25 | 12 | 956 | 26 |

Parameters Used in DEM Elastic Analysis | Value |
---|---|

PBM tensile strength (MPa) | 30 |

PBM cohesion strength (MPa) | 11 |

PBM modulus (GPa) | 0.6 |

PBM friction angle (°) | 20 |

Linear elastic modulus of “filled soil” and “coarse sand” (MPa) | 2 |

Friction angle of “filled soil” and “coarse sand” (°) | 20 |

Linear elastic modulus of “silty clay” (MPa) | 2.5 |

Friction angle of “silty clay” (°) | 30 |

Ratio of normal to shear stiffness | 1.2 |

Particle friction coefficient | 0.57 |

Density of particles (kg/m^{3}) | 1850~2500 |

Mean particle radius (mm) | 0.42 |

Damping coefficient | 0.5 |

Parameters used in DEM time-dependent analysis | |

Kelvin viscosity coefficient (10^{6}) | 1 |

Maxwell viscosity coefficient (10^{6}) | 2.5 |

Kelvin modulus of rock (GPa) | 0.6 |

Kelvin modulus of “filled soil” and “coarse sand” (MPa) | 2 |

Kelvin modulus of “filled soil” and “coarse sand” (MPa) | 2 |

Maxwell modulus of rock (GPa) | 0.6 |

Maxwell modulus of “silty clay” (MPa) | 2 |

Maxwell modulus of “silty clay” (MPa) | 2 |

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## Share and Cite

**MDPI and ACS Style**

Lin, C.; Xia, C.; Zhang, H.; Liu, Z.; Zhou, C.
Simulated Short- and Long-Term Deformation in Coastal Karst Caves. *J. Mar. Sci. Eng.* **2022**, *10*, 1315.
https://doi.org/10.3390/jmse10091315

**AMA Style**

Lin C, Xia C, Zhang H, Liu Z, Zhou C.
Simulated Short- and Long-Term Deformation in Coastal Karst Caves. *Journal of Marine Science and Engineering*. 2022; 10(9):1315.
https://doi.org/10.3390/jmse10091315

**Chicago/Turabian Style**

Lin, Chunxiu, Chang Xia, Hong Zhang, Zhen Liu, and Cuiying Zhou.
2022. "Simulated Short- and Long-Term Deformation in Coastal Karst Caves" *Journal of Marine Science and Engineering* 10, no. 9: 1315.
https://doi.org/10.3390/jmse10091315