# Stability Analysis of Karst Tunnels Based on a Strain Hardening–Softening Model and Seepage Characteristics

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{3D}numerical simulation by the programming language FISH to analyze the stability and water inrush characteristics of karst tunnels in overlying confining caves. The results showed that rock masses between the cave and tunnel were prone to overall sliding instability. Confined water in the karst cave intruded into the tunnel through the shear-slip rupture zone on both sides instead of the shortest path. Two water inrush points existed on the tunnel surface. The variation law of the permeability coefficients of the surrounding rocks could more truly reflect whether there was a seepage channel between the tunnel and karst cave, as well as the permeable area and water inrush speed of the seepage channel. The work provides a new idea for the stability control of karst tunnels.

## 1. Introduction

## 2. Establishment and Verification of Strain Hardening–Softening Model of Carbonatite

#### 2.1. Analysis of the Results of the Triaxial Compression Test of Carbonatite

#### 2.2. Establishment of the Strain Hardening–Softening Model

- (1)
- Linear elastic stage (O–A): The cohesive and internal friction angle of rocks are c
_{e}and φ_{e}, respectively. - (2)
- Strain hardening stage (A–B): The cohesive and internal friction angle of rocks change from c
_{e}and φ_{e}to c_{p}and φ_{p}, respectively. - (3)
- Strain softening stage (B–C): The cohesive and internal friction angle of rocks change from c
_{p}and φ_{p}to c_{r}and φ_{r}, respectively. - (4)
- Residual strength stage (C–D): The cohesion and internal friction angle of rocks are still c
_{r}and φ_{r}, respectively.

_{3}is the confining pressure; ${\sigma}_{1}^{elas}$, ${\sigma}_{1}^{peak}$, and ${\sigma}_{1}^{res}$ are the axial stress at yield point A, peak point B, and residual point C under the current confining pressure, respectively.

_{ep}and M

_{pr}of rocks decay exponentially with the change in confining pressure σ

_{3}[45], expressed as

_{e}is the elastic modulus of rocks at stage O–A, which has nothing to do with the confining pressure if the initial internal damage is ignored; α

_{1}, β

_{1}, γ

_{1}, β

_{2}, and γ

_{2}are the fitting constants of the triaxial compression test.

- (1)
- Elastic expansion stage (O–A): Rocks are elastically compacted, and volumetric strain increases linearly with axial strain.
- (2)
- Slow expansion stage (A–B): The new micro-fractures appear in rocks and plastic deformation occurs. Volumetric strain decreases slowly, and the dilatancy angles of rocks are small, which can be approximated to keep ψ
_{ep}unchanged. - (3)
- Rapid expansion stage (B–C): Micro-fractures in rocks gradually penetrate and macro-fractures appear. Volumetric strain decreases rapidly; the dilatancy angles of rocks can be approximated to keep ψ
_{pr}unchanged, and ψ_{pr}> ψ_{ep}. - (4)
- Stable expansion stage (C–D): After rocks enter the residual strength stage, volumetric strain decreases steadily, and rocks’ dilatancy angle ψ
_{a}_{r}tends to be constant.

_{ep}, ψ

_{pr}, and ψ

_{ar}are the dilatancy angles of rocks at different confining pressures at stages A–B, B–C, and C–D, respectively; θ

_{0}, θ

_{1}, θ

_{2}, θ

_{3}, ω

_{1}, ω

_{2}, and ω

_{3}are the test fitting constants.

_{v}is the volumetric strain; μ is the Poisson’s ratio; ε

_{1}is the axial strain; ${\epsilon}_{1}^{p}$ is the plastic axial strain; ${\epsilon}_{3}^{p}$ is the plastic radial strain; ψ is the dilatancy angle.

#### 2.3. Verification of the Strain Hardening–Softening Model

## 3. Mechanical Properties and Permeability of Tunnel Surrounding Rocks Based on the Strain Hardening–Softening Model

#### 3.1. Analysis of Mechanical Properties of Surrounding Rocks of the Karst Tunnel

- (1)
- Cohesion and internal friction angle

- (2)
- Elastic modulus

- (3)
- Poisson’s ratio

- (4)
- Tensile strength

_{t}and plastic strain ${\epsilon}_{1}^{p}$.

- (5)
- Dilatancy angle

#### 3.2. Analysis of Caracteristic Parameters of Surrounding-Rock Seepage in Karst Tunnels

_{v}is expressed as

_{v}; n

_{0}is the initial porosity of rocks.

_{0}and k

_{0}are the initial permeability and permeability coefficient of rocks, respectively; b is the performance parameter of rock materials, which is obtained from the test.

## 4. Application of the Strain Hardening–Softening Model

#### 4.1. Engineering Background

_{0}of the surrounding rocks is 0.1; the initial permeability coefficient k is 0.0027 cm/s; the seepage-characteristic fitting parameter b is 7.65. Table 2 shows the permeability parameters under different statuses.

#### 4.2. Numerical-Simulation Scheme Design

^{3D}(Figure 7). The width and height of the model were 75 and 65 m, respectively, with a total of 119,748 elements and 239,370 nodes. The boundary conditions of the model were set to apply the stress of 3.5 MPa on the top surface, normal constraints on the bottom surface and surrounding areas, and a water pressure of 1.5 MPa in the vaulted cave. The surrounding rocks adopted the strain hardening–softening constitutive model and the isotropic permeability model. When the parameters of the strain hardening–softening model of the surrounding rocks were input, the equivalent plastic strain parameter r

_{p}in the software and ${\epsilon}_{1}^{p}$ in the model transformed as follows [52].

^{3D}was used to change the surrounding-rock mechanics and seepage parameters (Figure 8). The specific implementation process is as follows.

- (1)
- According to the size and location of the tunnel and karst cave, a numerical simulation model of karst tunnel excavation was established.
- (2)
- Set the boundary conditions of the model, and define the mechanical constitutive model and seepage model of the surrounding rocks of the tunnel. After inputting initial material parameters, perform fluid–solid coupling calculations up to equilibrium.
- (3)
- Calculate and analyze the excavation of karst caves and tunnels, and traverse all units every 40 time steps. According to the equivalent plastic strain parameter value r
_{p}of each unit body, the values of each mechanical parameter and seepage parameter are changed by Equations (8)–(16). - (4)
- Repeat Step (3) until the stress field and seepage field of the surrounding rocks reach equilibrium or the maximum displacement of the surrounding rocks exceeds 0.4 m. Then, stop the calculation.

#### 4.3. Analysis of Numerical Simulation Results

#### 4.3.1. Change Law of Surrounding-Rock Displacement

#### 4.3.2. Change Law of the Surrounding-Rock Plastic Zone

#### 4.3.3. Change Law of Permeability Coefficients of Surrounding Rocks

#### 4.4. Comparison with the Test Results of the Physical Model

^{3}. According to the similarity theory criterion, the geometric similarity ratio, time similarity ratio, and bulk density similarity ratio of the physical simulation test were 52.3, 7.23, and 1.2, respectively, and the size of the actual surrounding rock was 62.8 m × 5.2 m × 28.8 m. The uniaxial compressive strength of rock was taken as the main index to prepare the model material. The material ratios were as follows: quartz sand: light calcium carbonate: heavy crystal powder = 7:3:1, white cement: gypsum = 3:7, aggregate: cementing material = 17:1. The mass of silicone oil is 3% of the total mass of aggregate and cementing material.

## 5. Discussion

## 6. Conclusions

- (1)
- The results of triaxial compression tests of two kinds of carbonate rocks by other scholars were cited. The simplified stress hardening–softening model of rocks was established by analyzing the deviator stress and volumetric strain curves of two kinds of carbonate rocks. The total stress–strain curve of rocks was simplified into four linear stages: the linear elastic stage, strain hardening stage, strain-softening stage, and residual stage. The volumetric strain–axial strain curve was simplified into four corresponding linear stages: the elastic expansion stage, slow expansion stage, rapid expansion stage, and stable expansion stage.
- (2)
- The stress hardening–softening model was used to deduce the relationship between the rocks’ mechanical parameters such as cohesion, internal friction angle, dilatancy angle, and plastic strain, as well as the relationship between seepage characteristic parameters such as the permeability coefficient and porosity, and volumetric strain.
- (3)
- The stress-hardening–softening constitutive model and seepage characteristic parameters were applied to the FLAC numerical simulation using programming language FISH to analyze the stability and water inrush characteristics of karst tunnels in overlying confining karst caves. Rock masses between the cave and tunnel were prone to overall sliding instability. Confined water in the karst cave intruded into the tunnel through the shear-slip fracture zones on both sides instead of the shortest path. Two water inrush points existed on the tunnel surface. The variation law of the permeability coefficients of the surrounding rocks could more truly reflect whether there was a seepage channel between the tunnel and karst cave, as well as the permeable area and water inrush speed of the seepage channel.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Li, J.; Hong, A.H.; Yuan, D.X.; Jiang, Y.J.; Deng, S.J.; Cao, C.; Liu, J. A new distributed karst-tunnel hydrological model and tunnel hydrological effect simulations. J. Hydrol.
**2021**, 593, 125639. [Google Scholar] [CrossRef] - Xu, Z.H.; Lin, P.; Xing, H.L.; Pan, D.D.; Huang, X. Hydro-mechanical Coupling Response Behaviors in Tunnel Subjected to a Water-Filled Karst Cave. Rock Mech. Rock Eng.
**2021**, 54, 3737–3756. [Google Scholar] [CrossRef] - Zhang, L.K.; Qin, X.Q.; Tang, J.S.; Liu, W.; Yang, H. Review of arsenic geochemical characteristics and its significance on arsenic pollution studies in karst groundwater, Southwest China. Appl. Geochem.
**2017**, 77, 80–88. [Google Scholar] [CrossRef] - Wang, Z.M.; Rawal, K.; Hu, L.B.; Yang, R.D.; Yang, G.L. A study of dissolution and water-bearing characteristics of the restricted platform dolomite facies in the karst areas of Guizhou, China. Environ. Earth Sci.
**2017**, 76, 124. [Google Scholar] [CrossRef] - Yu, L.; Lv, C.; Wang, Z.H.; Sun, Y.; Yang, N.; Wang, Z.L.; Wang, M.N. Upper Bound Analysis of Collapse Failure in Deep Buried Tunnel Under Upper Cave. China J. Highw. Transp.
**2021**, 34, 209–219. [Google Scholar] - Ma, G.M.; Zhang, X.L.; Yang, H.Q. Study on Water Inrush Mechanism and Safety Critical Conditions of Karst Tunnels. Saf. Environ. Eng.
**2022**, 29, 64–70. [Google Scholar] - Zhang, L.W.; Fu, H.; Wu, J.; Zhang, X.Y.; Zhao, D.K. Effects of Karst Cave Shape on the Stability and Minimum Safety Thickness of Tunnel Surrounding Rock. Int. J. Geomech.
**2021**, 21, 04021150. [Google Scholar] [CrossRef] - Song, J.; Chen, D.Y.; Wang, J.; Bi, Y.F.; Liu, S.; Zhong, G.Q.; Wang, C. Evolution Pattern and Matching Mode of Precursor Information about Water Inrush in a Karst Tunnel. Water
**2021**, 13, 1579. [Google Scholar] [CrossRef] - Tian, Q.Y.; Zhang, J.T.; Zhang, Y.L. Similar simulation experiment of expressway tunnel in karst. Constr. Build. Mater.
**2018**, 178, 1–13. [Google Scholar] [CrossRef] - Yang, W.M.; Fang, Z.D.; Wang, H.; Li, L.P.; Shi, S.S.; Ding, R.S.; Bu, L.; Wang, M.X. Analysis on Water Inrush Process of Tunnel with Large Buried Depth and High Water Pressure. Processes
**2019**, 7, 134. [Google Scholar] [CrossRef] [Green Version] - He, Y.W.; Fu, H.L.; Luo, L.F.; Liu, Y.S.; Rao, J.Y. Theoretical Solution of the Influence of Karst Cavern Beneath Tunnel on the Stability of Tunnel Structure. China Civ. Eng. J.
**2014**, 47, 128–135. [Google Scholar] - Fraldi, M.; Guarracino, F. Limit Analysis of Collapse Mechanisms in Cavities and Tunnels According to the Hoek-Brown Failure Criterion. Int. J. Rock. Mech. Min.
**2009**, 46, 665–673. [Google Scholar] [CrossRef] - Fraldi, M.; Guarracino, F. Evaluation of Impending Collapse in Circular Tunnels by Analytical and Numerical Approaches. Tunn. Undergr. Sp. Technol.
**2011**, 26, 507–516. [Google Scholar] [CrossRef] - Fraldi, M.; Guarracino, F. Analytical Solutions for Collapse Mechanisms in Tunnels with Arbitrary Cross Sections. Int. J. Solids Struct.
**2010**, 47, 216–223. [Google Scholar] [CrossRef] - Yang, X.L.; Li, Z.W. Collapse Analysis of Tunnel Floor in Karst Area Based on Hoek-Brown Rock Media. J. Cent. South Univ.
**2017**, 24, 957–966. [Google Scholar] [CrossRef] - Huang, F.; Zhao, L.H.; Ling, T.H.; Yang, X.L. Rock Mass Collapse Mechanism of Concealed Karst Cave Beneath Deep Tunnel. Int. J. Rock Mech. Min.
**2017**, 91, 133–138. [Google Scholar] [CrossRef] - Li, Z. Study on Design Methodology and Key Technologies of Railway Tunnels in Low Mountains and Hills. Ph.D. Thesis, Southwest Jiaotong University, Chengdu, China, 2018. [Google Scholar]
- Sun, J.L.; Wang, F.; Wang, X.L.; Wu, X. A Quantitative Evaluation Method Based on Back Analysis and the Double-Strength Reduction Optimization Method for Tunnel Stability. Adv. Civ. Eng.
**2021**, 2021, 8899685. [Google Scholar] [CrossRef] - Liu, Z.W.; He, M.C.; Wang, S.R. Study on karst water burst mechanism and prevention countermeasures in Yuanliangshan tunnel. Rock Soil Mech.
**2006**, 2, 228–232. [Google Scholar] - Wan, F.; Xu, P.W.; Zhang, P.; Qu, H.F.; Wang, L.H.; Zhang, X. Quantitative Inversion of Water-Inrush Incidents in Mountain Tunnel beneath a Karst Pit. Adv. Civ. Eng.
**2021**, 2021, 9971944. [Google Scholar] [CrossRef] - Yu, J.X. Risk Assessment of Water Inrush and Mud in Karst Tunnel and Influence of Surrounding Rock Stability. Ph.D. Thesis, Chang’an University, Xi’an, China, 2018. [Google Scholar]
- Yang, T.H.; Tang, C.A.; Tan, Z.H.; Zhu, W.C.; Feng, Q.Y. State of The Art of Inrush Models in Rock Mass Failure and Developing Trend for Prediction and Forecast of Groundwater Inrush. Chin. J. Rock Mech. Eng.
**2007**, 2, 268–277. [Google Scholar] - Huang, M.L.; Wang, F.; Lu, W.; Tan, Z.S. Numerical study on the process of water inrush in Karst caves with hydraulic pressure caused by tunnel excavation. Strateg. Study CAE
**2009**, 11, 93–96. [Google Scholar] - Zhang, Q.; Huang, B.X.; He, M.C.; Guo, S. A Numerical Investigation on the Hydraulic Fracturing Effect of Water Inrush during Tunnel Excavation. Geofluids
**2020**, 2020, 6196327. [Google Scholar] [CrossRef] - Sun, F. Study on The Key Technique of Composite Grouting for Water Blockage in Weathered Slot of Subsea Tunnel. Ph.D. Thesis, Beijing Jiaotong University, Beijing, China, 2010. [Google Scholar]
- Wang, Y.; Lu, Y.G.; Ni, X.D.; Li, D.T. Study on mechanism of water burst and mud burst in deep tunnel excavation. J. Hydraul. Eng.
**2011**, 42, 595–601. [Google Scholar] - Gao, Y. Research on the Safety Distance That Water Inrush Disaster Prevention of DeJiang Tunnel Roof under Complicated hydrogeological Conditions. Ph.D. Thesis, Guizhou University, Guiyang, China, 2016. [Google Scholar]
- Zhu, W.X. Study on Mechanism of Water Inrush in Muddy Limestone Karst Tunnel after Wetting and Drying Cycles. Ph.D. Thesis, China University of Mining and Technology, Beijing, China, 2018. [Google Scholar]
- Liu, H.Y.; Qin, S.Q.; Li, H.E.; Ma, P.; Sun, Q.; Yang, J.H. Simulation of rock slope seepage failure by numerical manifold method. Hydrogeol. Eng. Geol.
**2007**, 1, 66–69. [Google Scholar] - Yan, C.Z.; Zheng, H.; Sun, G.H.; Ge, X.R. Combined finite-discrete element method for simulation of hydraulic fracturing. Rock Mech. Rock Eng.
**2016**, 49, 1389–1410. [Google Scholar] [CrossRef] - Yan, C.Z.; Jiao, Y.Y. A 2D fully coupled hydro-mechanical finite discrete element model with real pore seepage for simulating the deformation and fracture of porous medium driven by fluid. Comput. Struct.
**2018**, 196, 311–326. [Google Scholar] [CrossRef] - Yan, C.Z.; Jiao, Y.Y.; Zheng, H. A fully coupled three-dimensional hydro-mechanical finite discrete element approach with real porous seepage for simulating 3D hydraulic fracturing. Comput. Geotech.
**2018**, 96, 73–89. [Google Scholar] [CrossRef] - Zhang, R.; Yan, L.; Qian, Z.W.; Sun, X.W.; Liu, W. Study on the Evolution Relationship of the Deformation Mechanism and Permeability of Underground Rock. Chin. J. Undergr. Space Eng.
**2022**, 18, 129–135. [Google Scholar] - Wang, J.A.; Park, H.D. Fluid permeability of sedimentary rock in a complete stress-strain process. Eng. Geol.
**2002**, 63, 291–300. [Google Scholar] [CrossRef] - Peng, S.P.; Meng, Z.P.; Wang, H.; Ma, C.L.; Pan, J.N. Testing study on pore ration and permeability of sandstone under different confining pressures. Chin. J. Rock Mech. Eng.
**2003**, 22, 742–746. [Google Scholar] - Peng, S.P.; Qu, H.L.; Luo, L.P.; Wang, L.; Duan, Y.E. An experimental study on the penetrability of sedimentary rock during the complete stress-strain path. J. China Coal Soc.
**2000**, 25, 113–116. [Google Scholar] - Yang, Y.J.; Chu, J.; Huan, D.Z.; Li, L. Experimental of coal’s strain-permeability rate under solid and liquid coupling condition. J. China Coal Soc.
**2008**, 33, 760–764. [Google Scholar] - Li, S.P.; Li, Y.S.; Wu, Z.Y. The permeability-strain equations relating to complete stress-strain path of the rock. Chin. J. Geotech. Eng.
**1995**, 17, 13–19. [Google Scholar] - Zhang, C.H.; Yue, H.L.; Wang, L.G.; Guo, X.K.; Wang, Y.J. Strain softening and permeability evolution model based on brittle modulus coefficient. J. China Coal Soc.
**2016**, 41, 255–264. [Google Scholar] - Liu, C.Y.; Du, L.Z.; Zhang, X.P.; Wang, Y.; Hu, X.M.; Han, Y.L. A New Rock Brittleness Evaluation Method Based on the Complete Stress-Strain Curve. Lithosphere
**2021**, 2021, 4029886. [Google Scholar] [CrossRef] - Yuan, S.C.; Harrison, J.P. An empirical dilatancy index for the dilatant deformation of rock. Int. J. Rock Mech. Min.
**2004**, 41, 679–686. [Google Scholar] [CrossRef] - Yuan, S.C.; Harrison, J.P. Development of a hydro-mechanical local degradation approach and its application to modelling fluid flow during progressive fracturing of heterogeneous rocks. Int. J. Rock Mech. Min.
**2005**, 42, 961–984. [Google Scholar] [CrossRef] - Wang, D.; Han, X.G.; Zhou, X.M. Limestone failure law and post-failure constitutive relation in the control of lateral deformation. J. China Coal Soc.
**2010**, 35, 2022–2027. [Google Scholar] - Jing, W.; Xue, W.P.; Yao, Z.S. Variation of the internal friction angle and cohesion of the plastic softening zone rock in roadway surrounding rock. J. China Coal Soc.
**2018**, 43, 2203–2210. [Google Scholar] - Lu, Y.D.; Ge, X.R.; Jiang, Y.; Ren, J.X. Study on Conventional Triaxial Compression Test of Complete Process for Marble and Its Constitutive Equation. Chin. J. Rock Mech. Eng.
**2004**, 23, 2489–2493. [Google Scholar] - Zhao, X.G.; Cai, M.; Cai, M.F. A rock dilation angle model and its verification. Chin. J. Rock Mech. Eng.
**2010**, 29, 970–981. [Google Scholar] - Meng, Q.B.; Wang, J.; Han, L.J.; Sun, W.; Qiao, W.G.; Wang, G. Physical and mechanical properties and constitutive model of very weakly cemented rock. Rock Soil Mech.
**2020**, 41, 19–29. [Google Scholar] - Zhao, X.G.; Li, P.F.; Ma, L.K.; Su, R.; Wang, J. Damage and dilation characteristics of deep granite at Beishan under cyclic loading-unloading conditions. Chin. J. Rock Mech. Eng.
**2014**, 33, 1740–1748. [Google Scholar] - Zhang, N.X.; Sheng, Z.P.; Li, X.; Li, S.D.; Hao, J.M. Study of relationship between poisson’s ratio and angle of internal friction for rocks. Chin. J. Rock Mech. Eng.
**2011**, 30, 2599–2609. [Google Scholar] - Zhang, N.X.; Li, S.D.; Sheng, Z.P. The method for estimating tensile strength by using shear strength parameters and discussions. J. Eng. Geol.
**2018**, 26, 446–456. [Google Scholar] - Zhang, B.Y.; Bai, H.B.; Zhang, K. Research on permeability characteristics of karst collapse column fillings in complete stress-strain process. J. Min. Saf. Eng.
**2016**, 33, 734–740. [Google Scholar] - Sun, C.; Zhang, S.G.; Jia, B.X.; Wu, Z.Q. Physical and numerical model tests on post-peak mechanical properties of granite. Chin. J. Geotech. Eng.
**2015**, 37, 847–852. [Google Scholar]

**Figure 2.**Variation curves of deviatoric stress and volumetric strain of dolomite and limestone with axial strain under triaxial compression. (

**a**) Gebdykes dolomite. (

**b**) Daqing limestone.

**Figure 3.**Simplified triaxial stress–strain and volumetric–axial strain curves of rocks. (

**a**) Total stress–strain curve. (

**b**) Volumetric strain–axial strain curve.

**Figure 4.**Comparison between the total stress–strain experimental curves and the strain hardening–softening model curves under rocks’ triaxial compression. (

**a**) Gebdykes dolomite. (

**b**) Daqing limestone.

**Figure 5.**Comparison between the volumetric strain–axial strain experimental curves and strain hardening–softening model curves under rocks’ triaxial compression. (

**a**) Gebdykes dolomite. (

**b**) Daqing limestone.

**Figure 9.**Displacements of karst-tunnel surrounding rocks under different solution steps. (

**a**) Step 800, (

**b**) Step 4800, (

**c**) Step 9800, and (

**d**) Step 14,800.

**Figure 10.**Distribution of surrounding-rock plastic zones of the karst tunnel under different solution steps. (

**a**) Step 800, (

**b**) Step 4800, (

**c**) Step 9800, and (

**d**) Step 14,800.

**Figure 11.**Changes for permeability coefficients of karst-tunnel surrounding rocks under different solution steps. (

**a**) Step 800, (

**b**) Step 4800, (

**c**) Step 9800, and (

**d**) Step 14,800.

**Figure 13.**Physical test results of the karst tunnel. (

**a**) Total displacement nephogram. (

**b**) Surrounding-rock damage.

Status Point | ε_{1}/% | E/GPa | μ | c/MPa | φ/° | σ_{t}/MPa |
---|---|---|---|---|---|---|

Yield point A | 0.30 | 2.19 | 0.33 | 0.53 | 29.2 | 0.31 |

Peak point B | 0.49 | 3.0 | 0.30 | 0.6 | 35 | 0.31 |

Residual point C | 0.60 | 2.13 | 0.31 | 0.19 | 32.3 | 0.11 |

Status Point | ε_{v}/% | ψ/° | n | k/(cm·s^{−1}) |
---|---|---|---|---|

Initial point O | 0 | 0 | 0.10 | 0.0027 |

Yield point A | 0.10 | 16 | 0.099 | 0.0025 |

Peak point B | −0.05 | 25.1 | 0.100 | 0.0027 |

Residual point C | −0.23 | 22.9 | 0.102 | 0.0032 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, H.; Lin, Z.; Liu, C.; Zhang, B.; Wang, C.; Liu, J.; Liang, H.
Stability Analysis of Karst Tunnels Based on a Strain Hardening–Softening Model and Seepage Characteristics. *Sustainability* **2022**, *14*, 9589.
https://doi.org/10.3390/su14159589

**AMA Style**

Liu H, Lin Z, Liu C, Zhang B, Wang C, Liu J, Liang H.
Stability Analysis of Karst Tunnels Based on a Strain Hardening–Softening Model and Seepage Characteristics. *Sustainability*. 2022; 14(15):9589.
https://doi.org/10.3390/su14159589

**Chicago/Turabian Style**

Liu, Hongyang, Zhibin Lin, Chengwei Liu, Boyang Zhang, Chen Wang, Jiangang Liu, and Huajie Liang.
2022. "Stability Analysis of Karst Tunnels Based on a Strain Hardening–Softening Model and Seepage Characteristics" *Sustainability* 14, no. 15: 9589.
https://doi.org/10.3390/su14159589