Abstract
In this study, a mechanical model for tunnels excavated in a nonuniform stress field is developed. A new strainsoftening model simultaneously considers the weakening of cohesion and internal friction angle is proposed. Then, an analytical solution for the postpeak region radii, stresses, and displacements is deduced. Taking a tunnel in Taoyuan coal mine as an engineering example, the postpeak region radii, surface displacement, and stresses distribution are determined. The effects of the horizontaltovertical stress ratio, intermediate principal stress, residual cohesion, and residual internal friction angle on tunnel deformations are discussed. The results show that the postpeak region radii and stresses distribution around the tunnel varies with direction due to the nonuniform stress field. The postpeak region radii and surface displacement are larger with consideration of intermediate principal stress. Tunnels surrounded by rock masses with a higher residual cohesion and internal friction angle experience lower postpeak region radii and surface displacement.
Keywords:
Tunnels; nonuniform stress field; analytical solution; intermediate principal stress; strainsoftening model
Graphical Abstract
Keywords:
Tunnels; nonuniform stress field; analytical solution; intermediate principal stress; strainsoftening model
1 INTRODUCTION
Tunnels are common structures in civil engineering, underground traffic engineering, and mining engineering. After tunnel excavation, the original stress state is disrupted, which causes deformation, displacement, and even destruction of rock masses around the tunnel (Brown et al. 1983Brown, E.T., Bray, J.W., Ladanyi, B., Hoek, E. (1983). Ground response curves for rock tunnels. Journal of Geotechnical Engineering 109: 1539.). Accurate calculations for stresses and deformations of the surrounding rock play an important role in evaluating tunnel stability and designing the support system (Cui et al. 2015Cui, L., Zheng, J., Zhang, R., Dong, Y. (2015). Elastoplastic analysis of a circular opening in rock mass with confining stressdependent strainsoftening behaviour. Tunnelling and Underground Space Technology 50: 94108., Mohammad and Ahmad 2016Mohammad, R.Z., Ahmad, F. (2016). Analytical solutions for the stresses and deformations of deep tunnels in an elasticbrittleplastic rock mass considering the damaged zone. Tunnelling and Underground Space Technology 58: 186196.). In early work, Park (2015Park, K. (2015). Large strain similarity solution for a spherical or circular opening excavated in elasticperfectly plastic media. International Journal for Numerical and Analytical Methods in Geomechanics 39: 724737.), Park and Kim (2006)Park, K., Kim, Y. (2006). Analytical solution for a circular opening in an elasticbrittleplastic rock. International Journal of Rock Mechanics and Mining Science 43: 616622., Sharan (2003Sharan, S.K. (2003). Elasticbrittleplastic analysis of circular openings in HoekBrown media. International Journal of Rock Mechanics and Mining Science 40: 817824., ^{2008}Sharan, S.K. (2008). Analytical solutions for stresses and displacements around a circular opening in a generalized HoekBrown rock. International Journal of Rock Mechanics and Mining Science 45: 7885.), and Wang et al. (2019Wang, X.F., Jiang, B.S., Zhang, Q., Lu, M.M., Chen, M. (2019). Analytical solution of circular tunnel in elasticviscoplastic rock mass. Latin American Journal of Solids and Structures 16: 119.) analyzed the stresses and strains of the surrounding rock by regarding the stress condition of tunnels as a uniform stress field for simplicity. However, considerable field measurements have indicated that the insitu stress in the horizontal and vertical directions vary due to the faults, folds and other geological structures in underground coal mines (Wang et al. 2000Wang, C., Wang, Y., Lu, S. (2000). Deformational behaviour of roadways in soft rocks in underground coal mines and principles for stability control. International Journal of Rock Mechanics and Mining Science 37: 937946., Zhao et al. 2013Zhao, Z.; Wang, W.; Wang, L. (2013). Response models of weakly consolidated soft rock roadway under different interior pressures considering dilatancy effect. Journal of Central South University 20: 37363744.). Tunnels are thus subjected to a nonuniform stress field with a horizontaltovertical stress ratio not equal to one, which has a significant impact on the stresses distribution and displacement of the surrounding rock (Shen 2013Shen, B. (2013). Coal mine roadway stability in soft rock: a case study. Rock Mechanics and Rock Engineering 47: 22252238.). Therefore, it is expected that the theoretical analysis of tunnels should take the horizontaltovertical stress ratio into consideration (Detournay and Fairhurst 1987Detournay, E., Fairhurst, C. (1987). Twodimensional elastoplastic analysis of a long, cylindrical cavity under nonhydrostatic loading. International Journal of Rock Mechanics and Mining Science and Geomechanics Abstracts 24:197211., Detournay and St. John 1988Detournay, E., St. John, C. (1988). Design charts for a deep circular tunnel under nonuniform loading. Rock Mechanics and Rock Engineering 21: 119137., Galin 1946Galin, L.A. (1946). Plane elasticplastic problem: plastic regions around circular holes in plates and beams. Prikladnaia Matematika i Mechanika 10: 365386., Simanjuntak et al. 2014Simanjuntak, T.D.Y.F., Marence, M., Mynett, A.E., Schleiss, A.J. (2014). Pressure tunnels in nonuniform in situ stress conditions. Tunnelling and Underground Space Technology 42: 227236.).
Strength criteria are another crucial component of tunnels analysis. In past decades, research was carried out by using the linear MohrCoulomb strength criterion (Alejano et al. 2009Alejano, L.R., RodriguezDono, A., Alonso, E., Fdez.Manín, G. (2009). Ground reaction curves for tunnels excavated in different quality rock masses showing several types of postfailure behaviour. Tunnelling and Underground Space Technology 24: 689705., Mohammad and Ahmad 2015Mohammad, R.Z., Ahmad, F. (2015). Elasticbrittleplastic analysis of circular deep underwater cavities in a MohrCoulomb rock mass considering seepage forces. International Journal of Geomechanics 15: 110.), nonlinear HoekBrown strength criterion (Alejano et al. 2010Alejano, L.R., Alonso, E., RodríguezDono, A., FernándezManín, G. (2010). Application of the convergenceconfinement method to tunnels in rock masses exhibiting HoekBrown strainsoftening behaviour. International Journal of Rock Mechanics and Mining Science 47: 150160., Sharan 2003Sharan, S.K. (2003). Elasticbrittleplastic analysis of circular openings in HoekBrown media. International Journal of Rock Mechanics and Mining Science 40: 817824.), and generalized HoekBrown strength criterion (Chen and Tonon 2011Chen, R., Tonon, F. (2011). Closedform solutions for a circular tunnel in elasticbrittleplastic ground with the original and generalized HoekBrown failure criteria. Rock Mechanics and Rock Engineering 44: 169178., Sharan 2008); however, the influence of the intermediate principal stress on the rock mass strength was ignored, which had an unfavorable effect on the tunnel support design. In practice, the rock mass is in a true triaxial stress state, and numerous experiments have confirmed that the intermediate principal stress has a significant effect on rock strength (Jiang et al. 2019Jiang, B., Gu, S., Wang, L., Zhang, G., Li, W. (2019). Strainburst process of marble in tunnelexcavationinduced stress path considering intermediate principal stress. Journal of Central South University 26: 984999., Li et al. 2019Li, Z., Wang, L., Lu, Y., Li, W., Wang, K., Fan, H. (2019). Experimental investigation on True Triaxial Deformation and Progressive Damage Behaviour of Sandstone. Scientific Reports 9: 3386., Mogi 1981Mogi, K. (1981). Flow and fracture of rocks under general triaxial compression. Applied Mathematics and Mechanics 2: 635651.). Thus, the intermediate principal stress is essential to engineering applications. The MogiCoulomb strength criterion reasonably considers the influence of intermediate principal stress and better reflects the properties of rock under true triaxial stress (AlAjmi and Zimmerman 2005AlAjmi, A.M., Zimmerman, R.W. (2005). Relation between the Mogi and the Coulomb failure criteria. International Journal of Rock Mechanics and Mining Science 42: 431439., Benz and Schwab 2008Benz, T., Schwab, R. (2008). A quantitative comparison of six rock failure criteria. International Journal of Rock Mechanics and Mining Science 45: 11761186., Zhang et al. 2010Zhang, L., Cao, P., Radha, K.C. (2010). Evaluation of rock strength criteria for wellbore stability analysis. International Journal of Rock Mechanics and Mining Science 47: 13041316., Singh et al. 2017Singh, A., Rao, K.S., Ayothiraman, R. (2017). Effect of intermediate principal stress on cylindrical tunnel in an elastoplastic rock mass. Procedia Engineering 173: 10561063., ^{2018}Singh, A., Kumar, C., Kannan, L.G., Rao, K.S., Ayothiraman, R. (2018). Engineering properties of rock salt and simplified closedform deformation solution for circular opening in rock salt under the true triaxial stress state. Engineering Geology 243:218230.). Therefore, the MogiCoulomb criterion is selected as the strength criterion for the rock mass in this study.
It is known that most of the rock mass undergoes strength attenuation after peaking (Lu et al. 2010Lu, Y., Wang, L., Yang, F., Li, Y., Chen, H. (2010). Postpeak strain softening mechanical properties of weak rock. Chinese Journal of Rock Mechanics and Engineering 29: 640648. (In Chinese), Wang et al. 2010aWang, S., Wang, W., Wu, Z. (2010a). Study of relationship between evolution of postpeak strength parameters and stressstrain curves of geomeaterials. Chinese Journal of Rock Mechanics and Engineering 29: 15241529. (In Chinese)), which is called “strainsoftening.” Previous analytical solutions for tunnels in strainsoftening materials have primarily considered a decrease in cohesion, whereas variations in internal friction angle have been neglected (Han et al. 2013Han, J., Li, S., Li, S., Yang, W. (2013). A procedure of strainsoftening model for elastoplastic analysis of a circular opening considering elastoplastic coupling. Tunnelling and Underground Space Technology 37: 128134., Wang et al. 2010bWang, S., Yin, X., Tang, H., Ge, X. (2010b). A new approach for analyzing circular tunnel in strainsoftening rock masses. International Journal of Rock Mechanics and Mining Science, 47, 170178., Zhang et al. 2012Zhang, Q., Jiang, B., Wang, S., Ge, X., Zhang, H. (2012). Elastoplastic analysis of a circular opening in strainsoftening rock mass. International Journal of Rock Mechanics and Mining Science 50: 3846.). The internal friction angle also shows a decreasing trend in the postpeak stage (Li et al. 2015Li, Y., Cao, S., Fantuzzi, N., Liu, Y. (2015). Elastoplastic analysis of a circular borehole in elasticstrain softening coal seams. International Journal of Rock Mechanics and Mining Science 80: 316324.).
In the present study, a mechanical model for a circular tunnel subjected to a nonuniform stress field is established. A strainsoftening model that simultaneously considers the weakening of cohesion and internal friction angle is proposed. Based on the strainsoftening model and MogiCoulomb criterion, a unified solution for the stresses and displacement of the surrounding rock is determined. Taking a tunnel in the Taoyuan coal mine as an engineering example, the postpeak region radii, surface displacement, and stresses distribution are determined by the newly developed theoretical solution. Finally, the sensitivity of the geomechanical parameters on the tunnel deformations is analyzed.
2 DEFINITION OF THE PROBLEM
2.1 Mechanical model of a circular tunnel in a nonuniform stress field
A circular tunnel of radius R _{0} was excavated in an infinite rock mass (Figure 1). The vertical and horizontal stresses are p _{0} and λp _{0}, respectively, where λ is the horizontaltovertical stress ratio. A support pressure (p _{s} ) is uniformly distributed along the excavation surface. The surrounding rock of the tunnel is subdivided into elastic region (“e”), plastic softening region (“p”), and broken region (“b”). The radii of the plastic softening and broken regions are denoted by R _{p} and R _{b} , respectively.
2.2 Strainsoftening model
Once the stress exceeds the peak strength, the rock enters a postpeak softening stage. Both the cohesion (c) and internal friction angle (φ) of the rock mass gradually decrease to the residual value in this stage. Assuming that the cohesion and internal friction angle linearly decrease with the shear strain (ε _{θp} ) in the strainsoftening stage (Figure 2), the cohesion (c _{p} ) and internal friction angle (φ _{p} ) in the plastic softening region can be expressed as:
where S
_{c} and S
_{φ} are the softening coefficients of the cohesion and internal friction angle in the plastic softening region, respectively; c
_{0} and φ
_{0} are the initial cohesion and initial friction angle, respectively; and
Softening model of the cohesion and internal friction angle (I: Elastic stage; II: Strainsoftening stage; III: Damage stage).
2.3 MogiCoulomb strength criterion
Mogi (1971Mogi, K. (1971). Fracture and flow of rocks under high triaxial compression. Journal of Geophysical Research 76: 12551269.) and Chang and Haimson (2012Chang, C., Haimson, B. (2012). A failure criterion for rocks based on true triaxial testing. Rock Mechanics and Rock Engineering 45: 10071010.) considered that the MogiCoulomb criterion can be expressed as a function between the octahedral shear stress (τ_{oct}) and the average effective normal stress (σ_{m,2}):
where σ _{1}, σ _{2}, and σ _{3} are the maximum, intermediate, and minimum principal stresses, respectively. For axisymmetric plane strain problems, the radial stress (σ _{r} ), axial stress (σ _{z}), and tangential stress (σ _{θ} ) in the surrounding rock of a tunnel can be regarded as σ _{1}, σ _{2}, and σ _{3}, respectively. The relationship among the three principal stresses can be expressed as:
According to Eqs. (2)(5), the MogiCoulomb criterion can be rewritten as:
where
3 UNIFIED SOLUTION OF THE CIRCULAR TUNNELS IN A NONUNIFORM STRESS FIELD
3.1 Basic equations
The equilibrium differential equation for the different regions of surrounding rock can be written as:
where σri and σθi are the radial and tangential stresses in different regions, respectively.
The geometric equation can be expressed as:
where εri and εθi are the radial and tangential strains in different regions, respectively, and ui represents the displacement in different regions.
In addition, the strains in different regions should satisfy the constitutive equations:
where μ and E are the Poisson’s ratio and Young’s modulus of the rock mass, respectively.
Generally, rock mass volume varies in the plastic softening and broken regions, and the relationship between the radial and tangential strains can be established by adopting a nonassociated linear flow rule as:
where
3.2 Stresses and displacement in the elastic region (R _{ p } ≤ r < ∞)
The nonuniform stress field can be decomposed into two stress states, as shown in Figure 3. In state I, the surrounding rock is subjected to a uniform pressure (0.5(1+λ)p _{0}) and a support pressure (p _{s} ), and the stresses in the elastic region can be written as:
where
In state II, the surrounding rock is subjected to a horizontal tension (0.5(1λ)p _{0}) and a vertical pressure (0.5(1λ)p _{0}). The boundary condition at r = R _{0} can be expressed as:
At r = R _{s} , σ _{x} = 0.5(1λ)p _{0}, σ _{y} = 0.5(1λ)p _{0}, and τ _{rθ} = 0. Thus, the boundary condition at r = R _{s} can be obtained by the coordinate transformation:
where
The biharmonic equation is expressed as:
By substituting Eq. (14) into Eq. (15), the following can be obtained:
The general solution of Eq. (16) is:
where A, B, C, and D are integral constants. The stress function can be rewritten as:
Therefore, the stresses can be deduced as:
The integral constants can be determined by the boundary conditions Eq. (12) and Eq. (13) as:
Therefore, Eq. (19) can be rewritten as:
In summary, the stresses in the elastic region under a nonuniform stress field can be obtained by superimposing Eq. (11) and Eq. (21):
The radial and tangential stresses in the elastic region should satisfy Eq. (6) at r = Rp. Thus,
The radial and tangential strains in the elastic region can be derived by substituting Eq. (22) into Eq. (9):
Using Eq. (8), the displacement in the elastic region can be easily obtained as:
3.3 Stresses and displacement in the plastic softening region (R _{ b } ≤ r < R _{ p } )
Combined with the boundary condition of
In the plastic softening region, the total strains can be decomposed into elastic and plastic parts as follows:
where
The displacement differential equation can be given by integrating Eqs. (8), (9), and (27):
The displacement in the plastic softening region can be derived by solving Eq. (28) with the boundary condition u _{p} = u _{e} at r = R _{p} :
where
The strains in the plastic softening region can be determined by substituting Eq. (29) into Eq. (8):
3.4 Stresses and displacement in the broken region (R _{0} ≤ r < R _{ b } )
The stresses in the broken region can be solved by substituting Eq. (6) into Eq. (7) and considering the boundary condition σ _{r} = p _{s} at r = R _{0}:
In the broken region, the total strains of the surrounding rock are also composed of two parts as
where
The displacement in the broken region can be derived by integrating Eqs. (8), (10), and (32) with the boundary condition of u _{b} = u _{p} at r = R _{b} :
where
3.5 Postpeak region radii (R _{ p } and R _{ b } )
To determine the stresses and deformations of the surrounding rock, the postpeak region radii (R _{p} and R _{b} ) should be solved first.
When r = R
_{b} , c
_{p} = c
_{b} , and
With the boundary condition of σ _{rp} =σ _{rb} at r = R _{b} , the following equation can be obtained by integrating Eq. (26) and Eq. (31):
The postpeak region radii can be derived by substituting Eq. (34) into Eq. (35):
Thus, the stresses distribution and surface displacement can be obtained by substituting Eq. (36) into Eqs. (22), (26), (31), and (33).
4 EXAMPLE STUDY
4.1 Validation for the present solution
The haulage tunnel in the Taoyuan coal mine in China is buried approximately 730 m underground with an approximate vertical stress of 19.4 MPa and an average horizontal stress of 21.7 MPa. The horizontaltovertical stress ratio is 1.12. The equivalent excavation radius of the tunnel is 3 m, and the rock mass parameters are E = 1550 MPa, μ= 0.28, c _{0} = 4.3 MPa, c _{b} = 0.8 MPa, S _{c} = 420 MPa, φ _{0} = 34°, φ _{b} = 19°, S _{φ} = 1800°, and ψ _{i} = 10°.
The finite element program COMSOL, based on MohrCoulomb criterion and the strainsoftening model, was introduced to study the stresses distribution of the surrounding rock. To satisfy the same conditions as the numerical simulation, the analytical results based on the MohrCoulomb criterion in the present study were also calculated. The stresses distribution obtained by numerical simulation and the comparison between the analytical and numerical results are shown in Figure 4 and Figure 5, respectively. The stresses distribution around the circular tunnel is consistent with the numerical simulation results, which validates the correctness of the present solution.
4.2 Stress and deformation of the surrounding rock of a tunnel under a nonuniform stress field
Figure 6 shows the postpeak region radii and surface displacement of the surrounding rock. The postpeak region radii and the surface displacement (u _{0}) vary with direction because of the nonuniform stress field. For instance, the R _{p} , R _{b} , and u _{0} values at θ = 0 are 5.00 m, 3.95 m, and 0.14 m, respectively; however, the R _{p} , R _{b} , and u _{0} values at θ = π/2 are 6.43 m, 5.24 m, and 0.28 m, with an increment of 28.60%, 32.66%, and 100.00%, respectively. Additionally, the stresses distribution at different locations of the tunnel in a nonuniform stress field is also different (Figure 7). The peak tangential stress at the tunnel crown is 41.79 MPa, which is approximately 1.29 times that at the tunnel side. Therefore, the tunnel crown experiences a more severe rupture than tunnel sides.
Postpeak region radii and surface displacement around the tunnel in a nonuniform stress field.
4.3 Geomechanical parameters analysis
4.3.1 Horizontaltovertical stress ratio
To analyze the sensitivity of the horizontaltovertical stress ratio (λ) to the deformations of the surrounding rock, the λ value is selected as 0.9, 1.0, 1.1, and 1.2 while keeping all the other parameters constant at their basic values in this case. The postpeak region radii and surface displacement around the tunnel with different stress ratios are shown in Figure 8. The shapes of the postpeak regions are determined by the stress ratio. Only when λ = 1 are the postpeak regions circular and the surface displacements in different directions in the tunnel equal. In addition, when λ < 1, the postpeak region radii and surface displacement show a nonlinear decrease as θ changes from 0 to π/2. The deformation at the tunnel side is most severe. On the contrary, when λ > 1, the postpeak region radii and surface displacement show an increase as θ changes from 0 to π/2. The tunnel crown suffers from the most severe damage and needs more support. Therefore, the horizontaltovertical stress ratio exerts a significant effect on the tunnel deformations and the support design of a tunnel should consider the horizontaltovertical stress ratio.
Postpeak region radii and surface displacement around the tunnel under different horizontaltovertical stress ratio.
4.3.2 Intermediate principal stress
To discuss the influence of intermediate principal stress on the postpeak region radii and surface displacement, the analytical solution in this study is compared with the results obtained using the MohrCoulomb criterion (Figure 9). The R _{p} , R _{b} , and u _{0} values computed by the MohrCoulomb criterion are all larger than those computed by the MogiCoulomb criterion. For example, the R _{p} , R _{b} , and u _{0} values at θ = π/2 from the MogiCoulomb criterion are 6.43 m, 5.24 m, and 0.28 m, respectively; whereas, the results from the MohrCoulomb criterion are 11.02 m, 8.75 m, and 0.81 m, with an increment of 71.38%, 66.98%, and 189.29%, respectively. The primary reason for these diffirent results is that the intermediate principal stress is completely ignored by the MohrCoulomb criterion, which causes an increased unnecessary support design. Therefore, it is more reasonable to take the intermediate principal stress into consideration in engineering applications.
Postpeak region radii and surface displacement around the tunnel based on different strength criteria.
4.3.3 Residual cohesion and residual internal friction angle
Figures 10 and 11 show the sensitivity of the residual cohesion and residual internal friction angle to the postpeak region radii and surface displacement. The R _{p} , R _{b} , and u _{0} values all decrease with increasing c _{b} and φ _{b} . For example, as c _{b} increases from 0.8 MPa to 2.6 MPa, R _{p} , R _{b} , and u _{0} decrease by 1.44 m, 0.70 m, and 0.08 m, with a reduction of 22.42%, 13.39%, and 28.42%, respectively. As φ _{b} increases from 19° to 28°, R _{p} , R _{b} , and u _{0} decrease by 1.51 m, 0.76 m, and 0.12 m, with a reduction of 23.46%, 14.48%, and 42.09%, respectively. Therefore, both the residual cohesion and residual internal friction angle exert a crucial influence on the postpeak region radii and surface displacement. The rock mass bearing capacity gradually increases with the increasing c _{b} and φ _{b} . As a result, some measures, such as grouting, can be used to increase the residual cohesion and residual internal friction angle of the rock mass and ensure tunnel stability.
Influence of residual internal friction angle on the postpeak region radii and surface displacement.
5 CONCLUSIONS
Based on the strainsoftening model and MogiCoulomb criterion, a new unified solution for the stresses distribution and displacement of the surrounding rock of a tunnel in a nonuniform stress field was deduced. The conclusions can be summarized as follows:

(1) Due to the nonuniform stress field, the postpeak region radii and stresses distribution around the tunnel vary with direction. When λ < 1, the postpeak region radii and surface displacement show a nonlinear decrease as θ changes from 0 to π/2. The deformation at the tunnel side is most severe. On the contrary, when λ > 1, the postpeak region radii and surface displacement show an increase as θ changes from 0 to π/2. The tunnel crown suffers from the most damage and needs more support. Therefore, the support parameter design should account for the influence of the horizontaltovertical stress ratio.

(2) The postpeak region radii and surface displacement under the MohrCoulomb criterion are larger than those under the MogiCoulomb criterion, which is the result of neglecting the intermediate principal stress in the MohrCoulomb criterion. Therefore, consideration of the intermediate principal stress with an appropriate strength criterion can lead to a more reasonable tunnel support design.

(3) Tunnels surrounded by rock masses with a higher residual cohesion and residual internal friction angle have lower postpeak region radii and surface displacement. Therefore, deformation of the tunnel surrounding rock can be effectively controlled by increasing the residual cohesion and residual internal friction angle.
Acknowledgments
This study was supported by National Key Research and Development Program of China (No. 2017YFC0603004) and the support is gratefully acknowledged.
References
 AlAjmi, A.M., Zimmerman, R.W. (2005). Relation between the Mogi and the Coulomb failure criteria. International Journal of Rock Mechanics and Mining Science 42: 431439.
 Alejano, L.R., Alonso, E., RodríguezDono, A., FernándezManín, G. (2010). Application of the convergenceconfinement method to tunnels in rock masses exhibiting HoekBrown strainsoftening behaviour. International Journal of Rock Mechanics and Mining Science 47: 150160.
 Alejano, L.R., RodriguezDono, A., Alonso, E., Fdez.Manín, G. (2009). Ground reaction curves for tunnels excavated in different quality rock masses showing several types of postfailure behaviour. Tunnelling and Underground Space Technology 24: 689705.
 Benz, T., Schwab, R. (2008). A quantitative comparison of six rock failure criteria. International Journal of Rock Mechanics and Mining Science 45: 11761186.
 Brown, E.T., Bray, J.W., Ladanyi, B., Hoek, E. (1983). Ground response curves for rock tunnels. Journal of Geotechnical Engineering 109: 1539.
 Chang, C., Haimson, B. (2012). A failure criterion for rocks based on true triaxial testing. Rock Mechanics and Rock Engineering 45: 10071010.
 Chen, R., Tonon, F. (2011). Closedform solutions for a circular tunnel in elasticbrittleplastic ground with the original and generalized HoekBrown failure criteria. Rock Mechanics and Rock Engineering 44: 169178.
 Cui, L., Zheng, J., Zhang, R., Dong, Y. (2015). Elastoplastic analysis of a circular opening in rock mass with confining stressdependent strainsoftening behaviour. Tunnelling and Underground Space Technology 50: 94108.
 Detournay, E., Fairhurst, C. (1987). Twodimensional elastoplastic analysis of a long, cylindrical cavity under nonhydrostatic loading. International Journal of Rock Mechanics and Mining Science and Geomechanics Abstracts 24:197211.
 Detournay, E., St. John, C. (1988). Design charts for a deep circular tunnel under nonuniform loading. Rock Mechanics and Rock Engineering 21: 119137.
 Galin, L.A. (1946). Plane elasticplastic problem: plastic regions around circular holes in plates and beams. Prikladnaia Matematika i Mechanika 10: 365386.
 Han, J., Li, S., Li, S., Yang, W. (2013). A procedure of strainsoftening model for elastoplastic analysis of a circular opening considering elastoplastic coupling. Tunnelling and Underground Space Technology 37: 128134.
 Jiang, B., Gu, S., Wang, L., Zhang, G., Li, W. (2019). Strainburst process of marble in tunnelexcavationinduced stress path considering intermediate principal stress. Journal of Central South University 26: 984999.
 Li, Y., Cao, S., Fantuzzi, N., Liu, Y. (2015). Elastoplastic analysis of a circular borehole in elasticstrain softening coal seams. International Journal of Rock Mechanics and Mining Science 80: 316324.
 Li, Z., Wang, L., Lu, Y., Li, W., Wang, K., Fan, H. (2019). Experimental investigation on True Triaxial Deformation and Progressive Damage Behaviour of Sandstone. Scientific Reports 9: 3386.
 Lu, Y., Wang, L., Yang, F., Li, Y., Chen, H. (2010). Postpeak strain softening mechanical properties of weak rock. Chinese Journal of Rock Mechanics and Engineering 29: 640648. (In Chinese)
 Mogi, K. (1971). Fracture and flow of rocks under high triaxial compression. Journal of Geophysical Research 76: 12551269.
 Mogi, K. (1981). Flow and fracture of rocks under general triaxial compression. Applied Mathematics and Mechanics 2: 635651.
 Mohammad, R.Z., Ahmad, F. (2015). Elasticbrittleplastic analysis of circular deep underwater cavities in a MohrCoulomb rock mass considering seepage forces. International Journal of Geomechanics 15: 110.
 Mohammad, R.Z., Ahmad, F. (2016). Analytical solutions for the stresses and deformations of deep tunnels in an elasticbrittleplastic rock mass considering the damaged zone. Tunnelling and Underground Space Technology 58: 186196.
 Park, K. (2015). Large strain similarity solution for a spherical or circular opening excavated in elasticperfectly plastic media. International Journal for Numerical and Analytical Methods in Geomechanics 39: 724737.
 Park, K., Kim, Y. (2006). Analytical solution for a circular opening in an elasticbrittleplastic rock. International Journal of Rock Mechanics and Mining Science 43: 616622.
 Sharan, S.K. (2003). Elasticbrittleplastic analysis of circular openings in HoekBrown media. International Journal of Rock Mechanics and Mining Science 40: 817824.
 Sharan, S.K. (2008). Analytical solutions for stresses and displacements around a circular opening in a generalized HoekBrown rock. International Journal of Rock Mechanics and Mining Science 45: 7885.
 Shen, B. (2013). Coal mine roadway stability in soft rock: a case study. Rock Mechanics and Rock Engineering 47: 22252238.
 Simanjuntak, T.D.Y.F., Marence, M., Mynett, A.E., Schleiss, A.J. (2014). Pressure tunnels in nonuniform in situ stress conditions. Tunnelling and Underground Space Technology 42: 227236.
 Singh, A., Rao, K.S., Ayothiraman, R. (2017). Effect of intermediate principal stress on cylindrical tunnel in an elastoplastic rock mass. Procedia Engineering 173: 10561063.
 Singh, A., Kumar, C., Kannan, L.G., Rao, K.S., Ayothiraman, R. (2018). Engineering properties of rock salt and simplified closedform deformation solution for circular opening in rock salt under the true triaxial stress state. Engineering Geology 243:218230.
 Wang, C., Wang, Y., Lu, S. (2000). Deformational behaviour of roadways in soft rocks in underground coal mines and principles for stability control. International Journal of Rock Mechanics and Mining Science 37: 937946.
 Wang, S., Wang, W., Wu, Z. (2010a). Study of relationship between evolution of postpeak strength parameters and stressstrain curves of geomeaterials. Chinese Journal of Rock Mechanics and Engineering 29: 15241529. (In Chinese)
 Wang, S., Yin, X., Tang, H., Ge, X. (2010b). A new approach for analyzing circular tunnel in strainsoftening rock masses. International Journal of Rock Mechanics and Mining Science, 47, 170178.
 Wang, X.F., Jiang, B.S., Zhang, Q., Lu, M.M., Chen, M. (2019). Analytical solution of circular tunnel in elasticviscoplastic rock mass. Latin American Journal of Solids and Structures 16: 119.
 Zhang, L., Cao, P., Radha, K.C. (2010). Evaluation of rock strength criteria for wellbore stability analysis. International Journal of Rock Mechanics and Mining Science 47: 13041316.
 Zhang, Q., Jiang, B., Wang, S., Ge, X., Zhang, H. (2012). Elastoplastic analysis of a circular opening in strainsoftening rock mass. International Journal of Rock Mechanics and Mining Science 50: 3846.
 Zhao, Z.; Wang, W.; Wang, L. (2013). Response models of weakly consolidated soft rock roadway under different interior pressures considering dilatancy effect. Journal of Central South University 20: 37363744.
Publication Dates

Publication in this collection
24 Jan 2020 
Date of issue
2020
History

Received
19 Sept 2019 
Reviewed
13 Dec 2019 
Accepted
19 Dec 2019 
Published
07 Jan 2020