An analytical solution for stresses and deformations of tunnels in a non-uniform stress field based on strain-softening model and Mogi-Coulomb criterion

In this study, a mechanical model for tunnels excavated in a non-uniform stress field is developed. A new strain-softening model simultaneously considers the weakening of cohesion and internal friction angle is proposed. Then, an analytical solution for the post-peak region radii, stresses, and displacements is deduced. Taking a tunnel in Taoyuan coal mine as an engineering example, the post-peak region radii, surface displacement, and stresses distribution are determined. The effects of the horizontal-to-vertical stress ratio, intermediate principal stress, residual cohesion, and residual internal friction angle on tunnel deformations are discussed. The results show that the post-peak region radii and stresses distribution around the tunnel varies with direction due to the non-uniform stress field. The post-peak 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 post-peak region radii and surface displacement.


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. 1983). 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. 2015, Mohammad andAhmad 2016). In early work, Park (2015), Park and Kim (2006), Sharan (2003Sharan ( , 2008, and Wang et al. (2019) 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. 2000, Zhao et al. 2013. Tunnels are thus subjected to a non-uniform stress field with a horizontal-to-vertical stress ratio not equal to one, which has a significant impact on the stresses distribution and displacement of the surrounding rock (Shen 2013). Therefore, it is expected that the theoretical analysis of tunnels should take the horizontal-to-vertical stress ratio into consideration (Detournay and Fairhurst 1987, Detournay and St. John 1988, Galin 1946, Simanjuntak et al. 2014. Strength criteria are another crucial component of tunnels analysis. In past decades, research was carried out by using the linear Mohr-Coulomb strength criterion (Alejano et al. 2009, Mohammad andAhmad 2015), nonlinear Hoek-Brown strength criterion (Alejano et al. 2010, Sharan 2003, and generalized Hoek-Brown strength criterion Tonon 2011, 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 , Mogi 1981 Zhang et al. 2010, Singh et al. 2017, 2018. Therefore, the Mogi-Coulomb 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. 2010, Wang et al. 2010a, which is called "strain-softening." Previous analytical solutions for tunnels in strain-softening materials have primarily considered a decrease in cohesion, whereas variations in internal friction angle have been neglected (Han et al. 2013, Wang et al. 2010b, Zhang et al. 2012. The internal friction angle also shows a decreasing trend in the post-peak stage (Li et al. 2015).
In the present study, a mechanical model for a circular tunnel subjected to a non-uniform stress field is established. A strain-softening model that simultaneously considers the weakening of cohesion and internal friction angle is proposed. Based on the strain-softening model and Mogi-Coulomb 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 post-peak 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.

Mechanical model of a circular tunnel in a non-uniform stress field
A circular tunnel of radius R0 was excavated in an infinite rock mass (Figure 1). The vertical and horizontal stresses are p0 and λp0, respectively, where λ is the horizontal-to-vertical stress ratio. A support pressure (ps) 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 Rp and Rb, respectively.

Strain-softening model
Once the stress exceeds the peak strength, the rock enters a post-peak 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 (εθp) in the strain-softening stage (Figure 2), the cohesion (cp) and internal friction angle (φp) in the plastic softening region can be expressed as: where Sc and Sφ are the softening coefficients of the cohesion and internal friction angle in the plastic softening region, respectively; c0 and φ0 are the initial cohesion and initial friction angle, respectively; and p R   is the shear strain at the interface between the elastic and plastic softening regions.
According to Eqs.
(2)-(5), the Mogi-Coulomb criterion can be rewritten as: where , and φi and ci are the cohesion and internal friction angle in different regions, respectively. The subscript symbol "i" can be replaced by "e," "p," and "b."

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: Latin American Journal of Solids and Structures, 2020, 17(1), e246 5/16 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 non-associated linear flow rule as: , which is the dilation coefficient in different regions, and i  is the dilation angle.

Stresses and displacement in the elastic region (Rp ≤ r < ∞)
The non-uniform 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)p0) and a support pressure (ps), and the stresses in the elastic region can be written as: is the radial stress on the interface between the elastic and plastic softening regions. In state II, the surrounding rock is subjected to a horizontal tension (0.5(1-λ)p0) and a vertical pressure (0.5(1-λ)p0). The boundary condition at r = R0 can be expressed as: At r = Rs, σx = -0.5(1-λ)p0, σy = 0.5(1-λ)p0, and τrθ = 0. Thus, the boundary condition at r = Rs can be obtained by the coordinate transformation: where 2      . An assumed stress function is given as: An analytical solution for stresses and deformations of tunnels in a non-uniform stress field based on strain-softening model and Mogi-Coulomb criterion Hao Fan et al.
Latin American Journal of Solids and Structures, 2020, 17(1), e246 6/16 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: In summary, the stresses in the elastic region under a non-uniform stress field can be obtained by superimposing Eq. (11) and Eq. (21): (1 ) 1 (1 4 3 )cos 2 2 2 The radial and tangential stresses in the elastic region should satisfy Eq. (6) 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:

Stresses and displacement in the plastic softening region (Rb ≤ r < Rp)
Combined with the boundary condition of p R r r    at r = Rp, the stresses in the plastic softening region can be derived by substituting Eq. (6) into Eq. (7): In the plastic softening region, the total strains can be decomposed into elastic and plastic parts as follows: are the radial and tangential strains at the interface between the elastic and plastic softening regions, respectively.
The displacement differential equation can be given by integrating Eqs. (8), (9), and (27): An analytical solution for stresses and deformations of tunnels in a non-uniform stress field based on strain-softening model and The displacement in the plastic softening region can be derived by solving Eq. (28) with the boundary condition up = ue at r = Rp: is the displacement at the interface between the elastic and plastic softening regions. The strains in the plastic softening region can be determined by substituting Eq. (29) into Eq. (8)

Stresses and displacement in the broken region (R0 ≤ r < Rb)
The stresses in the broken region can be solved by substituting Eq. (6) into Eq. (7) and considering the boundary condition σr = ps at r = R0: In the broken region, the total strains of the surrounding rock are also composed of two parts as is the displacement at r = Rb.

Post-peak region radii (Rp and Rb)
To determine the stresses and deformations of the surrounding rock, the post-peak region radii (Rp and Rb) should be solved first.

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 horizontal-to-vertical 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, c0 = 4.3 MPa, cb = 0.8 MPa, Sc = -420 MPa, φ0 = 34°, φb = 19°, Sφ = -1800°, and ψi = 10°.
The finite element program COMSOL, based on Mohr-Coulomb criterion and the strain-softening 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 Mohr-Coulomb 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.   Figure 6 shows the post-peak region radii and surface displacement of the surrounding rock. The post-peak region radii and the surface displacement (u0) vary with direction because of the non-uniform stress field. For instance, the Rp, Rb, and u0 values at θ = 0 are 5.00 m, 3.95 m, and 0.14 m, respectively; however, the Rp, Rb, and u0 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 non-uniform 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.

Horizontal-to-vertical stress ratio
To analyze the sensitivity of the horizontal-to-vertical 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 post-peak region radii and surface displacement around the tunnel with different stress ratios are shown in Figure 8. The shapes of the post-peak regions are determined by the stress ratio. Only when λ = 1 are the post-peak regions circular and the surface displacements in different directions in the tunnel equal. In addition, when λ < 1, the post-peak 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 post-peak 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 horizontal-to-vertical stress ratio exerts a significant effect on the tunnel deformations and the support design of a tunnel should consider the horizontal-to-vertical stress ratio.

Intermediate principal stress
To discuss the influence of intermediate principal stress on the post-peak region radii and surface displacement, the analytical solution in this study is compared with the results obtained using the Mohr-Coulomb criterion (Figure 9). The Rp, Rb, and u0 values computed by the Mohr-Coulomb criterion are all larger than those computed by the Mogi-Coulomb criterion. For example, the Rp, Rb, and u0 values at θ = π/2 from the Mogi-Coulomb criterion are 6.43 m, 5.24 m, and 0.28 m, respectively; whereas, the results from the Mohr-Coulomb 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 Mohr-Coulomb criterion, which causes an increased unnecessary support design. Therefore, it is more reasonable to take the intermediate principal stress into consideration in engineering applications.

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 post-peak region radii and surface displacement. The Rp, Rb, and u0 values all decrease with increasing cb and φb. For example, as cb increases from 0.8 MPa to 2.6 MPa, Rp, Rb, and u0 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 φb increases from 19° to 28°, Rp, Rb, and u0 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 post-peak region radii and surface displacement. The rock mass bearing capacity gradually increases with the increasing cb and φb. 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.

CONCLUSIONS
Based on the strain-softening model and Mogi-Coulomb criterion, a new unified solution for the stresses distribution and displacement of the surrounding rock of a tunnel in a non-uniform stress field was deduced. The conclusions can be summarized as follows: 1. (1) Due to the non-uniform stress field, the post-peak region radii and stresses distribution around the tunnel vary with direction. When λ < 1, the post-peak 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 horizontal-to-vertical stress ratio. 2.
(2) The post-peak region radii and surface displacement under the Mohr-Coulomb criterion are larger than those under the Mogi-Coulomb criterion, which is the result of neglecting the intermediate principal stress in the Mohr-Coulomb criterion. Therefore, consideration of the intermediate principal stress with an appropriate strength criterion can lead to a more reasonable tunnel support design. 3.
(3) Tunnels surrounded by rock masses with a higher residual cohesion and residual internal friction angle have lower post-peak 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.