ANewUnified Solution for Circular Tunnel Based on a Four-Stage Constitutive Model considering the Intermediate Principal Stress

Based on the triaxial test, the elasto-perfectly plastic strain-softening damage model (EPSDM) is proposed as a new four-stage constitutive model. Compared with traditional models, such as the elasto-brittle-plastic model (EBM), elasto-strain-softening model (ESM), elasto-perfectly plastic model (EPM), and elasto-peak plastic-brittle plastic model (EPBM), this model incorporates both the plastic bearing capacity and strain-softening characteristics of rockmass. Moreover, a new closed-form solution of the circular tunnel is presented for the stress and displacement distribution, and a plastic shear strain increment is introduced to define the critical condition where the strain-softening zone begins to occur.+e new analysis solution obtained in this paper is a series of results rather than one specific solution; hence, it is suitable for a wide range of rock masses and engineering structures. +e numerical simulation has been used to verify the correctness of the EPSDM. +e parametric studies are also conducted to investigate the effects of supporting resistance, residual cohesion, dilation angle, strain-softening coefficient, plastic shear strain increment, and yield parameter on the result. It is shown that when the supporting resistance is fully released, both the post-peak failure radii and surface displacement could be summarized as EBM>EPBM>ESM>EPSDM>EPM; the dilation angle in the damage zone had the highest influence on the surface displacement, whereas the dilation angle in the perfectly plastic zone had the lowest influence; the strainsoftening coefficient had themost significant effect on the damage zone radii; the EPSDM is recommended as the optimummodel for support design and stability evaluation of the circular tunnel excavated in the perfectly plastic strain-softening rock mass.


Introduction
Although the plane strain of circular holes is a relatively simple problem, it can provide effective theoretical basis for the support parameter design and stability evaluation of surrounding rock in underground engineering.erefore, it has been widely applied in the tunnel, coal mine shaft construction, oil extraction, coal gas penetration, and other projects.
In the early stage, the elastoplastic analysis of the circular tunnel was first proposed by Fenner and then corrected by Kastner.However, both of them regarded the rock mass as a perfectly elastoplastic material.In recent years, a series of studies have been carried out by using the linear Mohr-Coulomb (M-C) criterion, nonlinear Hoek-Brown (H-B) criterion, and the generalized Hoek-Brown (GHB) criterion with the associated and nonassociated flow rule.However, the influence of intermediate principal stress on the surrounding rock state was ignored [1][2][3][4][5][6][7][8][9][10][11].A large number of studies have shown that rock strength is not only related to its own characteristics but also closely related to its stress state [12][13][14].In practice, the shallow rock mass is still in the triaxial stress state under the effect of the supporting structure after tunnel excavation.en, the yield strength of rock mass increased due to the influence of intermediate principal stress, which in turn would affect the stress and deformation of surrounding rock.erefore, intermediate principal stress is crucial for tunnel design [15,16].e unified strength theory (UST) not only considers the impact of all stress components on material yield failure under different stress states but also is applicable to a variety of material mediums at good accuracy.us, it was chosen as the failure criterion of the rock material [17][18][19][20][21][22][23][24].
It is known that the elastoplastic analysis problems of surrounding rock are mainly dependent on the constitutive relations of rock mass.As a natural geological body, the rock material is easily affected by internal fissures, joints, components, and external environment.erefore, its constitutive relation is extremely complicated.As shown in Figure 1, the elasto-brittle-plastic model (EBM), elasto-strain-softening model (ESM), and elasto-perfectly plastic model (EPM) were usually used to research this problem [1, 4-10, 22, 25-30].However, based on the geotechnical quality, Hoek and Brown pointed out that the EBM was applied only to the poor-quality rock mass, while EPM and ESM were suitable for the highquality and average-quality rock masses, respectively [31].
en, Zhang et al. [11] and Jiang et al.'s [32,33] experiment showed that a plastic bearing zone before brittle failure for the rock material under high confining pressures could exist.
us, the elasto-peak plastic-brittle plastic model (EPBM) was presented (Figure 1(d)) and applied to the Jinping II hydropower station engineering.Nevertheless, when the confining pressures were 5 MPa and 40 MPa in Zhang's triaxial test, the brittle rock material showed obvious strain-softening characteristics from the peak plastic state to the brittle plastic state.erefore, it may not be reasonable to simplify the EPBM.As shown in Figure 2, a large number of rock masses firstly showed the strain-softening characteristics after the peak plastic zone and then entered the residual flow stage.erefore, according to the total stress-strain curve, the elasto-perfectly plastic strain-softening damage model (EPSDM) was proposed in this paper and then applied to the engineering practice.Actually, the EPSDM includes all the features of the EPM, EBM, ESM, and EPBM and can be transformed into the above four models under certain conditions.Hence, it can be regarded as a unified constitutive model.
Based on the triaxial test, the elasto-perfectly plastic strain-softening damage model (EPSDM) is proposed as a new four-stage constitutive model.en, a new closedform solution of the circular opening is deduced based on UST and EPSDM, and a plastic shear strain increment is introduced to determine the critical condition that the strain-softening zone begins to develop.e correctness of the solution has been verified by making a comparative analysis of Pan's numerical simulation results [23].Finally, the effect of the parameters is also discussed.

Problem Description
2.1.e Establishment of EPSDM. Figure 3 shows a circular tunnel being excavated in a unified, continuous, isotropic UST rock mass subjected to a hydrostatic pressure (σ 0 ) at infinity boundary and a supporting resistance (p in ) at inner radius (a).As p in gradually reduces, the rock mass begins to enter the perfectly plastic stage ("AB") when the maximum and the minimum principal stresses satisfy the initial yield conditions.
is stage is not an infinite extension whose range should be restricted by some factors.Assuming the plastic shear strain increment of the perfectly plastic zone reaches a certain value, the rock mass will begin to enter the strain-softening stage ("BC") where the rock mass cohesion gradually decreases.Until a residual value is reached, the rock mass starts to enter the damage stage and finally achieves a equilibrium state under the supporting resistance.Meanwhile, the radii of the perfectly plastic zone, strainsoftening zone, and damage zone are denoted as R p , R s , and R c , respectively.

Unified Strength eory (UST).
Based on the twin shear yield criterion, the unified strength theory is established by considering the influence of all the stress components and intermediate principal stress on the material yield failure [19][20][21][22]24]. e UST is a series of yield criteria and failure criteria, which can be adopted for most materials.In geotechnical engineering, the cohesion (c) and the internal friction angle (φ) are usually used to represent this yield theory.In general, the compressive stress is positive and the tensile stress is negative.erefore, the UST yield function can be expressed as follows [24]: where F and F ′ represent the yield function; σ Elasto-brittle-plastic model (EBM) Elasto-strain-so ening model (ESM) Elasto-peak plastic-brittle plastic model (EPBM)

Advances in Civil Engineering
Under axisymmetric plane strain conditions, the relationship among three principal stresses is generally expressed as the parameter n [15,22]: For rock materials, n generally satisfies 2] ≤ n ≤ 1.When n → 1, σ 2 becomes closer to the average of maximum and minimum principal stresses; ] is Poisson's ratio.In order to simplify the calculation, we take n � 1. σ 2 ≥ ((σ 1 + σ 3 )/2) − ((σ 1 − σ 3 )/2)sin φ can be judged by substituting σ 2 � (σ 1 + σ 3 )/2 into (1) or (2).erefore, UST can be rewritten as where For axisymmetric plane strain problems, when p in < σ 0 , the circumferential and radial stresses are, respectively, the maximum and minimum principal stresses.In other words, σ 1 � σ θ and σ 3 � σ r , and substituting them into (4), the UST can be rewritten as ) and ε p 1 represent the compressive strength and the maximum principal strain in the strain-softening zone, respectively; c r represents residual cohesion.It should be noted that this paper assumes that the strain-softening process of rock mass is only related to the cohesion attenuation, which is independent of the internal friction angle.

Definition of Dilatancy Coefficient.
As a natural geological body, many microcracks exist in the rock material, which make its post-peak stage appear nonlinear and dilatant, and then, the volume will also change.It is generally believed that the plastic deformation satisfies the nonassociated flow rule, determined by the plastic potential function [8,10,11,22].e plastic potential function f and the yield function F ′ have the same expression form.erefore, this paper assumes that the plastic potential function is as follows: where , in which ψ i represents the dilation angle in the "i" zone.
According to the plastic potential theory, the relationship between stress and strain increment is satisfied: where dε p ij and σ ij represent the plastic strain increment and stress tensor, respectively, and dλ represents the nonnegative proportional constant.From ( 8) and ( 9), the plastic principal strain increment can be obtained as e dilatancy coefficient of the post-peak failure zone can be determined by the ratio of the minimum plastic principal strain increment to the maximum plastic principal strain increment.en, the relationship between the plastic principle strain and the dilatancy coefficient is obtained as follows: where ε ri and ε θi are, respectively, the radial and tangential strain.

e Basic Equation.
e equilibrium differential equation for the axisymmetric plane strain problem in the "i" zone can be expressed as (ignoring the body force of rock mass) e geometric equation, based on the small deformation assumption, can be denoted as where u ri is the radial displacement of surrounding rock.e stress boundary conditions can be described as

Stresses and Displacement in the Elastic Zone.
Based on the elasticity theory, the stress function for the axisymmetric plane strain problem in the elastic zone can be expressed as where A and B are the unknown constants.e radial and tangential stresses are then given by Substituting the boundary conditions σ re � σ 0 at r → ∞ and σ re � σ R p at r � R p into (5), where σ R p is the radial stress on the interface between the elastic zone and the perfectly 4 Advances in Civil Engineering plastic zone, the specific expressions of ( 16) and ( 17) can be obtained as e radial and tangential stresses in the elastic zone should satisfy (5) at r � R p .us, σ R p can be deduced by substituting (18) and ( 19) into (5): According to the small deformation theory, the displacement and strain can also be expressed as follows [28]: where

Stresses and Displacement in the Perfectly Plastic
Zone.Combined with the boundary condition σ rp � σ R p at r � R p , the stresses in the perfectly plastic zone can be derived by substituting ( 5) into (12): en, the radial contact stress (σ R s ) on the interface between the perfectly plastic zone and the strain-softening zone can be easily obtained by substituting r � R s into (23): Substituting ( 13) into (11), the differential equation of the radial displacement in the perfectly plastic zone should be given as Solving (25), the displacement and strain can be deduced by considering the boundary condition u rp � u R p at r � R p : When σ 0 ≥ p in , the tangential strain is the maximum principal strain.Consequently, the displacement and maximum principal strain on the interface of the perfectly plastic strain-softening zone can be obtained by substituting r � R s into (26) as follows:

Stresses and Displacement in the Strain-Softening
Zone.Based on the calculation method of the perfectly plastic zone, combined with the boundary condition u rs � u R s at r � R s , we can easily get the expressions of the displacement and strain in the strain-softening zone: According to (29), the radial displacement on the interface between the strain-softening zone and the damage zone can also be obtained easily: In the strain-softening zone, it is assumed that the strength attenuation of rock mass is only related to cohesion (c).
en, the compressive strength at any point can be expressed as where β � tan θ/E, which can be defined as a strainsoftening coefficient.tan θ may be called the strain-softening modulus, which can be determined by the slope of the "BC" segment in Figure 3. E is Young's modulus, which can be determined by the slope of the "OA" segment.erefore, the parameter β can be determined by the ratio of the slope of the "BC" segment to the slope of the "OA" segment in Figure 3. Introducing ( 28) and ( 29) into (31), it can be rewritten as Combined with the boundary condition σ rs � σ R s at r � R s , the stresses in the strain-softening zone can be derived by substituting (32) and ( 6) into the equilibrium equation (12): (33)

Stresses and Displacement in the Damage
Zone.Introducing (7) into the equilibrium equation ( 12), the stresses in the damage zone can be solved by considering the boundary condition σ rc � p in at r � a: As shown in (29), combined with the boundary condition (u rc ) r�R c � u R c , the displacement and strain in the damage zone can also be expressed as According to (35a), when r � a, the surface displacement of surrounding rock is as follows: 3.6.e Radius of the Post-Peak Failure Zone (R p , R s , and R c ).In order to obtain the closed solution of the stresses and deformation of the EPSDM, the post-peak failure zone radii (R c , R s , and R p ) should be firstly determined.According to the relationship among R p , R s , and R c , the EPSDM can be converted to the following 5 cases including its original form.Case 1. From (32), when β � 0 and σ c � σ c � σ R c , the post-peak failure radii satisfy the relation a � R c � R s < R p , and then, the surrounding rock only consists of the elastic zone and the perfectly plastic zone.erefore, the EPSDM's solution can be converted to the EPM's result.Combining with (23) and the stress boundary condition (σ rp ) r�a � p in , the perfectly plastic zone radius R p can be obtained: Case 2. When a < R c < R s < R p , in order to get the closed solution of R p , R s , and R c , it is required to establish three linear independent equations to solve the problem.According to the stress contact condition (σ rs ) r�R c � (σ rc ) r�R c , one of the relationships can be established by combining with (33) and (34): en, the slope of the line "BC" in the strain-softening zone can be expressed as Apart from (37) and (38), a new condition is still needed to make the EPSDM form a closed solution.It is generally believed that the extension of the perfectly plastic zone is not infinite and will be restricted by other conditions.Zhang et al. [11] consider that when the plastic shear strain or the equivalent plastic shear strain reaches a critical value, the 6 Advances in Civil Engineering rock mass begins to enter the damage zone; Wei et al. and Jiang et al. [32,33] believe that another relationship between R p and R c may be established by the radial strain continuous condition (ε rc ) r�R c � (ε rp ) r�R c .In this paper, it is assumed that when the plastic shear stress increment Δc * between the yield points "A" and "B" (Figure 3) reaches a certain value, the rock mass starts to enter the strain-softening state.Accordingly, the other relationship among R p , R s , and R c can be expressed as where c * s and c * p represent the plastic shear strain at points "B" and "A," respectively.As shown in Figure 3, in the triaxial test, the maximum and minimum principal strain of rock mass can be easily obtained, and then, In addition, ε Peak ss and ε Peak pp represent the minimum principal strain at the points "B" and "A," respectively.Substituting ( 26) into (41), it can be rewritten as where T � [(Δc * /(A 0 (1 + η p ))) + 1] 1/(1+η p ) .Substituting (40) into (38), the following expression can also be obtained: . Substituting (40) and ( 41) into (37), the damage zone radius is expressed as Subsequently, the radii R s and R p can also be obtained by substituting (42) into (41) and (40), respectively.
Case 3. From (40), when Δc * � 0, the limit of T is equal to one.
en, the post-peak failure radii satisfy the relation a < R c < R s � R p , and the surrounding rock is composed of the elastic zone, strain-softening zone, and damage zone.
erefore, the EPSDM's solution is converted to the ESM's result.Combining with (41) and (42), the radius expression of R c and R s can be deduced: where 41), when β → ∞, the limit of t is equal to one.en, the post-peak failure zone radii satisfy the relation and the surrounding rock is parted into three zones: the elastic zone, perfectly plastic zone, and damage zone.erefore, the EPSDM's solution is converted to the EPBM's result.Combining with (40) and (42), the post-peak failure radii R p and R c can be obtained as follows: Case 5. When Δc * � 0 and β → ∞, the limit of T and t is equal to one.en, the post-peak failure radii satisfy the relation a < R s � R c � R p , and the surrounding rock only includes the elastic zone and damage zone.erefore, the EPSDM's solution is converted to the EBM's result.According to (42), the damage zone radius R c will be given as (45)

Transformation Relationship between EPSDM and the Other Models.
Compared with the traditional solution, the influences of intermediate principal stress and the dilatancy coefficient on the surrounding rock state are considered.e solutions based on M-C and GTSS yield criteria can be generated from the new closed-form solution using b � 0 and b � 1.0, respectively.In addition, compared with the traditional EBM, EPM, ESM, and EPBM, the EPSDM includes all the features of the above models.When β � 0, the model is transformed into the EPM.When Δc * � 0 and β → ∞, the model is transformed into the EBM.Only when Δc * � 0, the model is transformed into the ESM.Only when β → ∞, the model is transformed into the EPBM.erefore, the EPSDM has more extensive practicality and theoretical values than other models.Detournay [1], Xu and Yu [15], and Li et al. [28] all studied the same issue, but Detournay's solution based on the M-C yield criterion neglected the influence of intermediate principal stress, whereas the solutions by Xu and Li neglected the influence of rock dilatancy on the radial displacement.
erefore, the solution may have a large deviation between the calculation results and the practical situation.
is paper takes account of the intermediate principal stress and the dilation.When b � 0, η c � η s � η p � 1.0, and Δc * � 0, the EPSDM's solution is Li et al.'s solution [28].When β � 0 and η c � η s � η p � 1.0, the EPSDM's solution is Xu and Yu's result [15].erefore, the solutions by Xu and Li are all special cases of this paper.

Validation of Model Correctness.
Based on the elastoperfectly plastic constitutive model, Pan et al. [23] introduced the UST into the finite element program ABAQUS and then researched the stress distribution of surrounding rock in the deep circular tunnel with b � 0, 0.5, and 1.0.
e simulated rock material and model geometry parameters are shown in Table 1.To satisfy the numerical simulation conditions, the calculation uses the associated flow law (ψ i � φ i ), β � 0, and η i � (1 + sin ψ i )/(1 − sin ψ i ). e comparison between the calculation and simulation results is shown in Figure 4.
It can be seen from Figure 4 that the stress distribution around the circular tunnel is basically consistent with the Pan's numerical simulation results (b � 0, 0.5, and 1.0).erefore, it confirms the correctness of the stress solution.

Case I: Stress and Deformation Evolution Law of the EPSDM.
e stresses and deformation are the important basis for evaluating the stability of surrounding rock and the reliability of support design.In order to study the evolution law of stresses and displacement under different constitutive models, the mechanical and geometrical parameters of soft rock are shown in Table 2, which is available by Ogawa and Lo [2].In addition, b � 0, Δc * � 2.5 × 10 −5 , and β � 2.5.
e stress distribution of the EPSDM under different critical support resistance is shown in Figure 5.As shown in Table 3 and Figure 5, it can be seen that when p in /σ 0 ≥ 0.200, the rock mass only consists of the elastic zone.When 0.152 ≤ p in /σ 0 < 0.200, the perfectly plastic zone begins to occur, and then, the rock mass is composed of the elastic zone and the perfectly plastic zone.While 0.028 ≤ p in /σ 0 < 0.152, the strain-softening zone is gradually formed and the rock mass is composed of three parts: the elastic zone, perfectly plastic zone, and strain-softening zone.Once p in /σ 0 < 0.028, the damaged zone begins to develop and the rock mass finally displays four zones: elastic zone, perfectly plastic zone, strainsoftening zone, and damage zone.
Figures 6 and 7 illustrate the influence of support resistance on the post-peak failure radii (R p , R s , and R c ) and surface displacement.In addition, when p in � 0, the postpeak failure radii and surface displacement are as shown in  30 8 Advances in Civil Engineering Table 4. From the above two gures, it can be seen that once the support resistance satis es 0.152 ≤ p in /σ 0 < 0.200, the calculated results based on the EPSDM are the same as those of the EPM and EPBM.However, once the plastic shear strain increment is larger than the critical plastic shear strain increment, Δc * 2.5 × 10 −5 , and then, the strain-softening zone begins to develop.When the residual cohesion reaches a residual value, p in /σ 0 0.028 and the rock mass begins to enter the damage stage.As the support resistance is completely released, then R p /a Compared with the ESM and EPBM, the EPSDM represents the in uence of the plastic bearing properties and strain-softening characteristics on the surrounding rock state.In fact, most of the fractured and jointed rocks are prone to strain-softening characteristics after perfectly plastic.erefore, in the deep underground engineering, the design of support parameters by using this paper's EPSDM may be more reasonable.Advances in Civil Engineering 9   e excavation radius a is 3.0 m, Young's modulus E is 30.0MPa, Poisson's ratio ] is 0.25, strain-softening coe cient β is 1.25, initial internal friction angle φ is 30 °, and initial cohesion c and residual cohesion c R are 5.85 MPa and 2.0 MPa, respectively.e plastic shear strain increment Δc * is 0.0001.In addition, the yield parameter b is zero, and the initial dilation angle are all ψ p ψ s ψ c 15 °. 8 and 9 indicate the e ect of residual cohesion on the post-peak failure radii and surface displacement.With the continuous increase of the residual cohesion, the post-peak failure radii and surface displacement depict a nonlinear decrease characteristic.For instance, when c r transforms from 1.0 MPa to 1.6 MPa, the dimensionless radii R p /a, R s /a, and R c /a decrease by 0.222, 0.217, and 0.159, with reduction of 11.78%, 11.74%, and 10.25%, respectively; the surface displacement decreases by 1.1 mm, with reduction of 28.21%.e above data show that the residual cohesion has an important in uence on the surrounding rock state, especially for the extremely broken rock mass.

e In uence of Residual Cohesion. Figures
In addition, as shown in Figure 9, it can be seen that the surface displacement gradually increases with the increase of the dilatancy angle. is is mainly because the larger the dilatancy angle is, the higher the dilatancy coe cient is and the greater the volume expansion of surrounding rock is.Meanwhile, the in uence of the dilatancy angle in the damage zone on the surface displacement is the most signi cant, whereas the dilatancy angle in the perfectly plastic zone has the lowest in uence.
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e In uence of Plastic Shear Strain Increment.
e e ect of plastic shear strain increment on the post-peak failure radii and surface displacement is presented in Figures 10 and 11, respectively.As Δc * gradually increases, the dimensionless radii R p /a, R s /a, and R c /a and the displacement u 0 /a all present the approximate linear decrease characteristics.Compared with R p /a and R c /a, the reduction rate of R s /a is the largest.For example, when Δc * increases from 0.0001 to 0.0005, R p /a, R s /a, and R c /a are, respectively, reduced to 5.72%, 11.92%, and 10.08%; u 0 /a is reduced to 16.67%.e above data show that Δc * has a crucial in uence on the post-peak failure radii and surface displacement.e plastic bearing capacity of rock mass gradually increases with the increasing Δc * .As a result, the supporting resistance to maintain the stability of surrounding rock gradually decreases.12 and 13 indicate the e ect of strain-softening coe cient on the post-peak failure radii and surface displacement, respectively.It can be seen that the change of β exerts important e ects on R p , R s , R c , and u 0 .As β increases, the dimensionless radii R p /a, R s /a, and R c /a and the displacement u 0 /a all increase.For instance, β increases from 2/3 to 2 and R p /a, R s /a, R c /a, and u 0 /a increase by 0.133, 0.130, 0.300, and 0.0005, respectively.Nevertheless, β increases from 2 to 10/3 and R p /a, R s /a, R c /a, and u 0 /a only increase by 0.026, 0.026, 0.077, and 0.0002, respectively.e above data also show that the strain-softening coe cient has the most signi cant in uence on the damage zone radii.12 Advances in Civil Engineering R s /a, R c /a, and u 0 /a, respectively, decrease by 0.205, 0.197, 0.175, and 0.0005.Moreover, as shown in Figure 14, it is indicated that the parameter b has important in uences on the stress distribution of surrounding rock as well.As parameter b gradually decreases, the maximum circumferential stress σ θ gradually keeps a decreasing change and transfers to the deep rock mass.erefore, the design of the support scheme should take into account the parameter b.

Conclusion
Taking into account the plastic bearing capacity of strainsoftening rock mass, a new four-stage constitutive model was proposed.Meanwhile, a plastic shear strain increment was introduced to determine the critical condition where the strain-softening zone begins to develop.en, based on the uni ed strength theory and four-stage constitutive model, a new closed-form solution of the circular opening was proposed.Compared with the numerical simulation results, the validity of the solution has been veri ed and several conclusions could therefore be drawn: (1) As the supporting resistance is fully released, both the post-peak failure radii and surface displacement could be summarized as EBM > EPBM > ESM > EPSDM > EPM.Meanwhile, once the plastic shear strain increment is zero, the calculated results of the EPSDM could be converted to the ESM's solution.
Similarly, when strain-softening coe cient was zero or was large enough, the EPSDM turned to the EPM and EPBM, respectively.As the plastic shear strain increment was zero and the strain-softening coe cient was large enough, the calculated results of the EPSDM was found to be in accordance with the closed-form solution of the EBM.(2) Compared with the above four models, the EPSDM re ects all the features of them, which represent the in uence of the plastic bearing properties and strainsoftening coe cient on the surrounding rock state.erefore, the design of support parameters by using the EPSDM may be more reasonable for the perfectly plastic strain-softening rock mass.
(3) e residual cohesion, plastic shear strain increment, and yield parameter b are negatively correlated with the post-peak failure radii and surface displacement; however, the dilatancy angle and strain-softening coe cient are positively correlated.On the other hand, it was also clear that the dilatancy angle in the damage zone had the highest in uence on the surface displacement, whereas the dilatancy angle in the perfectly plastic zone had the lowest in uence.e strain-softening coe cient had the most signi cant in uence on the damage zone radii.In conclusion, the research can provide an important theoretical basis for the design and stability analysis of the circular tunnel excavated in the perfectly plastic strain-softening rock mass.

Figure 5 :
Figure 5: Stress distribution law of the EPSDM under di erent critical support resistance.

Figure 6 :
Figure 6: e in uence of support resistance on the post-peak failure radii.

igure 9 :
e values of u 0 under di erent c r and ψ i .

Figure 8 :Figure 7 :
Figure 8: e values of R p , R s , and R c under di erent c r .

Figure 10 :Figure 11 :Figure 12 :
Figure 10: e values of R p , R s , and R c under di erent Δc * .

Figure 14 :
Figure 14: e in uence of the parameter b on the stresses of surrounding rock.

Table 1 :
Geometrical and physical parameters of the circular tunnel.
1.165 for the EPM; R p /a 1.602 for the EBM; R p /a 1.418 and R c /a 1.218 for the ESM; R p /a 1.540 and R c /a 1.492 for the EPBM; and R p /a 1.165, R s /a 1.295, and R c /a 1.122 for the EPSDM.In addition, the dimensionless radial displacement 2u 0 G/[a(σ 0 − σ R p )] of the EPM, EBM, ESM, EPBM, and EPSDM is 1.357, 2.565, 2.010, 2.371, and 1.786, respectively.

Table 4 :
e post-peak failure radii and surface displacement when

Table 5 :
e in uence of the parameter b on the post-peak failure radii and surface displacement.