# A New Unified Solution for Circular Tunnel Based on a Four-Stage Constitutive Model considering the Intermediate Principal Stress

Authors: Liang Chen; Xianbiao Mao; Yanlong Chen; Ming Li; Yang Hao; Ding Liu
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

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 rock mass. 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. The 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. The numerical simulation has been used to verify the correctness of the EPSDM. The 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 strain-softening coefficient had the most significant effect on the damage zone radii; the EPSDM is recommended as the optimum model for support design and stability evaluation of the circular tunnel excavated in the perfectly plastic strain-softening rock mass.


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## Body

## 1. 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. Therefore, 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 [1–11]. 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 [12–14]. In practice, the shallow rock mass is still in the triaxial stress state under the effect of the supporting structure after tunnel excavation. Then, 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. Therefore, intermediate principal stress is crucial for tunnel design [15, 16]. The 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. Thus, it was chosen as the failure criterion of the rock material [17–24].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. Therefore, its constitutive relation is extremely complicated. As shown in Figure1, 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 high-quality and average-quality rock masses, respectively [31]. Then, 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. Thus, 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. Therefore, 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. Therefore, 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.Figure 1
Previous research model.
(a)
(b)
(c)
(d)Figure 2
Total stress-strain curve of rock mass in the local area of China: (a) Yangzhuang coal mine in Henan Province; (b) Wushan area of Chongqing; (c) Ya’an area of Sichuan.
(a)
(b)
(c)Based on the triaxial test, the elasto-perfectly plastic strain-softening damage model (EPSDM) is proposed as a new four-stage constitutive model. Then, a new closed-form 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. The 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.


## 2. Problem Description

### 2.1. The Establishment of EPSDM

Figure3 shows a circular tunnel being excavated in a unified, continuous, isotropic UST rock mass subjected to a hydrostatic pressure (σ0) at infinity boundary and a supporting resistance (pin) at inner radius (a). As pin 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. This 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, strain-softening zone, and damage zone are denoted as Rp, Rs, and Rc, respectively.Figure 3
EPSDM of the circular tunnel.


### 2.2. Unified Strength Theory (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 [19–22, 24]. The 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. Therefore, the UST yield function can be expressed as follows [24]:whenσ2≤σ1+σ3/2−σ1−σ3/2sinφ,(1)F=1−sinφ1+sinφσ1−bσ2+σ31+b=2ccosφ1+sinφ.Whenσ2≥σ1+σ3/2−σ1−σ3/2sinφ,(2)F′=1−sinφ1+b1+sinφσ1+bσ2−σ3=2ccosφ1+sinφ,where F and F′ represent the yield function; σ1, σ2, and σ3 represent the maximum, intermediate, and minimum principal stresses, respectively; b represents the yield parameter related to the intermediate principal stress, which can reflect the effect of the intermediate principal shear stress and the positive stress on the yield failure of the rock material, and 0≤b≤1. When b=0, UST translates into the M-C strength criterion; when b=1, UST translates into the general twin shear strength (GTSS) criterion; when 0<b<1, UST translates into a series of other ordered new strength criteria.Under axisymmetric plane strain conditions, the relationship among three principal stresses is generally expressed as the parametern [15, 22]:(3)σ2=nσ1+σ32.For rock materials,n generally satisfies 2ν≤n≤1. When n→1, σ2 becomes closer to the average of maximum and minimum principal stresses; ν is Poisson’s ratio. In order to simplify the calculation, we take n=1. σ2≥σ1+σ3/2−σ1−σ3/2sinφ can be judged by substituting σ2=σ1+σ3/2 into (1) or (2). Therefore, UST can be rewritten as(4)σ1=Npσ3+σc,where Np=2+b+2+3bsinφ/2+b1−sinφ and σc=41+bccosφ/2+b1−sinφ.For axisymmetric plane strain problems, whenpin<σ0, the circumferential and radial stresses are, respectively, the maximum and minimum principal stresses. In other words, σ1=σθ and σ3=σr, and substituting them into (4), the UST can be rewritten as(5)σθ=Npσr+σcfor  perfectly  plastic  stage,(6)σθ=Npσr+σc¯ε1pfor  strain  softening  stage,(7)σθ=Npσr+σcRfor  damage  stage,where σcR=41+bcrcosφ/2+b1−sinφ; σc¯ε1p and ε1p represent the compressive strength and the maximum principal strain in the strain-softening zone, respectively; cr 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.


### 2.3. 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]. The plastic potential function f and the yield function F′ have the same expression form. Therefore, this paper assumes that the plastic potential function is as follows:(8)fσθi,σri=σθi−ηiσri,where ηi=2+b+2+3bsinψi/2+b1−sinψi, in which ψi represents the dilation angle in the “i” zone.According to the plastic potential theory, the relationship between stress and strain increment is satisfied:(9)dεijp=dλ∂f∂σij,where dεijp and σij 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(10)dεθip=dλ∂f∂σθi=dλ,dεrip=dλ∂f∂σri=−ηidλ.The 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. Then, the relationship between the plastic principle strain and the dilatancy coefficient is obtained as follows:(11)εrip+ηiεθip=0,where εri and εθi are, respectively, the radial and tangential strain.


## 2.1. The Establishment of EPSDM

Figure3 shows a circular tunnel being excavated in a unified, continuous, isotropic UST rock mass subjected to a hydrostatic pressure (σ0) at infinity boundary and a supporting resistance (pin) at inner radius (a). As pin 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. This 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, strain-softening zone, and damage zone are denoted as Rp, Rs, and Rc, respectively.Figure 3
EPSDM of the circular tunnel.


## 2.2. Unified Strength Theory (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 [19–22, 24]. The 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. Therefore, the UST yield function can be expressed as follows [24]:whenσ2≤σ1+σ3/2−σ1−σ3/2sinφ,(1)F=1−sinφ1+sinφσ1−bσ2+σ31+b=2ccosφ1+sinφ.Whenσ2≥σ1+σ3/2−σ1−σ3/2sinφ,(2)F′=1−sinφ1+b1+sinφσ1+bσ2−σ3=2ccosφ1+sinφ,where F and F′ represent the yield function; σ1, σ2, and σ3 represent the maximum, intermediate, and minimum principal stresses, respectively; b represents the yield parameter related to the intermediate principal stress, which can reflect the effect of the intermediate principal shear stress and the positive stress on the yield failure of the rock material, and 0≤b≤1. When b=0, UST translates into the M-C strength criterion; when b=1, UST translates into the general twin shear strength (GTSS) criterion; when 0<b<1, UST translates into a series of other ordered new strength criteria.Under axisymmetric plane strain conditions, the relationship among three principal stresses is generally expressed as the parametern [15, 22]:(3)σ2=nσ1+σ32.For rock materials,n generally satisfies 2ν≤n≤1. When n→1, σ2 becomes closer to the average of maximum and minimum principal stresses; ν is Poisson’s ratio. In order to simplify the calculation, we take n=1. σ2≥σ1+σ3/2−σ1−σ3/2sinφ can be judged by substituting σ2=σ1+σ3/2 into (1) or (2). Therefore, UST can be rewritten as(4)σ1=Npσ3+σc,where Np=2+b+2+3bsinφ/2+b1−sinφ and σc=41+bccosφ/2+b1−sinφ.For axisymmetric plane strain problems, whenpin<σ0, the circumferential and radial stresses are, respectively, the maximum and minimum principal stresses. In other words, σ1=σθ and σ3=σr, and substituting them into (4), the UST can be rewritten as(5)σθ=Npσr+σcfor  perfectly  plastic  stage,(6)σθ=Npσr+σc¯ε1pfor  strain  softening  stage,(7)σθ=Npσr+σcRfor  damage  stage,where σcR=41+bcrcosφ/2+b1−sinφ; σc¯ε1p and ε1p represent the compressive strength and the maximum principal strain in the strain-softening zone, respectively; cr 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.


## 2.3. 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]. The plastic potential function f and the yield function F′ have the same expression form. Therefore, this paper assumes that the plastic potential function is as follows:(8)fσθi,σri=σθi−ηiσri,where ηi=2+b+2+3bsinψi/2+b1−sinψi, in which ψi represents the dilation angle in the “i” zone.According to the plastic potential theory, the relationship between stress and strain increment is satisfied:(9)dεijp=dλ∂f∂σij,where dεijp and σij 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(10)dεθip=dλ∂f∂σθi=dλ,dεrip=dλ∂f∂σri=−ηidλ.The 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. Then, the relationship between the plastic principle strain and the dilatancy coefficient is obtained as follows:(11)εrip+ηiεθip=0,where εri and εθi are, respectively, the radial and tangential strain.


## 3. Analytical Solution of EPSDM

### 3.1. The Basic Equation

The equilibrium differential equation for the axisymmetric plane strain problem in the “i” zone can be expressed as (ignoring the body force of rock mass)(12)dσridr+σri−σθir=0.The geometric equation, based on the small deformation assumption, can be denoted as(13)εri=duridr,εθi=urir,where uri is the radial displacement of surrounding rock. The stress boundary conditions can be described as(14)σre=σ0,r→∞σre=σrp,r=Rpσrp=σrs,r=Rsσrs=σrc,r=Rcσrs=pin,r=a.


### 3.2. 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(15)Φ=Alnr+Br2,where A and B are the unknown constants. The radial and tangential stresses are then given by(16)σre=1rdϕdr=Ar2+2B,(17)σθe=d2ϕdr2=−Ar2+2B.Substituting the boundary conditionsσre=σ0 at r→∞ and σre=σRp at r=Rp into (5), where σRp is the radial stress on the interface between the elastic zone and the perfectly plastic zone, the specific expressions of (16) and (17) can be obtained as(18)σre=σ0+σRp−σ0Rp2r2,(19)σθe=σ0−σRp−σ0Rp2r2.The radial and tangential stresses in the elastic zone should satisfy (5) at r=Rp. Thus, σRp can be deduced by substituting (18) and (19) into (5):(20)σRp=2σ0−σc1+Np.According to the small deformation theory, the displacement and strain can also be expressed as follows [28]:(21)ure=A0Rp2r,εre=−A0Rp2r2,εθe=A0Rp2r2,where A0=1+νσ0−σRp/E. When r=Rp, the displacement on the interface is(22)uRp=A0Rp.


### 3.3. Stresses and Displacement in the Perfectly Plastic Zone

Combined with the boundary conditionσrp=σRp at r=Rp, the stresses in the perfectly plastic zone can be derived by substituting (5) into (12):(23)σrp=2σ01−Np−2σc1−Np2rRpNp−1+σc1−Np,σθp=Np2σ01−Np−2σc1−Np2rRpNp−1+σc1−Np.Then, the radial contact stress (σRs) on the interface between the perfectly plastic zone and the strain-softening zone can be easily obtained by substituting r=Rs into (23):(24)σRs=2σ01−Np−2σc1−Np2RsRpNp−1+σc1−Np.Substituting (13) into (11), the differential equation of the radial displacement in the perfectly plastic zone should be given as(25)durpdr+ηpurpr=0.Solving (25), the displacement and strain can be deduced by considering the boundary condition urp=uRp at r=Rp:(26)urp=A0Rp1+ηpr−ηp,εrp=−A0ηpRp1+ηpr−ηp−1,εθp=A0Rp1+ηpr−ηp−1.Whenσ0≥pin, 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=Rs into (26) as follows:(27)uRs=A0Rp1+ηpRs−ηp,(28)εsPeak=A0Rp1+ηpRs−ηp−1.


### 3.4. Stresses and Displacement in the Strain-Softening Zone

Based on the calculation method of the perfectly plastic zone, combined with the boundary conditionurs=uRs at r=Rs, we can easily get the expressions of the displacement and strain in the strain-softening zone:(29)urs=A0Rp1+ηpRsηs−ηpr−ηs,εrs=−A0ηsRp1+ηpRsηs−ηpr−ηs−1,εθs=A0Rp1+ηpRsηs−ηpr−ηs−1.According to (29), the radial displacement on the interface between the strain-softening zone and the damage zone can also be obtained easily:(30)uRc=A0Rp1+ηpRsηs−ηpRc−ηs.In the strain-softening zone, it is assumed that the strength attenuation of rock mass is only related to cohesion (c). Then, the compressive strength at any point can be expressed as(31)σc¯ε1p=σc−β1+νσ0−σRpεθsr−εsPeak,where β=tanθ/E, which can be defined as a strain-softening coefficient. tanθ 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. Therefore, 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(32)σc¯=σc−β1+νσ0−σRpRp1+ηpRs−ηp−1Rsr1+ηs−1.Combined with the boundary conditionσrs=σRs at r=Rs, the stresses in the strain-softening zone can be derived by substituting (32) and (6) into the equilibrium equation (12):(33)σrs=2σ01−Np−2σc1−Np2Rpr1−Np+σc1−Np+β1+νσ0−σRpηs+NpRpRs1+ηpRsr1+ηs−Rsr1−Np+β1+νσ0−σRp1−NpRpRs1+ηp1−Rsr1−Np,σθs=Np2σ01−Np−2σc1−Np2Rpr1−Np+σc1−Np+β1+νσ0−σRpηs+NpRpRs1+ηpRsr1+ηs−Rsr1−Np+β1+νσ0−σRp1−NpRpRs1+ηp1−Rsr1−Np+σc−β1+νσ0−σRpRpRs1+ηpRsr1+ηs−1.


### 3.5. 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 σrc=pin at r=a:(34)σrc=1−Nppin−σcR1−NpraNp−1+σcR1−Np,σθc=Np1−Nppin−σcR1−NpraNp−1+σcR1−Np.As shown in (29), combined with the boundary condition urcr=Rc=uRc, the displacement and strain in the damage zone can also be expressed as(35a)urc=A0Rp1+ηpRsηs−ηpRcηc−ηsr−ηc,εrc=−ηcA0Rp1+ηpRsηs−ηpRcηc−ηsr−ηc−1,εθc=A0Rp1+ηpRsηs−ηpRcηc−ηsr−ηc−1.According to (35a), when r=a, the surface displacement of surrounding rock is as follows:(35b)u0=A0Rp1+ηpRsηs−ηpRcηc−ηsa−ηc.


### 3.6. The Radius of the Post-Peak Failure Zone (Rp, Rs, and Rc)

In order to obtain the closed solution of the stresses and deformation of the EPSDM, the post-peak failure zone radii (Rc,Rs,andRp) should be firstly determined. According to the relationship among Rp, Rs, and Rc, the EPSDM can be converted to the following 5 cases including its original form.Case 1. From (32), when β=0 and σc¯=σc=σcR, the post-peak failure radii satisfy the relation a=Rc=Rs<Rp, and then, the surrounding rock only consists of the elastic zone and the perfectly plastic zone. Therefore, the EPSDM’s solution can be converted to the EPM’s result. Combining with (23) and the stress boundary condition σrpr=a=pin, the perfectly plastic zone radius Rp can be obtained:(36)Rp=apin−σc/1−Np2σ01−Np−2σc/1−Np21/1−Np.Case 2. When a<Rc<Rs<Rp, in order to get the closed solution of Rp, Rs, and Rc, it is required to establish three linear independent equations to solve the problem. According to the stress contact condition σrsr=Rc=σrcr=Rc, one of the relationships can be established by combining with (33) and (34):(37)2σ01−Np−2σc1−Np2RpRc1−Np+σc1−Np+β1+νσ0−σRpηs+NpRpRs1+ηpRsRc1+ηs1−RsRc−ηs−Np+β1+νσ0−σRp1−NpRpRs1+ηp1−RsRc1−Np−1−Nppin−σcR1−NpRcaNp−1−σcR1−Np=0.
Then, the slope of the line “BC” in the strain-softening zone can be expressed as(38)tanθ=σc−σcRεcR−εsPeak=σc−σcRA0Rp/Rs1+ηpRs/Rc1+ηs−1.
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 rock mass begins to enter the damage zone; Wei et al. and Jiang et al. [32, 33] believe that another relationship between Rp and Rc may be established by the radial strain continuous condition εrcr=Rc=εrpr=Rc. In this paper, it is assumed that when the plastic shear stress increment Δγ∗ 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 Rp, Rs, and Rc can be expressed as(39)Δγ∗=γs∗−γp∗=εθp−εrpr=Rs−εθp−εrpr=Rp,where γs∗ and γp∗ 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, γs∗=εsPeak−εssPeak and γp∗=εpPeak−εppPeak. In addition, εssPeak and εppPeak represent the minimum principal strain at the points “B” and “A,” respectively.
Substituting (26) into (41), it can be rewritten as(40)Rp=Δγ∗A01+ηp+11/1+ηpRs=TRs,where T=Δγ∗/A01+ηp+11/1+ηp.
Substituting (40) into (38), the following expression can also be obtained:(41)Rs=σc−σcRβ1+νσ0−σRpT1+ηp+11/1+ηsRc=tRc,where t=σc−σcR/β1+vσ0−σRpT1+ηp1/1+ηs.
Substituting (40) and (41) into (37), the damage zone radius is expressed as(42)Rc=a2σ01−Np−2σc/1−Np2T1−Np−β1+νσ0−σRp/ηs+Np+β1+νσ0−σRp/1−NpT1+ηpt1−Np+β1+νσ0−σRp/ηs+Npt1+ηs+β1+νσ0−σRp/1−NpT1+ηp+σc−σcR/1−Np×1−Nppin−σcR/1−Np−11/Np−1.
Subsequently, the radiiRs and Rp can also be obtained by substituting (42) into (41) and (40), respectively.Case 3. From (40), when Δγ∗=0, the limit of T is equal to one. Then, the post-peak failure radii satisfy the relation a<Rc<Rs=Rp, and the surrounding rock is composed of the elastic zone, strain-softening zone, and damage zone. Therefore, the EPSDM’s solution is converted to the ESM’s result. Combining with (41) and (42), the radius expression of Rc and Rs can be deduced:(43)Rc=aδ1σc−σcR/β1+νσ0−σRp+11−Np/1+ηs+β1+νσ0−σRp/ηs+Npσc−σcR/β1+νσ0−σRp+1+δ21−Nppin−σcR/1−Np1/Np−1,Rs=aσc−σcRβ1+νσ0−σRp+11/1+ηsδ1σc−σcR/β1+νσ0−σRp+11−Np/1+ηs+β1+νσ0−σRp/ηs+Npσc−σcR/β1+νσ0−σRp+1+δ21−Nppin−σcR/1−Np1/Np−1,where δ1=2σ01−Np−2σc/1−Np2−β1+νσ0−σRp/ηs+Np−β1+νσ0−σRp/1−Np and δ2=σc−σcR+β1+νσ0−σRp/1−Np.Case 4. From (41), when β→∞, the limit of t is equal to one. Then, the post-peak failure zone radii satisfy the relation a<Rc=Rs<Rp, and the surrounding rock is parted into three zones: the elastic zone, perfectly plastic zone, and damage zone. Therefore, the EPSDM’s solution is converted to the EPBM’s result. Combining with (40) and (42), the post-peak failure radii Rp and Rc can be obtained as follows:(44)Rc=a2σ01−Np−2σc/1−Np2T1−Np+σc−σcR/1−Np1−Nppin−σcR/1−Np1/Np−1,Rp=aT2σ01−Np−2σc/1−Np2T1−Np+σc−σcR/1−Np1−Nppin−σcR/1−Np1/Np−1.Case 5. When Δγ∗=0 and β→∞, the limit of T and t is equal to one. Then, the post-peak failure radii satisfy the relation a<Rs=Rc=Rp, and the surrounding rock only includes the elastic zone and damage zone. Therefore, the EPSDM’s solution is converted to the EBM’s result. According to (42), the damage zone radius Rc will be given as(45)Rc=a2σ0−σc−1+NpσcR/1−Np1+Np1−Nppin−σcR/1−Np1/Np−1.


## 3.1. The Basic Equation

The equilibrium differential equation for the axisymmetric plane strain problem in the “i” zone can be expressed as (ignoring the body force of rock mass)(12)dσridr+σri−σθir=0.The geometric equation, based on the small deformation assumption, can be denoted as(13)εri=duridr,εθi=urir,where uri is the radial displacement of surrounding rock. The stress boundary conditions can be described as(14)σre=σ0,r→∞σre=σrp,r=Rpσrp=σrs,r=Rsσrs=σrc,r=Rcσrs=pin,r=a.


## 3.2. 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(15)Φ=Alnr+Br2,where A and B are the unknown constants. The radial and tangential stresses are then given by(16)σre=1rdϕdr=Ar2+2B,(17)σθe=d2ϕdr2=−Ar2+2B.Substituting the boundary conditionsσre=σ0 at r→∞ and σre=σRp at r=Rp into (5), where σRp is the radial stress on the interface between the elastic zone and the perfectly plastic zone, the specific expressions of (16) and (17) can be obtained as(18)σre=σ0+σRp−σ0Rp2r2,(19)σθe=σ0−σRp−σ0Rp2r2.The radial and tangential stresses in the elastic zone should satisfy (5) at r=Rp. Thus, σRp can be deduced by substituting (18) and (19) into (5):(20)σRp=2σ0−σc1+Np.According to the small deformation theory, the displacement and strain can also be expressed as follows [28]:(21)ure=A0Rp2r,εre=−A0Rp2r2,εθe=A0Rp2r2,where A0=1+νσ0−σRp/E. When r=Rp, the displacement on the interface is(22)uRp=A0Rp.


## 3.3. Stresses and Displacement in the Perfectly Plastic Zone

Combined with the boundary conditionσrp=σRp at r=Rp, the stresses in the perfectly plastic zone can be derived by substituting (5) into (12):(23)σrp=2σ01−Np−2σc1−Np2rRpNp−1+σc1−Np,σθp=Np2σ01−Np−2σc1−Np2rRpNp−1+σc1−Np.Then, the radial contact stress (σRs) on the interface between the perfectly plastic zone and the strain-softening zone can be easily obtained by substituting r=Rs into (23):(24)σRs=2σ01−Np−2σc1−Np2RsRpNp−1+σc1−Np.Substituting (13) into (11), the differential equation of the radial displacement in the perfectly plastic zone should be given as(25)durpdr+ηpurpr=0.Solving (25), the displacement and strain can be deduced by considering the boundary condition urp=uRp at r=Rp:(26)urp=A0Rp1+ηpr−ηp,εrp=−A0ηpRp1+ηpr−ηp−1,εθp=A0Rp1+ηpr−ηp−1.Whenσ0≥pin, 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=Rs into (26) as follows:(27)uRs=A0Rp1+ηpRs−ηp,(28)εsPeak=A0Rp1+ηpRs−ηp−1.


## 3.4. Stresses and Displacement in the Strain-Softening Zone

Based on the calculation method of the perfectly plastic zone, combined with the boundary conditionurs=uRs at r=Rs, we can easily get the expressions of the displacement and strain in the strain-softening zone:(29)urs=A0Rp1+ηpRsηs−ηpr−ηs,εrs=−A0ηsRp1+ηpRsηs−ηpr−ηs−1,εθs=A0Rp1+ηpRsηs−ηpr−ηs−1.According to (29), the radial displacement on the interface between the strain-softening zone and the damage zone can also be obtained easily:(30)uRc=A0Rp1+ηpRsηs−ηpRc−ηs.In the strain-softening zone, it is assumed that the strength attenuation of rock mass is only related to cohesion (c). Then, the compressive strength at any point can be expressed as(31)σc¯ε1p=σc−β1+νσ0−σRpεθsr−εsPeak,where β=tanθ/E, which can be defined as a strain-softening coefficient. tanθ 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. Therefore, 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(32)σc¯=σc−β1+νσ0−σRpRp1+ηpRs−ηp−1Rsr1+ηs−1.Combined with the boundary conditionσrs=σRs at r=Rs, the stresses in the strain-softening zone can be derived by substituting (32) and (6) into the equilibrium equation (12):(33)σrs=2σ01−Np−2σc1−Np2Rpr1−Np+σc1−Np+β1+νσ0−σRpηs+NpRpRs1+ηpRsr1+ηs−Rsr1−Np+β1+νσ0−σRp1−NpRpRs1+ηp1−Rsr1−Np,σθs=Np2σ01−Np−2σc1−Np2Rpr1−Np+σc1−Np+β1+νσ0−σRpηs+NpRpRs1+ηpRsr1+ηs−Rsr1−Np+β1+νσ0−σRp1−NpRpRs1+ηp1−Rsr1−Np+σc−β1+νσ0−σRpRpRs1+ηpRsr1+ηs−1.


## 3.5. 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 σrc=pin at r=a:(34)σrc=1−Nppin−σcR1−NpraNp−1+σcR1−Np,σθc=Np1−Nppin−σcR1−NpraNp−1+σcR1−Np.As shown in (29), combined with the boundary condition urcr=Rc=uRc, the displacement and strain in the damage zone can also be expressed as(35a)urc=A0Rp1+ηpRsηs−ηpRcηc−ηsr−ηc,εrc=−ηcA0Rp1+ηpRsηs−ηpRcηc−ηsr−ηc−1,εθc=A0Rp1+ηpRsηs−ηpRcηc−ηsr−ηc−1.According to (35a), when r=a, the surface displacement of surrounding rock is as follows:(35b)u0=A0Rp1+ηpRsηs−ηpRcηc−ηsa−ηc.


## 3.6. The Radius of the Post-Peak Failure Zone (Rp, Rs, and Rc)

In order to obtain the closed solution of the stresses and deformation of the EPSDM, the post-peak failure zone radii (Rc,Rs,andRp) should be firstly determined. According to the relationship among Rp, Rs, and Rc, the EPSDM can be converted to the following 5 cases including its original form.Case 1. From (32), when β=0 and σc¯=σc=σcR, the post-peak failure radii satisfy the relation a=Rc=Rs<Rp, and then, the surrounding rock only consists of the elastic zone and the perfectly plastic zone. Therefore, the EPSDM’s solution can be converted to the EPM’s result. Combining with (23) and the stress boundary condition σrpr=a=pin, the perfectly plastic zone radius Rp can be obtained:(36)Rp=apin−σc/1−Np2σ01−Np−2σc/1−Np21/1−Np.Case 2. When a<Rc<Rs<Rp, in order to get the closed solution of Rp, Rs, and Rc, it is required to establish three linear independent equations to solve the problem. According to the stress contact condition σrsr=Rc=σrcr=Rc, one of the relationships can be established by combining with (33) and (34):(37)2σ01−Np−2σc1−Np2RpRc1−Np+σc1−Np+β1+νσ0−σRpηs+NpRpRs1+ηpRsRc1+ηs1−RsRc−ηs−Np+β1+νσ0−σRp1−NpRpRs1+ηp1−RsRc1−Np−1−Nppin−σcR1−NpRcaNp−1−σcR1−Np=0.
Then, the slope of the line “BC” in the strain-softening zone can be expressed as(38)tanθ=σc−σcRεcR−εsPeak=σc−σcRA0Rp/Rs1+ηpRs/Rc1+ηs−1.
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 rock mass begins to enter the damage zone; Wei et al. and Jiang et al. [32, 33] believe that another relationship between Rp and Rc may be established by the radial strain continuous condition εrcr=Rc=εrpr=Rc. In this paper, it is assumed that when the plastic shear stress increment Δγ∗ 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 Rp, Rs, and Rc can be expressed as(39)Δγ∗=γs∗−γp∗=εθp−εrpr=Rs−εθp−εrpr=Rp,where γs∗ and γp∗ 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, γs∗=εsPeak−εssPeak and γp∗=εpPeak−εppPeak. In addition, εssPeak and εppPeak represent the minimum principal strain at the points “B” and “A,” respectively.
Substituting (26) into (41), it can be rewritten as(40)Rp=Δγ∗A01+ηp+11/1+ηpRs=TRs,where T=Δγ∗/A01+ηp+11/1+ηp.
Substituting (40) into (38), the following expression can also be obtained:(41)Rs=σc−σcRβ1+νσ0−σRpT1+ηp+11/1+ηsRc=tRc,where t=σc−σcR/β1+vσ0−σRpT1+ηp1/1+ηs.
Substituting (40) and (41) into (37), the damage zone radius is expressed as(42)Rc=a2σ01−Np−2σc/1−Np2T1−Np−β1+νσ0−σRp/ηs+Np+β1+νσ0−σRp/1−NpT1+ηpt1−Np+β1+νσ0−σRp/ηs+Npt1+ηs+β1+νσ0−σRp/1−NpT1+ηp+σc−σcR/1−Np×1−Nppin−σcR/1−Np−11/Np−1.
Subsequently, the radiiRs and Rp can also be obtained by substituting (42) into (41) and (40), respectively.Case 3. From (40), when Δγ∗=0, the limit of T is equal to one. Then, the post-peak failure radii satisfy the relation a<Rc<Rs=Rp, and the surrounding rock is composed of the elastic zone, strain-softening zone, and damage zone. Therefore, the EPSDM’s solution is converted to the ESM’s result. Combining with (41) and (42), the radius expression of Rc and Rs can be deduced:(43)Rc=aδ1σc−σcR/β1+νσ0−σRp+11−Np/1+ηs+β1+νσ0−σRp/ηs+Npσc−σcR/β1+νσ0−σRp+1+δ21−Nppin−σcR/1−Np1/Np−1,Rs=aσc−σcRβ1+νσ0−σRp+11/1+ηsδ1σc−σcR/β1+νσ0−σRp+11−Np/1+ηs+β1+νσ0−σRp/ηs+Npσc−σcR/β1+νσ0−σRp+1+δ21−Nppin−σcR/1−Np1/Np−1,where δ1=2σ01−Np−2σc/1−Np2−β1+νσ0−σRp/ηs+Np−β1+νσ0−σRp/1−Np and δ2=σc−σcR+β1+νσ0−σRp/1−Np.Case 4. From (41), when β→∞, the limit of t is equal to one. Then, the post-peak failure zone radii satisfy the relation a<Rc=Rs<Rp, and the surrounding rock is parted into three zones: the elastic zone, perfectly plastic zone, and damage zone. Therefore, the EPSDM’s solution is converted to the EPBM’s result. Combining with (40) and (42), the post-peak failure radii Rp and Rc can be obtained as follows:(44)Rc=a2σ01−Np−2σc/1−Np2T1−Np+σc−σcR/1−Np1−Nppin−σcR/1−Np1/Np−1,Rp=aT2σ01−Np−2σc/1−Np2T1−Np+σc−σcR/1−Np1−Nppin−σcR/1−Np1/Np−1.Case 5. When Δγ∗=0 and β→∞, the limit of T and t is equal to one. Then, the post-peak failure radii satisfy the relation a<Rs=Rc=Rp, and the surrounding rock only includes the elastic zone and damage zone. Therefore, the EPSDM’s solution is converted to the EBM’s result. According to (42), the damage zone radius Rc will be given as(45)Rc=a2σ0−σc−1+NpσcR/1−Np1+Np1−Nppin−σcR/1−Np1/Np−1.


## 4. Comparative Analysis

### 4.1. 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. The solutions based on M-C and GTSS yield criteria can be generated from the new closed-form solution usingb=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 Δγ∗=0 and β→∞, the model is transformed into the EBM. Only when Δγ∗=0, the model is transformed into the ESM. Only when β→∞, the model is transformed into the EPBM. Therefore, 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. Therefore, the solution may have a large deviation between the calculation results and the practical situation. This paper takes account of the intermediate principal stress and the dilation. When b=0, ηc=ηs=ηp=1.0, and Δγ∗=0, the EPSDM’s solution is Li et al.’s solution [28]. When β=0 and ηc=ηs=ηp=1.0, the EPSDM’s solution is Xu and Yu’s result [15]. Therefore, the solutions by Xu and Li are all special cases of this paper.


### 4.2. Validation of Model Correctness

Based on the elasto-perfectly 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. The 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 (ψi=φi), β=0, and ηi=1+sinψi/1−sinψi. The comparison between the calculation and simulation results is shown in Figure 4.Table 1
Geometrical and physical parameters of the circular tunnel.
Parameters
Values from Pan’s research [23]
Radius of the opening,a (m)
10
Initial stress,σ0 (MPa)
30
Internal pressure,pin (MPa)
0
Young’s modulus,E (MPa)
2000
Dilation coefficient,ηi
3.0
Poisson’s ratio,ν
0.3
Initial cohesion,c (MPa)
6
Residual cohesion,cr (MPa)
6
Strain-softening factor,β (MPa)
0
Internal friction angle,φ (°)
30Figure 4
Contrast of stress distribution.
(a)
(b)
(c)It can be seen from Figure4 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). Therefore, it confirms the correctness of the stress solution.


## 4.1. 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. The solutions based on M-C and GTSS yield criteria can be generated from the new closed-form solution usingb=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 Δγ∗=0 and β→∞, the model is transformed into the EBM. Only when Δγ∗=0, the model is transformed into the ESM. Only when β→∞, the model is transformed into the EPBM. Therefore, 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. Therefore, the solution may have a large deviation between the calculation results and the practical situation. This paper takes account of the intermediate principal stress and the dilation. When b=0, ηc=ηs=ηp=1.0, and Δγ∗=0, the EPSDM’s solution is Li et al.’s solution [28]. When β=0 and ηc=ηs=ηp=1.0, the EPSDM’s solution is Xu and Yu’s result [15]. Therefore, the solutions by Xu and Li are all special cases of this paper.


## 4.2. Validation of Model Correctness

Based on the elasto-perfectly 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. The 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 (ψi=φi), β=0, and ηi=1+sinψi/1−sinψi. The comparison between the calculation and simulation results is shown in Figure 4.Table 1
Geometrical and physical parameters of the circular tunnel.
Parameters
Values from Pan’s research [23]
Radius of the opening,a (m)
10
Initial stress,σ0 (MPa)
30
Internal pressure,pin (MPa)
0
Young’s modulus,E (MPa)
2000
Dilation coefficient,ηi
3.0
Poisson’s ratio,ν
0.3
Initial cohesion,c (MPa)
6
Residual cohesion,cr (MPa)
6
Strain-softening factor,β (MPa)
0
Internal friction angle,φ (°)
30Figure 4
Contrast of stress distribution.
(a)
(b)
(c)It can be seen from Figure4 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). Therefore, it confirms the correctness of the stress solution.


## 5. Example Study

### 5.1. Case I: Stress and Deformation Evolution Law of the EPSDM

The 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 Table2, which is available by Ogawa and Lo [2]. In addition, b=0, Δγ∗=2.5×10−5, and β=2.5.Table 2
Geometrical and mechanical parameters by Ogawa and Lo [2].
Parameters
Values
Radius of the opening,a (m)
1
Initial stress,σ0 (MPa)
1
Young’s modulus,E (MPa)
5000
Dilation angle,ψi
0
Poisson’s ratio,ν
0.2
Initial cohesion,c (MPa)
0.276
Residual cohesion,cr (MPa)
0.055
Internal friction angle,φ (°)
35The stress distribution of the EPSDM under different critical support resistance is shown in Figure5. As shown in Table 3 and Figure 5, it can be seen that when pin/σ0≥0.200, the rock mass only consists of the elastic zone. When 0.152≤pin/σ0<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≤pin/σ0<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 pin/σ0<0.028, the damaged zone begins to develop and the rock mass finally displays four zones: elastic zone, perfectly plastic zone, strain-softening zone, and damage zone.Figure 5
Stress distribution law of the EPSDM under different critical support resistance.
(a)
(b)
(c)
(d)Table 3
Critical support resistance (pin/σ0).
State
Elasto-perfectly plastic
Perfectly plastic strain softening
Strain-softening damage
Value
0.200
0.152
0.028Figures6 and 7 illustrate the influence of support resistance on the post-peak failure radii (Rp,Rs,andRc) and surface displacement. In addition, when pin=0, the post-peak failure radii and surface displacement are as shown in Table 4. From the above two figures, it can be seen that once the support resistance satisfies 0.152≤pin/σ0<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, Δγ∗=2.5×10−5, and then, the strain-softening zone begins to develop. When the residual cohesion reaches a residual value, pin/σ0=0.028 and the rock mass begins to enter the damage stage. As the support resistance is completely released, then Rp/a=1.165 for the EPM; Rp/a=1.602 for the EBM; Rp/a=1.418 and Rc/a=1.218 for the ESM; Rp/a=1.540 and Rc/a=1.492 for the EPBM; and Rp/a=1.165, Rs/a=1.295, and Rc/a=1.122 for the EPSDM. In addition, the dimensionless radial displacement 2u0G/aσ0−σRp of the EPM, EBM, ESM, EPBM, and EPSDM is 1.357, 2.565, 2.010, 2.371, and 1.786, respectively.Figure 6
The influence of support resistance on the post-peak failure radii.
(a)
(b)Figure 7
The influence of support resistance on the surface displacement.Table 4
The post-peak failure radii and surface displacement whenpin=0.
Model
Transformation
R
p
/
a
R
s
/
a
R
c
/
a
2
u
0
G
/
a
σ
0
−
σ
R
p
EPM
a
=
R
c
=
R
s
<
R
p
1.165
1.000
1.000
1.357
EPSDM
a
<
R
c
<
R
s
<
R
p
1.336
1.295
1.122
1.786
ESM
a
<
R
c
<
R
s
=
R
p
1.418
1.418
1.218
2.010
EPBM
a
<
R
c
=
R
s
<
R
p
1.540
1.492
1.492
2.371
EBM
a
<
R
c
=
R
s
=
R
p
1.602
1.602
1.602
2.565Compared with the ESM and EPBM, the EPSDM represents the influence 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. Therefore, in the deep underground engineering, the design of support parameters by using this paper’s EPSDM may be more reasonable.


### 5.2. Case II: Parameter Analysis of EPSDM

A case of tunnel excavation in coal mine was used to study the influence of rock mass parameters on the post-peak failure radii and surface displacement. The typical rock mass parameters of the haulage roadway by Tang Kou coal mine in China were utilized. The haulage roadway is buried at about 1028 m underground, and the vertical stress and horizontal stress are 23.77 MPa and 23.04 MPa, respectively. Therefore, the in situ stressσ0 is 23.4 MPa. The excavation radius a is 3.0 m, Young’s modulus E is 30.0 MPa, Poisson’s ratio ν is 0.25, strain-softening coefficient β is 1.25, initial internal friction angle φ is 30°, and initial cohesion c and residual cohesion cR are 5.85 MPa and 2.0 MPa, respectively. The plastic shear strain increment Δγ∗ is 0.0001. In addition, the yield parameter b is zero, and the initial dilation angle are all ψp=ψs=ψc=15°.


#### 5.2.1. The Influence of Residual Cohesion

Figures8 and 9 indicate the effect 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 cr transforms from 1.0 MPa to 1.6 MPa, the dimensionless radii Rp/a, Rs/a, and Rc/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%. The above data show that the residual cohesion has an important influence on the surrounding rock state, especially for the extremely broken rock mass.Figure 8
The values ofRp,Rs,andRc under different cr.Figure 9
The values ofu0 under different cr and ψi.In addition, as shown in Figure9, it can be seen that the surface displacement gradually increases with the increase of the dilatancy angle. This is mainly because the larger the dilatancy angle is, the higher the dilatancy coefficient is and the greater the volume expansion of surrounding rock is. Meanwhile, the influence of the dilatancy angle in the damage zone on the surface displacement is the most significant, whereas the dilatancy angle in the perfectly plastic zone has the lowest influence.


#### 5.2.2. The Influence of Plastic Shear Strain Increment

The effect of plastic shear strain increment on the post-peak failure radii and surface displacement is presented in Figures10 and 11, respectively. As Δγ∗ gradually increases, the dimensionless radii Rp/a,Rs/a,andRc/a and the displacement u0/a all present the approximate linear decrease characteristics. Compared with Rp/a and Rc/a, the reduction rate of Rs/a is the largest. For example, when Δγ∗ increases from 0.0001 to 0.0005, Rp/a, Rs/a, and Rc/a are, respectively, reduced to 5.72%, 11.92%, and 10.08%; u0/a is reduced to 16.67%. The above data show that Δγ∗ has a crucial influence on the post-peak failure radii and surface displacement. The plastic bearing capacity of rock mass gradually increases with the increasing Δγ∗. As a result, the supporting resistance to maintain the stability of surrounding rock gradually decreases.Figure 10
The values ofRp,Rs,andRc under different Δγ∗.Figure 11
The values ofu0 under different Δγ∗.


#### 5.2.3. The Influence of Strain-Softening Coefficient

Figures12 and 13 indicate the effect of strain-softening coefficient on the post-peak failure radii and surface displacement, respectively. It can be seen that the change of β exerts important effects on Rp,Rs,Rc, and u0. As β increases, the dimensionless radii Rp/a, Rs/a, and Rc/a and the displacement u0/a all increase. For instance, β increases from 2/3 to 2 and Rp/a, Rs/a, Rc/a, and u0/a increase by 0.133, 0.130, 0.300, and 0.0005, respectively. Nevertheless, β increases from 2 to 10/3 and Rp/a, Rs/a, Rc/a, and u0/a only increase by 0.026, 0.026, 0.077, and 0.0002, respectively. The above data also show that the strain-softening coefficient has the most significant influence on the damage zone radii.Figure 12
The values ofRp,Rs,andRc under different β.Figure 13
The values ofu0 under different β.


#### 5.2.4. The Influence of the Yield Parameterb

The influences of the yield parameterb on the post-peak failure radii and surface displacement are displayed in Table 5. It can be seen that the change of the parameter b exerts important effects on Rp,Rs,Rc, and u0. As b increases, the dimensionless radii Rp/a, Rs/a, and Rc/a and the displacement u0/a all decrease. For instance, b increases from 0.2 to 0.8 and Rp/a, Rs/a, Rc/a, and u0/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 influences on the stress distribution of surrounding rock as well. As parameter b gradually decreases, the maximum circumferential stress σθ gradually keeps a decreasing change and transfers to the deep rock mass. Therefore, the design of the support scheme should take into account the parameter b.Table 5
The influence of the parameterb on the post-peak failure radii and surface displacement.
b
R
p
/
a
R
s
/
a
R
c
/
a
u
0
/
a (%)
0.0
1.579
1.549
1.339
0.24
0.2
1.448
1.422
1.226
0.20
0.4
1.358
1.336
1.150
0.18
0.6
1.293
1.273
1.094
0.16
0.8
1.243
1.225
1.051
0.15
1.0
1.203
1.186
1.017
0.14Figure 14
The influence of the parameterb on the stresses of surrounding rock.


## 5.1. Case I: Stress and Deformation Evolution Law of the EPSDM

The 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 Table2, which is available by Ogawa and Lo [2]. In addition, b=0, Δγ∗=2.5×10−5, and β=2.5.Table 2
Geometrical and mechanical parameters by Ogawa and Lo [2].
Parameters
Values
Radius of the opening,a (m)
1
Initial stress,σ0 (MPa)
1
Young’s modulus,E (MPa)
5000
Dilation angle,ψi
0
Poisson’s ratio,ν
0.2
Initial cohesion,c (MPa)
0.276
Residual cohesion,cr (MPa)
0.055
Internal friction angle,φ (°)
35The stress distribution of the EPSDM under different critical support resistance is shown in Figure5. As shown in Table 3 and Figure 5, it can be seen that when pin/σ0≥0.200, the rock mass only consists of the elastic zone. When 0.152≤pin/σ0<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≤pin/σ0<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 pin/σ0<0.028, the damaged zone begins to develop and the rock mass finally displays four zones: elastic zone, perfectly plastic zone, strain-softening zone, and damage zone.Figure 5
Stress distribution law of the EPSDM under different critical support resistance.
(a)
(b)
(c)
(d)Table 3
Critical support resistance (pin/σ0).
State
Elasto-perfectly plastic
Perfectly plastic strain softening
Strain-softening damage
Value
0.200
0.152
0.028Figures6 and 7 illustrate the influence of support resistance on the post-peak failure radii (Rp,Rs,andRc) and surface displacement. In addition, when pin=0, the post-peak failure radii and surface displacement are as shown in Table 4. From the above two figures, it can be seen that once the support resistance satisfies 0.152≤pin/σ0<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, Δγ∗=2.5×10−5, and then, the strain-softening zone begins to develop. When the residual cohesion reaches a residual value, pin/σ0=0.028 and the rock mass begins to enter the damage stage. As the support resistance is completely released, then Rp/a=1.165 for the EPM; Rp/a=1.602 for the EBM; Rp/a=1.418 and Rc/a=1.218 for the ESM; Rp/a=1.540 and Rc/a=1.492 for the EPBM; and Rp/a=1.165, Rs/a=1.295, and Rc/a=1.122 for the EPSDM. In addition, the dimensionless radial displacement 2u0G/aσ0−σRp of the EPM, EBM, ESM, EPBM, and EPSDM is 1.357, 2.565, 2.010, 2.371, and 1.786, respectively.Figure 6
The influence of support resistance on the post-peak failure radii.
(a)
(b)Figure 7
The influence of support resistance on the surface displacement.Table 4
The post-peak failure radii and surface displacement whenpin=0.
Model
Transformation
R
p
/
a
R
s
/
a
R
c
/
a
2
u
0
G
/
a
σ
0
−
σ
R
p
EPM
a
=
R
c
=
R
s
<
R
p
1.165
1.000
1.000
1.357
EPSDM
a
<
R
c
<
R
s
<
R
p
1.336
1.295
1.122
1.786
ESM
a
<
R
c
<
R
s
=
R
p
1.418
1.418
1.218
2.010
EPBM
a
<
R
c
=
R
s
<
R
p
1.540
1.492
1.492
2.371
EBM
a
<
R
c
=
R
s
=
R
p
1.602
1.602
1.602
2.565Compared with the ESM and EPBM, the EPSDM represents the influence 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. Therefore, in the deep underground engineering, the design of support parameters by using this paper’s EPSDM may be more reasonable.


## 5.2. Case II: Parameter Analysis of EPSDM

A case of tunnel excavation in coal mine was used to study the influence of rock mass parameters on the post-peak failure radii and surface displacement. The typical rock mass parameters of the haulage roadway by Tang Kou coal mine in China were utilized. The haulage roadway is buried at about 1028 m underground, and the vertical stress and horizontal stress are 23.77 MPa and 23.04 MPa, respectively. Therefore, the in situ stressσ0 is 23.4 MPa. The excavation radius a is 3.0 m, Young’s modulus E is 30.0 MPa, Poisson’s ratio ν is 0.25, strain-softening coefficient β is 1.25, initial internal friction angle φ is 30°, and initial cohesion c and residual cohesion cR are 5.85 MPa and 2.0 MPa, respectively. The plastic shear strain increment Δγ∗ is 0.0001. In addition, the yield parameter b is zero, and the initial dilation angle are all ψp=ψs=ψc=15°.


### 5.2.1. The Influence of Residual Cohesion

Figures8 and 9 indicate the effect 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 cr transforms from 1.0 MPa to 1.6 MPa, the dimensionless radii Rp/a, Rs/a, and Rc/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%. The above data show that the residual cohesion has an important influence on the surrounding rock state, especially for the extremely broken rock mass.Figure 8
The values ofRp,Rs,andRc under different cr.Figure 9
The values ofu0 under different cr and ψi.In addition, as shown in Figure9, it can be seen that the surface displacement gradually increases with the increase of the dilatancy angle. This is mainly because the larger the dilatancy angle is, the higher the dilatancy coefficient is and the greater the volume expansion of surrounding rock is. Meanwhile, the influence of the dilatancy angle in the damage zone on the surface displacement is the most significant, whereas the dilatancy angle in the perfectly plastic zone has the lowest influence.


### 5.2.2. The Influence of Plastic Shear Strain Increment

The effect of plastic shear strain increment on the post-peak failure radii and surface displacement is presented in Figures10 and 11, respectively. As Δγ∗ gradually increases, the dimensionless radii Rp/a,Rs/a,andRc/a and the displacement u0/a all present the approximate linear decrease characteristics. Compared with Rp/a and Rc/a, the reduction rate of Rs/a is the largest. For example, when Δγ∗ increases from 0.0001 to 0.0005, Rp/a, Rs/a, and Rc/a are, respectively, reduced to 5.72%, 11.92%, and 10.08%; u0/a is reduced to 16.67%. The above data show that Δγ∗ has a crucial influence on the post-peak failure radii and surface displacement. The plastic bearing capacity of rock mass gradually increases with the increasing Δγ∗. As a result, the supporting resistance to maintain the stability of surrounding rock gradually decreases.Figure 10
The values ofRp,Rs,andRc under different Δγ∗.Figure 11
The values ofu0 under different Δγ∗.


### 5.2.3. The Influence of Strain-Softening Coefficient

Figures12 and 13 indicate the effect of strain-softening coefficient on the post-peak failure radii and surface displacement, respectively. It can be seen that the change of β exerts important effects on Rp,Rs,Rc, and u0. As β increases, the dimensionless radii Rp/a, Rs/a, and Rc/a and the displacement u0/a all increase. For instance, β increases from 2/3 to 2 and Rp/a, Rs/a, Rc/a, and u0/a increase by 0.133, 0.130, 0.300, and 0.0005, respectively. Nevertheless, β increases from 2 to 10/3 and Rp/a, Rs/a, Rc/a, and u0/a only increase by 0.026, 0.026, 0.077, and 0.0002, respectively. The above data also show that the strain-softening coefficient has the most significant influence on the damage zone radii.Figure 12
The values ofRp,Rs,andRc under different β.Figure 13
The values ofu0 under different β.


### 5.2.4. The Influence of the Yield Parameterb

The influences of the yield parameterb on the post-peak failure radii and surface displacement are displayed in Table 5. It can be seen that the change of the parameter b exerts important effects on Rp,Rs,Rc, and u0. As b increases, the dimensionless radii Rp/a, Rs/a, and Rc/a and the displacement u0/a all decrease. For instance, b increases from 0.2 to 0.8 and Rp/a, Rs/a, Rc/a, and u0/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 influences on the stress distribution of surrounding rock as well. As parameter b gradually decreases, the maximum circumferential stress σθ gradually keeps a decreasing change and transfers to the deep rock mass. Therefore, the design of the support scheme should take into account the parameter b.Table 5
The influence of the parameterb on the post-peak failure radii and surface displacement.
b
R
p
/
a
R
s
/
a
R
c
/
a
u
0
/
a (%)
0.0
1.579
1.549
1.339
0.24
0.2
1.448
1.422
1.226
0.20
0.4
1.358
1.336
1.150
0.18
0.6
1.293
1.273
1.094
0.16
0.8
1.243
1.225
1.051
0.15
1.0
1.203
1.186
1.017
0.14Figure 14
The influence of the parameterb on the stresses of surrounding rock.


## 5.2.1. The Influence of Residual Cohesion

Figures8 and 9 indicate the effect 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 cr transforms from 1.0 MPa to 1.6 MPa, the dimensionless radii Rp/a, Rs/a, and Rc/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%. The above data show that the residual cohesion has an important influence on the surrounding rock state, especially for the extremely broken rock mass.Figure 8
The values ofRp,Rs,andRc under different cr.Figure 9
The values ofu0 under different cr and ψi.In addition, as shown in Figure9, it can be seen that the surface displacement gradually increases with the increase of the dilatancy angle. This is mainly because the larger the dilatancy angle is, the higher the dilatancy coefficient is and the greater the volume expansion of surrounding rock is. Meanwhile, the influence of the dilatancy angle in the damage zone on the surface displacement is the most significant, whereas the dilatancy angle in the perfectly plastic zone has the lowest influence.


## 5.2.2. The Influence of Plastic Shear Strain Increment

The effect of plastic shear strain increment on the post-peak failure radii and surface displacement is presented in Figures10 and 11, respectively. As Δγ∗ gradually increases, the dimensionless radii Rp/a,Rs/a,andRc/a and the displacement u0/a all present the approximate linear decrease characteristics. Compared with Rp/a and Rc/a, the reduction rate of Rs/a is the largest. For example, when Δγ∗ increases from 0.0001 to 0.0005, Rp/a, Rs/a, and Rc/a are, respectively, reduced to 5.72%, 11.92%, and 10.08%; u0/a is reduced to 16.67%. The above data show that Δγ∗ has a crucial influence on the post-peak failure radii and surface displacement. The plastic bearing capacity of rock mass gradually increases with the increasing Δγ∗. As a result, the supporting resistance to maintain the stability of surrounding rock gradually decreases.Figure 10
The values ofRp,Rs,andRc under different Δγ∗.Figure 11
The values ofu0 under different Δγ∗.


## 5.2.3. The Influence of Strain-Softening Coefficient

Figures12 and 13 indicate the effect of strain-softening coefficient on the post-peak failure radii and surface displacement, respectively. It can be seen that the change of β exerts important effects on Rp,Rs,Rc, and u0. As β increases, the dimensionless radii Rp/a, Rs/a, and Rc/a and the displacement u0/a all increase. For instance, β increases from 2/3 to 2 and Rp/a, Rs/a, Rc/a, and u0/a increase by 0.133, 0.130, 0.300, and 0.0005, respectively. Nevertheless, β increases from 2 to 10/3 and Rp/a, Rs/a, Rc/a, and u0/a only increase by 0.026, 0.026, 0.077, and 0.0002, respectively. The above data also show that the strain-softening coefficient has the most significant influence on the damage zone radii.Figure 12
The values ofRp,Rs,andRc under different β.Figure 13
The values ofu0 under different β.


## 5.2.4. The Influence of the Yield Parameterb

The influences of the yield parameterb on the post-peak failure radii and surface displacement are displayed in Table 5. It can be seen that the change of the parameter b exerts important effects on Rp,Rs,Rc, and u0. As b increases, the dimensionless radii Rp/a, Rs/a, and Rc/a and the displacement u0/a all decrease. For instance, b increases from 0.2 to 0.8 and Rp/a, Rs/a, Rc/a, and u0/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 influences on the stress distribution of surrounding rock as well. As parameter b gradually decreases, the maximum circumferential stress σθ gradually keeps a decreasing change and transfers to the deep rock mass. Therefore, the design of the support scheme should take into account the parameter b.Table 5
The influence of the parameterb on the post-peak failure radii and surface displacement.
b
R
p
/
a
R
s
/
a
R
c
/
a
u
0
/
a (%)
0.0
1.579
1.549
1.339
0.24
0.2
1.448
1.422
1.226
0.20
0.4
1.358
1.336
1.150
0.18
0.6
1.293
1.273
1.094
0.16
0.8
1.243
1.225
1.051
0.15
1.0
1.203
1.186
1.017
0.14Figure 14
The influence of the parameterb on the stresses of surrounding rock.


## 6. Conclusion

Taking into account the plastic bearing capacity of strain-softening 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. Then, based on the unified 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 verified 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 coefficient 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 coefficient 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 reflects all the features of them, which represent the influence of the plastic bearing properties and strain-softening coefficient on the surrounding rock state. Therefore, the design of support parameters by using the EPSDM may be more reasonable for the perfectly plastic strain-softening rock mass.(3)
The residual cohesion, plastic shear strain increment, and yield parameterb are negatively correlated with the post-peak failure radii and surface displacement; however, the dilatancy angle and strain-softening coefficient are positively correlated. On the other hand, it was also clear that the dilatancy angle in the damage zone had the highest influence on the surface displacement, whereas the dilatancy angle in the perfectly plastic zone had the lowest influence. The strain-softening coefficient had the most significant influence 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.



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*Source: 7912062-2018-04-19.xml*