#### Abstract

Geomaterials generally show strain-softening characteristics after peak-load. Based on the triaxial test for sandy mudstone, a simple elastopeak plastic-strain-softening-damage model (EPSDM) was proposed. Compared with the traditional strain-softening model, EPSDM shows obvious plastic bearing characteristics before strain softening. Then, the closed-formed solution of circular opening was deduced based on the newly proposed model. A plastic shear strain increment was introduced as the extension constraint condition of peak plastic zone. The solution correctness of EPSDM was also verified by comparing with other research results. In addition, the solution based on EPSDM could degenerate for a series of results obtained by elastobrittle plastic model (EBM), elasto-strain-softening model (ESM), and elasto-perfectly plastic model (EPM) under certain conditions. Hence, it could be regarded as a unified solution. Finally, the research results denoted that when the inner pressure was fully released, the maximum postpeak failure radii and surface displacement of surrounding rock indicated the characteristics of EBM>ESM>EPSDM>EPM. Therefore, the plastic bearing behavior could effectively decrease the postpeak failure zone radii and surface displacement. The dilation coefficient noticeably influenced postpeak failure range and surface displacement, particularly the damage zone radii and tunnel wall convergence. The research results can provide very important theoretical bases for evaluating the tunnel stability and support design reliability for underground engineering.

#### 1. Introduction

The stresses and deformation of surrounding rock are an important basis for evaluating the tunnel stability and support design reliability for underground engineering. The ground reaction curve (GRC) of circular tunnel is generally required to be given to optimize support design. Originally, it was researched by regarding the rock mass as the elastic material. However, Placidi L and Altenbach H et al. pointed out that the inelastic phenomena like damage, brittle fracture, and plasticity are also extremely important for the mechanical behavior of materials [1–3]. Therefore, many studies began to predict the evolution law of surrounding rock states by using different constitutive models, such as elasto-perfectly plastic (EPM) model [4–8], elastobrittle plastic (EBM) model [9–16], elastopeak plastic-brittle plastic (EPBM) model [17, 18], and elasto-strain-softening (ESM) model [19–28] (see Figure 1). Nevertheless, each constitutive model has its own application scope. Based on the geological quality, the application scope of the above models was discussed by Hoek and Brown [29]. The results show that EBM applies only to the poor quality rock mass, while EPM is suitable for high quality rock mass and ESM for the average quality rock mass. Moreover, the EPBM is suitable for brittle rock mass with plastic bearing behavior [18]. In fact, a number of strain-softening rock masses also obviously presented plastic bearing behavior. Since the early 1970s till now, the study on strain-softening model can be divided into two categories. As shown in Figure 1(c), one is that the strain-softening zone is simplified into a continuous slash of which the slope can be obtained by the stress-strain curve [25, 26]. The other is that the strain-softening zone is considered to be composed of a series of straight lines which show the instantaneous drop characteristics (see Figure 1(d)) [19–24, 27, 28]. In addition, for the first model, when the slope of slash is equal to zero, the strain-softening models will be converted into elasto-perfectly plastic model; when the slope of slash is up to infinity, the strain-softening models will become elastobrittle plastic model. Geomaterials are easily affected by internal fissures, joints, components, and external environment. Therefore, its constitutive relation shows obvious diversity and complexity.

**(a)**

**(b)**

**(c)**

**(d)**

As shown in Figure 2, the No. 4 coal mine of Saier Group in China is located between Altai mountain and Junger basin. The sandy mudstone is taken from the haulage roadway of the No. 4 coal mine, and the triaxial test result was also given. From the total stress-strain curve, it can be seen that the sandy mudstone firstly shows obvious peak plastic state (“AB”) when the initial yield occurs; then, as the strain continues to increase, the strain-softening stage will begin to develop (“BC”) with the rock mass parameter gradually decreasing. Until a certain residual value is reached, the surrounding rock starts to enter the residual flow stage. And, so, the stress-strain curve approximately experienced four phases during the triaxial test. However, above four models (EPM, EBM, and ESM) cannot be consistent with this experiment result. In other words, if the test results are simplified to the above four models, the calculation result deviation may be larger compared with the actual engineering. Therefore, the elastopeak plastic-strain-softening-damage model (EPSDM) was proposed and applied to the underground engineering. The EPSDM, including all the features of the EPM, EBM, and ESM, represents the influence of the plastic bearing behavior and strain softening on surrounding rock state and can be also transformed into the above three models under certain conditions. Hence, it can be regarded as a unified constitutive model.

This current study mainly deals with a new constitutive model of strain-softening rock mass with consideration given to the plastic bearing behavior and strain-softening characteristics. Moreover, a uniform analytical solution for circular tunnel based on the EPSDM is also derived.

#### 2. Problem Descriptions

##### 2.1. Constitutive Model

As shown in Figure 2, the completed stress-strain curve of sandy mudstone was obtained in MTS816 experimental system under different confining pressures. It can be seen that the rock mass firstly experiences elastic stage (“OA”), and then the plastic stage begins to occur when the principal stress satisfies the initial yield function. This stage represents the plastic bearing capacity of sandy mudstone. However, its extension restricted by some factors is not infinite. This paper assumes that once the plastic shear strain increment of “AB” segment reaches a certain value, the strain-softening phase (“BC”) begins to happen. As the deformation continues to increase, the parameters of rock mass will gradually decrease. Until a residual parameter is reached, the surrounding rock starts to enter the damage stage.

In other words, the mechanical behavior for sandy mudstone can be represented by four phases as follows: elastic stage, peak plastic stage, strain-softening stage, and damage stage. It is obvious that the proposed EPSDM combines the plastic bearing characteristics of EPM, strain-softening feature of ESM, and residual flow characteristics of EBM as well as ESM. Hence, it can be regarded as a unified constitutive model.

##### 2.2. Establishment of Mechanical Model

Figure 3 shows a circular opening being excavated in an infinite, homogeneous, isotropic EPSDM rock mass subjected to a hydrostatic pressure () at the infinity boundary and inner pressure () at inner radii . As gradually reduced, the displacement occurs and the peak plastic zone firstly develops around the circular opening when is less than the initial yield stress. Then, the change of surrounding rock state is consistent with the material postpeak behavior of the elastopeak plastic-strain-softening-damage model. When the inner pressure is completely released, there are four different zones around the circular opening: elastic zone (“”), peak plastic zone (“”), strain-softening zone (“”), and damage zone (“”). The radii of the peak plastic zone, strain-softening zone, and damage zone are denoted as , , and respectively. The mechanical model should meet the following assumption conditions:(i)The tunnel is excavated in an infinite geological body, so the problem can be regarded as plane strain problem to be solved.(ii)The maximum and minimum principal strains are only composed of plastic strain at the postpeak stage, respectively.

For axisymmetric plane strain problems, when , the tangential and radial stresses, and , are, respectively, the maximum and minimum principal stresses. and are the maximum and minimum principal strain, respectively. Accordingly, the linear Mohr-Coulomb (M-C) criterion as the rock mass yield condition can be expressed aswhere ; is the internal friction angle; and are the initial and residual uniaxial compressive strength of rock mass; ; and . and represent initial and residual cohesion; represents the softening strength in strain-softening zone. It should be noted that this paper assumes that the decrease of stress-strain curve is only related to the cohesion attenuation, which is independent of the internal friction angle [25].

#### 3. Analytical Solution of EPSDM

##### 3.1. Basic Equations

The equilibrium differential equation for axisymmetric plane strain problem in the “” zone can be expressed as (ignoring the body force of rock mass) [4, 9, 12, 16, 22, 25]where and are radial stress and tangential stress in the “” zone, respectively. The subscript symbol “” represents different zone of surrounding rock, which can be replaced by the symbols “,” “,” “,”and “.”

Based on the small deformation assumption, the geometric equation can be denoted as [9, 17, 24]where and are, respectively, radial and circumferential strain in different zones and represents the radial displacement.

Supposing that the volume of the rock mass is changing in the postpeak failure zone, the plastic-strain relationships can be established by adopting a nonassociated linear flow rule and small strain theory as follows [11, 14, 22]:where which is the dilation coefficient in the “” zone. is the dilation angle.

##### 3.2. Elastic Zone

According to the elasticity theory, the stress function can be expressed as [30]where and are the constants. The radial and tangential stresses are then given by

Substituting the boundary conditions at and at into (8), where is the radial stress at the elastic-peak plastic zone interface, (8) and (9) can be rewritten as

When , (10) and (11) should satisfy (1). Thus, can be deduced as follows:

The displacement and strains can be easily obtained as follows [25]:where . and are Poisson’s ratio and Young’s modulus. When , the radial displacement at the elastic-peak plastic zone interface is

##### 3.3. Peak Plastic Zone

Substituting (1) into (4), the equilibrium differential equation in peak plastic zone can be rewritten as follows [25]:

Solving (15), the radial stress in peak plastic zone can be derived by considering the boundary conditions at .

And then, introducing (16) into (1), the circumferential stress will be also be deduced as

The radial contact stresses at the peak plastic-strain-softening zone interface can be easily obtained by substituting into (16):

Then, the displacement differential equation can be also given by integrating (5) and (6) [15, 24].

Solving (19), the radial displacement will be deduced in peak plastic zone by considering the boundary condition at .

According to (5), the strains in peak plastic zone can be also obtained as follows:

According to (20), the radial displacement at the peak plastic-strain-softening zone interface can be achieved as follows:

In addition, the maximum principal strain at the peak plastic-strain-softening zone interface can be also deduced by substituting into (20) and (22).

##### 3.4. Strain-Softening Zone

Based on the calculation method of peak plastic zone, the radial displacement in strain-softening zone is easily acquired as follows [24]:

Then, the radial and tangential strain in strain-softening zone are also deduced by integrating (5) and (25).

Meanwhile, according to (25), the radial displacement at the strain-softening-damage zone interface can be obtained as well.

In strain-softening zone, it is assumed that the strength attenuation of rock mass is only related to cohesion (). Then, the compressive strength at any point can be expressed aswhere is the maximum principal strain at the point “C” in Figure 3. may be called strain-softening modulus and can be determined by the slope of “BC” segment in Figure 3. Introducing (24) and (27) into (29), it can be rewritten aswhere , which can be defined as a strain-softening coefficient.

Substituting (2) and (30) into (4), the equilibrium differential equation in strain-softening zone can be expressed as

Combined with the boundary condition at , the radial stress in strain-softening zone can be derived by solving (31).

The circumferential stress is also obtained by substituting (30) and (32) into (2).

##### 3.5. Damage Zone

Substituting (3) into (4), the equilibrium differential equation in damage zone can be rewritten as follows:

Solving (34) and considering the boundary conditions at , the radial stress in damage zone can be derived as

Then, introducing (35) into (3), the circumferential stress in damage zone will be also obtained as

As shown in (25), the radial displacement of damage zone can be deduced by considering the boundary condition .

The radial and circumferential strains in damage zone can be also obtained by substituting (37) into (5).

According to (37), when , the surface displacement of surrounding rock is as follows:

##### 3.6. Radius of Postpeak Failure Zone

In order to obtain the closed-form solution of the stresses and deformation of EPSDM, the postpeak failure zone radius () should be firstly determined. According to the stress contact condition at , one of the relationship among , , and can be established by combining with (32) and (35).

According to (30), when , the residual strength can be expressed as follows:

In addition to (41) and (42), a new condition is still needed to make the EPSDM form a closed solution. Zhang and Jiang et al. assumed that the surrounding rock begins to enter the next stage when the plastic shear strain or the equivalent plastic shear strain in peak plastic zone reaches a critical value [17, 18]. This paper assumes that when the plastic shear stress increment between the yield points “A” and “B” (see Figure 3) satisfies certain conditions, the surrounding rock will enter the strain-softening state. Thus, the other relationship among , , and can be expressed aswhere and are the plastic shear strain at points “B” and “A,” respectively. is the maximum principal strain at the point “A.” and are the minimum principal strain at the points “B” and “A.” As shown in Figure 3, the maximum and minimum principal strain of rock mass can be easily obtained in the triaxial test. Substituting (21) and (22) into (43), it can be rewritten as

Introducing (44) into (42), the following expression can be obtained as follows:

Then, the damage zone radius can be deduced by substituting (44) and (45) into (41).

Subsequently, the radiuses and can be also derived by substituting (46) into (45) and (44), respectively.

##### 3.7. Critical Inner Pressure under EPSDM

According to (44),(45), and (46), when the relationship between postpeak failure zone radius () and inner radius () satisfies certain condition, the rock mass will be in the critical state.(i)When , the rock mass is in the critical state that the peak plastic is just not arisen yet. Then, the critical inner pressure at this state will be deduced according to (12)(ii)When , the rock mass is in the critical state that the peak plastic zone has reached the maximum and the stain softening zone has not yet appeared. Consequently, the critical radii of peak plastic zone can be obtained by (44). Then, introducing (48) into (16), the critical inner pressure at this state will be deduced as follows:(iii)When , the rock mass is in the critical state that the damage zone begins to form. According to (45), the strain-softening zone radius can be expressed as

The peak plastic zone radius can be calculated by (44) at this state. Then, when , the critical inner pressure can be also obtained by substituting (44) and (50) into (32).

##### 3.8. The Special Cases

###### 3.8.1. Li’s Formula

When , , and , the EPSDM degenerates for ESM. By solving (45) and (46), the radii of damage zone and strain-softening zone will be, respectively, rewritten as

Equations (52a) and (52b) are Li’s formula [25].

Then, the critical inner pressure between strain-softening and damage zone can be also rewritten as follows:

###### 3.8.2. Kastner’s Formula

When , , and , the EPSDM degenerates for EPM. Combining with (16) and the stress boundary condition at , the postpeak failure zone radii can be determined by the following equation:Equation (54) is Kastner’s formula [8]

###### 3.8.3. Wilson’s Formula

When , , , and , the EPSDM degenerates for EBM. According to (46), the damage zone radius will be rewritten as

Equation (55) is the formula of Wilson [13].

#### 4. Example Study

##### 4.1. Case I: Verification for EPSDM

In this paper, the analytical solution for the circular opening based on the EPSDM can be regarded as an extension of strain-softening model. For the contraction problem of the circular hole excavated in the ESM rock mass, Li et al. [25] also carried out relevant studies. However, the influence of plastic bearing behavior and dilation coefficient on rock mass was neglected. The geometrical and physical parameters of circular opening are shown in Table 1. For guaranteeing the same condition as Li’s research, the plastic shear strain increment is assumed as 0.

The stress and displacement of surrounding rock along the radius direction are shown in Figure 4 and Figure 5. In addition, Table 2 also lists the calculated results under EPSDM, in accordance with Li’s closed solution. Therefore, the solution by Li et al. [25] is a special case of this paper. In other words, the EPSDM has more extensive practicality.

##### 4.2. Case II: Compared with the Other Constitutive Models

As previously mentioned, the constitutive model of EPSDM reflects all the features of the EPM, EBM, EPBM, and ESM. Therefore, the closed solution of EPSDM is different from the above four models. In order to study the evolution law of surrounding rock states under different constitutive models, the input data for soft rock mass is shown in Table 3.

The influences of constitutive model on the postpeak failure zone radii, surface displacement, and stresses distribution are presented in Figures 6, 7, and 8, respectively. According to (47), (49), and (51), there is only elastic zone around the circular opening when . Then, the surrounding rock will consist of peak plastic zone and elastic zone when . Next, the strain-softening zone will be gradually formed when the inner pressure satisfies . Finally, the surrounding rock will be composed of damage zone, strain-softening zone, peak plastic zone, and elastic zone with the continuous decrease of inner pressures. From Figures 6, 7, and 8, it can be seen that the closed solution of EPSDM is in accordance with EPM for and when . In addition, once the inner pressure is fully released, the surface displacement of surrounding rock will show the characteristics of EBM>EPBM>ESM>EPSDM>EPM (see Figure 7). Therefore, the small plastic bearing capacity can effectively decrease the postpeak failure zone radii as well as surface displacement.

**(a)**The radii evolution law (; )

**(b)**The radii evolution law (; )

**(a)**The displacement evolution law (; )

**(b)**The displacement evolution law (; )

**(a)**The stresses evolution law (; )

**(b)**The stresses evolution law (; )The support parameter design is closely related to the rock mass properties in deep underground engineering. The rock mass with certain plastic bearing capacity can effectively resist the influence of external load on the tunnel deformation. Therefore, the support strength designs for the above different characteristics tunnel should satisfy the relationship with EBM>EPBM>ESM>EPSDM>EPM.

In general, most jointed and fractured rock masses show a certain plastic bearing capacity before strain softening. Therefore, the support parameters design by using EPSDM may be more reasonable.

When , the postpeak failure zone radius and surface displacement of EPSDM are shown in Table 4. It can be seen that the postpeak failure zone radius and surface displacement increase with an increasing dilation coefficient, respectively. For instance, when is changed from 1.0 to 3.0, the dimensionless radius (, , and ) and surface displacement (), respectively, increase 12.65%, 15.29%, 23.62%, and187.57%. Therefore, the dilation coefficient noticeably influences postpeak failure zone radius and surface displacement, particularly the damage zone radii and tunnel wall convergence.

##### 4.3. Case III: Application in No. 4 Coal Mine of Saier Group

The haulage roadway by the No. 4 coal mine of Saier Group in China was excavated in the EPSDM rock mass. It is buried at about 320m underground, the maximum vertical stress is 15.16MPa, and two horizontal principal stresses are, respectively, 18.67MPa and 22.43MPa. Thus, the average value of the in situ stress () is 18.75MPa. The equivalent excavation radius () is 3.54m, and Poisson’s ratio () is 0.23. According to Figure 2, the relationship between maximum and minimum principal stress for the sandy mudstone can be obtained (see Figure 9). Meanwhile, the parameters variation ( and ) with confining pressure is also shown in Figure 10. From the above two figures, It can be seen that the peak plastic strength () and residual strength () are, respectively, 20.68MPa and 5.15MPa, the parameter () is about 3.38, Young’s modulus () is 1.99GPa, and strain-softening coefficient (*α*) is 3.56. In addition, it is assumed that the plastic shear strain increment () is 0.0001, and the dilation coefficient is 1.0.

Figure 11 shows the failure region tested by JL-IDOI (A) intelligent optical borehole imager at the haulage roadway. In the test, nine boreholes were arranged around the tunnel. According to the test result, the average failure range and surface displacement are 1.48m and 6.28cm, respectively. In addition, the calculated peak plastic zone, strain-softening zone, and damage zone radius by EPSDM are m, m, and , respectively, with equivalent excavation radii . Meanwhile, the calculated surface displacement is cm. It should be noted that the postpeak failure region should be determined by the strain-softening and damage zones radius. Hence, the calculated maximum failure region is 1.49m and is in good accordance with the field test results.

#### 5. Conclusion

Considering the influence of plastic bearing capacity on surrounding rock state, a new strain-softening constitutive model was proposed. Then, a closed-form solution for circular opening based on the newly proposed constitutive model was deduced and the validity of the solution is also verified. Moreover, the postpeak failure zone radius, surface displacement, and stresses distribution of circular opening were analyzed under different constitutive models. Finally, an engineering case shows that this paper’s solution is consistent with the field test results. The primary conclusions can be summarized as follows:(1)The peak plastic zone represents the plastic bearing behavior of strain-softening rock mass, and once plastic shear strain increment reaches the critical failure condition, the strain-softening zone begins to occur. Compared with other solutions, only when the plastic shear strain increment was zero, the closed solution of the EPSDM can be converted into ESM’s solution. Similarly, when the plastic shear strain increment was zero and strain-softening coefficient was large enough, the EPSDM’s solution will transform into EBM’s solution; only when strain-softening coefficient was zero, the calculated results of the EPSDM was found to be in accordance with the closed-form solution of the EPM.(2)When the inner pressure is fully released, the maximum postpeak failure radii and surface displacement of surrounding rock satisfy the relationship with EBM>ESM>EPSDM>EPM. It is shown that the plastic bearing behavior of strain-softening rock mass can effectively decrease the postpeak failure zone radii as well as surface displacement. Therefore, the design of support parameters by using this paper’s EPSDM may be more reasonable for peak plastic-strain-softening rock mass.(3)The rock mass with certain plastic bearing capacity can effectively resist the influence of external load on the tunnel deformation. Therefore, the support strength designs should satisfy the relationship with EBM tunnel>EPBM tunnel>ESM tunnel>EPSDM tunnel>EPM tunnel.(4)The dilation coefficient noticeably influences postpeak failure zone radius and surface displacement, particularly the damage zone radii and tunnel wall convergence. Therefore, the support parameters design should take into account the influence of the dilation coefficient.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The authors acknowledge financial support by the National Natural Science Foundation of China (51574222 & 51704281).