Evolution Characteristics of Plastic Zone in Jointed Rock Mass of High-Temperature Hydraulic Tunnel

Abstract

The evolution characteristics and the extent of the plastic zone in rock mass can reflect the failure characteristics and destruction degree of a hydraulic tunnel. In this research, we derived the equation of a plastic zone range based on a gateway-like structure high-temperature tunnel through a theoretical analysis. On-site monitoring and discrete element model simulation were combined to analyze the temperature field law and plastic zone evolution characteristics of the jointed rock mass at high temperatures. The results show that the joints affect the temperature field variation in rock mass, and the vertical and horizontal joint groups pose significantly greater influence than the inclined joint group on temperature field. The sensitivity of the joint internal friction angle and thermal expansion coefficient to the range of the plastic zone is relatively small. Under various joint spacings, the influence of horizontal and inclined joint groups on the plastic zone morphology decreases, while the vertical joint group exhibits an incremental influence on the plastic zone morphology. Similar to the influence of the temperature field, the vertical and horizontal joint groups have a significantly greater influence on the plastic zone range than the inclined joint group. Under various rock mass temperatures, the plastic zone in rock mass results in the occurrence of uneven expansions along the direction of the joint dip angle, which changes the potential failure direction of rock mass and increases the potential destruction degree of rock mass, whereas it exhibits a smaller uniform expansion perpendicular to the joint dip angle, and the boundary of the plastic zone coincides with the joint surface, relatively hindering the potential destruction degree of rock mass. The research results of this study have certain reference values for the stability control of jointed rock mass in high-temperature hydraulic tunnels.

1. Introduction

There exists a series of burning issues in rock mass failure and instability in high-temperature tunnels. Both theory and practice have proven that the failure and instability of rock mass in hydraulic tunnels are essentially caused by the evolution of plastic zones, and the characteristics and range of the plastic zone can reflect the failure characteristics and destruction degree of the tunnel [1,2,3]. Understanding the evolution characteristics of the plastic zone in tunnel rock mass is one of the key problems to be solved in engineering [4,5]. Researchers have conducted a large number of studies on the characteristics of plastic zones in intact rock mass. For instance, Wang et al. [6] combined the Mohr Coulomb strength criterion and derived an approximate solution of the plastic zone boundary equation of tunnel rock mass considering the effect of support. They obtained the effect of support resistance on the range and morphology of the plastic zone in rock mass through a theoretical analysis and numerical simulation. Guo et al. [7] revealed that the plastic zone morphology of homogeneous rock mass under different confining pressure conditions can be circular-, elliptical-, and butterfly-shaped. A formula was derived to calculate the morphology coefficient, and the morphology characteristics of the plastic zone can be distinguished with this coefficient. Eugie et al. [8] used the convergence constraint method (CCM) to evaluate the convergence of tunnels and the rationality of support, obtaining 10% and 4% higher predicted values of plastic zone and tunnel convergence range than those obtained using conventional methods. Jiang et al. [9] derived analytical formulas for the range of the fracture zone and plastic zone of the rock mass under a uniform stress field using the elastic–plastic theory. Molladavoodi et al. [10] found, based on the finite difference software simulation, that the plastic zone expansion angle under non-uniform in situ stress conditions is approximately constant. These achievements have contributed considerably to understanding the evolution characteristics of plastic zones in an intact rock mass. However, natural rock mass often has joint fissures, and the distribution pattern of plastic zones in it is inevitably different from intact rock mass [11,12]. Zhang et al. [13] discovered through indoor experiments, on-site measurements, and numerical simulations that the tensile strength of rock mass is related to the joint dip angle, and the joint dip angle has an impact on the development degree of the plastic zone in the rock mass. In addition, because rock masses are often in a complex mechanical environment, studying the plastic zone evolution characteristics of the jointed rock mass under a multi-field coupling effect is of great importance [14,15,16], especially for high ground temperature disease, which is one of the causes of instability in rock mass. González-Gómez et al. [17] studied the effects of thermal degradation on the compressive strength, ultimate compressive strain, color, and mass loss of four limestone blocks from the Yucatan Peninsula. Chen et al. [18] studied the influence of ground temperature on the characteristics of rock mass and conducted conventional triaxial compression tests under different temperatures and confining pressures using acoustic emission technology to analyze the damage and stress. The results showed that the peak strength, loudest emission energy, brittleness, and other parameters of granite increased at a temperature of lower than 60 °C, while those all decreased when the temperature raised to higher than 60 °C. Based on physical model experiments and FLAC3D numerical simulation, Su et al. [19] analyzed the influence of a high geothermal gradient and high internal water pressure on the safe bearing capacity of rock mass and revealed that the damage degree and cracking depth of tunnel rock mass increased with the increase in the temperature gradient, internal water pressure, and linear expansion coefficient of the rock mass. In order to better understand how periodic fluctuations in surface temperature can lead to rock failure, Marmoni et al. [20] conducted thermal and strain monitoring activities on selected quarry walls and conducted experimental testing sites on connected rock blocks. These studies either focus on the plastic zone of jointed rock mass under the action of a single stress field or on the plastic zone of an intact rock mass under the thermal–mechanical coupling effect, whereas there are few studies focused on the plastic zone of the jointed rock mass under the thermal–mechanical coupling effect.

Therefore, this study creatively derives a prediction equation for the plastic zone range of rock mass in gateway-like structure tunnels under thermo-mechanical coupling effects. A theoretical analysis and numerical simulation were combined to analyze the influence of joint groups on the temperature field, and the plastic zone evolution characteristics of the three types of jointed rock mass were analyzed. The research results are of great significance for the stability control of jointed rock mass tunnel engineering in high-temperature environments.

2. Theoretical Prediction of Plastic Zone Range in High-Temperature Tunnels

2.1. Analysis of Temperature Field in Gateway-like Structure Tunnel

In general, under the impact of temperature, rock mass will generate a steady-state temperature field related to radial distance along the tunnel wall when ignoring the convective heat transfer. In this study, a circular tunnel model was established to analyze the temperature field, and the thermal conductivity differential equation is [21]:

$$\lambda \frac{1}{r}\frac{d}{d_r}(r\frac{d_T}{d_r})=0$$

(1)

where λ is the thermal conductivity, T is the temperature, and r is the diameter length.

By conducting continuous integration on Equation (1), the general solution is:

$$T=c_1\ln r+c_2$$

(2)

where c₁ and c₂ are the indeterminate constants.

The calculation model is shown in Figure 1. Based on the influence range of rock mass temperature, it is assumed that the temperature at the inner boundary radius R₁ is t₁, and the rock mass temperature at the maximum influence radius R₂ is t₂, that is, the boundary conditions of the rock mass temperature field are determined.

By substituting the boundary conditions into Equation (2), the indeterminate constants are obtained as:

$$\{\begin{array}{c}{c}_1=\frac{t_2-t_1}{\ln \frac{R_2}{R_1}}\\ c_2=t_1-\ln r_1\frac{t_2-t_1}{\ln \frac{R_2}{R_1}}\end{array}$$

(3)

By substituting Equation (3) into Equation (2), the temperature field distribution equation of the circular tunnel rock mass is obtained as:

$$T=t_1+\ln \frac{r}{R_1}\frac{t_2-t_1}{\ln \frac{R_2}{R_1}}$$

(4)

Since the influence radius of tunnel excavation disturbance on the temperature field of the rock mass is twice about the excavation diameter [22], R₂ = 4R₁ is taken. The area of the gateway-like structure tunnel is set to A_S, which is equivalent to a circular tunnel of the same area.

$$\{\begin{array}{c}{R}_1=\sqrt{\frac{A_s}{\pi }}\\ R_2=4\sqrt{\frac{A_s}{\pi }}\end{array}$$

(5)

Then, the temperature field distribution equation of rock mass in the equivalent gateway-like structure tunnel can be obtained as:

$$T=t_1+\ln \frac{r}{\sqrt{\frac{A_s}{\pi }}}\frac{t_2-t_1}{\ln 4}$$

(6)

According to elastic–plastic mechanics, the temperature stress equation of rock mass in the circular tunnel is obtained as [23]:

$$\{\begin{array}{c}{\sigma }_{r1}=-\frac{\alpha E\Delta t}{2\left(1-v\right)}\left(\frac{\ln \frac{R_2}{r}}{\ln \frac{R_2}{R_1}}-\frac{{\left(\frac{R_2}{r}\right)}^2-1}{{\left(\frac{R_2}{R_1}\right)}^2-1}\right)\\ {\sigma }_{\theta 1}=-\frac{\alpha E\Delta t}{2\left(1-v\right)}\left(\frac{\ln \frac{R_2}{r}-1}{\ln \frac{R_2}{R_1}}+\frac{{\left(\frac{R_2}{r}\right)}^2+1}{{\left(\frac{R_2}{R_1}\right)}^2-1}\right)\end{array}$$

(7)

where α is the thermal expansion coefficient, E is the elastic modulus, Δt is the temperature difference, v is the Poisson’s ratio, and r is a specific point in the tunnel.

Let $-\frac{\alpha E\Delta t}{2 \left(1-v\right)}=k$ and substitute Equation (5) into Equation (7), and then the temperature stress associated with the gateway-like structure tunnel can be obtained as:

$$\{\begin{array}{c}{\sigma }_{r2}=k\left(\frac{1}{15}\left(1-\frac{16A_s}{\pi r^2}\right)+\frac{\ln \frac{4\sqrt{A_s}}{\sqrt{\pi }r}}{\ln 4}\right)\\ {\sigma }_{\theta 2}=k\left(\frac{1}{15}\left(1+\frac{16A_s}{\pi r^2}\right)+\frac{\ln \frac{4\sqrt{A_s}}{\sqrt{\pi }r}-1}{\ln 4}\right)\end{array}$$

(8)

2.2. Prediction of Plastic Zone Range of Gateway-like Structure Tunnel under Thermo-Mechanical Coupling Effect

2.2.1. Criteria of Plastic Zone

The broad Mohr’s strength theory is only applicable to the mechanical failure laws of homogeneous rocks. When there exist joints in the rock mass, the actual strength value is significantly lower than the theoretical strength value [24]. Assuming that the joint is a flat elliptical shape, the three principal stresses can be ranked as ${\sigma }_1>{\sigma }_2>{\sigma }_3$, the compressive stress is positive, the tensile stress is negative, and the angle β is the included angle between the joint and the maximum principal stress. The schematic diagram of element stresses in the rock mass based on this assumption is shown as Figure 2 [25].

According to the theory of elasticity, the strength curve equation is obtained as [25]:

$${\tau }_1\ge 4{\sigma }_t\left({\sigma }_t-{\sigma }_y\right)$$

(9)

However, Equation (9) is workable only when the crack is open. Actually, cracks tend to close when the rock mass is under pressure, and only shear stress causes the concentration of stress of the cracks. Therefore, this study assumes that the joint surface is only subjected to shear failure under the condition of biaxial stresses. The relationship between the strength curve and the stress circle is shown in Figure 3 [26].

From the geometric relationship in Figure 3, it can be obtained that:

AB = BD − DA

(10)

$$DB=\frac{1}{2}\left({\sigma }_1-{\sigma }_3\right)$$

(11)

$$DA=\frac{1}{2}\left({\sigma }_1+{\sigma }_3\right)\sin \phi$$

(12)

$$AB=-2{\sigma }_t\cos \phi$$

(13)

By substituting Equations (11)–(13) into Equation (10), it can be obtained that

$$-2{\sigma }_t\cos \phi =\frac{1}{2}\left({\sigma }_1-{\sigma }_3\right)-\frac{1}{2}\left({\sigma }_1+{\sigma }_3\right)\sin \phi$$

(14)

Multiply $\frac{2}{{\sigma }_t\cos \phi }$ with both sides of Equation (14) and it can be obtained that

$${\sigma }_t=\frac{-4{\sigma }_t}{\left(1-\frac{{\sigma }_3}{{\sigma }_1}\right)\sqrt{1+{\tan }^2\phi }-\left(1+\frac{{\sigma }_3}{{\sigma }_1}\right)\tan \phi }$$

(15)

Let ${\sigma }_1={\sigma }_t$, ${\sigma }_3=0$ [26], and substitute them into Equation (15), and then it can be obtained that

$${\sigma }_t=-\frac{{\sigma }_c\left(\sqrt{1+{\tan }^2\phi }-\tan \phi \right)}{4}$$

(16)

By substituting Equation (16) into Equation (15), the strength condition containing the joints can be obtained as:

$${\sigma }_1\left(\sqrt{1+{\tan }^2\phi }-\tan \phi \right)-{\sigma }_3\left(\sqrt{1+{\tan }^2\phi }+\tan \phi \right)={\sigma }_c\left(\sqrt{1+{\tan }^2\phi }-\tan \phi \right)$$

(17)

2.2.2. Prediction of Plastic Zone Range of Rock Mass under Thermo-Mechanical Coupling Effect

The calculation of deep buried tunnels can be regarded as a plane strain problem. Given that the vertical load of the circular hydraulic tunnel is P₀, symmetrical along the transverse axis; the horizontal load is $\lambda P_0$, symmetrical along the vertical axis; $\mathsf{\lambda }$ is the lateral pressure coefficient; and R₁ is the radius of the circular tunnel, the stress solution of rock mass in the circular tunnel can be obtained as [24]:

$$\{\begin{array}{c}{\sigma }_{r3}=\frac{1}{2}\left(1+\lambda \right)P_0\left(1-\frac{R_1{}^2}{r^2}\right)-\frac{1}{2}\left(1-\lambda \right)P_0\left(1-4\frac{R_1{}^2}{r^2}+3\frac{R_1{}^4}{r^4}\right)\cos 2\theta \\ {\sigma }_{\theta 3}=\frac{1}{2}\left(1+\lambda \right)P_0\left(1+\frac{R_1{}^2}{r^2}\right)+\frac{1}{2}\left(1-\lambda \right)P_0\left(1+3\frac{R_1{}^4}{r^4}\right)\cos 2\theta \\ {\tau }_{r\theta 3}=\frac{1}{2}\left(1-\lambda \right)P_0\left(1+2\frac{R_1{}^2}{r^2}-3\frac{R_1{}^4}{r^4}\right)\sin 2\theta \end{array}$$

(18)

If under a uniformly distributed load, the lateral pressure coefficient is $\mathsf{\lambda }=1$ [27]. By superposing the stresses of Equations (8) and (18), it can be obtained that

$$\{\begin{array}{c}{\sigma }_r=k\left(\frac{1}{15}\left(1-\frac{16A_S}{\pi r^2}\right)+\frac{\ln \frac{4\sqrt{A_s}}{\sqrt{\pi }r}}{\ln 4}\right)+P_0\left(1-\frac{A_s}{\pi r^2}\right)\\ {\sigma }_{\theta }=k\left(\frac{1}{15}\left(1+\frac{16A_S}{\pi r^2}\right)+\frac{\ln \frac{4\sqrt{A_s}}{\sqrt{\pi }r}-1}{\ln 4}\right)+P_0\left(1+\frac{A_s}{\pi r^2}\right)\end{array}$$

(19)

The equations of large principal stress and small principal stress under the plane strain state in a polar coordinate system are [25]:

$$\begin{array}{c}{\sigma }_1\\ {\sigma }_3\end{array}=\frac{{\sigma }_r+{\sigma }_{\theta }}{2}\pm \sqrt{{\left(\frac{{\sigma }_r-{\sigma }_{\theta }}{2}\right)}^2+{\tau }_{r\theta }{}^2}$$

(20)

By combining Equations (17), (19), and (20), the calculation equation of the plastic zone of the rock mass in the high-temperature gateway-like structure tunnel can be obtained as:

$$\begin{array}{l}\left\{k\left[\frac{1}{15}\left(1+\frac{16A_s}{\pi r^2}\right)+\frac{\ln \frac{4\sqrt{A_s}}{\sqrt{\pi }r}-1}{\ln 4}\right]+P_0\left(1+\frac{A_s}{\pi r^2}\right)\right\}\left(\sqrt{1+{\tan }^2\phi }-\tan \phi \right)\\ -\left\{k\left[\frac{1}{15}\left(1-\frac{16A_s}{\pi r^2}\right)+\frac{\ln \frac{4\sqrt{A_s}}{\sqrt{\pi }r}}{\ln 4}\right]+P_0\left(1-\frac{A_s}{\pi r^2}\right)\right\}\left(\sqrt{1+{\tan }^2\phi }+\tan \phi \right)\\ ={\sigma }_c\left(\sqrt{1+{\tan }^2\phi }-\tan \phi \right)\end{array}$$

(21)

3. Analysis of High Ground Temperature Monitoring Results

3.1. Project Overview

A deep buried hydraulic tunnel in Xinjiang was selected as the studied object. It has a total length of 18 km and a general elevation of 3200~5000 m. The joint development is complex, mainly consisting of three groups of joints, namely J₁:71°NW ∠ 89°, J₂:55°NW ∠ 42°, and J₃:64°NW ∠ 68°. The joint surfaces are mostly straight without fillers and penetrate the tunnel. During the branch tunnel construction of this project, a high-temperature tunnel section with a total length of about 4111 m was found, with the pile number ranging from 2 + 688 m to 6 + 799 m. This section is located in the upstream and downstream sections of the 2#, 3#, and 4# construction branch tunnels. During the excavation process, the highest temperature inside the tunnel reached around 80 °C. The high-temperature zone with the pile number ranging from 3 + 500 m to 5 + 500 m between the 2# and 3# branch tunnels was selected for analysis in this study. The profile of the tunnel section is shown in Figure 4.

3.2. Monitoring Plan

Monitoring points were set according to the on-site situation. The temperature of the jointed rock mass was measured using temperature monitoring probes, and the temperature monitoring range was 9 m. The monitoring instruments were set at the arch waist of the tunnel so that the probes can monitor all the temperatures of the rock mass at the positions of 0 m, 3 m, 6 m, and 9 m away from the tunnel wall. A total of eight monitoring groups were arranged. Prior to use, the monitoring probes and cables were subjected to high-temperature and waterproof treatments. The on-site monitoring layout is shown in Figure 5.

3.3. Analysis of Temperature Monitoring Results at Different Rock Mass Depths

The 60-day temperature data of the typical surface of the tunnel during the construction period were taken for analysis, and the monitoring results are shown in Figure 6.

From Figure 6, it can be seen that after tunnel excavation, the temperature of the jointed rock mass varies with the depth of rock mass. When the depth from the tunnel axis is less than 8.3 m, the rock mass temperature increases radially from the tunnel wall to the deep rock mass. The closer to the tunnel wall, the greater the amplitude of temperature change. For instance, on the 10th day, the temperatures of probes 1#, 2#, 3#, and 4# were 19.03 °C, 62.4 °C, 72.8 °C, and 79.9 °C, respectively. In addition, the temperature of the three temperature sensors varies over time, and the temperature of jointed rock mass at different depths shows a simultaneous increase or decrease. At the depth of 11.3 m from the tunnel axis, the temperature of the jointed rock mass stabilizes at around 80 °C, indicating that at this depth it is in a constant temperature zone of jointed rock mass.

4. Evolution Characteristics of Plastic Zone in Jointed Rock Mass

4.1. Model and Calculation Parameters

Relying on a section of a high-temperature hydraulic tunnel project in Xinjiang, the three-dimensional model was established using 3DEC software, as shown in Figure 7. Block discrete element calculation software was used to simulate the jointed rock mass, mainly because it can more effectively and accurately simulate complex structural planes [28,29,30,31,32,33,34]. Due to the presence of joints in the model establishment and the involvement of coupled calculations, which require a long computational time, the model was simplified into a size of X × Y × Z = 4 m × 60 m × 60 m, 4.6 m in tunnel width, and 5.3 m in tunnel height. The front, rear, left, right, and bottom boundaries of the model were constrained by fixed displacement, and the upper boundary was loaded according to the actual engineering burial depth of 800 m. The mechanical parameters of jointed rock mass are shown in

Table 1. Mechanical parameters of rock mass.

Parameter	Value
Uniaxial compressive strength (MPa)	28.7
Density (g/cm−3)	2.7
Cohesion (MPa)	1.3
Poisson’s ratio	0.25
Elastic modulus (GPa)	18.0
Internal friction angle (°)	37
Uniaxial tensile strength (MPa)	5.1
Thermal conductivity (w/(m·k))	1.49
Specific heat capacity (J/(kg−1/k))	985
Thermal expansion coefficient (10−5/kw/(m·k))	1

. After the comprehensive consideration of the three joint groups J₁, J₂, and J₃ distributed throughout the entire model based on the on-site data, the joints adopted the Mohr Coulomb sliding yield criterion, and the joint parameters obtained from the survey and design data are shown in

Table 2. Parameters of joint structure.

Parameters of Joint Group				Mechanical Parameters of Joint Surfaces
Inclination (°)	71	55	64	Normal stiffness (GPa)	9.0
Dip angle (°)	89	42	68	Tangential stiffness (GPa)	3.5
Average spacing (m)	13.3	8.6	10.4	Cohesion (KPa)	0.25
Inclination (°)	71	55	64	Internal friction angle (°)	30

. According to the on-site monitoring results, the rock mass temperature in the model was set to 80 °C, and the temperature in the tunnel was set to 20 °C.

4.2. Impact of Joints on Rock Mass Temperature

In order to reveal the influence of jointed rock mass on high ground temperature, an intact rock mass model was introduced to compare and describe the temperature distribution. The typical surface was selected for analysis. Figure 8a,b show the cloud charts of the temperature field of the intact and jointed rock mass, respectively, and Figure 9 shows the temperature variation curve of the rock mass.

From Figure 8, it can be seen that the varied temperature field of the intact rock mass is much larger than the jointed rock mass. From Figure 9, it can be seen that the temperature variation shows a nonlinear increasing trend along the thickness direction of rock mass, which gradually tends to stabilize. The simulated and on-site measured temperature values of the jointed rock mass and the intact rock mass, as well as the calculated values of Equation (6) at depths of 11.7 m, 22.8 m, 11.3 m, and 10.6 m, respectively, show a stable trend. The simulated and on-site measured values of the jointed rock mass are basically consistent with the calculated values of Equation (6), which to some extent confirms the accuracy of the numerical simulation and theoretical formula derivation. However, it was far less than the simulated temperature value of the intact rock mass. For instance, when the rock mass depth was 5.3 m from the tunnel axis, the simulated value, the on-site measured value, and the theoretical calculation value of the jointed rock mass were 58.6 °C, 57.0 °C, and 51.5 °C, respectively, which were far lower than the simulated value of 45.4 °C of the intact rock mass temperature. When the rock mass depth was 11.3 m from the tunnel axis, the simulated value, the on-site measured value, and the theoretical value of the jointed rock mass were 79.5 °C, 79.8 °C, and 80.0 °C, respectively, which were far lower than the simulated values of 64.8 °C of the intact rock mass temperature. In addition, the average temperature change rate of the jointed rock mass was 5.13 °C/m, while that of the intact rock mass was 2.76 °C/m, indicating that the temperature variation area in the jointed rock mass was significantly reduced and the constant temperature area was increased.

To facilitate the establishment of the simulation models, the three joint groups in practical engineering were simplified as follows in the model: inclination dd = 0°, spacing space = 2 m, and dip angles = 0°, 45°, and 90° corresponding to the horizontal joint group, inclined joint group, and vertical joint group, respectively. The temperature field distribution cloud charts of the typical surfaces of the three joint groups are shown in Figure 10.

From Figure 10, it can be seen that joint groups with different dip angles generate different sizes of temperature variation areas. The radii of the temperature variation area of the horizontal joint group (dip angle of 0°), inclined joint group (dip angle of 45°), and vertical joint group (dip angle of 90°) were 22.8 m, 13.4 m, and 24.9 m, respectively. This indicates that the influence of vertical and horizontal joint groups on the temperature variation area is significantly greater than that of the inclined joint group, and the different dip angles of joint groups affect the temperature field distribution in the jointed rock mass.

4.3. Impact of Joint Spacing on Plastic Zone Range

At the joint spacings of 0.5 m, 1.0 m, 1.5 m, 2.0 m, and 2.5 m, the effects of the horizontal joint group (dip angle of 0°), inclined joint group (dip angle of 45°), and vertical joint group (dip angle of 90°) on the plastic zone range of the rock mass were analyzed. The model parameters are the same as those within Table 1 and Table 2, and the initial stress was set to 5 Mpa.

Table 3. Calculation results of plastic zone range at different joint spacings for three joint groups.

Dip Angle/°	Joint Spacing/m	Radius of Plastic Zone/m
		Maximum Radius/m	Minimum Radius/m	Ratio of Minimum Radius to Maximum Radius
0	0.5	4.80	2.86	0.60
	1.0	6.81	4.15	0.61
	1.5	4.26	2.64	0.62
	2.0	10.95	7.06	0.64
	2.5	8.25	5.62	0.68
45	0.5	5.21	3.57	0.70
	1.0	4.76	3.43	0.72
	1.5	4.26	3.15	0.74
	2.0	3.82	2.87	0.75
	2.5	3.34	2.54	0.76
90	0.5	4.29	4.05	0.94
	1.0	6.28	5.27	0.84
	1.5	8.55	6.67	0.78
	2.0	11.98	9.22	0.77
	2.5	9.23	6.97	0.76

lists the calculation results of plastic zone range at different joint spacings for the three joint groups. Figure 11 shows the morphological characteristics of the three joint groups at different joint spacings.

From Table 3, in the horizontal joint group, the plastic zone radius first increases and then decreases with the increase in joint spacing, and then increases and decreases. In the inclined joint group, the plastic zone radius gradually decreases with the increase in joint spacing. In the vertical joint group, the plastic zone radius first increases and then decreases with the increase in joint spacing. Under different joint spacings, the ratios of the minimum radius to the maximum radius of the plastic zone of three joint groups ranged from 0.6 to 0.68, 0.70 to 0.76, and 0.76 to 0.94, respectively. At the same time, in the horizontal and inclined joint groups, the ratios of the minimum radius to the maximum radius of the plastic zone increased with the increase in joint spacing, while the ratio of the vertical joint group decreased with the increase in joint spacing. This indicates that with the increase in joint spacing, the influence of horizontal and inclined joint groups on the plastic zone morphology decreases, while the influence of the vertical joint group on the plastic zone morphology increases. As shown in Figure 11, in the vertical joint group, the plastic zone radius overall shows the most changes with the increase in joint spacing, and the plastic zone mainly develops in the vertical direction; followed by the horizontal joint group, with the plastic zone mainly developing in the horizontal direction; and the radius of the plastic zone in the inclined joint group changes the least, with the plastic zone mainly developing at an angle of 45° to the horizontal direction.

4.4. Impact of Joint Internal Friction Angle on Plastic Zone Range

The influence of the joint internal friction angle on the plastic zone range of high-temperature rock mass was analyzed. Based on Equation (21) and other parameters which remained unchanged, the plastic zone radii of the high-temperature tunnel at joint internal friction angles φ of 15°, 20°, 25°, 30°, 35°, 40°, and 45° were calculated using MATLAB software. In addition, when the joint spacing of the three joint groups was set to 1.0 m, the joint internal friction angles were set to 15°, 20°, 25°, 30°, 35°, 40°, and 45°, respectively, other parameters remained unchanged, and the variation in the plastic zone range of rock mass in the high-temperature tunnel was studied. The relationship between the joint internal friction angle and the plastic zone radius is shown in Figure 12.

As shown in Figure 12, as the joint internal friction angle continues to increase, the plastic zone range of the high-temperature rock mass decreases gradually. The plastic zone range of the vertical joint group in the rock mass is the largest, followed by the horizontal joint group, and finally the inclined joint group. For instance, when the internal friction angle φ was 15°, the plastic zone range of the vertical joint group, the horizontal joint group, and the inclined joint group in the rock mass was 6.85 m, 5.21 m, and 4.49 m, respectively. The radius value of the plastic zone of the inclined joint group is close to the theoretical derivation, indicating that this derivation formula is more applicable for calculating the plastic zone range of inclined joint groups. Meanwhile, as the internal friction angle increases, the sensitivity of the plastic zone radius to the internal friction angle gradually decreases. Referring to the literature [1,35], highly similar conclusions were also obtained in this study. However, references [1,35] were focused on the influence of the internal friction angle on the evolution of the plastic zone in homogeneous rock mass. Through comparison, it was found that the presence of joint groups mainly changed the morphological characteristics of the plastic zone.

4.5. Impact of Thermal Expansion Coefficient on Plastic Zone Range

An inclined joint group with a dip angle of 45° and a joint spacing of 1 m was selected for analysis. Based on Equation (21) and other parameters which remained unchanged, the variation in the plastic zone radius at thermal expansion coefficients α of 1 × 10⁻⁵ kw/(m·k), 2 × 10⁻⁵ kw/(m·k), 3 × 10⁻⁵ kw/(m·k), 4 × 10⁻⁵ kw/(m·k), and 5 × 10⁻⁵ kw/(m·k) was calculated using MATLAB and 3DEC software simulation. The relationship between the thermal expansion coefficient and plastic zone radius is shown in Figure 13.

From Figure 13, it can be seen that as the thermal expansion coefficient continues to increase, the plastic zone range of the high-temperature tunnel increases gradually, indicating that the sensitivity of the plastic zone radius to the thermal expansion coefficient is relatively weak. For instance, when the thermal expansion coefficient increased from 4 × 10⁻⁵ kw/(m·k) to 5 × 10⁻⁵ kw/(m·k), the plastic zone radius only increased by 0.11 cm. Similarly, the theoretical value obtained by Equation (21) was also closer to the simulated value of the inclined joint group.

4.6. Impact of Temperature on Plastic Zone Range and Morphological Characteristics

After tunnel excavation, the distribution of the temperature field changes, resulting in temperature stress having a significant influence on the plastic zone. For calculation, the model parameters were set to be consistent with Table 1 and Table 2. Considering the impact of temperature on the plastic zone morphological characteristics, different sizes of rock mass temperature model groups were set for analysis. Figure 14a–c describe the relationship between the rock mass temperature and plastic zone distribution characteristics of the three joint groups at a joint spacing of 2 m.

From Figure 14, it can be seen that as the rock mass temperature increases, the area of the plastic zone gradually increases. Under the same rock mass temperature, the plastic zone area of the vertical joint group is the largest, followed by the horizontal joint group, and finally the inclined joint group featured by a far smaller plastic zone area than the other groups. For instance, when the rock mass temperature was 50 °C, the plastic zone area of the vertical joint group, the horizontal joint group, and the inclined joint group was 54.06 m², 51.13 m², and 15.95 m², respectively, indicating that the plastic zone of the inclined joint group showed the least sensitivity to temperature stress. In addition, in the plastic zone of the horizontal joint group, non-uniform plastic zone boundaries with significant changes formed along the horizontal direction, while uniform plastic zone boundaries with small changes formed along the vertical direction. In the plastic zone of the inclined joint group, non-uniform plastic zone boundaries with significant changes formed along the direction with a 45° angle to the horizontal direction, while uniform plastic zone boundaries with small changes formed along the direction perpendicular to the horizontal direction with a 45° angle to each other. In the plastic zone of the vertical joint group, non-uniform plastic zone boundaries with significant changes formed along the vertical direction, while uniform plastic zone boundaries with small changes formed along the horizontal direction. The above results suggest that with the varying rock mass temperature, non-uniform expansion along the direction of the joint dip angle will be generated, which can change the potential failure direction of the rock mass and increase the potential failure range of the rock mass. In contrast, in the direction perpendicular to the joint dip angle, there is less uniform expansion generated, and the boundary of the plastic zone often coincides with the joint surface, which relatively hinders the potential failure range of rock mass.

5. Conclusions and Suggestion

(1) The equation for predicting the plastic zone range of rock mass in the gateway-like structure tunnel under the thermo-mechanical coupling effect is derived, and the accuracy of the equation is verified to a certain extent via numerical simulation. In addition, compared with the intact rock mass, the joints in the rock mass significantly reduce the temperature variation area of the rock mass and increase the constant temperature area. The vertical joint group and horizontal joint group have a far greater influence on the temperature field than the inclined joint group.

(2) At various joint spacings, the influence of the horizontal and inclined joint groups on the plastic zone morphology decreases, while the influence of the vertical joint group on the plastic zone morphology increases. Similar to the influence of temperature field, the influence of the vertical and horizontal joint groups on the plastic zone range is far greater than the inclined joint group.

(3) The influence of the joint internal friction angle and thermal expansion coefficient on the plastic zone range of jointed rock mass is relatively small, mainly manifested as decreasing with the increase in the internal friction angle and increasing with the increase in the thermal expansion coefficient.

(4) For the three types of joint groups, under the same rock mass temperature, the plastic zone range of the vertical joint group is the largest, followed by the horizontal joint group, and finally the inclined joint group. In terms of the plastic zone morphology, the plastic zone shows uneven expansion along the direction of the joint dip angle, which changes the potential failure direction of rock mass and intensifies the potential destruction degree of rock mass. However, in the direction perpendicular to the joint dip angle, smaller uniform expansions occur, and the boundary of the plastic zone also coincides with the joint surface, which relatively hinders the potential destruction degree of the rock mass.