#### 2. Mechanical Model

##### 2.2. Fundamental Assumptions

We make the following fundamental assumptions:(1)Both the frozen wall and surrounding rock consist of linear elastic material under plane-strain conditions, and their physical and mechanical parameters depend on the radius.(2)The initial geostress field is fixed before and after freezing, and the initial principal stresses are and . Switching to polar coordinates giveswhere is the angular coordinate (Figure 1). and give the directions of minimum and maximum crustal stress (x and y directions), respectively.

##### 2.3. Boundary Conditions

The analytical solution of the problem is the superposition of solutions for the initial stratum stress or displacement and the solutions of the mechanical model I (Figure 1). The frozen wall and surrounding rock are divided into n concentric layered cylinders, which are regarded as homogeneous, isotropic, and linear elastic. These cylinders remain in the range of small deformations. The number n of cylinder layers depends on the radial temperature profile . For model I, the loads on the inner () and outer () surfaces are

The displacement condition on the external boundary of surrounding rock () is

The stress boundary condition on the inner edge of the frozen wall () iswhere is the unloading ratio of the internal surface of the frozen wall caused by the excavation (, when completely unloaded and when not excavated). The value of the “unloading ratio” is determined by the construction scheme, the lining structure, and the mechanical properties of surrounding ground (such as stiffness and Poisson’s ratio) [23].

#### 3. Solution of Model

##### 3.1. Basic Solution

The mechanical model I can be decomposed into many single-layer models, which we call model II (Figure 2).

The analytical solution [17, 18] of model II iswhere , , , , , , and is Poisson’s ratio. The coefficients to are related to the boundary conditions. When , the values of to are as follows:

When , we have and . The coefficients to can be obtained by using (3). The result is

When , we obtain

##### 3.2. Solution for Mechanical Model I

The contact condition between each single layer in mechanical model I involves two different limited states: complete contact and smooth contact [17, 2427]. Obviously, in the analysis of the heterogeneous frozen wall, the contact condition is complete contact. In addition, to make the solution of model I more universal and applicable to more engineering problems, we also solve model I for the condition of smooth contact.

###### 3.2.1. Complete Contact

For complete contact, the radial stresses at the interfaces between contact thin cylinders are equal and displacements at the contact interface are continuous. Thus, according to the compatibility condition for displacements, we obtainwhere is the contact-surface number (from interior to exterior). The above boundary conditions are applicable for arbitrary , so we get solutions simultaneously with (5)–(9):

Arranging (14)–(16), we getwhere is the state parameter of the interface and when contact is complete. When , and ; when , and :

###### 3.2.2. Smooth Contact

For smooth contact, normal stresses and radial displacements at the contact interface are continuous and the tangential stress is zero (). We can still use (17)–(19), but the state parameter of the interface is zero (). Also, under these conditions.

###### 3.2.3. Solving Equations of Model I

There are contact interfaces in an n-layered cylinder. Substituting into (17)–(19), we obtain a -degree linear equation with , , , …, , , as variables. This can be solved directly by using mathematical calculation software. Next, substituting the result into (5)–(9) gives the analytical solutions for stress and displacement of each layer in model I.

##### 3.3. Stress and Displacement Solutions for Frozen Wall and Surrounding Rock

The stress and displacement solutions for the frozen wall and surrounding rock are the sum of the solutions in Section 3.2 and the initial stress field and displacement field (initial displacement field is zero). The total solution is expressed by adding an asterisk () above the variable:

#### 4. Comparison with Homogeneous Unloading Model

The homogeneous unloading model (abbreviated as “HF model”) regards the frozen wall and surrounding rock as homogeneous materials and thus uses average temperatures instead of considering radial variations in temperature. We compare the HF model with the inhomogeneous model (abbreviated as “IF model”) and analyze the stress and displacement given by the two models under three freezing modes (single-circle pipes, double-circle pipes, and triple-circle pipes) and two formation conditions (rock freezing and soil freezing). For rock freezing, the elastic modulus varies only slightly upon freezing. However, the elastic modulus of soil increases significantly upon freezing.

##### 4.1. Calculation Parameters

Figure 3 shows the distribution of temperature for different freezing modes. The thicknesses of the frozen wall for single-circle, double-circle, and triple-circle freezing are , , and , respectively. Horizontal lines ①, ②, and ③ give the average temperatures for each of these three freezing modes. Points A, B, and C show the location of the inner-, middle-, and outer-ring freezing pipes.

The basic parameters in the comparison include the depth (150 m), excavating radius (), exterior boundary (100 m), soil specific weight (0.02 MN/m3), unloading rate (), and nonuniform coefficient of ground press (). The frozen wall is divided into several coaxial cylinders of identical thickness (0.1 m). The temperature of each cylinder is obtained from linear interpolation in Figure 3. The nonfrozen stratum is treated as a single outer cylinder, and the temperature of this region is assumed to be 25°C.

For the IF model, the elastic modulus and Poisson’s ratio in frozen rock can be expressed as

In frozen soil, they are

For the HF model, Table 1 gives the elastic modulus and Poisson’s ratio corresponding to the average temperatures.

##### 4.2. Verification of Analytical Solutions

To verify the analytical solution, we compare its predictions with the results of a finite-element model of a quarter of an ice wall () for single-cycle pipe freezing (Figure 4).

Figure 5 shows the distribution of radial stress , tangential stress , and radial displacement along the inner and outer edges of the frozen wall. The scatter plot shows the results of the finite-element calculation, and the curve shows the analytical results. The result of the analytical solution is consistent with the result of the finite-element calculation, which verifies the analytical solution.

##### 4.3. Relationships between Stress and Displacement of Inhomogeneous Frozen Wall and and

For single-cycle pipe freezing, Figures 6(a) and 6(b) show the tangential stress and radial displacement , respectively, along the inner edge of the inhomogeneous frozen wall as a function of the nonuniform coefficient of ground press and of the excavation unloading ratio . and are linearly correlated with and , which is consistent with the accumulative theory of linear elasticity. Thus, we deduce that the stress and displacement of any point in the frozen wall are also linear in and .

According to the elastic stress solution for a thick-walled cylinder, the weakest point is at the inner edge under the limit state, so we focus on the magnitudes of and at the inner edge of the inhomogeneous frozen wall. For the orientation , Figure 6(a) shows the circumferential compressive stress concentration, and the inner edge of the frozen wall expands outward ( is positive) when is relatively small (). As increases, the circumferential compressive stress gradually decreases and the inner edge of the frozen wall begins to move inward instead of expanding. However, the dependencies are opposite when . The inner edge of the frozen wall incurs tensile stress with small , which degrades the safety of the frozen wall. To solve this problem, we can add auxiliary freezing holes in the direction or adopt the freezing model of oval-ring pipes to avoid tensile damage at the inner edge of the frozen wall.

Figure 6(b) shows the tangential stress and radial displacement of the inner edge as a function of the excavation unloading ratio . As increases, the circumferential compressive stress concentrates significantly at , and the inner edge shrinks much more in the direction .

##### 4.4. Stress Variation at Outer Edge of Frozen Wall due to Excavation Unloading

Initially, we solve for the stress variation. For a homogeneous frozen wall and surrounding rock, we have and in (17)–(19). Thus, the interface mechanical parameters , , and can be obtained and then substituted into (21) to get the interaction force on the interface:where

Comparing (24) with (1), we see that the unloading rates differ for the radial and tangential pressure.(1)For the radial pressure , the unloading proportions of the uniform term and nonuniform term are and , respectively.(2)The unloading rate for the tangential pressure is .

After numerous calculations, we find that the interface pressure of a radially inhomogeneous frozen wall can also be expressed by (24), except that the parameters , , and differ slightly between the two models (Table 2). Figure 7 shows the parameters , , and as functions of the contact-surface number j for single-circle soil freezing. The stress state in the frozen wall is complicated; the parameters related with nonuniform load ( and ) vary nonmonotonically, unlike which decreases monotonically along the radius. initially increases and then decreases along the radius, whereas decreases sharply and then rises gradually. In addition, calculations indicate that , , and gradually approach zero with increasing radius when the unfrozen stratum is inhomogeneous.

From Figure 7, we see that and are always positive along the radius, but is negative for . This indicates that the excavation should decrease the radial force exerted by the frozen wall on the surrounding rock and increase the tangential pressure. To explain this phenomenon, we extract the orientations of greatest principal stress in the frozen wall before and after excavation. The initial direction of maximal principal stress is (y direction); after excavation, the direction deflects by varying degrees (Figure 8), which is a sign of stress redistribution. Figure 8 shows the deviation angles of principal stress at different locations. The greatest deflection occurs at the intersection of the inner edge of the frozen wall and in the direction , where the orientation changes 90° from the y direction to the x direction. The redistribution of stress may be the reason for the increase in the tangential interaction force between the frozen wall and surrounding rock.

Curve 1 shows the initial directions of the maximal principal stress at different locations. Curves 2–5 represent a quarter of a ring of radius of 4.2, 6.3, 8.2, and 10.2 meters, respectively, and show the deflection angles of the principal stress in the circumferential direction.

Table 2 shows , , and for different thicknesses of the frozen wall and for the two calculation models. With increasing thickness, the absolute values of all three coefficients decrease. The excavation unloading has less of an effect on the interaction between the frozen wall and surrounding rock, which means that the stress at the outer edge of the frozen wall approaches the initial ground stress.

To summarize, considering the heterogeneity of the frozen wall is helpful to understand the complex state of the interior stress. After excavation, the directions of principal stress change because of the nonuniform crustal stress field and the tangential interaction force between the frozen wall and surrounding rock increases. These factors should be considered when designing frozen walls.

##### 4.5. Comparison of Radial Distributions of Circumferential Stress and Radial Displacement in Both Mechanical Models

The radial distributions of circumferential stress for both the mechanical models are shown in Figure 9. and represent the circumferential stress for the IF model and HF model, respectively. The decrease in , which differs from , has inflection points near the freezing pipes. Combined with Figure 3, we find that the discrepancy between and is closely related to the temperature distribution in the two models. For below-average temperature, , whereas with above-average temperature, . This explains why at the inner and outer boundaries of the frozen wall and near freezing pipes. Furthermore, upon increasing the thickness of the frozen wall, the temperature in the HF model approaches the minimum temperature of the IF model, and the gap between and decreases outside the innermost freezing pipes as the lower-temperature region of the IF model widens. This analysis illustrates that the homogeneous model cannot accommodate the frozen-wall design because of the significant discrepancy between the distributions of and .

Copious calculations indicate that the radial displacement in all cases and models decreases nonlinearly in the radial direction, with only minor differences occurring in magnitude. Therefore, we show them in Figure 10 as functions of radius only for single-circle freezing.

##### 4.6. Comparison between Models of Stress and Displacement at Inner Edge of Frozen Wall

Because the mechanical state of the inner edge is the key factor in the design of frozen walls, we calculate and sum the stress and the displacement of the inner edge for different freezing modes (Table 3). The elastic modulus depends on the temperature. The results of the comparison indicate that, at the inner edge, the absolute value of the circumferential stress of the IF model is less than that of the HF model in any freezing mode, and the discrepancy increases with thickness: in going from single-circle freezing to triple-circle freezing, the discrepancy increases from 12.41% to 18.03% at . The reason behind the variation in discrepancy between the two circumferential stresses may be the increasing temperature difference at the inner edge. In this example calculation, the temperature of the inner margin for the IF model is independent of the thickness of the frozen wall, whereas the average temperature for the HF model decreases with thickness. Thus, the discrepancy in the temperature of the inner margin between the two models correlates positively with frozen-wall thickness.

Table 3 also shows the results for calculating the displacements under different conditions. For small frozen-wall thickness, the inner edge for the IF model is less displaced than for the HF model. In addition, the deviation decreases with increasing thickness. Finally, for the IF model, the displacement increases for triple-circle freezing.

For the IF model and frozen rock, the absolute value of the displacement of the inner edge decreases with increasing frozen-wall thickness. However, the relationship is different in frozen soil: at , the absolute value of the displacement increases with thickness, whereas the opposite is true at . These calculation results coincide with actual physical phenomena. Frozen rock is relatively strong and can strongly interact with unfrozen ground, which could restrict the deformation of the frozen wall. However, for frozen soil, the lower strength and weaker interaction between frozen and unfrozen strata should weaken the constraints for frozen soil.

The circumferential stress of the inner edge is a primary concern in the frozen-wall design. Based on the analysis above, we see that the circumferential stress is smaller for the IF model. Consequently, the HF model cannot completely describe the bearing property and the deformation performance of the frozen wall. This will result in a significant waste of refrigeration power, particularly for triple-circle freezing.

#### 5. Conclusions

This study presents a plane-strain model that accounts for the inhomogeneity of a frozen wall and excavation unloading under a nonuniform ground stress. The model is based on the superposition of thin concentric cylinders with two types of contact conditions: complete contact and smooth contact. The analytical solution is derived and verified by comparing it with the result of a finite-element simulation. The proposed model is applicable for any case with radial variation of the parameters and accurately reflects the characteristics of a frozen-soil wall, the surrounding earth mass, and the stress state.

For the nonuniform ground stress field, unlike the uniform geostress, the principal stress rotates after excavation, which increases the tangential interaction force between the frozen wall and surrounding rock.

The tangential stress and the radial displacement at the inner edge of the frozen wall correlate linearly with the nonuniform coefficient of ground press and the excavation unloading ratio . The inner edge of the frozen wall may incur tensile stress and large inward pulling when is relatively small (). Consequently, measurements should be made to ensure the safety of the shaft.

The difference in the results of the two models is mainly embodied in circumferential stress. (), when the temperature of the IF model is below (above) average.

When the frozen wall is relatively thick, the inner-edge circumferential stress of the frozen wall for the IF model is less than that for the HF model; the decrease is 8.12%∼9.32% for frozen rock and 13.41%∼18.03% for frozen soil (Table 3). Therefore, it is economical and reasonable to use the IF model to design frozen walls.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

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

#### Acknowledgments

The work presented in this paper was supported by the National Natural Science Foundation of China (Grant no. 41472224) and the National Key Research and Development Program of China (Grant no. 2016YFC0600904).