Research Article | Open Access

Minjie Wen, Jinming Xu, Houren Xiong, "Thermal Diffusion Effects in a Tunnel with a Cylindrical Lining and Soil System under Explosive Loading", *Mathematical Problems in Engineering*, vol. 2019, Article ID 2535980, 11 pages, 2019. https://doi.org/10.1155/2019/2535980

# Thermal Diffusion Effects in a Tunnel with a Cylindrical Lining and Soil System under Explosive Loading

**Academic Editor:**Francesco Marotti de Sciarra

#### Abstract

An analytical method is employed to study the thermoelastic dynamic response of deep-buried circular tunnel lining-soil system under explosion load, considering thermal diffusion effects. The soil and lining are analyzed as homogeneous elastic media. Based on the generalized thermal diffusion theory and the classical thermal elasticity theory, the thermoelastic dynamic response of a soil-lining system in the event of an explosion is solved using the Laplace transform and Helmholtz decomposition. By using continuity boundary conditions, the corresponding numerical solution is obtained through an inverse Laplace transform. The calculated results are compared to those without the lining and without consideration for the diffusion effect. The effects are analyzed under thermal, mechanical, and chemical coupling of the lining and soil properties, and their geometric parameters on the temperature gradient, displacement, stress, and chemical potential of the system. It provides significant guidance for theoretical calculations and antiexplosion design of the lining tunnel.

#### 1. Introduction

Underground lining structures, such as subway tunnels, heat distribution pipelines, petroleum pipelines, and gas pipelines can generate high temperatures and pressures when subjected to endogenous explosive loads, which often cause extensive damage to the lining structure and the surrounding geomaterials, such as crushing, cracking, and deformation. In previous research, the problem of analyzing the underground lining structure under explosive loading was equivalent to the endogenous force load, which ignored the impact of high temperature caused by explosion [1â€“5]. At the moment of the explosion, the temperature and stress in the soil and lining interact with each other. Therefore, considering the thermal coupling effect, it is of great engineering significance to study the dynamic response of the lining structure and soil under loads from explosions.

At the present time, there are relatively few studies on the thermoelastic dynamic response of soil and underground structures that include coupling between thermal effects and stresses. In the 1960s, Nowacki et al. obtained a thermoelastic diffusion behavioral model based on a thermoelastic coupling model [6â€“9]. However, in this theory, the heat propagation rate was considered to be infinite. In order to avoid this problem, many scholars extended that analysis to introduce the thermal relaxation time to obtain a generalized thermoelastic theory [10â€“12]. Zhao Weitao et al. [13] used G-L generalized thermoelastic theory with two thermal relaxation times, to study the thermoelastic response of solid spheres that were subjected to a suddenly applied load with a homogeneous distribution on the spheresâ€™ external surfaces and analyzed the effects of the thermal relaxation time and coupling coefficient on their displacement and temperature distribution. Wang et al. [14] employed a Fourier series expansion method to study the one-dimensional thermoelastic problem. Chandrasekharaiah et al. [15] studied the thermoelastic response problem of a half-space elastic body under thermal shock and derived the analytical expressions for the temperature gradient, displacements, and stresses, using the Laplace transform and inverse transform. Based on the generalized thermoelastic theory, Dhaliwal and Sherief [16] obtained the thermoelastic dynamic response of a half-space elastic body seeing a point heat source load and analyzed the temperature distribution in the depth direction. Sherief et al. [17] obtained the thermoelastic response of an infinite elastic body around a spherical cavity using a Laplace transform approach and found the corresponding numerical solution by means of the Laplace inverse transformation. It has been modified by Sherief based on the Nowacki thermoelastic diffusion theory. Singh et al. [18] studied the propagation characteristics of thermoelastic waves by introducing a displacement potential function and analyzed the reflections of P waves and SV waves on the free surfaces of an elastic body. Aouadi [19] studied the thermoelastic diffusion problem of spherical cavities in infinite elastic bodies and obtained the thermoelastic response of elastomers under thermal, mechanical, and chemical coupling. Zheng Rongyue et al. [20] considered the thermal diffusion effect to study the dynamic response of an infinite soil mass with a circular tunnel considering thermoelastic coupling and analyzed the temperature distribution, stress, displacement, and chemical potential. Liu Ganbin et al. [21] and Lu Zheng et al. [22] obtained the dynamic response of porous elastic media under the thermo-hydro-mechanical coupling effect, based on the thermoelastic theory and the generalized thermoelastic theory. The integral form and numerical solutions were obtained by using the Fourier transform and inverse Fourier transform, respectively. Sherief et al. [23, 24] obtained the fractional heat conduction equation according to the definitions by Caputo and Riemann-Liouville, and Youssef [25] obtained the fractional heat conduction equation by means of fractional Taylor expansion method. This latter method was successfully employed by many scholars to analyze problems such as low temperature, thermal shock, and responses to explosions [26â€“31]. However, these results did not take into account the combination of shock waves and high temperatures caused by explosions and ignored the effect of linings. This paper considers the interaction between linings and soils. The thermoelastic dynamic response of the deeply buried circular tunnel lining-soil system under explosion loads is studied by employing more realistic force load and thermal shock load formulas to simulate an explosionâ€™s effects. Based on generalized thermal diffusion theory and the classical thermoelastic theory, the thermoelastic dynamic response of the soil-lining system under the action of an explosion is obtained. Using continuity boundary conditions, the corresponding numerical solution can be obtained by a Laplace inverse transformation. The effects of lining and soil properties and geometric parameters on the temperature gradients, displacements, stresses, and chemical potential of the system are analyzed under thermal, mechanical, and chemical coupling. Also, compared with the existing calculated results, the correctness of calculated results is verified, which provides guiding significance for the theoretical calculation and antiexplosion design of the lining tunnel.

#### 2. Mathematical Model and Basic Assumptions

The system diagram which the calculations are based on is shown in Figure 1. A cylindrical tunnel lining is deeply buried in the infinite soil body. The inner and outer diameters of the lining are and , respectively, and the lining wall thickness is* h* = -. The inner wall of the lining structure is subjected to a radial uniformly distributed load from an explosion . According to prior work [32], an explosionâ€™s load is a combination of force and heat source loads. When an explosion occurs in the center of the tunnel, the explosionâ€™s shock wave is assumed to be emitted from the tunnelâ€™s center. After encountering the liningâ€™s inner wall an explosionâ€™s shock waves will be reflected many times, and the maximum pressure shows the tendency towards attenuation of vibrations. Thus, the simplified force source load pattern in Figure 2(a) is used as a high-pressure load source simulating the inner wall of the lining. Due to the high temperatures seen while subjected to high pressure shock waves and the studies showing that the temperatures decrease exponentially, the simplified heat source load pattern in Figure 2(b) is used to simulate high temperatures. Assuming that the surrounding soil and lining are homogeneous elastic media, the lining-soil system undergoes small deformations only. If the lining and tunnel are infinitely long, the problem can be regarded as a plane strain model for analysis. The interface between the lining and the soil is assumed to be bonded (without relative slip), which satisfies the continuity boundary condition.

**(a) Force source load**

**(b) Heat source load**

For the axisymmetric problem, regardless of physical force, the kinematic equation, heat conduction equation, and thermal diffusion equation of soil are all expressed by where denotes the volumetric strain, denotes the temperature gradient, and denote the absolute temperature and the initial temperature, respectively, , denote the thermoelastic modulus, denotes the linear thermal expansion coefficient, and are the Lame constants, denotes the Poissonâ€™s ratio of the soil, and denotes the shear modulus. In , denotes the expansion coefficient of linear thermal diffusion, denotes the intensity of diffusion flow, denotes the density of the soil, and denotes the radial displacement. Parameters denote the thermal relaxation time and the diffusion relaxation time, respectively, denotes the specific heat of the soil, denote the thermal diffusion influence coefficients, denotes the heat transfer coefficient, denotes the diffusion coefficient, and denotes a differential operator.

The stress-strain constitutive relationship of the soil with consideration of temperature effects is defined bywhere denotes the chemical potential. Variables denote the radial and hoop stresses of the soil.

Considering the explosion load as the combination of a force source and a heat source load, the Laplace transform form of the force load and heat source is as follows:where denotes the dimensionless loading time. The Laplace transform of iswhere denote the maximum force and heat load amplitude values.

#### 3. Solution of Soil Control Equation

Dimensionless processing was carried out for values in Equations (1)-(3), which is expressed by

For convenient calculation, the asterisk is omitted. By substituting Equation (9) into Equation (1)-(3) and performing Laplace transform, the following expressions s can be obtained:whereBy combining Equations (10)-(12), we can obtain the following:whereAccording to Helmholtz decomposition theory, (14) can be further decomposed to formwhereBased on the properties of Bessel function and the variables at infinity when the boundary conditions are zero, the volumetric strain, temperature gradient, and diffusion flow intensity can be obtained as follows: where are the undetermined coefficients.

Thus, the following expression for radial displacement can be obtained:The Laplace transform was conducted on (4)-(6), and the (18) and (19) are substituted into it. Then, the expressions of radial stress, hoop stress, and chemical potential can be obtained as follows:where .

Because are linear independent constants, (18) and (19) can be substituted into (10) and (11), and the following expression is obtained:

#### 4. Lining Control Equation and Its Solution

The lining is regarded as a homogeneous elastic medium. Under these axisymmetric conditions, according to the classical thermoelastic theory, the motion equation and heat conduction equation are expressed aswhere and denotes the Lame constants, denotes the Poissonâ€™s ratio of the lining, denotes the shear modulus, denotes the volumetric strain, and denotes the radial displacement of the lining. The grouping denotes the linear heat expansion coefficient and denotes the density of lining.

The heat and force coupling constitutive relationship in consideration of temperature effects isTo solve (24) and (25), the following dimensionless values are introduced:After the Laplace transform was performed on Equations (24) and (25), they can be manipulated to formwhereEquations (23) and (24) are combined and the expression for volumetric strain and temperature gradient can be expressed according to the solution method of (16):where , , , and are undetermined coefficients.

Using the basic properties of the Bessel function, the radial displacement of the lining can be obtained from Equation (31):Since are linear independent constants, (31) and (32) are substituted into (28), and the following expressions can be obtained:The Laplace transform is used on Equation (26). By substituting Equations (31)-(33) into the s-domain expressions, the radial and hoop stresses of the tunnel lining are expressed as follows:where

#### 5. Boundary Conditions

Assuming that the lining outer diameter is entirely in close contact with the soil and there is no relative slip, the continuity condition is satisfied at the interface, there is no reflection of the elastic wave, and the contact thermal resistance is ignored. It is assumed that the chemical potential of the lining and soil contact surface is zero. A continuous model is employed to simulate the tunnel and soil. For the surface contact between the lining and soil (), the following can be obtained:

Since the explosion load is equivalent to the combination of temperature and force load, for the inner boundary of the lining (), there is

By substituting (18)-(22), (32), (33), (35), and (36) into (38) and (39), a set of seven expressions with seven unknowns can be found. Therefore, the solution of the thermoelastic dynamic response for the deep-buried circular tunnel lining-soil system under explosion load can be obtained by solving these equations.

#### 6. Theoretical Degenerate Solution to the Problem

##### 6.1. Solution without considering the Diffusion Effect

When , the model can be simplified into a thermoelastic dynamic response of the system without considering the diffusion effect. At this time, the model degenerates into a generalized thermoelastic model, and the equation of motion (1) of the soil can be written asThe heat conduction Equation (2) can be written asTheir constitutive relationship is changed as follows:

##### 6.2. Solution of the Problem without Lining

When the thickness of the lining is zero, that is, , the material parameters of the lining are equal to the material parameters of the soil, and it can be simplified to a cylindrical hole without a lining. At this time, the boundary condition at should be

By substituting Equations (20)-(22) into (44)

Therefore, the expression for the undetermined coefficient can be obtained, getting the thermoelastic response of the infinite elastic body around the cylindrical cavity under load from the explosion.

##### 6.3. The Comparison of Results with Existing Literature

In [20], the thermoelastic coupling dynamic response of an infinite elastic body around a circular tunnel under a step thermal shock load was explored. The thickness of the lining is zero and the parameters of the lining are equal to those of the soil. The following boundary conditions are satisfied, which can be simplified to the solution seen in [20]:where .

#### 7. Numerical Analysis and Numerical Results

It is difficult to directly perform Laplace inverse transformation on expressions such as (35) and (36). In this paper, numerical methods are used to perform inverse Laplace transforms to develop closed-form results for the temperature gradient, radial displacement, stress, and chemical potential distribution of lining and soil system under explosion load. The following Crump numerical inversion method is used [33].

If we let the function be the Laplace transform of the function , then the Crump inversion algorithm of Laplace inverse transform isIf , the error is , where, .

The parameters of the soil in case study refer to [20], and the specific parameters are as follows: , , , , , , , , , , , , , .

The parameters of the lining are as follows: , , , , , , , , .

##### 7.1. Comparative Analysis

###### 7.1.1. The Comparison of Calculated Results without Lining

As shown in Figure 3, the calculated results for the temperature gradients and radial displacements with and without a lining are analyzed with respect to the change of the radius of the tunnel. According to the abovementioned parameters, the dimensionless time is . It can be seen that the thickness of the lining has a great deal of influence on the temperature gradient and temperature within the lining. For the peak value, the temperature gradient values without the lining are obviously larger than those seen with the lining. With increased radius, the influence of the lining on the temperature gradient decreases gradually. This is because the thermal conduction of the lining material is considered in the explosion, and the physical parameters such as the heat transfer coefficient and the linear thermal expansion coefficient of the lining are substantially different from those of soil. When considering the influence of lining, the radial displacement of the soil is smaller than it is without the lining. This is due to the lining having greater rigidity than the soil.

**(a) Temperature gradient**

**(b) Radial displacement**

###### 7.1.2. Comparison of Calculation Results with Results from Existing Literature

Figure 4 shows that the dimensionless time is , and the solution (A) of this paper is simplified to compare it to the solution described in [20], and, compared with the calculated result (B) of [20], the temperature gradients and radial displacements of the soil are found to be consistent, which verifies the correctness and accuracy of the calculated results.

**(a) Temperature gradient**

**(b) Radial displacement**

##### 7.2. Parameter Analysis

###### 7.2.1. Time-Domain Analysis Procedure

Figure 5 shows set plots of the variation of the temperature gradient in the soil with respect to the radius, when the dimensionless time is , , and . It can be seen that, under the action of the explosion, the temperature gradient fluctuates, and the heat wave propagates at a finite speed, and multiple reflections occur after the force source reaches the inner wall of the lining. For and , the peak value of the temperature gradient has a positive value. As the radius increases, the temperature gradient gradually decreases and finally approaches zero. The peak magnitude is negative for and . As the radius increases, a positive peak value appears first and then gradually decreases. This is caused by the interaction and mutual influence of temperature, stress, and chemical field under the explosive load, that is, the coupling effect of heat, force, and chemical.

Figure 6 shows the attenuation rule for radial displacement of the soil with radii, at , , and . It can be seen that, at the interface between the lining and the soil under the action of the explosion load, the radial displacement of the soil reaches its maximum positive value at and , and the radial displacement at the time is greater than the displacement at . As the radius increases, the displacement amplitude gradually decreases. At time , the radial displacement of the soil reaches its maximum magnitude at a negative value. As the radius increases, the displacement shows a positive peak value first and then gradually decreases. This is because the explosion wave is emitted from the center, and multiple reflections will occur after encountering the inner wall of the lining. The maximum pressure is caused by the tendency of oscillating attenuation.

Figure 7 shows a set of curves for the variation of the radial stress of the soil with radius at the time , , and . It can be seen that, at , the radial stress at the interface between the lining and the soil is positive, while, at the time and , the stress has a negative value at the interface between the lining and the soil. This is due to the lining being subjected to shock waves and high temperatures and pressures during the explosion.

Figure 8 shows a curve for variation of the diffusion flow intensity with respect to radius at times , , and . In the process of heat transfer, there is an interinfiltration between substances, that is, a thermal diffusion effect. It can be seen that, under the action of explosive load, like heat conduction, heat diffusion propagates at a finite speed. At the time , the diffusion flow intensity at the interface between the lining and the soil reaches its maximum magnitude at a positive value, and the value at is larger than that at . As the radius increases, a negative value occurs first and then gradually decreases. At the time , the diffusion flow intensity at the interface between the lining and the soil reaches its maximum value at a negative value and gradually decreases with increasing radius.

###### 7.2.2. The Impact of the Diffusion Coefficient

Figures 9 and 10 show the effect of the diffusion coefficient on the temperature gradient and chemical potential. When these are evaluated at time , the relevant coefficients are , , and . It can be seen that the thermal diffusion effects of the coupled temperature field, material diffusion, and strain field are more obvious in the elastic medium, and the diffusion coefficient has a great deal of effect on the peak value of the temperature gradient. As the diffusion coefficient increases, the peak value increases significantly, especially within the radius range . The diffusion coefficient has also a great deal of influence on the chemical potential of the soil. As the diffusion coefficient increases, the negative peak value of the chemical potential gradually increases. As the radius increases, the chemical potential gradually decreases.

#### 8. Conclusions

The effect of thermal diffusion is considered in this paper based on the generalized thermal diffusion theory and the classical thermal elasticity theory. An analytical solution is used to study the thermoelastic dynamic response of a deeply buried circular tunnel lining-soil system under a sudden load from an explosion within the tunnelâ€™s cavity. The thermoelastic dynamic response of a soil-lining system under the explosionâ€™s load is solved using the Laplace transform and the Helmholtz decomposition techniques. By using continuity boundary conditions, the corresponding numerical solution is obtained via inverse Laplace transformation. The effects of lining and soil parameters on the thermoelastic dynamic response of the system are analyzed, and the conclusions can be summarized as follows.

(1) Due to the large difference in physical parameters such as heat transfer coefficient and linear thermal expansion coefficient between the lining and the soil, the lining thickness has a significant impact on the temperature gradient and displacement. At the peak value of the temperature gradient, the temperature gradient without the lining is significantly larger than that with lining, and the radial displacement of the soil is smaller than the calculated result for the case without the lining.

(2) At different times, such as , and , the rules for the changes in the temperature gradient, displacement, stress, and diffusion flow intensity with radius are significantly different. As the radius increases, these values gradually decrease.

(3) The diffusion coefficient has a significant effect on the peak values of the temperature gradients and chemical potential. As the diffusion coefficient increases, the peak value increases significantly within the range , and the negative peak value of the chemical potential gradually increases. As the radius increases, the chemical potential value gradually decreases.

(4) This study shows the propagation of heat and diffusion concentration at finite speed, rather than the infinite speed propagation under the action of explosive load.

#### Data Availability

All data used to support the findings of this study were supplied by the Natural Sciences Foundation Committee of China under Grant no. 41472254. Requests for access to these data should be made to the corresponding author Jinming Xu at Department of Civil Engineering, Shanghai University, Shanghai, 200072, China, xjming@shu.edu.cn. Phone number is +86 13166143882 and fax number is +86 021 56382059.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

Financial supports for the study were provided by the Natural Sciences Foundation Committee of China under Grant no. 41472254, the National Special Projects for International Science and Technology Cooperation of China under Grant no. 2014DFE90040, the Zhejiang Provincial Natural Science Foundation of China under Grant no. LGF18E080010, and the Zhejiang Key R&D Program of China under Grant no. 2019C03120. These supports are gratefully acknowledged.

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#### Copyright

Copyright © 2019 Minjie Wen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.