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

Authors: Minjie Wen; Jinming Xu; Houren Xiong
Journal: Mathematical Problems in Engineering
 (2019)

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## 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.


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## Body

## 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 [1–5]. 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 [6–9]. 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 [10–12]. 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 spheres’ 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 [26–31]. 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 explosion’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 Figure1. A cylindrical tunnel lining is deeply buried in the infinite soil body. The inner and outer diameters of the lining are R1 and R2, respectively, and the lining wall thickness ish = R2-R1. The inner wall of the lining structure is subjected to a radial uniformly distributed load from an explosion T(t). According to prior work [32], an explosion’s load is a combination of force and heat source loads. When an explosion occurs in the center of the tunnel, the explosion’s shock wave is assumed to be emitted from the tunnel’s center. After encountering the lining’s inner wall an explosion’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.Figure 1
The interaction model between lining and soil.Figure 2
Simplified explosion load pattern.
(a)
Force source load (b)
Heat source loadFor the axisymmetric problem, regardless of physical force, the kinematic equation, heat conduction equation, and thermal diffusion equation of soil are all expressed by(1)λS+2μS∂eS∂r-β1∂θS∂r-β2∂M∂r=ρS∂2urS∂t2(2)∂∂t+τ0∂2∂t2ρScsθS+β1T0eS+c1T0M=κS∇2θS(3)Dβ2∇2eS+Dc1∇2θS+∂M∂t+τ∂2M∂t2=Db∇2Mwhere eS=∂(rurS)/r∂r denotes the volumetric strain, θS=T-T0 denotes the temperature gradient, and T,T0 denote the absolute temperature and the initial temperature, respectively, T-T0/T0≪1, β1=3λS+2μSαt denote the thermoelastic modulus, αt denotes the linear thermal expansion coefficient, λS=2vSμS/(1-2vS) and μS=GS are the Lame constants, vS denotes the Poisson’s ratio of the soil, and GS denotes the shear modulus. In β2=3λS+2μSαc, αc denotes the expansion coefficient of linear thermal diffusion, M denotes the intensity of diffusion flow, ρS denotes the density of the soil, and urS denotes the radial displacement. Parameters τ0,τ denote the thermal relaxation time and the diffusion relaxation time, respectively, cs denotes the specific heat of the soil, c1,b denote the thermal diffusion influence coefficients, κS denotes the heat transfer coefficient, D denotes the diffusion coefficient, and ∇2 denotes a differential operator.The stress-strain constitutive relationship of the soil with consideration of temperature effects is defined by(4)σrS=2μS∂urS∂r+λSeS-β1θS-β2M(5)σθS=2μSurSr+λSeS-β1θS-β2M(6)P=-β2eS+bM-c1θSwhere P denotes the chemical potential. Variables σrS,σθS 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 loadq(t) and heat source θ(t) is as follows:(7)q-s=q0s+q0t-s2e-st--1+q02t-e-3st-s2+5t-s-1s2e-2st-+q04t-e-5st-s2+9t-s-1s2e-4st-where t- denotes the dimensionless loading time. The Laplace transform of θ(t) is(8)θ-s=θ0s+0.5where q0,θ0 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(9)r∗=Vηr,urS∗=VηurS,t∗=V2ηt,τ0∗=V2ητ,θS∗=β1θSλS+2GS,M∗=β1MλS+2GS,σijS∗=σijSλS+2GS,P∗=Pβ2,V=λS+2GSρS,η=ρScsκS.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:(10)∂e-S∂r-∂θ-S∂r-∂M-∂r=s2u-rS(11)s1+τ0sθ-S+εe-S+εα1M-=∇2θ-S(12)∇2e-S+α1∇2θ-S+α2s1+τsM-=α3∇2M-where(13)α1=c1ρSV2β1β2,α2=ρSV2β22Dη,α3=bρSV2β22,ε=T0β12ρS2csV2By combining Equations (10)-(12), we can obtain the following:(14)∇6-ξ1∇4+ξ2∇2-ξ3e-S,θ-S,M-=0where(15)x1=s1+τ0s,x2=s1+τ0sε,x3=s1+τ0sεα1,x4=s1+τsα2,ξ1=1x3α3-1x3+α3x2x3-x1-x2x4+x3+x2α3x1+x4+x3α3+s2x3α3ξ2=1x3α3-1x2x4x3-x1+x2+x3x1x4+x3s2α3x1+x4+x3α1ξ3=1x3α3-1x1x3x4s2According to Helmholtz decomposition theory, (14) can be further decomposed to form(16)∇2-k12∇2-k22∇2-k32e-S,θ-S,M-=0where(17)k12=132psin⁡q+ξ1,k22=13ξ1-p3cos⁡q+sin⁡q,k32=13ξ1+p3cos⁡q-sin⁡q,p=ξ12-3ξ2,q=13arsin⁡-2ξ13-9ξ1ξ2+27ξ32p3Based 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: (18)e-S=∑i=13K0kirAiθ-S=∑i=13K0kirBiM-=∑i=13K0kirMiwhere Ai,Bi,Mii=1,2,3 are the undetermined coefficients.Thus, the following expression for radial displacement can be obtained:(19)u-rS=-∑i=13AikiK1kirThe 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:(20)σ-rS=∑i=13β2rkiK1kirAi+Ai-Bi-MiK0kir(21)σ-θS=∑i=131-β2Ai-Bi-MiK0kir-β2rkiK1kirAi(22)P-=∑i=13-Ai-α1Bi+α3MiK0kirwhere β2=2GS/λS+2GS.BecauseAi,Bi,Mii=1,2,3 are linear independent constants, (18) and (19) can be substituted into (10) and (11), and the following expression is obtained:(23)χi=ki2s1+τ0sAi=ki2εα1+χi-1εα1ki2-s2+εki2BiMi=χi-1ki2-s2-ki2εεα1ki2-s2+εki2Bi


## 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 as(24)λL+2μL∂eL∂r-βL∂θL∂r=ρL∂2urL∂t2(25)κL∇2θL=ρLcL∂θL∂t+T0βL∂eL∂twhere λL=2vLμL/(1-2vL) and μL=GL denotes the Lame constants, vL denotes the Poisson’s ratio of the lining, GL denotes the shear modulus, eL=∂(rurL)/r∂r denotes the volumetric strain, and urL denotes the radial displacement of the lining. The grouping βL=3λL+2μLαL denotes the linear heat expansion coefficient and ρL denotes the density of lining.The heat and force coupling constitutive relationship in consideration of temperature effects is(26)σrL=2μL∂urL∂r+λLeL-βLθLσθL=2μLurLr+λLeL-βLθLTo solve (24) and (25), the following dimensionless values are introduced:(27)θL∗=β1θLλS+2GS,urL∗=VηurL,σijL∗=σijLλS+2GSAfter the Laplace transform was performed on Equations (24) and (25), they can be manipulated to form(28)∂e-L∂r-m1∂θ-L∂r=s2m2u-rL(29)∇2θ-L=sn1θ-L+sn2e-Lwhere(30)n1=ρLcLηκL,n2=βLT0β1ηκLλS+2GS,m1=βLλS+2GSβ1λS+2GS,m2=ρLV2λL+2GL.Equations (23) and (24) are combined and the expression for volumetric strain and temperature gradient can be expressed according to the solution method of (16):(31)e-L=∑i=12K0δirDi+I0δirEi(32)θ-L=∑i=12K0δirGi+I0δirHiwhere δ1,22=d1±d12-4d2/2, d2=s2n1sm2, d1=s2m2+n1s+n2sm1, and Di,Ei,Gi,Hi are undetermined coefficients.Using the basic properties of the Bessel function, the radial displacement of the lining can be obtained from Equation (31):(33)u-rL=-∑i=12K1δirδiDi-I1δirδiEiSince Di,Ei,Gi,Hi are linear independent constants, (31) and (32) are substituted into (28), and the following expressions can be obtained:(34)Gi=δi2-s2m2δi2m1DiHi=δi2-s2m2δi2m1EiThe 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:(35)σ-rL=∑i=12n3-βLβ1δi2-s2m2m1δi2K0δir+n4K1δirrδiDi+∑i=12n3-βLβ1δi2-s2m2m1δi2I0δir-n4I1δirrδiEi(36)σ-θL=∑i=12n6-βLβ1δi2-s2m2m1δi2K0δir-n4K1δirrδiDi+∑i=12n6-βLβ1δi2-s2m2m1δi2I0δir+n4I1δirrδiEiwhere(37)n3=λL+2GLλS+2GS,n4=2GLλS+2GS,n6=λLλS+2GS


## 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 (r=R2), the following can be obtained:(38)u-rS=u-rL,σ-rS=σ-rL,θ-S=θ-L,σ-θS=σ-θL,P-=0Since the explosion load is equivalent to the combination of temperature and force load, for the inner boundary of the lining (r=R1), there is (39)σ-rL=q-s,θ-L=θ-sBy 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

Whenβ2=0,c1=0,b=0, 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 as(40)λS+2μS∂eS∂r-β1∂θS∂r=ρS∂2urS∂t2The heat conduction Equation (2) can be written as(41)∂∂t+τ0∂2∂t2ρScsθS+β1T0eS=κS∇2θSTheir constitutive relationship is changed as follows:(42)σrS=2μS∂urS∂r+λSeS-β1θS(43)σθS=2μSurSr+λSeS-β1θS


### 6.2. Solution of the Problem without Lining

When the thickness of the lining is zero, that is,h=0, 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 r=R2 should be (44)σ-rS=-q-s,θ-S=θ-s,P-=0By substituting Equations (20)-(22) into (44)(45)∑i=13β2R2kiK1kiR2Ai+Ai-Bi-MiK0kiR2=-q-s(46)∑i=13K0kirBi=θ-s(47)∑i=13-Ai-α1Bi+α3MiK0kir=0Therefore, the expression for the undetermined coefficientBi 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]:(48)σ-rS=0,θ-S=θ-s,P-=0where θ-(s)=1-e-s/s2θ0.


## 6.1. Solution without considering the Diffusion Effect

Whenβ2=0,c1=0,b=0, 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 as(40)λS+2μS∂eS∂r-β1∂θS∂r=ρS∂2urS∂t2The heat conduction Equation (2) can be written as(41)∂∂t+τ0∂2∂t2ρScsθS+β1T0eS=κS∇2θSTheir constitutive relationship is changed as follows:(42)σrS=2μS∂urS∂r+λSeS-β1θS(43)σθS=2μSurSr+λSeS-β1θS


## 6.2. Solution of the Problem without Lining

When the thickness of the lining is zero, that is,h=0, 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 r=R2 should be (44)σ-rS=-q-s,θ-S=θ-s,P-=0By substituting Equations (20)-(22) into (44)(45)∑i=13β2R2kiK1kiR2Ai+Ai-Bi-MiK0kiR2=-q-s(46)∑i=13K0kirBi=θ-s(47)∑i=13-Ai-α1Bi+α3MiK0kir=0Therefore, the expression for the undetermined coefficientBi 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]:(48)σ-rS=0,θ-S=θ-s,P-=0where θ-(s)=1-e-s/s2θ0.


## 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 functionF(s) be the Laplace transform of the function F(t), then the Crump inversion algorithm of Laplace inverse transform is(49)Ft≈eatT∗12Fa+∑k=1∞ReFa+kπiT∗cos⁡kπtT∗-ImFa+kπiT∗sin⁡kπtT∗If F(t)<Meαt, the error is ς≤Meυe-2T∗(a-υ), where, T∗>t/2.The parameters of the soil in case study refer to [20], and the specific parameters are as follows: GS=3.86×107Pa, vS=0.3, ρS=1800kg/m3, τ0=0.02s, τ=0.2s, D=0.85×10-8, T0=293K, αt=1.78×10-5Co-1, αc=1.2×10-4Co-1, cs=2000m2s-2Co-1, κS=3.8W, c1=12, b=9×105, t-=1.The parameters of the lining are as follows:ρL=2440kg/m3, GL=3.5×108Pa, vL=0.55, h=0.2m, R1=3m, R2=3.2m, αL=4.78×10-5K-1, κL=5.6W, cL=2710m2·s-2C°-1.


### 7.1. Comparative Analysis

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

As shown in Figure3, 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 t=0.5. 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.Figure 3
Comparison of calculation results with or without lining.
(a)
Temperature gradient (b)
Radial displacement


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

Figure4 shows that the dimensionless time is t=0.5, 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.Figure 4
Comparison of calculated results with results from [20].
(a)
Temperature gradient (b)
Radial displacement


### 7.2. Parameter Analysis

#### 7.2.1. Time-Domain Analysis Procedure

Figure5 shows set plots of the variation of the temperature gradient in the soil with respect to the radius, when the dimensionless time is t=0.1, t=0.5, and t=1. 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 t=0.5 and r/R2<2.8, 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 t=0.1 and t=1. 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 5
Variations of the temperature gradient with respect to the radius at different times.Figure6 shows the attenuation rule for radial displacement of the soil with radii, at t=0.1, t=0.5, and t=1. 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 t=0.1 and t=0.5, and the radial displacement at the time t=0.1 is greater than the displacement at t=0.5. As the radius increases, the displacement amplitude gradually decreases. At time t=1, 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 6
Curves showing variation of the radial displacement with radius at different times.Figure7 shows a set of curves for the variation of the radial stress of the soil with radius at the time t=0.1, t=0.5, and t=1. It can be seen that, at t=0.5, the radial stress at the interface between the lining and the soil is positive, while, at the time t=0.1 and t=1, 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 7
Curves showing variation of the radial stress with respect to radius at different times.Figure8 shows a curve for variation of the diffusion flow intensity with respect to radius at times t=0.1, t=0.5, and t=1. 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 t=0.5, 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 t=0.1 is larger than that at t=1. As the radius increases, a negative value occurs first and then gradually decreases. At the time t=1, 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.Figure 8
Curve showing variation of the diffusion flow intensity with respect to radius at different times.


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

Figures9 and 10 show the effect of the diffusion coefficient on the temperature gradient and chemical potential. When these are evaluated at time t=0.5, the relevant coefficients are D=0.5×10-8, D=0.85×10-8, and D=1.15×10-8. 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 1<r/R2<3. 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.Figure 9
Curves showing the influence of the diffusion coefficient on the temperature gradient.Figure 10
Curves showing the influence of the diffusion coefficient on the chemical potential.


## 7.1. Comparative Analysis

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

As shown in Figure3, 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 t=0.5. 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.Figure 3
Comparison of calculation results with or without lining.
(a)
Temperature gradient (b)
Radial displacement


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

Figure4 shows that the dimensionless time is t=0.5, 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.Figure 4
Comparison of calculated results with results from [20].
(a)
Temperature gradient (b)
Radial displacement


## 7.1.1. The Comparison of Calculated Results without Lining

As shown in Figure3, 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 t=0.5. 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.Figure 3
Comparison of calculation results with or without lining.
(a)
Temperature gradient (b)
Radial displacement


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

Figure4 shows that the dimensionless time is t=0.5, 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.Figure 4
Comparison of calculated results with results from [20].
(a)
Temperature gradient (b)
Radial displacement


## 7.2. Parameter Analysis

### 7.2.1. Time-Domain Analysis Procedure

Figure5 shows set plots of the variation of the temperature gradient in the soil with respect to the radius, when the dimensionless time is t=0.1, t=0.5, and t=1. 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 t=0.5 and r/R2<2.8, 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 t=0.1 and t=1. 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 5
Variations of the temperature gradient with respect to the radius at different times.Figure6 shows the attenuation rule for radial displacement of the soil with radii, at t=0.1, t=0.5, and t=1. 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 t=0.1 and t=0.5, and the radial displacement at the time t=0.1 is greater than the displacement at t=0.5. As the radius increases, the displacement amplitude gradually decreases. At time t=1, 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 6
Curves showing variation of the radial displacement with radius at different times.Figure7 shows a set of curves for the variation of the radial stress of the soil with radius at the time t=0.1, t=0.5, and t=1. It can be seen that, at t=0.5, the radial stress at the interface between the lining and the soil is positive, while, at the time t=0.1 and t=1, 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 7
Curves showing variation of the radial stress with respect to radius at different times.Figure8 shows a curve for variation of the diffusion flow intensity with respect to radius at times t=0.1, t=0.5, and t=1. 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 t=0.5, 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 t=0.1 is larger than that at t=1. As the radius increases, a negative value occurs first and then gradually decreases. At the time t=1, 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.Figure 8
Curve showing variation of the diffusion flow intensity with respect to radius at different times.


### 7.2.2. The Impact of the Diffusion Coefficient

Figures9 and 10 show the effect of the diffusion coefficient on the temperature gradient and chemical potential. When these are evaluated at time t=0.5, the relevant coefficients are D=0.5×10-8, D=0.85×10-8, and D=1.15×10-8. 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 1<r/R2<3. 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.Figure 9
Curves showing the influence of the diffusion coefficient on the temperature gradient.Figure 10
Curves showing the influence of the diffusion coefficient on the chemical potential.


## 7.2.1. Time-Domain Analysis Procedure

Figure5 shows set plots of the variation of the temperature gradient in the soil with respect to the radius, when the dimensionless time is t=0.1, t=0.5, and t=1. 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 t=0.5 and r/R2<2.8, 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 t=0.1 and t=1. 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 5
Variations of the temperature gradient with respect to the radius at different times.Figure6 shows the attenuation rule for radial displacement of the soil with radii, at t=0.1, t=0.5, and t=1. 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 t=0.1 and t=0.5, and the radial displacement at the time t=0.1 is greater than the displacement at t=0.5. As the radius increases, the displacement amplitude gradually decreases. At time t=1, 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 6
Curves showing variation of the radial displacement with radius at different times.Figure7 shows a set of curves for the variation of the radial stress of the soil with radius at the time t=0.1, t=0.5, and t=1. It can be seen that, at t=0.5, the radial stress at the interface between the lining and the soil is positive, while, at the time t=0.1 and t=1, 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 7
Curves showing variation of the radial stress with respect to radius at different times.Figure8 shows a curve for variation of the diffusion flow intensity with respect to radius at times t=0.1, t=0.5, and t=1. 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 t=0.5, 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 t=0.1 is larger than that at t=1. As the radius increases, a negative value occurs first and then gradually decreases. At the time t=1, 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.Figure 8
Curve showing variation of the diffusion flow intensity with respect to radius at different times.


## 7.2.2. The Impact of the Diffusion Coefficient

Figures9 and 10 show the effect of the diffusion coefficient on the temperature gradient and chemical potential. When these are evaluated at time t=0.5, the relevant coefficients are D=0.5×10-8, D=0.85×10-8, and D=1.15×10-8. 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 1<r/R2<3. 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.Figure 9
Curves showing the influence of the diffusion coefficient on the temperature gradient.Figure 10
Curves showing the influence of the diffusion coefficient on the chemical potential.


## 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 tunnel’s cavity. The thermoelastic dynamic response of a soil-lining system under the explosion’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 ast=0.1, t=0.5 and t=1, 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 range1<r/R2<3, 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.



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*Source: 2535980-2019-03-20.xml*