Thermo-hydro-mechanical dynamic response of a cylindrical lined tunnel in a poroelastic medium with fractional thermoelastic theory

Thermo-hydro-mechanical dynamic response of a cylindrical lined tunnel in a poroelastic medium with fractional thermoelastic theory

Soil Dynamics and Earthquake Engineering 130 (2020) 105960 Contents lists available at ScienceDirect Soil Dynamics and Earthquake Engineering journa...

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Soil Dynamics and Earthquake Engineering 130 (2020) 105960

Contents lists available at ScienceDirect

Soil Dynamics and Earthquake Engineering journal homepage: http://www.elsevier.com/locate/soildyn

Thermo-hydro-mechanical dynamic response of a cylindrical lined tunnel in a poroelastic medium with fractional thermoelastic theory Minjie Wen a, *, Jinming Xu a, Houren Xiong b a b

Department of Civil Engineering, Shanghai University, Shanghai, 200444, PR China College of Engineering and Architecture, Jiaxing University, Jiaxing, 314001, PR China

A R T I C L E I N F O

A B S T R A C T

Keywords: Saturated soil Cylindrical lined tunnel Fractional thermoelastic theory Coupled thermo-hydro-mechanical dynamic response Laplace transform

This study deals with the coupled thermo-hydro-mechanical dynamic response of a cylindrical lined tunnel in a poroelastic medium when subjected to the joint action of a thermal and mechanical source. With the fractionalorder thermoelasticity theory, the fully coupled thermo-hydro-mechanical dynamic model is presented based on the equations of motion, fluid flux, and fractional heat conductivity. The soil is regarded as a saturated poro­ thermoelastic medium and solved by the generalized energy equation of porothermoelasticity with fractional derivative. Furthermore, the lining structure is equivalent to the elastic material by the theory of thermoelastic shell. The study derives the temperature increment, displacement, stresses, and pore water pressure by using the differential operator decomposition method and the Laplace transform method. The influences of the fractional derivative parameter on the responses are discussed and the numerical results compared with the results of the theory of saturated porous medium and the cavity without lining.

1. Introduction The coupled thermo-hydro-mechanical dynamic response of a lining structure in a saturated porous medium is important in the field of subway tunnels, thermal pipelines, petroleum and natural gas trans­ portation pipelines, ocean, and river tunnels. These lining structures are always designed in a context of harsh environments, e.g., high temper­ atures or impact loads, which may result in the interaction among pore water pressure, stress, and temperature in soil, leading to deformation of soil and lining [1]. Therefore, the dynamic response of a lined tunnel in a porous or elastic medium has been widely studied. Previous researches mainly dealt with the lined tunnel surrounded by a thermoelastic me­ dium without pore water or a fluid-saturated poroelastic medium, neglecting the influence of the temperature. Ezzat et al. [2], Kundu and Mukhopadhyay [3], Sharma et al. [4], Sherief and Saleh [5], and Xia et al. [6] developed the thermoelastic theory and thermo-viscoelasticity theory with pore water to study the dynamic response of the under­ ground structures. Besides, Chen et al. [7], Gao et al. [8], Hasheminejad and Komeili [9], and Liu et al. [10] studied the dynamic response of the underground structure by the theory of saturated porous medium, which neglect the influence of the heat source. Nevertheless, due to complexity in control equation and coupling mechanism, the dynamic response

considering the coupling among heat (temperature field), fluid (seepage field), and force (stress and displacement field) is poorly investigated for an underground structure in saturated porous medium subjected to a thermal source and impact load. Biot [11] developed the classic coupled theory of thermoelasticity based on the classical Fourier heat conduction law. Based on this model, many scholars modified the heat conduction law and established a full coupled thermo-hydro-mechanical model. For example, Bai et al. [12] considered the compressibility of saturated porous thermoelastic media and developed a coupled thermo-hydro-mechanical responses model which analyzes the irre­ versible thermal consolidation problem of a two-phase saturated poro­ thermoelastic spherical cavity under different boundary conditions. Wang and Dong [13] developed a numerical solution with the finite element method (FEM) to couple the thermo-elastic-plastic dynamic response of saturated soil. Liu et al. [14] presented a fully coupled thermo-hydro-elastodynamic model based on the equations of motion, fluid flow, feat flow, and constitutive equation and the temperature change, and investigated the thermo-hydro-elastodynamic response of a spherical cavity in isotropic saturated poroelastic medium. Tao et al. [15] analyzed the characteristics of wave propagation in a saturated thermoelastic porous medium. Singh et al. [16] established a thermo­ elastic wave theory for saturated porous media, and then, Singh and

* Corresponding author. E-mail address: [email protected] (M. Wen). https://doi.org/10.1016/j.soildyn.2019.105960 Received 19 January 2019; Received in revised form 4 November 2019; Accepted 4 November 2019 Available online 22 November 2019 0267-7261/© 2019 Published by Elsevier Ltd.

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Soil Dynamics and Earthquake Engineering 130 (2020) 105960

influences of the fractional derivative parameter and lining structure on the temperature, displacement, and stresses distribution seem still un­ clear. Therefore, this paper applies the equation of fractional derivative heat conduction to the saturated porothermoelastic dynamic model to address this deficiency and analyzes the transient response of a cylin­ drical lined tunnel. Also, the lining structure is considered by applying the theory of thermoelastic shell, and the Laplace transform method is adopted to the analysis. By numerical implementation, the influences of the fractional derivative parameter and lining structure on the poro­ thermoelastic responses are evaluated. The rest of the paper is organized as follows. Section 2 describes the background of the problem and points out the development of the mathematical model. Section 3 illustrates the control equations of the saturated soil and their solutions. In Section 4, the control equations of the lining and solutions are obtained, while the numerical calculation and analysis are provided in Section 5. Finally, conclusions are drawn.

Fig. 1. Calculation model.

2. Mathematical model

Tomar [17] examined the reflection and propagation characteristics of the transverse wave on the interface of saturated porous medium. Levy et al. [18,19] investigated the propagation rules of thermoelastic waves in saturated porous thermoelastic media. Based on Biot thermoelastic wave theory, Darcy law, and Fourier heat conduction law, Liu et al. [20] established a thermo-hydro-elastodynamic model for saturated poro­ thermoelastic media and analyzed the coupled thermo-hydro-elastodynamic response of saturated porous media with cylindrical cavities subjected to a time-dependent thermal/mechanical source. By considering heat conduction between fluid and solid, Sherief and Hussein [21] developed the thermo-hydro-elastodynamic model of saturated porous materials and analyzed the dynamic response of a half-space saturated porothermoelastic medium under thermal shock. Yang [22] proposed a mathematical model of the dynamic-thermo-hydro-mechanical coupling in a non-local thermal equilibrium fluid-saturated porous medium and presented the several Gurtin-type variational principles. He et al. [23] established the thermo-hydro-elastodynamic model based on Biot’s theory and the concept of weighted average temperature and analyzed the thermo­ elastic dynamic response of saturated porous media in a cylindrical or spherical cavity. In the mentioned research, the conventional poro­ thermoelasticity is based on the principles of the classical theory of heat conductivity and generalized thermoelastic theory. Due to the different physical properties of porous materials, thermal waves show different heat transfer phenomena while propagating. However, the studies fail to describe the transient temperature field in situations involving short times, high frequencies, and small wavelengths [24]. In order to solve the above problems, Sherief et al. [24,25] derived the equation of generalized thermoelasticity with fractional derivative, and Youssef et al. [26] proposed a damped version of Fourier’s law by taking the fractional Taylor-Riemann series of time-fractional order. Afterward, many scholars extensively applied the heat conduction equation to analyze the thermoelastic response of half-space formed of a material, spherical cavity, and infinitely long hollow cylinder [27–30], and the effects of the fractional derivative parameter on the dynamic response of single-phase medium. It appears that the mentioned re­ searches mainly deal with an underground structure surrounded by a thermoelastic medium; the soil should be more properly modeled as a porothermoelastic medium in practical applications. Moreover, frac­ tional calculus was successfully used to modify many existing models of physical processes and served as mathematical objects to describe many systems in the real world. Also, the fractional derivative parameter was regarded as an influencing factor of delay time. In other words, different orders can be used to describe different heat conduction phenomena, and the effect of heat conduction can be varied by changing parameters. The coupled thermo-hydro-mechanical dynamic response of a cy­ lindrical lined tunnel in a porothermoelastic medium with fractional derivative heat conduction has not been deeply investigated yet. The

Consider an infinite, homogeneous, isotropic porothermoelastic medium in which a cylindrical lined tunnel with a radius of R and thickness of h (see Fig. 1) exists. It can be regarded as a plane strain axisymmetric problem in polar coordinates. In Fig. 1, R0 is the distance between the tunnel center and the middle surface of the lining structure. The lining is a kind of uniform elastomer. Since the thickness h of the lining is far smaller than the radius of tunnel in practical engineering (i. e., h « R), one can assume that R0 is the distance between the tunnel center and the contact surface between lining and saturated soil. The thickness of liner is assumed to be very small to the radius of tunnel that there is no need to distinguish whether time-dependent thermal source T (t) or mechanical source q(t) acts on the internal or external boundary of the lining [31]. This study also assumes that both lining and saturated porous medium deform a little, and no temperature difference exists between soil skeleton and pore fluid under the action of an external thermal source, i.e., no heat conduction occurs between them. The interface between the lining and soil satisfies the continuity condition, no relative slip can be observed, and the thermophysical properties of both lining and soil are invariant with the temperature change. This work adopts the new fractional Taylor-Riemann series with time-fractional order to expand and retaining terms up to order in the thermal relaxation time [25]: qS þ

τγ0

∂γ qS ¼ kr2 θ Γð1 þ γÞ ∂tγ

(1)

where k is the thermal conductivity, θ ¼ T-T0 is the temperature incre­ ment, T is the absolute temperature of the medium, T0 is the reference uniform temperature of the body chosen such that |(T- T0)/T0|«1, γ is a fractional derivative parameter (0 < γ � 1), qS is the heat flux, Γ(1þγ) is the coefficient of thermal relaxation and denotes the Gamma function of the independent variable (1þγ); t is the time. Z ∞ Γð1 þ γÞ ¼ tγ e ð1þγÞ dt (2) 0

3. Control equation of saturated soil and solutions According to the Fourier heat conduction law, the equation of heat conduction can be written as: � � � � ∂ τγ0 ∂1þγ ∂ τγ0 ∂1þγ ’ m T (3) θ þ λ e ¼ kr2 θ þ þ 0 ∂t Γð1 þ γÞ ∂t1þγ ∂t Γð1 þ γÞ ∂t1þγ where m¼(1-n)ρsCsþ nρwCw, Cs and Cw are specific heat of soil mass, soil particles, and water, respectively. ρs and ρw are the densities of soil skeleton and water, respectively; n is the porosity of the soil; k¼(1-n)ksþ nkw, ks and kw denote the heat conduction coefficients of soil mass, soil 2

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particles, and water, respectively; the thermal modulus λ’ ¼ Kac, K ¼ λ þ 2G/3 is the drained bulk modulus of the soil medium, and ac is thermal expansion coefficient G is the shear modulus; e¼(∂ur/∂r)þ(ur/r) is the volume strain, ur is the radial displacement, r is polar coordinates; r2 ¼ ∂2 ∂r 2

r6

þ is the Laplace operator. By considering the compressibility of a fluid-solid two-phase me­ dium, the motion equation of saturated soil can be written as [14,15]: ðλ þ 2μÞr2 e

αr2 p

λ’ r2 θ ¼ ðρ

ρw α Þ

2

2

∂e ∂θ þ ρw au 2 ∂t2 ∂t

ρw ∂ p M ∂t2

ξ1 ¼ Z1 þ x1 þ Z1 ψ 1 ;

Z1 ¼ s þ

(4)

where λ ¼ 2vμ=ð1 2vÞ and μ ¼ G are Lam�e constants; v and G are the Poisson’s ratio and the shear modulus, respectively; α ¼ 1 Kb = Ks are the compressibility coefficient of soil skeleton; Kb and Ks are the volume modulus of soil skeleton and soil particles, respectively; M is Biot modulus (1=M ¼ n=Kw þ ð1 nÞ=Ks );Kw is the volume modulus of pore water; ρ ¼ ð1 nÞρs þ nρw is the total density of soil; au ¼ naw þ ð1 nÞas ac ð1 αÞ, as and aw are the thermal expansion coefficients of soil particles and pore fluid, respectively; p is the pore water pressure. The equation of fluid motion can be written as [14,15]: κS r2 p ¼ α

∂e ∂t

au

∂θ 1 ∂p S �α þ þ κ ρw ∂t M ∂t n

� ∂2 e 1 ∂t 2

κS ρw au ∂2 θ κS ρw ∂2 p þ n ∂t 2 nM ∂t2

.

ðλ þ 2GÞ p* ¼ αp

. ðλ þ 2GÞ;

σ *ij ¼ σ ij

. ðλ þ 2GÞ;



h

� sþ

� i � � τγ0 τγ0 s1þγ þ r2 θ ¼ s þ s1þγ ψ 1 e Γð1 þ γÞ Γð1 þ γÞ

(9)

where s is Laplace transform parameter; e ¼ R∞ p ¼ 0 e st pdt; φ1 ¼

ρ

R∞ 0

e

st

edt; θ ¼

ρw α ρ au ðλ þ 2GÞ ρw ðλ þ 2GÞ ; φ2 ¼ w ; φ3 ¼ ρ ραM λ’ ρ

R∞ 0

e

st

ϕ1 s

x2 ¼ ϕ3 φ1 s3 þ φ1 ϕ6 s4

φ 3 ϕ 1 s3

α2 ðλ þ 2GÞκS η

; ϕ2 ¼

� 1 ; ϕ5 ¼

k22 ¼

(12)

ρw ðλ þ 2GÞ T0 ðλ’ Þ ; ψ1 ¼ nρM mðλ þ 2GÞ

2

ϕ3 φ2 s3

(20)

ϕ6 φ2 s4

1h ξ 3 1

p1

(22)

�pffiffi �i 3 cosðq1 Þ þ sinðq1 Þ

(23)

(6)

τ*0 ¼ V 2 ητ0

k23 ¼

�pffiffi 1h ξ1 þ p1 3 cosðq1 Þ 3

p1 ¼

� qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ξ21 3ξ2 ; q1 ¼ arc sin 3

�i sinðq1 Þ 2ξ31

(24) 9ξ1 ξ2 þ 27ξ3 2p31

� (25)

According to the three kinds of waves P1, P2, and T, in saturated porothermoelastic medium, propagation in pore water and soil skeleton, respectively, which were denoted by Liu et al. [14]. The solution of Eq. (21) can be written as θ ¼ θ1 þ θ2 þ θ3 ; p ¼ p1 þ p2 þ p3 ; e ¼ e1 þ e2 þ e3 , and. So, Eq. (21) can be decomposed as follow: � � � (26) r2 k21 ðθ; p; eÞ ¼ 0; r2 k22 ðθ; p; eÞ ¼ 0; r2 k23 ðθ; p; eÞ ¼ 0

(10)

ρw au αðλ þ 2GÞ nρλ’

ϕ6 ¼

(19)

ϕ5 s2

ϕ2 s

For solving Eq. (26):

(11)

ρw α �α ρ n

(18)

φ3 ϕ4 s4

1 k21 ¼ ð2p1 sinðq1 Þ þ ξ1 Þ 3

θdt;

αau 1 ; ϕ3 ¼ S κ ηM λ’ κ S η

ϕ4 ¼

φ2 s2

(17)

ϕ4 s2

Where k1 ; k2 and k3 are the characteristic roots of Eq. (14), denoting the velocities of propagation of three possible waves, i.e., the compressional (P1 and P2) wave, and thermal (T) wave. Roots k1 ; k2 and k3 are given by

3 X

θ¼

3 X

Ai K0 ðki rÞ þ i¼1

ϕ1 ¼

(16)

x1 ¼ ϕ3 s þ ϕ6 s2 þ φ1 s2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððλ þ 2GÞÞ=ρ;

(8)

(15)

According to the differential operation decomposition theory, pro­ cessing Eq. (14) leads to the following expression: � � � r2 k21 r2 k22 r2 k23 ðθ; p; eÞ ¼ 0 (21)

(5)

� � � ϕ6 s2 p ¼ ϕ1 s þ ϕ4 s2 e þ ϕ2 s þ ϕ5 s2 θ

sϕ3

ξ3 ¼ Z1 x2 þ Y2 Z1 ψ 1

s1þγ Γð1 þ γÞ

Y2 ¼ φ3 ϕ2 s3 þ φ3 ϕ5 s4

Substituting Eq. (6) into Eqs. (3)–(5) and performing the Laplace transform, the following expressions can be derived: � � � r2 s2 φ1 e ¼ r2 þ φ2 s2 θ þ r2 þ φ3 s2 p (7) r2

ξ2 ¼ Z1 x1 þ x2 þ Y1 Z1 ψ 1 ;

τγ0

Y1 ¼ ϕ3 s þ ϕ6 s2

where κS ¼ kl =ρw g is the coefficient of fluid motion, kl is the intrinsic permeability (m/s); g is the gravitational acceleration (m/s2). In order to solve Eqs. (3)–(5), this paper introduces the following dimensionless quantities: � r* ¼ V ηr; u*r ¼ V ηur ; t* ¼ V 2 ηt η ¼ m k;

θ* ¼ ðλ’ θÞ

(14)

where

1 ∂ r ∂r

2

� ξ3 ðθ; p; eÞ ¼ 0

ξ1 r4 þ ξ2 r2

Di I0 ðki rÞ

(27)

i¼1

where K0(x) is the modified Bessel function of the second kind and order 0, I0(x) is the modified Bessel function of the first kind and order 0; ki (i ¼ 1, 2, 3), is characteristic roots of Eq. (14). Note that I0(x)→∞ when x→∞, so constants Di should equal to zero. Then, the solutions of Eq. (26), which is bounded at infinity, are given by 3 X

(13)

θ¼

Ai K0 ðki rÞ i¼1

By substituting Eq. (8) and Eq. (9) into Eq. (7), the following expression can be got: 3

(28)

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3 X

θ¼

Bi K0 ðki rÞ

(29)

Ci K0 ðki rÞ

(30)

The thickness of the liner is assumed to be so small to the radius of the tunnel that there is no need to distinguish whether a thermal source or a mechanical source is applied at the middle surface (r ¼ R0 ) or at the contact surface between lining and soil [33]. Also, this study assumes that the interface between lining and soil satisfies the continuity con­ dition. Then, the boundary condition in Laplace transform can be expressed as

i¼1 3 X

e¼ i¼1

where Ai, Bi, Ci (i ¼ 1, 2, 3) are arbitrary constants. By substituting Eqs. 28–30 into Eq. (7) and Eq. (8) yields the relation: Bi ¼ ω1i Ai ;

L

uLr ¼ ur ; θ ¼ θ; QðsÞ ¼ qðsÞ

(31)

Ci ¼ ω2i Ai

(1) For a suddenly applied constant load TðsÞ ¼

Integrating both sides of Eq. (30) from zero to infinity, and assuming that ur vanishes at infinity, one obtains ur ¼ i¼1

Ci K1 ðki rÞ ki

β2

ur r 2

T θ

σ ¼ 1

θ

(32)

TðsÞ ¼



β eþβ

TðsÞ ¼ 2 ur

θ

r

(34)

p

2

σTr ,

σ Tr ¼

i¼1

σ Tθ ¼

β Ci K1 ðki rÞ þ ðCi rki

Bi

� � 3 � X � β2 Ci K1 k i r þ 1 rki i¼1

� Ai ÞK0 ðki rÞ � β 2 Ci

Bi

(36)

The lining can be treated as a uniformly elastic circular thin shell. According to Timoshenko shell thermodynamic theory, after Laplace transform, the motion equation of the lining can be written as [31,32]: Z

h 2 h 2

Δ1 ¼

Δ2 ¼

Δ3 ¼

R20 ð1

ð1

e s Þ2 θ0 ð1 ; qðsÞ ¼ s2

e s Þ2 q0 s2

(47)

i¼1 3 X ½K0 ðki R0 Þ�Ai ¼ TðsÞ

(39)

EL α L R0 ð1 vL Þλ’

(40)

(48)

i¼1

(38)

ρL hðλ þ 2GÞη2 ρ2

(44)

(45)

3 X ½ki K1 ðki R0 Þω1i �Ai ¼ 0

where EL h v2L Þðλ þ 2GÞ

(43)

h 2

(37)

L

θ dr ¼ QðsÞ

e s Þq0 s2

Substituting Eqs. (28), (29), (32) and (35) into Eqs. (37), (41) and (45), one obtains � � 2 β ω2i 2 3 Δ1 Δ2 s2 K1 ðki R0 Þþ R0 ki 6 7 7 h 3 6 X 6 7 6 7Ai ¼ qðsÞ (46) Z2 6 7 7 i¼1 6 4 ðω2i ω1i 1ÞK0 ðki R0 Þ þ Δ3 K0 ðki rÞdr5

4. Control equation of lining and solutions

Δ1 uLr þ Δ2 s2 uLr þ Δ3

e s Þθ0 ð1 ; qðsÞ ¼ s2

∂p ¼0 ∂r

(35) � �� �� Ai K 0 k i r

ð1

Where θ0 and q0 are the maximum load and heat shock acting on inner surface of the lining structure, respectively. Assuming that the contact surface between lining and soil is imper­ meable, then:

σ Tθ

where β ¼ 2G=ððλ þ 2GÞ Þ, and are radial stress and hoop stress of a porous medium, respectively. By substituting Eqs. (28)–(30) and (32) into Eqs. 33 and 34, both radial stress and circumferential stress can be expressed as: 3 � 2 X

(42)

(3) For a triangular pulse load

(33)

p

θ0 q0 ; qðsÞ ¼ s s

(2) For a gradually applied step load

By considering the effect of temperature, the following stress-strain constitutive relation can be established:

σ Tr ¼ e

(41)

where qðsÞ and TðsÞ are the load and heat shock in Laplace transform domain, respectively, and satisfy the following expressions:

where ω1i and ω2i are the two terms of constants; � � k2 þ φ2 s2 ðϕ1 s þ ϕ4 s2 Þ þ k2i φ1 s2 ðϕ2 s þ ϕ5 s2 Þ � 2 � � ; ω2i ω1i ¼ 2 i k2i þ φ3 s2 ðϕ1 s þ ϕ4 s2 Þ ki φ1 s2 ki ϕ3 s ϕ6 s2 � � � k2 þ φ2 s2 k2i ϕ3 s ϕ6 s2 þ k2i þ φ3 s2 ðϕ2 s þ ϕ5 s2 Þ � � � ; i ¼ 1; 2; 3 ¼ i2 k2i þ φ3 s2 ðϕ1 s þ ϕ4 s2 Þ ki φ1 s2 k2i ϕ3 s ϕ6 s2

3 X

σTr ; θ ¼ TðsÞ

The solution of the linear system of Eqs. 46–48, gives the undeter­ mined constants A1, A2, and A3: A3 ¼

A2 ¼

ða3 b1 qðsÞb1

where EL is elastic modulus, h is thickness, vL is Poisson ratio, ρL is L

density, uLr is normalized radial displacement, θ is normalized tem­ perature increment, and αL thermal expansion coefficient and all these variables relate to the lining; QðsÞ is the net outward radial pressure.

A1 ¼ where 4

qðsÞb1 ðb1 d2 a1 b3 Þðb1 d2

b2 d1 Þ b2 d1 Þ

ða3 b1 a1 b3 ÞA3 a2 b1 a1 b2

b2 A2 þ b3 A3 b1

TðsÞb1 ða2 b1 a1 b2 Þ ða2 b1 a1 b2 Þðb1 d3 b3 d1 Þ

(49) (50) (51)

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M. Wen et al.



β2 Δ1 Δ2 s 2 R0 Z h2 þ Δ3 K0 ðki rÞdr



ai ¼

ω2i ki

K1 ðki R0 Þ þ ðω2i

ω1i

1ÞK0 ðki R0 Þ (52)

h 2

bi ¼ ki K1 ðki R0 Þω1i

(53)

di ¼ K0 ðki R0 Þ

(54)

Accordingly, the coupled thermo-hydro-mechanical dynamic response of the system of soil and cylindrical lined tunnel under the joint action of different forms of thermal shocks and force load can be obtained. 5. Numerical results and discussions It is difficult to directly obtain analytical solutions of temperature increment, radial displacement, stress, and pore water pressure and examine the influence of fractional derivative parameter γ on the dy­ namic response. A numerical inverse transform appears a better solu­ tion; currently, there exist many Laplace inverse transform methods, among which the application of Crump’s numerical inversion of Laplace transform is the most accurate. Assuming that F(s) is the Laplace transform of the function F(t), Crump’s inversion algorithm of the Laplace inverse transform can be written as:

Fig. 2. Time history of radial displacement under a sudden constant mechan­ ical source. THMD and TMD cases.

8 > > > > > > > > at <

9 1 FðaÞþ > > > 2 > > > �� � � > > 2 3 = k π i k π t e Re F a þ cos * * FðtÞ � * T T ∞ 6 7 X > T > > 6 7> > > > 6 7> > > � � �� > 4 5> > k¼1 > > kπ i kπ t > : Im F a þ * sin * ; T T

(55)

*

If jFðtÞj < Meαt , the error jςj � Meυ e 2T ða υÞ , where T* > t=2. According to Ref. [14], the parameters of saturated soil in the calculation can be set to: � v ¼ 0:35; G ¼ 1 � 106 Pa; ρs ¼ 2610 kg m3 ; Kw ¼ 3:3GPa; γ ¼ 0:25; ks � ¼ 3:29 J ðs � m � � CÞ; T0 ¼ 300K; Ks ¼ 59GPa; Cw � � ¼ 4186m2 ⋅s 2 ⋅� C 1 ; ac ¼ 3 � 10 5 C 1 ; g ¼ 9:8 m s; as

Fig. 3. Time history of radial displacement for THMD and TMD cases under a sudden constant heat source.

5�



C 1 ; aw ¼ 3 � 10 4 C 1 ; n ¼ 0:4; kl � � ¼ 1 � 10 8 m s; kw ¼ 0:582 J s⋅m⋅� C; Cs � � ¼ 937 m2 s 2 ⋅� C 1 ; ρw ¼ 1000 ​ kg m3 ; Kb ¼ 3 � 10

¼ 2:95GPa; and τ0 ¼ 0:02s; θ0 ¼ 1; q0 ¼ 1: (56) According to Ref. [8], the parameters of the lining in the calculation can be set as below: � ρL ¼ 2440kg m3 ; EL ¼ 2:5 � 108 Pa; vL ¼ 0:55; h ¼ 0:2m; R ¼ 6m; R0 ¼ 6:1m and αL ¼ 1:78 � 10 5 K

1

(57) 5.1. Comparative analysis 5.1.1. Case 1: Thermo-elastodynamic response For an ideal thermoelastic medium, there is no fluid in the soil for ρw ¼ 0 and n ¼ 0, and the governing equation of the coupled thermo-hydromechanical response case described in Section 2 can also be reduced to that of a general thermoelastic medium.

Fig. 4. Comparison between numerical results presented in this paper and those in Ref. [8].

ðλ þ 2μÞr2 e

5

λ’ r2 θ ¼ ρ

∂2 e ∂t2

(58)

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Soil Dynamics and Earthquake Engineering 130 (2020) 105960

Fig. 5. (a) Time history of radial displacement under the joint action of a sudden constant heat source and mechanical source. Lining and no-lining cases (b) Time history of pore water pressure under the joint action of a sudden constant heat source and mechanical source. Lining and no-lining cases.

Fig. 6. (a) Influence of γ on temperature increment along the radius under the joint action of a sudden constant heat source and mechanical source. (b) Influence of γ on temperature increment along the radius under the joint action of a gradually applied step heat source and mechanical source. (c) Influence of γ on temperature increment along the radius under the joint action of a triangular pulse heat source and mechanical source.

6

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Soil Dynamics and Earthquake Engineering 130 (2020) 105960

Fig. 7. (a) Influence of γ on radial displacement with normalized time under the joint action of a sudden constant heat source and mechanical source. (b) Influence of γ on radial displacement with normalized time under the joint action of a gradually applied step heat source and mechanical source. (c) Influence of γ on radial displacement with normalized time under the joint action of triangular pulse heat source and mechanical source.



ρC E





the TMD model has a faster heat transfer rate. The above comparisons indicate that the derivation in Section 3 is correct, and the numerical method is also efficient.



∂ τγ0 ∂1þγ ∂ τγ0 ∂1þγ θ þ λ’ T0 e ¼ kr2 θ þ þ 1þγ ∂t Γð1 þ γÞ ∂t ∂t Γð1 þ γÞ ∂t1þγ

(59)

Where CE is the specific heat of the thermoelastic medium. To compare with the coupled thermo-hydro-mechanical dynamic response medium, ρCE is defined as ð1 nÞρs Cs þ nρw Cw and ρ ¼ ð1 nÞρs þ nρw , k ¼ ð1 nÞks þ nkw . When γ ¼ 1, Eq. (59) can also be reduced to the LordShulman model of linear thermoelasticity [34]. Ezzat [35] established a new mathematical model of porothermoelasticity with fractional order avoiding the negative temperature defect of the Lord-Shulman model. Figs. 2 and 3 give the time history of radial displacement at the surface between soil and lining for the coupled thermo-hydromechanical dynamic model (THMD) and thermoelastic dynamic model (TMD), respectively, under a suddenly applied constant mechanical source and heat source. The calculation results for the THMD and TMD models do not have big differences in terms of the effect of mechanical source (Fig. 2). The peak value of radial displacement calculated by the THMD model is slightly higher than that of the TMD model because there is no external heat source on the inner surface of lining tunnel under the mechanical source. However, for the suddenly applied con­ stant heat source, radial displacement of the THMD model is signifi­ cantly larger than that of thermoelastic model because the coefficient of thermal expansion of the water is much larger than that of soil skeleton. When the inner surface of the lining tunnel is subjected to thermal shock,

5.1.2. Case 2: Hydro-mechanical dynamic response Without considering the effect of thermal source, the governing equations of the thermo-hydro-mechanical dynamic response case in Section 2 can be reduced to those of the saturated porous medium. ðλ þ 2μÞr2 e

κ S r2 p ¼ α

σ Tr ¼ e σ Tθ ¼ 1

αr2 p ¼ ðρ

ρw αÞ

∂e 1 ∂p S �α þ þ κ ρw ∂t M ∂t n

β2

ur r

1

ρw ∂2 p M ∂t 2

� ∂2 e κS ρ ∂2 p w þ ∂t2 nM ∂t2

(60) (61) (62)

p

� ur β2 e þ β2 r

∂2 e ∂t2

p

(63)

Gao et al. [8] developed the transient response of a cylindrical lined cavity in a poroelastic medium and investigated the influences of different parameters of the saturated soil and lining on dynamic response. In order to establish a comparison with the coupled thermo-hydro-mechanical dynamic response medium, this paper ig­ nores the influences of the temperature of soil and lining and the 7

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Fig. 8. (a) Influence of γ on pore water pressure along the radius under the joint action of a sudden constant heat source and mechanical source. (b) Influence of γ on pore water pressure along the radius under the joint action of a gradually applied step heat source and mechanical source. (c) Influence of γ on pore water pressure along the radius under the joint action of a triangular pulse heat source and mechanical source.

additional mass of poroelastic medium. Fig. 4 gives the time history of radial displacement at the surface between soil and lining under a sudden constant mechanical source. The calculated results of the degradation of the model in this paper are consistent with those in Ref. [8], but slightly larger than the displacement response because this paper regards the lining as the shell structure, and ignores one-half of thickness of the lining during the boundary processing.

lining is larger than that of soil. Since the cavity boundary is imper­ meable, the lining structure less influences the pore water pressure. 5.2. Effect of fractional derivative parameter The fractional derivative parameter γ is an impact factor of the relaxation effects of the saturated porothermoelastic model and de­ scribes the propagation characteristics of the thermal wave in the saturated porothermoelastic medium. This paper investigates the in­ fluence of γ on the dynamic responses of a cylindrical lined cavity in a saturated porothermoelastic medium under three different types of heat source and mechanical source. The study considers the variations of temperature increments of the soil with the radial distance for three types of loads, i.e., a suddenly applied load of constant magnitude, a gradually applied step load, and a triangular pulse load (Fig. 6.) Ac­ cording to the figure, the maximum values of temperature increments of the soil occur at the surface between soil and lining, and with the in­ crease of the radius that attenuates gradually. This behavior suggests that the distribution of temperature increments is related to the type of load; furthermore, with the increase of γ, the temperature increment gradually increases. Fig. 7 shows the time history of radial displacement at the surface between soil and lining varying γ. The fractional derivative parameter little influences the radial displacement of soil. Moreover, note that the

5.1.3. Case 3: No lining structure Without the influence of lining structure, the boundary conditions of the dynamic response of saturated porothermoelastic medium in the Laplace transform domain are:

σ Tr ¼ qðsÞ; θ ¼ TðsÞ;

∂p ∂p ¼ 0; θ ¼ TðsÞ; ¼0 ∂r ∂r

(64)

By substituting Eqs. (28), (29) and (33) into Eq. (64), a solution for the arbitrary constants Ai, Bi, and Ci for the present problem from the boundary conditions can be obtained. This paper compares the calculation results of the lined tunnel with that of the unlined tunnel to analyze the effect of the lining. Fig. 5 shows the time history of radial displacement and pore water pressure at the cavity surface under the joint action of a sudden constant heat source and mechanical source. The displacement of the lined tunnel is signifi­ cantly smaller than that of unlined tunnel because the stiffness of the 8

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Fig. 9. (a) Influence of γ on radial stress along the radius under the joint action of a sudden constant heat source and mechanical source. (b) Influence of γ on radial stress along the radius under the joint action of a gradually applied step heat source and mechanical source. (c) Influence of γ on radial displacement along the radius under the joint action of a triangular pulse heat source and mechanical source.

Fig. 10. (a) Distributions of temperature increment with radial distance for a sudden constant heat source and mechanical source. (b) Distributions of temperature increment with radial distance for triangular pulse heat source and mechanical source.

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Fig. 11. (a) Distributions of radial displacement with radial distance for a sudden constant heat source and mechanical source. (b) Distributions of radial displacement with radial distance for a triangular pulse heat source and mechanical source.

6. Conclusions

solution corresponding to a gradually-applied step load is nearly equal to that at a constant load. For the triangular pulse load, the radial displacement has obvious peak value and the fractional derivative parameter affects the displacement at the peak. Fig. 8 plots the variation of pore water pressure of the soil with the radial distance for t ¼ 1. Note that the pressure of soil under a gradually applied step load decays much faster than that under a sudden constant load. For a triangular pulse load, however, the pore water pressure is negative at the surface between soil and lining; also, a peak value of the pore water pressure appears along the radius. Furthermore, the pore water pressure gradually increases with the increase of γ. This behavior suggests that the heat transfer due to the temperature increase in­ fluences the seepage. Fig. 9 shows the variation of soil radial stress with radial distance for t ¼ 1. The fractional derivative parameter influences the radial stress significantly for a sudden constant load, while the influence on radial stress for a gradually applied step load is minimal. Also, the influence of γ on radial stress is the maximum at the interface.

Based on the theories of fractional thermoelastic and thermoelastic shell, this paper presents the analytical solutions of the thermo-hydromechanical dynamic response in the Laplace domain for a cylindrical lined tunnel in an infinite saturated porothermoelastic medium sub­ jected to the heat source and mechanical source on the cavity surface. The influence of the lining structure and fractional derivative on the responses is discussed using the Crump’s numerical inversion method of Laplace transform. The following conclusions can be drawn. With the THMD and TMD models, the radial displacement under the action of a sudden constant mechanical source is smaller than that under the action of a sudden constant heat source. The radial displacement of the soil calculated by the THMD model is larger than that by the TMD model. The radial displacement of the unlined circular tunnel is larger than that of the lined circular tunnel under the joint action of the sudden constant heat source and mechanical source. Moreover, the appropriate increase in the thickness of lining can reduce the radial displacement of soil. The lining structure affects little the pore water pressure under the impermeable condition at the cavity interface. The fractional derivative parameter influences the relaxation effects in the saturated porothermoelastic model and describes the propagation characteristics of thermal waves in the saturated porothermoelastic medium. The fractional derivative parameter has a significant influence on temperature increment and pore water pressure, but a small one on radial displacement and radial stress. The saturated porothermoelastic model with fractional derivative is adopted to describe the mechanical behavior of the soil more accurately than the thermoelastic and the saturated porothermoelastic models do. The temperature increment of the soil at different instants under the joint action of a sudden constant heat source and a mechanical source is significantly different from that under triangular pulse heat source and mechanical source. Under triangular pulse heat source and mechanical source, the temperature increment of the soil fluctuates significantly; also, the peak of the shock wave gets smaller and moves to the radial direction gradually with the increase of the time. The radial displacement of the soil at different instants under the joint action of a sudden constant heat source and mechanical source differs significantly from the one under the triangular pulse heat source and mechanical source. Under the joint action of a triangular pulse heat source and mechanical source, the radial displacement of the soil rea­ ches a peak, and as the dimensionless time increases, the peak postpones

5.3. Distribution of temperature and radial displacement with radial distance Fig. 10 shows the variations of temperature increment θ with radial distance for a sudden constant and triangular pulse loads (heat source and mechanical source), respectively. Under a sudden constant load (heat source and mechanical source), the temperature increment significantly increases with the increase of time while the one increases slowly, and the attenuation gradually slows down after t ¼ 5. The temperature increment fluctuates significantly under a triangular pulse load (heat source and mechanical source), and with the increase of time, the peak value of temperature increment moves along the radius and gradually reduces. Fig. 11 shows the variations of radial displacement with radial dis­ tance for a sudden constant and triangular pulse load (heat source and mechanical source, respectively). Depending on the type of load, radial displacements of soil have significant differences; the radial displace­ ment for a sudden constant load attenuates with the increasing of the radial distance, while in the case of triangular pulse load, θ has an obvious peak value that postpones with the increase of time. When the dimensionless time t ¼ 10, the radial displacement of soil is negative at the interface.

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and gradually decreases. At the dimensionless time t ¼ 10, the radial displacement of the soil is negative.

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Acknowledgments This research was supported by the Natural Science Foundation of China (No. 41472254), the Zhejiang Provincial Natural Science Foun­ dation of China (No. LGF18E080010), and the Primary Research and Development Plant of Zhejiang Province (No. 2019C03120). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.soildyn.2019.105960. References [1] Yousefi H, Kani AT, Kani IM. Response of a spherical cavity in a fully-coupled thermo-poro-elastodynamic medium by cell-adaptive second-order central high resolution schemes. Undergr Space 2018;3:206–17. https://doi.org/10.1016/j. undsp.2018.04.003. [2] Ezzat MA, Othman MI, El-Karamany AMS. State space approach to twodimensional generalized thermo-viscoelasticity with two relaxation times. Int J Eng Sci 2002;40:1251–74. https://doi.org/10.1016/S0020-7225(02)00012-5. [3] Kundu RM, Mukhopadhyay B. A thermoviscoelastic problem of an infinite medium with a spherical cavity using generalized theory of thermoelasticity. Math Comput Model 2005;41:25–32. https://doi.org/10.1016/j.mcm.2004.07.009. [4] Sharma JN, Kumari N, Sharma KK. Diffusion in a generalized thermoelastic solid in an infinite body with a cylindrical cavity. J Appl Mech Tech Phys 2013;54:819–31. https://doi.org/10.1134/S0021894413050155. [5] Sherief HH, Saleh HA. A half-space problem in the theory of generalized thermoelastic diffusion. Int J Solids Struct 2005;42:4484–93. https://doi.org/ 10.1016/j.ijsolstr.2005.01.001. [6] Xia RH, Tian XG, Shen YP. The influence of diffusion on generalized thermoelastic problems of infinite body with a cylindrical cavity. Int J Eng Sci 2009;47:669–79. https://doi.org/10.1016/j.ijengsci.2009.01.003. [7] Chen G, Cai YQ, Liu FY, Sun HL. Dynamic response of a pile in a transversely isotropic saturated soil to transient torsional loading. Comput Geotech 2008;35: 165–72. https://doi.org/10.1016/j.compgeo.2007.05.009. [8] Gao M, Wang Y, Gao GY, Yang J. An analytical solution for the transient response of a cylindrical lined cavity in a poroelastic medium. Soil Dyn Earthq Eng 2013;46: 30–40. https://doi.org/10.1016/j.soildyn.2012.12.002. [9] Hasheminejad SM, Komeili M. Effect of imperfect bonding on axisymmetric elastodynamic response of a lined circular tunnel in poroelastic soil due to a moving ring load. Int J Solids Struct 2009;46:398–411. https://doi.org/10.1016/j. ijsolstr.2008.08.040. [10] Liu GB, Xie KH, Liu XH. Dynamic response of a partially sealed tunnel in porous rock under inner water pressure. Tunn Undergr Space Technol 2010;25:407–14. https://doi.org/10.1016/j.tust.2010.02.005. [11] Biot MA. Thermoelasticity and irreversible thermodynamics. J Appl Phys 1955;27: 240–53. https://doi.org/10.1063/1.1722351. [12] Bai B. Thermal response of saturated porous spherical body containing a cavity under several boundary conditions. J Therm Stress 2013;36:1217–32. https://doi. org/10.1080/01495739.2013.788389. [13] Wang X, Dong J. Formulation and study of thermal-mechanical coupling for saturated porous media. Comput Struct 2003;81:1019–29. https://doi.org/ 10.1016/S0045-7949(02)00476-5. [14] Liu GB, Xie KH, Zheng RY. Thermo-elastodynamic response of a spherical cavity in saturated poroelastic medium. Appl Math Model 2010;34:2203–22. https://doi. org/10.1016/j.apm.2009.10.031. [15] Tao HB, Liu GB, Xie KH, Zheng RY, Deng YB. Characteristics of wave propagation in the saturated thermoelastic porous medium, transp. Porous Media 2014;103: 47–68. https://doi.org/10.1007/s11242-014-0287-6. [16] Singh B. On propagation of plane waves in generalized porothermoelasticity. Bull Seismol Soc Am 2011;101:756–62. https://doi.org/10.1785/0120100091. [17] Singh J, Tomar SK. Reflection and transmission of transverse waves at a plane interface between two different porous elastic solid half-spaces. Appl Math Comput 2006;176:364–78. https://doi.org/10.1016/j.amc.2005.09.027. [18] Levy A, Sorek S, Ben-Dor G, Bear J. Evolution of the balance equations in saturated thermoelastic porous media following abrupt simultaneous changes in pressure and temperature. Transp Porous Media 1995;21:241–68. https://doi.org/10.1007/ bf00617408. [19] Sorek S, Levy A, Ben-Dor G, Smeulders D. Contributions to theoretical experimental developments in shock waves propagation in porous media. Transp Porous Media 1999;34:63–100. https://doi.org/10.1023/A:1006553206369. [20] Liu GB, Xie KH, Zheng RY. Model of nonlinear coupled thermo-hydroelastodynamics response for a saturated poroelastic medium. Sci China Ser E Technol Sci 2009;52:2373–83. https://doi.org/10.1007/s11431-008-0220-8. [21] Sherief HH, Hussein EM. A mathematical model for short-time filtration in poroelastic media with thermal relaxation and two temperatures. Transp Porous Media 2012;91:199–223. https://doi.org/10.1007/s11242-011-9840-8.

Nomenclature R: radius of long cylindrical lined tunnel R0 : distance between tunnel center and the contact surface between lining and saturated porothermoelastic medium h: thickness of lining structure TðtÞ: thermal source qðtÞ: mechanical source k: thermal conductivity T: T0 absolute temperature and reference uniform temperature θ: θ* temperature increment and normalized value γ: fractional derivative parameter qS : heat flux Γð1 þ γÞ: ð1 þγÞ coefficient of thermal relaxation and Gamma function of the independent variable t: t* time and normalized value m: specific heat of soil mass n: soil porosity ρ: ρs , ρw densities of total density, soil skeleton and water Cs : Cw specific heat of soil particles and water k: ks , kw heat conduction coefficients of soil mass, soil particles and water λ’ : thermal modulus K: drained bulk modulus of the soil medium ac : thermal expansion coefficient e: volume strain ur : , u*r radial displacement of the soil and normalized value r: r* polar coordinates and normalized value r2 : Laplace operator λ: μ Lame constants v: Poisson’ ratio G: shear modulus α: compressibility coefficient of soil skeleton Kb : Ks volume modulus of soil skeleton and soil particles M: Biot’s modulus Kw : volume modulus of pore water as : aw thermal expansion coefficients of soil particles and pore fluid p: p* pore water pressure and normalized value κS : coefficient of fluid motion kl : intrinsic permeability g: gravitational acceleration k1 ; k2 : k3 characteristic roots of equation K0 ðxÞ: I0 ðxÞ modified Bessel function of the second kind and order 0, modified Bessel function of the first kind and order 0

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Ai ​ ; Bi ; ​ Ci ​ ; Di : arbitrary constants ω1i : ω2i two terms of constants σTr : σTθ radial stress and hoop stress of a porous medium EL : elastic modulus of the lining h: thickness of the lining vL : Poisson’s ratio of the lining ρL : density of the lining

uLr : normalized radial displacement in Laplace transform L

θ : normalized temperature increment of the lining in Laplace transform αL : thermal expansion coefficient of the lining QðsÞ: net outward radial pressure qðsÞ: TðsÞ load and heat shock in Laplace transform domain θ0 : q0 maximum load and heat shock acting on the inner surface of the lining structure

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