Coupled analytical solutions for deep-buried circular lined tunnels considering tunnel face advancement and soft rock rheology effects

Tunnelling and Underground Space Technology 94 (2019) 103111

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Coupled analytical solutions for deep-buried circular lined tunnels considering tunnel face advancement and soft rock rheology effects

T

⁎

Zhaofei Chua, Zhijun Wua, , Baoguo Liub, Quansheng Liua a b

School of Civil Engineering, Wuhan University, Wuhan 430072, China School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Rock rheology Analytical solution Circular lined tunnel Tunnel face Viscoelastic model

For tunnelling in a deep soft rock mass, rock rheology and the advancing process significantly affect the induced pressures on the support system and the long-term safety of the tunnel. In this study, coupled analytical solutions, which take both the rock rheology and tunnel face advancement effects into consideration, are proposed to predict the mechanical behavior of deep-buried circular lined tunnels in a soft rock mass. To account for different rheological behaviors of rocks, five types of viscoelastic models are adopted in the theoretical derivations. The analytical solutions are validated by comparing the predicted results with corresponding existing solutions and those predicted by finite difference simulations. According to parameter degradation, the proposed solutions can be simplified to the classic viscoelastic solution without consideration of tunnel face advancement and the elastic solution with consideration of tunnel face advancement. Based on the proposed coupled solutions, the characteristics of convergent displacements of unlined tunnels with or without consideration of the tunnel face advancement, are studied. In addition, for lined tunnels, the influences of the initial stress release parameter, influence radius of the tunnel face, excavation rate and liner installation time on rock stress and displacement and support pressure are systematically investigated. Finally, the proposed solution is successfully applied to analyze the time-dependent behavior of the Lyon-Turin Base Tunnel, and good agreements are achieved between the theoretically predicted values and corresponding field monitoring data.

1. Introduction

spatial support effect. In this area, the radial stress of the surrounding rock is partially released, and the closer to the tunnel face, the smaller the proportion of stress that is released is Graziani et al. (2005), Meguid et al. (2003), Paraskevopoulou and Diederichs (2018), Shalabi (2005). However, during tunneling, the tunnel face will constantly advance forward, and the spatial support effect a certain distance away from the tunnel face will gradually disappear, which causes the in situ stress of the surrounding rock at this part to gradually be released (Weng et al., 2017). As a result, the pressure on the liner behind the tunnel face will gradually increase with the tunnel face advancement. For soft rock, owing to its poor quality and strong rheological characteristics, the tunnel should be supported in time, and the liner is usually installed within such a constrained area (Roateşi, 2014; Sulem et al., 1987). As a result, for tunneling in soft rock, the liner is subjected not only to the pressure resulting from the stress release induced by tunnel face advancement but also to the pressures resulting from rock rheology, which means that both the tunnel convergence and the mechanic interaction between rock and liner are affected by the coupled effect of rock rheology and tunnel face advancement (Kontogianni et al., 2006).

Soft rock generally has obvious rheological characteristics, even at low stress levels, displaying obvious viscoelasticity (Zhao et al., 2018). Many creep tests on soft rocks show that the rheological deformation generally accounts for more than 30% of their total deformation, which may, in some cases, be up to 70%. (Maranini and Brignoli, 1999; Tomanovic, 2006; Zhang et al., 2015, 2012). In practical projects, when tunnels are driven through such soft rocks, such as the Lyon-Turin Base tunnel (Bonini and Barla, 2012), Meuse/Haute-Marne URL (Guayacán et al., 2016) and Shibli tunnel (Sharifzadeh et al., 2013), the tunnel convergences exhibit strong time-dependent characteristics, which causes the pressures on liners to continuously increase with time. Therefore, taking the time-dependent behavior into consideration is of great importance for the long-term safety of tunnels especially deepburied tunnels constructed in soft rock (Hasanpour et al., 2018; Xing et al., 2017; Zhang and Zhou, 2017). After the tunnel cross-section is excavated, there is a longitudinal arch constrained area in front and behind of the tunnel face due to its

⁎

Corresponding author. E-mail addresses: [email protected] (Z. Chu), [email protected] (Z. Wu).

https://doi.org/10.1016/j.tust.2019.103111 Received 24 April 2019; Received in revised form 31 July 2019; Accepted 8 September 2019 0886-7798/ © 2019 Elsevier Ltd. All rights reserved.

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effect. For Zhao’s solution, the displacement expression contains an unknown parameter, which makes it only applicable to prediction of the tunnel ultimate displacement. As can be found from the above discussion, although there have been many viscoelastic analytical solutions proposed for deep-buried circular tunnels, only a few works have considered the tunnel face advancement. Moreover, the coupled effect of tunnel face advancement and rock rheology is seldom realistically considered in previous viscoelastic solutions. Therefore, in this study, coupled analytical solutions that take both tunnel face advancement and rock rheology effects into consideration are proposed for predicting the mechanical behavior of deep-buried lined circular tunnels in soft rheological rock. To account for different rheological behaviors of rocks, five types of viscoelastic models are adopted in the theoretical derivation. The analytical solutions are compared with previous corresponding solutions for particular viscoelastic models. Moreover, the effectiveness of the analytical solutions is further validated by comparing their results with corresponding finite difference simulation results. Then, based on the analytical solutions, a series of parametric analyses are carried out to investigate their influences on the stress and deformation of the surrounding rock, as well as pressure on support. Finally, the presented solutions are applied to predict the time-dependent behavior of the Lyon-Turin Base Tunnel.

To analyze the time-dependent behavior of tunnels in soft rheological rock, numerous tools have been used, including analytical solutions (Bui et al., 2014; Ladanyi and Gill, 1988; Phienwej et al., 2007), empirical approaches (Sakurai, 1978; Sulem et al., 1987), and numerical simulations (Paraskevopoulou and Diederichs, 2018; Pellet et al., 2009; Roateşi, 2014; Shalabi, 2005). Among them, the analytical solutions have the advantages of low cost and fast calculation and can be used directly to study the mechanism of long-term interaction of rockliner and make the parametric analyses very convenient (Fei et al., 2018). However, for tunneling problems it is actually a three-dimensional (3D) problem when taking the tunnel face effect into account. Additionally, it is still very challenging to solve the analytical solution by directly considering the 3D stress release process. Therefore, to solve the problem analytically, the stress release process caused by 3D excavation is usually assumed to be equivalent to a fictional support pressure acting on the inner wall of the tunnel and changing with the tunnel face movement, so that the actual 3D problem can be transformed into two dimensional (2D) plane strain analysis (CarranzaTorres et al., 2013; Kielbassa and Duddeck, 1991; Sulem et al., 1987). The deduction of an analytical solution is closely related to the constitutive model used. Although various rheological models for the soft rocks have been developed (Debernardi and Barla, 2009; Guan et al., 2008; Yang and Cheng, 2011; Yang et al., 2014), most of the constitutive equations of the models are implicit and in incremental forms, which limit the derivation of the corresponding explicit closedform solutions. In contrast, the constitutive relations of the linear viscoelastic models are expressed by a set of operator equations, which has advantages in theoretical derivation. Moreover, the linear viscoelastic models consisting of a sequence of springs and dashpots in parallel and in series can well capture the simple or complex rheological behavior of various rocks under different stress levels (Nomikos et al., 2011; Wang et al., 2014). Therefore, in this paper, linear viscoelastic models are adopted to account for the rheological behaviors of soft rocks. For circular tunnels in deep soft rock, a series of viscoelastic analytical solutions have been developed to account for the rock-liner interaction (Chu et al., 2017; Ladanyi and Gill, 1988; Lo and Yuen, 1981; Mogilevskaya and Lecampion, 2018; Nomikos et al., 2011; Pan and Dong, 1991; Phienwej et al., 2007; RE, 1989; Song et al., 2018; Wang et al., 2013, 2018). However, few of these works have taken the effect of tunnel face advancement into consideration. When tunneling at a constant rate, the stress release coefficient is a function of time related to the tunnel face advancement, which makes the corresponding solutions very hard to deduce when simultaneously taking the rock viscoelasticity into consideration. To solve these problems, recently, with certain simplifications and assumptions several viscoelastic analytical solutions with respect to the tunnel face effect have been proposed. Fahimifar et al. (2010) proposed a closed-form solution for the circular lined tunnel in Burgers viscoelastic rock, which considered the effect of tunnel face and the stress history prior to tunnel construction. However, in this solution, the rate of tunnel face advancement was assumed to be infinitely large, and the rock behavior was initially assumed to be independent of the rock support pressure (Nomikos et al., 2011). Also based on the Burgers model, Birchall and Osman (2012) deduced the 3D stress and deformation expressions for surrounding viscoelastic rock taking the tunnel face effect into consideration. However, this solution was only for unlined tunnels. Wang et al. (2014) proposed comprehensive analytical solutions for circular tunnels in Maxwell, Kelvin and Kelvin-Voigt rocks with consideration of the influences of excavation sequence and tunnel face advancement. However, in their study, the main focus was on the effect of excavation sequence, while the stress release effect, especially the stress release process effect, was insufficiently studied. Based on the Kelvin-Voigt model, Roateşi (2013) and Zhao et al. (2016) both successfully deduced analytical solutions for lined tunnels with consideration of the tunnel face effect. However, for Roateşi’s solution, the effects of tunnel face and rock rheology were regarded as two independent parts without considering their coupled

2. Definition of the problem This study addresses the problem of a deep-buried circular lined tunnel excavated in rheological soft rock. The following assumptions are made in this regard: (1) A circular tunnel with the radius of r0 is excavated by a full-size method under a hydrostatic stress field p0. (2) The rock is a homogeneous, isotropic and viscoelastic material, and the liner is elastic; (3) The tunnel face advances at a constant rate v along the longitudinal direction, and rock deformation is assumed to be elastic before the tunnel face reaches the analysis cross-section. When the effect of tunnel face advancement is accounted for, as shown in Fig. 1, there exists a longitudinal influence area in front and behind of the tunnel face due to its space restraint effect. As a result, the radial stress release of the surrounding rock exhibits three levels along the tunnel axis direction. Farther in front of or behind the tunnel face, the radial stress of surrounding rock is in the in situ state (pa = p0, where pa is the fictitious support pressure) or the fully released state (pa = 0) because it is beyond the influence range of the tunnel face. Therefore, the stress release coefficients for these two parts are λ(X) = 0 and 1, respectively. However, within the influence range of the tunnel face, the radial stress of the surrounding rock is just partially released (0 ≤ pa ≤ 1), which makes λ(X) vary in the range of 0 ~ 1. Nevertheless, λ(X) can generally be expressed as a negative exponential function of the distance X to the tunnel face (Bian et al., 2013; Sun and Zhu, 1994; Zhao et al., 2016), i.e.,

λ (X ) = 1 − αe−X /RL

(1)

Obviously, the initial stress release coefficient at X = 0 is λ(0) = 1−α, where α is defined as the initial stress release parameter. The previous studies based on elastic and elastoplastic analyses show that the initial stress release coefficient λ(0) is usually in the range of 0.25–0.45 (Graziani et al., 2005; Kielbassa and Duddeck, 1991), and thus α equals 0.55–0.75. RL is the influence radius of the tunnel face, whose value depends on the rock quality and tunnelling method (Panet and Guenot, 1982; Sakurai, 1978). When tunnelling continuously at a constant rate v, X = vt, and the stress release coefficient λ(X) in Eq. (1) can be rewritten as 2

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The influence range of tunnel face

0

( X )=0

( X )=1

( X) 1

Rock Tunnel axis

Tunnel face

v

r0

0 (0)

1 Ȝ(X)

X

X0

pa

p0

Liner

Support starting point

pa+q

q

Stress release coefficient curve

Fig. 1. Illustration of the construction sequence of the tunnel along the longitudinal direction.

λ (t ) = 1 − αe−mt

(2)

modulus of the rock; and J(t) and K are both related to rock properties. sij and eij are the deviator tensors of the stress σij and strain εij, respectively. εm is the mean strain corresponding to the mean stress σm. By definition,

where m = v/RL. Therefore, when α = 0 or m → ∞ in Eq. (2), λ(t) = 1, indicating no stress release effect. As mentioned previously, the problem of tunnel 3D excavation can be equivalent to the problem in 2D plane strain analysis. The radial stress release process of the surrounding rock induced by tunnel face advancement can be implemented by introducing the fictitious support pressure applied on the tunnel wall in the plane-strain model (CarranzaTorres et al., 2013; Graziani et al., 2005). By definition, the fictitious support pressure pa(t) is a function of the stress release coefficient, which is given by

pa (t ) = [1 − λ (t )] p0

sij = σij − σm ⎫ eij = εij − εm ⎬ ⎭

In Eq. (4), the asterisk (*) represents the convolution operation and is given as,

f1 (t ) ∗ df2 (t ) = f1 (t ) f2 (0) +

∫0

t

f1 (t − ξ )

∂f2 (ξ ) dξ ∂ξ

(6)

(3) Soft rocks with different qualities and types, as well as the same soft rock subjected to different stress levels, generally exhibit different viscosities. Rock with relatively good quality of mechanical properties or at low stress levels always shows limited viscosity, i.e. only the attenuated rheology stage exists. This type of rheological behavior can be well described by the Kelvin, Kelvin-Voigt and Poynting-Thomson models (Dai et al., 2004; Zhao et al., 2016; Zhifa et al., 2001). Instead, for soft rock with poor quality or soft rock under high stress, remarkable viscosity may arise, and the characteristic of steady creep is obvious. In this case, the Maxwell and Burgers models can be commonly employed (Asadollahpour et al., 2014; Paraskevopoulou and Diederichs, 2018; Sharifzadeh et al., 2013). In this study, the above five typical viscoelastic models, which are shown in Fig. 3, will be adopted in the analytical solution. In addition, the corresponding creep compliances J(t) for the five models (see Eq. (4)) are also presented in Fig. 3, where td is the retardation time of each model, and td = ηk/Gk for figures (a), (b), (c) and (e), but td=(1 + Gm/Ge)ηm/Gm for figure (d). From Fig. 3, it can be clearly noted that the Burgers model is a relative compounded model that can degenerate to the other simple viscoelastic models. When Gk→∞ or ηk → ∞, the Burgers model degenerates to the Maxwell model; when ηm→∞, it degenerates to the Kelvin-Voigt model; and when Gm→∞ and ηm→∞, it can degenerate to the Kelvin model. Moreover, because the forms of the constitutive equations of the Kelvin-Voigt model and the Poynting-Thomson model are basically the same (Dai et al., 2004; Zhifa et al., 2001), the parameters of the two models can be transformed into each other. Therefore, in the following analytical derivations, the problem of tunnel excavation and support in the Burgers model will first be solved. Then, considering the parameter degeneration, the corresponding solutions for the Maxwell, Kelvin and Kelvin-Voigt models can subsequently be obtained. Meanwhile, the solution of the Poynting-Thomson model can

Therefore, as shown in Fig. 2, what we studied here becomes a 2D plane-strain problem for the axisymmetric lined tunnel in the hydrostatic stress field p0. The wall of surrounding rock is loaded by the fictitious support pressure pa(t) and support pressure q(t). As mentioned above, the rock is assumed as linear viscoelastic materials in this study. However, for the linear viscoelastic analysis, the volume deformation of rock is usually assumed to be linear elastic, but the rock rheology is considered to be caused only by deviatoric stress (Cristescu ND, 1998). Therefore, the three-dimensional viscoelastic integral constitutive of rock can be written as (Findley et al., 1977):

eij = J (t ) ∗ dsij ⎫ εm = σm/3K ⎬ ⎭

(4)

where the function J(t) is called the creep compliance; K is the bulk

p0 Rock

q(t)

uR(t)

p0

(5)

pa(t)+q(t) r0

r1 r 0

p0 Liner

p0 Fig. 2. The 2D plane strain analysis model of a tunnel. 3

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(a) Maxwell model

J

(c) Kelvin-Voigt model

(b) Kelvin model

J

J

(d) Poynting-Thomson model

(e) Burgers model Gm

J

J Fig. 3. Five typical viscoelastic models and the corresponding creep compliances J(t).

be derived based on the parameter transforms of the Kelvin-Voigt solution. In addition, when the rheological property of rock is not considered (i.e., the parameters ηm → ∞ in the Maxwell model), the elastic solution accounting for the tunnel face effect can be obtained as well.

Δsr (t ) = ⎡1 − λ (t ) ⎣

r 02 ⎤p r2 ⎦ 0

Δsθ (t ) = ⎡1 + λ (t ) ⎣

r 02 ⎤p r2 ⎦ 0

Δσm = σm − σm = 3. Derivation of analytical solutions

− σm − (p0 − σm) = −λ (t ) − σm − (p0 − σm) = λ (t ) 2(1 + μ) p0 3

−

2(1 + μ) p0 3

r 02 p⎫ r2 0⎪

r 02 p r2 0

=0

⎪ ⎬ ⎪ ⎪ ⎭

(12)

3.1. Solution for unlined tunnels

Obviously, for the axisymmetric plane strain analysis, because σm is invariable during excavation, no volume deformation occurs, i.e., εm(t) = 0. Hence, from Eqs. (4)–(6), the tangential strain deviation caused by excavation is equal to

In this paper, the tunnel is assumed to be excavated at t = 0, and the liner makes contact with the rock mass at time t = t0. Therefore, as shown in Fig. 2, stress conditions at the boundary of the rock mass are,

εθ (t ) = J (t )Δsθ (0) +

σr (r → ∞, t ) = p0

(7)

pa (t ), t < t0 σr (r0, t ) = ⎧ ⎨ p t ( ⎩ a ) + q (t ), t ⩾ t0

(8)

σθ (t ) = ⎡1 + ⎣

⎭

uR (t ) = (9)

−

∫0

t

J (t − ξ ) λ′ (ξ ) p0 dξ

(14)

αr 0 p0 2

r 0 p0 2

1

⎡G + ⎣ m

1 t ηm

⎡ 1 −( 1 − mηm ⎣ mηm

+

1 Gk

(1 − e−t / td)⎤

1 ) e−mt Gm

⎦

+

( e−mt − e−t / td ) Gk (1 − mtd)

⎤ ⎦

(15)

Subsequently, the rock displacements for the other four viscoelastic rocks can be derived by the degeneration of Eq. (15), and are all given as follows: (a) For the Maxwell rock (Gk → ∞ or ηk → ∞ in Eq. (15)),

(10)

where μ is the Poisson's ratio of the rock. The mean stress σm of the rock before and after excavation is also the same and can be calculated as follows:

σm = [σr (t ) + σθ (t ) + σz (t )]/3 = 2(1 + μ) p0 /3

(13)

Therefore, when we substitute the expressions of λ(t) (see Eq. (2)) and J(t) into Eq. (14), uR(t) will be easily obtained. Here, we first consider the Burgers model, and thus the corresponding expression of uR(t) can be obtained as follows:

where r is the radial distance from the center of the tunnel. For the plane strain problem (εz(t) = 0), the stresses in the z direction before and after excavation are the same, which can be expressed as

σz = μ (σr + σθ ) = 2μp0

J (t − ξ )[Δsθ (ξ )]′dξ

uR (t ) = J (t ) λ (0) r0 p0 + r0

r 02 ⎤p ⎫ r2 ⎦ 0 ⎪

⎬ r 2 λ (t ) 02 ⎤ p0 ⎪ r ⎦

t

From Eq. (12), it can be seen that the increment of the tangential deviatoric stress Δsθ(ξ) at the tunnel boundary (r = r0) is λ(ξ)p0 and that the corresponding initial value is λ(0)p0. According to the geometrical relationship εθ(t) = uR(t)/r0, the rock displacement uR(t) at the tunnel boundary is

where q(t) is the support pressure. Based on the plane strain solution of a thick-walled cylinder, the radial and tangential stresses of the surrounding rock can be obtained through the above boundary conditions, which are given by

σr (t ) = ⎡1 − λ (t ) ⎣

∫0

uR (t ) =

αr0 p0 ⎡ 1 r0 p0 ⎛ 1 1 ⎞ 1 1 −mt ⎤ )e t⎟ − + −( − ⎜ ⎥ 2 ⎝ Gm 2 ⎢ mη mη G ηm ⎠ m m ⎦ ⎣ m

(16)

(b) For the Kelvin rock (Gm → ∞ and ηm → ∞ in Eq. (15)),

(11)

uR (t ) =

As a result, the increments of deviatoric stress induced by excavation are given by

r0 p0 ⎡ α (e−mt − e−t / td ) ⎤ 1 − e−t / td − ⎥ 2G k ⎢ 1 − mtd ⎣ ⎦

(17)

(c) For the Kelvin-Voigt and Poynting-Thomson rocks (ηm → ∞ in 4

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Laplace space is given by

Eq. (15)),

uR (t ) = −

r 0 p0

αr 0 p0 2

2

1

⎡G + ⎣ 0

⎡ 1 e−mt G0 ⎣

(

) (1 − e + ( − )( 1 GL

1 G0

−

1 GL

1 G0

−t / td

)⎤

e−mt − e−t / td 1 − mtd

) ⎤⎦

Furthermore, note that for α = 0 or m → ∞ in Eqs. (15)–(19), the corresponding solutions of unlined tunnels without consideration of the tunnel face effect can be derived as well.

b1 = α

τ

J (τ − ξ )

dq (ξ ) ⎤ dξ ⎥ dξ ⎦

∫0

dJ (τ − ξ ) ⎤ dξ q (ξ ) d (τ − ξ ) ⎥ ⎦

J (s ) =

1 k − ηk m

(

1 s

1 mηm

−

)e ) −t0 / td

)e ( −mt0

1 s + 1 / td

)e

1 s

−

1 s+m

)⎤⎥ ⎥ ⎥ ⎦

−t0 / td

(28)

1 Gk (1 − mtd)

(

1 Gk

−

+

1 Gm

−

α Gk (1 − mtd)

1 mηm

) exp(−mt )⎫⎪ 0

⎬ ⎪ ⎭

) exp (− ) t0 td

(29)

⎟

(30)

(31)

1 (s ηm

{

+ m)(s + 1

s (m + s ) ( G

m

+

1 ) td

2 ) s2 Ks

+ b1 ms (s + 1

+ ⎡( G + ⎣ k

1 Gm

1 ) td

+ b2 s (s + m) t

1

+

2 1 ) K s td

+

1 ⎤s ηm ⎦

d

+

1 ηm td

} (32)

To further simplify the above equation,

(s + m)(s + 1/ td )/ ηm + b1 ms (s + 1/ td ) + b2 s (s + m)/ td q (s ) = p0 a1 s (s + m)(s − x1)(s − x2)

(33)

where 1 2 + K Gm s 1 2 1 + ) Gm K s td

a1 = 1 (G k

b3 =

(23)

+

c1 =

⎫ ⎪

+

1 ⎪ ηm

1 ηm td

Δ = (b3/ a1)2 − 4c1/ a1 x1,2 =

(24)

−b3 / a1 ± Δ 2

⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(34)

Then, use of the inverse Laplace transformation technique on both sides of Eq. (33) yields

q (τ ) = q1 (τ ) + q2 (τ ) + q3 (τ )

(35)

where (25)

q1 (τ ) =

where Gs and μs are the shear modulus and Poisson's ratio of liner, respectively. r0 and r1 are the outer and the inner radius of the liner. Substituting Eqs. (21), (22) and (24) into Eq. (23), yields:

∫0

−G

−

1 s + 1 / td

−

1 ⎡ 1 1 1 ⎤ + + ⎥ 2s ⎢ G η s η ( s + 1/ t ) m d m k ⎦ ⎣

q (s ) = p0

(21)

where Ks is the support stiffness and its expression is given as below (Ladanyi and Gill, 1988):

g (τ ) − ⎧q (τ ) J (0) + ⎨ ⎩

1 G k − ηk m

1 s

Then, the substitution of Eqs. (30) and (31) into Eq. (27) yields

Furthermore, because the liner is an elastic material, the relationship between the support pressure q(τ) and liner radial displacement us(τ) can be expressed as:

2Gs (1 − r12/ r0 2) 1 − 2μs + r12/ r0 2

+

(

Meanwhile, the J(s) of the Burgers model is expressed as (20)

where g(τ) is an undetermined function, which can be easily determined by the expression of uR(t). Assume that the liner displacement caused by the support pressure q(τ) is us(τ). Hence, there exists a compatibility condition between liner and rock at the boundary after the tunnel is supported (t≥t0):

q (τ ) r0 us (τ ) = Ks

1 Gm

1 Gk

+

⎜

(22)

uR (τ + t0) − uR (t0) − uR '(τ ) = us (τ )

1 ηm s 2

g (s ) 1 b 1 1 ⎞ b 1 1 ⎞ = + 1⎛ − + 2⎛ − p0 2ηm s 2 2 ⎝s s + m⎠ 2 ⎝s s + 1/ td ⎠

Because the rock displacement for the unlined tunnel has been obtained in the above section, from time t0 to time τ + t0, the increment of rock displacement caused by excavation can be written as

uR (τ + t0) − uR (t0) = r0 g (τ )

(

(

Eq. (28) can then be simplified as

where q(0) = 0. Based on the Stieltjes convolution operation (Findley et al., 1977), the above equation can be rewritten as τ

(

b2 =

As mentioned earlier, liner and rock begin to interact at the moment t = t0. For ease of analysis, it is assumed here that τ = t−t0. Combining Eqs. (4)–(6) with Eq. (13), the displacement induced by the support pressure q(τ) can be expressed as

Ks =

2

⎡ ⎢ ⎢ ⎢ ⎣

2

Assuming that,

3.2. Solution for lined tunnels

uR '(τ ) = r0 ⎡q (τ ) J (0) + ⎢ ⎣

αp0

(19)

∫0

p0

g (s ) =

+

r0 p0 (1 − αe−mt ) uR (t ) = 2Gm

(27)

where q(s), g(s) and J(s) are the expressions of q(τ), g(τ) and J(τ) in the Laplace space, respectively. Apparently, Eq. (27) can be used universally for different viscoelastic models if the corresponding J(s) is substituted into it. Therefore, for the Burgers model, according to Eqs. (15) and (22), the expression of g(τ) in Laplace space is given by

(18)

In Eq. (18), G0 and GL are the instantaneous and long-term shear moduli of rock, respectively. For the Kelvin-Voigt rock, G0 = Gm, GL = GmGk/(Gm + Gk) and td = ηk/Gk; while for the Poynting-Thomson rock, G0 = Ge + Gm, GL = Ge and td = (1 + Gm/Ge)ηm/Gm. In addition, according to ηm → ∞ in the Maxwell solution (c.f. Eq. (16)), the elastic displacement solution of an unlined tunnel with consideration of the tunnel face effect is given by

uR '(τ ) = r0 ⎡q (0) J (τ ) + ⎢ ⎣

g (s ) sJ (s ) + 1/ Ks

q (s ) =

⎦

τ

q (ξ )

q (τ ) dJ (τ − ξ ) ⎫ dξ = Ks d (τ − ξ ) ⎬ ⎭

p0 ⎡ e x1 τ − e x2 τ 1 ⎛ x e x1 τ − x1 e x2 τ ⎞ ⎤ 1+ 2 + a1 ηm ⎢ x x t x x x1 − x2 − 1 2 d 1 2 ⎝ ⎠⎥ ⎦ ⎣ ⎜

⎟

(36)

q2 (τ ) =

(26)

Using the Laplace transform technique on parameter τ of both sides of the above equation, the expression of the support pressure q(τ) in

b1 mp0 ⎡ (x1 + 1/ td ) e x1 τ (x2 + 1/ td ) e x2 τ − ⎢ a1 ⎣ (x1 + m)(x1 − x2) (x2 + m)(x1 − x2) +

5

(1/ td − m) e−mτ ⎤ ⎥ (x1 + m)(x2 + m) ⎦

(37)

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Z. Chu, et al.

q3 (τ ) =

b2 p0 (e x1 τ − e x2 τ ) a1 td (x1 − x2)

uR (t ) =

(38)

It can be obviously noted that the values of x1 and x2 in Eq. (34) are both negative. As a result, for τ → ∞ in Eqs. (36)–(38), q1(∞) = p0, q3(∞) = 0, while q2(∞) takes different values for m = 0 and m ＞ 0, namely:

q2 (∞, m > 0) = 0 ⎫ q2 (∞, m = 0) = αp0 ⎬ ⎭

uR (t ) =

σθ (t ) = ⎡1 + λ (t ) ⎣

r 02 ⎤p r2 ⎦ 0

− q (t − t0)

r 02 ⎬ r2 ⎪

⎭

f1 = f2 =

(42)

τ

)

1 + td x 1 ( 1 + ηm / ηk e−t 0/ td ) ⎫ x 1 ηm td (1 / G m + 2 / K s ) ⎪ 1 + td x2 ( 1 + ηm / ηk e−t 0/ td ) ⎬ x2 ηm td (1 / G m + 2 / K s )

⎪ ⎭

f e x1 τ + f2 e x2 τ ⎤ q (τ ) = ⎡1 + 1 p0 ⎢ ⎥ x1 − x2 ⎣ ⎦

(49)

(50)

where

f1 = −

Ladanyi and Gill (1988) reported solutions without the tunnel face effect for Maxwell and Kelvin-Voigt rocks. They provided the following expressions of uR(t) and q(τ) for the two rocks (in our notations): (a) for the Maxwell rock (corresponding to Eqs. (21) and (24) in their paper)

(

(48)

Mogilevskaya and Lecampion (2018) presented the closed-form expressions of support pressure for the four viscoelastic models a), b), c), and e) of Fig. 3, where the tunnel face effect was not considered and the tunnel was assumed to be supported at t0 = 0. Here we chose their Burgers solution for comparative analysis, and the expression of q(τ) (see Eq. (60) in their paper) can be re-written in our notation as

4.1. Comparison with Ladanyi and Gill

q (τ ) = p0 1 − e− ηm a1

(47)

4.3. Comparison with Mogilevskaya and Lecampion

From the above analytical solutions, it can be seen that the solution of each viscoelastic model can degenerate into the corresponding classic solution without considering the tunnel face effect when α = 0 or m → ∞. Therefore, in this section the solutions presented in this study are compared with previously published relevant solutions for particular viscoelastic models, and their similarities and differences are investigated. For the convenience of comparison, the rock displacement uR(t) of an unlined tunnel and support pressure q(τ) of a lined tunnel are chosen as comparison items.

r0 p0 ⎛ 1 1 ⎞ t⎟ + ⎜ 2 ⎝ Gm ηm ⎠

)

and the roots x1 and x2 are the same as those in Eq. (34). It also can be seen that Eqs. (47)–(49) also have the same structures as the Burgers solution with α = 0 or m → ∞ in this paper. When τ → ∞ in Eq. (48), the ultimate value of the support pressure is q(∞) = p0, which is the same as that when considering the tunnel face effect.

4. Comparisons with previous solutions

uR (t ) =

(

where

In addition, the internal stresses of the surrounding rock after supporting can be written as: r 02 ⎫ r2 ⎪

r0 p0 ⎡ 1 1 1 t+ 1 − e−t / td ⎤ + ⎥ 2 ⎢ G η G m k m ⎦ ⎣

f e x1 τ − f2 e x2 τ ⎤ q (τ ) = ⎡1 + 1 p ⎢ ⎥ 0 x1 − x2 ⎣ ⎦

(41)

+ q (t − t0)

(46)

Nomikos et al. (2011) reported solutions for Burgers rock without the tunnel face effect. The expressions of uR(t) and q(τ) can be rewritten in our notation as

As mentioned before, according to the parameter degradation of Gk, ηm and Gm in Eqs. (28)–(38), the closed-form solutions of the lined tunnel for the Maxwell, Kelvin, Kelvin-Voigt and Poynting-Thomson rocks can be obtained as well, which are presented in Appendices A, B and C. Similarly, when the rheology of the rock is not considered, the elastic solution with reference to the tunnel face effect is as presented in Appendix D. Finally, the displacement of the liner us(τ) can also be derived by Eq. (24). Meanwhile, the total displacement of rock at time t is

r 02 ⎤p r2 ⎦ 0

(45)

4.2. Comparison with Nomikos et al

(40)

σr (t ) = ⎡1 − λ (t ) ⎣

⎟

Compared with the solutions in this paper, it can be observed that the uR(t) terms in Eqs. (43) and (45) are the first items of Eqs. (15) and (18), respectively. In addition, the expressions of q(τ) in the above equations are the same as those in Eqs. (A1) and (C1) with α = 0 or m → ∞ in Appendices A and C. Moreover, from Eqs. (44) and (A1), note that, regardless of whether the effect of the tunnel face is considered or not, the ultimate support pressures for τ → ∞ for the Maxwell rock are q (∞) = p0. However, the q(∞) terms for the Kelvin-Voigt rock are different when comparing Eqs. (46) and (A1).

However, m = 0 indicates that neither the tunnel face advancement nor the liner installation can be carried out, which is not true in practice. Therefore, when the tunnel face advances continuously (m＞0), the ultimate support pressure q(∞) = p0, which means that the support pressure for the Burgers rock will finally converge to p0 and will not be affected by other parameters. Correspondingly, the limit tunnel displacement is obtained as

uR (t ) = uR (t0) + us (t − t0)

⎜

b3 p0 Ks (1 − G L / G0 ) e−t0/ td ⎛1 − e− a1 τ ⎞ 2G L + Ks ⎝ ⎠

q (τ ) =

(39)

uR (∞) = uR (t0) + p0 r0/ Ks

r0 p0 ⎡ G − G L ⎞ −t / td⎤ 1−⎛ 0 e ⎥ 2G L ⎢ G0 ⎠ ⎝ ⎣ ⎦

f2 =

2G m (x 1 + Gk / ηk ) ⎫ 2G m + K s

2G m (x2 + Gk / ηk ) 2G m + K s

⎬ ⎭

(51)

where x1 and x2 are the roots as those shown in Eq. (34). For t0 = 0 in Eq. (49), the expressions of f1 and f2 in Eqs. (49) and (51) are different. Furthermore, for Eq. (49), the initial support pressure at τ = 0 is q (0) = 0, while for Eq. (50), q(0) = Ks/(2Gm + Ks). The main reason for this difference is that Eq. (50) assumes that there is no instantaneous elastic deformation of the rock before the tunnel is supported so that the larger pressure will be temporarily exerted on the liner (Mogilevskaya and Lecampion, 2018). Therefore, those solutions are obviously different from those of Ladanyi and Gill (1988), Nomikos et al. (2011) and ours.

(43) (44)

(b) for the Kelvin-Voigt rock (corresponding to Eqs. (14) and (18) in their paper) 6

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Table 1 Parameters for validating the Kelvin-Voigt solution. Kelvin-Voigt rock

Table 2 Parameters for validating the Burgers solution.

Gm = 2.07 GPa; Gk = 2.07 GPa; ηk = 20.69 GPa·day; td = 10 day r0 = 4 m; r1 = 3.65 m; Es = 70.9 GPa; νs = 0.2; Ks = 6.9 GPa α = 0.73;m = 0.5,+∞; RL = 2 m p0 = 7.6 MPa; t0 = 2 day; v = 2 m/day

Liner Stress release parameters Others parameters

Gm = 3447 MPa; ηm = 1.59 × 106 MPa·month; Gk = 345 MPa; ηk = 7.98 × 103 MPa·day; td = 23 year r0 = 4.57 m; r1 = 3.96 m; Es = 16.55 GPa; νs = 0.2; Ks = 2542 MPa α = 0.68;m = 0.6,+∞; RL = 0.84r0 p0 = 5.89 MPa; t0 = 2 month; v = 2.28 m/month

Burgers rock

Liner Stress release parameters Others parameters

4.4. Comparison with Fahimifar et al expressions for the Kelvin-Voigt model were all determined without containing any unknown parameters, and the corresponding support pressure was also presented. Therefore, the solutions proposed in this paper are more extensive and comprehensive for practical application.

Fahimifar et al. (2010) proposed a closed-form solution for a circular lined tunnel in Burgers viscoelastic rock. In their solution, the rock displacement uR(t) of an unlined tunnel and support pressure q(τ) of a lined tunnel were expressed as follows (in our notations):

uR (t ) =

q (τ ) =

λr0 p0 ⎡ 1 1 1 t+ 1 − e−t / td ⎤ + ⎥ 2 ⎢ G η G m k m ⎦ ⎣

(

)

Ks λp0 [J (τ + t0) − J (t0)] Ks J (τ + t0) + 1

5. Validations and comparisons with numerical simulations (52)

In this section, numerical simulations provided by the finite difference software FlAC3D5.01 (Itasca, 2012) will be used to further validate the exactness of the analytical solutions proposed by this paper. Considering that the solutions of the Kelvin-Voigt and Burgers models are relatively complicated and representative, numerical validations will take these two models as examples in the following analyses. Therefore, referring to the previous literatures (Ladanyi and Gill, 1988; RE, 1989), two tunnel cases associated with the two models are selected. Both of the tunnels are circular sections with radius r0 and fullface excavated in an infinite rock mass subjected to a hydrostatic pressure p0. The liners of the two tunnels are both elastic with inner radius r1. Tables 1 and 2 summarized the basic parameters for validating the Kelvin-Voigt and Burgers solutions. The parameters of the rock, liner, geometry and loading are all suggested by Ladanyi and Gill (1988) and Goodman (1989). Because the analytical solutions proposed by this paper are applied to solve the problem of tunnel longitudinal excavation, the three-dimensional numerical simulations are taken into account. As shown in Fig. 4, numerical models are axisymmetric with a size of 6r0×6r0×14 l, where l is the length of each excavation step per cycle. The liners in the numerical models are simulated by the elastic shell element and installed at 2 l behind the tunnel face. To make the meshes match the excavation steps, for Kelvin-Voigt rock: l = 1.92 m, and the creep time in each excavation step is 0.96 day; meanwhile, for Burgers rock: l = 2.3 m, and the creep time in each excavation step is 1 month. Also, in simulation, the creep timestep for the two models are both set as the fixed values of 1×10−4 day−1 and 1×10−4 month−1, respectively. To reduce the boundary effect, the analysis section in the simulation is

(53)

where the stress release coefficient λ was independent of time. Obviously, the two influence factors, i.e., the rock rheology and tunnel face effect, were simultaneously considered in their solution. However, comparing Eq. (52) with Eq. (15), we find that the two factors in Eq. (52) are assumed to affect uR(t) separately, regardless of their coupled effects. In addition, for Eq. (53), the rock behavior is initially assumed to be independent of the rock support pressure (Nomikos et al., 2011), which actually is the ‘‘ageing’’ theory solution (Ladanyi and Gill, 1988). Therefore, when λ = 1, q(τ) in Eq. (53) is significantly different from that in Eq. (48), which has been discussed by Nomikos et al. (2011). 4.5. Comparison with Zhao et al Zhao et al. (2016) also reported an analytical solution to consider the coupling effect of tunnel face advancement and rock rheology, where the concept of “initial equivalent stress” was introduced to concern the spatial effect of tunnel face. However, in their solution, only the Kelvin-Voigt model was taken into account and only the solution of displacement was deduced, whereas the support pressure was not involved. Moreover, since the initial equivalent stress employed in their study was a negative exponential function of the distance X to the tunnel face, the displacement expressions before and after support (i.e. Eqs. (20) and (27) in Zhao’s paper) can be respectively rewritten in our notations as:

uR (t ) =

r0 p0 αr0 p0 e−mt ⎛ 1 1 ⎞ C − + + 1 e−t / td 2G L 2(1 − mtd ) ⎝ G0 GL ⎠ r0

(54)

uR (t ) =

1 (A + Be−mt + C2 e−Mt ) r0

(55)

⎜

(a)

⎟

Rock 6r0 Analysis section

where

A=

B =

GL p0 (b1 + b2) 1 + 2GL / K s

Liner

⎫ ⎪

(b)

M = b3/ a1 ⎬ αp0 K s (td GL / G m − 1) (M − m)(2GL + K s )

⎪ ⎭

6r0

14l Liner

(56)

and C1 and C2 are unknown parameters. Obviously, the above displacement expressions are both in a parametric form with an unknown parameter contained, which make it unable to observe the displacement variation. Also, when t → ∞ in Eqs. (54) and (55), the ultimate values of uR(t) are given as uR(∞) = r0p0/(2GL) and A/r0, which are identical with Eqs. (18) and (C3). Nevertheless, Eqs. (54) and (55) are not comprehensive enough, which are only suitable to predict the ultimate displacements of tunnel. By contrast, in our paper, the displacement

r0 Tunnel axis

1

2

3

4 7l

Tunnel face 5

6

7

v

Rock

Analysis section

Fig. 4. (a) Numerical model (b) sequence of the tunnel excavation and location of the analysis section. 7

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Rock displacement, uR(t) (mm)

Kelvin-vogit model: =0.73,m=0.5,RL=0.5r0 Burgers model: =0.68,m=0.6,RL=0.84r0

0.6

Numerical simulation:

0.4

Kelvin-vogit model Burgers model Fitting curve:

0.2

Kelvin-vogit model Burgers model

m=0.5,uR(t) m=+’,u (t) R

8

1

Numberical simulation Analytical solution Landanyi and Gill (1988)

6

m=+’,q(t)

4

-2

-1

0

1

2

3

4

5

3

0.8 mm

2

m=0.5,q(t)

t0=0.96 day

0

-3

2

0

20

40

60

80

4 100

Time, t/day

X/r0

(a)

Fig. 5. Stress release coefficient λ with X/r0, and determination of the stress release parameters.

0

18

m=+’,uR(t)

16

Rock displacement, uR(t) (mm)

chosen at the middle position (see Fig. 4), where the distance from the outer boundary of the model is 7 l. Moreover, to be consistent with the analytical solution, assume that the time when the tunnel face arrive at the analysis section is t = 0 in the following analysis. Finally, after the entire tunnel is completely excavated, the numerical model will continue to creep until the final time. To be as consistent as possible between analytical solutions and numerical simulations with regard to the stress release coefficient, first, the parameters in the expression of λ (see Eq. (2)) for analytical solutions should be determined by numerical simulations. Fig. 5 shows the distribution of λ within the influence range of the tunnel face. Meanwhile, the fitting curves of λ for the two tunnels are derived by the least-square method, which are also drawn in Fig. 5. The parameters of α, RL and m associated with λ are listed in Tables 1 and 2, where, as described above, m → ∞ means that the tunnel is excavated instantly without the tunnel face effect. Fig. 6 shows results of rock displacement and support pressure in the two rocks calculated by the analytical solution and the numerical simulation. For comparison, the results without the tunnel face effect (m → ∞) calculated by Ladanyi and Gill (1988) and Pavlos Nomikos (2011) are also presented in Fig. 6. Apparently, when m → ∞, the three calculations are practically identical with each other. However, when m = 0.5 for the Kelvin-Voigt rock and m = 0.6 for the Burgers rock, there are some minor differences in rock displacement between the analytical solutions and simulations. For example, at time t = 0, deviations of approximately about 0.8 mm and 0.91 mm can be found for the two cases (see Fig. 6). The main reasons for the gaps between the analytical solutions and the numerical simulations are: although the ratio of the initial stress release at t = 0 is the same for the two methods, it takes some time for the tunnel face to reach the analysis section in the simulations (prior to t = 0), so the initial rock displacement uR(0) for the simulations is composed of the elastic portion induced by the stress release and the viscous portion caused by rock rheology. However, for the analytical solution, the initial position of the tunnel face (at t = 0) is located at the analysis section, which makes uR(0) only comprise the elastic displacement included by the stress release. Therefore, uR(t) in the simulation is slightly larger than that in the analytical solution due to the deviations of uR(0). However, when m → ∞, uR(0) for the two methods are the same, which indicates that their deviations decrease gradually with the increase in m. Furthermore, it can also be observed from Fig. 6 that the support pressures q(t) calculated by the two methods for each model are still consistent, and thus the deviations of the initial rock displacement have a minor effect on support pressure. By and large,

1

14

m=0.6,uR(t)

12

2

Numberical simulation Analytical solution Nomikos et al (2011)

10 8

3

6 4

m=+’,q(t)

4

0.91 mm t0=2 month

2 0 0

20

m=0.6,q(t) 40

60

80

100

Support pressure q(t) (MPa)

0.8

0

Support pressure, q(t) (MPa)

1.0

5 120

Time, t/month

(b) Fig. 6. Rock displacement and liner pressure of the analytical solution and comparison with the numerical simulation results and other existing solutions: (a) Kelvin-Voigt rock; (b) Burgers rock.

some good agreements are achieved between the proposed solutions and numerical simulation. 6. Parametric investigation Based on the provided analytical solutions, it can be seen that the initial stress release parameter α, parameter m, liner installation time t0, liner stiffness Ks and viscoelastic parameters of rocks have effects on the results of solutions. In this section, the influences of those parameters on rock displacement and stress, as well as support pressure, will be investigated. However, as defined, the parameter m is equal to v/RL, and thus, the influences of the excavation rate v and the influence radius of the tunnel face RL should be discussed separately. To account for the two different viscous behaviors of rocks (see Section 2 in this paper), here we will take the Kelvin-Vogit and Burgers models as examples to illustrate this in detail. Moreover, because the displacement convergence laws of unlined and lined tunnels are different, parametric analyses will be conducted for the two cases. 6.1. Parametric investigation for unlined tunnels Fig. 7 shows the rock displacements of unlined tunnels in the KelvinVogit and Burgers rocks with or without consideration of the tunnel face effect. The coordinate axes of the two diagrams in Fig. 7 are the 8

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uR/ue

1+

6.2. Parametric investigation for lined tunnels

Gm Gk

Gm Gk

Gm Gk

(1

v td

)

Because the influences of the support stiffness Ks and the constitutive parameters for the lined tunnels in two viscoelastic rocks have been discussed in previous studies (Cristescu and Duda, 1989; Ladanyi and Gill, 1988; Lo and Yuen, 1981; Nomikos et al., 2011), in this paper, only the influences of the parameters α, v, RL and t0 will be deeply investigated. For each parameter, we focus primarily on its effect on rock stresses (σr(t) and σθ(t)) at the inner boundary of the tunnel (r = r0), rock displacement uR(t) and support pressure q(t), where σr(t), σθ(t) and q(t) are normalized as σr(t)/p0, σθ(t)/p0 and q(t)/p0. Additional parameters besides α, v, RL and t0 are listed in Tables 1 and 2. Moreover, because the time of liner installation is comparatively small relative to that of rock creep, the logarithmic time scale will be adopted in the following analysis.

RL

Viscoelasticity: Without tunnel face effect v / RL

1

With tunnel face effect Elasticity

1 0

t/td

(a)

uR/ue

Gm Gk

1+

+

td

td / t m

tm Viscoelasticity:

Gm

Without tunnel tunn face effect

Gk 1

Gk 0

With tunnel face effect

v / RL

Gm

1

6.2.1. Influence of the initial stress release parameter To better investigate the influence of the initial stress release parameter α, it is assumed to be 0.3, 0.5 and 0.7, i.e., the coefficient of the initial stress release is 70%, 50% and 30%, respectively. The other parameters are still as listed in Tables 1 and 2. Fig. 8 illustrates the influences of α on the temporal evolutions of normalized rock stresses and displacement, as well as normalized support pressure, for the two viscoelastic rocks. Apparently, for a tunnel in Burgers rock (see Fig. 8(c) and (d)), the variation characteristics of each set of curves within approximately t < 103 months are essentially similar to the overall variation characteristics of the corresponding curves in the Kelvin-Voigt rock (see Fig. 8(a) and (b)). In such cases, the same conclusions can be obtained for the two rocks. From Fig. 8(a) and (c), it can be observed that σr(t)/p0 first increases and then decreases gradually until a steady value is reached, while σθ(t)/p0 varies conversely. The changes in the stress path of rock are mainly due to the radial stress release and the increase in support resistance. Fig. 8(b) and (d) show the changes in uR(t) and q(t)/ p0 under different α. At the same t, the larger α is, the larger q(t)/p0 will be and the smaller uR(t) will be. Further analysis indicates that, because the excavation rate v is constant, the larger α is, the smaller the radial stress released during 0 ~ t0 is. Therefore, lower tunnel deformation and larger rock loads are exerted on liners after the tunnel is supported. In addition, for the Kelvin-Voigt rock, it can be seen from Fig. 8(a) and (b) that the ultimate rock stresses (σr(∞),σθ(∞)) and the ultimate support pressure q(∞) are all dependent on the parameter α. Additionally, the larger α is, the larger q(∞) is. However, for the Burgers rock (see Fig. 8(c) and (d)), the rock stresses and the support pressure are all relatively insensitive to the parameter α after the tunnel is supported. For long periods of time, the above three quantities will all converge to the in-situ stress p0, which corresponds to the analysis in Section 3.2 above.

+

td tm

(1

)

v td

Elasticity

RL

(b)

t/td

Fig. 7. Schematics of rock displacements for unlined tunnels in the (a) KelvinVoigt rock; (b) Burgers rock.

normalized displacements uN = uR/ue versus the normalized creep time t/td, where ue = p0r0/(2Gm). As shown in Fig. 7(a), it can be seen that, when accounting for the effect of tunnel face advancement in the Kelvin-Voigt rock, the initial slope of uN at t = 0 is not only proportional to the stiffness ratio Gm/Gk, but also relates to the term vαtd/RL. Additionally, the normalized displacements uN are temporarily affected by tunnel face advancement, while this effect will gradually disappear due to the radial stress release of the surrounding rock. Finally, regardless of consideration of the effect of tunnel face advancement or not, the ultimate uN of the KelvinVoigt rock will eventually converge to 1 + Gm/Gk. For the Burgers rock in Fig. 7(b), it can be seen that the initial slope of uN without the tunnel face effect is only proportional to Gm/Gk and td/tm, where tm = ηm/Gm is relaxation time. However, when the tunnel face advancement effect is considered, the initial slope of uN is also an increasing function of the terms of 1-α and vαtd/RL. Similar to the Kelvin-Voigt rock, the tunnel displacement with consideration of the tunnel face effect is smaller than that without consideration of the tunnel face effect in a short period time. Over longer periods of time, the normalized displacements with or without the effect of tunnel face advancement both continue to increase along an asymptotic line whose intercept is 1 + Gm/Gk and slope is td/tm, without converging to a fixed value. In addition, for comparative analysis, the elastic normalized displacements considering the tunnel face effect (see Eq. (19)) are also drawn in Fig. 7. Note that uN in elasticity varies gradually from 1-α to 1 and that its initial slope is equal to vα/RL. The ultimate normalized displacement and corresponding convergent time in elasticity are both lower than those of viscoelasticity. Therefore, the selection of tunnel liner and the determination of liner installation time for the rheological rock cannot be simply analogous to the elastic analysis.

6.2.2. Influence of the influence radius of the tunnel face In this section, the influences of the influence radius of the tunnel face RL on rock stresses and displacement as well as support pressure are investigated. According to Eq. (1), when α and v are constant, the larger RL is, the broader the range of stresses released is. Therefore, supposing that α and v are constant, which are listed in Tables 1 and 2, the RL parameters of the two rocks are both set to three levels:2.5r0, r0 and 0.5r0. Fig. 9(a) and (c) illustrate the temporal evolutions of normalized rock stresses calculated with different values of RL for the two rocks. It emerges from the figures that, the larger RL is, the gentler the rock stresses change and the smaller the deviatoric stress (σθ−σr) is. Fig. 9(b) and (d) show the changes in uR(t) and q(t)/p0 for the two types of rocks. It can be seen that, in a relatively short period after the liner is installed, the larger RL is, the smaller the support pressure exhibited is. This is because the stress release induced by the tunnel face advancement has not yet completely vanished, and it still plays a dominant role in 9

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10

2.0 1.8

Support starting point

0.1 uR(t)

8

1.4 r

(t)/p0

(t)/p0

0.6

0.2 0.3

6

q(t)/p0

1.6

uR(t) (mm)

(t)/p0

0.0

Support starting point

0.2

4

t0=0.96 day

2 0.1

0.0 0.1

1

10

100

0.6 100

10

t (day)

(a)

(b) 20

r

(t)/p0

0.0

(t)/p0 0.2

1.5

15

Support starting point

uR(t) (mm)

(t)/p0

1

t (day)

2.0

1.0

0.4 q(t)/p0

uR(t)

10

Support starting point

0.6

0.5 5

r

(t)/p0

0.5

q(t)/p0 t0=0.96 day

q(t)/p0

(t)/p0 r

0.4 0.4

0.8

t0=2 month 0.0 10-1

10

0

10

1

t0=2 month

10

2

10

3

10

4

10

5

10

-1

10

0

101

102

t (month)

t (month)

(c)

(d)

103

104

1.0 105

Fig. 8. Normalized stresses (σr/p0 and σθ/p0), rock displacement (uR(t)) and normalized support pressure (q(t)/p0) calculated with different α for tunnels in the two rheological rocks: (a) and (b) Kelvin-Voigt rock; (c) and (d) Burgers rock.

is, the larger the rock displacement prior to the liner installation is, resulting in larger tunnel convergence. Meanwhile, the smaller v is, the smaller the support pressure finally enforced on the liner will be. Moreover, for the Burgers rock (see Fig. 10(d)), the same conclusions can be obtained within approximately t < 103 months. However, at the ultimate limit state, q(∞) in the Kelvin-Voigt rock is basically similar to that in the Burgers rock and is not appreciably affected by the excavation rate v.

support pressure as compared to the rock rheology effect. However, with the gradual completion of the stress release, the larger RL is, the larger the remaining load from the surrounding rock exerted on the liners is. Therefore, when RL = 2.5r0, the ultimate support pressure is the largest. Moreover, similar to the influence of the parameter α, q(∞) and uR(∞) in the Kelvin-Voigt rock are both related to RL. The larger RL is, the larger q(∞) will be and the smaller uR(∞) will be. However, for the Burgers rock, the two quantities are independent of RL because q (∞) = p0.

6.2.4. Influence of the liner installation time To investigate the influences of the liner installation time t0 on rock stresses and displacement as well as support pressure, assume that t0 = 0, 2, and 5 days for the Kelvin-Voigt rock and that t0 = 0, 2 and 5 months for the Burgers rock. The other parameters are still as presented in Tables 1 and 2. Fig. 11(a) and (c) illustrate the results of σr(t)/p0 and σθ(t)/p0 of the two rocks calculated under the three levels of t0. As observed, the larger t0 is, the greater the proportion of radial stress release that has occurred before the tunnel is supported is, and thus the smaller σr(t)/p0 is and the largerσθ(t)/p0 is at the support starting position. Furthermore, from Fig. 11(b), it can be seen that uR(t) and q(t)/p0 for the Kelvin-Voigt rock are both significantly influenced by the liner installation time t0. In addition, the larger t0 is, the larger uR(∞) will be and the smaller q(∞) will be. Similarly, the same conclusions for the Burgers rock can be obtained from Fig. 11(d), except that q(∞) = p0 in this case.

6.2.3. Influence of the excavation rate To explore the influence of the excavation rate v on the rock stresses and displacement, as well as support pressure, v is set to 1, 2 and 4 m/ day for the Kelvin-Voigt rock, and v = 1.14, 2.28 and 4.57 m/month for the Burgers rock. To ensure the same ratio of stress release at the support starting point under different v, the distances between the tunnel face and support starting position have to be constant, which are set to be 0.5r0 and r0 for the two rocks, respectively. As a result, the time of liner installation is different here. Here, the parameters α and RL are both constant, as presented in Tables 1 and 2. Fig. 10(a) and (c) illustrate the influences of v on rock stresses for the two rocks. Obviously, due to the same ratio of stress release at the support staring point, σr(t)/p0 or σθ(t)/p0 for each rock under different v is constant at this position. However, after liner installation, the differences of radial or tangential rock stresses under different v are very small, which indicates that the excavation rate v has a minor influence on rock stresses. Fig. 10(b) shows the influences of v on q(t)/p0 and uR(t) for the Kelvin-Voigt rock. It can be clearly observed that, the smaller v 10

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RL=2.5r0

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RL=2.5r0 (t)/p0

RL=r0

RL=r0

RL=0.5r0

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RL=2.5r0 RL=r0

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RL=0.5r0 16

Support starting point

r

(t)/p0

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RL=2.5r0 q(t)/p0

RL=r0

0.6

q(t)/p0

uR(t) (mm)

1.2

RL=0.5r0

8

RL=2.5r0 0.4

r

(t)/p0

RL=r0

t0=2 month 0.0 10-1

0

10

1

2

10

0.8

4

RL=0.5r0 10

3

10

4

10

t0=2 month 0 10-1

5

10

100

101

Support starting point 102

t (month)

t (month)

(c)

(d)

103

104

1.0 105

Fig. 9. Normalized stresses (σr/p0 and σθ/p0), rock displacement (uR(t)) and normalized support pressure (q(t)/p0) calculated with different RL for tunnels in the two rheological rocks:(a) and (b) Kelvin-Voigt rock; (c) and (d) Burgers rock.

final concrete liner (Barla, 2010; Barla et al., 2012). However, previous researches have released that the pre-reinforcement and primary support are both with high deformability (Barla et al., 2012; G. Barla, 2008), which can withstand a large deformation of surrounding rock. In particular, the primary support is a low stiffness and elastic-perfectly plastic structure, which allows the maximum radial convergence up to 1000 mm (Barla et al., 2012). Also, the monitoring data of stress gauges ever showed that, when this primary support was emplaced, it was easily to yield in a short time with the yield limit of 0.283 MPa. As a consequence, the curves of tunnel convergence were nearly continuous between stage 1 and stage 2, and the convergences prior to the final concrete liner installation were almost up to 450 mm − 850 mm (see Fig. 12). For this reason, a certain degree of simplifications with respect to the pre-reinforcement and primary support were made in previous numerical simulation and analytical solution (Barla et al., 2011; TranManh et al., 2016). Similarly, to simplify the analysis, in this study we assume that the pre-reinforcement and primary support does not exist, where the tunnel will be entirely supported by the concrete liner. However, this assumption does not mean that the effects of the pre-reinforcement and primary support will be ignored, but reflected by strengthening the parameters of surrounding rock. In other words, when determining the parameters of surrounding rock, it is assumed that the tunnel is not supported by the two structures within t≤145 days, and that their influences can be considered in rock parameters. Moreover, because of the remarkable tunnel convergences and the rapid increase of convergence curves (see Fig. 12), in this case, the Burgers model was chosen to describe the rock rheology. Further, prior to the analysis, the

7. Application in the Lyon-Turin Base Tunnel In this section, to better illustrate the feasibility in practical application of the solutions provided in this paper, a time-dependent analysis of the Saint Martin La Porte access adit of the Lyon-Turin Base Tunnel will be carried out. The tunnel was built in a series of soft rocks (like the marlstone, coal schist and limestone), which all processed a strong rheological property and were subjected to a high in situ stress (Barla et al., 2012; Bonini and Barla, 2012). As a result, the strongly extreme squeezing effect and time-dependent deformation were encountered in construction, which made the liner bear a heavy load varying with time. In this study, to facilitate comparative analysis with monitoring data, the analysis sections of the tunnel are selected at Chainage 1443 m and 1457 m. Given that the depth at these positions is up to 400 m and the in-situ stress field within this range is nearly close to the hydrostatic state (Barla et al., 2012), in this paper the initial stress p0 is assumed to be 9.8 MPa. Also, since the cross section of the tunnel is quasi-circular, here we assume it is a circular cross section with the equivalent radius of 6 m (Tran-Manh et al., 2016). In addition, according to Barla et al. (2012), the tunnel from Chainage 1443 m to 1457 m was excavated by the full-face method and the rate of tunnel face advancement was 0.54 m/day. As aforementioned, owing to the obvious rheology of the rock mass, a composite ‘‘yield-control’’ support system was adopted in this tunnel. The construction sequence of this support system was divided into four stages, where stages 0–2 were the installation of the pre-reinforcement and primary support while the stage 3 was for the installation of the 11

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7

r

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1 m/day 2 m/day 4 m/day

(t)/p0

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uR(t)

q(t)/p0

v=1 m/day v=2 m/day v=4 m/day

6 5

0.2

v=1 m/day v=2 m/day v=4 m/day

4

0.3 0.4

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v=1 m/day v=2 m/day v=4 m/day

uR(t) (mm)

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Z. Chu, et al.

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0.5

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0.2 1

10

2

100

0.1

t (day)

1

t (day)

(a)

(b)

r

1.6

(t)/p0

q(t)/p0

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(t)/p0

v =1.14 m/month v =2.28 m/month v =4.57 m/month

20

1.2

r

0.0

24 v =1.14 m/month v =2.28 m/month v =4.57 m/month

uR(t) (mm)

(t)/p0

2.0

(t)/p0

100

101

102

v =1.14 m/month v =2.28 m/month v =4.57 m/month

103

104

0.2

16

0.4 uR(t)

12 8

0.4 0.0 10-1

0.6 100

10

v =1.14 m/month 0.6 v =2.28 m/month v =4.57 m/month

q(t)/p0

0.1

0.8 4

105

10-1

Support starting point 100

101

102

t (month)

t (month)

(c)

(d)

103

104

1.0 105

Fig. 10. Normalized stresses (σr/p0 and σθ/p0), rock displacements (uR(t)) and normalized support pressure (q(t)/p0) calculated with different v for tunnels in the two rheological rocks: (a) and (b) Kelvin-Voigt rock; (c) and (d) Burgers rock.

predicted curves are basically consistent with those of the actual measurement. The discrepancies between the calculation and measurement are mainly exhibited in deep rock (≥12 m), where the calculated displacements are slightly larger than the measurements. In fact, there may exist a certain range of plastic zone or crack zone in the surrounding rock due to excavation. However, the rock in the calculation is assumed to be a viscoelastic material, resulting in the larger range of the strained zone. Finally, the circumferential membrane stresses in the concrete liner at Chainage 1,457 m are shown in Fig. 14. From the figure, it can be observed that the liner stresses measured at different positions have a wide range of distribution due to the anisotropy of the rock mass. Fig. 14 also presents the corresponding mean values of the monitoring data and the calculated values. Apparently, good agreement has been achieved between them. Therefore, the proposed solution can be successfully used for analyzing the time-dependent behavior of this type of deep-buried circular lined tunnel in soft rocks.

basic parameters (α, RL and m) associated with stress release and face advancement are first determined according to the range of empirical values and excavation way. After that, to account for the influences of the pre-reinforcement and primary support, the Burgers parameters (Gm, ηm, Gk and ηk) can be determined according to the back analysis of monitoring convergences (Asadollahpour et al., 2014; Sharifzadeh et al., 2013). However, since the Burgers displacement expression of the unlined tunnel has been deduced in this study (see Eq. (15)), the Burgers parameters can be easily obtained by fitting the mean values of the monitoring convergences (see Fig. 12), where the fit function is given as f(Gm, ηm, Gk, ηk) = uR(t)−uR(0), and uR(0) is the initial elastic displacement. In addition, the liner elastic parameters and its installation time were determined referring to Barla et al. (2011). Finally, the parameters employed in calculation of this tunnel were presented in Table 3. Based on this, the tunnel convergence, rock radial displacement as well as the liner stress were calculated and predicted, which were also compared with monitoring data. Fig. 12 shows calculated versus monitoring convergences with time at Chainage 1,443 m. It emerges that monitoring data are relatively dispersed, reflecting the intrinsic inhomogeneity of the problem. However, the analytical solution can accurately describe the tunnel convergence, capturing the average response of the surrounding rock. In addition, the radial displacements versus distance from the tunnel wall are measured by the multi-position borehole extensometers at 36 days and 166 days after excavation, which are shown in Fig. 13(a) and (b), respectively. The calculated displacements are also presented in Fig. 13. Obviously, the calculated values are basically within the range of the monitoring values, and the variation tendencies of the

8. Conclusions In this paper, coupled analytical solutions for deep-buried lined circular tunnels that take tunnel face advancement and the rheology of soft rock into account simultaneously have been proposed. To account for different rheological behaviors of soft rocks, five types of viscoelastic models were adopted in the theoretical derivations. The provided analytical solutions were first compared with previous corresponding solutions for particular viscoelastic models. After that, the effectiveness of the analytical solutions was further validated by comparing their 12

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(t)/p0

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(t)/p0

t0=2 days

t0=2 days t0=5 days

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(b)

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Support starting point

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r

t(day)

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q(t)/p0 t0 =0 months

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t0 =2 months

8

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(t)/p0

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t0=5 months

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0.7 100

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q(t)/p0

(t)/p0

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1.8

q(t)/p0

2.0

105

t (month)

0 10-1

100

101

102

103

104

1.0 105

t (month)

(c)

(d)

Fig. 11. Normalized stresses (σr/p0 and σθ/p0), rock displacement (uR(t)) and normalized support pressure (q(t)/p0) calculated with different t0 for tunnels in the two rheological rocks: (a) and (b) Kelvin-Voigt rock; (c) and (d) Burgers rock. 900 800

4 5

1

700

Convergence (mm)

results with those predicted by finite difference simulation. Based on the proposed coupled solutions, the characteristics of convergent displacements of unlined tunnels with or without consideration of the tunnel face effect, were studied. In addition, for lined tunnels, the influences of the initial stress release parameter α, influence radius of the tunnel face RL, excavation rate v and liner installation time t0 on rock stress and displacement, as well as support pressure, were systematically investigated. Finally, the time-dependent behavior of the LyonTurin Base Tunnel was studied. Based on the results, the following conclusions can be drawn:

3 2

7

6

600 500 400

1-5 6-7 3-4

300 200

2-4 1-3 2-3

(1) When the excavation rate v → ∞ or the initial stress release parameter α = 0, the proposed solutions can be degenerated to the existing viscoelastic solutions without consideration of the tunnel face advancement. When the rock rheology is not taken into account, the proposed solutions can be simplified to the elastic solutions with consideration of the tunnel face advancement. (2) For unlined tunnels, the tunnel displacements are temporarily affected by tunnel face advancement, though this effect will gradually disappear. When the time is large enough, the tunnel displacement in the Kelvin-Voigt rock will converge to a fixed value regardless of whether the effect of tunnel face advancement is considered or not. However, for the Burgers rock, due to its fluid property, the tunnel displacement with or without the effect of tunnel face advancement continues to increase without converging to a fixed value. (3) For the lined tunnel in the Kelvin-Voigt rock, the rock stress and displacement, as well as the support pressure, are all significantly affected by the initial stress release parameter α, influence radius of

Mean values

100 stage 1

0 0

20

Fitting curve

stage 2 40

60

80

100

120

stage3 140

160

Time (day) Fig. 12. Fitting versus monitoring convergences with time at Chainage 1,443 m. Table 3 Parameters used for calculation of the Lyon-Turin Base Tunnel. Rock (Burgers model) Liner Others parameters

Gm = 265 MPa; ηm = 500 GPa·day; Gk = 48.5 MPa; ηk = 3.58 GPa·day r0 = 6 m; r1 = 5 m; Es = 30 GPa; νs = 0.3; Ks = 5.9 GPa α = 0.73; RL = 0.75r0; m = 0.12; t0 = 145 day

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350 ext3

Radial displacement (mm)

300

ext4

ext2

250

ext5

ext1 ext6

200

the excavation rate v. In addition, the larger of α, RL and t0 will result in smaller ultimate support pressure but larger ultimate rock displacement. (4) For the lined tunnel in the Burgers rock, the variation characteristics of rock stress and displacement as well as support pressure, within a certain range are basically similar to those in the KelvinVoigt rock. However, the ultimate support pressure always equals the in situ stress p0, which is not influenced by the parameters α, RL, t0, and v. (5) Good agreements are achieved between the predicted values and the monitoring data in the Lyon-Turin Base Tunnel, which indicates that the proposed solutions can be successfully used for predicting the time-dependent behavior of deep-buried circular lined tunnels in soft rocks.

ext1 ext2 ext5 ext6 ext3 Mean values Calculated values

150 100 50 0 0

4

8

12

16

20

24

Distance (m)

(a) 700

ext3

ext1 ext2 ext3 ext4 ext5 ext6 Mean values Calculated values

ext4

ext2

600

Radial displacement (mm)

Acknowledgements

ext1

500

ext5 ext6

400 300

The research work was financially supported by the National Natural Science Foundation of China (Grant Nos. 51908431, 41772309), for which the authors are grateful.

200 100 0

0

2

4

6

8

10

12

14

16

Distance (m)

(b) Fig. 13. Calculated versus monitoring radial displacements at depth, at Chainage 1,443 m, (a) 33 days after excavation (b) 166 days after excavation.

12

6 8 7

Stress in the liner (MPa)

10

10

52

CV

99

3 1

4

cv1 cv10

2

8

cv2 cv9 cv8 cv6 cv3,7 cv4 cv5

6 4 2 0

Mean values Calculated values 0

100

200

300

400

500

Time (day) Fig. 14. Calculated versus monitoring circumferential membrane stress in the final concrete liner, at Chainage 1,457 m.

the tunnel face RL, and liner installation time t0 but less affected by Appendix A. Lined tunnel solution for Maxwell rock If Gk → ∞ or ηk → ∞ in Eqs. (28)–(38), the support pressure q(τ) for Maxwell rock can be easily derived, which is given by

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τ

(

)

q (τ ) = p0 1 − e− ηm a1 +

ηm b1 mp0 − τ ⎡e−mτ − e ηm a1 ⎤ ⎦ 1 − ηm a1 m ⎣

(A1)

where 1 Gm

a1 = b1 = α

(

1 Gm

1 mηm

−

2 Ks

+

⎫ ⎪ exp(−mt0) ⎬ ⎪ ⎭

)

(A2)

When τ → ∞ in Eq. (A1), the ultimate support pressure q(∞) also converges to p0, which is the same as in Burgers rock. Appendix B. Lined tunnel solution for Kelvin rock When ηm → ∞ and Gm → ∞ in Eqs. (28)–(38), the support pressure q(τ) for Kelvin rock can be derived, which is given by

q (τ ) =

b K (b1 + b2) Ks G k p0 b1 p0 K s (m − 1/ td ) −mτ p b m (2 − td b3 K s) −⎛ 3 s ⎞ τ e + 0 ⎡ 1 − b2 ⎤ e ⎝ 2 ⎠ + ⎥ 2G k + Ks 2 b3 K s − 2m td b3 ⎢ b K − m 3 s ⎣ ⎦

(B1)

where b1, b2, b3 are as follows, αe−mt 0 Gk (1 − mtd)

b1 =

(

b2 = − G

α k (1 − mtd)

(

b3 =

1 Gk

) exp (− ) )

+

1 Gk

+

2 Ks

t0 td

1 td

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(B2)

Here, for τ → ∞, the ultimate support pressure q(∞) and the ultimate rock displacement uR(∞) are expressed as Gk p0 (b1 + b2)

q (∞) = uR (∞) =

1 + 2Gk / K s

(

r 0 p0 2

1 Gk

b1 + b2 1 + 2Gk / K s

−

)

⎫ ⎪ ⎬ ⎪ ⎭

(B3)

Appendix C. Lined tunnel solutions for Kelvin-Voigt and Poynting-Thomson rocks When ηm → ∞ in Eqs. (28)–(38), the support pressure q(τ) for Kelvin-Voigt rock can be easily derived. Furthermore, as mentioned previously, the forms of the constitutive equations of the Kelvin-Voigt model and the Poynting-Thomson model are basically the same. Therefore, q(τ) of the two rocks can be expressed in a unified manner, which is given by

q (τ ) =

b3 (b1 + b2) Ks G L p0 b1 p0 (m − 1/ td ) −mτ p b m (a1 − td b3) e + 0 ⎡ 1 − b2 ⎤ e− a1 τ + ⎥ 2G L + Ks b3 − a1 m td b3 ⎢ b − a m 3 1 ⎣ ⎦

(C1)

where the expressions of G0, GL and td for the two rocks are the same as those in Eq. (18); and a1, a2, b1 and b2 are shown as follows:

a1 =

( =(

b1 = α ⎡ ⎣

1 GL

b2

1 GL

+

2 Ks

⎫ ⎪ exp(−mt0 ) ⎪ ⎪ t ⎬ exp − t 0 ⎪ d ⎪ ⎪ ⎭

+ ) ) (1 − ) ( ) =( + )

−

1 G0

−

1 G0

b3

1 G0

1 (1 − mtd)

1 ⎤ G0 ⎦

α 1 − mtd

1 GL

2 Ks

1 td

(C2)

Substituting the corresponding G0, GL and td of each rock into the above two equations, the closed-form solution for each rock will be obtained. Moreover, for τ → ∞, q(∞) and uR(∞) are expressed as

q (∞) = uR (∞) =

r 0 p0 2

GL p0 (b1 + b2) 1 + 2GL / K s

(

1 GL

−

b1 + b2 1 + 2GL / K s

)

⎫ ⎪ ⎬ ⎪ ⎭

(C3)

Appendix D. Lined tunnel solution for elastic rock As mentioned above, if ηm, ηk and Gk all tend to infinity in Eqs. (28)–(38), the support pressure for the elastic solution with reference to the tunnel face effect is given as

q (τ ) =

αK s p0 [e−mt0 − e−m (t0+ τ )] 2Gm + K s

(D1)

Therefore, for τ → ∞, q(∞) and uR(∞) are given as

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q (∞) = uR (∞) =

r 0 p0 2G m

(1 −

αK s p0 (2G m + K s)

αe−mt0 )

e−mt0

+

αK s r 0 p0 (2G m + K s)

⎫ e−mt0 ⎬

(D2)

⎭

Appendix E. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.tust.2019.103111.

far-field stress. Int. J. Rock Mech. Min. Sci. 106, 350–363. Nomikos, P., Rahmannejad, R., Sofianos, A., 2011. Supported axisymmetric tunnels within linear viscoelastic burgers rocks. Rock Mech. Rock Eng. 44, 553–564. Pan, Y.W., Dong, J.J., 1991. Time-dependent tunnel convergence—I. Formulation of the model. Int. J. Rock Mech. Min. Sci. 28, 469–475. Panet, M., Guenot, A., 1982. Analysis of convergence behind the face of a tunnel. In: Proc. of the International Symposium Tunnelling, IMM, London, pp. 197–204. Paraskevopoulou, C., Diederichs, M., 2018. Analysis of time-dependent deformation in tunnels using the convergence-confinement method. Tunn. Undergr. Space Technol. 71, 62–80. Pellet, F., Roosefid, M., Deleruyelle, F., 2009. On the 3D numerical modelling of the timedependent development of the damage zone around underground galleries during and after excavation. Tunn. Undergr. Space Technol. 24, 665–674. Phienwej, N., Thakur, P., Cording, E., 2007. 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