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Research Paper

Dynamic response of a non-circular lined tunnel with visco-elastic imperfect interface in the saturated poroelastic medium Xue-Qian Fang ⇑, He-Xin Jin Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China

a r t i c l e

i n f o

Article history: Received 21 June 2016 Received in revised form 20 October 2016 Accepted 4 November 2016 Available online xxxx Keywords: Non-circular lined tunnel Saturated poroelastic medium Visco-elastic interface Conformal transformation method Dynamic stress

a b s t r a c t A semi-analytical method is proposed to investigate the dynamic response of a non-circular tunnel with visco-elastic imperfect interface in poroelastic medium. Biot’s dynamic theory is used to simulate the saturated poroelastic medium, and the governing equations are solved by reducing them into three Helmholtz equations that the potential functions satisfy. The visco-elastic interface model with elastic and viscosity coefficients is adopted to analyze the interface effect around the non-circular lined tunnel. The analytic solutions of displacements and stresses are expanded in terms of wave functions. Some numerical examples are provided to analyze the effect of visco-elastic interface on the dynamic stress around the tunnel. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, the tunnels constructed in water-rich ground are crucial components of the transportation and utility networks in urban and seabed areas. To improve the strength of tunnels serving in the harsh environments, the dynamic response of tunnels and underground structures in saturated poroelastic soil is of high interest, especially when the underground tunnels are built in areas subjected to strong earthquakes [1]. To evaluate the response of underground tunnels under many kinds of loadings, several analytical and numerical methods were presented. For example, Glazanov and Shenderov [2] studied the plane wave scattering by a cylindrical cavity in an isotropic elastic medium. By using the series solution method, Lee and Trifunac [3] analyzed the twodimensional scattering of SH-waves by a circular tunnel in a homogeneous elastic half-space. Krutin et al. [4] studied the propagation of an elastic harmonic wave in the neighborhood of a fluid-filled cylindrical cavity embedded in a saturated porous medium. Liu and Han [5] investigated the dynamic response of an underground circular cavity in a fluid-saturated porous medium. Shiba and Okamoto [6] proposed a simple method for the earthquake response of cylindrical tunnels in homogeneous soft ground. Mei et al. [7] applied a boundary layer approximation to study the scattering of waves by a circular cavity in a boundless porous solid. Senjuntichat and Rajapakse [8] employed Biot’s equations for poroelastic ⇑ Corresponding author. E-mail address: [email protected] (X.-Q. Fang). http://dx.doi.org/10.1016/j.compgeo.2016.11.001 0266-352X/Ó 2016 Elsevier Ltd. All rights reserved.

medium in combination with the Laplace transform technique to analyze the transient response of a pressurized long cylindrical cavity in an infinite poroelastic medium. Based on Biot’s theory, Hu et al. [9] studied the scattering and refraction of plane strain waves by a cylindrical cavity in the fluid-saturated soil. To solve the harmonic wave diffraction by unlined or lined cavities of arbitrary geometry in fully saturated poroelastic soil medium, Kattis et al. [10] developed an efficient and accurate boundary element method based on Biot’s theory. Kumar et al. [11] used the Laplace transform technique to solve the radial displacement fields in a liquid-saturated porous medium with a cylindrical cavity subjected to an arbitrary time-dependent force. Recently, a partially debonded pipeline embedded in a saturated poroelastic medium has been proposed, and the dynamic response of this model to harmonic plane waves was theoretically investigated [12]. In practical engineering, the micro-cracks or interstitial media often exist around the tunnels, and so the interface between the tunnel and its surrounding soil is imperfect. To simulate the discontinuous boundary conditions, the elastic interface model [13,14] and time-dependent visco-elastic model [15] are introduced to simulate the imperfect interface. In these models, the interface exhibits extremely complicated mechanical behavior. The main object of this paper is to evaluate the scattering of elastic waves by a non-circular tunnel in a saturated poroelastic medium. The medium is described by Biot’s theory and the lining is treated as homogenous isotropic and elastic material. By introducing three potential functions, the governing equations for Biot’s theory are decoupled and reduced into three Helmholtz equations.

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X.-Q. Fang, H.-X. Jin / Computers and Geotechnics 83 (2017) 98–105

The wave function expansion method is used to solve the Helmholtz equations. The imperfect interface is simulated by a viscoelastic interface model. By using the complex variables method and the wave function expansion method together, the analytical solutions of displacements and stresses resulting from the viscoelastic interface are obtained. Some numerical examples are given to analyze the dynamic responses of the tunnel under different elastic and viscosity properties of interfaces in detail.

f½u1f /1pf ;jj þ ðg5 u1pf g4 Þ/1pf þ ½u1s /1ps;jj þ ðg5 u1ps g4 Þ/1ps g;kk ¼ 0; ð10Þ where g2 ¼ a þ q1f x =g1 , g3 ¼ q1 x þ q x =g1 , g4 ¼ ag1 þ q1f x2 , 2

2

2 1f

4

g5 ¼ g1 =M. Then, u1pf and u1ps are the solutions of the following equation

u21pf ;1ps þ

g3 ðk1 þ 2l1 Þg5 g2 g4 ðk þ 2l1 Þg4 u1pf ;1ps þ 1 ¼ 0; g2 g5 g2 g5 ð11Þ

2. Governing equations of saturated poroelastic soil This problem involves a lined non-circular tunnel with viscoelastic interface embedded in the saturated porous medium. The material properties are characterized by physical parameters, such as lame constants k1 , l1, the porosity of the porous medium v, the fluid viscosity j, the permeability kd, the Biot’s parameters a and M. The pore pressure is P. The densities of porous medium, solid skeleton and pore fluid are denoted by q1, q1s, q1f, and q1 = (1 v)q1s + q1f. The concrete lining of the tunnel is assumed to be homogenous and isotropic material with properties characterized by lame constants k2 , l2 and density q2. It is assumed that a plane strain wave propagates in the saturated porous medium. Based on Biot’s theory, the constitutive relations for the homogeneous saturated poroelastic medium are expressed as [16]

where u1pf,1ps denotes the two variables of u1pf and u1ps. The displacement potentials /1pf, /1ps and wk must satisfy the following Helmholtz equation

r2 /1pf þ k21pf /1pf ¼ 0;

ð12Þ

r2 /1ps þ k21ps /1ps ¼ 0;

ð13Þ

r2 wk þ k21sh wk ¼ 0;

ð14Þ

and 2

ð15Þ

2

ð16Þ

2

ð17Þ

k1pf ¼ g3 =ðk1 þ 2l1 g2 u1pf Þ ¼ ðg5 u1pf g4 Þ=u1pf ;

rij ¼ 2l1 eij þ k1 dij e adij pp ;

ð1Þ

k1ps ¼ g3 =ðk1 þ 2l1 g2 u1ps Þ ¼ ðg5 u1ps g4 Þ=u1ps ;

P ¼ aMe þ M#;

ð2Þ

k1sh ¼ g3 =l1 ;

ð3Þ

where k1pf, k1ps and k1sh denote, respectively, the complex wave numbers for the Pf wave, Ps wave and S wave propagating in the porous medium. In order to ensure that Pf wave is faster than Ps wave, we should take that Re(k1pf) < Re(k1ps). Moreover, the inequality Im(k1pf,1ps,sh) > 0 should be taken to ensure that the propagating waves attenuate within the porous medium. For the lining structure, the governing equation can be expressed as [14]

e ¼ ui;i ;

# ¼ wi;i ;

i; j ¼ x; y;

where rij denotes the total stress components of the bulk material, eij is the strain component of the solid matrix, e is the dilatation of the solid matrix, dij denotes the Kronecker delta, # represents the variation of fluid content per unit reference volume, P is the excess pore pressure, ui and wi denote the average solid displacement and the fluid displacement relative to the solid frame, respectively. The governing equations for the bulk material and the pore fluid can be expressed as

l2 ui;jj þ ðk2 þ l2 Þuj;ji ¼ q2 u€ :

ð18Þ

ð4Þ

The scalar potentials /2 and vector potential w2 in the lining can be given as

ð5Þ

r2 /2 þ k22p /2 ¼ 0;

ð19Þ

where the over-dots denote the derivatives of field variables with respect to time t. Following Helmholtz decoupling theorem [18], the displacements u can be expressed as the sum scalar field /1p and the vector field wk, i.e.,

r2 w2 þ k22s w2 ¼ 0;

ð20Þ

ui ¼ /1p;i þ eijk wk;j ¼ /1pf ;i þ /1ps;i þ eijk wk;j ;

ð6Þ

3. The non circular geometry of the tunnel lining

P ¼ u1pf /1pf ;ii þ u1ps /1ps;ii ;

ð7Þ

The plane incident waves are considered in the paper. Due to the geometric characters and the characters of applied loadings, this problem can be simplified into a plane strain problem. To obtain the analytical solution of the wave fields around the noncircular tunnel, the complex function theory is introduced. Let g (f) be the conformal mapping function which can transform the non-circular tunnel in the z-plane into an annular region with unit and R0 radii in the f-plane, as shown in Fig. 2. The conformal mapping function is expressed as

€ l1 ui;jj þ ðk1 þ a M þ l1 Þuj;ji þ aMwj;ji ¼ q1 u€ þ q1f w; 2

aMuj;ji þ Mwj;ji ¼ q1f u€ þ

wi ¼

1

g1

P;i

q1f j _ € þ w; w i; j ¼ x; y; v kd

q1f x2 u; g1 i

ð8Þ

where /1pf and /1ps are the displacement potentials corresponding to Pf and Ps waves in the poroelastic medium, respectively. wk (k = 1, 2, 3) is the vector of displacement potential, u1pf and u1ps are the amplitudes of Pf and Ps waves, eijk is the Levi-Civita symbol and g1 = q1fx2/v ijx/kd. x is the frequency of incident waves. The caret represents the Fourier transform with respect to time. Substituting Eqs. (6)–(8) into (4) and (5), one can obtain f½ðk1 þ 2l1 g2 u1pf Þ/1pf ;jj þ g3 /1pf þ ½ðk1 þ 2l1 g2 u1ps Þ/1ps;jj þ g3 /1ps g;i þ eimk ½l1 wk;jj þ g3 wk ;m ¼ 0;

ð9Þ

where k2p and k2s denote the complex wave numbers of P and S waves in the concrete lining.

! n X k ; z ¼ gðfÞ ¼ R f þ ck f

f ¼ n þ ig ¼ qeih ;

ð21Þ

k¼0

where R is a real constant that reflecting the size of the tunnel. Due to the symmetry of the tunnel, the coefficients ck must be real numbers and they determine the shape of the tunnel. All coefficients in

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the mapping function can be determined when the shape of the tunnel cross-section and support thickness are known. The number of n is determined by the optimization technique of mixing penalty function. To improve the accuracy, the inner and outer boundaries (doubly-connected domain) are both considered in this paper. So, n = 9 is defined in the following numerical examples. 4. Wave propagation in the fluid, the matrix and the lining 4.1. Wave fields in the saturated poroelastic medium and the concrete lining In the cylindrical coordinate system, the incident Pf, Ps, and S waves propagating along the x axis in the poroelastic medium can be expressed as ðinÞ

/1pf ;1ps

ðinÞ

w1

n 1 X gðfÞ n ¼ upf 0;ps0 i J n ðk1pf ;1ps jgðfÞjÞ eixt ; jgðfÞj n¼1

¼ w10

n gðfÞ n i J n ðk1sh jgðfÞjÞ eixt ; jgðfÞj n¼1 1 X

ðscÞ

/1pf

ðscÞ /1ps

ðscÞ

w1sh

n gðfÞ ¼ An Hnð1Þ ðk1pf jgðfÞjÞ eixt ; jgðfÞj n¼1 ¼

1 X

Bn Hð1Þ n ðk1ps jgðfÞjÞ n¼1

gðfÞ jgðfÞj

e

;

n gðfÞ ¼ C n Hnð1Þ ðk1sh jgðfÞjÞ eixt ; jgðfÞj n¼1 1 X

ðrrÞ

w2

ð26Þ

ð28Þ

where En and Nn are the undermined coefficients of the refracted waves, Hnð2Þ ðÞ are the nth Hankel function of the second kind, and denotes the inwards propagating waves. There are reflected waves that propagate outwards from the inner boundary of the concrete lining. The displacement fields of reflected waves are given as ðrlÞ

/2

ðrlÞ

n 1 X gðfÞ ¼ Dn Hnð1Þ ðk2p jgðfÞjÞ eixt ; jgðfÞj n¼1

ð29Þ

n gðfÞ Mn Hnð1Þ ðk2s jgðfÞjÞ eixt ; jgðfÞj n¼1

ð30Þ

w2 ¼

1 X

ðinÞ

ðscÞ

ð32Þ

ðtÞ

ðinÞ

ðscÞ

ð33Þ

w1sh ¼ w1sh þ w1sh :

The total wave fields in the concrete lining are produced by the superposition of the refracted waves and the reflected waves resulting from the inner boundary of lining, ðtÞ

ðrrÞ

ðrlÞ

ð34Þ

ðtÞ

ðrrÞ

ðrlÞ

ð35Þ

/2 ¼ /2 þ /2 ;

The displacements, stresses and pore pressure in the poroelastic medium can be solved by using the following formula

u1r þ iu1h ¼

2f @ ðtÞ ðtÞ ðtÞ /1pf þ /1ps iw1sh ; 0 jg ðfÞj @f

ð36Þ

u1r iu1h ¼

2n @ ðtÞ ðtÞ ðtÞ /1pf þ /1ps þ iw1sh ; 0 g ðfÞ @f

ð37Þ

w1r þ iw1h ¼

2f @ ðtÞ ðtÞ ðtÞ 1 / þ 1 / i 1 w ; 5 6 3 1ps 1pf 1sh jg 0 ðfÞj @f

ð38Þ

w1r iw1h ¼

2n @ ðtÞ ðtÞ ðtÞ 1 / þ 1 / þ i 1 w ; 5 6 3 1ps 1pf 1sh g 0 ðfÞ @f

ð39Þ

ð25Þ

n gðfÞ ¼ N n Hð2Þ eixt ; n ðk2s jgðfÞjÞ jgðfÞj n¼1 1 X

ðtÞ

/1ps ¼ /1ps þ /1ps ;

ð24Þ

ð27Þ

1 X

ð31Þ

4.2. Stresses, pore pressure, and displacements of saturated poroelastic medium and concrete lining

n gðfÞ En Hnð2Þ ðk2p jgðfÞjÞ eixt ; jgðfÞj n¼1

ðrrÞ

ðscÞ

ð23Þ

where An, Bn, Cn are the mode coefficients of scattered waves, Hð1Þ n ðÞ is the nth Hankel function of the first kind that denotes the outgoing waves. The refracted waves being confined inside the concrete lining are expressed as

/2 ¼

ðinÞ

w2 ¼ w2 þ w2 :

n ixt

ðtÞ

/1pf ¼ /1pf þ /1pf ;

ð22Þ

where u1pf0,1ps0 are the amplitudes of Pf and Ps waves, w10 is the amplitude of S wave, and Jn ðÞ is the nth Bessel function of the first kind. The multiple scattering of incident waves will come into being when the incident waves run into the lined tunnel. The displacement fields of scattered waves are given by 1 X

The total wave fields in the poroelastic soil are produced by the superposition of the incident waves and the scattered waves resulting from the interface of the tunnel and its surrounding medium,

where Dn and Mn are the mode coefficients of reflected waves in the concrete lining.

2 ðtÞ ðtÞ r1rr þ r1hh ¼ 2ðk1 þ l1 Þ k21pf /1pf þ k1ps /1sh ;

ðtÞ ðtÞ r1rr þ ir1rh ¼ 11 /1pf þ 12 /1ps þ

4l 1 f 2 g 0 ðfÞ

" # @2 1 @ ðtÞ ðtÞ ðtÞ /1pf þ /1ps iw1sh ; @f2 g 0 ðfÞ @f

ðtÞ ðtÞ r1rr ir1rh ¼ 11 /1pf þ 12 /1ps þ

ðtÞ

2

ðtÞ

P1 ¼ u1pf k1pf /1pf u1sh k1ps /1ps ; where

ð41Þ

4l 1 f 2

g 0 ðfÞ @ 1 @ ðtÞ ðtÞ ðtÞ / þ / þ iw ; 1ps 1sh @f g 0 ðfÞ @f 1pf

2

ð40Þ

ð42Þ ð43Þ

11 ¼ au1pf k21pf ðk1 þ l1 Þk21pf , 12 ¼ au1ps k21ps ðk1 þ l1 Þk21ps , 2

13 ¼ q1fg1x , 14 ¼ g11 , 15 ¼ 13 12 u1pf k21pf , 16 ¼ 13 12 u1ps k21ps . The displacements and stresses in the lining can be solved by using the formula as follows

u2r þ iu2h ¼

2f @ ðtÞ ðtÞ / iw2 ; R0 jg 0 ðfÞj @f 2

ð44Þ

u2r iu2h ¼

2n @ ðtÞ ðtÞ / þ iw2 eih ; R0 g 0 ðfÞ @f 2

ð45Þ

r2rr þ r2hh ¼ 2ðk2 þ l2 Þk22p /ðtÞ 2 ;

ð46Þ

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r2rr þ ir2rh ¼ ðk2 þ l2 Þk22p /2ðtÞ þ

4l2 f2 @ 2 R20 g 0 ðfÞ @f2

"

# 1 @ ðtÞ ðtÞ ; /2 iw2 g 0 ðfÞ @f

y

ð47Þ

6

4l f @ 1 @ ðtÞ ðtÞ / r2rr ir2rh ¼ ðk2 þ l2 Þk22p /2ðtÞ þ 2 20 þ iw : 2 2 R0 g ðfÞ @f g 0 ðfÞ @f 2

3

5. The visco-elastic interface boundary conditions A visco-elastic model is introduced to analyze the imperfect interface effect. In this model, the tractions at the outer boundary of the concrete are continuous, but the displacements are discontinuous across the interface, which means that the tunnel and the poroelastic soil are bonded by a system, in which a linear spring and a linear dashpot are connected in parallel, as shown in Fig. 1 Using this model, the boundary conditions at the interface of the concrete lining (L2 in Fig. 1) can be described as follows

r1rr ¼ r2rr ;

ð49Þ

r1rh ¼ r2rh ;

ð50Þ

u1r u2r þ v cr u1h u2h þ v ch

@ðu1r u2r Þ r1rr ¼ ; @t spr

ð51Þ

@ðu1h u2h Þ r1rh ; ¼ @t sph

ð52Þ

where spr and sph are the radial and circumferential stiffness parameters of the imperfect interface, vcr and vch are the radial and circumferential viscosities of visco-elastic interface. They can be used to tailor the mechanical properties of visco-elastic interface. If vcr = vch = 0, the boundary conditions degenerate to the linear spring model, in such case the responses of interface are independent of the time variable. Further, if spr ? 0 and sph ? 0, no waves are transmitted from the saturated poroelastic medium to the noncircular tunnel. At the inner boundary of the concrete lining (L1 in Fig. 1), it is assumed that the tractions are free. It can be expressed as

r2rr ¼ 0;

ð53Þ

y Visco-elastic interface

o

3

θ

O

6

L1

ρ

1

x

R0

L2

(a)

ξ

(b)

Fig. 2. Conformal mapping of non-circular tunnel into ring-shaped region. (a) Lined non-circular tunnel in the z plane; (b) Ring-shaped region in the f plane.

r2rh ¼ 0:

ð54Þ

For impermeable boundary conditions, the relative displacement between the solid skeleton and the pore fluid should vanish, one can obtain

w1r ¼ 0:

ð55Þ

Substituting Eqs. (36)–(48) into (49)–(54), a set of linear algebra equation can be obtained as follows

2

X 11

6 6 X 21 6 6 6 X 31 6 6 6 X 41 6 6 6 X 51 6 6 6X 4 61 X 71

X 17

32

An

3

2

Y 11

3

X 12

X 13

X 14

X 15

X 16

X 22

X 23

X 24

X 25

X 26

X 32

X 33

X 34

X 35

X 36

X 42

X 43

X 44

X 45

X 46

X 52

X 53

X 54

X 55

X 56

X 62

X 63

X 64

X 65

X 66

76 7 6 7 X 27 76 Bn 7 6 Y 21 7 76 7 6 7 76 7 6 7 6 C n 7 6 Y 31 7 X 37 7 76 7 6 7 76 7 6 7 6 Dn 7 ¼ 6 Y 41 7: X 47 7 76 7 6 7 76 7 6 7 6 En 7 6 Y 51 7 X 57 7 76 7 6 7 76 7 6 7 7 6 7 6 X 67 7 54 Mn 5 4 Y 61 5

X 72

X 73

X 74

X 75

X 76

X 77

Nn

ð56Þ

Y 71

The elements in this matrix equation can be found in Appendix A. The coefficients An, Bn, Cn, Dn, En, Mn, Nn can be determined by solving Eq. (56). 6. Numerical examples and analysis In really earthquakes, an incident horizontal Pf wave is the most interesting (and most common) situation of loading. In this section, some numerical examples under an incident plane Pf wave are given to analyze the interaction between the tunnel and its surrounding saturated porous medium with different interface conditions. In order to analyze the dynamic response of the non-circular tunnel under different loading frequencies, the dimensionless circumferential stress is introduced. According to the definition of dynamic stress concentration factor (DSCF), the DSCF around the non-circular tunnel is expressed as [17]

Saturated porous medium

L1

5m

3 O 3 6

6

ð48Þ

η

L2

DSCF ¼ rhh ¼ jr1hh =r0 j;

x

where r1hh is the circumferential stress around the outer boundary of the lining, r0 is the maximum magnitude of the stress in the incident direction, and

Incident waves Fig. 1. Two lined non-circular tunnels with imperfect interfaces in saturated porous medium.

ð57Þ

r0 ¼ ðk1 þ 2l1 Þk21pf u1pf þ 13 u21pf k21pf .

The material properties used in Biot’s model [8] and concrete lining [19] are illustrated in Tables 1 and 2, respectively. The geological nature of the surrounding medium is water-saturated sandstone [20]. The coefficients used in conformal mapping function (Eq. (21)) are given in Table 3. In the following analysis, the dimensionless parameters have been chosen for computation: the thickness of concrete tunnel is 1 m, the stiffness parameter is

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Table 1 Input data for the material properties of saturated porous medium in Biot’s model [8].

v

10

2

1.0 10

0.4

q1f (kg/m3)

j (Pas)

kd (m/s)

1.0 10

q1s (kg/m3)

1000

a

2500

8

90

l2 (Pa)

k2

2000

1.2 109

2.3 109

4 60

1

3

and the tunnel is circular. By defining sp ¼ 1030 and v c ¼ 0, the interface effect can be ignored. The reduced model is consistent with that in Refs. [17,19]. Excellent agreement with the existing results can be seen. According to the typical span of seismic wave frequencies, 10 Hz and 60 Hz frequencies correspond to the weak and strong earthquakes, respectively. So, they are considered as ‘‘low” and ‘‘high” frequencies in the following numerical examples. Fig. 4 shows the dynamic stress distribution under low frequency (x = 10 Hz) around the tunnel, and only the stiffness parameter of the visco-elastic interface is considered. It can be seen that DSCF increases significantly if a small value of sp is selected, especially at the positions near the vault and floor of tunnel. This phenomenon can be interpreted as follows: the weak scattering of waves resulting from the elastic interface can reduce the dynamic stress. As expect, the incident waves counteract some scattering waves. So, the effect of elastic interface is small, and the DSCF in the shadow and illuminate sides decreases. Figs. 5 and 6 show the dynamic stress distribution around the non-circular tunnel with visco-elastic interface in the region of low frequency (x = 10 Hz). In Fig. 5, a small value of stiffness parameter is selected, and a large value of stiffness parameter is adopted. Different viscosity coefficients are selected in Figs. 5 and 6. It can be seen that the DSCF near the vault and floor of tunnel increases with decreasing viscosity coefficient of interfaces. However, the DSCF near the straight wall decreases with decreasing viscosity coefficient of interfaces. If a large stiffness parameter is selected, the effect of viscosity coefficient on the dynamic stress decreases significantly. It can be concluded that the effect of viscosity coefficient of interface will become evident with the degradation of soil mass at the interface. Figs. 7 and 8 show the distribution of DSCF around the tunnel when the wave frequency is high (x = 60 Hz). It can be seen that the effect of interface viscosity coefficient on the dynamic stress

1.Obtained from this paper 2.Obtained from Ref. [19]

2 2

150

sp ¼ spr l1 =jgðfÞj ¼ sph l1 =jgðfÞj, and the viscosity coefficient is v c ¼ v cr =jgðfÞj ¼ v ch =jgðfÞj. The normalized stiffness and viscosity coefficients are introduced to analyze the interface effect on the dynamic stress distribution around the tunnel. The normalized interface coefficients correspond to a nonuniform interface. For convenience, it is assumed that h1pf = 0. In the following examples, the range of sp value is 107–109, and the range of v c value is 105–107. They are physically acceptable for the rock mass with micro-cracks and interstitial media. To validate this dynamic model, comparison with the existing results is given in Fig. 3. In this figure, the interface is perfect,

1.5 108

1.0 10

120

q2 (kg/m )

k1 8

1.0 10

0.999

Table 2 Input data for the material properties of concrete lining [19]. 3

l1 (Pa)

M (Pa)

30

1

180

0

330

210

300

240 270

Fig. 3. Distribution of DSCF around the circular tunnel with perfect interface (x = 2 Hz).

1. sp = 10 *

2. sp = 10 *

90

6

120

7

60

8

3. sp = 10 *

9

4 1

150

30 3

2

180

0

2

210

330

240

300 270

Fig. 4. Distribution of DSCF around the tunnel (x = 10 Hz, v c ¼ 0).

near the vault and floor of tunnel decreases with increasing wave frequency. However, the viscosity coefficient effect on the dynamic stress near the straight wall increases if the wave frequency is high. The interface property effect decreases with increasing wave frequency.

Table 3 Coefficients in conformal mapping function. R

R0

c1

c2

c3

c4

c5

c6

c7

c8

c9

7.54561

0.93

0.07499

0.12076

0.04536

0.06513

0.02083

0.00118

0.00285

0.00096

0.00221

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X.-Q. Fang, H.-X. Jin / Computers and Geotechnics 83 (2017) 98–105

90

5

120

60 4

150

1. vc = 10 *

5

2. vc = 10 *

6

3. vc = 10

7

*

*

120

3

150

30

3

2.5 60

2

5

2. vc = 10 *

6

3. vc = 10

7

*

2

3

1

1. vc = 10

90

1.5 30 1

2

0.5 1

1

180

180

0

0

210

210

330

2

240

300

240

330

300 270

270 Fig. 5. Distribution of DSCF around the tunnel (x = 10 Hz, sp ¼ 108 ).

Fig. 8. Distribution of DSCF around the tunnel (x = 60 Hz, sp ¼ 109 ).

7. Conclusions 90

4

120

60 1 3 2

150

*

5

2. vc = 10 *

6

3. vc = 10

7

*

3 2

1. vc = 10

30

1

180

0

330

210

300

240 270

Fig. 6. Distribution of DSCF around the tunnel (x = 10 Hz, sp ¼ 109 ).

90

2.5

120

60 2

1. vc = 10 *

5

*

2. vc = 10

6

3. vc = 10

7

*

1.5 150

Combining the Biot’s dynamic theory and the visco-elastic interface model, the interface dynamic stress around a noncircular tunnel in the saturated porous medium is predicted accurately. The stiffness parameter and viscosity coefficient are introduced to take into account the visco-elastic character of interface. Some numerical solutions are illustrated to analyze the interface effect on the dynamic stress around the tunnel, and the following important conclusions are found, a. The effect of viscosity coefficient of interface on the dynamic stress around the non-circular tunnel decreases with increasing stiffness of interface. b. The effect of the viscosity coefficient of interface on the dynamic stress in the region of high frequency is smaller than that in the region of low frequency. c. In the region of low frequency, the visco-elastic interface effect on the dynamic stress at the vault and bottom is greater than that at other positions; however, the viscoelastic interface effect on the dynamic stresses around the non-circular tunnel shows little variation in the region of high frequency.

Acknowledgements The paper is supported by National Natural Science Foundation of China (No. 11472181).

30 1

Appendix A

0.5 1 3

180

0

2

330

210

240

300 270

Fig. 7. Distribution of DSCF around the tunnel (x = 60 Hz, sp ¼ 108 ).

X 11 ¼ 11 Hð1Þ n k1pf jgðfÞj n ð1Þ o 1 2 ð1Þ þ l1 k1pf Hnþ2 k1pf jgðfÞj þHn2 k1pf jgðfÞj ; 2 X 12 ¼ 12 Hð1Þ n k1ps jgðfÞj n ð1Þ o 1 2 ð1Þ þ l1 k1ps Hnþ2 k1ps jgðfÞj þHn2 k1ps jgðfÞj ; 2 X 13 ¼

l1 2

n o 2 ð1Þ ð1Þ ik1sh Hn2 ½k1sh jgðfÞjHnþ2 ½k1sh jgðfÞj ;

104

X.-Q. Fang, H.-X. Jin / Computers and Geotechnics 83 (2017) 98–105

2 X 14 ¼ ðk2 þ l2 Þk2p Hnð1Þ k2p jgðfÞj n ð1Þ o 1 2 ð1Þ l2 k2p Hn2 k2p jgðfÞj þHnþ2 k2p jgðfÞj ; 2

vs ð1Þ o xv sh n ð1Þ r X 41 ¼ þ k1pf Hnþ1 k1pf jgðfÞj þHn1 k1pf jgðfÞj 2i 2 ð1Þ o l1 2 n ð1Þ k1pf Hnþ2 k1pf jgðfÞj Hn2 k1pf jgðfÞj ; 2i

2 X 15 ¼ ðk2 þ l2 Þk2p Hnð2Þ k2p jgðfÞj n ð2Þ o 1 2 ð2Þ l2 k2p Hn2 k2p jgðfÞj þHnþ2 k2p jgðfÞj ; 2

vs ð1Þ o xv sh n ð1Þ r X 42 ¼ þ k1ps Hnþ1 k1ps jgðfÞj þHn1 k1ps jgðfÞj 2i 2 ð1Þ o l1 2 n ð1Þ k1ps Hnþ2 k1ps jgðfÞj Hn2 k1ps jgðfÞj ; 2i

X 16 ¼ X 17 ¼

l2 2

l2 2

n o 2 ð1Þ ð1Þ ik2s Hn2 ½k2s jgðfÞjHnþ2 ½k2s jgðfÞj ;

X 43 ¼

n o 2 ð2Þ ð2Þ ik2s Hn2 ½k2s jgðfÞjHnþ2 ½k2s jgðfÞj ;

n o ð1Þ ð1Þ ik1sh Hnþ1 ½k1sh jgðfÞjHn1 ½k1sh jgðfÞj 2 o l 2 n ð1Þ ð1Þ þ 1 k1sh Hn2 ½k1sh jgðfÞjþHnþ2 ½k1sh jgðfÞj ; 2i

v s

r

2i

xv sh

n ð1Þ o 2 ð1Þ X 21 ¼ l1 k1pf Hnþ2 k1pf jgðfÞj Hn2 k1pf jgðfÞj ;

vs ð1Þ o xv sh n ð1Þ r X 44 ¼ þ k2p Hnþ1 k2p jgðfÞj Hn1 k2p jgðfÞj ; 2i 2

n ð1Þ o 2 ð1Þ X 22 ¼ l1 k1ps Hnþ2 k1ps jgðfÞj Hn2 k1ps jgðfÞj ;

X 45 ¼

n ð1Þ o 2 ð1Þ X 23 ¼ il1 k1ps Hn2 k1ps jgðfÞj þHnþ2 k1ps jgðfÞj ;

o vs xv sh n ð1Þ r ð1Þ X 46 ¼ þ ik2s Hnþ1 ½k2s jgðfÞjHn1 ½k2s jgðfÞj ; 2i 2

n ð1Þ o 2 ð1Þ X 24 ¼ l2 k2p Hnþ2 k2p jgðfÞj Hn2 k2p jgðfÞj ; n ð2Þ o 2 ð2Þ X 25 ¼ l2 k2p Hnþ2 k2p jgðfÞj Hn2 k2p jgðfÞj ;

o vs xv sh n ð2Þ r ð2Þ X 47 ¼ þ ik2s Hnþ1 ½k2s jgðfÞjHn1 ½k2s jgðfÞj ; 2i 2 n ð1Þ o ð1Þ X 51 ¼ 15 k1pf Hn1 k1pf jgðfÞj Hnþ1 k1pf jgðfÞj ;

n o 2 ð1Þ ð1Þ X 26 ¼ l2 ik2s Hnþ2 ½k2s jgðfÞjþHn2 ½k2s jgðfÞj ;

n ð1Þ o ð1Þ X 52 ¼ 16 k1ps Hn1 k1ps jgðfÞj Hnþ1 k1ps jgðfÞj ;

n o 2 ð2Þ ð2Þ X 27 ¼ l2 ik2s Hnþ2 ½k2s jgðfÞjþHn2 ½k2s jgðfÞj ;

n ð1Þ o ð1Þ X 53 ¼ i13 k1ps Hn1 k1ps jgðfÞj þHnþ1 k1ps jgðfÞj ;

n ð1Þ o spr ixsph ð1Þ k1pf Hn1 k1pf jgðfÞj Hnþ1 k1pf jgðfÞj 2 2 11 Hnð1Þ k1pf jgðfÞj n ð1Þ o l 2 ð1Þ 1 k1pf Hnþ2 k1pf jgðfÞj þHn2 k1pf jgðfÞj ; 2 n ð1Þ o spr ixsph ð1Þ ¼ k1ps Hn1 k1ps jgðfÞj Hnþ1 k1ps jgðfÞj 2 2 11 Hnð1Þ k1ps jgðfÞj n ð1Þ o l 2 ð1Þ 1 k1ps Hnþ2 k1ps jgðfÞj þHn2 k1ps jgðfÞj ; 2 n o spr ixsph ð1Þ ð1Þ ¼ ik1sh Hn1 ½k1sh jgðfÞjþHnþ1 ½11sh jgðfÞj 2 2 o l1 2 n ð1Þ ð1Þ ik1sh Hn2 ½k1sh jgðfÞjHnþ2 ½k1sh jgðfÞj ; 2 n ð1Þ o sp ixsph ð1Þ ¼ rþ k2p Hn1 k2p jgðfÞj Hnþ1 k2p jgðfÞj ; 2 2

X 31 ¼

X 32

X 33

X 34

X 35

n ð2Þ o sp ixsph ð2Þ ¼ rþ k2p Hn1 k2p jgðfÞj Hnþ1 k2p jgðfÞj ; 2 2

X 36

n o isp xsph ð1Þ ð1Þ ¼ r k2s Hn1 ½k2s jgðfÞjþHnþ1 ½k2s jgðfÞj ; 2 2

X 37 ¼

n o isp xsph ð2Þ ð2Þ r k2s Hn1 ½k2s jgðfÞjþHnþ1 ½k2s jgðfÞj ; 2 2

v s

r

2i

n ð2Þ o ð2Þ k2p Hnþ1 k2p jgðfÞj þHn1 k2p jgðfÞj ;

xv sh 2

X 54 ¼ X 55 ¼ X 56 ¼ X 57 ¼ 0; X 61 ¼ X 62 ¼ X 63 ¼ 0; X 64 ¼ ðk2 þ l2 Þk2p Hnð1Þ ðk2p R0 Þ h i 1 2 ð1Þ ð1Þ þ l2 k2p Hn2 ðk2p R0 Þ þ Hnþ2 ðk2p R0 Þ ; 2 2

X 65 ¼ ðk2 þ l2 Þk2p Hnð2Þ ðk2p R0 Þ h i 1 2 ð2Þ ð2Þ þ l2 k2p Hn2 ðk2p R0 Þ þ Hnþ2 ðk2p R0 Þ ; 2 2

X 66 ¼

h i 1 ð1Þ ð1Þ l2 ik22s Hn2 ðk2s R0 Þ Hnþ2 ðk2s R0 Þ ; 2

X 67 ¼

h i 1 ð2Þ ð2Þ l2 ik22s Hn2 ðk2s R0 Þ Hnþ2 ðk2s R0 Þ ; 2

X 71 ¼ X 72 ¼ X 73 ¼ 0; h i 2 ð1Þ ð1Þ X 74 ¼ l2 k2p Hnþ2 ðk2p R0 Þ Hn2 ðk2p R0 Þ ; h i 2 ð2Þ ð2Þ X 75 ¼ l2 k2p Hnþ2 ðk2p R0 Þ Hn2 ðk2p R0 Þ ; h i 2 ð1Þ ð1Þ X 76 ¼ ik2s Hnþ2 ðk2s R0 Þ þ Hn2 ðk2s R0 Þ ; h i 2 ð2Þ ð2Þ X 77 ¼ ik2s Hnþ2 ðk2s R0 Þ þ Hn2 ðk2s R0 Þ ;

X.-Q. Fang, H.-X. Jin / Computers and Geotechnics 83 (2017) 98–105 2

n

l1 k1pf i n Y 11 ¼ 11 i J n k1pf jgðfÞj fJ nþ2 k1pf jgðfÞj 2 þ J n2 k1pf jgðfÞj g; 2 n Y 21 ¼ l1 k1pf i fJ nþ2 k1pf jgðfÞj J n2 k1pf jgðfÞj g; Y 31 ¼

ixspr sph n k1pf i J n1 ðk1pf jgðfÞjÞ Jnþ1 ðk1pf jgðfÞjÞ 2 2 n

þ 11 i J n ðk1pf jgðfÞjÞ l 2 n þ 1 k1pf i Jnþ2 ðk1pf jgðfÞjÞ þ J n2 ðk1pf jgðfÞjÞ ; 2 vs xv sh r n Y 41 ¼ þ k1pf i J nþ1 ðk1pf jgðfÞjÞ J n1 ðk1pf jgðfÞjÞ 2i 2 l 2 n þ 1 k1pf i Jnþ2 ðk1pf jgðfÞjÞ J n2 ðk1pf jgðfÞjÞ ; 2i n Y 51 ¼ 15 k1pf i J n1 ðk1pf jgðfÞjÞ J nþ1 ðk1pf jgðfÞjÞ ; Y 61 ¼ Y 71 ¼ 0: References [1] Fotieva NN, Bulychev NS. Principles of tunnel design for seismic regions. Proc Int Conf Soil Mech Found Eng 1981;1:291–2. [2] Glazanov VE, Shenderov EL. Plane wave scattering by cylindrical cavity in isotropic elastic medium. Sov Phys Acoust 1971;17:141–3. [3] Lee VW, Trifunac MD. Response of tunnels to incident SH-waves. J Eng Mech (ASCE) 1979;105:643–59. [4] Krutin VN, Markov MG, Yumatov AY. Normal waves in a fluid-filled cylindrical cavity located in a saturated porous medium. J Appl Math Mech 1988;52:67–74.

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