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Scattering of a plane wave by a lined cylindrical cavity in a poroelastic half-plane Ling-Fa Jiang a,*, Xiang-Lian Zhou b, Jian-Hua Wang b a b

State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, CAS, Wuhan 430071, PR China Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China

a r t i c l e

i n f o

Article history: Received 25 March 2008 Received in revised form 25 December 2008 Accepted 2 January 2009 Available online 23 February 2009 Keywords: Complex variable function Poroelastic half-plane Lining Stress concentration

a b s t r a c t The focus of this contribution is to develop a complex variable function method to solve the two-dimensional scattering of plane waves by a lined cylindrical cavity in the poroelastic half-plane. The poroelastic half-plane is based on Biot’s dynamic theory, and the governing equations are solved by reduction to three Helmholtz equations. The lining structure can be treated as an elastic material and decoupled into two Helmholtz equations. Here, the large circle assumption is applied to simulate the half-plane boundary. By using appropriate boundary conditions and continuity conditions, the unknown coefﬁcients in the potentials can be determined. Selected numerical results are presented in this paper. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Analysis of scattering of elastic waves by a cavity is one of the most important topics in several engineering disciplines, such as civil engineering, earthquake engineering, and mechanical engineering. In recent years, due to the requirements of practical engineering, the study of dynamic stress concentration around a cavity is attracting increasing attention from researchers [1]. For example, the dynamic stress concentration factor at the surface of a semi-circular cavity was studied by Gamer [2] using the wave function expansion method. The dynamic response of multiple scattering problems was investigated by Lee and Mal [3] using the volume integral equation technique. The valley response was studied by Bard and Bouchon [4] by using discrete wave number approximations. The dynamic stresses and displacements around cylindrical cavities of various shapes were studied by Datta et al. [5] using a combination of a ﬁnite element method and eigenfunction expansions. Zeng and Cakmak [6] used the series expansion method to investigate the scattering of plane SH waves by multiple cavities in both an inﬁnite space and a half-space. Mow and Workman [7] investigated the dynamic stresses around a ﬂuid-ﬁlled cavity and presented the formal solution of the steady-state problem of elasticity theory in the form of Rayleigh waves propagating on a free convex or concave cylindrical cavity. The scattering of elastic waves by cylindrical cavities buried in a two-dimensional elastic medium was investigated by Tadeu and Paulo [8] using the BEM. The transverse response of underground cylindrical cav* Corresponding author. Tel./fax: +86 27 87198142. E-mail addresses: [email protected], [email protected] (L.-F. Jiang). 0266-352X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2009.01.001

ities to incident shear waves was studied by Davis et al. [9] using the Fourier–Bessel series. The ground motion on stratiﬁed alluvial valleys under an incident plane SH wave from an elastic half-space was studied by Gil-Zepeda et al. [10] using a hybrid indirect boundary element method. Several scholars have also studied the scattering of an elastic wave by a cavity embedded in a poroelastic medium. For example, a circular cavity of arbitrary radius in a poroelastic medium was investigated for both P and SV incident waves by Mei and Foda [11] and Mei et al. [12] using the boundary layer approximation. The propagation of an elastic harmonic wave in the neighborhood of a ﬂuid-ﬁlled cylindrical cavity embedded in a saturated porous medium was studied by Krutin et al. [13]. The problem of wave diffraction by a spherical cavity in an inﬁnite poroelastic soil medium was studied by Zimmerman [14] using the boundary element method. The dynamic response of an underground circular cylindrical cavity in a ﬂuid-saturated porous medium was investigated by Liu and Han [15]. The problem of incident harmonic P and SV plane waves in tunnels in an inﬁnite poroelastic saturated soil was studied by Katties et al. [16] using the boundary element method. The scattering of harmonic plane waves by multiple elliptic cavities in a saturated soil medium was studied by Wang et al. [17] using the potential function and the complex function. The dynamic interaction of time harmonic plane waves with a pair of parallel circular cylindrical cavities of inﬁnite length buried in a boundless porous elastic ﬂuid-saturated medium was investigated by Hasheminejad and Avazmohammadi [18]. The aim of this study is to develop a semi-analytical method for addressing the two-dimensional scattering of a plane wave by a lined cylindrical cavity in a poroelastic half-plane. The poroelastic

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medium is described by Biot’s poroelastic theory [19,20], while the lining can be treated as an elastic material. By introducing potentials, the governing equations are decoupled and reduced to Helmholtz equations satisﬁed by the potentials. The series solutions for Helmholtz equations are obtained via the wave function expansion method. By using the complex variables method and utilizing the boundary conditions and continuity conditions between the saturated poroelastic medium and the lining structure, the unknown coefﬁcients in the potentials can be determined. Select numerical results under an incident plane Pf wave are presented in the paper. The methodology and analytical solution developed in this paper may provide a new method for further analysis of two-dimensional scattering of a transient wave in a ﬁnite poroelastic medium. 2. Governing equations and general solutions of a saturated poroelastic medium The model for this problem is illustrated in Fig. 1. The shape of the lined circular cavity is characterized by its inner radius r3, outer radius r1, and depth h, as shown in Fig. 1. The half-plane is assumed to be a saturated medium with properties characterized by physical parameters, such as Lamé constants kI ; lI , and the densities of porous medium, solid skeleton, and pore ﬂuid, qI, qIs, qIf. The lining is assumed to be an elastic medium with properties characterized by kII ; lII , and qII. Based on Biot’s theory for a two-phased material, the constitutive relations for the saturated poroelastic medium are expressed as [19,20]

rij ¼ 2lI eij þ kI dij e adij pf ði; j ¼ x; yÞ pf ¼ aMe þ M#

ð2Þ

e ¼ ui;i ;

ð3Þ

ð1Þ

# ¼ wi;i

where rij denotes the total stress components of the bulk material, eij and e are the strain component and dilatation of the solid matrix,

respectively, kI and lI represent Lamé constants, dij represents the Kronecker delta, # is the variation of ﬂuid content per unit reference volume, a and M are Biot’s parameters, respectively, pf is the pore pressure, and ui and wi denote the average solid displacement and the ﬂuid displacement relative to the solid frame, respectively. The equations of motion for the poroelastic medium can be expressed in terms of displacements ui and wi, as

lI ui;jj þ ðkI þ a2 M þ lI Þuj;ji þ aMwj;ji ¼ qI ui þ qIf wi q g aMuj;ji þ Mwj;ji ¼ qIf ui þ If wi þ wi n

k

ui ¼ uI;i þ eijk wIk;j ¼ uIf ;i þ uIs;i þ eijk wIk;j

ð6Þ

pf ¼ Af uIf ;ii þ As uIs;ii

ð7Þ

where Af and As are the amplitude of fast and slow waves, respectively, and eijk denotes the Livi-Civita symbol. When considering the time harmonic vibration of frequency x pﬃﬃﬃﬃﬃﬃﬃ by the term eixt, where i ¼ 1 for brevity, the term eixt is suppressed henceforth from all expressions in the sequence. After substituting Eqs. (6) and (7) into (4) and (5), the following formula is obtained:

y2 o

o2

x

x2

r2 y1

y

o

o1 r3

x

y1

x1 r1

β

r3

Fig. 1. Model of lining in saturated poroelastic half-plane.

ð5Þ

where qI and qIf denote the bulk density of the porous medium and the density of the pore ﬂuid, qI = (1 n)qIs + qIf, qIs is the density of the solid skeleton, n is the porosity of the porous medium, k is the permeability, and g is the ﬂuid viscosity. Over-dots denote the derivatives of ﬁeld variables with respect to time t. In all manipulations, a subscript I is used to denote the parameters of the poroelastic medium. In order to decouple the equations of motion of Biot’s theory, two scalar potentials uf, us and one vector potential w are introduced to express the displacement and the pore pressure of the porous medium. The displacement and the pore pressure are expressed by the potentials in the following form [14]:

y

h

ð4Þ

o1

r1

x1

Fig. 2. Approximate model of lining in saturated poroelastic half-plane.

775

L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786 5 4

6

Re(N=5) Im(N=5) Re(N=7) Im(N=7) Re(N=12) Im(N=12)

2 1

5

4

σθθ/ωρβUP

3

σ*

Present result Luco (1994)

0 -1

3

-2

2

-3 0

50

100

150

θ

200

250

300

1

350

-100

-50

0

50

100

150

200

250

300

θ

Fig. 3. A convergence test for lining in a saturated half-plane.

Fig. 5. Comparison present result with Luco and De Barros [22].

½ðkI þ 2lI b2 Af ÞuIf ;jj þ b3 uIf ;i þ ½ðkI þ 2lI b2 As ÞuIs;jj þ b3 uIs ;i þ eiml ½lI wIi;jj þ b3 wIi ;m ¼ 0

ð8Þ

Fulﬁllment of the above equation requires that the expressions in braces vanish independently, which gives the following equations for the potentials:

The following equations can guarantee the fulﬁllment of the above equation:

ðkI þ 2lI b2 Af ÞuIf ;jj þ b3 uIf ¼ 0

ð9Þ

Af uIf ;ii þ ðb5 Af b4 ÞuIf ¼ 0

ð15Þ

ðkI þ 2lI b2 As ÞuIs;jj þ b3 uIs ¼ 0

ð10Þ

As uIs;ii þ ðb5 As b4 ÞuIs ¼ 0

ð16Þ

lI wIi;jj þ b3 wIi ¼ 0

ð11Þ

where

where

b1 ¼

b4 ¼ ab1 þ qIf x2 ;

qIf x

2

n

igx ; k

b2 ¼ a þ

qIf x b1

2

and b3 ¼ qI x2 þ

;

2 If

4

q x

b1 ð12Þ

b pf;ii 1 pf ðab1 þ qIf x2 Þui;i ¼ 0 M

ð13Þ

Likewise, substituting Eqs. (6) and (7) into (13), the following equation is obtained:

½Af uIf ;ii þ ðb5 Af b4 ÞuIf ;jj þ ½As uIs;ii þ ðb5 As b4 ÞuIs ;jj ¼ 0

ð14Þ

ð17Þ

Using Eqs. (9) and (10) and (15) and (16), the following equation determining Af and As is derived:

A2f ;s þ

Substitution of Eqs. (2) and (3) into (5) yields

b5 ¼ b1 =M

b3 ðkI þ 2lI Þb5 b2 b4 ðkI þ 2lI Þb4 Af ;s þ ¼0 b2 b5 b2 b5

ð18Þ

For the plane strain wave involved in this paper, only one component, w3 of the vector potential w, remains. For simplicity, the symbol w is used to denote w3. From Eqs. (9)–(11) and (15) and (16), each component uIf, uIs, and wI must satisfy the Helmholtz equation of the following form:

8 7

7

6

6

5

5

4

Im(r2/r3=50)

3

Re(r2/r3=100)

2

2

Im(r2/r3=100)

1

Re(r2/r3=200)

1

0

Im(r2/r3=200)

4

σ

*

3

σ*

Re(r2/r3=50)

0

Re(r2/r3=50)

-1

Im(r2/r3=50)

-1

-2

Re(r2/r3=100)

-2

-3

Im(r2/r3=100)

-3

-4

Re(r2/r3=200)

-4

-5

Im(r2/r3=200)

0

50

100

150

200

250

a. Re(kfr1)=0.5; h=1.5r1

300

350

400

0

50

100

150

200

250

300

b. Re(kfr1)=0.5; h=30r1

Fig. 4. Comparison present result with different cases: r2/r3 = 50, 100, 200.

350

400

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L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786

r2 uIf þ k2If uIf ¼ 0

ð19Þ

r2 uIs þ k2Is uIs ¼ 0

ð20Þ

r2 wI þ k2It wI ¼ 0

ð21Þ

if introducing 2

ð22Þ

2

ð23Þ

2

ð24Þ

kIf ¼ b3 =ðkI þ 2lI b2 Af Þ ¼ ðb5 Af b4 Þ=Af kIs ¼ b3 =ðkI þ 2lI b2 As Þ ¼ ðb5 As b4 Þ=As kIt ¼ b3 =lI

where kIf, kIs, and kIt denote the complex wave numbers for the fast wave, the slow wave, and the shear wave of the porous medium, respectively. In order to guarantee that the waves attenuate when propagating within the porous medium, the inequality Im (kIf,Is,It) > 0 should always hold. Moreover, to ensure the Pf wave is faster than the Ps wave, we take Re (kIf) < Re (kIs). If introducing complex variables as follows z = x + iy, z ¼ x iy, the scattered wave ﬁeld is determined by the Helmholtz equations (19)–(21). The general solutions for Eqs. (19)–(21) can be expressed in terms of Hankel functions as follows:

n z jzj n¼1 n 1 X z uIs ¼ bn Hnð1Þ ðkIs jzjÞ jzj n¼1 n 1 X z cn Hð1Þ ðk jzjÞ wI ¼ It n jzj n¼1 1 X

uIf ¼

an Hð1Þ n ðkIf jzjÞ

ð25Þ ð26Þ ð27Þ

where Hð1Þ n ðÞ denotes the ﬁrst kind of Hankel function of order n, and an, bn, cn are arbitrary coefﬁcients to be determined by the boundary conditions.

3. Governing equations and general solutions of the lining structure In this section, we will derive the governing equation and general solution for the lining structure. For this purpose, the following constitutive relations for the lining structure can be expressed as:

lII ui;jj þ ðkII þ lII Þuj;ji ¼ qII u€i ði; j ¼ x; yÞ

7

ð28Þ

6

6 4

5 4

Re(n=0.35) Im(n=0.35) Re(n=0.4) Im(n=0.4)

3

1

0

σ

σ

2

2

0

-2

-1 -2

Re(n=0.35) Im(n=0.35) Re(n=0.4) Im(n=0.4)

-4

-3 -4

-6

0

50

100

150

200

250

300

350

0

50

θ a. Re(kfr1)=0.5; h=1.5r1

100

150

200

250

300

350

θ b. Re(kfr1)=2.0; h=1.5r1

7 6

6 5

4

4 3

2

1

σ

σ

2

0

0

-1

Re(n=0.35) Im(n=0.35) Re(n=0.4) Im(n=0.4)

-2 -3 -4

-2

Re(n=0.35) Im(n=0.35) Re(n=0.4) Im(n=0.4)

-4

-5 0

50

100

150

200

250

θ c. Re(kfr1)=0.5; h=30r1

300

350

0

50

100

150

200

250

300

350

θ d. Re(kfr1)=2.0; h=30r1

Fig. 6. Distribution of dynamic stress concentration around the lining with different porosity under the frequency Re (kfr1) = 0.5, 2.0 and the depth h = 1.5r1, 30r1.

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L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786

where qII denotes the density of the lining, and kII and lII represent the Lamé constants of the lining. First, we introduce the scalar potentials uII and vector potential wII to express the displacement of the lining structure. Then, from Eq. (28), we know that each component of uII and wII must satisfy the Helmholtz equations of the following form:

r2 uII þ k2IIp uII ¼ 0 2

r wII þ

2 kIIs wII

¼0

2

2

ð29Þ ð30Þ

where kIIp ¼ , Vp ¼ V2 P

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kII þ2lII

qII

2

2

, Vs ¼ , kIIs ¼ V2 s

qﬃﬃﬃﬃ lII q , and kIIp and kIIs II

half-plane. In this paper, we set r2 = 100r3 to insure that displacements on the curved surface approach accurately enough to those of a ﬂat surface in the free ﬁeld. Convergence of the solutions for various large ratios is guaranteed by [9]. For scattering of elastic waves by a circular cavity with lining in a poroelastic half-plane, the total wave ﬁeld is composed of the incident wave, the reﬂected wave, and the scattered wave: ðiÞ ðrÞ ðsÞ uðtÞ If ¼ uIf þ uIf þ uIf

ð31Þ

ðtÞ Is ðtÞ wI

ð32Þ

u

ðiÞ ðrÞ ðsÞ ¼ Is þ Is þ Is ðiÞ ðrÞ ðsÞ ¼ wI þ wI þ wI

u

u

u

ð33Þ

denote the complex wave numbers of the compressional wave and the shear wave, respectively. In all manipulations, a subscript II is used to denote the parameters of the lining.

By introducing complex variables, the incident plane harmonic Pf, Ps, and SV waves, can be written as follows:

4. Total waves of the poroelastic half-plane and lining

uðiÞ If ¼ uIf 0 exp

As shown in Fig. 2, the half-plane surface is approximated as a convex circular surface centered at o2 with large radius r2 r3. The curved surface of the large circle is then used as an approximation of the ﬂat surface of the inﬁnite half-plane. It is now obvious that when the radius of the large circle approaches inﬁnity, this model approaches that of the circular cavity with a lining in the

uðiÞ Is ðiÞ wI

p p i ikIf h z exp i b þ z exp i b ð34Þ 2 2 2 h p p i ikIs ¼ uIs0 exp z exp i b0 þ z exp i b0 2 2 2

p p i ikIt h ¼ wI0 exp z exp i c þ z exp i c 2 2 2

ð35Þ ð36Þ

6

1.0 0.8

4

0.6

Re(n=0.35) Im(n=0.35) Re(n=0.4) Im(n=0.4)

0.4 0.2

2

0

σ

0.0

-2

-0.2

Re(n=0.35) Im(n=0.35) Re(n=0.4) Im(n=0.4)

-0.4 -4

-0.6 -6

-0.8 0

50

100

150

200

250

300

0

350

50

100

150

200

250

300

350

θ b. Re(kfr1)=2.0; h=1.5r1

θ a. Re(kfr1)=0.5; h=1.5r1 1.5

1.0 0.8

1.0

0.6 0.4

0.5

0.2 0.0

0.0

-0.2 -0.4

-0.5

Re(n=0.35) Im(n=0.35) Re(n=0.4) Im(n=0.4)

-0.6 -0.8

Re(n=0.35) Im(n=0.35) Re(n=0.4) Im(n=0.4)

-1.0

-1.0 0

50

100

150

200

250

θ c. Re(kfr1)=0.5; h=30r1

300

350

0

50

100

150

200

250

θ d. Re(kfr1)=2.0; h=30r1

300

350

Fig. 7. Distribution of pore pressure concentration around the lining with different porosity under the frequency Re (kfr1) = 0.5, 2.0 and the depth h = 1.5r1, 30r1.

778

L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786

where b, b0, and c are the incident angles of the incident harmonic Pf, Ps, and SV waves, respectively, and uIf0, uIs0, wI0 are the amplitude ratios of the three incident waves. If there is no lining, the incident wave reﬂected from the halfplane will generate a reﬂected Pf and Ps as well as a SV wave to satisfy the stress-free boundary conditions [21].

p p i ikIf h z exp i b0 þ z exp i b0 2 2 2 p p i ikIs h ðrÞ 0 uIs ¼ A2 exp z exp i b0 þ z exp i b00 2 2 2 p p i ikIt h ðrÞ 0 þ z exp i c0 z exp i c wI ¼ A3 exp 2 2 2

uðrÞ If ¼ A1 exp

ð37Þ ð38Þ ð39Þ

where A1, A2, and A3 are the amplitude ratios of the reﬂected waves; b0 , b00 , c0 are the reﬂected angles of the incident harmonic Pf, Ps, and SV waves, respectively. The amplitude ratios of the potentials A1, A2, and A3 are obtained by the straight boundary conditions of the poroelastic half-plane. Based on Snell’s law, these reﬂected angles can be expressed as:

kIf sin b0 ¼ kIs sin b00 ¼ kIt sin c0

ð40Þ

In the half-plane, because of the presence of both the plane-free boundary and the lining, the incident P wave and the reﬂected P and SV waves from the ground surface will be scattered around the lining in the half-plane, and the total potentials of the harmonic plane Pf, Ps, and SV waves generated at the lining are represented by ðsÞ ðsÞ uðsÞ , uIs1 , and wI1 . The scattered cylindrical waves from the lining If 1 will be reﬂected back into the half-plane from the plane-free surface. The cylinder vibrations are reﬂected off the half-plane free surðsÞ ðsÞ ðsÞ face generating new waves represented by uIf 2 , uIs2 , and wI2 .

n zij ain Hð1Þ ðk jz jÞ ij If n jzij j i¼1 n¼1 n 2 1 X X zij ðsÞ ðsÞ ð1Þ uðsÞ ¼ u þ u ¼ b H ðk jz jÞ in Is ij n Is Is1 Is2 jzij j i¼1 n¼1 n 2 1 X X zij ðsÞ ðsÞ ðsÞ cin Hð1Þ ðk jz jÞ wI ¼ wI1 þ wI2 ¼ It ij n jzij j i¼1 n¼1 ðsÞ ðsÞ uðsÞ If ¼ uIf 1 þ uIf 2 ¼

2 1 X X

ð41Þ ð42Þ ð43Þ

where zij = z dj(j = 1, 2), and dj(j = 1, 2) is the complex coordinate between the origin of jth circular and the origin of total coordinate system.

7 4

6 5

2 1

0

σ

3

σ

2

Re(η=0.8) Im(η=0.8) Re(η=0.85) Im(η=0.85) Re(η=0.9) Im(η=0.9)

4

-2

0

Re(η=0.8) Im(η=0.8) Re(η=0.85) Im(η=0.85) Re(η=0.9) Im(η=0.9)

-1 -4

-2 -3 -4

-6

0

50

100

150

200

250

300

350

0

50

θ a. Re(kfr1)=0.5; h=1.5r1

100

150

200

250

300

350

300

350

θ b. Re(kfr1)=2.0; h=1.5r1 Re(η=0.8) Im(η=0.8) Re(η=0.85) Im(η=0.85) Re(η=0.9) Im(η=0.9)

8

6

7 6

4

5 4 3

2

1

σ

σ

2

0

0

Re(η=0.8) Im(η=0.8) Re(η=0.85) Im(η=0.85) Re(η=0.9) Im(η=0.9)

-1 -2 -3 -4

-2

-4

-5 0

50

100

150

200

250

θ c. Re(kfr1)=0.5; h=30r1

300

350

0

50

100

150

200

250

θ d. Re(kfr1)=2.0; h=30r1

Fig. 8. Distribution of dynamic stress concentration around the lining with different g under the frequency Re (kfr1) = 0.5, 2.0 and the depth h = 1.5r1, 30r1.

779

L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786

For the lining, the total wave ﬁeld is composed of the refracted wave and the scattered wave:

n z ð2Þ ½d1n Hð1Þ ðk jzjÞ þ d H ðk jzjÞ ð44Þ IIp 2n IIp n n jzj n¼1 n 1 X z ð2Þ ¼ ½e1n Hð1Þ ð45Þ n ðkIIs jzjÞ þ e2n H n ðkIIs jzjÞ jzj n¼1

ðsÞ ðf Þ uðtÞ II ¼ uII þ uII ¼

ðtÞ

ðsÞ

ðf Þ

wII ¼ wII þ wII

1 X

where Hð2Þ n ðÞ denotes the second kind of Hankel function of order n, and d1n, d2n, e1n, and e2n are arbitrary coefﬁcients to be determined by the boundary conditions. 5. Expressions of stress, pore pressure, and displacement of the poroelastic half-plane

@ ðtÞ g uðtÞ þ g2 uðtÞ expðihÞ Is ia1 wI @z 1 If @ ðtÞ expðihÞ g uðtÞ þ g2 uðtÞ wIr iwIh ¼ 2 Is þ ia1 wI @z 1 If

2 ðtÞ rIr þ rIh ¼ 2 kI þ lI kIf uIf þ k2Is uðtÞ Is wIr þ iwIh ¼ 2

@ ðtÞ ðtÞ ðtÞ u þ uIs iwI expðihÞ If @z @ ðtÞ ðtÞ uIr iuIh ¼ 2 uIf þ uðtÞ expðihÞ Is þ iwI @z

uIr þ iuIh ¼ 2

ð49Þ ð50Þ

@ ðtÞ ðtÞ ðtÞ ðtÞ expð2ihÞ rIr þ irIrh ¼ af uðtÞ If þ as uIs þ 4lI 2 uIf þ uIs iwI @z ð51Þ 2 @ ðtÞ ðtÞ ðtÞ rIr irIrh ¼ af uðtÞ uðtÞ expð2ihÞ If þ as uIs þ 4lI If þ uIs þ iwI @z2 ð52Þ 2

2

2

ðtÞ

ðtÞ

pf ¼ Af kIf uIf As kIs uIs 2

Introducing the complex variables z = x + iy and z ¼ x iy, the expressions of stress, pore pressure, and displacement of the poroelastic half-plane can be expressed as:

ð48Þ

ð53Þ 2

2

2

where af ¼ aAf kIf ðkI þ lI ÞkIf ; as ¼ aAs kIs ðkI þ lI ÞkIs ;

g1 ¼ a1 a2 Af k2If ; g2 ¼ a1 a2 As k2Is ; q x2 1 a1 ¼ If ; and a2 ¼

ð54Þ

b1

b1

ð46Þ ð47Þ

1.0 1.0 0.8 0.6

p

0.2 0.0

p

0.5

Re(η=0.8) Im(η=0.8) Re(η=0.85) Im(η=0.85) Re(η=0.9) Im(η=0.9)

0.4

-0.2

0.0

Re(η=0.8) Im(η=0.8) Re(η=0.85) Im(η=0.85) Re(η=0.9) Im(η=0.9)

-0.5

-0.4 -0.6

-1.0

-0.8 0

50

100

150

200

250

300

0

350

50

100

150

200

250

300

350

θ b. Re(kfr1)=2.0; h=1.5r1

θ a. Re(kfr1)=0.5; h=1.5r1 1.5

Re(η=0.8) Im(η=0.8) Re(η=0.85) Im(η=0.85) Re(η=0.9) Im(η=0.9)

1.2 1.0 0.8 0.6

1.0

0.5

0.2

p

p

0.4

0.0

0.0 -0.2

Re(η=0.8) Im(η=0.8) Re(η=0.85) Im(η=0.85) Re(η=0.9) Im(η=0.9)

-0.5

-0.4 -0.6

-1.0

-0.8 -1.0 0

50

100

150

200

250

θ c. Re(kfr1)=0.5; h=30r1

300

350

0

50

100

150

200

250

300

θ d. Re(kfr1)=2.0; h=30r1

Fig. 9. Distribution of pore pressure concentration around cavity with different g under the frequency Re (kfr1) = 0.5, 2.0 and the depth h = 1.5r1, 30r1.

350

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L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786

6. Expressions of stress and displacement of the lining

b1n, c1n, d1n, e1n, a2n, b2n, c2n, d2n, and e2n can be determined. The continuity conditions at the interface can be written as:

Let z = x + iy, z ¼ x iy. Then, the expressions of stress and displacement of lining can be expressed as:

rIr1 irIrh1 ¼ rIIr1 irIIrh1 rIr1 þ irIrh1 ¼ rIIr1 þ irIIrh1

ð61Þ

uIr1 iuIh1 ¼ uIIr1 iuIIh1

ð62Þ

@ ðtÞ expðihÞ uII iwðtÞ II @z @ ðtÞ expðihÞ uIIr iuIIh ¼ 2 uðtÞ II þ iwII @z rIIr þ rIIh ¼ 2ðkII þ lII Þk2IIp uðtÞ II uIIr þ iuIIh ¼ 2

rIIr þ irIIrh

rIIr irIIrh

ð55Þ

ð60Þ

uIr1 þ iuIh1 ¼ uIIr1 þ iuIIh1

ð63Þ

ð56Þ

wIr1 ¼ 0

ð64Þ

ð57Þ

The zero-stress boundary conditions at the free ground surface within the half-plane can be expressed as:

@ 2 ðtÞ 2 ðtÞ ðtÞ ¼ ðkII þ lII ÞkIIp uII þ 4lII 2 uII iwII expð2ihÞ @z ð58Þ 2 @ 2 ðtÞ ðtÞ ðtÞ ¼ ðkII þ lII ÞkIIp uII þ 4lII 2 uII þ iwII expð2ihÞ @z ð59Þ

rIr2 irIrh2 ¼ 0 rIr2 þ irIrh2 ¼ 0

ð65Þ ð66Þ

For a permeable boundary condition, the pore pressure should vanish. Consequently, using Eq. (53), the boundary condition has the form: 2

ðtÞ

2

ðtÞ

pf 2 ¼ Af kIf uIf As kIs uIs ¼ 0

7. The boundary value problem Applying the boundary conditions at the free surface within the lining and the half-plane, and the continuity conditions at the interface between the lining and half-plane, the constants a1n,

ð67Þ

For impermeable boundary conditions, the normal displacement of the ﬂuid relative to the solid skeleton should vanish. Consequently, using Eqs. (48) and (49), the boundary condition has the form:

7

4

6 5

2

4

Re(νII /νI=0.8)

3

Im(νII /νI=0.8)

0

Re(νII /νI=1.2) Im(νII /νI=1.2)

1

σ

σ

2

-2

0

Re(νII /νI=0.8)

-1

Im(νII /νI=0.8)

-4

-2

Re(νII /νI=1.2) Im(νII /νI=1.2)

-3

-6

-4 0

50

100

150

200

250

θ a. Re(kfr1)=0.5; h=1.5r1

300

0

350

50

100

150

200

250

300

350

θ b. Re(kfr1)=2.0; h=1.5r1

7 6

6 5

4

4 3

2

2

σ

σ

1 0

0

-1

Re(νII /νI=0.8)

-2 -3

Im(ν II /ν I=0.8)

Re(νII /νI=1.2)

-4

Re(ν II /ν I=0.8)

-2

Im(νII /νI=0.8)

Re(ν II /ν I=1.2)

-4

Im(νII /νI=1.2)

Im(ν II /ν I=1.2)

-5 0

50

100

150

200

250

θ c. Re(kfr1)=0.5; h=30r1

300

350

0

50

100

150

200

250

300

350

θ d. Re(kfr1)=2.0; h=30r1

Fig. 10. Distribution of dynamic stress concentration around the lining with different mII/mI under the frequency Re (kfr1) = 0.5, 2.0 and the depth h = 1.5r1, 30r1.

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L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786

wIr2 ¼

@ ðtÞ g uðtÞ þ g2 uðtÞ expðihÞ Is þ ia1 wI @z 1 If @ ðtÞ ðtÞ expðihÞ ¼ 0 g1 uðtÞ þ If þ g2 uIs ia1 wI @z

ies. This can be checked by assessing the extent to which the boundary conditions are satisﬁed.

ð68Þ

The zero-stress boundary conditions at the free ground surface within the lining can be expressed as:

rIIr3 irIIrh3 ¼ 0 rIIr3 þ irIIrh3 ¼ 0

8. Numerical results and discussion Dynamic stress concentration factor r* is deﬁned as:

ð69Þ ð70Þ

Obviously, Eqs. (60)–(66), (68)–(70) form a set of inﬁnite algebraic equations for determining the coefﬁcients a1n, b1n, c1n, d1n, e1n, a2n, b2n, c2n, d2n, and e2n. Also, it should be noted that Eqs. (60)–(67) and (69), (70) are used to solve the permeable problem, while Eqs. (60)–(66) and (68)–(70) are used to solve the impermeable problem. The derivations of Eqs. (60)–(66), (68)–(70) are given in Appendix A. It should also be noted that the above equations are all in the form of inﬁnite sums; therefore, the system of equations must be solved by truncating the inﬁnite terms into ﬁnite terms. The accuracy of the solution depends upon the truncation of the inﬁnite ser-

r1 ¼

r0Ih r00

ð71Þ

where

2 ðtÞ r0Ih ¼ ðkI þ lI Þ k2If uðtÞ If þ kIs uIs

@ 2 ðtÞ ðtÞ expð2ihÞ uIf þ uðtÞ Is þ iwI 2 @z @ 2 ðtÞ ðtÞ ðtÞ 2lI 2 uIf þ uIs iwI expð2ihÞ apf @z

2lI

ð72Þ

r00 ¼ ðkI þ 2lI Þk2If uIf 0 aI pf 0

ð73Þ

1.0 1.0

0.8 0.6

0.5

0.4

Re(νII /νI=0.8) Im(νII /νI=0.8)

0.2

Re(νII /νI=1.2) 0.0

0.0

Im(νII /νI=1.2)

-0.2

Re(νII /νI=0.8)

-0.5

Im(νII /νI=0.8)

-0.4

Re(νII /νI=1.2)

-0.6

Im(νII /νI=1.2)

-1.0

-0.8 0

50

100

150

200

250

300

350

0

50

θ a. Re(kfr1)=0.5; h=1.5r1

100

150

200

250

300

350

θ b. Re(kfr1)=2.0; h=1.5r1 1.5

1.0 0.8

1.0

0.6 0.4

0.5

0.2 0.0

0.0

-0.2

Re(νII /νI=0.8)

-0.4

Re(νII /νI=0.8)

-0.5

Im(νII /νI=0.8)

-0.6

Im(νII /νI=0.8)

Re(νII /νI=1.2)

-0.8

Re(νII /νI=1.2)

-1.0

Im(νII /νI=1.2)

Im(νII /νI=1.2)

-1.0 0

50

100

150

200

250

θ c. Re(kfr1)=0.5; h=30r1

300

350

0

50

100

150

200

250

300

350

θ d. Re(kfr1)=2.0; h=30r1

Fig. 11. Distribution of pore pressure concentration around the lining with different mII/mI under the frequency Re (kfr1) = 0.5, 2.0 and the depth h = 1.5r1, 30r1.

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L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786

Pore stress concentration factor p* is deﬁned as:

pf p ¼ pf 0

ð74Þ 2

where pf 0 ¼ Af kIf uf 0 :

ð75Þ

First, the convergence of the proposed scheme will be veriﬁed. The material parameters for the saturated half-plane are: qIs = 2500 kg/m3, qIf = 1000 kg/m3, lI = 1.0 107 Pa, mI = 0.35, a = 0.999, M = 1.0 108 Pa, g = 1.0 102 Pa s, and k = 1.0 1010 m/s. The material parameters for the lining are: qII = 0.8qIs, mII = 0.8mI, lII = 0.8lI, g = r1/r3 = 1.0, b = 0°, Re (kfr1) = 0.25, and h = 1.5r1. Fig. 3 shows that the stresses have good convergence for a truncation constant of N take 7 and 12. In the following parameter analyses, the truncation constant of N is taken as 12. Fig. 4 shows the stress concentration factor when wave number Re (kfr1) = 0.5, h = 1.5r1 and 30r1 for three cases: r2/r3 = 50, 100, and 200. Fig. 4 indicates that the stresses converge very well when r2/r3 = 50, 100, and 200. In order to guarantee calculation precision, we determined that the ratio of the radius r2/r3 should be 100, which assures convergence at the typical span of seismic wave frequency range 10 6 f 6 100 (Hz). The approach and numerical scheme are checked by comparing our solution with that reported by other scholars. Fig. 5 shows a comparison of the present results for r* with those reported by

Luco and De Barros [22]. Our results in Fig. 5 are obtained by reducing the lining in the poroelastic medium to the cavity in an ideal single-phase elastic medium. This ﬁgure shows that the two solutions agree with each other very well. This paper discusses the effects of the non-dimensional wave number, the porosity, the ratio of the inner radius to outer radius of the lining, and the embedding depth on the dynamic response under incident plane Pf waves. The material parameters for the saturated medium are: qIs = 2750 kg/m3, qIf = 1000 kg/m3, nI = 0.35 and 0.4, lI = 1.0 107 Pa, mI = 0.3, a = 0.999, M = 1.0 108 Pa, g = 1.0 102 Pa s, and k = 1.0 1010 m/s. The material parameters for the lining are: qII = 0.8qIs, mII = 0.8mI, lII = 0.8lI, b = 0°, Re (kfr1) = 0.5 and 2.0, h = 1.5r1 and 30r1. The ratio of the inner radius to outer radius of the lining is 0.8. Figs. 6 and 7 show the effect of porosity n on the dynamic stress concentration and the pore pressure concentration. The effects of porosity on the stresses and the pore pressures are clearly illustrated in Figs. 6 and 7. The dynamic stresses and the pore pressures exhibit small changes with increasing porosity, especially at the condition of low depth h = 1.5r1 and low frequency Re (kfr1) = 0.5. In general, the effects of porosity on the stresses and pore pressures under the condition of deeper depth (h = 30r1) are greater than those at condition of lower depth (h = 1.5r1). Likewise, the effects of porosity on the stresses and

7 4

6 5

2

Re(μII /μI=0.8)

3

Im(μII /μI=0.8)

2

Re(μII /μI=2.0)

1

Im(μII /μI=2.0)

0

σ

σ

4

0

-2

Re(μII /μI=0.8)

-1 -2

Im(μII /μI=0.8)

-4

Re(μII /μI=2.0)

-3

Im(μII /μI=2.0)

-4 -6 0

50

100

150

200

250

300

350

0

50

θ a. Re(kfr1)=0.5; h=1.5r1

100

150

200

250

300

350

300

350

θ b. Re(kfr1)=2.0; h=1.5r1

7

Re(μII /μI=0.8)

6

6

Im(μII /μI=0.8)

5

Re(μII /μI=2.0) 4

4

Im(μII /μI=2.0)

3 2

2

0

σ

σ

1 0

-1 -2

Re(μII /μI=0.8)

-3

-2

Im(μII /μI=0.8)

-4

Re(μII /μI=2.0)

-5

Im(μII /μI=2.0)

-4

-6 0

50

100

150

200

250

θ c. Re(kfr1)=0.5; h=30r1

300

350

0

50

100

150

200

250

θ d. Re(kfr1)=2.0; h=30r1

Fig. 12. Distribution of dynamic stress concentration around the lining with different lII/lI under the frequency Re (kfr1) = 0.5, 2.0 and the depth h = 1.5r1, 30r1.

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L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786

the pore pressures under the condition of higher frequency (Re (kfr1) = 2.0) are greater than those for the condition of lower frequency (Re (kfr1) = 0.5). In Figs. 8 and 9, the effects of the ratio of inner radius to outer radius of the lining on the dynamic stress concentration factors r* and pore pressure concentration factors pf along the boundary of the lining are plotted versus g = r1/ r3 = 1.1, 1.15, and 1.2; Re (kfr1) = 0.5 and 2.0; h = 1.5r1 and 30r1; n = 0.4 and b = 0°. The effects of lining thickness on the stresses and the pore pressures are clearly illustrated in Figs. 8 and 9. For the condition of low frequency (Re (kfr1) = 0.5), the dynamic stresses and the pore pressures show a small decrease with increases in lining thickness g, while for the condition of high frequency (Re (kfr1) = 2.0), the dynamic stresses and the pore pressures increase with increases in lining thickness g. At the lower frequency (Re (kfr1) = 0.5), the effects of lining thickness on the stresses and the pore pressures show small changes, while for the condition of high frequency (Re (kfr1) = 2.0, the stresses and the pore pressures undergo significant changes. Figs. 8 and 9 show that the lining thickness has a large inﬂuence on the dynamic stresses and pore pressures. In general, the stresses and the pore pressures decrease with increases in the lining thick-

ness under the condition of low frequency (Re (kfr1) = 0.5). In contrast, the stresses and the pore pressures increase with increases in the lining thickness at the condition of high frequency (Re (kfr1) = 2.0). Because the greatest stresses and greatest pore pressures at lower frequencies are greater than those at higher frequencies, from the point of view of safety, an increasing lining thickness can decrease the greatest stresses and the greatest pore pressures for the saturated medium. In Figs. 10 and 11, the effects of the Poisson ratio on the dynamic stress concentration factors r* and the pore pressure concentration factors pf along the boundary of the lining are plotted versus mII = 0.8mI, 1.2mI; g = r1/r3 = 1.2; Re (kfr1) = 0.5 and 2.0; h = 1.5r1 and 30r1; n = 0.4 and b = 0°. The effects of the Poisson ratio on the dynamic stresses and pore pressures are clearly illustrated in Figs. 10 and 11. The dynamic stresses and the pore pressures show small changes with changes in the Poisson ratio, especially at the condition of low frequency Re (kfr1) = 0.5. The effects of the shear modulus ratio on the stresses and pore pressures are clearly illustrated in Figs. 12 and 13. Under the condition of low frequency (Re (kfr1) = 0.5), the dynamic stresses and pore pressures increase with increases in the shear modulus ratio.

1.0

1.0 0.8 0.6

0.5 0.4

Re( μII /μI=0.8) Im(μII /μI=0.8)

0.2

Re( μII /μI=2.0)

0.0

0.0

Im(μII /μI=2.0)

-0.2

-0.5 -0.4

Re( μII /μI=0.8)

-0.6

Re( μII /μI=2.0)

Im(μII /μI=0.8) -1.0

Im(μII /μI=2.0)

-0.8 0

50

100

150

200

250

300

350

0

θ a. Re(kfr1)=0.5; h=1.5r1

50

100

150

200

250

θ b. Re(kfr1)=2.0; h=1.5r1

300

350

1.5

1.2

Re(μII /μI=0.8)

1.0

Im(μII /μI=0.8)

0.8

Re(μII /μI=2.0)

1.0

Im(μII /μI=2.0)

0.6

0.5

0.4 0.2 0.0

0.0

-0.2 -0.4

Re( μII /μI=0.8)

-0.5

-0.6

Im(μII /μI=0.8)

-0.8

Re( μII /μI=2.0) Im(μII /μI=2.0)

-1.0

-1.0 0

50

100

150

200

250

θ c. Re(kfr1)=0.5; h=30r1

300

350

0

50

100

150

200

250

300

350

θ d. Re(kfr1)=2.0; h=30r1

Fig. 13. Distribution of pore pressure concentration around the lining with different lII/lI under the frequency Re (kfr1) = 0.5, 2.0 and the depth h = 1.5r1, 30r1.

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L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786

The greatest concentration factor increases with increases in the lining rigidity. Under the condition of high frequency (Re (kfr1) = 2.0), the dynamic stresses and pore pressures decrease with increases in the shear modulus ratio. The concentration factors at low frequency are greater than those at high frequency.

E115in

9. Conclusion Based on Biot’s theory and the complex variable function method, a new approach for solving the two-dimensional scattering of plane waves by a cylindrical cavity with lining in a poroelastic half-plane is developed in this paper. The methodology suggested in this paper is more advantageous than conventional methods, such as the eigenfunction expansion method, the BEM, and the FEM. Based on the derivation and numerical examples presented above, the following conclusions can be drawn: (1) The thickness of the lining has a great inﬂuence on the dynamic stress and pore pressure. In general, the stresses and pore pressures decrease with increases in the thickness of the lining at the condition of low frequency. From the point of view of safety, increases in the thickness of the lining can decrease the greatest stresses and pore pressures of the saturated medium. (2) The shear module ratio also has a great inﬂuence on the stresses and pore pressures. Under the condition of low frequency, the dynamic stresses and the pore pressures increase with increases in the shear module ratio. (3) The porosity and Poisson ratio have a small inﬂuence on the stresses and pore pressures. (4) The methodology and analytical solution developed in this paper may provide a new method for further analysis of the two-dimensional scattering of a transient wave in a ﬁnite poroelastic medium.

n zi jzi j n2 zi 2 ð1Þ lII kIIp Hn2 ðkIIp jzi jÞ expð2ihÞ ð77dÞ jzi j n2 zi 2 ð1Þ ¼ ilII kIIs Hn2 ðkIIs jzi jÞ expð2ihÞ ð77eÞ jzi j n nþ2 zi zi 2 ð1Þ ¼ af Hð1Þ þ lI kIf Hnþ2 ðkIf jzi jÞ expð2ihÞ n ðkIf jzi jÞ jzi j jzi j 2

E114in ¼ ðkII þ lII ÞkIIp Hnð1Þ ðkIIp jzi jÞ

E121in

E122in

E123in

ð77fÞ n nþ2 zi zi 2 ð1Þ ¼ as Hnð1Þ ðkIs jzi jÞ þ lI kIs Hnþ2 ðksf jzi jÞ expð2ihÞ jzi j jzi j

zi 2 ð1Þ ¼ ilI kIt Hnþ2 ðkIt jzi jÞ jzi j

ð77gÞ

nþ2

expð2ihÞ n

ð77hÞ

zi jzi j nþ2 zi 2 ð1Þ lII kIIp Hnþ2 ðkIIp jzi jÞ expð2ihÞ jzi j nþ2 zi 2 ð1Þ ¼ ilII kIIs Hnþ2 ðkIIs jzi jÞ expð2ihÞ jzi j 2

E124in ¼ ðkII þ lII ÞkIIp Hnð1Þ ðkIIp jzi jÞ

E125in

ðiÞ f

r11 ¼ af ðu þ u

ðrÞ f Þ

ð77iÞ ð77jÞ

ðrÞ as ðuðiÞ s þ us Þ

2

4lI

o ðiÞ ðrÞ ðiÞ ðrÞ ðrÞ ½u þ uf þ uðiÞ s þ us þ iðw þ w Þ expð2ihÞ oz2 f ð77kÞ ðiÞ

ðrÞ

ðrÞ r12 ¼ af ðuf þ uf Þ as ðuðiÞ s þ us Þ

4lI

o2 ðiÞ ðrÞ ðiÞ ðrÞ ðrÞ ½u þ uf þ uðiÞ s þ us iðw þ w Þ expð2ihÞ oz2 f ð77lÞ

in which

x1in ¼ ain ;

Acknowledgements

zi ¼ r 1 e This investigation was supported by Key Laboratory of Geotechnical Mechanics and Engineering of the Ministry of Water Resources (Project No.: G07-09). The research was also ﬁnancially supported by Shanghai Leading Academic Discipline Project (Project No.: B208) and State Key Laboratory of Geomechanics and Geotechnical Engineering (Project Nos.: SKLQ015 and SKLZ0803).

ih

x2in ¼ bin ;

x3in ¼ cin ;

x4in ¼ din ;

x5in ¼ ein

ði ¼ 1Þ

zi ¼ r 2 eih þ d2 d1

ð77mÞ ð77nÞ

ði ¼ 2Þ

ð77oÞ

Multiplying both sides of Eq. (76) with eish and integrating over the interval [p, p], one obtains: 5 X 2 1 X X

1s E1s kpin xpin ¼ r k

ðk ¼ 1; 2Þ ðs ¼ 0; 1; . . .Þ

ð78Þ

p¼1 i¼1 n¼1

where Appendix A Substituting Eqs. (51), (52), (58), and (59) into (60) and (61), one obtains: 5 X 2 1 X X

E1kpin xpin ¼ r 1k

ðk ¼ 1; 2Þ

ð76Þ

p¼1 i¼1 n¼1

E112in

ð79aÞ

r1s k

ð79bÞ

Likewise, by substitution of Eqs. (46), (47), (55), and (56) into (62) and (63), one obtains: 5 X 2 1 X X

where

E111in

Z p 1 E1 eish dh 2p p kpin Z p 1 ¼ r1 eish dh 2p p k

E1s kpin ¼

n n2 zi zi 2 ð1Þ ¼ af Hð1Þ ðk jz jÞ þ l k H ðk jz jÞ expð2ihÞ i i If If I If n2 n jzi j jzi j ð77aÞ n n2 z z 2 ð1Þ i i ¼ as Hð1Þ þ lI kIs Hn2 ðksf jzi jÞ expð2ihÞ n ðkIs jzi jÞ jzi j jzi j

2

ð1Þ

E113in ¼ ilI kIt Hn2 ðkIt jzi jÞ

zi jzi j

n2 expð2ihÞ

E2kpin xpin ¼ r2k

ðk ¼ 1; 2Þ ðs ¼ 0; 1; . . .Þ

ð80Þ

p¼1 i¼1 n¼1

where

n1 zi jzi j n1 zi ð1Þ ¼ kIs Hn1 ðkIs jzi jÞ jzi j n1 zi ð1Þ ¼ ikIt Hn1 ðkIt jzi jÞ jzi j ð1Þ

E211in ¼ kIf Hn1 ðkIf jzi jÞ

ð77bÞ

E212in

ð77cÞ

E213in

ð81aÞ ð81bÞ ð81cÞ

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L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786

n1 zi ð1Þ E214in ¼ kIIp Hn1 ðkIIp jzi jÞ jzi j n1 zi ð1Þ E215in ¼ ikIIs Hn1 ðkIIs jzi jÞ jzi j nþ1 zi ð1Þ E221in ¼ kIf Hnþ1 ðkIf jzi jÞ jzi j nþ1 zi ð1Þ E222in ¼ kIs Hnþ1 ðkIs jzi jÞ jzi j nþ1 zi ð1Þ E223in ¼ ikIt Hnþ1 ðkIt jzi jÞ jzi j nþ1 zi ð1Þ E224in ¼ kIIp Hnþ1 ðkIIp jzi jÞ jzi j nþ1 zi ð1Þ E225in ¼ ikIIs Hnþ1 ðkIIs jzi jÞ jzi j i o h ðiÞ ðrÞ ðiÞ ðrÞ r 21 ¼ 2 u þ uf þ uðiÞ þ uðrÞ i wI þ wI s s f oz i o h ðiÞ ðrÞ 2 ðiÞ ðrÞ r 2 ¼ 2 uf þ uðrÞ f þ us þ us þ iwI oz ish

Multiplying both sides of Eq. (80) by e interval [p, p], one obtains: 5 X 2 1 X X

2s E2s kpin xpin ¼ r k

ð81dÞ

ð87aÞ

r3s k

ð87bÞ

ð81fÞ ð81gÞ

Z p 1 E3 eish dh 2p p pin Z p 1 ¼ r3 eish dh 2p p

E3s pin ¼ ð81eÞ

Substituting Eqs. (51) and (52) into (65) and (66), one obtains: 3 X 2 1 X X

ð81hÞ ð81iÞ

ð81kÞ

E412in

ð81lÞ

ð82Þ

ð83aÞ

r 2s k

ð83bÞ

ð89bÞ

n2

ð89dÞ n nþ2 zi zi 2 ð1Þ ¼ as Hnð1Þ ðkIs jzi jÞ þ lI kIs Hnþ2 ðkIs jzi jÞ expð2ihÞ jzi j jzi j

ð1Þ

E423in ¼ ilI kIt Hnþ2 ðkIt jzi jÞ

zi jzi j

ð89eÞ

nþ2 expð2ihÞ

ði ¼ 1; 2Þ ðs ¼ 0; 1; . . .Þ

ð84Þ

ð89gÞ

x1in ¼ ain ;

x2in ¼ bin ;

z1 ¼ r 2 eih þ d2 d1 z2 ¼ r 2 e

ih

x3in ¼ cin

ð89hÞ

ði ¼ 1Þ

ð89iÞ

ði ¼ 2Þ

ð89jÞ ish

where

g1 kIf

n1

zi expðihÞ jzi j nþ1 g kIf ð1Þ zi 1 Hnþ1 ðkIf jzi jÞ expðihÞ ð85aÞ 2 jzi j n1 g kIs ð1Þ zi E32in ¼ 2 Hn1 ðkIs jzi jÞ expðihÞ 2 jzi j nþ1 g kIs ð1Þ zi expðihÞ ð85bÞ 2 Hnþ1 ðkIs jzi jÞ 2 jzi j n1 iakIt ð1Þ zi E33in ¼ expðihÞ Hn1 ðkIt jzi jÞ 2 jzi j nþ1 iakIt ð1Þ zi þ expðihÞ ð85cÞ Hnþ1 ðkIt jzi jÞ 2 jzi j i

o 2 h ðiÞ ðrÞ ðrÞ r 3 ¼ 2 g1 uf þ uf þ g2 uðiÞ þ ia1 wðrÞ expðihÞ s þ us oz

o2 h ðiÞ ðrÞ ðrÞ ia1 wðrÞ expðihÞ 2 g1 uf þ uf þ g2 uðiÞ s þ us oz ð85dÞ 2

ð1Þ

Hn1 ðkIf jzi jÞ

Multiplying both sides of Eq. (84) by eish and integrating over the interval [p, p], one obtains: 2 X

1 X

p¼1 i¼1 n¼1

3s E3s pin xpin ¼ r k

ð89fÞ

r41 ¼ r 42 ¼ 0

p¼1 i¼1 n¼1

3 X

ð1Þ

2

Likewise, substituting Eqs. (48) and (49) into (64), one obtains:

E31in ¼

zi expð2ihÞ ð89cÞ jzi j n nþ2 zi zi 2 ð1Þ ¼ af Hnð1Þ ðkIf jzi jÞ þ lI kIf Hnþ2 ðkIf jzi jÞ expð2ihÞ jzi j jzi j 2

E421in

E2s kpin ¼

ð89aÞ n n2 z z 2 ð1Þ i i ¼ as Hnð1Þ ðkIs jzi jÞ þ lI kIs Hn2 ðkIs jzi jÞ expð2ihÞ jzi j jzi j

E413in ¼ ilI kIt Hn2 ðkIt jzi jÞ

E422in

E3pin xpin ¼ r 3

ð88Þ

n n2 zi zi 2 ð1Þ E411in ¼ af Hnð1Þ ðkIf jzi jÞ þ lI kIf Hn2 ðkIf jzi jÞ expð2ihÞ jzi j jzi j

ð81jÞ

where

3 X 2 1 X X

ðk ¼ 1; 2Þ

where

p¼1 i¼1 n¼1

Z p 1 E1 eish dh 2p p kpin Z p 1 ¼ r 2 eish dh 2p p k

E4kpin xpin ¼ r 4k

p¼1 i¼1 n¼1

and integrating over the

ðk ¼ 1; 2Þ ðs ¼ 0; 1; . . .Þ

where

ðk ¼ 1; 2Þ ðs ¼ 0; 1; . . .Þ

ð86Þ

Multiplying both sides of Eq. (88) with e the interval [p, p], one obtains: 3 X 2 1 X X

4s E4s kpin xpin ¼ r k

and integrating over

ðk ¼ 1; 2Þ ðs ¼ 0; 1; . . .Þ

ð90Þ

p¼1 i¼1 n¼1

where

Z p 1 E4 eish dh 2p p kpin Z p 1 ¼ r4 eish dh 2p p k

E4s kpin ¼

ð91aÞ

r4s k

ð91bÞ

Likewise, by substitution of Eq. (53) into (67), one obtains: 3 X 2 1 X X

E6pin xpin ¼ r 6

ð92Þ

p¼1 i¼1 n¼1

where

n1 zi ð1Þ expðihÞ Hn1 ðkIf jzi jÞ 2 jzi j nþ1 g kIf ð1Þ zi I1 Hnþ1 ðkIf jzi jÞ expðihÞ 2 jzi j n1 g kIs ð1Þ zi ¼ I2 Hn1 ðkIs jzi jÞ expðihÞ 2 jzi j nþ1 g kIs ð1Þ zi I2 Hnþ1 ðkIs jzi jÞ expðihÞ 2 jzi j

E61in ¼

E62in

gI1 kIf

ð93aÞ

ð93bÞ

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L.-F. Jiang et al. / Computers and Geotechnics 36 (2009) 773–786

E63in

n1 iaI1 kIt ð1Þ zi ¼ expðihÞ Hn1 ðkIt jzi jÞ 2 jzi j nþ1 iaI1 kIt ð1Þ zi þ expðihÞ Hnþ1 ðkIt jzi jÞ 2 jzi j

ð93cÞ

r6 ¼ 0 Likewise, multiplying both sides of Eq. (91) with e grating over the interval [p, p], one obtains: 3 X 2 1 X X

6s E6s pin xpin ¼ r

ðs ¼ 0; 1; 2; . . .Þ

and inte-

ð94Þ

p¼1 i¼1 n¼1

where

Z p 1 E6 eish dh 2p p pin Z p 1 ¼ r 6 eish dh 2p p

E6s pin ¼

ð95aÞ

r 6s

ð95bÞ

Substituting Eqs. (48), (49) into (68), one obtains: 5 X 2 1 X X

E8kpin xpin ¼ r 8k

ðk ¼ 1; 2Þ

ð96Þ

p¼4 i¼1 n¼1

where

n z jzj n2 z 2 ðiÞ þ lII kIIp Hn2 ðkIIp jzjÞ expð2ihÞ jzj n2 z 2 ðiÞ ¼ ilII kIIs Hn2 ðkIIs jzjÞ expð2ihÞ jzj n z 2 ¼ ðkII þ lII ÞkIIp HðiÞ n ðkIIp jzjÞ jzj nþ2 z 2 ðiÞ þ lII kIIp Hnþ2 ðkIIp jzjÞ expð2ihÞ jzj nþ2 z 2 ðiÞ ¼ ilII kIIs Hnþ2 ðkIIs jzjÞ expð2ihÞ jzj 2

E811in ¼ ðkII þ lII ÞkIIp HðiÞ n ðkIIp jzjÞ

E821in

E822in

3

3

r 81 ¼ f13 if 2 ; r 82 ¼ f13 þ if 2 x4in ¼ din ;

ð97aÞ ð97bÞ

ð97cÞ ð97dÞ ð97eÞ

x5in ¼ ein

ð97fÞ

z ¼ r 3 eih

ð97gÞ

Likewise, multiplying both sides of Eq. (96) with eish and integrating over the interval [p, p], one obtains: 5 X 2 1 X X p¼4 i¼1 n¼1

where

ð99aÞ

r8s k

ð99bÞ

ð93dÞ ish

E812in

Z p 1 E8 eish dh 2p p kpin Z p 1 ¼ r8 eish dh 2p p k

E8s kpin ¼

8s E8s kpin xpin ¼ r k

ðk ¼ 1; 2Þ ðs ¼ 0; 1; . . .Þ

ð98Þ

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