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Wave scattering in cross-anisotropic porous media around the cavities and inclusions
ARTICLE IN PRESS
Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027 www.elsevier.com/locate/soildyn
Wave scattering in cross-anisotropic porous media around the cavities and inclusions Behrooz Gatmiria,b,, Hashem Eslamib a
University of Tehran, Tehran, Iran Ecole Nationale des Ponts et Chausse´es, 6-8 Av. Blaise pascal, 77455 Marne la Vallee cedex 2, France
b
Received 31 August 2007; received in revised form 25 October 2007; accepted 5 November 2007
Abstract A method is presented for evaluating dynamic response of a cylindrical cavity or inclusion in an infinite cross-anisotropic porous medium. The basis of the method is the Biot’s consolidation equations in the complex plane. Employing two groups of potential functions for solid skeleton and pore fluid (each group includes three functions), the u-w formulation of Biot’s equations are solved. The stress, displacement, and pore pressure fields are evaluated in the vicinity of the cylinder in complex plane. Also the effects of many parameters ðkh =kv ; E h =E v ; G v =E v ; nhh Þ on the responses are evaluated by numerical examples. r 2007 Elsevier Ltd. All rights reserved. Keywords: Wave scattering; Analytical solution; Porous media; Complex functions; Piles; Tunnels
1. Introduction The problem of wave propagation and scattering in the porous media is described by consolidation theory of Biot [1,2]. In the most previous researches, the isotropic condition was considered for porous media. In this way, Gatmiri [3] studied the wave propagation problem in the quasi-harmonic case by use of finite element method. For the wave propagation problem in dynamic case, it can be mentioned the works by Degrande and Roeck [4,5], Kaynia [6], Dominguez [7], Gatmiri and Kamalian [8], Gatmiri and Nguyen [9], Nguyen and Gatmiri [10], Kamalian et al. [11], and Gatmiri et al. [12] on the basis of the finite element and boundary element methods. An analytical method for calculating dynamic response of a circular cavity in an infinite isotropic poroelastic medium was presented by the authors [13]. Mei et al. [14] and Zimmerman and Stern [15] studied the problem of wave scattering in the poroelastic media by other methods such as boundary-layer approximation. A few studies have been performed on the crossCorresponding author at: Ecole Nationale des Ponts et Chausse´es, 6-8 Av. Blaise Pascal, 77455 Marne la Valle´e cedex 2, France. Tel.: +33 1 64 15 35 66; fax: +33 1 64 15 35 62. E-mail address: [email protected] (B. Gatmiri).
0267-7261/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2007.11.003
anisotropic porous media. Gatmiri [16] and Jeng [17] achieved the solution for the problem of wave propagation in porous media (quasi-harmonic case) on the basis of finite element method and an analytical method, respectively. Honarvar and Sinclair [18] solved the problem of acoustic wave scattering from transversely isotropic cylinders. Also Fan et al. [19] solved this problem for the case that cylinder encased in a solid elastic medium. Both of them are on the basis of normal mode expansion method. Ahmad and Rahman [20] solved the problem of acoustic scattering by transversely isotropic cylinders with the same method but with different potential functions. Carcione [21] and Sharma and Gogna [22] studied wave propagation in anisotropic saturated porous media, also Sun et al. [23] studied this problem for the case of layered media. Yang and Zhang [24] studied the problem of wave propagation in the medium including Biot/squirt mechanism and the solid/ fluid anisotropy by means of Fourier transform method. Gatmiri and Jabbari [25,26] have studied and developed the fundamental solutions for unsaturated soils for the first time. This paper presents an analytical solution for the problem of scattering of harmonic waves in an infinite cross-anisotropic porous medium on the basis of complex functions theory. The use of complex functions in
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
Nomenclature sij total stress tensor component rbi ; rf bi body forces ui ; v i displacement vector components of solid skeleton and pore fluid respectively wi relative displacement vector components of pore fluid n porosity of the soil p pore fluid pressure kh permeability parameter of the soil in the plane ðx; yÞ kv permeability parameter of the soil in the z direction r; rf ; rs mass density of mixture, pore fluid and solid skeleton, respectively eij solid strain tensor component C 11 ; C 12 ; C 13 ; C 33 ; C 44 anisotropic parameters E h ; E v ; G v ; Gh ; nhh ; nvh ; nhv elastic parameters a Biot coefficient Q Biot modulus dij Kronecker delta Fs ; Cs ; X s potential functions for solid skeleton Ff ; Cf ; X f potential functions for pore fluid r2 two-dimensional Laplace operator o wave frequency d1 quasi-P1 wave number (qP1) d2 quasi-P2 wave number (qP2)
elastostatic problems was expanded by Muskhelishvili [27]. This method was used for solving the problem of wave scattering by a cavity in infinite elastic media by Liu et al. [28] and Han et al. [29]. Nowinski [30] solved the problem of static stress concentration around holes subjected to uniaxial tension, using complex functions theory. The use of this method for solving the problem of wave scattering is presented in previous writers’ paper for the first time and now is extended to the cross-anisotropic case. In this way, first two groups of potential functions (Fs , s C , X s and Ff , Cf , X f ) are applied into u-w formulation of Biot’s theory. Then, the governing equations are transformed into the complex plane. Solving an eigenvalue problem, the relation between potential functions Fs , Ff and Cs , Cf is determined. Also the relation between functions X s , X f is found directly. In the complex plane, solution of the resulting partial differential equations (calculation of the potential functions) is found in series of the Hankel functions (complex sum of the Bessel functions) with unknown coefficients. These Hankel functions satisfy the radiation condition. Applying appropriate boundary conditions of the problem, a set of algebraic equations are achieved. Solving these equations, the unknown coefficients and consequently potential functions are calculated. After evaluating the potential functions, all desirable parameters such as stress, pore pressure, and
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quasi-SV wave number (qSV) shear wave number complex variable ðz ¼ x þ iyÞ Bessel function of first kind and order n Bessel function of second kind and order n Hankel function of first kind and order n displacements of solid skeleton in Cartesian coordinates vx ; vy ; vz displacements of pore fluid in Cartesian coordinates wx ; wy ; wz relative displacements of pore fluid in Cartesian coordinates ur ; uy ; uz displacements of solid skeleton in polar coordinates sx ; sy ; sz ; sxy ; sxz ; syz components of stress tensor in Cartesian coordinates sr ; sy ; sz ; sry ; srz ; syz components of stress tensor in polar coordinates a radius of the cylinder fin incident potential function fsc scattered potential function K incident wave number Kx incident wave number in direction x Kz incident wave number in direction z g angle of incident wave with axis of cylinder q1 ; q2 ; q3 ; q01 ; q02 ; q03 ; q001 ; q002 ; q003 ; q000 coefficients X n ; Y n ; Zn ; W n unknown coefficients
d3 b z ¼ reiy J n ð‘rÞ Y n ð‘rÞ H ð1Þ n ð‘rÞ ux ; uy ; uz
displacement can be calculated in any point of the medium. Numerical results are presented including stress, displacement, and pore pressure fields in the vicinity of the cylinder in an infinite cross-anisotropic porous medium subjected to P1 harmonic waves. Also the effect of important parameters such as kh =kv , E h =E v , G v =E v , and nhh are mentioned. 2. Governing equations Based on the Biot theory, equilibrium equation and modified Darcy equation of the saturated porous media are written as sij;j þ rbi ¼ ru€ i þ rf w€ i ; r 1 p;i þ rf bi ¼ rf u€ i þ f w€ i þ w_ i n kðiÞ
with
r ¼ ð1 nÞrs þ nrf ; wi ¼ nðvi ui Þ
(1) In these equations, sij is the total stress tensor component, rbi and rf bi are the body forces, and ui , vi are the displacement vector components of solid skeleton and pore fluid, respectively. n is the porosity of the soil, p is the pore fluid pressure, and kðiÞ is the permeability parameter of the soil in the direction ðiÞ. r, rf , and rs are the mass density of
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
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mixture, pore fluid, and solid skeleton, respectively. The dots ð: Þ denote the differentiation respect to the time ðq=qtÞ. The constitutive equations in a cross-anisotropic porous medium are written as below: 8 quy qux quz > > sx ¼ C 11 þ C 12 þ C 13 ap; > > > qx qy qz > > > > quy qux quz > > > sy ¼ C 12 þ C 11 þ C 13 ap; > > qx qy qz > > > > quy qux quz > > > > < sz ¼ C 13 qx þ C 13 qy þ C 33 qz ap; C 11 C 12 qux quy > > þ ¼ s ; > xy > 2 qy qx > > > > > qux quz > > > þ ¼ C ; s xz 44 > > qz qx > > > > > > > syz ¼ C 44 quy þ quz ; > : qz qy
qux quy quz qwx qwy qwz p ¼ aQ þ þ þ þ þQ , qx qy qz qx qy qz
(2)
(3)
sx ; sy ; sz are the normal components of stress tensor and sxy , sxz , syz are the shear components of it. ux , uy , uz are the components of displacements in direction x, y and z, respectively. a is the Biot coefficient and Q is the Biot modulus. The relation between coefficients C ij and elastic parameters is written as follows: 8 C 11 > > > > > > < C 12 C 13 > > > C 33 > > > : C 44
¼ E h ð1 nhv nvh Þ=D; ¼ E h ðnhv nvh þ nhh Þ=D; ¼ E h ð1 þ nhh Þ nvh =D; ¼ E v ð1 ¼ Gv :
(4)
n2hh Þ=D;
E h and E v are the Young’s modulus in the vertical and any horizontal direction. nhh and nvh are the Poisson’s ratio as the corresponding operations of lateral expansion due to horizontal direct stress in horizontal direction and due to vertical direct stress in vertical direction, respectively, and G v is the modulus of shear deformation in a vertical plane. The other parameters are defined as below:
are derived as 8 > q2 ux ðC 11 C 12 Þ q2 ux > > ðC 11 þ a2 QÞ 2 þ > > > 2 qx qy2 > > 2 > 2 > > q uy q ux C 11 þ C 12 > 2 > þa Q þC 44 2 þ > > qz 2 qxqy > > > ! > > 2 2 2 2 > q w q u q w q w > y z x z > > þ aQ þ þðC 13 þ C 44 þ a2 QÞ þ > > qxqz qx2 qxqy qxqz > > > > > > ¼ ru€ x þ rf w€ x ; > > > > > ðC 11 C 12 Þ q2 uy q2 uy > 2 > þ ðC þ a QÞ > 11 > > 2 qx2 qy2 > > > 2 > 2 > > < þC 44 q uy þ C 11 þ C 12 þ a2 Q q ux qz2 2 qxqy ! > > 2 2 2 2 > > q w q u q w q w y z x z > > þðC 13 þ C 44 þ a2 QÞ þ aQ þ þ > > qyqz qxqy qy2 qyqz > > > > > > ¼ ru€ y þ rf w€ y ; > > > > > q2 uz q2 uz q2 uz > > > C 44 2 þ C 44 2 þ ðC 33 þ a2 QÞ 2 > > qx qy qz > > > > 2 > q ux > > > þðC 13 þ C 44 þ a2 QÞ > > qxqz > > ! > > 2 2 2 2 > q u q w q w q w > y y x z > þðC 13 þ C 44 þ a2 QÞ þ aQ þ þ 2 > > > qyqz qxqz qyqz qz > > > > : ¼ ru€ z þ r w€ z ; f
! 8 2 2 2 > q u q u q u y x z > > aQ þ þ þQ > > qx2 qxqy qxqz > > > > > > r 1 > > ¼ rf u€ x þ f w€ x þ w_ x ; > > > kh n > > ! > > 2 2 2 > q u q u q u > y x z > þ þ þQ > aQ < qxqy qy2 qyqz > r 1 > > > ¼ rf u€ y þ f w€ y þ w_ y ; > > kh n > > ! > > 2 2 > > q ux q uy q2 uz > > > > aQ qxqz þ qyqz þ qz2 þ Q > > > > > > rf 1 > > : ¼ rf u€ z þ w€ z þ w_ z : kv n
q2 wx q2 wy q2 wz þ þ qx2 qxqy qxqz
q2 wx q2 wy q2 wz þ þ qxqy qy2 qyqz
q2 wx q2 wy q2 wz þ þ 2 qxqz qyqz qz
(6) !
!
!
(7) kh and kv are the permeability parameter of the soil in the in plane ðx; yÞ and z direction, respectively.
E h nhv ¼ , E v nvh
3. Potential functions
Eh , 2ð1 þ nhh Þ D ¼ ð1 þ nhh Þð1 nhh 2nvh nhv Þ.
The relation between displacements and potential functions ðFs ; Cs ; X s for solid skeleton and Ff , Cf , X f for pore fluid) are defined as below:
Gh ¼
ð5Þ
From Eqs. (1)–(3), the equilibrium equations in the absence of body forces in term of displacement vector components
~ ez Þ þ ar r ðCs~ ez Þ, u ¼ rFs þ r ðX s~ ~ ¼ rFf þ r ðX f ~ ez Þ þ ar r ðCf ~ ez Þ. w
ð8Þ
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
In which ‘‘a’’ is the radius of Cartesian coordinates: 8 > qFs qX s q2 C s > > þ þa ; > ux ¼ > qx qy qx qz > > > > < qFs qX s q2 C s u þ a ; ¼ y > qy qx qy qz > > > > > > qFs > > ar2 Cs ; : uz ¼ qz
the cylindrical cavity. In the 8 qFf qX f q2 Cf > > > w þ þ a ; ¼ > x > qx qy qx qz > > > > < qFf qX f q2 Cf wy ¼ þa ; > qy qx qy qz > > > > > > qFf > > ar2 Cf ; : wz ¼ qz ð9Þ
where ux , uy , uz and wx , wy , wz are the Cartesian components of the vectors ui and wi , respectively. Therefore Eqs. (6) and (7) are converted to the two groups of equations (F ¼ f eiot eiK z z , C ¼ c eiot eiK z z , and X ¼ w eiot eiK z z Þ: 8 q2 fs > > > ðC 11 þ a2 QÞr2 fs þ ðC 13 þ 2C 44 þ a2 QÞ 2 > > qz > > ! > > 2 f > > > > þaQ r2 ff þ q f þ ro2 fs þ rf o2 ff > > > qz2 > > > > > > > q q2 c s > 2 s > ðC þa C C Þr c þ C > 11 13 14 44 > > qz qz2 > > > > > > > 2 s 2 f > > c þ r o c þro ¼ 0; f > > > > > " > > > > q q2 fs > > ðC 13 þ 2C 44 þ a2 QÞr2 fs þ ðC 33 þ a2 QÞ 2 > > > qz qz > > > > ! # > > > > q2 ff > 2 f > þaQ r f þ þ ro2 fs þ rf o2 ff > > qz2 > > > > > > > > q2 cs > 2 2 s > < ar C 44 r c þ ðC 33 C 13 C 44 Þ 2 qz ð10Þ > > > > > þro2 cs þ rf o2 cf ¼ 0; > > > > > > ! > > > q2 fs q2 ff > 2 s 2 f > > aQ r f þ 2 þ Q r f þ 2 þ rf o2 fs > > qz qz > > > > > > > rf 2 io f > > o þ f > þ > > kh n > > > > > q rf 2 io f > > 2 s > r o þa o c þ þ c ¼ 0; > > qz f kh n > > > > " ! > > 2 s 2 f > q q f q f > > aQ r2 fs þ 2 þ Q r2 ff þ 2 þ rf o2 fs > > > qz qz qz > > > > > > > r io f > > þ f o2 þ f > > kv n > > > > > > rf 2 io 2 f > s 2 2 > > : a rf o r c þ n o þ k r c ¼ 0; v
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8 > C 11 C 12 2 s q2 ws > > r w þ C þ ro2 ws þ rf o2 wf ¼ 0; 44 > < 2 qz2 ð11Þ > rf 2 io f > 2 s > > : rf o w þ n o þ kh w ¼ 0; where r2 is the two-dimensional Laplace operator, r2 ¼ q2 =qx2 þ q2 =qy2 , and o is the frequency of the waves. Also it is possible to use below formulation for potential functions: s qC qFs ~ ~ ez Þ þ u ¼ rFs þ r ðX s~ ez , qz qz f qC qFf f f ~ ¼ rF þ r ðX ~ ~ ð12Þ ez Þ þ w ez . qz qz It is simpler than the mentioned, but it has not any powerful physical meaning. Our formulation is in relation with the P, SV, and SH waves that is very useful for seeing the contribution of each component of the wave. 4. Complex functions The complex variables are introduced: z ¼ x þ iy;
z¯ ¼ x iy;
z ¼ r eiy .
(13)
Using relations below: q q q q q q ¼ þ ; ¼i , qx qz qz¯ qy qz qz¯ q2 q2 q2 q2 ¼ þ 2 ; þ qx2 qz2 qzqz¯ qz¯ 2
(14)
! q2 q2 q2 q2 ¼ 2 þ , qy2 qzqz¯ qz¯ 2 qz2 (15)
the Laplace operator in the complex plane is obtained as r2 ¼ 4
q2 . qz qz¯
Also the potential functions 8 þ1 P s ð1Þ > s > an H n ðdrÞeiny ; >f ¼ > > n¼1 > > > < þ1 P s ð1Þ cs ¼ bn H n ðdrÞeiny ; > n¼1 > > > þ1 > P s ð1Þ > s > cn H n ðbrÞeiny ; > :w ¼ n¼1
(16) are considered as below: 8 þ1 P f ð1Þ > f > an H n ðdrÞeiny ; >f ¼ > > n¼1 > > > < þ1 P f ð1Þ cf ¼ bn H n ðdrÞeiny ; > n¼1 > > > þ1 > P f ð1Þ > f > cn H n ðbrÞeiny : > :w ¼ n¼1
(17) H ð1Þ n ð‘rÞ is the Hankel function of first kind and order n: H ð1Þ n ð‘rÞ ¼ J n ð‘rÞ þ iY n ð‘rÞ,
(18)
where J n ð‘rÞ and Y n ð‘rÞ are the Bessel functions of first and second kind, respectively, and of order n. This type of the Hankel function satisfies the radiation boundary condition (Sommerfield condition). Also asn , bsn , csn , afn , bfn , cfn are the unknown coefficients. Based on Eqs. (16) and (17) and
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
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relations below: q ‘ ½H ð1Þ ð‘rÞeiny ¼ H ð1Þ ð‘rÞeiðn1Þy , qz n 2 n1
(19)
q ‘ ð‘rÞeiðnþ1Þy . ½H ð1Þ ð‘rÞeiny ¼ H ð1Þ 2 nþ1 qz¯ n
(20)
(qP1), d2 for quasi-P2 wave (qP2), d3 for quasi-SV wave (qSV) and ½q1 q01 q001 T for d1 , ½q2 q02 q002 T for d2 , and ½q3 q03 q003 T for d3 Þ. Also from Eq. (11),the relation between potential functions of solid skeleton and pore fluid, also the wave number b are achieved directly as wf ¼ q000 ws , q000 ¼
Eq. (10) is converted to an eigenvalue problem as 2
a11 6 a21 6 6 4 a31 a41
a12 a22
a13 a23
a32 a42
a33 a43
rf o2 rf 2 io o þ kh n
,
1=2 ro2 C 44 K 2z þ rf o2 q000 . b¼ 2 ðC 11 C 12 Þ
38 s 9 f > a14 > > > > > > = < ff > a24 7 7 ¼ 0. 7 s c > a34 5> > > > > > > a44 : cf ;
ð23Þ
(21) Finally, the potential functions are written as below: 8 þ1 þ1 P P > iny > fs ¼ X n H ð1Þ ðd1 rÞ einy þ Y n H ð1Þ > n n ðd2 rÞ e > > n¼1 n¼1 > > > > þ1 > P > iny > þq Z n H ð1Þ > 3 n ðd3 rÞ e ; <
The coefficients aij are a11 ¼ ðC 11 þ a2 QÞd2 þ ðC 13 þ 2C 44 þ a2 QÞK 2z ro2 , a12 ¼ aQðd2 þ K 2z Þ rf o2 ,
n¼1
2
a13 ¼ aiK z ½ðC 11 C 13 C 44 Þd þ
C 44 K 2z
2
ro ,
2
a14 ¼ aiK z rf o ,
ð22aÞ
þ1 þ1 P P > iny > > ff ¼ q01 X n H ð1Þ ðd1 rÞ einy þ q02 Y n H ð1Þ > n n ðd2 rÞ e > > n¼1 n¼1 > > > þ1 > P > 0 iny > þq Z n H ð1Þ > 3 n ðd3 rÞ e ; : n¼1
(24)
a21 ¼ iK z ½ðC 13 þ 2C 44 þ a2 QÞd2 þ ðC 33 þ a2 QÞK 2z ro2 , a22 ¼ iK z ½aQðd2 þ K 2z Þ rf o2 , a23 ¼ ad2 ½C 44 d2 þ ðC 33 C 13 C 44 ÞK 2z ro2 , a24 ¼ ad2 rf o2 ,
ð22bÞ
a31 ¼ aQðd2 þ K 2z Þ rf o2 , rf 2 io o þ a32 ¼ Qðd2 þ K 2z Þ , kh n a33 ¼ aiK z rf o2 , r io , a34 ¼ aiK z f o2 þ kh n
n¼1 þ1 P > iny > > cf ¼ q001 X n H ð1Þ þ > n ðd1 rÞ e > > n¼1 > > > þ1 > P > iny > þq003 Z n H ð1Þ > n ðd3 rÞ e ; : n¼1
ð22cÞ
a41 ¼ iK z ½aQðd2 þ K 2z Þ rf o2 , rf 2 io o þ a42 ¼ iK z aQðd2 þ K 2z Þ , kv n a43 ¼ ad2 rf o2 , r io a44 ¼ ad2 f o2 þ . kv n
8 þ1 þ1 P P > iny > cs ¼ q1 X n H ð1Þ ðd1 rÞ einy þ q2 Y n H ð1Þ > n n ðd2 rÞ e > > n¼1 n¼1 > > > > þ1 > P > iny > þ Z n H ð1Þ > n ðd3 rÞ e ; < q002
þ1 P n¼1
iny Y n H ð1Þ n ðd2 rÞ e
(25)
8 þ1 P > s iny > W n H ð1Þ > n ðbrÞ e ;
> f > > :w ¼ q
n¼1
(26)
iny W n H ð1Þ n ðbrÞ e :
ð22dÞ
Solving this eigenvalue problem, three values for d and three eigenvectors will be found. (d1 for quasi-P1 wave
The unknown coefficients, X n , Y n , Z n , W n are calculated from boundary conditions of the problem. Now, knowing the potential functions, displacements, stresses, and pore
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pressure can be calculated based on these functions and then the boundary value problem can be solved.
5. Displacements, stresses, and pore pressure From Eq. (9) the displacement field is 8 þ1 P > iny > > ur þ iuy ¼ d1 ð1 þ aiK z q1 Þ X n H ð1Þ > nþ1 ðd1 rÞ e > > n¼1 > > > þ1 > P > iny > d2 ð1 þ aiK z q2 Þ Y n H ð1Þ > nþ1 ðd2 rÞ e > > n¼1 > > > > þ1 P > > iny > d ðq þ aiK Þ Z n H ð1Þ 3 z > 3 nþ1 ðd3 rÞ e > > n¼1 > > > þ1 > P > iny > > þib W n H ð1Þ > nþ1 ðbrÞ e ; > > n¼1 > > > þ1 > P > iny > ur iuy ¼ d1 ð1 þ aiK z q1 Þ X n H ð1Þ > n1 ðd1 rÞ e > > n¼1 > > > < þ1 P iny þd2 ð1 þ aiK z q2 Þ Y n H ð1Þ n1 ðd2 rÞ e > n¼1 > > > þ1 > P > iny > þd3 ðq3 þ aiK z Þ Z n H ð1Þ > n1 ðd3 rÞ e > > n¼1 > > > > þ1 > P > iny > þib W n H ð1Þ > n1 ðbrÞ e ; > > n¼1 > > > þ1 > P > > iny > uz ¼ ðiK z þ aq1 d21 Þ X n H ð1Þ > nþ1 ðd1 rÞ e > > n¼1 > > > þ1 > P > iny > þðiK z þ aq2 d22 Þ Y n H ð1Þ > nþ1 ðd2 rÞ e > > n¼1 > > > > þ1 P > > 2 iny > þðiK q þ ad Þ Z n H ð1Þ > z 3 3 nþ1 ðd3 rÞ e : :
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functions are 8 ! 2 s 2 f > > q f q f > > p ¼ aQ r2 fs þ 2 Q r2 ff þ 2 ; > > qz qz > > > > > > > ðC 11 þ C 12 Þ 2 s q2 f s > > r is ¼ f þ C s > r y 13 > 2 qz2 > > > > a q > > > þ ðC 11 þ C 12 2C 13 Þ r2 cs > > 2 qz > > > 2 > q qcs 2iy > s > s > þ2ðC 11 C 12 Þ 2 f þ iw þ a e ; > > qz > qz > > < ðC 11 þ C 12 Þ 2 s q2 f s r f þ C 13 2 sr þ isy ¼ > 2 qz > > > > a q > > þ ðC 11 þ C 12 2C 13 Þ r2 cs > > > 2 qz > > > 2 > q qcs 2iy > s s > þ2ðC 11 C 12 Þ 2 f iw þ a e ; > > > qz > qz¯ > > > > > qfs q2 c s > 2 s iy q iy q > þe þ a 2 ar c srz ¼ C 44 e 2 > > > qz qz qz qz¯ > > > s > q q qw > > > þi eiy eiy : > : qz qz¯ qz (29) (The second and third relations were constructed from subtracting the second and third relations of Eq. (28)). Finally Eq. (29) is written in term of the Hankel functions as below: þ1 X iny p ¼ Qðd21 þ K 2z Þða þ q01 Þ X n H ð1Þ n ðd1 rÞe þ Qðd22 þ
n¼1
(27)
Eqs. (2) and (3) results: 8 qux quy quz qwx qwy qwz > > > p ¼ aQ þ þ þ þ Q ; > > qx qy qz qx qy qz > > > > > qux quy quz > > þ þ s ¼ ðC þ C Þ þ 2C 13 ; s > 11 12 y < r qx qy qz q q > > > i s sr þ 2isry ¼ ðC 11 C 12 Þ ðux iuy Þ e2iy ; > > y qx qy > > > > > quy quz qux quz > > þ þ ¼ C s cos y þ sin y ; > rz 44 : qz qx qz qy (28)
where sr , sy are the radial and hoop components of stress tensor and sry , srz are the shear components of it. These relations in complex plane and in term of potential
þ Qðd23 þ
n¼1 þ1 X iny K 2z Þða þ q02 Þ Y n H ð1Þ n ðd2 rÞe n¼1 þ1 X iny K 2z Þðaq3 þ q03 Þ Zn H ð1Þ n ðd3 rÞe , n¼1
ð30aÞ
ðC 11 þ C 12 Þ 2 a d1 C 13 K 2z iK z q1 d21 sr isy ¼ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ X n H ð1Þ n ðd1 rÞe n¼1 þ1 X ðC 11 C 12 Þ 2 iny d1 ð1 þ aiK z q1 Þ þ X n H ð1Þ n2 ðd1 rÞe 2 n¼1 ðC 11 þ C 12 Þ 2 a d2 C 13 K 2z iK z q2 d22 þ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ Y n H ð1Þ n ðd2 rÞe n¼1 þ1 X ðC 11 C 12 Þ 2 iny d2 ð1 þ aiK z q2 Þ þ Y n H ð1Þ n2 ðd2 rÞe 2 n¼1 ðC 11 þ C 12 Þ 2 a q3 d3 C 13 q3 K 2z iK z d23 þ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ Z n H ð1Þ n ðd3 rÞe n¼1
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
1020
þ
þ1 X ðC 11 C 12 Þ 2 iny d3 ðq3 þ aiK z Þ Z n H ð1Þ n2 ðd3 rÞe 2 n¼1
þi
þ1 X
ðC 11 C 12 Þ 2 ð1Þ b W n H n2 ðbrÞeiny , 2 n¼1
ð30bÞ
ðC 11 þ C 12 Þ 2 a d1 C 13 K 2z iK z q1 d21 sr þ isy ¼ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ X n H ð1Þ n ðd1 rÞe þ1 X ðC 11 C 12 Þ 2 iny d1 ð1 þ aiK z q1 Þ X n H ð1Þ nþ2 ðd1 rÞe 2 n¼1 ðC 11 þ C 12 Þ 2 a 2 d2 C 13 K z iK z q2 d22 þ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ Y n H ð1Þ n ðd2 rÞe
þ
n¼1 þ1 X ðC 11 C 12 Þ 2 iny d2 ð1 þ aiK z q2 Þ Y n H ð1Þ nþ2 ðd2 rÞe 2 n¼1 ðC 11 þ C 12 Þ a 2 q3 d3 C 13 q3 K 2z iK z d23 þ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ Z n H ð1Þ n ðd3 rÞe
þ
n¼1
ðC 11 C 12 Þ 2 iny b W n H ð1Þ nþ2 ðbrÞe , 2 n¼1
at r ¼ a,
(31)
8 in sc > : w ¼ win þ wsc ;
ð30cÞ
h i a a srz ¼ C 44 d1 iK z K 2z q1 þ q1 d21 2 2 þ1 X ð1Þ iny X n ½H n1 ðd1 rÞ H ð1Þ nþ1 ðd1 rÞe
(32)
where fin , cin , win are the incident and fsc , csc , wsc are the scattered components of the potential functions. Therefore the boundary conditions at the cavity surface are 8 in p þ psc ¼ 0; > > > > < ðsr isry Þin þ ðsr isry Þsc ¼ 0; at r ¼ a, (33) ðsr þ isry Þin þ ðsr þ isry Þsc ¼ 0; > > > > : sin þ ssc ¼ 0: rz
þ1 X ðC 11 C 12 Þ 2 ð1Þ d3 ðq3 þ aiK z Þ þ Z n H nþ2 ðd3 rÞeiny 2 n¼1
i
8 p ¼ 0; > > > < sr isry ¼ 0; > sr þ isry ¼ 0; > > : srz ¼ 0:
All of the wave variables are the sum of incident and scattering components, therefore the potential functions are written as
n¼1
þ1 X
be satisfied at the cavity surface:
rz
Incident components of the stresses and pore pressure are derived from incident potential functions using Eq. (29). Also scattering components of these variables are found in Eqs. (30a)–(30d). In the case of dilatational incident wave (P1 wave), the incident potential function is written as fin ¼ f0 eiK x x eiK z z ,
(34)
n¼1
h i a a þ C 44 d2 iK z K 2z q2 þ q2 d22 2 2 þ1 X ð1Þ iny Y n ½H ð1Þ n1 ðd2 rÞ H nþ1 ðd2 rÞe n¼1
h
z
i
a a þ C 44 d3 iK z q3 K 2z þ d23 2 2 þ1 X ð1Þ iny Zn ½H ð1Þ n1 ðd3 rÞ H nþ1 ðd3 rÞe
a θ
Incident Wave
n¼1 þ1 K zb X ð1Þ iny C 44 W n ½H ð1Þ n1 ðbrÞ þ H nþ1 ðbrÞe . ð30dÞ 2 n¼1
γ
y x
6. Boundary value problem Cross-Anisotropic Porous Medium
Now consider a cylindrical cavity in an infinite porous medium is subjected to a harmonic incident wave. The radius of the cavity is ‘‘a’’ and on its surface the pressure and stress are vanished. These boundary conditions should
Fig. 1. Cylinder in an infinite cross-anisotropic porous medium subjected to the harmonic wave.
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
f0 is a coefficient and
and consequently the incident pore pressure and stresses are found
K x ¼ K cos g, K z ¼ K sin g.
8 þ1 P n > > > pin ¼ f0 aQðK 2x þ K 2z Þ i J n ðK x rÞeiny ; > > > n¼1 > > þ1 > > P n ðC 11 þ C 12 Þ 2 > in 2 > is Þ ¼ f þ C K i J n ðK x rÞeiny ðs K > r ry 13 z 0 x > > 2 n¼1 > > > > > P n ðC 11 C 12 Þ 2 þ1 > > þf0 i J n2 ðK x rÞeiny ; Kx > < 2 n¼1 þ1 P n ðC 11 þ C 12 Þ 2 > > > ðsr þ isry Þin ¼ f0 i J n ðK x rÞeiny K x þ C 13 K 2z > > 2 > n¼1 > > > > P n ðC 11 C 12 Þ 2 þ1 > > þf0 i J nþ2 ðK x rÞeiny ; Kx > > > 2 n¼1 > > > > þ1 P n > > in > i ½J n1 ðK x rÞ J nþ1 ðK x rÞeiny : > srz ¼ f0 iC 44 K x K z : n¼1
ð35Þ
K is the wave number and is related to the wave frequency as following: o¼K
1=2 C 11 þ a2 Q . r
(36)
Also g is the angle of incident wave with axis of cylinder. The incident potential function can be written as fin ¼ f0
þ1 X
1021
in J n ðK x rÞeiny eiK z z
(37)
(38)
n¼1
Table 1 Parameters used in applications n
rs ðkg=m3 Þ
rf ðkg=m3 Þ
a
Q (MPa)
a (m)
f0
E v (MPa)
kv (m/s)
0.3
2700
1000
1
6:67 103
1
1
5:32 103
1 104
1.0
1.0 5
Radial Stress
Pore Pressure
5 3
0.5
4 1
4
0.5
3 1
2
2
0.0
0.0 2
1
3 r/a
4
5
3
4
5
r/a
1.0
1.0
3
Shear Stress
Hoop Stress
2
1
1
0.5 5
4
0.5
0.0 2
3 r/a
4
5
1
2
5
1
0.0 1
4
3
2
2
3 r/a
4
5
Fig. 2. Values of normalized pore pressure and effective stresses versus distance from the cylinder (1: y ¼ 0, 2: y ¼ p=4, 3: y ¼ p=2, 4: y ¼ 3p=4, 5: y ¼ p). Solid line: present study, point line: adapted from Mei et al. [14].
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
1022
Finally our boundary value problem (Eq. (33)) is written as an algebraic equation: 9 8 9 2 38 m11 m12 m13 m14 > X n > > n1 > > > > > > > > > = < 6 m21 m22 m23 m24 7< n2 = 6 7 Yn ¼ . (39) 6 7 > Zn > n3 > 4 m31 m32 m33 m34 5> > > > > > > > > ; : ; : m41 m42 m43 m44 Wn n4
ðC 11 þ C 12 Þ 2 a d1 C 13 K 2z iK z q1 d21 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd1 aÞ
m21 ¼
þ
ðC 11 þ C 12 Þ 2 a d2 C 13 K 2z iK z qz d22 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd2 aÞ
The coefficients mij are defined as
m22 ¼
m11 ¼ Qðd21 þ K 2z Þða þ q01 ÞH ð1Þ n ðd1 aÞ, m12 ¼ Qðd22 þ K 2z Þða þ q02 ÞH ð1Þ n ðd2 aÞ, m13 ¼ Qðd23 þ K 2z Þðaq3 þ q03 ÞH ð1Þ n ðd3 aÞ, m14 ¼ 0,
þ
ð40aÞ
kh/kv=1 Ratio of Effective Hoop Stress to K2 (N)
ðC 11 C 12 Þ 2 d1 ð1 þ aiK z q1 ÞH ð1Þ n1 ðd1 aÞ, 2
kh/kv=5
ðC 11 C 12 Þ 2 d2 ð1 þ aiK z q2 ÞH ð1Þ n2 ðd2 aÞ, 2
kh/kv=10
kh/kv=100
4.0E+10 3.5E+10 3.0E+10 2.5E+10 nu=0.33 n=0.30 G=2000MPa r=a
2.0E+10 1.5E+10 1.0E+10 5.0E+09 0.0E+00 0.0
0.5
1.0
1.5
2.0 Ka
2.5
3.0
3.5
4.0
Fig. 3. Ratio of effective hoop stress to the parameter K 2 with respect to the ratio of kh =kv and dimensionless wave number ðy ¼ p=2Þ.
kh/kv=1
kh/kv=5
kh/kv=10
kh/kv=100
Ratio of Pore Pressure to K2 (N)
1.2E+10 1.0E+10 8.0E+09 6.0E+09 nu=0.33 n=0.30 G=2000MPa r=2a
4.0E+09 2.0E+09 0.0E+00 0.0
0.5
1.0
1.5
2.0 Ka
2.5
3.0
3.5
4.0
Fig. 4. Ratio of pore pressure to the parameter K 2 with respect to the ratio of kh =kv and dimensionless wave number ðy ¼ p=2Þ.
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
m23
ðC 11 þ C 12 Þ a q3 d23 C 13 q3 K 2z iK z d23 ¼ 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd3 aÞ þ
m24 ¼ i
ðC 11 þ C 12 Þ 2 a d2 C 13 K 2z iK z qz d22 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd2 aÞ
m32 ¼
ðC 11 C 12 Þ 2 d3 ðq þ aiK z ÞH ð1Þ n2 ðd3 aÞ, 2
þ
ðC 11 C 12 Þ 2 ð1Þ b H n2 ðbaÞ, 2
(40b)
ðC 11 þ C 12 Þ 2 a d1 C 13 K 2z iK z q1 d21 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd1 aÞ
m31 ¼
þ
m34 ¼ i Eh/Ev=0.5
ðC 11 C 12 Þ 2 d2 ð1 þ aiK z q2 ÞH ð1Þ n2 ðd2 aÞ, 2
ðC 11 þ C 12 Þ a q3 d23 C 13 q3 K 2z iK z d23 m33 ¼ 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd3 aÞ þ
ðC 11 C 12 Þ 2 d1 ð1 þ aiK z q1 ÞH ð1Þ nþ2 ðd1 aÞ, 2
Ratio of Effective Hoop Stress to K2 (N)
1023
Eh/Ev=0.8
ðC 11 C 12 Þ 2 d3 ðq3 þ aiK z ÞH ð1Þ nþ2 ðd3 aÞ, 2 ðC 11 C 12 Þ 2 ð1Þ b H nþ2 ðbaÞ, 2
(40c)
Eh/Ev=1.0
1.0E+11 9.0E+10
nu hh=0.80 nu vh=0.20 Gv=0.5Ev kh=5kv r=a
8.0E+10 7.0E+10 6.0E+10 5.0E+10 4.0E+10 3.0E+10 2.0E+10 1.0E+10 0.0E+00 0.0
0.5
1.0
1.5
2.0 Ka
2.5
3.0
3.5
4.0
Fig. 5. Ratio of effective hoop stress to the parameter K 2 with respect to the ratio of E h =E v and dimensionless wave number ðy ¼ p=2Þ.
Eh/Ev=0.5
Eh/Ev=0.8
Eh/Ev=1.0
Ratio of Pore Pressure to K2 (N)
1.2E+10 1.0E+10 8.0E+09 6.0E+09 nu hh=0.80 nu vh=0.20 Gv=0.5Ev kh=5kv r= 2a
4.0E+09 2.0E+09 0.0E+00 0.0
0.5
1.0
1.5
2.0 Ka
2.5
3.0
3.5
4.0
Fig. 6. Ratio of pore pressure to the parameter K 2 with respect to the ratio of E h =E v and dimensionless wave number ðy ¼ p=2Þ.
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1024
h i a a ð1Þ m41 ¼ C 44 d1 iK z K 2z q1 þ q1 d21 ½H ð1Þ n1 ðd1 aÞ H nþ1 ðd1 aÞ, 2 2 h i a a ð1Þ m42 ¼ C 44 d2 iK z K 2z q2 þ q2 d22 ½H ð1Þ n1 ðd2 aÞ H nþ1 ðd2 aÞ, 2 2
m43
h
a a i ð1Þ ¼ C 44 d3 iK z q3 K 2z þ d23 ½H ð1Þ n1 ðd3 aÞ H nþ1 ðd3 aÞ, 2 2
m44 ¼ C 44
K z b ð1Þ ½H n1 ðbaÞ þ H ð1Þ nþ1 ðbaÞ. 2
(40d)
Also the coefficients ni are written as 8 > n1 ¼ faQðK 2x þ K 2z Þin J n ðK x aÞ; > > > > > ðC 11 þ C 12 Þ 2 > 2 n > K ¼ f þ C K n 13 z i J n ðK x aÞ > 2 0 x > 2 > > > > > ðC 11 C 12 Þ 2 n > > K x i J n2 ðK x aÞ; f0 < 2 ðC 11 þ C 12 Þ 2 > 2 n > > K n ¼ f þ C K 3 13 z i J n ðK x aÞ > 0 x > 2 > > > > > ðC 11 C 12 Þ 2 n > > K x i J nþ2 ðK x aÞ; f0 > > > 2 > > : n4 ¼ f in C 44 in K x K z ½J n1 ðK x aÞ J nþ1 ðK x aÞ: 0
(41)
7. Numerical results Fig. 1 represents a cylinder in an infinite crossanisotropic porous medium. The cylinder is subjected to the P1 incident wave and its surface is free of stress and pressure. In the following examples, the parameters given in Table 1 are assumed.
Eh/Ev=0.5 Eh/Ev=0.8 Eh/Ev=1.0 Ratio of Effective Hoop Stress to K2 (N)
2.0x1010 1.5x1010
7.1. Validation
1.0x1010 5.0x109 0.0 5.0x109 1.0x1010 1.5x1010 2.0x1010
Fig. 7. Ratio of effective hoop stress to the parameter K 2 for various points of circumference and the ratio of E h =E v (Ka ¼ 1, r ¼ a).
To verify the proposed solution, a comparison between the solution obtained by Mei et al. [14] in an isotropic poroelastic medium (by using a boundary-layer approximation) and the result of this solution reduced to an isotropic poroelastic case is made. Fig. 2 shows the variations of normalized pore pressure and effective stresses for the cylinder in a poroelastic medium versus the normalized radial distance (r=a) for various angle (y) of P1 incident wave. In this example, the following values are chosen: kh =kv ¼ 1, E h =E v ¼ 1, nhh ¼ 0:33, Ka ¼ 1. As it can be observed, there is an excellent agreement between these results and those from Mei et al. [14] for hoop and shear stresses for all values of r=a.
Ratio of Effective Radial Stress to K2 (N)
NU hh=0.3 1.75E+10
NU hh=0.5
NU hh=0.8
nu vh=0.20 Eh=0.5Ev Gv=0.5Ev kh=5kv Ka=1
1.50E+10 1.25E+10 1.00E+10 7.50E+09 5.00E+09 2.50E+09 0.00E+00 1.0
3.0
5.0
7.0
9.0
11.0
13.0
15.0
r/a Fig. 8. Ratio of effective radial stress to the parameter K 2 with respect to nhh and radial distance from the cylinder ðy ¼ pÞ.
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
7.2. Parametric study To investigate the effect of various parameters of the cross-anisotropic porous medium on the solution, a complete study is performed. Fig. 3 gives the ratio of effective hoop stress to the parameter K 2 with respect to the ratio of kh =kv and dimensionless wave number, Ka (g ¼ 0 , r ¼ a, y ¼ p=2Þ. It is seen that this parameter has very small effect on the stress response (only in the case of kh =kv ¼ 100 this parameter has slightly significant effect). Also Fig. 4 shows the ratio of pore pressure to the parameter K 2 with respect to the ratio of kh =kv and dimensionless wave number, Ka (g ¼ 0 , r ¼ 2a, y ¼ p=2Þ. It is seen that the effect of this parameter is much more when compared with its effect on stress response especially inthe case of kh =kv ¼ 100. (the case of kh =kv ¼ 1 is concerned to the isotopic porous medium). In above examples, the isotropic conditions for solid skeleton were
considered. Fig. 5 gives the ratio of effective hoop stress to the parameter K 2 with respect to the ratio of E h =E v and dimensionless wave number, Ka (g ¼ 0 , r ¼ a, y ¼ p=2Þ. It is seen that with increasing the ratio of E h =E v the hoop stress increases. Fig. 6 represents the ratio of pore pressure to the parameter K 2 with respect to the ratio of E h =E v and dimensionless wave number, Ka (g ¼ 0 , r ¼ 2a, y ¼ p=2). Itis seen the influence of this parameter on the pore pressure response is very small. Fig. 7 gives the ratio of effective hoop stress to the parameter K 2 for various points of circumference of cylinder and the ratio of E h =E v (g ¼ 0 , Ka ¼ 1, r ¼ a and other parameters are as Fig. 5). Fig. 8 shows the ratio of effective radial stress to the parameter K 2 with respect to nhh and radial distance from the cylinder (g ¼ 0 , Ka ¼ 1, y ¼ pÞ. It is seen that with increasing the values of nhh the values of radial stress increases. Fig. 9 shows the ratio of pore pressure to the parameterK 2 with respect to nhh and radial distance from the cylinder (g ¼ 0 ,
NU hh=0.3
NU hh=0.5
NU hh=0.8
Ratio of Pore Pressure to K2 (N)
1.25E+10
1.00E+10
7.50E+09
5.00E+09 nu vh=0.20 Eh=0.5Ev Gv=0.5Ev kh=5kv Ka=1
2.50E+09
0.00E+00 1.0
3.0
5.0
7.0
9.0
11.0
13.0
15.0
r/a Fig. 9. Ratio of pore pressure to the parameter K 2 with respect to nhh and radial distance from the cylinder ðy ¼ pÞ.
NU hh=0.3
NU hh=0.5
NU hh=0.8
Ratio of Radial Displacement to K (m2)
3.0 nu vh=0.20 Eh=0.5Ev Gv=0.5Ev kh=5kv Ka=1
2.5 2.0 1.5 1.0 0.5 0.0 1.0
3.0
5.0
1025
7.0
9.0
11.0
13.0
15.0
r/a Fig. 10. Ratio of radial displacement to the parameter K with respect to nhh and radial distance from the cylinder (g ¼ 10, y ¼ p).
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
1026
Ratio of Effective Shear Stress (rz) to K2 (N)
Gv/Ev=0.5
Gv/Ev=1.0
Gv/Ev=1.25
2.50E+09 nu hh=0.80 nu vh=0.20 Eh=0.5Ev kh=5kv Ka=1
2.00E+09
1.50E+09
1.00E+09
5.00E+08
0.00E+00 1.0
3.0
5.0
7.0
9.0
11.0
13.0
15.0
r/a Fig. 11. Ratio of effective out of plane shear stress ðsrz Þ to the parameter K 2 with respect to ratio of Gv =E v and radial distance from the cylinder (g ¼ 10, y ¼ p=2).
Ka ¼ 1, y ¼ p). It is seen that this parameter has no any important effect on the pore pressure response. Fig. 10 shows the ratio of effective radial displacement to the parameter K with respect to nhh and radial distance from the cylinder (g ¼ 10 , Ka ¼ 1, y ¼ p). In this case also the value of nhh has no any important effect in the displacement response. Fig. 11 gives the ratio of effective out of plane shear stress (srz ) to the parameter K 2 with respect to the ratio of G v =E v and radial distance from the cylinder (g ¼ 10 , Ka ¼ 1, y ¼ p=2). It is seen that with increasing the ratio of Gv =E v , the graph of srz is shifted to the right. 8. Conclusions An analytical method for solving the problem of wave scattering in a cross-anisotropic porous medium was presented. This method includes the solution of Biot’s consolidation equations by means of complex functions theory. Using potential functions, the solution is obtained in series of the Hankel functions with unknown coefficients, calculated from boundary conditions of the problem. Many numerical results including pore pressure, effective stresses, and displacement were obtained for a cylinder in a cross-anisotropic porous medium subjected to the P1 incident waves. Also the effect of many parameters such as kh =kv , E h =E v , G v =E v , and nhh were studied. It was found that solid skeleton anisotropy has more effect on the results when compared with the permeability anisotropy. Also the stress field is much more sensitive to solid skeleton anisotropy when compared with pore pressure. Since in this method the radiation condition is satisfied completely in a very simple manner, it is one of the most powerful analytical schemes for solving the wave propagation and scattering problems in the unbounded domains.
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ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027 [15] Zimmerman Ch, Stern M. Scattering of plane compressional wave by spherical inclusions in a poroelastic medium. J Acoust Soc Am 1993; 94(1):527–36. [16] Gatmiri B. Response of a cross-anisotropic seabed to ocean waves. J Geotech Eng 1992;118(9):1295–314. [17] Jeng DS. Soil response in cross-anisotropic seabed due to standing waves. J Geotech Geoenvironmental Eng 1997;123(1):9–19. [18] Honarvar F, Sinclair AN. Acoustic wave scattering from transversely isotropic cylinders. J Acoust Soc Am 1996;100(1):57–63. [19] Fan Y, Sinclair AN, Honarvar F. Scattering of a plane acoustic wave from a transversely isotropic cylinder encased in a solid elastic medium. J Acoust Soc Am 1999;106(3):1229–36. [20] Ahmad F, Rahman A. Acoustic scattering by transversely isotropic cylinders. Int J Eng Sci 2000;38:325–35. [21] Carcione JM. Wave propagation in anisotropic, saturated porous media: plane-wave theory and numerical simulation. J Acoust Soc Am 1996;99(5):2655–66. [22] Sharma MD, Gogna ML. Wave propagation in anisotropic liquidsaturated porous solids. J Acoust Soc Am 1991;90(2):1068–73. [23] Sun F, Banks-Lee P, Peng H. Wave propagation theory in anisotropic periodically layered fluid-saturated porous media. J Acoust Soc Am 1993;93(3):1277–85.
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[24] Yang D, Zhang Z. Poroelastic wave equations including the Biot/ squirt mechanism and the solid/fluid coupling anisotropy. Wave Motion 2002;35:223–45. [25] Gatmiri B, Jabbari E. Time-domain Green’s functions for unsaturated soils, part I: two dimensional solution. Int J Solid Struct 2005;42(23):5971–90. [26] Gatmiri B, Jabbari E. Time-domain Green’s functions for unsaturated soils, part II: three dimensional solution. Int J Solid Struct 2005;43(23):5991–6002. [27] Muskhelishvili NI. Some basic problems of the mathematical theory of elasticity. [Radok JRM, Trans. from 4th ed.] University of Groningen, Netherlands, Noordhoff, New York, 1963 [In Russian]. [28] Liu D, Gai B, Tao G. Applications of the method of complex functions to dynamic stress concentrations. Wave Motion 1982;4: 293–304. [29] Han R, Yeh K, Liu G, Liu D. Scattering of plane SH-waves by a cylindrical canyon of arbitrary shape in anisotropic media. Int J Eng Sci 1992;30(12):1773–87. [30] Nowinski JL. Stress concentration around holes in a class of rheological materials displaying a poroelastic structure. Dev Mech 1982;6:445–58.
Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027 www.elsevier.com/locate/soildyn
Wave scattering in cross-anisotropic porous media around the cavities and inclusions Behrooz Gatmiria,b,, Hashem Eslamib a
University of Tehran, Tehran, Iran Ecole Nationale des Ponts et Chausse´es, 6-8 Av. Blaise pascal, 77455 Marne la Vallee cedex 2, France
b
Received 31 August 2007; received in revised form 25 October 2007; accepted 5 November 2007
Abstract A method is presented for evaluating dynamic response of a cylindrical cavity or inclusion in an infinite cross-anisotropic porous medium. The basis of the method is the Biot’s consolidation equations in the complex plane. Employing two groups of potential functions for solid skeleton and pore fluid (each group includes three functions), the u-w formulation of Biot’s equations are solved. The stress, displacement, and pore pressure fields are evaluated in the vicinity of the cylinder in complex plane. Also the effects of many parameters ðkh =kv ; E h =E v ; G v =E v ; nhh Þ on the responses are evaluated by numerical examples. r 2007 Elsevier Ltd. All rights reserved. Keywords: Wave scattering; Analytical solution; Porous media; Complex functions; Piles; Tunnels
1. Introduction The problem of wave propagation and scattering in the porous media is described by consolidation theory of Biot [1,2]. In the most previous researches, the isotropic condition was considered for porous media. In this way, Gatmiri [3] studied the wave propagation problem in the quasi-harmonic case by use of finite element method. For the wave propagation problem in dynamic case, it can be mentioned the works by Degrande and Roeck [4,5], Kaynia [6], Dominguez [7], Gatmiri and Kamalian [8], Gatmiri and Nguyen [9], Nguyen and Gatmiri [10], Kamalian et al. [11], and Gatmiri et al. [12] on the basis of the finite element and boundary element methods. An analytical method for calculating dynamic response of a circular cavity in an infinite isotropic poroelastic medium was presented by the authors [13]. Mei et al. [14] and Zimmerman and Stern [15] studied the problem of wave scattering in the poroelastic media by other methods such as boundary-layer approximation. A few studies have been performed on the crossCorresponding author at: Ecole Nationale des Ponts et Chausse´es, 6-8 Av. Blaise Pascal, 77455 Marne la Valle´e cedex 2, France. Tel.: +33 1 64 15 35 66; fax: +33 1 64 15 35 62. E-mail address: [email protected] (B. Gatmiri).
0267-7261/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2007.11.003
anisotropic porous media. Gatmiri [16] and Jeng [17] achieved the solution for the problem of wave propagation in porous media (quasi-harmonic case) on the basis of finite element method and an analytical method, respectively. Honarvar and Sinclair [18] solved the problem of acoustic wave scattering from transversely isotropic cylinders. Also Fan et al. [19] solved this problem for the case that cylinder encased in a solid elastic medium. Both of them are on the basis of normal mode expansion method. Ahmad and Rahman [20] solved the problem of acoustic scattering by transversely isotropic cylinders with the same method but with different potential functions. Carcione [21] and Sharma and Gogna [22] studied wave propagation in anisotropic saturated porous media, also Sun et al. [23] studied this problem for the case of layered media. Yang and Zhang [24] studied the problem of wave propagation in the medium including Biot/squirt mechanism and the solid/ fluid anisotropy by means of Fourier transform method. Gatmiri and Jabbari [25,26] have studied and developed the fundamental solutions for unsaturated soils for the first time. This paper presents an analytical solution for the problem of scattering of harmonic waves in an infinite cross-anisotropic porous medium on the basis of complex functions theory. The use of complex functions in
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
Nomenclature sij total stress tensor component rbi ; rf bi body forces ui ; v i displacement vector components of solid skeleton and pore fluid respectively wi relative displacement vector components of pore fluid n porosity of the soil p pore fluid pressure kh permeability parameter of the soil in the plane ðx; yÞ kv permeability parameter of the soil in the z direction r; rf ; rs mass density of mixture, pore fluid and solid skeleton, respectively eij solid strain tensor component C 11 ; C 12 ; C 13 ; C 33 ; C 44 anisotropic parameters E h ; E v ; G v ; Gh ; nhh ; nvh ; nhv elastic parameters a Biot coefficient Q Biot modulus dij Kronecker delta Fs ; Cs ; X s potential functions for solid skeleton Ff ; Cf ; X f potential functions for pore fluid r2 two-dimensional Laplace operator o wave frequency d1 quasi-P1 wave number (qP1) d2 quasi-P2 wave number (qP2)
elastostatic problems was expanded by Muskhelishvili [27]. This method was used for solving the problem of wave scattering by a cavity in infinite elastic media by Liu et al. [28] and Han et al. [29]. Nowinski [30] solved the problem of static stress concentration around holes subjected to uniaxial tension, using complex functions theory. The use of this method for solving the problem of wave scattering is presented in previous writers’ paper for the first time and now is extended to the cross-anisotropic case. In this way, first two groups of potential functions (Fs , s C , X s and Ff , Cf , X f ) are applied into u-w formulation of Biot’s theory. Then, the governing equations are transformed into the complex plane. Solving an eigenvalue problem, the relation between potential functions Fs , Ff and Cs , Cf is determined. Also the relation between functions X s , X f is found directly. In the complex plane, solution of the resulting partial differential equations (calculation of the potential functions) is found in series of the Hankel functions (complex sum of the Bessel functions) with unknown coefficients. These Hankel functions satisfy the radiation condition. Applying appropriate boundary conditions of the problem, a set of algebraic equations are achieved. Solving these equations, the unknown coefficients and consequently potential functions are calculated. After evaluating the potential functions, all desirable parameters such as stress, pore pressure, and
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quasi-SV wave number (qSV) shear wave number complex variable ðz ¼ x þ iyÞ Bessel function of first kind and order n Bessel function of second kind and order n Hankel function of first kind and order n displacements of solid skeleton in Cartesian coordinates vx ; vy ; vz displacements of pore fluid in Cartesian coordinates wx ; wy ; wz relative displacements of pore fluid in Cartesian coordinates ur ; uy ; uz displacements of solid skeleton in polar coordinates sx ; sy ; sz ; sxy ; sxz ; syz components of stress tensor in Cartesian coordinates sr ; sy ; sz ; sry ; srz ; syz components of stress tensor in polar coordinates a radius of the cylinder fin incident potential function fsc scattered potential function K incident wave number Kx incident wave number in direction x Kz incident wave number in direction z g angle of incident wave with axis of cylinder q1 ; q2 ; q3 ; q01 ; q02 ; q03 ; q001 ; q002 ; q003 ; q000 coefficients X n ; Y n ; Zn ; W n unknown coefficients
d3 b z ¼ reiy J n ð‘rÞ Y n ð‘rÞ H ð1Þ n ð‘rÞ ux ; uy ; uz
displacement can be calculated in any point of the medium. Numerical results are presented including stress, displacement, and pore pressure fields in the vicinity of the cylinder in an infinite cross-anisotropic porous medium subjected to P1 harmonic waves. Also the effect of important parameters such as kh =kv , E h =E v , G v =E v , and nhh are mentioned. 2. Governing equations Based on the Biot theory, equilibrium equation and modified Darcy equation of the saturated porous media are written as sij;j þ rbi ¼ ru€ i þ rf w€ i ; r 1 p;i þ rf bi ¼ rf u€ i þ f w€ i þ w_ i n kðiÞ
with
r ¼ ð1 nÞrs þ nrf ; wi ¼ nðvi ui Þ
(1) In these equations, sij is the total stress tensor component, rbi and rf bi are the body forces, and ui , vi are the displacement vector components of solid skeleton and pore fluid, respectively. n is the porosity of the soil, p is the pore fluid pressure, and kðiÞ is the permeability parameter of the soil in the direction ðiÞ. r, rf , and rs are the mass density of
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
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mixture, pore fluid, and solid skeleton, respectively. The dots ð: Þ denote the differentiation respect to the time ðq=qtÞ. The constitutive equations in a cross-anisotropic porous medium are written as below: 8 quy qux quz > > sx ¼ C 11 þ C 12 þ C 13 ap; > > > qx qy qz > > > > quy qux quz > > > sy ¼ C 12 þ C 11 þ C 13 ap; > > qx qy qz > > > > quy qux quz > > > > < sz ¼ C 13 qx þ C 13 qy þ C 33 qz ap; C 11 C 12 qux quy > > þ ¼ s ; > xy > 2 qy qx > > > > > qux quz > > > þ ¼ C ; s xz 44 > > qz qx > > > > > > > syz ¼ C 44 quy þ quz ; > : qz qy
qux quy quz qwx qwy qwz p ¼ aQ þ þ þ þ þQ , qx qy qz qx qy qz
(2)
(3)
sx ; sy ; sz are the normal components of stress tensor and sxy , sxz , syz are the shear components of it. ux , uy , uz are the components of displacements in direction x, y and z, respectively. a is the Biot coefficient and Q is the Biot modulus. The relation between coefficients C ij and elastic parameters is written as follows: 8 C 11 > > > > > > < C 12 C 13 > > > C 33 > > > : C 44
¼ E h ð1 nhv nvh Þ=D; ¼ E h ðnhv nvh þ nhh Þ=D; ¼ E h ð1 þ nhh Þ nvh =D; ¼ E v ð1 ¼ Gv :
(4)
n2hh Þ=D;
E h and E v are the Young’s modulus in the vertical and any horizontal direction. nhh and nvh are the Poisson’s ratio as the corresponding operations of lateral expansion due to horizontal direct stress in horizontal direction and due to vertical direct stress in vertical direction, respectively, and G v is the modulus of shear deformation in a vertical plane. The other parameters are defined as below:
are derived as 8 > q2 ux ðC 11 C 12 Þ q2 ux > > ðC 11 þ a2 QÞ 2 þ > > > 2 qx qy2 > > 2 > 2 > > q uy q ux C 11 þ C 12 > 2 > þa Q þC 44 2 þ > > qz 2 qxqy > > > ! > > 2 2 2 2 > q w q u q w q w > y z x z > > þ aQ þ þðC 13 þ C 44 þ a2 QÞ þ > > qxqz qx2 qxqy qxqz > > > > > > ¼ ru€ x þ rf w€ x ; > > > > > ðC 11 C 12 Þ q2 uy q2 uy > 2 > þ ðC þ a QÞ > 11 > > 2 qx2 qy2 > > > 2 > 2 > > < þC 44 q uy þ C 11 þ C 12 þ a2 Q q ux qz2 2 qxqy ! > > 2 2 2 2 > > q w q u q w q w y z x z > > þðC 13 þ C 44 þ a2 QÞ þ aQ þ þ > > qyqz qxqy qy2 qyqz > > > > > > ¼ ru€ y þ rf w€ y ; > > > > > q2 uz q2 uz q2 uz > > > C 44 2 þ C 44 2 þ ðC 33 þ a2 QÞ 2 > > qx qy qz > > > > 2 > q ux > > > þðC 13 þ C 44 þ a2 QÞ > > qxqz > > ! > > 2 2 2 2 > q u q w q w q w > y y x z > þðC 13 þ C 44 þ a2 QÞ þ aQ þ þ 2 > > > qyqz qxqz qyqz qz > > > > : ¼ ru€ z þ r w€ z ; f
! 8 2 2 2 > q u q u q u y x z > > aQ þ þ þQ > > qx2 qxqy qxqz > > > > > > r 1 > > ¼ rf u€ x þ f w€ x þ w_ x ; > > > kh n > > ! > > 2 2 2 > q u q u q u > y x z > þ þ þQ > aQ < qxqy qy2 qyqz > r 1 > > > ¼ rf u€ y þ f w€ y þ w_ y ; > > kh n > > ! > > 2 2 > > q ux q uy q2 uz > > > > aQ qxqz þ qyqz þ qz2 þ Q > > > > > > rf 1 > > : ¼ rf u€ z þ w€ z þ w_ z : kv n
q2 wx q2 wy q2 wz þ þ qx2 qxqy qxqz
q2 wx q2 wy q2 wz þ þ qxqy qy2 qyqz
q2 wx q2 wy q2 wz þ þ 2 qxqz qyqz qz
(6) !
!
!
(7) kh and kv are the permeability parameter of the soil in the in plane ðx; yÞ and z direction, respectively.
E h nhv ¼ , E v nvh
3. Potential functions
Eh , 2ð1 þ nhh Þ D ¼ ð1 þ nhh Þð1 nhh 2nvh nhv Þ.
The relation between displacements and potential functions ðFs ; Cs ; X s for solid skeleton and Ff , Cf , X f for pore fluid) are defined as below:
Gh ¼
ð5Þ
From Eqs. (1)–(3), the equilibrium equations in the absence of body forces in term of displacement vector components
~ ez Þ þ ar r ðCs~ ez Þ, u ¼ rFs þ r ðX s~ ~ ¼ rFf þ r ðX f ~ ez Þ þ ar r ðCf ~ ez Þ. w
ð8Þ
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
In which ‘‘a’’ is the radius of Cartesian coordinates: 8 > qFs qX s q2 C s > > þ þa ; > ux ¼ > qx qy qx qz > > > > < qFs qX s q2 C s u þ a ; ¼ y > qy qx qy qz > > > > > > qFs > > ar2 Cs ; : uz ¼ qz
the cylindrical cavity. In the 8 qFf qX f q2 Cf > > > w þ þ a ; ¼ > x > qx qy qx qz > > > > < qFf qX f q2 Cf wy ¼ þa ; > qy qx qy qz > > > > > > qFf > > ar2 Cf ; : wz ¼ qz ð9Þ
where ux , uy , uz and wx , wy , wz are the Cartesian components of the vectors ui and wi , respectively. Therefore Eqs. (6) and (7) are converted to the two groups of equations (F ¼ f eiot eiK z z , C ¼ c eiot eiK z z , and X ¼ w eiot eiK z z Þ: 8 q2 fs > > > ðC 11 þ a2 QÞr2 fs þ ðC 13 þ 2C 44 þ a2 QÞ 2 > > qz > > ! > > 2 f > > > > þaQ r2 ff þ q f þ ro2 fs þ rf o2 ff > > > qz2 > > > > > > > q q2 c s > 2 s > ðC þa C C Þr c þ C > 11 13 14 44 > > qz qz2 > > > > > > > 2 s 2 f > > c þ r o c þro ¼ 0; f > > > > > " > > > > q q2 fs > > ðC 13 þ 2C 44 þ a2 QÞr2 fs þ ðC 33 þ a2 QÞ 2 > > > qz qz > > > > ! # > > > > q2 ff > 2 f > þaQ r f þ þ ro2 fs þ rf o2 ff > > qz2 > > > > > > > > q2 cs > 2 2 s > < ar C 44 r c þ ðC 33 C 13 C 44 Þ 2 qz ð10Þ > > > > > þro2 cs þ rf o2 cf ¼ 0; > > > > > > ! > > > q2 fs q2 ff > 2 s 2 f > > aQ r f þ 2 þ Q r f þ 2 þ rf o2 fs > > qz qz > > > > > > > rf 2 io f > > o þ f > þ > > kh n > > > > > q rf 2 io f > > 2 s > r o þa o c þ þ c ¼ 0; > > qz f kh n > > > > " ! > > 2 s 2 f > q q f q f > > aQ r2 fs þ 2 þ Q r2 ff þ 2 þ rf o2 fs > > > qz qz qz > > > > > > > r io f > > þ f o2 þ f > > kv n > > > > > > rf 2 io 2 f > s 2 2 > > : a rf o r c þ n o þ k r c ¼ 0; v
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8 > C 11 C 12 2 s q2 ws > > r w þ C þ ro2 ws þ rf o2 wf ¼ 0; 44 > < 2 qz2 ð11Þ > rf 2 io f > 2 s > > : rf o w þ n o þ kh w ¼ 0; where r2 is the two-dimensional Laplace operator, r2 ¼ q2 =qx2 þ q2 =qy2 , and o is the frequency of the waves. Also it is possible to use below formulation for potential functions: s qC qFs ~ ~ ez Þ þ u ¼ rFs þ r ðX s~ ez , qz qz f qC qFf f f ~ ¼ rF þ r ðX ~ ~ ð12Þ ez Þ þ w ez . qz qz It is simpler than the mentioned, but it has not any powerful physical meaning. Our formulation is in relation with the P, SV, and SH waves that is very useful for seeing the contribution of each component of the wave. 4. Complex functions The complex variables are introduced: z ¼ x þ iy;
z¯ ¼ x iy;
z ¼ r eiy .
(13)
Using relations below: q q q q q q ¼ þ ; ¼i , qx qz qz¯ qy qz qz¯ q2 q2 q2 q2 ¼ þ 2 ; þ qx2 qz2 qzqz¯ qz¯ 2
(14)
! q2 q2 q2 q2 ¼ 2 þ , qy2 qzqz¯ qz¯ 2 qz2 (15)
the Laplace operator in the complex plane is obtained as r2 ¼ 4
q2 . qz qz¯
Also the potential functions 8 þ1 P s ð1Þ > s > an H n ðdrÞeiny ; >f ¼ > > n¼1 > > > < þ1 P s ð1Þ cs ¼ bn H n ðdrÞeiny ; > n¼1 > > > þ1 > P s ð1Þ > s > cn H n ðbrÞeiny ; > :w ¼ n¼1
(16) are considered as below: 8 þ1 P f ð1Þ > f > an H n ðdrÞeiny ; >f ¼ > > n¼1 > > > < þ1 P f ð1Þ cf ¼ bn H n ðdrÞeiny ; > n¼1 > > > þ1 > P f ð1Þ > f > cn H n ðbrÞeiny : > :w ¼ n¼1
(17) H ð1Þ n ð‘rÞ is the Hankel function of first kind and order n: H ð1Þ n ð‘rÞ ¼ J n ð‘rÞ þ iY n ð‘rÞ,
(18)
where J n ð‘rÞ and Y n ð‘rÞ are the Bessel functions of first and second kind, respectively, and of order n. This type of the Hankel function satisfies the radiation boundary condition (Sommerfield condition). Also asn , bsn , csn , afn , bfn , cfn are the unknown coefficients. Based on Eqs. (16) and (17) and
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relations below: q ‘ ½H ð1Þ ð‘rÞeiny ¼ H ð1Þ ð‘rÞeiðn1Þy , qz n 2 n1
(19)
q ‘ ð‘rÞeiðnþ1Þy . ½H ð1Þ ð‘rÞeiny ¼ H ð1Þ 2 nþ1 qz¯ n
(20)
(qP1), d2 for quasi-P2 wave (qP2), d3 for quasi-SV wave (qSV) and ½q1 q01 q001 T for d1 , ½q2 q02 q002 T for d2 , and ½q3 q03 q003 T for d3 Þ. Also from Eq. (11),the relation between potential functions of solid skeleton and pore fluid, also the wave number b are achieved directly as wf ¼ q000 ws , q000 ¼
Eq. (10) is converted to an eigenvalue problem as 2
a11 6 a21 6 6 4 a31 a41
a12 a22
a13 a23
a32 a42
a33 a43
rf o2 rf 2 io o þ kh n
,
1=2 ro2 C 44 K 2z þ rf o2 q000 . b¼ 2 ðC 11 C 12 Þ
38 s 9 f > a14 > > > > > > = < ff > a24 7 7 ¼ 0. 7 s c > a34 5> > > > > > > a44 : cf ;
ð23Þ
(21) Finally, the potential functions are written as below: 8 þ1 þ1 P P > iny > fs ¼ X n H ð1Þ ðd1 rÞ einy þ Y n H ð1Þ > n n ðd2 rÞ e > > n¼1 n¼1 > > > > þ1 > P > iny > þq Z n H ð1Þ > 3 n ðd3 rÞ e ; <
The coefficients aij are a11 ¼ ðC 11 þ a2 QÞd2 þ ðC 13 þ 2C 44 þ a2 QÞK 2z ro2 , a12 ¼ aQðd2 þ K 2z Þ rf o2 ,
n¼1
2
a13 ¼ aiK z ½ðC 11 C 13 C 44 Þd þ
C 44 K 2z
2
ro ,
2
a14 ¼ aiK z rf o ,
ð22aÞ
þ1 þ1 P P > iny > > ff ¼ q01 X n H ð1Þ ðd1 rÞ einy þ q02 Y n H ð1Þ > n n ðd2 rÞ e > > n¼1 n¼1 > > > þ1 > P > 0 iny > þq Z n H ð1Þ > 3 n ðd3 rÞ e ; : n¼1
(24)
a21 ¼ iK z ½ðC 13 þ 2C 44 þ a2 QÞd2 þ ðC 33 þ a2 QÞK 2z ro2 , a22 ¼ iK z ½aQðd2 þ K 2z Þ rf o2 , a23 ¼ ad2 ½C 44 d2 þ ðC 33 C 13 C 44 ÞK 2z ro2 , a24 ¼ ad2 rf o2 ,
ð22bÞ
a31 ¼ aQðd2 þ K 2z Þ rf o2 , rf 2 io o þ a32 ¼ Qðd2 þ K 2z Þ , kh n a33 ¼ aiK z rf o2 , r io , a34 ¼ aiK z f o2 þ kh n
n¼1 þ1 P > iny > > cf ¼ q001 X n H ð1Þ þ > n ðd1 rÞ e > > n¼1 > > > þ1 > P > iny > þq003 Z n H ð1Þ > n ðd3 rÞ e ; : n¼1
ð22cÞ
a41 ¼ iK z ½aQðd2 þ K 2z Þ rf o2 , rf 2 io o þ a42 ¼ iK z aQðd2 þ K 2z Þ , kv n a43 ¼ ad2 rf o2 , r io a44 ¼ ad2 f o2 þ . kv n
8 þ1 þ1 P P > iny > cs ¼ q1 X n H ð1Þ ðd1 rÞ einy þ q2 Y n H ð1Þ > n n ðd2 rÞ e > > n¼1 n¼1 > > > > þ1 > P > iny > þ Z n H ð1Þ > n ðd3 rÞ e ; < q002
þ1 P n¼1
iny Y n H ð1Þ n ðd2 rÞ e
(25)
8 þ1 P > s iny > W n H ð1Þ > n ðbrÞ e ;
> f > > :w ¼ q
n¼1
(26)
iny W n H ð1Þ n ðbrÞ e :
ð22dÞ
Solving this eigenvalue problem, three values for d and three eigenvectors will be found. (d1 for quasi-P1 wave
The unknown coefficients, X n , Y n , Z n , W n are calculated from boundary conditions of the problem. Now, knowing the potential functions, displacements, stresses, and pore
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pressure can be calculated based on these functions and then the boundary value problem can be solved.
5. Displacements, stresses, and pore pressure From Eq. (9) the displacement field is 8 þ1 P > iny > > ur þ iuy ¼ d1 ð1 þ aiK z q1 Þ X n H ð1Þ > nþ1 ðd1 rÞ e > > n¼1 > > > þ1 > P > iny > d2 ð1 þ aiK z q2 Þ Y n H ð1Þ > nþ1 ðd2 rÞ e > > n¼1 > > > > þ1 P > > iny > d ðq þ aiK Þ Z n H ð1Þ 3 z > 3 nþ1 ðd3 rÞ e > > n¼1 > > > þ1 > P > iny > > þib W n H ð1Þ > nþ1 ðbrÞ e ; > > n¼1 > > > þ1 > P > iny > ur iuy ¼ d1 ð1 þ aiK z q1 Þ X n H ð1Þ > n1 ðd1 rÞ e > > n¼1 > > > < þ1 P iny þd2 ð1 þ aiK z q2 Þ Y n H ð1Þ n1 ðd2 rÞ e > n¼1 > > > þ1 > P > iny > þd3 ðq3 þ aiK z Þ Z n H ð1Þ > n1 ðd3 rÞ e > > n¼1 > > > > þ1 > P > iny > þib W n H ð1Þ > n1 ðbrÞ e ; > > n¼1 > > > þ1 > P > > iny > uz ¼ ðiK z þ aq1 d21 Þ X n H ð1Þ > nþ1 ðd1 rÞ e > > n¼1 > > > þ1 > P > iny > þðiK z þ aq2 d22 Þ Y n H ð1Þ > nþ1 ðd2 rÞ e > > n¼1 > > > > þ1 P > > 2 iny > þðiK q þ ad Þ Z n H ð1Þ > z 3 3 nþ1 ðd3 rÞ e : :
1019
functions are 8 ! 2 s 2 f > > q f q f > > p ¼ aQ r2 fs þ 2 Q r2 ff þ 2 ; > > qz qz > > > > > > > ðC 11 þ C 12 Þ 2 s q2 f s > > r is ¼ f þ C s > r y 13 > 2 qz2 > > > > a q > > > þ ðC 11 þ C 12 2C 13 Þ r2 cs > > 2 qz > > > 2 > q qcs 2iy > s > s > þ2ðC 11 C 12 Þ 2 f þ iw þ a e ; > > qz > qz > > < ðC 11 þ C 12 Þ 2 s q2 f s r f þ C 13 2 sr þ isy ¼ > 2 qz > > > > a q > > þ ðC 11 þ C 12 2C 13 Þ r2 cs > > > 2 qz > > > 2 > q qcs 2iy > s s > þ2ðC 11 C 12 Þ 2 f iw þ a e ; > > > qz > qz¯ > > > > > qfs q2 c s > 2 s iy q iy q > þe þ a 2 ar c srz ¼ C 44 e 2 > > > qz qz qz qz¯ > > > s > q q qw > > > þi eiy eiy : > : qz qz¯ qz (29) (The second and third relations were constructed from subtracting the second and third relations of Eq. (28)). Finally Eq. (29) is written in term of the Hankel functions as below: þ1 X iny p ¼ Qðd21 þ K 2z Þða þ q01 Þ X n H ð1Þ n ðd1 rÞe þ Qðd22 þ
n¼1
(27)
Eqs. (2) and (3) results: 8 qux quy quz qwx qwy qwz > > > p ¼ aQ þ þ þ þ Q ; > > qx qy qz qx qy qz > > > > > qux quy quz > > þ þ s ¼ ðC þ C Þ þ 2C 13 ; s > 11 12 y < r qx qy qz q q > > > i s sr þ 2isry ¼ ðC 11 C 12 Þ ðux iuy Þ e2iy ; > > y qx qy > > > > > quy quz qux quz > > þ þ ¼ C s cos y þ sin y ; > rz 44 : qz qx qz qy (28)
where sr , sy are the radial and hoop components of stress tensor and sry , srz are the shear components of it. These relations in complex plane and in term of potential
þ Qðd23 þ
n¼1 þ1 X iny K 2z Þða þ q02 Þ Y n H ð1Þ n ðd2 rÞe n¼1 þ1 X iny K 2z Þðaq3 þ q03 Þ Zn H ð1Þ n ðd3 rÞe , n¼1
ð30aÞ
ðC 11 þ C 12 Þ 2 a d1 C 13 K 2z iK z q1 d21 sr isy ¼ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ X n H ð1Þ n ðd1 rÞe n¼1 þ1 X ðC 11 C 12 Þ 2 iny d1 ð1 þ aiK z q1 Þ þ X n H ð1Þ n2 ðd1 rÞe 2 n¼1 ðC 11 þ C 12 Þ 2 a d2 C 13 K 2z iK z q2 d22 þ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ Y n H ð1Þ n ðd2 rÞe n¼1 þ1 X ðC 11 C 12 Þ 2 iny d2 ð1 þ aiK z q2 Þ þ Y n H ð1Þ n2 ðd2 rÞe 2 n¼1 ðC 11 þ C 12 Þ 2 a q3 d3 C 13 q3 K 2z iK z d23 þ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ Z n H ð1Þ n ðd3 rÞe n¼1
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1020
þ
þ1 X ðC 11 C 12 Þ 2 iny d3 ðq3 þ aiK z Þ Z n H ð1Þ n2 ðd3 rÞe 2 n¼1
þi
þ1 X
ðC 11 C 12 Þ 2 ð1Þ b W n H n2 ðbrÞeiny , 2 n¼1
ð30bÞ
ðC 11 þ C 12 Þ 2 a d1 C 13 K 2z iK z q1 d21 sr þ isy ¼ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ X n H ð1Þ n ðd1 rÞe þ1 X ðC 11 C 12 Þ 2 iny d1 ð1 þ aiK z q1 Þ X n H ð1Þ nþ2 ðd1 rÞe 2 n¼1 ðC 11 þ C 12 Þ 2 a 2 d2 C 13 K z iK z q2 d22 þ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ Y n H ð1Þ n ðd2 rÞe
þ
n¼1 þ1 X ðC 11 C 12 Þ 2 iny d2 ð1 þ aiK z q2 Þ Y n H ð1Þ nþ2 ðd2 rÞe 2 n¼1 ðC 11 þ C 12 Þ a 2 q3 d3 C 13 q3 K 2z iK z d23 þ 2 2 X þ1 iny ðC 11 þ C 12 2C 13 Þ Z n H ð1Þ n ðd3 rÞe
þ
n¼1
ðC 11 C 12 Þ 2 iny b W n H ð1Þ nþ2 ðbrÞe , 2 n¼1
at r ¼ a,
(31)
8 in sc >
ð30cÞ
h i a a srz ¼ C 44 d1 iK z K 2z q1 þ q1 d21 2 2 þ1 X ð1Þ iny X n ½H n1 ðd1 rÞ H ð1Þ nþ1 ðd1 rÞe
(32)
where fin , cin , win are the incident and fsc , csc , wsc are the scattered components of the potential functions. Therefore the boundary conditions at the cavity surface are 8 in p þ psc ¼ 0; > > > > < ðsr isry Þin þ ðsr isry Þsc ¼ 0; at r ¼ a, (33) ðsr þ isry Þin þ ðsr þ isry Þsc ¼ 0; > > > > : sin þ ssc ¼ 0: rz
þ1 X ðC 11 C 12 Þ 2 ð1Þ d3 ðq3 þ aiK z Þ þ Z n H nþ2 ðd3 rÞeiny 2 n¼1
i
8 p ¼ 0; > > > < sr isry ¼ 0; > sr þ isry ¼ 0; > > : srz ¼ 0:
All of the wave variables are the sum of incident and scattering components, therefore the potential functions are written as
n¼1
þ1 X
be satisfied at the cavity surface:
rz
Incident components of the stresses and pore pressure are derived from incident potential functions using Eq. (29). Also scattering components of these variables are found in Eqs. (30a)–(30d). In the case of dilatational incident wave (P1 wave), the incident potential function is written as fin ¼ f0 eiK x x eiK z z ,
(34)
n¼1
h i a a þ C 44 d2 iK z K 2z q2 þ q2 d22 2 2 þ1 X ð1Þ iny Y n ½H ð1Þ n1 ðd2 rÞ H nþ1 ðd2 rÞe n¼1
h
z
i
a a þ C 44 d3 iK z q3 K 2z þ d23 2 2 þ1 X ð1Þ iny Zn ½H ð1Þ n1 ðd3 rÞ H nþ1 ðd3 rÞe
a θ
Incident Wave
n¼1 þ1 K zb X ð1Þ iny C 44 W n ½H ð1Þ n1 ðbrÞ þ H nþ1 ðbrÞe . ð30dÞ 2 n¼1
γ
y x
6. Boundary value problem Cross-Anisotropic Porous Medium
Now consider a cylindrical cavity in an infinite porous medium is subjected to a harmonic incident wave. The radius of the cavity is ‘‘a’’ and on its surface the pressure and stress are vanished. These boundary conditions should
Fig. 1. Cylinder in an infinite cross-anisotropic porous medium subjected to the harmonic wave.
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
f0 is a coefficient and
and consequently the incident pore pressure and stresses are found
K x ¼ K cos g, K z ¼ K sin g.
8 þ1 P n > > > pin ¼ f0 aQðK 2x þ K 2z Þ i J n ðK x rÞeiny ; > > > n¼1 > > þ1 > > P n ðC 11 þ C 12 Þ 2 > in 2 > is Þ ¼ f þ C K i J n ðK x rÞeiny ðs K > r ry 13 z 0 x > > 2 n¼1 > > > > > P n ðC 11 C 12 Þ 2 þ1 > > þf0 i J n2 ðK x rÞeiny ; Kx > < 2 n¼1 þ1 P n ðC 11 þ C 12 Þ 2 > > > ðsr þ isry Þin ¼ f0 i J n ðK x rÞeiny K x þ C 13 K 2z > > 2 > n¼1 > > > > P n ðC 11 C 12 Þ 2 þ1 > > þf0 i J nþ2 ðK x rÞeiny ; Kx > > > 2 n¼1 > > > > þ1 P n > > in > i ½J n1 ðK x rÞ J nþ1 ðK x rÞeiny : > srz ¼ f0 iC 44 K x K z : n¼1
ð35Þ
K is the wave number and is related to the wave frequency as following: o¼K
1=2 C 11 þ a2 Q . r
(36)
Also g is the angle of incident wave with axis of cylinder. The incident potential function can be written as fin ¼ f0
þ1 X
1021
in J n ðK x rÞeiny eiK z z
(37)
(38)
n¼1
Table 1 Parameters used in applications n
rs ðkg=m3 Þ
rf ðkg=m3 Þ
a
Q (MPa)
a (m)
f0
E v (MPa)
kv (m/s)
0.3
2700
1000
1
6:67 103
1
1
5:32 103
1 104
1.0
1.0 5
Radial Stress
Pore Pressure
5 3
0.5
4 1
4
0.5
3 1
2
2
0.0
0.0 2
1
3 r/a
4
5
3
4
5
r/a
1.0
1.0
3
Shear Stress
Hoop Stress
2
1
1
0.5 5
4
0.5
0.0 2
3 r/a
4
5
1
2
5
1
0.0 1
4
3
2
2
3 r/a
4
5
Fig. 2. Values of normalized pore pressure and effective stresses versus distance from the cylinder (1: y ¼ 0, 2: y ¼ p=4, 3: y ¼ p=2, 4: y ¼ 3p=4, 5: y ¼ p). Solid line: present study, point line: adapted from Mei et al. [14].
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
1022
Finally our boundary value problem (Eq. (33)) is written as an algebraic equation: 9 8 9 2 38 m11 m12 m13 m14 > X n > > n1 > > > > > > > > > = < 6 m21 m22 m23 m24 7< n2 = 6 7 Yn ¼ . (39) 6 7 > Zn > n3 > 4 m31 m32 m33 m34 5> > > > > > > > > ; : ; : m41 m42 m43 m44 Wn n4
ðC 11 þ C 12 Þ 2 a d1 C 13 K 2z iK z q1 d21 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd1 aÞ
m21 ¼
þ
ðC 11 þ C 12 Þ 2 a d2 C 13 K 2z iK z qz d22 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd2 aÞ
The coefficients mij are defined as
m22 ¼
m11 ¼ Qðd21 þ K 2z Þða þ q01 ÞH ð1Þ n ðd1 aÞ, m12 ¼ Qðd22 þ K 2z Þða þ q02 ÞH ð1Þ n ðd2 aÞ, m13 ¼ Qðd23 þ K 2z Þðaq3 þ q03 ÞH ð1Þ n ðd3 aÞ, m14 ¼ 0,
þ
ð40aÞ
kh/kv=1 Ratio of Effective Hoop Stress to K2 (N)
ðC 11 C 12 Þ 2 d1 ð1 þ aiK z q1 ÞH ð1Þ n1 ðd1 aÞ, 2
kh/kv=5
ðC 11 C 12 Þ 2 d2 ð1 þ aiK z q2 ÞH ð1Þ n2 ðd2 aÞ, 2
kh/kv=10
kh/kv=100
4.0E+10 3.5E+10 3.0E+10 2.5E+10 nu=0.33 n=0.30 G=2000MPa r=a
2.0E+10 1.5E+10 1.0E+10 5.0E+09 0.0E+00 0.0
0.5
1.0
1.5
2.0 Ka
2.5
3.0
3.5
4.0
Fig. 3. Ratio of effective hoop stress to the parameter K 2 with respect to the ratio of kh =kv and dimensionless wave number ðy ¼ p=2Þ.
kh/kv=1
kh/kv=5
kh/kv=10
kh/kv=100
Ratio of Pore Pressure to K2 (N)
1.2E+10 1.0E+10 8.0E+09 6.0E+09 nu=0.33 n=0.30 G=2000MPa r=2a
4.0E+09 2.0E+09 0.0E+00 0.0
0.5
1.0
1.5
2.0 Ka
2.5
3.0
3.5
4.0
Fig. 4. Ratio of pore pressure to the parameter K 2 with respect to the ratio of kh =kv and dimensionless wave number ðy ¼ p=2Þ.
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
m23
ðC 11 þ C 12 Þ a q3 d23 C 13 q3 K 2z iK z d23 ¼ 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd3 aÞ þ
m24 ¼ i
ðC 11 þ C 12 Þ 2 a d2 C 13 K 2z iK z qz d22 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd2 aÞ
m32 ¼
ðC 11 C 12 Þ 2 d3 ðq þ aiK z ÞH ð1Þ n2 ðd3 aÞ, 2
þ
ðC 11 C 12 Þ 2 ð1Þ b H n2 ðbaÞ, 2
(40b)
ðC 11 þ C 12 Þ 2 a d1 C 13 K 2z iK z q1 d21 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd1 aÞ
m31 ¼
þ
m34 ¼ i Eh/Ev=0.5
ðC 11 C 12 Þ 2 d2 ð1 þ aiK z q2 ÞH ð1Þ n2 ðd2 aÞ, 2
ðC 11 þ C 12 Þ a q3 d23 C 13 q3 K 2z iK z d23 m33 ¼ 2 2 ðC 11 þ C 12 2C 13 Þ H ð1Þ n ðd3 aÞ þ
ðC 11 C 12 Þ 2 d1 ð1 þ aiK z q1 ÞH ð1Þ nþ2 ðd1 aÞ, 2
Ratio of Effective Hoop Stress to K2 (N)
1023
Eh/Ev=0.8
ðC 11 C 12 Þ 2 d3 ðq3 þ aiK z ÞH ð1Þ nþ2 ðd3 aÞ, 2 ðC 11 C 12 Þ 2 ð1Þ b H nþ2 ðbaÞ, 2
(40c)
Eh/Ev=1.0
1.0E+11 9.0E+10
nu hh=0.80 nu vh=0.20 Gv=0.5Ev kh=5kv r=a
8.0E+10 7.0E+10 6.0E+10 5.0E+10 4.0E+10 3.0E+10 2.0E+10 1.0E+10 0.0E+00 0.0
0.5
1.0
1.5
2.0 Ka
2.5
3.0
3.5
4.0
Fig. 5. Ratio of effective hoop stress to the parameter K 2 with respect to the ratio of E h =E v and dimensionless wave number ðy ¼ p=2Þ.
Eh/Ev=0.5
Eh/Ev=0.8
Eh/Ev=1.0
Ratio of Pore Pressure to K2 (N)
1.2E+10 1.0E+10 8.0E+09 6.0E+09 nu hh=0.80 nu vh=0.20 Gv=0.5Ev kh=5kv r= 2a
4.0E+09 2.0E+09 0.0E+00 0.0
0.5
1.0
1.5
2.0 Ka
2.5
3.0
3.5
4.0
Fig. 6. Ratio of pore pressure to the parameter K 2 with respect to the ratio of E h =E v and dimensionless wave number ðy ¼ p=2Þ.
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1024
h i a a ð1Þ m41 ¼ C 44 d1 iK z K 2z q1 þ q1 d21 ½H ð1Þ n1 ðd1 aÞ H nþ1 ðd1 aÞ, 2 2 h i a a ð1Þ m42 ¼ C 44 d2 iK z K 2z q2 þ q2 d22 ½H ð1Þ n1 ðd2 aÞ H nþ1 ðd2 aÞ, 2 2
m43
h
a a i ð1Þ ¼ C 44 d3 iK z q3 K 2z þ d23 ½H ð1Þ n1 ðd3 aÞ H nþ1 ðd3 aÞ, 2 2
m44 ¼ C 44
K z b ð1Þ ½H n1 ðbaÞ þ H ð1Þ nþ1 ðbaÞ. 2
(40d)
Also the coefficients ni are written as 8 > n1 ¼ faQðK 2x þ K 2z Þin J n ðK x aÞ; > > > > > ðC 11 þ C 12 Þ 2 > 2 n > K ¼ f þ C K n 13 z i J n ðK x aÞ > 2 0 x > 2 > > > > > ðC 11 C 12 Þ 2 n > > K x i J n2 ðK x aÞ; f0 < 2 ðC 11 þ C 12 Þ 2 > 2 n > > K n ¼ f þ C K 3 13 z i J n ðK x aÞ > 0 x > 2 > > > > > ðC 11 C 12 Þ 2 n > > K x i J nþ2 ðK x aÞ; f0 > > > 2 > > : n4 ¼ f in C 44 in K x K z ½J n1 ðK x aÞ J nþ1 ðK x aÞ: 0
(41)
7. Numerical results Fig. 1 represents a cylinder in an infinite crossanisotropic porous medium. The cylinder is subjected to the P1 incident wave and its surface is free of stress and pressure. In the following examples, the parameters given in Table 1 are assumed.
Eh/Ev=0.5 Eh/Ev=0.8 Eh/Ev=1.0 Ratio of Effective Hoop Stress to K2 (N)
2.0x1010 1.5x1010
7.1. Validation
1.0x1010 5.0x109 0.0 5.0x109 1.0x1010 1.5x1010 2.0x1010
Fig. 7. Ratio of effective hoop stress to the parameter K 2 for various points of circumference and the ratio of E h =E v (Ka ¼ 1, r ¼ a).
To verify the proposed solution, a comparison between the solution obtained by Mei et al. [14] in an isotropic poroelastic medium (by using a boundary-layer approximation) and the result of this solution reduced to an isotropic poroelastic case is made. Fig. 2 shows the variations of normalized pore pressure and effective stresses for the cylinder in a poroelastic medium versus the normalized radial distance (r=a) for various angle (y) of P1 incident wave. In this example, the following values are chosen: kh =kv ¼ 1, E h =E v ¼ 1, nhh ¼ 0:33, Ka ¼ 1. As it can be observed, there is an excellent agreement between these results and those from Mei et al. [14] for hoop and shear stresses for all values of r=a.
Ratio of Effective Radial Stress to K2 (N)
NU hh=0.3 1.75E+10
NU hh=0.5
NU hh=0.8
nu vh=0.20 Eh=0.5Ev Gv=0.5Ev kh=5kv Ka=1
1.50E+10 1.25E+10 1.00E+10 7.50E+09 5.00E+09 2.50E+09 0.00E+00 1.0
3.0
5.0
7.0
9.0
11.0
13.0
15.0
r/a Fig. 8. Ratio of effective radial stress to the parameter K 2 with respect to nhh and radial distance from the cylinder ðy ¼ pÞ.
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7.2. Parametric study To investigate the effect of various parameters of the cross-anisotropic porous medium on the solution, a complete study is performed. Fig. 3 gives the ratio of effective hoop stress to the parameter K 2 with respect to the ratio of kh =kv and dimensionless wave number, Ka (g ¼ 0 , r ¼ a, y ¼ p=2Þ. It is seen that this parameter has very small effect on the stress response (only in the case of kh =kv ¼ 100 this parameter has slightly significant effect). Also Fig. 4 shows the ratio of pore pressure to the parameter K 2 with respect to the ratio of kh =kv and dimensionless wave number, Ka (g ¼ 0 , r ¼ 2a, y ¼ p=2Þ. It is seen that the effect of this parameter is much more when compared with its effect on stress response especially inthe case of kh =kv ¼ 100. (the case of kh =kv ¼ 1 is concerned to the isotopic porous medium). In above examples, the isotropic conditions for solid skeleton were
considered. Fig. 5 gives the ratio of effective hoop stress to the parameter K 2 with respect to the ratio of E h =E v and dimensionless wave number, Ka (g ¼ 0 , r ¼ a, y ¼ p=2Þ. It is seen that with increasing the ratio of E h =E v the hoop stress increases. Fig. 6 represents the ratio of pore pressure to the parameter K 2 with respect to the ratio of E h =E v and dimensionless wave number, Ka (g ¼ 0 , r ¼ 2a, y ¼ p=2). Itis seen the influence of this parameter on the pore pressure response is very small. Fig. 7 gives the ratio of effective hoop stress to the parameter K 2 for various points of circumference of cylinder and the ratio of E h =E v (g ¼ 0 , Ka ¼ 1, r ¼ a and other parameters are as Fig. 5). Fig. 8 shows the ratio of effective radial stress to the parameter K 2 with respect to nhh and radial distance from the cylinder (g ¼ 0 , Ka ¼ 1, y ¼ pÞ. It is seen that with increasing the values of nhh the values of radial stress increases. Fig. 9 shows the ratio of pore pressure to the parameterK 2 with respect to nhh and radial distance from the cylinder (g ¼ 0 ,
NU hh=0.3
NU hh=0.5
NU hh=0.8
Ratio of Pore Pressure to K2 (N)
1.25E+10
1.00E+10
7.50E+09
5.00E+09 nu vh=0.20 Eh=0.5Ev Gv=0.5Ev kh=5kv Ka=1
2.50E+09
0.00E+00 1.0
3.0
5.0
7.0
9.0
11.0
13.0
15.0
r/a Fig. 9. Ratio of pore pressure to the parameter K 2 with respect to nhh and radial distance from the cylinder ðy ¼ pÞ.
NU hh=0.3
NU hh=0.5
NU hh=0.8
Ratio of Radial Displacement to K (m2)
3.0 nu vh=0.20 Eh=0.5Ev Gv=0.5Ev kh=5kv Ka=1
2.5 2.0 1.5 1.0 0.5 0.0 1.0
3.0
5.0
1025
7.0
9.0
11.0
13.0
15.0
r/a Fig. 10. Ratio of radial displacement to the parameter K with respect to nhh and radial distance from the cylinder (g ¼ 10, y ¼ p).
ARTICLE IN PRESS B. Gatmiri, H. Eslami / Soil Dynamics and Earthquake Engineering 28 (2008) 1014–1027
1026
Ratio of Effective Shear Stress (rz) to K2 (N)
Gv/Ev=0.5
Gv/Ev=1.0
Gv/Ev=1.25
2.50E+09 nu hh=0.80 nu vh=0.20 Eh=0.5Ev kh=5kv Ka=1
2.00E+09
1.50E+09
1.00E+09
5.00E+08
0.00E+00 1.0
3.0
5.0
7.0
9.0
11.0
13.0
15.0
r/a Fig. 11. Ratio of effective out of plane shear stress ðsrz Þ to the parameter K 2 with respect to ratio of Gv =E v and radial distance from the cylinder (g ¼ 10, y ¼ p=2).
Ka ¼ 1, y ¼ p). It is seen that this parameter has no any important effect on the pore pressure response. Fig. 10 shows the ratio of effective radial displacement to the parameter K with respect to nhh and radial distance from the cylinder (g ¼ 10 , Ka ¼ 1, y ¼ p). In this case also the value of nhh has no any important effect in the displacement response. Fig. 11 gives the ratio of effective out of plane shear stress (srz ) to the parameter K 2 with respect to the ratio of G v =E v and radial distance from the cylinder (g ¼ 10 , Ka ¼ 1, y ¼ p=2). It is seen that with increasing the ratio of Gv =E v , the graph of srz is shifted to the right. 8. Conclusions An analytical method for solving the problem of wave scattering in a cross-anisotropic porous medium was presented. This method includes the solution of Biot’s consolidation equations by means of complex functions theory. Using potential functions, the solution is obtained in series of the Hankel functions with unknown coefficients, calculated from boundary conditions of the problem. Many numerical results including pore pressure, effective stresses, and displacement were obtained for a cylinder in a cross-anisotropic porous medium subjected to the P1 incident waves. Also the effect of many parameters such as kh =kv , E h =E v , G v =E v , and nhh were studied. It was found that solid skeleton anisotropy has more effect on the results when compared with the permeability anisotropy. Also the stress field is much more sensitive to solid skeleton anisotropy when compared with pore pressure. Since in this method the radiation condition is satisfied completely in a very simple manner, it is one of the most powerful analytical schemes for solving the wave propagation and scattering problems in the unbounded domains.
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