Scientia Iranica A (2016) 23(6), 2469{2477
Sharif University of Technology Scientia Iranica Transactions A: Civil Engineering www.scientiairanica.com
Dispersion of surface waves in a transversely isotropic half-space underlying a liquid layer
A. Bagheria, A. Khojastehb, M. Rahimiana; and R. Attarnejada a. School of Civil Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran. b. School of Engineering Science, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran.
Received 23 November 2014; received in revised form 23 August 2015; accepted 28 November 2015
KEYWORDS Abstract. Surface waves dispersion is studied in a two-layer half-space consisting of a nite liquid layer overlying a transversely isotropic solid half-space. A couple of complete Surface wave; potential functions are utilized to uncouple the equation of motion of the transversely Dispersion; isotropic solid along with a displacement potential for the liquid. The frequency equation Liquid; and velocity dispersion curves are developed. Several solid materials are considered for Transversely isotropic; the bed, and both phase and group velocity curves are calculated. Higher modes are also Rayleigh wave; discussed and the respective curves are plotted. Various special cases are considered by Stoneley wave; letting the liquid layer depth take zero value, to form a solid half-space, or very large values Scholte wave; to form a full-space bi-material. Moreover, an isotropic bed material can be obtained as a Frequency equation; special case by appropriately setting the respective elastic constants. Reduced frequency Anisotropy. equations and numerical results are derived for each case to con rm the results with existing ones. © 2016 Sharif University of Technology. All rights reserved.
1. Introduction isotropy of the geologic materials [4]. This anisotropy can considerably a ect the dispersive properties of Propagation of elastic surface waves in various media surface waves. Studying wave propagation in trans- has been of importance and interest in di erent elds of versely isotropic solids is the research subject of many science such as seismology, earthquake engineering, ma- investigators due to its importance in di erent elds of terial engineering, geophysical studies, etc. The media science and technology. Stoneley [5] was the rst to consisting of a liquid layer lying over an isotropic solid point that transverse isotropy signi cantly in uences half-space has been studied by scientists interested in the wave propagation in a medium in comparison with elds such as ocean engineering, etc. [1-3]. isotropic one. Later, elastodynamics of transversely However, in a wide range of cases, there are isotropic media was subsequently studied by Synge [6], observations revealing the existence of anisotropy in Buchwald [7], and Payton [8]. Earth's crust. It is believed that the fact of following Rahimian et al. [9] presented an ecient ana- a preferred pattern instead of random orientations by lytical formulation to obtain the response of a three- materials depositing in the water and also existence of dimensional transversely isotropic half-space to a time- vertically aligned microcracks result in the transverse harmonic surface loading. Khojasteh et al. [10] extended the previous work to present explicit ex- pressions for the three-dimensional dynamic Green's *. Corresponding author. Tel.: +98 21 61112256; functions in a transversely isotropic half-space and Fax: +98 21 88078263 E-mail addresses: [email protected] (A. Bagheri); investigated the wave propagation. They continued the [email protected] (A. Khojasteh); [email protected] (M. work for a two-layer half-space [11] as well. Also, Kho- Rahimian); [email protected] (R. Attarnejad) jasteh et al. subsequently studied wave propagation in 2470 A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477
@2u 1 @u u 1 @2u 1 @u transversely isotropic bi-materials, tri-materials, and a c r + r r +c r 2c multilayered half-space [12-14]. 11 @r2 r @r r2 66 r2 @2 11 r2 @ Surface waves dispersion in a model of a liquid layer overlying a transversely isotropic half-space has @2u (c + c ) 1 @2u 1 @u + c r + 11 12 r + been investigated in some papers. Abubakar and 44 @z2 2 r @r@ r2 @ Hudson [15] studied the dispersive properties of sur- face waves in such a system in the two-dimensional @2u + (c + c ) z + !2u = 0; case. Sharma et al. [16,17] obtained the frequency 13 44 @r@z s r equation of a three-layer model consisting of a liquid 2 2 2 layer lying over poroelastic and anisotropic layers. @ u 1 @u u 1 @ u @ u c66 + + c11 + c44 Sharma [18] also tackled the dispersion of Stoneley @r2 r @r r2 r2 @2 @z2 waves in an oceanic crust model. In addition to 2 Kumar and Miglani [19], Kumar and Kumar [20] 1 @ur (c11 + c12) 1 @ ur 1 @ur + 2c11 + solved the problem for the proposed oceanic crust r2 @ 2 r @r@ r2 @ models. In this paper, the dispersive characteristics 2 of a three-dimensional earth crust model, consisting 1 @ uz 2 + (c13 + c44) + s! u = 0; of a homogeneous transversely isotropic elastic me- r @@z dia overlaid by a homogeneous compressible liquid @2u 1 @u 1 @2u @2u layer, are investigated utilizing the potential method c z + z + z + c z 44 2 2 2 33 2 presented by Rahimian et al. [9] and extended by @r r @r r @ @z Khojasteh et al. [10]. The velocity dispersion curves of 2 2 @ ur 1 @ur 1 @ u surface waves are plotted. Arrival times of the waves + (c13 + c44) + + are distinctly dependent on the elastic properties of @r@z r @z r @@z the bed. Analyzing this phenomenon to realize the 2 earth crust's characteristics by reverse calculation is + s! uz = 0; (1) of signi cant importance in marine seismology and where the time factor exp(i!t) is suppressed. In order underground geophysical explorations. The formula- to uncouple these equations, a couple of potential func- tion can also be utilized to derive Green's functions of tions introduced by Eskandari-Ghadi [21] are applied. stress and displacement elds for various load types These two potential functions, F and , are related to in such a system which is applicable to earthquake displacement components u , u , and u as: engineering. r z @2F (r; ; z) 1 @ (r; ; z) ur(r; ; z) = 3 ; 2. Statement of the problem and governing @r@z r @ equations 1 @2F (r; ; z) @ (r; ; z) u(r; ; z) = 3 + ; The geometry of the system and the coordinate axes r @@z @r selected for the problem are shown in Figure 1. For @2 !2 the homogeneous transversely isotropic half-space, the u (r; ; z)= (1+ )r2 + 2 + s F (r; ; z); z 1 r @z2 c time-harmonic equations of motion are expressed as [9]: 66 (2) where: @2 1 @ 1 @2 r2 = + + ; (3) r @r2 r @r r2 @2 and also:
c12 + c66 c44 c13 + c44 1 = ; 2 = ; 3 = : (4) c66 c66 c66 Rewriting the equation of motion (Eq. (1)) in terms of the potential functions yields to the two separate partial di erential equations as: @2 r2r2 + !2 F = 0; 1 2 @z2
2 Figure 1. Geometry of the system. r0 = 0; (5) A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477 2471
d2 de ning: r~ 2 r~ 2 + !2 F~m(z) = 0; 1m 2m dz2 m 1 @2 1 !2 r2 = r2 + + s ; i = 0; 1; 2; (6) i r 2 2 s @z i c66 ~ 2 m i r0m ~m = 0; (13) and: in which: = 1 0 !2 1 d2 r~ 2 = s 2 + ; i = 0; 1; 2: (14) c c im 2 2 44 11 ic66 s dz 1 = 2 = ; 2 = 1 + 1 = ; i c66 c66 The general solutions for equations in Eq. (13) are: delta 1 1 1 c = + 1 + 33 ; (7) 2 2 F~m(; z) =A ()e1z + B ()e 1z + C ()e2z s c44s2 c11s1 c11 c44 m m m m where: 2z + Dm()e ; 1 s = p : (8) 0 m 3z 3z 2 ~m(; z) = Em()e + Fm()e ; (15)
Note that s1 and s2 are not zero or pure imaginary introducing: numbers with respect to the positive de niteness of the r strain energy function [22]: 1p 2 4 2 1 = a + b + c + d + e; 4 2 2 2 c33c44s + c13 + 2c13c44 c11c33 s + c11c44 = 0: (9) r 1p = a2 + b c4 + d2 + e; The two real roots of this equation can be easily 2 2 obtained since no unknowns of odd order exist in s Eq. (9). Hence s1 and s2 are obtained by Eq. (10) as !2 2 s shown in Box I. With respect to the angular direction, 3 = s0 : (16) c66 Fourier series expansion is now applied as follows: X1 In the above terms: im [F (r; ; z); (r; ; z)] = [Fm(r; z); m(r; z)] e : 1 2 2 1 (11) a = (s + s ); 2 1 2 Also, with respect to the radial direction, mth order 1 2 1 1 Hankel transform may be expressed as: b = s! + ; h i 2 c33 c44 ~m m F (; z); ~ (; z) 2 2 2 c = (s2 s1) ; Z 1 = [F (r; z); (r; z)] rJ (r)dr; 1 1 m d = 2 !2 + (s2 + s2) 0 s 1 2 c33 c44 [F (r; z); (r; z)] c11 1 1 Z 2 + ; 1 h i c33 c11 c44 ~m m = F (; z); ~ (; z) Jm(r)d: (12) 0 2 2 4 1 1 Applying Hankel transform and Fourier series expan- e = s! : (17) c33 c44 sion, one may arrive at the two following ordinary di erential equations for transformed components of The unknown functions Am, Bm, Cm, Dm, Em, and and F and as [9]: Fm will be determined from the boundary conditions.
s p 2 2 2 2 (c13 + c13c44 c11c33) (c13 + c13c44 c11c33) 4c33c11c44 s1; s2 = (10) 2c33c44
Box I 2472 A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477
, , and are multi-valued functions and are made X1 1 2 3 '(r; ; z) = ' (r; z)eim; (21) single-valued by specifying the branch cuts emanating m p 1 from the branch points = ! s=c11, = p p1 2 Z ! =c , and = ! =c on the complex - 1 s 44 3 s 66 m '~ (; z) = '(r; z)rJm(r)dr; (22) plane, such that the real parts of 1, 2, and 3 remain 0 non-negative for all values of [10]. The positive sign of these branch points denotes the wave numbers of where the inverse transform can be used to return to the body waves, P, SV, and SH, respectively. Unlike the real domain as: an isotropic media, transverse isotropy of the medium Z 1 m causes di erent velocities for SV and SH waves in this '(r; z) = '~ (; z)Jm(r)d: (23) kind of solids [5]. The Fourier components of the 0 transformed stress and displacement can be expressed The same is followed for stress and displacement com- as [10]: ponents. The respective ordinary di erential equation m of motion in terms of' ~m can be obtained as: 2 2 m d s! 2 u~ = + (1 + ) F~m; 2 2 zm 2 2 1 m ! d dz c66 l 2 + '~m = 0: (24) K dz2 m dF~m m+1 m+1 m m A general solution to this equation is: u~r + iu~ = 3 i ~m; m m dz '~m = S ()e 4z + R ()e4z; (25) ~m m m m m 1 m 1 dFm m u~r iu~ = 3 i ~m; m m dz where: s d2 !2 m+1 m+1 2 2 ~zr +i~ = c44 ( 3 2) + (1 + 1) 4 = : (26) m zm dz2 2 cl p !2 d ~m s ~m m Here, cl = K=l is the velocity of the compressional Fm c44i ; c66 dz wave in the liquid, and Sm and Rm will be calculated from the boundary conditions, where Rm is associated d2 with the amplitude of the transmitting wave from ~m 1 i~m 1 = c ( ) + 2(1 + ) zrm z 44 3 2 2 1 m dz the bottom boundary, and Sm is associated with the re ecting wave amplitude from the free surface of the 2 m s! ~m d ~m liquid. Fm c44i : (18) c66 dz Fourier factors of stress and displacement compo- nent in the transformed mode are written as: For the compressible liquid, the governing equation of ~ 2 m time-harmonic motion in cylindrical coordinate system ~zz = P = l! '~m; is [1]: m + 1 m 1 u~m = '~m 1 '~m+1 ; K rm 2m m 2m m r2'(r; ; z) = !2'; (19) l m m @'~m u~z = ; where ', K, and l are the displacement potential, m @z bulk modulus, and density of the liquid, respectively. i The displacement components and pressure can now be u~m = '~m 1 +' ~m+1 : (27) expressed in terms of ' as: m 2 m m
@' @' 1 @' 3. Boundary conditions and frequency u = ; u = ; u = ; r @r z @z r @ equation
2 2 P = Kr ' = l! ': (20) The boundary conditions for the problem under con- sideration are given below. The expressions provided Similar to the solid substratum, the procedure of in Eqs. (18) along with Eqs. (27) are utilized. The Fourier series expansion is followed with respect to coecients Am, Cm, and Em are omitted according to the angular direction, and applying mth-order Hankel the radiation condition. transform is followed with respect to the radial direc- The free surface condition requires the vanishing tion: of the liquid pressure, P : A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477 2473