Two-dimensional deformation of an orthotropic elastic medium due to seismic sources

PHYSICS OFTHE EARTH

AND PLANETARY INTERIORS

ELSEVIER Physics of the Earth and Planetary Interiors 94 (1996) 43-62

Two-dimensional deformation of an orthotropic elastic medium due to seismic sources

Nat Ram Garg *, Dinesh Kumar Madan, Raj Kumar Sharma Department of Mathematics, Maharshi Dayanand University, Rohtak-124001, India

Received 28 September 1994; revision accepted 3 August 1995

Abstract

The integral representation of two-dimensional seismic sources causing the antiplane strain deformation of an orthotropic elastic infinite medium has been obtained. Using this integral representation, the analytical expressions for the deformation of an orthotropic layered elastic medium due to a very long inclined strike-slip fault have been calculated, where the interface is horizontal and parallel to one plane of elastic symmetry. These expressions have been used to study the effect of the source location, dip of the fault and anisotropy of the medium on the horizontal displacement parallel to the fault. The variation of the horizontal displacement with the distance from the inclined fault is studied for three source locations: a surface-breaking fault and two buried faults. Both the source location and the dip of the fault are found to influence significantly the horizontal displacement at any point of an orthotropic elastic medium. Further, the horizontal displacement of an orthotropic elastic layered medium may differ significantly from that of an isotropic elastic layered medium.

1. Introduction

The static deformations of semi-infinite elastic isotropic media due to very long str ike-sl ip faults have been studied by many researchers, e.g. Kasahara (1960, 1964), Rybicki (1971, 1978) and others. The results of these studies have been successfully applied to several earthquakes, e.g. 1906 San Andreas, 1927 Tango, 1948 Fukui and 1966 Parkf ie ld-Cholame. In the case of long faults, two-dimensional approximation is useful when one is considering the deformation away from the edges of the fault. In the two-dimensional approximation, the algebra is simplified to a great extent and one gets a closed-form analytical solution.

Most anisotropic media of interest in seismology have, at least approximately, a horizontal plane of symmetry. A plane of symmetry is a plane in which the elastic propert ies have reflection symmetry. A medium with three mutually orthogonal planes of symmetry is known as orthorhombic symmetry. The

* Corresponding author.

0031-9201/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0031-9201(95)03095-6

44 N.R. Garg et aL / Physics of the Earth and Planetary' Interiors 94 (1996) 43-62

orthorhombic symmetry of the upper mantle is believed to be caused by orthorhombic crystals of olivine aligned relative to the spreading centres (Hess, 1964). Orthorhombic symmetry is also expected to occur in sedimentary basins as a result of the combination of vertical cracks with a horizontal axis of symmetry, and periodic thin-layer anisotropy with a vertical symmetry axis (Bush and Crampin, 1987).

When one of the planes of symmetry in an orthorhombic symmetry is horizontal, the symmetry is termed as orthotropic symmetry (Crampin, 1989). Since the orientation of stress in the crust of the earth is usually orthotropic, most symmetry systems in the earth's crust also have orthotropic orientations. The orthotropy symmetry is also exhibited by olivine and orthoyroxenes, the principal rock-forming minerals of the deep crust and upper mantle.

A transverse isotropy has an axis of cylindrical symmetry. This is usually the result of parallel cracks, cracks with co-planer normals or aligned grains. Pan (1989) provided a unified solution of the static deformation of the transversely isotropic and layered elastic half-space by seismic sources.

In the present paper we have obtained the integral representation of two-dimensional seismic sources in an orthotropic elastic infinite medium. Using the integral representation, we have then obtained the analytical solution for the deformation of an orthotropic elastic layered half-space as a result of a very long inclined strike-slip fault situated either in the orthotropic elastic layer or in the orthotropic elastic half-space. Both the problems have been studied in detail. The layer of uniform thickness and the half-space are assumed to be in welded contact at the horizontal interface. The solution obtained here may find application to earth models consisting of a lithosphere (representing crust and upper mantle of the earth) lying over an asthenosphere.

In order to study the effect of the orthotropy of the elastic medium in comparison with the isotropy, we compute the horizontal displacements caused by a very long inclined strike-slip surface-breaking fault placed in the layer for both types of layered elastic medium - entirely orthotropic or completely isotropic. By considering three values of the dip angle, namely, a = ~-/6, rr/3, rr/2 and by computing the horizontal displacements for each a due to a very long surface-breaking inclined strike-slip fault in an orthotropic elastic layered medium, the effect of the dip of the angle is studied. To examine the effect of the fault location, surface horizontal displacements for an orthotropic elastic layered medium are computed for a = rr/6, ~r/2 and three source locations: a surface-breaking fault and two buried faults. When the source lies in the orthotropic elastic half-space, the effect of the dip angle is studied by computing the surface horizontal displacements for a = rr/6, ~'/3, rr/2. For a very long vertical strike-slip fault in the orthotropic elastic half-space, three curves showing the horizontal displacements for z = 0, H/2, H have been drawn.

2. Bas ic e q u a t i o n s

The equilibrium equations in the cartesian co-ordinate system (x 1, x2, X3) for zero body forces are

0Tll 0T12 0T13 - - + - - + - - = 0 (1 ) ~X 1 ~X 2 ~X 3

~"/" 21 ~'r22 ~T23 - - + - - + - - = 0 (2) OX 1 OX 2 ~X 3

0'/'31 ~T32 i)'/'33 - - + - - + = 0 ( 3 ) OX 1 ~X 2 ~X 3

N.R. Garg et al. / Physics of the Earth and Planetary Interiors 94 (1996) 43-62 45

where ~'ij is the stress tensor. Let (Ul, u2, u 3) denote the displacement components. The strain-displacement relations are

ll u, I e i j= 2 ~ a x j + ~ ] l < i , j < 3 . (4)

For an orthotropic elastic medium, with co-ordinate planes coinciding with the planes of symmetry and one plane of symmetry being horizontal, the stress-strain relation in matrix form is

-711 cll c12 c13 0 0 0 ell

T22 C12 C22 C23 0 0 0 e22

'/'33 C13 £23 C33 0 0 0 e33

,r23 0 0 0 c ~ 0 0 2e23 (5 )

"/'13 0 0 0 0 C55 0 2e13

T12 0 0 0 0 0 C66 2e12

where the two-suffix quantities %. are elastic constants of the medium. A transversely isotropic elastic medium, with the xa-axis coinciding with the axis of symmetry, is a

particular case of an orthotropic elastic medium for which

C22 ~ Cll, C23 ~ C13, C55 -~- C44,

1 C66 = "~ ( e l l -- C12 ) (6)

and the number of independent elastic constants reduces from nine to five. When the medium is isotropic

C11=C22 =C33=A + 21d,, C12=c13=C23=A,

C44 = C55 = C66 =/.g (7) where A and ~ are the Lamd constants.

We consider the antiplane strain problem in which the displacement vector is parallel to the xl-axis which is taken to be horizontal and O/Oxl -- O. u 1 = u l (x z, x3) is the only non-zero component of the displacement vector. In the following, we write u for Ul and (y, z) for (x z, x3).The non-zero stresses can be written as

Ou Ou 3"12 CO~2 ~--~ = z13 = c - - ( 8 ) ' i~Z

where

C66 = CO~ 2 , C ----- C55. (9)

The values of a and c depend upon the elastic constants. We assume that a and c are positive real numbers.

The equilibrium Eqs. (2)-(3) are identically satisfied for the antiplane deformation and (1) reduces to

02U 1 02U + - - - - -- 0. (10)

Oy 2 ot 2 i}Z 2

In case of an isotropic elastic medium, c =/z and a = 1.

46 N.R. Garget al. / Physics of the Earth and Planetary Interiors 94 (1996) 43-62

3. Line-source in an infinite medium

Let a force of magnitude F, per unit length, act at the point Q(~2, ~3 ) in an infinite homogeneous orthotropic elastic medium in the positive x-direction.

A suitable solution of (10) is of the type

u =A log[( y - ~2) 2 + a2( z - ~3) 2] (11)

where the coefficient A is to be determined. The solution (11) satisfies the equilibrium Eq. (10) except at the point Q(~:2, ~:3). The boundary condition to be satisfied by the resulting tractions is (Maruyama, 1966)

~' l ' lkV k dtr = F (12)

in which the integral is the line integral surrounding the point Q and b, k denotes the direction cosines of the exterior normal. From (11) and (12), it has been found that

- F A = 47rac (13)

Thus

- F u = 41ra-----~ log[( y - ~2) 2 + a2( z - ~3) 2] (14)

is the displacement, parallel to x-axis, at any point P(y, z) of an orthotropic elastic infinite medium due to a line-source, parallel to the x-axis, passing through the point Q(~2, Ca).

Eq. (14) coincides with the corresponding equation for an isotropic elastic medium obtained by Maruyama (1966) provided the rigidity of the isotropic medium is replaced by a c = (c55. c66) 1/2 and the depth co-ordinate is taken as a z = (c66/c55)1/2.z. This conclusion is similar to the earlier observations made by Anderson (1962) and Singh (1986) regarding the deformation of transversely isotropic elastic media.

4. Integral representation of two-dimensional buried sources in an infinite medium

The displacement u, given in (14), can be expressed in an integral form as (see Appendix)

u ~1~= - - e -k'~lz-¢31 cos k ( y - ~ 2 ) dk (15) 2 rr ac

where the superscript (1) denotes the displacement due to a concentrated line force F, per unit length, acting in the x-direction.

4.1. Single couples (12) and (13)

The single couple (12) is a couple in the xy-plane with equal and opposite forces in the x-direction with its arm in the y-direction. Similarly, (13) is a single couple in the xz-plane with forces in the x-direction and arm in the z-direction (Fig. 1).

N.R Garg et aL / Physics of the Earth and Planetary Interiors 94 (1996) 43-62 47

S x y .'--Y •

COUPLE (12) COUPLE (13)

Z Z

(a) (b} Fig. 1. The single couples (12) and (13).

We consider two forces of the same magnitude, F, acting at points (~:2 + 6/2, ~3 ) and (~2 - 6/2, ~:3) along the positive and negative directions of the x-axis, respectively. The resultant displacement u t12), parallel to the x-axis, due to these two forces is found to be

E F oo _ _

u(t2)= 2~'ac e k~lz e31 sin k ( Y - ~ 2 ) dk. (16)

Taking the limit as F ~ ~, ~ ~ 0 in such a way that the product ~F tends to a finite quantity, say F12. Then F12 is the moment of the couple (12). Therefore, the displacement u (12) at the point P(y, z) due to a couple (12) of moment FIE at the point Q(~:2, ~:3) is

F12 f e -k~'Iz-¢31 sin k ( Y - ~ 2 ) dk. (17) u02) = 27rca "0

Similarly, the displacement u ¢13), parallel to x-axis, due to a line source (parallel to x-axis and passing through Q) at every point of which a couple (13) of moment El3 acts, is found to be

oo

++-F13[ e -k"lz-~31 cos k ( y - ~ 2 ) dk (18) u°3)= 2~rc "0

where the upper sign (plus) is for z > ~3 and the lower sign (minus) is for z < ~:3.

Table 1 Source coefficients for various seismic sources

Source A 0 Bo

F Single concentrated force acting in positive x-direction 0

2or ack F~2

Single couple (12) 2~rca 0

+ Fl3 Single couple (13) 0

2¢rc

The upper sign is for z > f3 and the lower sign for z < ~3.

48 N.R. Garg et al. / Physics of the Earth and Planetary Interiors 94 (1996) 43-62

The displacement parallel to the x-axis, due to either a line source or single couple (12) or couple (13), can be unified into the following integral

+,0 = f : [ A 0 sin k ( y - ~ : 2 ) +Bo cos k ( y - ~2)]e -kalz-~31 dk. (19)

The source coefficients A 0 and B 0 for various two-dimensional buried sources are given by Table 1.

5. Deformation of a layered elastic half-space

We consider a semi-infinite medium consisting of a homogeneous orthotropic elastic horizontal layer of thickness H lying over a homogeneous orthotropic elastic half-space. The origin of the cartesian co-ordinate system (x, y, z) is placed at the upper boundary of the semi-infinite medium with the z-axis vertically downwards (Fig. 2). The layer, therefore, occupies the region 0 < z < H and half-space z ___ H. It is assumed that the layer and the half-space are in welded contact at the interface z = H which is parallel to one plane of elastic symmetry. This implies the continuity of the displacement u and the traction 7-13 across the horizontal plane z -- H. The boundary conditions at the interface z = H are

u ( z = H - ) = u ( z = H + ) ,

~13(z = H - ) = ~'13(z = H + ). (20)

The bounding surface z = 0 is a horizontal plane and is a plane of elastic symmetry. It is further assumed that the bounding plane z = 0 is traction-free. So that

713 = 0 at z = 0. (21)

We shall determine the deformation of the semi-infinite elastic medium due to a very long str ike-sl ip fault lying either in the layer or in the half-space. We note that the source coefficient B 0 (given in Table 1) changes sign with z ~ ~3. We write B01 for B 0 for z < ~3, therefore, B 0 = - B o 1 for z > ~3.

z-O 0 ) y

T Orthotroplc H elastic Layer

~¢1'Cl ,~ Z=H

,c2,c 2 Orthotrop¢ elastic half-sPace

Fig. 2. x = 0 section of the layered orthotropic elastic medium.

N.R Garg et al. ~Physics of the Earth and Planetary Interiors 94 (1996) 43-62 49

5.1. Line-source in the layer

When the line-source lies in the layer, suitable expressions for the horizontal displacement, parallel to the line source and satisfying (10), in the layer and the half-space, are

sin k ( y - ~ 2 ) + B 1 cos k ( y - ~ 2 ) ] e -'~lkz

+ [C 1 sin k ( y - ~ 2 ) +D1 cos k ( y - ~ 2 ) ] e "~kz} dk (22)

u (m = fo~[A2 sin k ( y - g 2 ) +B2 cos k ( y - ~2)]e -'~kz dk (23)

with u 0 given by (19). The superscript (L) is used to indicate the layer and (H) for the half-space. The coefficients A~, B~,

etc. are to be determined from the boundary conditions given in (20) and (21). We find

A x = A[e -k~'~3 + T e-k~l~2n-~3)]Ao

B 1 = A[e -k~le3 - T e-kal (2H-~9]B 1

61 = AT[ e-kam(2H+~3) + e-kal(2H-~a)]Ao

D l = AT[ e-k~(2n +#9 _ e-k,,l(2n-#9 ]B~

A 2 = A(1 + T)e -k (a l -a2 )H[e -kalIi3 4- eka~¢3]Ao

B 2 = A(1 + T)e-k('~'-'~2)n[e -kale3 -- ek'~e3]B 1 (24)

where

T = ( S - 1 ) / ( S + I ) , S=Qal/c~a2, A -1 --- (1 - Te-2k~n). (25)

The substitution of the values of A1, B 1, etc. from (24) and (25) in (22) and (23), gives the integral expressions for the displacement in the layer and in the half-space due to a line-source, parallel to the x-axis and passing through Q(g2, ~3), situated in the layer. Evaluating these integrals analytically (see Appendix), we find

u(L)= ( Y - ¢ 2 ) A o + a l l z - ¢ a l B o ~ ( y - ~ 2 ) a o + a t ( E n H + ~ 3 + z ) a ~ + ~.,T n ( Y - ~:2)2 + [a l (z - ~:3)] 2 n~O (Y - ~:2)2 + [a,(2nH+e3+z)] 2

+ ~ Tn[(y_--~2)AO-al(2nH-e3 +z)B~ ~=1 [ (Y-*2)2+[ai(2nH-~a+Z)] 2

+ (Y-~2)Ao+al(2nH+~3-z)B~ ( Y - ~ 2 ) A o - a i ( 2 n H - ~ 3 - z ) B 11

+ (y_~2)-----~+-[ai(2nH------~, 3 z)] 2 (26a)

. I

for 0 < z < H and for z > H

u(n)= ~ ( l + T)Tn[ ( ( y '2)A° + [al(2nH + H + '3) + a2( z - H)]B~ , , = 0 + [a l (2nH + H + + a2( z

+ (Y - ~:2)Ao - [al(2nH + H - ~¢3) + a2(z - n ) ] no 1 ] (26b)

( y - ~ 2 ) 2 + [al(2nH+H-!/,3) + a 2 ( z - H ) ] 2 ]"

50 N.R. Garg et al. / Physics o f the Earth and Planetary Interiors 94 (1996) 43-62

On putting a I = a 2 = 1, c 1 =/z I and c 2 = ttz 2 in (26a) and (26b), expressions for the displacements at any point of an isotropic layered half-space are obtained which coincide with the corresponding expressions obtained by Sharma and Garg (1993).

5.2. Line-source in the half-space

In this case, the suitable expressions for the horizontal displacement, u, in the different parts of the elastic media are

u (L) = fo®{{L1 sin k( y - ~2) + M1 cos k( y - ~2)}e -k'lz

+{P, sin k ( y - ¢ z ) + Q , cos k ( y - gz)}e k~'z} dk (27)

and

oo

u ( m = U o + f o [L z sin k ( y - ~ 2 ) +ME cos k ( y - ~ 2 ) ] e -k'~:z dk (28)

with u0 given in (19). Applying the boundary conditions given in (20) and (21) to (27) and (28), we find

L1 = P1 = A(1 - T)e-ka'Heka2(H-~3)Ao

M1 = QI = A(1 - T)e-k~lHeke'2(n-~3)B 1

L 2 = A[e -2k(~t-°t2)He-ka2~3 - - T eka2(ZH-¢s)]Ao

M 2 = A[e -2k(al-°t2)He-k°t2sc3 -- T eka2(2H-¢s)]B 1. (29)

The closed-form expressions for the displacement u in the semi-infinite elastic layered orthotropic medium due to a line source, parallel to the x-axis, passing through the point Q(62, ~3) and situated in the half-space are obtained by integrating the integrals given in (27) and (28) after putting in the values of various coefficients from (29). We obtain

u(C)= E ( 1 - T ) T ~ ( y - ~ : 2 ) A ° + [ a ' ( 2 n H + H + z ) + a 2 ( ~ 3 - H ) I B ~

n=0 ( y -- ~2) 2 + [al(2nH + H + z) + 0d2(~3 - n ) ] 2

(y - s~2 )Ao + [al(2nH + n - z ) + a2 (~3 - H ) ] a ~ + - -- - - ] (30a)

( y - s~2) 2 + [a, (2nH + H - z ) + a2(s~3 - H) ] 2


u(n) = ( y - ~2)Ao + a 2 ] z - ~31Bo _ T ( y - ~2)Ao - [ a2 (2H - z - ~3)] Bo 1

( y - s ~ 2 ) 2 + [ a 2 ( z - s~3)] 2 ( y - s~2)2 + [ a 2 ( 2 n - z - ¢ 3 ) ] 2

oo

+(1 - r 2) E Tn (y - {~2)A° + [2a , (n + 1 ) n - a 2 ( 2 n - z - ~:3)]a I (30b)

n=o (Y - ~:2)2 + [2Otl(n + 1 ) H - o t 2 ( 2 H - z - ~ 3 ) ] 2

The deformation of a uniform orthotropic elastic half-space due to a buried line-source can be obtained either from (26a,b) or from (30a,b) on putting a 1 = a2 and Cl = c2. The expressions for the displacement at any point of the layered isotropic medium due to a line-source in the half-space can be reduced from

N.R. Garg et al. /Physics of the Earth and Planetary Interiors 94 (1996) 43-62 51

above, on putting al = a2 = 1, c I ---/z 1 and c 2 =/z 2. We note that these reduced expressions for the displacement coincide with the corresponding expressions for an isotropic medium obtained by Sharma and Garg (1993).

6. Inclined strike-slip fault

Taking the x-axis along the strike of the fault and z-axis, vertically downwards (Fig. 3(a)), it has been shown by Maruyama (1966) that the displacement vector ~ ' - (u, 0, 0) due to a very long strike-slip fault of arbitrary orientation can be expressed as the line integral

u = fLAuG knk ds (31)

where the summation convention has been used (k = 2, 3 only). Au is the discontinuity of the displacement vector in the x-direction, n k is the unit normal to the fault section L and the functions G~k are given by (Maruyama, 1966)

OU (1) OU (1)

G~2 = c66 0~:2 G~3 : c55 0~3 (32)

in which u (t), given by (15), is the displacement due to a concentrated line-force of unit magnitude, per unit length, acting in the x-direction.

Writing Au = b, n 2 = - s i n 8, n 3 = COS 8, where 8 is the dip of the fault (Fig. 3(b)), (31) becomes

u : [b(Gl3 cos 8 - G ] 2 sin 8) ds. (33) "L

~-~y 0 , 0

Z ~ " ~'~ Z

(a) (b) Fig. 3. (a) Geometry of an inclined str ike-sl ip in block diagram. The displacement discontinuity (b) on the inclined fault is parallel to the x-axis. (b) x = 0 section of the semi-infinite elastic medium. The cartesian co-ordinates of a point on the inclined fault a r e

(s cos 8, s sin `5), where ,5 is the dip angle and s I ~ s < s 2. The sign • indicates displacement in the direction of the x-axis, the sign e in the opposite direction.


In the case of a line dislocation, we can do away with the integration in (33). The displacement u is then obtained simply by multiplying the relevant Green 's function by bds. Thus, for a line dislocation, we write

u = b d s ( G ~ 3 cos 6 - G ~ 2 sin 6). (34)

Thus, the Green 's function G~2 corresponds to a very long vertical right-lateral s tr ike-sl ip fault (6 = 90 °) and represents, up to a dimensional constant, the x-component of the displacement at the point P (y , z) due to a single couple (12) at the point Q(~2, ~3). Similarly G~3 corresponds to slip (6 = 0 °) on a very long horizontal plane and represents, up to a dimensional constant, the x-component of the displacement at the point P(y , z) due to a single couple (13) at the point Q(~2, ~3). The relation (34) also shows that the field due to a very long str ike-sl ip line fault of arbitrary dip 6 can be expressed in terms of two fields - one due to a very long vertical strike-slip fault and the other due to a slip on a very long horizontal fault.

6.1. Deformation due to a line-source o f strike-slip type on a vertical plane

The displacement u t~s) due to a str ike-sl ip line-source on a vertical plane is given by (32) and (34). We obtain

- a b d s ~ _ _ - - £ u(~') 2~r e k~l~ e31 sin k ( y - ~ 2 ) dk . (35)

0 . ~ L

O'l- A . . ~ j O, B

- 0 , l -

- 0 . 2 -

I - 0 . 3 -

- O . ~ - SOURCE ]'

-0 .5 - z = 0,~=~I'16

-0 .6-

-0.7-

- 08-

-0,9- , , -2H 0 2H l~H 6H

~ y , , z-

Fig. 4. Variation of dimensionless horizontal displacement U = u/b with the distance from the fault y for orthotropic and isotropic elastic media for Source I (s I = 0, s 2 = H/2) lying in the layer, z = 0 and 8 = ~r/6. A denotes the curve for the orthotropic elastic medium and B for the isotropic elastic medium.


Comparing (35) with (17), we conclude that the single couple (12) represents a very long vertical strike-slip fault provided the moment F12 of the single couple (12) is given by

F12 = -co t2bds (36)

6.2. Deformation due to a line-source slipping on a very long horizontal plane

Eqs. (32) and (34) determine the displacement u (Hs) due to a line source slipping on a very long horizontal plane, which is given by

u ( m ) = +OaSf=e-k~ ' l~ -e3 ' k ( y - ~ 2 ) dk. (37) 2zr -10 cos

On comparing (37) with (18), it is observed that a line source with slip on a very long horizontal plane is represented by the single couple (13) whose moment F13 is given by

Fl3 = cbds. (38)

6.3. Deformation due to an inclined strike-slip line-source

The deformation u (m at any point of an elastic medium due to an inclined line source of strike-slip type with arbitrary dip 8 is

u (m = cos 3u (m) + sin 8u (°~) (39)

0.11. 0.1" ~ I

0.8- SOURCE I 0.6" z=/'/12, ~ =.n'16

0.~-

'[ 0.2" " ~ 0 |

I -0.2- - 0 . ~ - B ~

-0.6- - 0.8- -01-

I l -2H 0 2H ~H 6H ~ y )

Fig. 5. Variation of U with y for an orthotropic and isotropic elastic media for Source I, z ffi 1-1/2 and 8 = ¢r/6. Notation as in Fig. 4.

54 N.R. Garg et aL /Physics of the Earth and Planetary Interiors 94 (1996) 43-62

in which u (m) and u (~) denote, respectively, the displacements due to line sources on very long horizontal and vertical planes.

7. Deformation due to a very long inclined strike-slip fault of finite width lying in the layer

The deformation of a layered orthotropic elastic medium as a result of a very long inclined strike-slip line source situated in the layer is to be determined from (26a,b), (36), (38), (39) and Table 1. We obtain

b°tl( )-~ T~[(2nH+z-'3) c ° s ~ - ( Y - ~ 2 ) s i n 6 (2nH+z+'a) COS~+(y-,z)sin~ ] u(/S) = 2"/'/" n=0 t ( Y -- e--2 )'-~ 7 [ 0t l-~2"-nH +S ~ ~--7) ] --~ - (-7 -- ~2"):2 7 [ t~ 1"-~ 2-n~/-~ ; + b¢3---) ] -'-~

+ ~P,T ~ (2ng - z -~a ) C°S6-(Y-~2) sin~ (2nI-I-z+~3) cos6+(y-~2)s in6 n=l ( y _ t 2 ) 2 + [al(2nH_z_~3)] z - ( y _ sCz)2 + [oq(2nH_z+s~3)]2 ds

(4Oa) for 0_<z < H and for z >_H

-b ~ [al(2nH + H + ~3) + aE(Z - H ) ] cos ~ + al( y - ~2) sin ~

u ( m = 2rr = ( y - ~--~ + [a,(2nH + H + ~3) +az(z-H)] 2

_ [a l (2nH+H-'3)+a2(z-H)] c o s 6 - a l ( Y - ~ z ) s i n 6 ] }

( y _ ~ 2 ) 2 + [a,(2nH+H_~3 ) +az(z_H)]2 ds. (40b)

0.0!

SOURCE I

z = H , 6 = X / 6

l ~ 0-

I-

- O.OB- , , -2H 0 2H ~H 6H

Fig. 6. VariationofU withy ~rano~hotropicandi~tropicelasticmedia~rSourcel, z= H andSf~/6 . Notation ~inFig. 4.

N.R. Garg et al. ~Physics of the Earth and Planetary Interiors 94 (1996) 43-62 55

TO obtain the deformation of the orthotropic elastic medium due to a very long inclined strike-slip fault of finite width L, we follow Rani and Singh (1992). Using the polar co-ordinates (s, 6) of a point on the fault, we substitute (Fig. 3(b))

~2 = S COS 8 , ~3 = S sin 6 (41)

into (40a,b), and then integrate the resulting equations over s between the limits (s 1, s2), so that L = s 2 - s 1. We find

b ( ~ Tn[tan_l (COs2 8 + a 2 sin 2 8 ) s - [ y cosS+a2(2nH+z)s in 8]

u(ts) = 2--~ al[(2nH + z ) cos 6 - y sin 6] n = 0

_ t a n _ l (cos 2 8 + a 2 sin 2 8 ) s - [ y cos 6 - a 2 ( 2 n H + z ) sin 8]

al[(2nH+z ) cos 8 + y sin 8]

+ E r ~ tan-' (c°s~ 8 + ~ sin ~ 8 ) ~ - [y cos 8 +~,~(2, ,~'-z) sin 81 .ffix L cq[(2nH-z) cos 8 - y sin 6]

(cos2 8+a21sinZ a ) s - [y cos S -a21(2nH-z ) sin 6] ]}ls~ (41a)

- t a n - I oq[(2nH-z) cos 8 + y sin 8] ~

0o ~

O . k - ~ 0.3- ~ =)112 SOURCE !

Z=O 0.2- 0.1-

O-

T -0.1- 0.2- 6:~

-0 .3 -

-0 .g-

-0 ,5 -

- 0 . 6 -

-0.7- :~/ ~='~/6 / -0.8-

-o.g. i 2H 0 2H ~H

~ y ) Fig. 7. Variation of U with y for an orthotropic elastic medium for three values of dip angle 8 (8 = ~r/2, ~r/3, ~'/6) when z = 0 and source is of type I.


for 0 <z < H and for z > H

- b { ~__0(1 [ (cos2 8 +a~ sin 2 6 ) s - [ y cos S - [ a l ( 2 n + l ) H + a 2 ( z - H ) ] a l sin 6] u(l~) = 2rr + T)Tn tan-X a l y sin 6 + [al(2n + 1 ) H + a2(z - H ) ] cos

\ - - I_

+ tan-~ ~ ~ 6--- [-a~-~n + 1 - ~ + ~ z --)-/--~ cos ~ (41b) $1

where

f(s) ;21~-f(s2)--f(Sl). ( 4 1 C )

The results for the corresponding problem for an isotropic elastic medium can be obtained as a particular case of the above results by taking a 1 -- a 2 = 1, c I =/Zl, c z --/~z and T = (/~x -/~2)/(/xl +/~2) which coincides with the results obtained by Singh and Rani (1994). When the fault is a very long vertical strike-slip type, the results for the corresponding problem for an isotropic elastic medium coincide with the results given by Sharma and Garg (1993) on taking 6 -- 90 °.

s00L%i

I

! '-2H 0 2H I,H

~ y •

Fig. 8. Variation of U with y for an orthotropic elastic medium for three values of dip angle 8 (8 ffi 7r/2, ~r/3, ~ ' /6 ) when z = H / 2 and source is of type I.


8. Deformation due to a very long inclined strike-slip fault of finite width lying in the half-space

The deformation of a layered orthotropic elastic medium as a result of a very long inclined strike-slip line-source situated in the half-space can be determined from (30a, b), (36), (38), (39) and Table 1. We obtain

- b ( ~ ° [ ° q ( 2 n H + H + z ) + a 2 ( ~ 3 - H ) ] c°s 3+a2(Y -~2 ) sin 8 I'l(Is)-- 2 H (l-- T)T" (y -£2 )2 + [a , (2nH+H+z) + o t2 (~3 -H) ] 2

[ a l ( 2 n H + H - z ) + a z ( ~ 3 - H ) ] cos 6 + a 2 ( Y - ~:2) sin B ]) + ( y - ~2) 2 + [ a - ~ n H ~ z-) + "a'~(-~3: H- ~ ds (42a)

for O_<z < H and for z_>H

ba 2 [ ( z - ~3) cos 6 - ( y - ~:2) sin 6 _ T ( 2 H - z - st3) cos 6 - ( y - ~2) sin 6 ]

u(m = 2--~-~ [ ?Y -~:~~ + [--a 2--( z --- ~33 ~ ~ (Y - ~:2)2 + [a~(2H- z - ~:3)] 2 a ds

b ( 1 - T)2[ ~ T n [2a l (n + 1 ) U - a z ( 2 H - z - e 3 ) ] cos 8 +a2(Y- , s¢2)s in 6 ] 2~" L.=o ' ( y - sO2)2+ [2oq(n + 1 ) H - a z ( 2 H - z - ~ 3 ) ] 2 ds. (42b)

0.2"

0 .1 -

0 . . . .

- 0.1- - 0 . 2 - U

T - 0 . 3 -

-0.5- -0.6- / -0.7- - 11.8- - o . g . '" I i i ,

- 2 H 0 2H ~H

7

Fig. 9. Variation of U with y for an orthotropic elastic medium for three types of sources (I, H, 1]1) when z = 0 and 8 = ~-/6.


The deformat ion of the layered or tho t rop ic elastic m e d i u m due to a very long s t r ike-s l ip fault of finite width L lying in the half-space is found to be

-b (n~= ° [ (c°s26+azsin26)s-[yc°s6-a2sinS[al(2nH+H+z)-azH]] u ( m = 27r ( 1 - T)T n tan -1 yot 2 sin 6 + [a~(2nH+H+z) - a z H ] cos 6

+ tan_l (COS26 +aZ sinZ6)s- [y cos 6-a2 sin 6[a,(2nH + H-z) -a2H]] ]) s;2 ya 2 sin 6 + [ai(2nH + H-z) -a2H ] cos 6 (43a)


-b[ (cos26+aZsin26)s-(ycos6+aZ zsin6) u ( m = 2~" t a n - 1 a 2 ( y sin 6 - z cos 6)

- r tan -1 (c°s26 + az? sinZ6)s- (y cos 6 + a~(2H-z)sin 6)

a2[Y sin 6 - ( 2 H - z ) cos 6] oo

+(1- T 2) E T n t an - l n=O

(cosZ6 + a2 z sin26)s - [ y cos 6 - [ 2 a l ( n + 1)H-az(2H-z)]a2 sin 61 ~ . × s, (43b) y a 2 sin 6 + [ 2 a l ( n + 1)H-az(2H-z)] cos 6

0.5

0.W- z=O,~=~/2

0.3-

0,2-

l 0.1-

~ O"

- 0 , 2 -

- 0 3 -

- OAe-

- 0 . 5 ' , -2H 0 2H itH

y ),

Fig. 10. Variation of U with y for an orthotropic elastic medium for three types of sources (I, II, III) when z = 0 and a = ~r/2.

N.K Garg et al. ~Physics of the Earth and Planetary Interiors 94 (1996) 43-62 59

The deformat ion of a uni form or thot ropic elastic half-space as a result o f a very long inclined s tr ike-sl ip fault of finite width L may be obta ined f rom ei ther (41a,b) or (43a,b) on put t ing a l = a2 = a (say), c 1 = c 2 = c (say) and T = 0.

9. Numerical results

We wish to examine the effect of the anisotropy of the elastic medium, the dip of the fault and the location of the fault upon the displacement u (Is) due to a very long inclined str ike-sl ip fault of finite width and infinite length. This fault lies ei ther in the layer or in the half-space. We examine both cases. For numerical computat ion, we define the dimensionless displacement U through the relat ion

u (Is) = b. U (44)

and use the values of the elastic constants given by Love (1944). For an or thotropic layered medium, the values of the constants ot 1 = 0.99, c I -- 13.24 × 1011 dyne cm -2 (Topaz) and o~ 2 = 0 . 9 8 , C 2 = 2.87 × 1 0 1 1

dyne cm -2 (Barytes) have been considered. For an isotropic layered medium, the values o f the constants a 1 = 1, c t = 4.47 × 1011 dyne cm -2 (Copper) and a 2 = 1, c 2 = 2.40 × 1011 dyne cm -2 (Glass) have been

used. W h e n the fault lies in the layer, we consider the following three positions of the fault, namely,

S o u r c e I : S l = 0 , s 2 = H / 2

S o u r c e I I : s l = H / 4 , s 2 = 3 H / 4

S o u r c e I I I s t = H / 2 , s 2 = H .

0.0~,-

0.03- ~ , / ~ 12 SOURCE:Sl= 2 H ,S2: 3H

0.02- ~ Z : 0

0 o . , .

- ° ' ° 1 - ' ~ ~ " ~

l - 0,02-

- 0 03- ~ : ~ / 2 - -

-~ -0.0~-

-0.06-

-0,07-

- 0.08-

- 0.09-

-o .o i -

-o.11-

- 0,12" J I

-2H 0 2H ~H

y

Fig. 11. Variation of U with y for an orthotropic elastic medium when the source (s 1 = 2H, s 2 = 3H) lies in the half-space for three values of the dip angle 8 (8 = ~r/2, ~r/3, ~r/6) when z = 0.


Source I is the surface-breaking fault whereas the remaining sources are the buried faults situated within the layer.

To examine the effect of the orthotropy of the elastic layered medium, we consider only the surface-breaking very long str ike-sl ip fault (Source I) and take 8 = ~-/6. In Figs. 4 to 6, the horizontal dimensionless displacements parallel to the fault at the surface level z = 0 and sub-surfaces z = H / 2 and z = H have been shown. Fig. 4 shows that there is a discontinuity of magnitude unity in the horizontal dimensionless displacement at the point y = 0. In all the figures, the displacement in magnitude for the orthotropic medium is greater than that for the isotropic case. When 8 is kept constant, we note that the difference between the displacements for the two elastic media increases as the depth of the observation point from the stress free surface increases in the layer. From these figures, it is concluded that the horizontal displacement for an orthotropic elastic layered medium can differ significantly, in general, from the corresponding displacement for an isotropic elastic layered medium.

Various curves representing the dimensionless horizontal displacement parallel to the fault for three values of the dip of the angle, namely, 8 = ~ / 6 , 7r/3, 7r/2 have been drawn in Figs. 7 and 8. In both these figures, the fault causing the deformation is assumed to be a surface-breaking fault of width H / 2 (Source I). The sub-surface z = H / 2 becomes the lower boundary of the strike-slip fault when 8 = ~-/2. Fig. 7 shows that there is a discontinuity of magnitude unity in the dimensionless horizontal surface displacement at the point y = 0 for all values of 8.

In Fig. 8, various curves representing the dimensionless horizontal displacement have been drawn at the sub-surface level z = H / 2 for three values of 8. This figure indicates that there is a discontinuity of magnitude 1 / 2 in the displacement at y = 0 corresponding to 8 = ~-/2, however the displacements are continuous at y = 0 when 8 = ~-/6 or zr/3. From Figs. 7 and 8, we conclude that the dimensionless

O.Ok

~ 1 - SOURCE: Slffi 2H ,$2= 3H z=H ~ =.IE / 2

0 02- , z = O ~

'[ 0.01-

= 0

- 0.01-

- 0.02-

Z=0 -0.03"

- 0.0~.. , , -2H 0 2H ~H OH

Y ; Fig. 12. Variation of U with y for an orthotropic elastic medium for three horizontal levels (z ffi 0, 14/2, H) when the source (sl = 2H, s2 = 3H) lies in the half-space and ~ = ~-/2.

N.R. Garg et aL ~Physics of the Earth and Planetary Interiors 94 (1996) 43-62 61

horizontal displacement parallel to the strike-slip fault for an orthotropic elastic layered medium is affected much by the value of the dip of the angle 8.

The effect of the location of the strike-slip fault on the surface deformation of an orthotropic elastic layered medium has been studied in Figs. 9 and 10. In both these figures, three locations of the fault strike have been considered, namely, Sources I, II and III. For Source I, there is a discontinuity of magnitude unity at y = 0, while, for the other two buried sources, the displacement is continuous at y = 0. For the vertical strike-slip fault the horizontal surface displacement is antisyrnmetric with respect to the distance from the fault for different locations of the source under consideration. From these figures we note that the horizontal surface displacement due to surface-breaking fault is altogether different from the corresponding horizontal surface displacement for the other two buried sources.

When the inclined strike-slip fault lies in the orthotropic elastic half-space (z >__ H), the effect of the dip angle (8 _> rr/6) of the fault on the dimensionless horizontal surface displacement is shown in Fig. 11. From this figure, we see that the displacement corresponding to ~ = rr/3 lies, at least in the region - 2 H _< y _< 6H, between the displacements for 8 - - r r / 6 and ~--~ ' /2 . For the vertical strike-slip fault situated in the orthotropic elastic half-space, the dimensionless horizontal displacements at the levels, z -- O, H / 2 , H has been drawn in Fig. 12. We see that the displacement at the middle level of the layer lies between the corresponding displacements at the boundaries of the layer. This figure also shows that the dimensionless horizontal displacement is antisymmetric for a vertical strike-slip fault situated in an orthotropic elastic half-space.

Appendix

® 1 2 k = o k - l e - k e cos k~7 dk = - -~log(s c + 7/2)

=° e k~: sin k~7 dk- - ~2+ ~7----~

e -k6 cos k~7 dk = ~2+r/2"

R e f e r e n c e s

Anderson, D.I., 1962. Love wave dispersion in heterogeneous anisotropic media. Geophysics, 27: 445-454. Bush, I. and Crampin, S., 1987. Observations of EDA and PTL anisotropy in shear-wave VSPs. In: 57th SEG meeting, New

Orleans, Expanded Abstracts, pp. 646-649. Crampin, S., 1989. Suggestions for a consistent terminology for seismic anisotropy. Geophys. Prospect. 37: 753-770. Hess, H., 1964. Seismic anisotropy of the uppermost mantle under oceans. Nature, 203: 629-631. Kasahara, K., 1960. Static and Dynamic Characteristics of Earthquake Faults. Earthquake Res. Inst. The University of Tokyo, pp.

74-75. Kasahara, K., 1964. A strike-slip fault buried in a layered medium. Bull. Earthq. Res. Inst., 42: 609-619. Love, A.E.H., 1944. A Treatise on the Mathematical Theory of Elasticity. Dover Publications, New York. Maruyama, T., 1966. On two-dimensional elastic dislocations in an infinite and semi-infinite medium. Bull. Earthq. Res. Inst., 44:

811-871. Pan, E., 1989. Static resDonse of a transversely isotropic layered half-space to general dislocation sources. Phys. Earth Planet Inter.,

58: 103-117. Rani, S. and Singh, S.J., 1992. Static deformation of a uniform half-space due to a long dip-slip fault. Geophys. J. Int., 109:

469-476.

62 N.R. Garg et aL / Physics of the Earth and Planetary Interiors 94 (1996) 43-62

Rybicki, K., 1971. The elastic residual field of a very long strike-slip fault in the presence of a discontinuity. Bull. Seismol. Soc. Am., 61: 79-92.

Rybicki, K., 1978. Static deformation of a laterally inhomogeneous half-space by a two-dimensional strike-slip fault. J. Phys. Earth. 26: 351-366.

Sharma, R.K. and Garg, N.R., 1993. Subsurface deformation of a layered half-space due to very long strike-slip faults. Proc. Indian Natn. Sci. Acad., 59: 175-187.

Singh, S.J., 1986. Static deformation of a transversely isotropic multilayered half-space by surface loads. Phys. Earth Planet. Inter., 42: 263-273.

Singh, S.J. and Rani, S., 1994. Lithospheric deformation associated with two-dimensional strike-slip faulting. J. Phys. Earth, 42: 197-220.

Two-dimensional deformation of an orthotropic elastic medium due to seismic sources

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