Chapter-4 Propagation of Shear Waves Due to Shearing ...shodhganga.inflibnet.ac.in/bitstream/10603/44774/10... · 4.1 Generation of SH-type waves due to Shearing stress discontinuity

Chapter-4

Propagation of Shear Waves Due to Shearing

Stress Discontinuity

Propagation of shear waves due to shearing stress discontinuity

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4.1 Generation of SH-type waves due to Shearing stress discontinuity in a Sandy layer

overlying an isotropic and inhomogeneous elastic half-space*

4.1.1 Introduction

The study of wave propagation in layered media has received considerable attention,

especially in the context of exploration geophysics, seismology and engineering.

Theoretical solutions for wave propagation in layered anelastic media provide the

theoretical basis for seismic wave propagation in layered soil and rock deposits. These

exact theoretical solutions predict that the physical characteristics of two and three

dimensional wave fields in layered anelastic media are distinct from those predicted by

elastic models or one dimensional anelastic models.

Love waves (SH-type waves) are seismic waves that cause horizontal shifting of

earth during the earthquake. The particle motion of SH-type waves forms a horizontal line

perpendicular to direction of propagation. During earthquake seismic waves such as Love

waves are generated from the interior of the earth. SH-type (Love) waves are transversely

propagated surface waves which we feel directly during earthquakes.

Shearing stress discontinuity occurs in many cases, e.g., i) inside the earth between

two layers, if there is a crack which is being filled up by liquid, then there will be a case of

discontinuity of shearing stress in that region whereas the normal stress will be continuous,

ii) when a layer tends to slide over another layer inside the earth, then shearing stress

discontinuity may occur and iii) shearing stress discontinuity may be associated with

propagation of cracks in earthquake.

* Acta Geophysica (Springer), 62 (1), 44-58.


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This Problem is based on a paper by Nag and Pal, (1977) who have obtained the

displacement of SH-type waves at the free surface of an isotropic layered media due to

occurrence of a shearing stress discontinuity. In another paper, Pal and Debnath (1979)

have extended the problem of Nag and Pal (1977) by considering the shearing stress

discontinuity at the interface of an anisotropic layered media. In the above two problems

the shearing stress discontinuity so formed are travelled with a uniform velocity. Pal (1983)

has extended the problem of Pal and Debnath (1979) by considering the creation and

movement of shearing stress discontinuity with non-uniform velocity. In all the above three

problems the method employed by the authors is due to Garvin (1956) which is based on

Cagniard’s technique (Cagniard, 1939). Chakraborty and Chandra (1984) studied the

reflection and refraction of plane SH-waves at a plane interface between a dry sandy layer

and sedimentary rock anisotropic of transversely isotropic type. (Pal, 1985) has discussed

the propagation of Love waves in a dry sandy layer. Pal and Kumar (2000) have considered

the generation of SH waves by a moving stress discontinuity in an anisotropic soil layer

over an elastic half-space by using the Cagniard-de Hoop (1960) special reduction

technique. Khurana and Vashisth (2001) have studied the Love wave propagating in an

elastic sandy layer over a fluid saturated porous solid half space, both the layer are taken as

pre-stressed medium. De Hoop (2002) has considered the Reflection and transmission

properties of an elastic interfacial bonding of two semi-infinite solids are investigated for

the simplest possible case of a line-source excited two-dimensional SH-wave. Tomar and

Kaur (2007) have studied the reflection and transmission of a plane SH-wave incident at a

corrugated interface between a dry sandy half-space and an anisotropic elastic half-space.

Recently Diaz and Ezziani (2010) have studied the analytical solution for wave propagation

in heterogeneous acoustic/porous media in 3D case using Cagniard’s De Hoop techniques.

Neither of the above authors has considered the effect of shearing stress discontinuity in

porous soil (sandy) materials.

It is our intention in the present problem, to investigate the two dimensional

problem of generation of SH-type waves at the free surface of a sandy layer (soil) due to an

impulsive stress discontinuity moving with uniform velocity along the interface of isotropic


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homogeneous and inhomogeneous medium. The displacement is calculated numerically for

two particular distances on the surface for two different types of the discontinuity in the

shearing stress for different value of sandiness parameter, looseness of sandiness and

inhomogeneity parameters. It involves Laplace and Fourier transform and the inversion is

based on Garvin’s (1956) method. The problem discussed may be of importance in

connection with the propagation of cracks in sandy layer. Two cases of stress discontinuity

are considered and the numerical results are shown graphically.

4.1.2 Formulation of Problem

We consider the sandy layer of thickness h over the semi infinite medium of an isotropic

and inhomogeneous elastic half space. The origin is taken at the interface; the x-axis is

taken along the direction of the Love wave propagation and z-axis is in vertically

downward direction. The discontinuity is assumed to occur suddenly at the interface (xz-

plane) and to move with constant velocity V (<1) in the x-direction. The elastic constants

and the coordinate axes are shown in Fig.1.

Fig. 4.1.1: Geometry of problem.


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The stress-strain relation is modified considering the case of decrease in rigidity only due to

the slippage of granules. The modified stress-strain relations in an isotropic sandy medium

are (cf. Weiskopf (1945))

2 , 1.ij ij ije

(4.1.1)

where, and are Lame’s constant and (>1) is a measure of sandiness. The case =1

correspond to the classical isotropic elastic solid material.

Since only the SH-type of wave is being considered, so we can take u=w=0 and v is a

function of the variable x, z and t. The equation of motions to be satisfied for the medium I

and II are

2 2 2

1 1 12 2 2 2

1

v v vx z t

(4.1.2)

2 2 2

2 2 2 22 2 2 2

2

1v v v vx z z t

(4.1.3)

where, 2 2 011 2

1 0

,

and is inhomogeneous parameter.

The boundary conditions are

1

0yz at z = -h (4.1.4)

1 2v v at z = 0 (4.1.5)

1 2

,yz yz S x t H t at z = 0 (4.1.6)

where, S(x, t) is a function of x and t; H(t) is the Heaviside unit function of time t.

4.1.3 Method of solution

The above problem can readily be solved by using the Laplace and Fourier transforms

combined with the modified Cagniard De Hoop (1960) method. The Laplace transform

with respect to t and the Fourier transform with respect to x are defined by


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0

, ;i x ptv e dx e v x z t dt

(4.1.7)

We can easily get the upper and lower layer with 2 0v as z in the form

1 1 1, ; cosh sinh i xv x z p A s z B s z e d

(4.1.8)

22

i x s zv Ce d

(4.1.9)

where, the constant A, B and C are to be determined from the boundary conditions (4.1.4)-

(4.1.5)

22 21

22 2 221 22

1

4,

2

pps s

(4.1.10)

It follows from the boundary conditions (4.1.4) and (4.1.5) that

A=C, B coshs1=A sinhs2 (4.1.11)

Case I Let,

,,

0, elsewhereP a x b Vt

S x t

(4.1.12)

where, P is constant.

This definition of stress discontinuity shows that it is created in the region x=a to x=b and

then expands with the uniform velocity V in the x-direction. In particular, when a=b=0, the

discontinuity is created at the origin and expands with uniform velocity V in the x-direction.


~ 101 ~

From the boundary condition (4.1.6) one get, with the help of (4.1.12),

11 0 2 2

i a i b i bP e e eB s C s pp i iV

(4.1.13)

Solving for A, B and C from (4.1.11) and (4.1.13) get the displacement function at the free

surface at z = -h in the form

1

1

1

11 1 0 2 1

12

1 1 0 2

, ;2 sinh cosh

1

i x i a i b i b

i x hs i a i b i bhs

P e e e ev x h p dpp i is s h s s h V

P e e e e Ke dpp s s i iV

(4.1.14)

where,

1 1 0 2

1 1 0 2

1s sKs s

(4.1.15)

represents the reflection coefficient of SH-waves incident from the sandy medium at the

interface between two half-spaces. The coefficients of different power of K in series of

equation (4.1.14) are associated with the pulses undergoing repeated reflection in the upper

layer.

Using the inverse Laplace transform, we can rewrite (4.1.14) in the convenient form

11 1 2 3, ;v x h p L I I I (4.1.16)

where,


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1

1 1

12

11 1 0 2

1 hsi x hsKePI e d

p i s s

(4.1.17)

1

2 1

12

21 1 0 2

1 hsi x hsKePI e d

p i s s

(4.1.18)

1

2 1

12

3

1 1 0 2

1 hsi x hsKePI e d

pp i s sV

(4.1.19)

with x1=x-a and x2=x-b.

In order to evaluate the Laplace inversion integral, we shall use the Garvin’s (1956) method

who discussed the contour integration and mapping in detail. It may be fair to avoid

duplication of mathematical analysis similar to Garvin, and to quote some necessary results

from that paper without proof.

Next for non-dimensionalisation, we substitute 02 11

1 1 1 1

, , , ,p p

,

(4.1.10) and (4.1.15) so that

12 2

11

2 2 2

22

4 1

2

ps

ps

(4.1.20)

and,


~ 103 ~

2 2 212 2

2 2 212 2

4 1

2

4 1

2

K

(4.1.21)

Thus we obtain

1 1 1 12 42

11 1 0 20

1 ........2 Imi x hs hs hse Ke K ePI d

p s s

(4.1.22)

The first term in I1 is

12 2

1 1

1,12 2 210

2 21 1

exp2 Im

4 1

2

p i x hPI dp

p

(4.1.23)

The integrand (4.1.23) has singularities at 10, ,i .

Let

1

2 21 1t i x h

(4.1.24)

Then by inversion

1

22 2 2 211 1 12 2

1

t itx h t x hx h

(4.1.25)

The mapping of the -plane into the t-plane in shown in fig. 4.1.2.


~ 104 ~

Fig. 4.1.2: t-plane showing the mapping and the contour of integration.

Making the reference to the Fig. 4.1.2 and the paper of Nag and Pal (1977), we find

11,1 1 1,1 1,1 1 1

1 1 0

2 tpL I t G d

(4.1.26)

where, 2

1L tH tp

and


~ 105 ~

1,1

12 2 2

12 12

1,1 1,1 1,1 1,1

121,1 2 2 1

1 1

4 1Im

2G t t

d tH t x h

dt

(4.1.27)

Since for

122 2

1

1 1

x hh t

,

1,1

2 2 21

1,12 121,1 1,1

4 1 ( )( )

2d t

tdt

is real.

In general

11, 1 1, 1, 1 1

1 1 0

2 t

n n npL I t G d

(4.1.28)

where,

1,

12 2 2

12 12

1, 1, 1, 1,

1 121, 2 2 12

1, 1 1

4 1Im

2n

n n n n

nn

n

G t t t

d tK t H t x h

dt

(4.1.29)

and

1

22 2 2 2 211, 1 1 12 2 2

1

, 1,3,5......n t itx nh t x n h nx n h

(4.1.30)

So that


~ 106 ~

1 11 1,

1,3,5,..n

nL I L I

(4.1.31)

Similarly

1 12 2,

1,3,5,..n

nL I L I

(4.1.32)

where, x1 is replaced by x2 and K is given by (4.1.21).

Proceeding in the same way, we get

1 13 3,

1,3,5,..n

nL I L I

(4.1.33)

where,

13, 1 3, 2, 1 1

1 1 0

2 t

n n npL I t G d

(4.1.34)

2,

12 2 2

12 2

3, 2, 2,

1 11 22, 2 2 2 121 2, 2, 2 1

4 1Re

2n

n n n

nn

n n

tG t t

d ti t K t H t x n h

dt

(4.1.35)

and 2,n t is given by (4.1.30) with x2 in place of x1.

Finally, a simple combination of the results (4.1.31)-(4.1.33) gives the exact value of the

surface displacement field 1 , ,v x h t at the free surface.

Case II Let,

,S x t Ph x Vt (4.1.36)


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where P is a constant and δ(x-Vt) is Dirac’s delta function of argument (x-Vt). A term h is

included on the right hand side of equation (4.1.36) so as to give S as the dimension of a

stress.

The boundary condition (4.1.6) gives

11 0 2

2

PhB s C spV iV

(4.1.37)

Solving for A, B and C from (4.1.11) and (4.1.37) one gets

1 1 12 42

1

1 1 0 2

1 ........, ;

i x hs hs hse Ke K ePhv x h p dpv i s sV

(4.1.38)

Proceeding as a similar of case I gives the solution

11

1,3,5,..1 0

2, ,t

n nn

Phv x h t G d

(4.1.39)

where,

12 2 21

2 2

1 11 22 2 2 121 1

4 1Re

2n

n n n

nn

n n

tG t t

d ti t K t H t x n h

dt

(4.1.40)

and

1

22 2 2 2 2112 2 2 , 1,3,5......n t itx nh t x n h n

x n h

(4.1.41)

If the stress discontinuity is taken as H(x)-H(x-Vt) in place of δ(x-Vt) the corresponding

expression on the right hand side of equation (4.1.38) will differ only by a constant factor

from I3 (with a=b=0) in equation (4.1.19).


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4.1.4 Numerical results and Discussion

For the numerical calculation the values 12, 2, 1.5 are taken (cf. Nag (1963)).

The values of 1 1 , ,K v x h t for x=6h and x=12h have been plotted against τ1= τ- τ0, where

τ0 denotes the time at which the disturbance arrives at the point of observation with

11 2

KPh

. 1th is the time of the disturbance to arrive from source to initial point. The

value of at x=6h is 1/ 22 26 n , n=1,3,5,….. and 0 6.03 at n=1 and the value of at

x=12h is 1/ 22 212 n , n=1,3,5,….. and 0 12.04 at n=1.

When x=6h, for six initial values, we have

0 11

0 13

0 15

1 1

0 17

0 19

0 111

cosh 3737

cosh 4545

cosh 6161

, ,

cosh 8585

cosh 117117

cosh 157157

A H

A H

A H

K v x h t

A H

A H

A H

when x=12h, for six initial values, we have


~ 109 ~

0 11

0 13

0 15

1 1

0 17

0 19

0 111

cosh 145145

cosh 153153

cosh 169169

, ,

cosh 193193

cosh 225225

cosh 265265

A H

A H

A H

K v x h t

A H

A H

A H

and

3

2

122 2 21

2 2

0

2 2 212 '2

4 1 coscos sin

2Re , 1,3,5,.....

4 1 coscos cos

2

n

n

nn n

n

nn n

A n

0nA represents the reflection coefficient of SH type waves incident from the sandy

medium to elastic inhomogeneous half-space.

Figs. 4.1.3, 4.1.4, 4.1.5, 4.1.6 show the variation of displacement with elapsed time

τ1for different values of sandiness parameter. Sandiness effect shows that as the different

pulses arrive after reflection, the disturbance point slightly changes. Figs. 4.1.3, 4.1.4, 4.1.5,

4.1.6 are drawn to show the effect of sandiness and inhomogeneous parameter on

displacement coefficient (with time). Figs.4.1.3 and 4.1.4 show the disturbance effect for


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x=6h and x=12h respectively for some values of sandiness parameter and inhomogeneity

parameter. For figs. 4.1.3 and 4.1.4 the curve no. 1 are drawn for comparison purpose for

the case of isotropic and homogeneous material. For all the cases, the curves show the

oscillations. Figs. 4.1.3 and 4.1.4 show the oscillation with respect to time (initially). After

sometimes the behaviour become smooth i.e. no oscillations observed. Figs. 4.1.5 and 4.1.6

are drawn to show the effect of loose sands for same values of sandiness and

inhomogeneity parameters. For loose sands oscillations are more in negative direction

while for sand (soil) the oscillations are more in positive direction. This behaviour justifies

the disturbance natures in sandy medium. It is inferred that jumping effects for x=6h are

larger than x=12h i.e. if the distance from the source of observation are more the jumping

effect are less. It is observed that in the case of loose sands the jumping effect is very low

for both the values of x=6h and x=12h. This is may be due to penetration of fluid into the

soil (sand) and extension of fracture for shearing stress and pressure into pores.

Fig. 4.1.3: Variation of 1 1 , ,K v x h t with 1 for x=6h.

0 5 10 15 20 25 30-3

-2

-1

0

1

2

3

4

1

2

3 4

5

1

K1v 1(x

,-h,t)

1. =1, =02. =1, =0.53. =1.2, =14. =1.4, =1.55. =1.6, =2


~ 111 ~

Fig. 4.1.4: Variation of 1 1 , ,K v x h t with 1 for x=12h.

Fig. 4.1.5: Variation of 1 1 , ,K V x h t with 1 for loose sand in x=6h

0 2 4 6 8 10 12 14 16 18 20-1

-0.5

0

0.5

1

1.5

2

1

2

3

4

5

1

K1v 1(x

,-h,t)

1. =1, =02. =1, =0.53. =1.2, =14. =1.4, =1.55. =1.6, =2

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1

2

3

4

1

K1v 1(x

,-h,t)

1. =0.2, =0.252. =0.4, =0.53. =0.6, =0.754. =0.8, =1


~ 112 ~

Fig. 4.1.6: Variation of 1 1 , ,K V x h t with 1 for loose sand in x=12h.

0 1 2 3 4 5 6 7 8 9 10-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1

2

3

4

1

K1v 1(x

,-h,t)

1. =0.2, =0.252. =0.4, =0.53. =0.6, =0.754. =0.8, =1


~ 113 ~

4.2 Disturbance of SH-type waves due to moving stress discontinuity in an

anisotropic soil layer overlying an inhomogeneous elastic half-space*

4.2.1 Introduction

The two main types of seismic waves are body waves and surface waves. The first kind of

body waves is the P-wave or primary waves. The second type of body wave is the S-wave

or secondary waves which have a transverse, shear vibration in a plane perpendicular to the

direction of propagation. Two polarizations of shear waves are possible (I) one in vertical

direction named as SV-waves (II) another one in horizontal direction named as SH-waves.

The anisotropy present in rock types of crust would affect the speed of propagation of SV-

waves as well as SH-waves. Individual grains of most solids are highly anisotropic and

direction and position conscious. Thus anisotropic and inhomogeneity affect the speed of

propagation of SH-types waves.

While both soil and rocks are concerned with pores, there is a considerable

difference in the nature of the medium: soil are usually regarded as consisting of discrete

particles touching at isolated points, while materials such as sedimentary rocks and

concrete are regarded as a solid skeleton traversed by a fine network of capillaries. Some

soils behave as anisotropic materials where swelling of some constituents on wetting, is

ignored.

Surface wave propagation involves transition layers which in general are both

anisotropic and heterogeneous. The effect of anisotropy on reflection and transmission of

elastic waves at a plane interface between two elastic media was studied by Daley and Hron

(1977). Brock (1982) has considered the effect of displacement discontinuity derivatives on

wave propagation prescribed along arbitrarily time-varying line segments normal to an

elastic half-plane surface or bimaterial interface.

*Accepted for publication in Sadhana ( Springer)


~ 114 ~

Nag and Pal (1977) have considered the disturbance of SH-type waves due to

shearing stress discontinuity in an elastic layered half-space and considered two type of

shearing stress discontinuity for finding the displacement at the free surface. Pal and

Debnath (1979) have studied the Generation of SH-type waves at the free surface of a

layered anisotropic elastic media due to an impulsive stress discontinuity moving with

uniform velocity along the interface of the layered medium.

The generation of SH-type wave due to non-uniformly moving stress discontinuity

in a layered half space has been considered by Mittal and Sidhu (1982). Romeo (1997) has

investigated the SH surface waves on a layered anisotropic half-space is having recourse to

a wave-splitting approach. The displacements have been evaluated using de Hoop’s version

of Cagniard’s technique. Sengupta and Nath (2001) have investigated the surface waves in

fibre-reinforced anisotropic elastic media. De Hoop (2002) has considered the Reflection

and transmission properties of an elastic interfacial bonding of two semi-infinite solids are

investigated for the simplest possible case of a line-source excited two-dimensional SH-

wave. Tomar and Kaur (2007) have studied the reflection and transmission of a plane SH-

wave incident at a corrugated interface between a dry sandy half-space and an anisotropic

elastic half-space. All the above authors have not considered the effect of shearing stress

discontinuity in porous soil (sandy) materials.

Shearing stress discontinuity occurs in many case e.g.: (i) inside the earth between

two layers, if there is a crack which is being filled up by liquid, (ii) when a layer tends to

slide or slip without friction over another layer inside the earth. For the above reasons the

stress discontinuity may be treated as a moving source either with uniform velocity or with

non-uniform velocity. Relating to stress discontinuity, Pal (1983) has considered the

problem of generation of SH-type waves due to non-uniformly moving stress discontinuity

in a layered anisotropic elastic half-space. It is our intention in the present problem is to

investigate the disturbance of SH-type waves at the free surface of a anisotropic soil layer

due to an impulsive stress discontinuity moving with non-uniform velocity along the


~ 115 ~

interface of isotropic homogeneous and inhomogeneous medium. The displacement is

calculated numerically at a particular distance on the surface for different types of the

discontinuity in the shearing stress for different value of inhomogeneous parameter. It

involves Laplace and Fourier transform and the inversion is based on Garvin’s (1956)

modification of Cagniard’s (1962) method. The problem discussed may be of importance in

connection with the propagation of cracks in layers and the numerical results are shown

graphically.

4.2.2 Formulation of the problem

Let us consider an anisotropic soil layer I of thickness h with Weiskopf anisotropy which is

lying over an inhomogeneous elastic half-space (layer II) with shear modulus 2 0bze

and density 2 0bze (cf. Sinha (1966)). The problem is two dimensional and is being

analyzed in xz-plane. The positive z-axis is directed vertically downwards and x-axis is

along the interface z=0. Here we consider the transient wave motion due to a shearing stress

discontinuity which is being created at a certain instant at the point (x,0) and then moves

along x-axis with time dependent speed (Fig. 4.2.1).

Fig. 4.2.1: Geometry of problem.


~ 116 ~

For SH-type wave which is transverse in horizontal plane propagating along x-axis, we

have u=w=0 and v=v(x, z, t) where u, v, w are the displacement components along x, y, z

directions respectively and independent of y-axis. As we are interested to calculate the

transverse displacement v1 of an element, so the equation of motion in anisotropic layer is

(cf. Pal (1983)).

2 2 2

1 1 11 1 12 2 2

v v vN Gx z t

(4.2.1)

Where N1, G1 are the anisotropic parameter for soil and ρ1 is the mass density of the soil.

For layer II the equation of motion for the transverse displacement v2 is (cf. Sinha (1966))

2 2 2

2 2 2 22 2 2 2

2

1v v v vbx z z t

(4.2.2)

where, 2 02

0

and b is inhomogeneous parameter.

The boundary conditions are, in the usual notations,

1

0yz at z = -h, t>0 i. e shearing stress vanishes at the free surface z=-h

1 2v v at z = 0, t>0 i.e. displacement are continuous at the interface (4.2.3)

1 2

,yz yz f x t H t at z = 0

where, f(x, t) will be chosen later on, and H(t) is Heaviside’s unit function of time t. This

definition shows that stress discontinuity at interface z=0 moves with non-uniform velocity.

For two dissimilar layer stress discontinuity in shearing stress occurs at the interface where

as the normal stress is continuous there. Since normal stress are continuous and these are

related to normal strains with displacement components, so displacements are also

continuous at z=0. When discontinuity moves naturally the displacement components in

transverse directions also travels (or moves) with certain velocity.

4.2.3 Solution of problem

The problem can readily be solved by using the Laplace and Fourier transforms combined

with the modified Cagniard- De Hoop (1960) technique and used by Pal (1983). The

Laplace transform of a function v(x, z, t) with respect to t will be defined by


~ 117 ~

0

, , , , , ,ptL v x z t V x z p e v x z t dt

(4.2.4)

where, p is real and positive.

The Fourier transform V(x, z, p) with respect to x will be written as

, , , ,i xV z p e v x z p dx

(4.2.5)

1, , , ,2

i xV x z p e V z p d

(4.2.6)

Now applying Laplace transform and Fourier transform to the equations of motion, we get

solutions in the upper and lower layers with 2 0V as z in the form:

1 1 1cosh sinh at 0V A a z B a z h z (4.2.7)

22

a zV Ce (4.2.8)

where,

22 21

22 222 2 2 21 1

1 1 2 1 121 1 1

4, , ,

2

pb bG Npa a

G

where, 1 and 2 are shear wave velocities in anisotropic soil layer and isotropic and

inhomogeneous medium, 1 is square root of ratio of anisotropic soil constant.

Taking the Laplace transform of boundary condition, we get

1 2V V at z=0

11 0dVG

dz at z=-h (4.2.9)

1 21 0

dV dVG Fdz dz

at z=0

Solving for A, B, C with the aid of Eq. (4.2.9), we have the displacement function at the

free surface z=-h given by


~ 118 ~

1 112

11 1 0 2

11, ,a h a hi xFe e Ke

V x h p dG a a

(4.2.10)

where,

1 1 0 2

1 1 0 2

G a aKG a a

As ξ is complex, in order to apply Cagniard-De Hoop technique, we put ξ=ikp and b=pγ (as

b is dimension of length and so it is always +ve), we find 1 1 2 2,a b p a b p , where

2 212222 2

1 1 221

141 ,

2

kb k b

Hence relation (4.2.10) can be written as

1 112

11 1 0 2

1, ,

phb kpx phbFe KeiV x h p dkG b b

(4.2.11)

where,

1 1 0 2

1 1 0 2

G b bKG b b

represents the reflection coefficient for SH waves incident from the anisotropic soil layer at

the interface between the two media.

4.2.4 Concentrated non-uniformly moving stress discontinuity

Let us now define the unknown function f(x, t) as follows as considered by Freund (1972)

and is given by

0, ,f x t P x l t (4.2.12)

where 0P is a constant and is Dirac’s delta function. Here l t will be assumed to be

a continuous, monotonically increasing function of time, i.e., it will never be acted at a

single point for finite time. Function l t is invertible, i.e. there exists a continuous and

monotonically increasing function t x such that

,l x x l t t .


~ 119 ~

Moreover, these function satisfy the following relations identically in x or t (after

differentiation)

. .

' '1, 1l x x l t l t

where ' /l t dl dt is the speed of propagation of stress discontinuity at any time t and

' dxdx is related to slowness of stress discontinuity at any place x.

Since 0 0l , assumption (4.2.12) shows that shearing stress discontinuity is at rest until

time t=0, then begins to move according to l t in positive x-direction.

Now applying Laplace transform to (4.2.12) we get

00

, ptF x t P x l t e dt

(4.2.13)

but 'x l t x t x . Hence

' '0 0

0

p xptF P x t x e dt P x e H x

(4.2.14)

Taking Fourier transform of relation (4.2.14) we get

'0

0

p x i xF P x e H x e dx

(4.2.15)

Putting ,ikp

'0

0

p s kpsF P s e e ds

(4.2.16)

where, we have changed the variable of integration from x to s and made use of the

following results:

1, 0, 0

0, 0, 0

x xH x

x x

(4.2.17)

4.2.5 Solid displacement on the free surface (z=-h)

From equations (4.2.11) and (4.2.16), we have


~ 120 ~

1

1 12 4' 201

1 1 0 20

, , 1 ....p b h kx ks

p s pb h pb hiP eV x h p s e Ke K e dkdsG b b

(4.2.18)

where, K is defined earlier in equation (4.2.11). The coefficients of different powers of K in

the series of equation (4.2.18) are associated with pulses undergoing repeated reflections in

the upper layer.

Now Laplace inversion of 1 , ,V x h p can be done by Cagniard’s technique modified by

De Hoop (1960).

The method to be adopted here is the same as that considered by Pal (1983), so to avoid

mere repetition; we only write the final result. Hence, we finally obtain the solid

displacement component on the free surface z=-h of the soil layer as follows:

11 1 2 3, , , 1,3,5,.....n n n nv x h t L I I I I n (4.2.19)

where,

'01 1

01

1

2 , for

0 for

nn

nn

P s D t s ds t tI

t t

1 '02 12 1

02

12 1

2 , if for

0 if for ,

n nn n

nn n

P f s D t s ds t t tI

t t t t

2 '03 12 1

03

12 1

2 , if for

0 if for ,

n nn n

nn n

P f s D t s ds t t tI

t t t t


~ 121 ~

The following symbols here used

12

1

2 212222 2

1 1 021

, with14

12

nn

n

n m

n

n

dk K kdD t s I t s

kG k

(4.2.20)

32

2

2 212222 2

1 1 021

, with14

12

nn

n

n m

n

n

dk K kdD t s I t s

kG k

(4.2.21)

3( , )nD t s is the same as 2( , )nD t s with nk replaced by nk , as defined according to the

equation

1/ 222

2 21 1

sincos n n

n nn n

i rkr r

(4.2.22)

where 22 2 2 , tan , 0n n nnhr x s n h

x s

, and

1/ 222

2 21 1 1 1

sincos n n

n nn n

i rk ir r

(4.2.23)

Where 1 2

1

0 cosn

for x>s and 0 ,


~ 122 ~

1 2

11

1 for cos

0 elsewheren

f

(4.2.24)

1 2

21

1 for cos

0 elsewheren

nf

(4.2.25)

1

1 1

( ) nn

rt s

(4.2.26)

and

1/ 2

12 2 2 32 1 1 2

cos 1 1( ) sinn n nn n

rt s r

(4.2.27)

The additional contour integration path (Cagniard) is shown in Fig. 4.2.2, where the

hyperbola intersects branch cuts and the transformation from the k-plane to τ-plane is

defined by equation 1b h kx ks , where τ is real and positive.

Fig. 4.2 2: τ -plane showing the contour integration.


~ 123 ~

4.2.6 Brief discussion of the solution

The term I1 as defined in equation (4.2.19) is independent of K and gives the contribution

due to the direct wave from the source. The terms for different values of n (n>1) in Eq.

(4.2.19) gives the contributions due to the pulses or waves that suffer n reflections at the

interface of anisotropic soil layer and inhomogeneous medium. More explicitly, the terms In

(for all n) represent the integral sums of the disturbances due to all displacement

discontinuities occurring at a given x=0, z=-h and at any instant t≥0.

For the problem considered here, the shearing stress discontinuity generates wavelet at each

point of the interface. A typical wavelet is emitted from the point (s,0) at the instant s

and thereafter propagates with the velocity 1 11 . The radius of the shear wavelet at some

later time is 1 1t s .The time at which a particular shear wavelet (for direct wave)

arrives at the interface (x,-h) is just 1

1 1

rs . Thus for any t and any place (x,-h),

those points on the interface for which 1

1 1

rt s are undetected. Similar cases

happen with a reflected S wave. The above discussion justifies the condition imposed on

l(t), and consequently on ( )s , i.e. ( )s is a monotonically increasing function of time t.

4.2.7 Numerical results and discussion

Consider a numerical example for a particular case of uniformly moving discontinuity.

We consider 0,f x t P h x Ut , which shows that stress discontinuity moves uniformly

with constant velocity U along x-axis. Hence in this case sUs and so ' 1

Us .

The values of alluvial soil 1( 1.34 and 2.56) are taken from (Boer 2000) and

correspond to incompressible binary model. The initial behaviour of the disturbance for few

pulses is evaluated at a place x=4h and x=8h for alluvial soil (ϕ1) as obtained by Pal and

Debnath (1979). According to this model, we have taken 1 2.54km/s , 2 2.45km/s ,


~ 124 ~

01

0.91 and 1 1.49U

.The value of 1 1( , , )K V x h t is plotted against 1 0 , where

0 denotes the time at which the disturbance arrives at the point of observation from the

source to initial point with 11 2

GKPU

and 1th (the non-dimensional time). The change

occurs for a certain station x=4h and x=8h at different time for fixed values of alluvial soil

at 1

22 2 214 n and

122 2 2

18 n , n=1, 3, 5,…. The values of τ0 for alluvial soil

1 1.34,2.56 at x=4h are 4.21 & 4.74 and at x=8h are 8.11 & 8.39 respectively.

32

1

12

22 22

2112 2 0 22

11 1

0

22 22

212 2 0 2 12

11 1

4 1 cos1 cos sin

2

Re

4 1 cos

1 cos cos2

1

n

n

n

n n

n

n

n n

A

U

n

,3,5,..........

0nA represents the reflection coefficient of SH type waves incident from the anisotropic

soil layer to elastic inhomogeneous half-space.

Now when x=4h, 1

ht

for initial six values we have


~ 125 ~

0 1 2 2 21 12 2 2

1

0 1 2 2 23 12 2 2

1

0 1 2 2 25 12 2 2

11 1

0 1 2 2 27 12 2 2

1

0 19 2 2 2

1

cosh 4 14 1

cosh 4 34 3

cosh 4 54 5

, ,

cosh 4 74 7

cosh4 9

A H

A H

A H

K V x h t

A H

A

2 2 21

0 1 2 2 211 12 2 2

1

4 9

cosh 4 114 11

H

A H

Now when x=8h, 1

ht

for initial six values we have

0 1 2 2 21 12 2 2

1

0 1 2 2 23 12 2 2

1

0 1 2 2 25 12 2 2

11 1

0 1 2 2 27 12 2 2

1

0 19 2 2 2

1

cosh 8 18 1

cosh 8 38 3

cosh 8 58 5

, ,

cosh 8 78 7

cosh8 9

A H

A H

A H

K V x h t

A H

A

2 2 21

0 1 2 2 211 12 2 2

1

8 9

cosh 8 118 11

H

A H


~ 126 ~

In order to see the effect of disturbance, we have considered the variation of K1V1

(i.e. displacement on the free surface) against the time of arrival of pulses (or the

disturbance). The variations are considered in two ways. In first case, the variations are

considered for fixed value of γ (inhomogeneity parameter) and varying values of soil

parameter ϕ1 for the fixed distances from the source i.e. x=4h and x=8h. In second case, the

variations are considered for fixed value of soil parameter and varying values of γ for same

fixed distances from the source i.e. x=4h and x=8h. All these variations are shown as 3D-

plots in figures 4.2.3 to 4.2.10. Figs 4.2.3 to 4.2.6 are drawn for variation of K1V1 with time

for fixed value of soil parameter for the distances x=4h and x=8h whereas figs 4.2.7 to

4.2.10 are drawn for variation of K1V1 with time for fixed value of inhomogeneity

parameter. It is observed from figs 4.2.3 to 4.2.6 that for both fixed distances x=4h and

x=8h the discontinuity effects are pronounced and these are due to the arrival of pulses after

repeated reflection and refraction from the source to the surface of the soil layer and back to

interface. The same conclusion may be derived from figs 4.2.7 to 4.2.10. It is also

concluded that as we increase inhomogeneity parameter (figs 4.2.3 to 4.2.6), the

disturbance with discontinuity occurs very fast. From figs 4.2.7 to 4.2.10, it is inferred that

as we increase the soil parameter the disturbances are observed not very fast but slowly.

Over all conclusion is that SH-type waves get disturbed when it crosses different layers.

Two dimensional curves are in well agreement with the curves (not surface plots) of Nag

and Pal (1977) (for isotropic case) and Pal and Debnath (1979) (for anisotropic case).


~ 127 ~

Fig. 4.2.3: Variation of displacement K1V1 against τ1 and γ for ϕ1=1.34 at x=4h.

Fig.4.2.4: Variation of displacement K1V1 against τ1 and γ for ϕ1=2.56 at x=4h.


~ 128 ~

Fig.4.2.5: Variation of displacement K1V1 against τ1 and γ for ϕ1=1.34 at x=8h.

Fig. 4.2.6: Variation of displacement K1V1 against τ1 and γ for ϕ1=2.56 at x=8h.


~ 129 ~

Fig. 4.2.7: Variation of displacement K1V1 against τ1 and ϕ1 for γ=1 at x=4h.

Fig. 4.2.8: Variation of displacement K1V1 against τ1 and ϕ1 for γ=1.4 at x=4h.


~ 130 ~

Fig. 4.2.9: Variation of displacement K1V1 against τ1 and ϕ1 for γ=1 at x=8h.

Fig. 4.2.10: Variation of displacement K1V1 against τ1 and ϕ1 for γ=1.4 at x=8h.


~ 131 ~

4.2.8. Conclusions

The theoretical development of the dynamic properties for SH-waves using Cagniard-De

Hoop technique in plane anisotropic soil layer overlying an inhomogeneous half-space is

presented with an emphasis put on having the final formulas in a form that are readily

programmable with Mathematica-7. A numerical discussion of a simple displacement time

structure (surface plots) is given which shows the variation of inhomogeneity parameter

and soil parameter on the disturbance of SH-waves. Surface plots give better visualization

of stress discontinuity or jumps.

Finally the shearing stress discontinuity is always associated with the propagation of

cracks in earthquakes, the present study is having direct applications to Geophysics and

Seismology. This problem serves as a convenient vehicle for introducing the analytical

method, which may be applied to more complicated problems.

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