Static Deformation of two Half-Spaces in Smooth Contact due to a vertical Tensile Fault of Finite...

Meenal Malik, Minakshi, Ravinder Kumar Sahrawat , Mahabir Singh| Static Deformation of two Half-Spaces in Smooth Contact due to a vertical Tensile Fault of Finite Width

IFRSA’s International Journal Of Computing |Vol 4|issue 1|Jan 2014 440

Static Deformation of two Half-Spaces in Smooth Contact

due to a vertical Tensile Fault of Finite Width

Meenal Malik1, Minakshi

2, Ravinder Kumar Sahrawat

2 and Mahabir Singh

3

1 Department of Mathematics, A. I. J .H .M. College, Rohtak-124001.

2 Department of Mathematics, Deenbandhu Chhotu Ram University of Science and Technology, Murthal, Sonepat

3 COE, Deenbandhu Chhotu Ram University of Science and Technology, Murthal, Sonepat.

ABSTRACT

The Airy stress function due to a vertical tensile fault of

finite width for two homogeneous, isotropic, perfectly

elastic half-spaces in smooth contact is obtained. This

Airy stress function is used to derive closed form

analytical expressions for the stresses and displacements

at an arbitrary point in one of the two half-spaces. The

variation of the stress and displacement fields with

distance from the fault and depth is studied numerically.

KEY-WORDS: Static deformation, Vertical Tensile

Fault, Airy Stress Function, Smooth Contact.

1. INTRODUCTION

Tensile fault representation has several important geophysical applications, such as modelling of the

deformation fields due to dyke injection in the volcanic

region, mine collapse and fluid-driven cracks (recently

occurred in Uttarakhand on July 17, 2013) . Rongved

(1955) obtained closed-form algebraic expressions for

the Neuber-Papkovich displacement potentials for an

arbitrary point force acting in an infinite medium

consisting of two elastic half-spaces in welded contact.

Dundurs and Hetenyi (1965) obtained these functions

when the half-spaces are in smooth contact and also

obtained the corresponding displacement field. Singh and Garg (1986) obtained the integral expression

for the Airy stress function in an unbounded medium

due to various two-dimensional seismic sources.

Beginning with these expressions, Rani et al. (1991)

obtained the closed form analytical expressions for the

Airy stress function , stresses and displacements in a

homogeneous, isotropic, perfectly elastic half-space due

to various two-dimensional sources by applying the

traction-free boundary conditions at the surface of the

half-space.

Singh and Singh (2000) derived closed form analytical expressions for the displacements and stresses at an

arbitrary point of the half-space caused by a long

vertical tensile fault of finite width. Kumar et al. (2005)

derived analytical expressions for the displacement and

stress fields at any point of the two homogeneous,

isotropic elastic half-spaces in welded contact caused by

a long tensile fault of arbitrary dip and finite width.

Recently, Singh et al. (2013) obtained closed-form

analytical expressions for the displacement and stress

fields at any point of either of the two homogeneous,

isotropic, perfectly elastic half-spaces in smooth contact

caused by various two-dimensional sources embedded in one of the half-spaces.

The aim of the present paper is to study the deformation

of two homogeneous, isotropic, elastic half-spaces that

are in smooth contact caused by a long tensile fault of

finite width. Beginning with the closed form expression

for Airy stress function for two media given by Singh et

al. (2013), we obtained Airy stress function for a long

tensile fault of arbitrary dip and finite width. Then the

expressions for stress and displacement fields of two

media caused by a vertical tensile fault are studied

algebraically and numerically.

2. THEORY

Let the Cartesian coordinates be denoted by ( 1 2 3, ,x x x )

with the 3x -axis vertically downwards. We consider

two homogeneous, isotropic, perfectly elastic half-

spaces that are in smooth contact along the plane

3 0x .The upper half-space ( 3 0x ) is called

medium I and the lower half space ( 3 0x ) is medium

II, with elastic constants 1 1, and 2 2,

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respectively. We consider the plane strain problem

( that can be solved in terms of the Airy stress

functionU .

As shown by Singh and Garg (1986), the Airy stress

function 0U for a line source parallel to

1x -axis

passing through the point ( 0,0,h ) in an infinite

medium can be expressed in the form

3| |1

0 0 0 3 2 0 0 3 2

0

| | sin | | cosk x h

U L M k x h kx P Q k x h kx k e dk

(1)

where the source coefficients 0 0 0, ,L M P and 0Q are independent of k.

For a line source parallel to the 1x -axis acting at the point ( 0,0,h ) of medium II, the expressions for the Airy stress

functions in the two half-spaces are of the form

3(1) 1

1 1 3 2 1 1 3 2

0

sin cos ,kx

U L M kx kx P Q kx kx k e dk

(2)

3(2) 1

0 2 2 3 2 2 2 3 2

0

sin cos ,kx

U U L M kx kx P Q kx kx k e dk

(3)

where U0 is given by equation (1) and 1 1 1 1 2 2 2, , , , , ,L M P Q L M P and 2Q are unknowns to be determined from the

boundary conditions. We assume that the half-spaces are in smooth contact along the plane x3 = 0. Therefore the

boundary conditions are

(1) (2)

33 33 , (1)

23 0 ,

(1) (2)

3 3u u ,

(2)

23 0 , at 3 0x

. (4) We have the Airy stress function for medium I and medium II given by Singh et al. (2013) as:

(1) 12 3 3 321 2 2

3

( )tan log

x x x h xxU C L P R

R h x R

22

2 3 3 3 32

2 4 2 4

2 ( ) 2 ( )hx x h x hx x hx h hM Q

R R R R

(5)

20 2 3(2) 1 0 32

0 0 2 2

3

( )tan log

M x x h Q x hxU L P R

x h R R

12 2 3 2 3 322 22 2

3

( )(1 ) tan (1 ) log

C x x C x x hxL C P C S

S x h S

2

2 22 3 3 2 3 323 2 3 22 2 2

2 ( ) 2 ( )( )

C hx x h C hx x hx Mx h hC Q x h h C

S S S

(6)

where

2 2 2

2 3( )R x x h , 2 2 2

2 3( ) ,S x x h 3x h

1

2

2C

, 2

1

2C

, 1

2

,

1 2

1

. (7)

Following Singh and Singh (2000) and Singh et al. (2013) , the Airy stress function due to a long tensile fault of

arbitrary dip in two mediums I and II as:



(1) 3 3 2 3 31 1 1 2

2 2 4

( ) 2 ( )log sin 2

x h x hx x h xC bds x hU R

R R R

22

3 3

2 4

2 ( )cos 2

hx x hh

R R

(8)

(2) 2 3 32 22 2

( )log (1 ) log

C x x hbdsU R C S

S

2 3 2 3 2 2 3 32 2

2 2 2 4

( ) 2 ( )(1 )sin 2

x x h x x C hx x x hh C x

R S S S

2 2 22

3 3 2 3 32

2 2 2 4

( ) ( ) 2 ( )cos 2

x h x h C hx x hh C

R S S S

(9)

where the values of coefficients and are

given in Appendix and have different values for

. Let and be the values of

and respectively valid for .

Equations (8) and (9) give the Airy stress function for a

long tensile dislocation at the point ( 0,h ). However, if

the line source is located at an arbitrary point ( 2 3,y y ),

2x and h in equations (8) and (9) should be replaced

by ( 2 2x y ) and 3y respectively. We will obtain

(1) 3 3 3 2 2 3 3 3 2 2 3 31 1 1

2 2 4

( ) ( ) 2 ( )( )log sin 2

x x y x y y x y x y x yC bdsU R

R R R

`

2 2

3 3 3 3 3

2 4

2 ( )cos 2

y x y x y

R R

(10)

(2) 2 3 3 3 2 2 3 32 22 2 2

( ) ( )( )log (1 ) log sin 2

C x x y x y x ybdsU R C S

S R

2

2 2 2 2

3 3 3 3 2 3 2 3 3 3 3

2 2 4

( ) ( ) 2 ( )cos 2

x y x y C y C x y x y

R S SS

(11)

where 2 2 2

2 2 3 3( ) ( )R x y x y , 2 2 2

2 2 3 3( ) ( )S x y x y .

x1

Medium I: 1 1, O

Medium II: 2 2, 2x

h

Line Source

(0,0,h)

3x

Fig.1. Two half-spaces in smooth contact with a line source in the lower half-space at (0,0,h)



O δ

x2

x3

(s1, δ )

(s2, δ )

n

Fig. 1(a) Geometry of a long tensile fault of width 2 1L s s

(the Cartesian coordinates of a point on the fault

are ( 2 3,y y ) and its polar coordinates are ,s , where is the dip angle and 1 2s s s )

From Figure 1(a), we put 2 3cos , siny s y s into equations (10) and (11) and integrate over s between the

limits 1 2( , )s s . We will obtain the following expression for the airy stress function of medium I and II for a long

tensile fault of finite width 2 1L s s ;

2

1

(1) 2 2 231 1 12 2 3 2 32

sin( cos ) log cos ( cos sin )

s

s

xC bU s x R s x x s x x

R

(12)

(2) 22 2

3 2 3 22 cos ( sin cos ) log ( sin cos ) logb

U s s x x R s x x S

2

1

2 2 232 2 2 3 3 22

sincos ( cos ) log ( sin cos )

s

s

xC s s x S x x s x x

S

(13)

Using equations (12) and (13) the expressions for stress and displacement components due to a vertical tensile fault in two mediums are as:

2

1

2 2 2(1) 2 3 3 2 3 31 1 122 2 4 6

2 ( ) 8 ( )s

s

s x x x s x x s x sC b s

R R R

(14)

2

1

3

(1) 2 3 2 31 1 1

23 4 6

6 8s

s

x x s x x sC b

R R

(15)

2

1

2 2 2(1) 2 3 3 2 3 31 1 133 2 4 6

2 ( ) 8 ( )s

s

s x x x s x x s x sC b s

R R R

(16)

2(2) 3 3 2 2 32 222 2 2 4

(1 ) 2 ( )x s x C s x x sb

R S R

2

1

2 2 2 2

2 2 3 3 2 3 2 2 3 3

4 6

2 ( ) 2 ( ) 8 ( )s

s

sC x x x s x x s C x x s x s

S S

(17)



2

1

2 2 2

(2) 2 3 2 3 2 3 2 2 3 32 2 2 2

23 2 2 4 4 6

2 ( ) 2 ( ( ) ) 8 ( )s

s

x x s x x sC x s C x x s x sb x x

R S R S S

(18)

2(2) 3 3 2 2 32 233 2 2 4

(1 ) 2 ( )x s x C s x x sb

R S R

2

1

2 2 2 2

2 2 3 3 2 3 2 2 3 3

4 6

2 ( ) 2 ( ) 8 ( )s

s

sC x x x s x x s C x x s x s

S S

(19)

and 2

1

(1) 2 3 31 1 1 22 2 4

1

2 ( )1

2

s

s

x x s x sC b x su

R R

(20)

(2) 2 3 2 2 3 32 22 3 22 2 4

2

( ) 2 ( )1( ) 1

2

x x s C x x s x sb xu x s C s

R S S

2

1

1 12 2

2 3 3

1tan tan

s

s

x x

x s x s

(21) 2

1

2(1) 3 2 3 31 13 2 4 2

1

2 ( )11

2

s

s

x s x x s s x sC bu

R R R

(22)

2

1

2 222 3 2 3 2(2) 2 2 32 2

3 2 2 2 4

2

(1 )(x ) 211 log /

2

s

s

C s x s x C x x sb xu C R S

R S S

(23)

3. NUMERICAL RESULTS

The study of the two-dimensional stress and displacement fields of two half-spaces in smooth contact due to a

vertical tensile fault of finite width L is being done. For numerical calculations, we assume that the half-spaces are

Poissonian ( i i ) so that 1 2 2 3 , 1 0s , 2s L , 1

2

1C

, 2

2

1C

.

We define the following dimensionless quantities;

2xY

L , 3x

ZL

, ( ) ( )k k

ij ij

k

LP

b

, ( ) ( )k k

i iU ub

( 1,2k ). (24)

From equations (14) to (24), we get the following expressions for the dimensionless stresses and displacements for

the two half-spaces;

2 2 2

(1)

22 1 2 4 6

1 2( ) 8 ( 1)(2 / 3)

Z Z Y Y Z ZP C

A A A

(25) 3

(1)

23 1 4 6

6 8(2 / 3)

YZ Y ZP C

A A

(26) 2 2 2

(1)

33 1 2 4 6

1 2( ) 8 ( 1)(2 / 3)

Z Z Y Y Z ZP C

A A A

(27) 2

(2) 2

22 2 2 4

(1 )1 2 ( 1)(2 / 3)

Z CZ Y ZP

A B A

(28)



2 2 2 2

2 2

4 6

2 ( ) 2 ( 1) 8 ( 1)C Y Z Z Y Z C Y Z Z

B B

2 22

(2) 2 223 2 2 4 4 6

2 ( Z ( 1) ) 8 ( 1)2 ( 1)(2 / 3)

Y C Z C YZ ZY Y Y ZP

A B A B B

(29) 2

(2) 233 2 2 4

(1 )1 2 ( 1)(2 / 3)

Z CZ Y ZP

A B A

2 2 2 2

2 2

4 6

2 ( ) 2Y ( 1) 8 ( 1)C Y Z Z Z C Y Z Z

B B

(30)

(1)

2 1 2 4

2 ( 1)(1/ 3)

2

Y YZ ZU C

A A

(31) 2

(1)

3 1 2 4

3 5 2(1/ 3)

2

Z Y ZU C

A A

(32)

2(2) 22 2 2 4

( 1) (1/ 2) 2 ( 1)( 1)(1/ 3)

Y Z C C YZ ZY ZU

A B B

1 13 3tan tan

2 1 2 1

Y Y

Z Z

(33)

2 22

(2) 2 2

3 2 2 4

(5 3) 2 21(1 / 3) log

2 2

C Z Y C Y ZYU A B

A B B

(34)

where 2 2 2( 1)A Y Z

, 2 2 2( 1)B Y Z

. 4. DISCUSSION

Figures 2(a) - 4(c), show the variation of the

dimensionless stresses 22 23,P P and 33P with distance

from the fault for the rigidity contrast

1 2 1 5,1,3 2 respectively for two half-spaces in

smooth contact. We find that all the stresses are zero for

some values of 2x near the origin. As the rigidity

contrast increases, maximum and minimum values of

stress components change accordingly. Variation of

stresses with depth from the fault is shown in figures 5(a) to 7(c). Here also, the variation is examined for the

same rigidity ratios 1 2 1 5,1,3 2 . It is observed

that 23 0P at 3 0x while 22P and 33P has non-

zero values at 3 0x .


dimensionless distances 2U and 3U with distance from

the fault for rigidity ratios 1 2 1 5,1,3 2

respectively for medium I and medium II. We find that

near the fault, the variation of horizontal displacement

2U with distance from the fault is conspicuous,

however, as we move away from the fault this variation

becomes smooth. As the rigidity contrast increases, 3U

converges rapidly to zero.


dimensionless distances 2U and 3U with depth for the

two half-spaces for rigidity contrasts

1 2 1 5,1,3 2 respectively. We find that near the

fault, 3U is positive for medium I and negative for

medium II. As we move away from the fault, the values

of 3U for medium I and medium II overlap and tend to

zero. As the rigidity ratio increases, the gap between the

values of 2U for medium I and medium II becomes

smaller and finally 2U tends to zero as 3x approaches

to infinity for all the cases.

5. CONCLUSION

In both the mediums, all the stresses and displacements

tend to zero as 2x and 3x approaches to infinity

respectively. We observe that a change in the value of

1 2 alters the magnitude of stresses and

displacements.



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-15

-10

-5

0

5

10

15

DIMENSIONLESS DISTANCE FROM THE FAULT

P22

Medium l

Medium ll

Fig.2 (a) Variation of the dimensionless normal stress

with distance from the fault for 1 2 =1 5 for

smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-40

-30

-20

-10

0

10

20

30

40


P22

Medium l

Medium ll

Fig.2 (b) Variation of the dimensionless normal stress

22P with distance from the fault for 1 2 =1 for

smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

-40

-30

-20

-10

0

10

20

30

40

50


P22

Medium l

Medium ll

Fig.2(c) Variation of the dimensionless normal stress

22P with distance from the fault for 1 2 = 3 2 for

smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20

-10

0

10

20

30

40

50


P23

Medium l

Medium ll

Fig.3 (a) Variation of the dimensionless shear stress

23P with distancefrom the fault for 1 2 =1 5 for

smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20

-10

0

10

20

30

40

50

DIMENSIOMLESS DISTANCE FROM THE FAULT

P23

Medium l

Medium ll

Fig.3 (b) Variation of the dimensionless shear stress

23P with distance from the fault for 1 2 1 for

smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20

-10

0

10

20

30

40

50

60


P23

Medium l

Medium ll

Fig.3(c) Variation of the dimensionless shear stress

23P with distance from the fault for 1 2 3 2 for

smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-10

-5

0

5

10

15


P33

Medium l

Medium ll



smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-30

-20

-10

0

10

20

30

40

50


P33

Medium l

Medium ll

Fig.4 (b) Variation of the dimensionless normal

stress 33P with distance from the fault for 1 2 1

for smooth contact



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-40

-30

-20

-10

0

10

20

30

40

50

60


P33

Medium l

Medium ll

Fig.4(c) Variation of the dimensionless normal stress


smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-10

-5

0

5

10

15

20

DEPTH

P22

Medium l

Medium ll


22P with depth for 1 2 1 5 for smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20

-10

0

10

20

30

40

50

60

DEPTH

P22

Medium l

Medium ll

Fig.5 (b) Variation of dimensionless normal stress

22P with depth for 1 2 1 for smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20

-10

0

10

20

30

40

50

60

70

DEPTH

P22

Medium l

Medium ll

Fig.5(c) Variation of dimensional normal stress

22P with depth for 1 2 3 2 for smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-16

-14

-12

-10

-8

-6

-4

-2

0

2

DEPTH

P23

Medium l

Medium ll

Fig.6 (a) Variation of dimensionless shear stress 23P

with depth for 1 2 1 5 for smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

-40

-30

-20

-10

0

10

DEPTH

P23

Medium l

Medium ll

Fig.6 (b) Variation of dimensionless shear stress 23P

with depth for 1 2 1 for smooth contact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-60

-50

-40

-30

-20

-10

0

10

DEPTH

P23

Medium l

Medium ll

Fig.6(c) Variation of dimensionless shear stress 23P


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-25

-20

-15

-10

-5

0

5

10

DEPTH

P33

Medium l

Medium ll

Fig.7 (a) Variation of dimensionless normal stress 33P




0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

DEPTH

P33

Medium l

Medium ll

Fig.7 (b) Variation of dimensionless normal stress 33P


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-100

-80

-60

-40

-20

0

20

DEPTH

P33

Medium l

Medium ll

Fig.7(c) Variation of dimensionless normal stress 33P


0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


U2

Medium l

Medium ll

Fig.8 (a) Variation of the dimensionless distance 2U

with distance from the fault for 1 2 1 5 for

smooth contact

0 1 2 3 4 5 6-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5


U2

Medium l

Medium ll

Fig.8 (b) Variation of the dimensionless distance 2U

with distance from the fault for 1 2 1 for smooth

contact

0 1 2 3 4 5 6-3

-2

-1

0

1

2

3


U2

Medium l

Medium ll

Fig.8(c) Variation of the dimensionless distance 2U


smooth contact

0 1 2 3 4 5 6-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6


U3

Medium l

Medium ll



smooth contact

0 1 2 3 4 5 6-4

-3

-2

-1

0

1

2


U3

Medium l

Medium ll


with distance from the fault for 1 2 1 for

smooth contact

0 1 2 3 4 5 6-5

-4

-3

-2

-1

0

1

2


U3

Medium l

Medium ll





smooth contact

0 1 2 3 4 5 6-0.2

0

0.2

0.4

0.6

0.8

1

1.2

DEPTH

U2

Medium l

Medium ll



0 1 2 3 4 5 6-0.5

0

0.5

1

1.5

2

2.5

3

DEPTH

U2

Medium l

Medium ll



0 1 2 3 4 5 6-0.5

0

0.5

1

1.5

2

2.5

3

3.5

DEPTH

U2

Medium l

Medium ll



0 1 2 3 4 5 6-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

DEPTH

U3

Medium l

Medium ll



0 1 2 3 4 5 6-1

0

1

2

3

4

5

6

7

DEPTH

U3

Medium l

Medium ll



0 1 2 3 4 5 6-1

0

1

2

3

4

5

6

7

8

DEPTH

U3

Medium l

Medium ll



6. APPENDIX

1. Vertical dip-slip fault

0 0 0 0L P Q , 0

2 (1 )

bdsM

2. Vertical tensile fault

0 0 0L M , 0 0

2 (1 )

bdsP Q

3. Horizontal tensile fault

0 0 0L M , 0 0

2 (1 )

bdsP Q

The positive sign is for 3x h and the negative sign is

for 3x h in case of vertical dip-slip fault, b is the

magnitude of the displacement dislocation and ds is

the width of the line fault.

4. Stresses and Displacements in terms of Airy

Stress Function

2 ( )

( )

22 2

3

ii U

x

,

2 ( )( )

23

2 3

ii U

x x

,

2 ( )( )

33 2

2

ii U

x



2 2 0U

where ( )i

jk are the components of stress

and

2 22

2 2

2 3x x

, ( 1,2i ; j, k = 2,3).

The displacement fields in terms of U are given by ( )

( ) 2 ( )

2 2

2

12

2

ii i

i

i

Uu U dx

x

,

( )( ) 2 ( )

3 3

3

12

2

ii i

i

i

Uu U dx

x

,

where 2

i ii

i i

=

1

2(1 )i.

REFERENCES

[1] Bonafede M. and Danesi S., 1997: ‘Near-field

modifications of stress induced by dyke

injection at shallow depth’, Geophys. J. Int.,

130, 435-448.

[2] Bonafede M. and Rivalta E., 1999a: ‘The

tensile dislocation problem in a layered elastic

medium’, Geophys. J. Int., 136, 341-356.

[3] Bonafede M. and Rivalta E., 1999b: ‘On

tensile cracks close to and across the interface

between two welded elastic half-spaces’, Geophys. J. Int., 138, 410-434.

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