当前位置:文档之家› Plane strain deformation of an initially stressed orthotropic

Plane strain deformation of an initially stressed orthotropic

Plane strain deformation of an initially stressed orthotropic
Plane strain deformation of an initially stressed orthotropic

Plane strain deformation of an initially stressed orthotropic elastic medium

M.M.Selim

a,*

,M.K.Ahmed

b

a Mathematical Department,Al-A?aj Community College,King Saud University,

P.O.Box 710,Al-A?aj 11912,Kingdom of Saudi Arabia

b

Department of Mathematics,Faculty of Science,South Valley University,Qena,Egypt

Abstract

The eigenvalue approach,using the Laplace and Fourier transforms,has been employed to ?nd the analytical expressions for displacements and stresses at any point,as a result of an inclined line load,of an initially stressed orthotropic elastic medium.A plain strain problem has been studied.The results in the form of displacement and stress components have been obtained and discussed graphically for a particular model.In this paper it is shown that the displacement and stress components are a?ected by initial stresses in the medium.

ó2005Elsevier Inc.All rights reserved.

Keywords:Plane strain;Initial stress;Orthotropic;Eigenvalue;Fourier transform

1.Introduction

The problems related to prestressed elastic medium has been a subject of continued interest due to its importance in various ?elds,such as earthquake

0096-3003/$-see front matter ó2005Elsevier Inc.All rights reserved.doi:10.1016/j.amc.2005.08.002

*

Corresponding author.

E-mail address:selim23@https://www.doczj.com/doc/c04234664.html, (M.M.

Selim).

Applied Mathematics and Computation 175(2006)

221–237

222M.M.Selim,M.K.Ahmed/https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237 engineering,seismology and geophysics.Signi?cant initial stress way develops in a medium as a result of several physical factors.In fact,the earth is an ini-tially stressed medium.An early e?ort by Cauchy[1]used assumption that stress was due to central forces between particles of the solid.Bromwich[2] examined the e?ects of gravity on surface waves.Southwell[3]discussed the case of uniform initial stress.Love[4]derived the equations for an incompress-ible solid under hydrostatic pressure.A de?nitive theory of the dynamics of prestressed solids,including second-order terms,was developed by Biot[5,6]; an elegant and in-depth exposition of this theory will be found in his treatise entitled Mechanics of Incremental Deformations[7].Recent studies by Tolstoy [8],Dey and Mahto[9],El-Naggar and Selim[10,11],Dey and Dutta[12],and others have considerably increased our understanding of stress di?erences in the lithosphere and upper mantle of the earth.Kumar et al.[13]used eigen-value approach to solve the plain strain problem of poroelasticity for an isotro-pic medium.The corresponding problem for a transversely isotropic medium has been discussed by Kumar et al.[14].Considering the earth model as ortho-tropic instead of isotropic for better approximation,Garg et al.[15]have stud-ied the general plane-strain problem of an in?nite orthotropic elastic-medium due to two-dimensional sources.The e?ect of initial stresses present in the med-ium is,however,not considered in the above study.

In this paper,using the eigenvalue approach,an attempt has been made to ?nd the closed-from expressions for the two-dimensional displacements and stresses at any point of an in?nite initially stressed orthotropic medium due to an inclined line load.

2.Fundamental equations

According to Biot[7,p.52],in the absence of external forces,the equilib-rium equations in the Cartesian co-ordinate system(x,y,z)for the unbounded medium with normal initial stress S11=àP along the horizontal direction (Fig.1)are

o s11 o x t

o s12

o y

t

o s13

o z

àP

o x Z

o y

tP

o x y

o z

?0;

o s21 o x t

o s22

o y

t

o s23

o z

àP

o x Z

o x

?0;

o s31 o x t

o s32

o y

t

o s33

o z

tP

o x y

o x

?0;

e1T

where s ij(i,j=1,2,3)are the incremental stress components and x x,x y,x z are the rational components given by

x x?1

2

o w

o y

à

o v

o z

;

x y?1

2

o u

o z

à

o w

o x

;

x z?1

2

o v

o x

à

o u

o y

;

e2T

where u,v and w are the displacement components.

The stress–strain relations for an initially stressed orthotropic elastic med-ium,with co-ordinate planes as plans,of elastic symmetry,are[7,p.83] S11?B11e11tB12ee22te13T;

S22?eB12àPTe11tB22e22tB23e33;

S33?eB12àPTe11tB23e22tB22e33;

S23?2Q1e23;

S31?2Q2e31;

S12?2Q3e12.

e3T

The incremental strain components e ij(i,j=1,2,3)are related with the dis-placement components(u,v,w)through the relations

e11?o u

o x

;e22?

o v

o y

;e33?

o w

o z

;

e12?e21?1

2

o v

o x

t

o u

o y

;

e13?e31?1

2

o u

o z

t

o w

o x

;

e23?e32?1

2

o w

o y

t

o v

o z

.

e4T

The equilibrium equations in terms of displacement components can be ob-tained from Eqs.(1)–(4).We get

M.M.Selim,M.K.Ahmed/https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237223

B11o2u

o x2

t?Q3tP=2

o2u

o y2

t?Q2tP=2

o2u

o z2

t?B12tQ3àP=2

o2v

o x o y

teB12tQ2àP=2To2w

o x o z

?0;

eB12tQ3àP=2 o2u

t?Q3àP=2

o2v

2

tB22

o2v

2

tQ1

o2v

2

teB23tQ1To2w

o y o z

?0;

eB12tQ2àP=2T

o2u

o x o z

t?Q1tB23

o2v

o y o z

t?Q2àP=2

o2w

o x2

tQ1o2w

o y2

tB22

o2w

o z2

?0.e5T

3.Formulation of the problem

Let us consider an in?nite orthotropic elastic medium with normal initial stress P=àS11along the horizontal direction(Fig.1).Suppose that an in-clined line load F0,per unit length,is acting on the z-axis and its inclination with x-direction is a.

We consider plane strain deformation,parallel to xy-plane,in which the dis-placement components are independent of z and are of the type u?uex;yT;v?vex;yT;w?0.e6T

Assuming the anisotropy induced by initial stresses as orthotropic(in two dimensions)where the principal axes of initial stresses are identi?ed with x, y axes,the stress–strain relations are taken as[7]

S11?B11e11tB12e22;

S22?eB12àPTe11tB22e22;

S12?2Q3e12;

e7T

where B ij(i,j=1,2,3)and Q3are the incremental elastic coe?cients and shear modulus,respectively.

These incremental elastic coe?cients are related to Lame?s coe?cients k,l of the isotropic unstressed state.For the present case[7,p.111],these are B11??kt2le1tfT ;B12?ekt2lfT;

B21?k;B22?kt2l;Q3?l;

e8Twhere f=P/2l is the initial stress parameter.

224M.M.Selim,M.K.Ahmed/https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237

In this case,the equilibrium equations are

B11o2u

o x2

tN1

o2u

o y2

tN2

o2v

o x o y

?0;

N2

o2u

o x o y

tN3

o2v

o x2

tB22

o2v

o z2

?0;

e9T

where N1=l(1+f),N2=[k+l(1+f)]and N3=l(1àf).

We de?ne Fourier transform fex;gTof fex;yT[16]as

fex;gT?F?fex;yT ?

Z1

à1

fex;yTe i g y d y;e10Tso that

fex;yT?

1

2p

Z1

à1

fex;gTeài g y d g;e11T

where g is the transformed Fourier parameter.It was known that

F

o

o y

fex;yT

?eài gTfex;gT;

F

o2

o y2

fex;yT

?àg2fex;gT.

e12T

Inserting relations(10)–(12)in(9),we?nd

B11o2u

o x2

àg2N1utN2eài gT

o v

o x

?0;

N2eài gTo u

o x

tN3

o2v

o x2

tB22eàg2vT?0.

e13T

Eq.(13)can be written in the vector-matrix di?erential equation form as,

A d2R

d x2

ài g C

d R

d x

àg2DR?0;e14T

where A?

B110

0N3

,C?

0N2

N20

,D?

N10

0B22

,R?

u

v

.

We try a solution of the matrix equation(14)of the form[15] Rex;gT?EegTe mx;e15Twhere m is a parameter and E(g)is a matrix of the type2·1.

Substitution of the value of R from Eq.(15)into Eq.(14)we get the follow-ing characteristic equation:

B11N3m4àeB2

11tN1N3àN3

2

Tg2m2tB11N1g4?0.e16T

M.M.Selim,M.K.Ahmed/https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237225

The solution of characteristic Eq.(16)gives the eigenvalues,in case of an orthotropic elastic medium,as

m2?n2

1g2;n2

2

g2;e17T

where

n2 1?

B0t

???????????????????

B2

à4C0

q

2

;n2

2

?

B0à

???????????????????

B2

à4C0

q

2

;

B0?B2

11

tN1N3àN2

2

B11N3

;C0?

N1

N3

.

We assume that n15n2for an orthotropic medium.Then,the eigenvalues, with real parts of{n1,n2}as positive,may be written as

m1?n1j g j;m2?n2j g j;m3?àn1j g j;m4?àn2j g j.e18TThe eigenvalues for the orthotropic elastic medium are obtained by solving the matrix equation

?m2Aài m g Càg2D EegT?0.e19TThe eigenvectors are found to be

E T k ??P k;1 ;E T

kt2

??àP k;1 ;e20T

where

P k?i

n k N2

n

k

B11àN1

?ài

n2

k

N1àB11

n k N2

for k?1;2.e21T

So,the solution of the matrix equation(14)for the case of an orthotropic elastic medium is

Rex;gT?

X2

k?1eC k E T

k

e n k j g j xtC kt2E T

kt2

eàn k j g j xT;e22T

where C1,C2,C3,C4are constants to be-determined from boundary conditions and they may be depend upon g.Solving the matrix equation(14)and using Eqs.(20)and(22),we can write

uex;gT?C1P1e n1j g j xtC2P2e n2j g j xàC3P1eàn1j g j xàC4P2eàn2j g j x;

vex;gT?C1e n1j g j xtC2e n2j g j xtC3eàn1j g j xtC4eàn2j g j x.

e23T

Applying Fourier transform in Eq.(23),we get the displacements for an ini-tially stressed orthotropic elastic medium due to plane strain deformation in the following integral forms:

226M.M.Selim,M.K.Ahmed/https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237

uex;yT?

1

2p

Z1

à1

eC1P1e n1j g j XtC2P2e n2j g j xàC3P1eàn1j g j xàC4P2eàn2j g j xTeài g y d g;

vex;yT?

1

2p

Z1

à1

eC1e n1j g j xtC2e n2j g j xtC3eàn1j g j xtC4eàn2j g j xTeài g y d g.

e24T

From Eqs.(7)and(24),the stress components in integral forms for an ini-tially stressed orthotropic elastic medium due to plane strain deformation are

s11?

1

2p

Z1

à1

eH1C1e n1j g j xtH2C2e n2j g j xtH1C3eàn1j g j xtH2C4eàn2j g j xTeài g y d g;

s12?

1

2p

Z1

à1

Q

3

eH0

1

C1e n1j g j xtH0

2

C2e n2j g j xàH0

1

C3eàn1j g j xàH0

2

C4eàn2j g j xTeài g y d g;

e25T

where

H r?B11P r n r j g jài B12g and H0

r

?n r j g jài P r g;r?1;2.

4.Solution of the problem

Let us consider the in?nite medium as consisting of Medium I(x>0)and Medium II(x<0)of identical elastic properties.

The displacement and stress components for the medium I are

u Iex;yT?

1

2p

Z1

à1

eàC3P1eàn1j g j xàC4P2eàn2j g j xTeài g y d g;

t Iex;yT?

1

2p

Z1

à1

eC3eàn1j g j xtC4eàn2j g j xTeài g y d g;

s I 11?

1

2p

Z1

à1

eH1C3eàn1j g j xtH2C4eàn2j g j xTeài g y d g;

s I 12?

1

2p

Z1

à1

Q

3

eàH0

1

C3eàn1j g j xàH0

2

C4eàn2j g j xTeài g y d g

e26T

and for the Medium II are

u IIex;yT?

1

2p

Z1

à1

eC1P1e n1j g j xtC2P2e n2j g j xTeài g y d g;

t IIex;yT?

1

2p

Z1

à1

eC1e n1j g j xtC2e n2j g j xTeài g y d g;

s II 11?

1

2p

Z1

à1

eH1C1e n1j g j xtH2C2e n2j g j xTeài g y d g;

s II 12?

1

2p

Z1

à1

Q

3

eH0

1

C1e n1j g j xtH0

2

C2e n2j g j xTeài g y d g.

e27TM.M.Selim,M.K.Ahmed/https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237227

5.Boundary conditions

5.1.Normal Line load

Consider a normal line load F1per unit length,acting in the positive x-direc-tion on the interface x=0along the z-axis(Fig.1).Then the boundary condi-tions at x=0are

u Iex;yTàu IIex;yT?0;

v Iex;yTàv IIex;yT?0;

s I 11ex;yTàS II

11

ex;yT?àF1deyT;

s I 12ex;yTàS II

12

ex;yT?0;

e28T

where d(y)is the Dirac delta function satisfying the following properties:

Z1à1deyTd y?1;deyT?

1

2p

Z1

à1

eài g y d g.e29T

From Eqs.(26)–(29),we?nd the values of coe?cients for a normal line load as below.

C4?àC3?àC2?C1?

àF1

2eH2àH1T

;e30T

where H2àH1=B11(P2n2àP1n1)j g j.

Inserting the values of constants C1,C2,C3and C4in Eqs.(26)and(27)and using the standard integrals,we?nd the following closed-form expression for the displacements and stresses at any point of an initially stressed orthotropic in?nite elastic medium as a result of a normal line load F1as

u Nex;yT?

F1

4p B11eP2n2àP1n1T

?eP1logey2tn2

1

x2TàP2logey2tn2

2

x2TT;

v Nex;yT??

F1

4p B11eP2n2àP1n1T

?elogey2tn2x2Tàlogey2tn2

2

x2TT;

s N 11ex;yT?

xF1

2peP2n2àP1n1T

?

y2eP1n2

1

àP2n2

2

Ttn2

1

n2

2

x2eP1àP2T

ey2tn

1

x2Tey2tn

2

x2T

!

?

B21en2

1

àn2

2

Tx2yF1

2p B11eP2n2àP1n1Tey2tn

1

x2Tey2tn

2

x2T

;

s N 12ex;yT?

yN

1

F1

2p B11eP2n2àP1n1T

?

y2eP1àP2T?xyen2

1

àn2

2

Ttx2eP1n2

2

àP2n2

1

T

ey2tn2

1

x2Tey2tn2

2

x2T

!

;

e31T

where the upper sign is for medium I and the lower sign for medium II and superscript(N)indicates the information due to a normal line load F1.

228M.M.Selim,M.K.Ahmed/https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237

5.2.Tangential line load

Assume that a line force F2per unit length,is acting at the origin in the po-sitive y-direction(Fig.1)then the boundary conditions at the horizontal plane x=0are

u Iex;yTàu IIex;yT?0;

v Iex;yTàv IIex;yT?0;

s I 11ex;yTàs II

11

ex;yT?0;

s I 12ex;yTàs II

12

ex;yT?àF2deyT;

e32T

where d(y)is the Dirac delta function.From Eqs.(26)–(29)and using the boundary conditions given in Eq.(32),we?nd the following values of coe?-cients C i for a tangential line load:

C1?C3?

F2P2

2N1eP2H0

1

àP1H0

2

T

;

C2?C4?

àF2P1

2N1eP2H0

1

àP1H0

2

T

.

e33T

Inserting the values of constants C1,C2,C3and C4from Eq.(33)in Eqs.(26) and(27),and then using the standard integrals,we?nd the following closed-form expression for the displacements and stresses at any point of an initially stressed orthotropic in?nite elastic medium as a result of tangential line load F2 as:

u Tex;yT??

F2P1P2

4p N1eP2n1àP1n2T

?elogey2tn2

1

x2Tàlogey2tn2

2

x2TT;

v Tex;yT?

àF2

4p N1eP2n1àP1n2T

?eP2logey2tn2

1

x2TàP1logey2tn2

2

x2TT;

s T 11ex;yT?

B21yF

2

2p N1eP2n1àP1n2T

?

y2eP1àP2Ttx2eP1n2

1

àP2n2

2

T

ey2tn2

1

x2Tey2tn2

2

x2T

!

?

B11P1P2en2

1

àn2

2

Txy2F2

2p N1eP2n1àP1n2Tey2tn2

1

x2Tey2tn2

2

x2T

;

s T 12ex;yT?

xF2

2peP2n1àP1n2T

?

x2n2

1

n2

2

eP1àP2Tty2eP1n2

2

àP2n2

1

T

ey2tn

1

x2Tey2tn

2

x2T

!

t

P1P2en2

1

àn2

2

Tx2yF2

ey2tn

1

x2Tey2tn

2

x2T

;e34T

where(T)indicates results due to tangential line load.

M.M.Selim,M.K.Ahmed/https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237229

5.3.Inclined line load

For an inclined line load F0per unit length,we have(Fig.1)[17] F1?F0cos a;F2?F0sin a.

The displacements and stresses subjected to inclined load can be obtained by superposition of the vertical and tangential cases.The?nal deformation of the formulated problem is given by

uex;yT?ueNTex;yTtueTTex;yT;

vex;yT?veNTex;yTtveTTex;yT;

s11ex;yT?seNT

11ex;yTtseTT

11

ex;yT;

s12ex;yT?seNT

12ex;yTtseTT

12

ex;yT.

e35T

6.Particular case

When f=0,i.e.the medium is free from initial compressive stresses,then the elastic coe?cients

B11?B22?e2ltkT;B12?B21?k;Q3?le36Tand the values of N1,N2and N3from Eq.(9)are

N1?N3?l and N2?ektlTe37Tand hence the analytic expressions for the displacements and stresses(31),(34) and(35)coincide with the expressions obtained by Garg et al.[15].

7.Numerical results

To show the e?ect of initial stresses on the displacements and stresses of the orthotropic elastic medium due to plane strain deformation,numerical compu-tations of Eq.(35)were performed for a particular model.We use the values of elastic constants given by Kebeasy et al.[18]for Aswan crustal structure.In the model considered,we have

l?1:90930?1011dyne=cm;k?2:22075?1011dyne=cm;

B11?kt2le1tfT;B21?k;B12?kt2fl;B22?kt2l;

Q

3

?l;N1?le1tfT;N2?ktle1tfT;N3?le1àfT.

The e?ect of the initial stresses on the displacements and stresses of the orthotropic elastic medium due to plane strain deformation results are pre-230M.M.Selim,M.K.Ahmed/https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237

sented in Figs.2–6for the variation of displacements and stresses against the horizontal distance y for a ?xed value of x =1.0.The variation of f (the

initial

0.00

2.00

4.00 6.008.0010.00

distance(y)

t a n g e n t i a l d i s p l a c e m e n t (v )

0.00

2.00

4.00 6.008.0010.00

distance(y)

n o r m a l d i s p l a c e m e n t (u )

-0.4-0.3

-0.2

-0.10

0.1

0.00

2.00 4.00 6.008.0010.00

distance(y)

n o r m a l s t r e s s

-0.6

-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5

0.60.70.80.910.00

2.00

4.00 6.008.0010.00

distance(y)

t a n g e n t a l s t r e s s

Fig.2.Variation of normal displacement (u ),tangential displacement (m ),normal stress (s 11)and tangential stresses (s 12)against the horizontal distance (y )on the plane x =1.0(n =0.0).

M.M.Selim,M.K.Ahmed /https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237231

stress parameter),has been taken from 0.0to 0.8at the interval 0.2.Each ?gure has four di?erent relations,Variation of normal displacement (u ),tangential displacement (m ),normal stress (s 11)and tangential stresses (s 12)against the horizontal distance (y )on the plane x =1.0,respectively,and each

relation

-0.006

-0.0045

-0.003

-0.0015

0.0015

0.00

2.00 4.00 6.008.0010.00

distance(y)

n o r m a l d i s p l a c e m e n t (u )

-0.006

-0.0045

-0.003

-0.0015

0.0015

0.00

2.00 4.00 6.008.0010.00

distance(y)

t a n g e n t i a l d i s p l a c e m e n t (v )

-0.4

-0.3

-0.2

-0.100.1

0.00

2.00 4.00 6.008.0010.00

distance(y)n o r m a l s t r e s s

-0.6

-0.5

-0.4-0.3-0.2

-0.100.10.20.30.40.5

0.60.70.8

0.910.00

2.00 4.00 6.008.0010.00

distance(y)

t a n g e n t a l l s t r e s s

Fig.3.Variation of normal displacement (u ),tangential displacement (m ),normal stress (s 11)and tangential stresses (s 12)against the horizontal distance (y )on the plane x =1.0(n =2.0).

232

M.M.Selim,M.K.Ahmed /https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237

has four curves corresponding to four di?erent value of a ,namely a =0°,45°,60°and 90°.The case a =0°corresponds to a normal line load and a =90°for a tangential line

load.

-0.006

-0.0045

-0.003

-0.0015

0.0015

0.00

2.00

4.00 6.008.0010.00

distance(y)

n o r m a l d i s p l a c e m e n t (u )

0.0015

0.00

2.00 4.00 6.008.0010.00

distance(y)

t a n g e n t i a l d i s p l a c e m e n t (v )

-0.4

-0.3

-0.2

-0.10

0.1

0.00

2.00 4.00 6.008.0010.00

distance(y)n o r m a l s t r e s s

distance(y)

t a n g e n t a l s t r e s s

Fig.4.Variation of normal displacement (u ),tangential displacement (m ),normal stress (s 11)and tangential stresses (s 12)against the horizontal distance (y )on the plane x =1.0(n =4.0).

M.M.Selim,M.K.Ahmed /https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237

233

Figs.2–6show that the displacements for a =45°,60°lie between the cor-responding displacements for a normal line load and tangential line load.

Also,

-0.006

-0.0045

-0.003

-0.0015

0.0015

0.00

2.00 4.00 6.008.0010.00

distance(y)

n o r m a l d i s p l a c e m e n t (u )

0.00

2.00 4.00 6.008.0010.00

distance(y)

t a n g e n t i a l d i s p l a c e m e n t (v )

-0.4-0.3

-0.2

-0.1

0.1

0.00

2.00 4.00 6.008.0010.00

distance(y)n o r m a l s t r e s s

-0.6-0.5

-0.4-0.3-0.2

-0.100.10.20.30.40.5

0.60.70.80.9

10.00

2.00 4.00 6.008.0010.00

distance(y)

t a n g e n t a l s t r e s s

Fig.5.Variation of normal displacement (u ),tangential displacement (m ),normal stress (s 11)and tangential stresses (s 12)against the horizontal distance (y )on the plane x =1.0(n =6.0).

234

M.M.Selim,M.K.Ahmed /https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237

?gures show that the variation of tangential and normal stresses for di?erent values of a change

steadily.

-0.006

-0.0045-0.003-0.0015

0.0015

distance(y)

n o r m a l d i s p l a c e m e n t (u )

distance(y)

t a n g e n t i a l d i s p l a c e m e n t (v )

-0.4

-0.3

-0.2

-0.1

00.1

distance(y)n o r m a l s t r e s s

-0.6-0.5-0.4-0.3-0.2

-0.100.10.20.30.40.5

0.60.70.8

0.91distance(y)

t a n g e n t a l s t r e s s

Fig.6.Variation of normal displacement (u ),tangential displacement (m ),normal stress (s 11)and tangential stresses (s 12)against the horizontal distance (y )on the plane x =1.0(n =8.0).

M.M.Selim,M.K.Ahmed /https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237

235

236M.M.Selim,M.K.Ahmed/https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237 Fig.2shows the displacements and stresses of the orthotropic elastic medium due to plane strain deformation in the absence of the initial stresses(n=0.0).

In Fig.3,the e?ect of initial stress at n=0.2is shown.It is observed that the values of displacements and stresses are smaller than those in Fig.2.

Fig.4shows that the e?ect of the initial stress at n=0.4.It is observed that the displacements and stresses are smaller than those in Fig.3.

Fig.5shows that the e?ect of the initial stress on the displacements and stresses at n=0.6.Also it is observed that the values of displacements and stresses are smaller than those in Fig.4.

Fig.6gives the change of the values of the displacements and stresses at n=0.8.All the observations made from Figs.3–5are supported by the results shown in this?gure.

In comparison with Fig.2(initial stress-free,i.e.n=0.0)it is observed from Figs.3–6that an increase in compressive initial stresses(n>0.0)in the medium decrease the values of displacements and stresses for the same angle a.It is can be seen from these?gures that the displacements and stresses are highly a?ected by the initial stresses present in the medium.

References

[1]A.L.Cauchy,Exercises de Mathematique,2,Bure Freres,Paris,1827.

[2]T.J.L.A.Bromwich,On the in?uence of gravity on the elastic waves and in particular on the

vibration of an elastic globe,Proc.London Math.Soc.30(1898)98–120.

[3]R.V.Southwell,On the general theory of elastic stability,Philos.Trans.R.Soc.London Ser.A

213(1913)187–244.

[4]A.E.H.Love,in:The Mathematical Theory of Elasticity,Cambridge University Press,1927,

pp.176–178.

[5]M.A Biot,Non linear theory of elasticity and the linearized case for a body under initial

stresses,Philos.Mag.27(7)(1939)468–489.

[6]M.A Biot,The in?uence of initial stresses on elastic waves,J.Appl.Phys.11(8)(1940)520–

530.

[7]M.A.Biot,Mechanics of Incremental Deformation,John Wiley&Sons Inc.,New York,1965.

[8]I.Tolstoy,On elastic waves in prestressed solids,J.Geophys.Res.87(1982)6823–6827.

[9]S.Dey,P.Mahto,Surface waves in a highly pre-stressed medium,Acta.Geophys.Pol.XXXVI

(2)(1988)89–99.

[10]A.M.El-Naggar,M.M.Selim,The propagation of SH-waves in composite medium under

initial stress,Bull.Fac.Sci.Assiut Univ.Egypt23(2-c)(1994)129–142.

[11]A.M.El-Naggar,M.M.Selim,Wave propagation in layered media under initial stresses,Appl.

https://www.doczj.com/doc/c04234664.html,put.74(1996)95–117.

[12]S.Dey,D.Dutta,Propagation and attenuation of seismic body waves in initially stressed

dissipative medium,Acta Geophys.Pol.XLVI(3)(1998)351–365.

[13]R.Kumar,A.Miglani,N.R.Garg,Plane strain problem of poroelastictity using eigenvalue

approach,Proc.Indian Acad.Sci(Earth.Planet Sci)109(2000)371–380.

[14]R.Kumar,A.Miglani,N.R.Garg,Response of an anisotropic liquid-saturated porous

medium due to two-dimensional sources,Proc.Indian Acad.Sci.(Earth.Planet Sci.)111 (2002)143–151.

M.M.Selim,M.K.Ahmed/https://www.doczj.com/doc/c04234664.html,put.175(2006)221–237237 [15]N.R.Garg,R.Kumar,A.Goel,A.Miglani,Plane strain deformation of an orthotropic elastic

medium using an eigenvalue approach,Earth Planet Space55(2003)3–9.

[16]L.Depnath,Integral Transforms and their Application,CRC Press Inc.,New York,1995.

[17]A.S.Saada,Elasticity—Theory and Application,Pergamon Press Inc.,New York,1974.

[18]R.M.Kebeasy,A.I.Bayoumy,A.A.Gharib,Crustal structure modeling for the northern part

of the Aswan Lake area using seismic waves generated by explosions and local earthquake,J.

Geodynam.14(1–4)(1991)159–182.

相关主题
相关文档 最新文档