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
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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).
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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.
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