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Plane strain deformation of an initially unstressed elastic medium
Applied Mathematics and Computation 217 (2011) 8683–8692
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Plane strain deformation of an initially unstressed elastic medium Shamta Chugh a, Dinesh Kumar Madan b,⇑, Kuldip Singh a a b
Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar 125001, India Department of Mathematics, The Technological Institute of Textile and Sciences, Bhiwani 127021, India
a r t i c l e
i n f o
Keywords: Eigenvalue Plane strain Initial stress Line load Fourier transform
a b s t r a c t Selim and Ahmed [1] used the eigenvalue approach by assuming distinct eigenvalues to calculate the elastic deformation due to an inclined load at any point as a result of an inclined line load of initially stressed orthotropic elastic medium. They studied the plane strain problem and obtained the corresponding results for an unstressed orthotropic medium as a particular case. In the present paper, it is shown that all the eigenvalues do not remain distinct, but become repeated when the elastic medium is free from the initial compressive stresses. Further, the displacements and stresses for an unstressed elastic medium have been independently obtained. The variation of the displacements and stresses due to normal and tangential line load are also shown graphically. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The behavior of elastic materials due to loading is of great interest in engineering, soil mechanics and geophysics. When the source surface is very long in one direction in comparison to the others, the use of two-dimensional approximation is justified and consequently calculations are simplified to a great extent and one gets a closed-form analytical solution. A very long strip-source and a very long line source are examples of such two-dimensional sources. The deformation due to loading such as inclined line load, strip-load, continuous line load, etc., is useful in analyzing the field around mining tremors and drilling into the crust of the earth. It can also contribute to the theoretical consideration of the seismic and volcanic sources since it can account for the deformation fields in the entire volume surrounding the source region. Love [2] obtained expressions for the displacements due to line source in an isotropic elastic medium. Maruyama [3] obtained displacement and stress fields corresponding to long strike-slip faults in a homogenous isotropic half-space. Okada [4,5] provided compact analytical expressions for the surface deformation and internal deformation due to inclined shear and tensile faults in a homogenous isotropic half-space. Using the body-force equivalent of dislocation source, as discussed by Burridge and Knopoff [6] and Aki and Richards [7], Pan [8] obtained the response of transversely isotropic layered medium to general dislocation sources. Garg et al. [9] obtained the representation of seismic sources causing anti-plane strain deformation of an orthotropic medium. Kumar et al. [10] used eigenvalue approach to solve the plane strain problem of poroelasticity for an isotropic medium. The corresponding problem for a transversely isotropic medium has been discussed by Kumar et al. [11]. Garg et al. [12] has studied the general plane strain problem of an infinite orthotropic elastic medium, due to two-dimensional sources, without considering the effect of the initial stress present in the medium. By considering distinct eigenvalues, they have used eigenvalue approach to obtain the deformation due to inclined line load. Selim and Ahmed [1] used the same technique to obtain analytical expressions for displacements and stresses at any point as a result of an inclined line load of initially stressed orthotropic elastic medium. It has also been discussed there that the corresponding deformation for an ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (D.K. Madan). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.112
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unstressed medium can be obtained as a particular case. In the present paper we have shown that all the eigenvalues do not remain distinct, rather become repeated when the elastic medium is free from the initial compressive stresses. Therefore, this case cannot be considered as a particular case of the results of Selim and Ahmed [1]. Here, we have obtained the displacements and stresses for an unstressed elastic medium by using the novel analytical eigenvalue method given by Ross [13] for repeated eigenvalues. The variation of the displacements and stresses due to normal and tangential line load are also shown graphically. 2. Theory Selim and Ahmed [1] used an eigenvalue approach and obtained the analytical expressions for the displacement and stresses at any point due to a normal and tangential line loading at the origin of the xy-plane of an initially stressed orthotropic infinite elastic medium. They have considered the plane strain problem. For ready reference the expressions are reproduced hereunder: 2.1. Normal line load
uN ðx; yÞ ¼
F1 P1 log y2 þ n21 x2 P2 log y2 þ n22 x2 ; 4pB11 ðP2 n2 P1 n1 Þ
v N ðx; yÞ ¼ SN11 ðx; yÞ ¼
SN12 ðx; yÞ
F1 log y2 þ n21 x2 log y2 þ n22 x2 ; 4pB11 ðP2 n2 P 1 n1 Þ
! y2 P1 n21 P 2 n22 þ n21 n22 ðP1 P2 Þ B21 x2 yF 1 n21 n22 xF 1 ; 2pðP2 n2 P1 n1 Þ y2 þ n21 x2 y2 þ n22 x2 2pB11 ðP2 n2 P1 n1 Þ y2 þ n21 x2 y2 þ n22 x2
! y2 ðP1 P2 Þ xy n21 n22 þ x2 P 1 n22 P2 n21 yN 1 F 1 ¼ ; 2pB11 ðP 2 n2 P1 n1 Þ y2 þ n21 x2 y2 þ n22 x2
ð1Þ
where the upper sign is for medium I and the lower sign is for medium II and superscript (N) indicates the results due to the normal line load F1. 2.2. Tangential line load
uT ðx; yÞ ¼
v T ðx; yÞ ¼ ST11 ðx; yÞ
F 2 P1 P2 log y2 þ n21 x2 log y2 þ n22 x2 ; 4pN1 ðP2 n1 P1 n2 Þ
F 2 P2 log y2 þ n21 x2 P1 log y2 þ n22 x2 ; 4pN1 ðP 2 n1 P1 n2 Þ
! y2 ðP 1 P2 Þ þ x2 P1 n21 P 2 n22 B11 P1 P2 xy2 F 2 n21 n22 B21 yF 2 ¼ ; 2pN1 ðP2 n1 P1 n2 Þ y2 þ n21 x2 y2 þ n22 x2 2pN1 ðP2 n2 P 1 n1 Þ y2 þ n21 x2 y2 þ n22 x2
ST12 ðx; yÞ ¼
! x2 n21 n22 ðP1 P2 Þ þ y2 P 1 n22 P2 n21 P1 P2 n22 n21 x2 yF 2 xF 2 þ ; 2pðP2 n1 P1 n2 Þ y2 þ n21 x2 y2 þ n22 x2 y2 þ n21 x2 y2 þ n22 x2
ð2Þ
where (T) indicates the results due to tangential line load F2 . The stress–strain relations for initially unstressed orthotropic elastic medium for plane strain problem are Biot [14]
S11 ¼ B11 e11 þ B12 e22 ; S22 ¼ ðB12 PÞe11 þ B22 e22 ; S12 ¼ 2Q 3 e12 :
ð3Þ
Here Sij (i, j = 1, 2) are the incremental stress components, Bij and Q3 are the incremental elastic coefficients and shear modulus respectively. These incremental elastic coefficients are related to Lame’s coefficients k and l of the isotropic unstressed state and are given by
B11 ¼ k þ 2lð1 þ fÞ; B21 ¼ k;
B12 ¼ k þ 2lf;
B22 ¼ k þ 2l;
Q 3 ¼ l;
ð4Þ
S. Chugh et al. / Applied Mathematics and Computation 217 (2011) 8683–8692
N1 ¼ lð1 þ fÞ;
N2 ¼ k þ lð1 þ fÞ and N3 ¼ lð1 fÞ;
8685
ð5Þ
P 2l
where f ¼ is the initial stress parameter. Here, n1 and n2 are the eigenvalues corresponding to eigenvectors P1 and P2, respectively and are given by-
n21
¼
B0 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B20 4C 0
; 2 B2 þ N1 N3 N22 ; B0 ¼ 11 B11 N3
n21
¼
B0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B20 4C 0 2
;
ð6Þ
N1 C0 ¼ N3
and
Pk ¼ i
nk N2 n2k B11 N 1
n2k N1 B11 nk N2
¼ i
for k ¼ 1; 2:
ð7Þ
In Section (6) (p. 230) [1], a particular case has been considered to obtain the corresponding expressions of the displacements and stresses for an initially unstressed orthotropic elastic media, i.e., when f = 0, the medium is free from initial compressive stresses and the elastic coefficients reduce to
B11 ¼ B22 ¼ 2l þ k;
Q ¼l
B12 ¼ B21 ¼ k;
ð8Þ
and the values of N1, N2 and N3 are
N1 ¼ N 3 ¼ l and N2 ¼ k þ l:
ð9Þ
On using these values the characteristic equation [1, Eq. 16], for the initially unstressed orthotropic elastic medium becomes
m4 2g2 m2 þ g4 ¼ 0;
ð10Þ
provided that l – 0 and k + 2l – 0. It is observed that this characteristic equation is independent of elastic modulii and solving, the repeated eigenvalues are obtained, i.e.,
m2 ¼ g2 ; g2 :
ð11Þ
Also from [1], (on using Eqs. (36) and (37) in Eq. (17)), we obtain
B0 ¼ 2;
C 0 ¼ 1;
n21 ¼ n22 ¼ 1
and
m2 ¼ g2 ; g2 ;
ð12Þ
which coincides with the eigenvalues obtained by using Eq. (11). Therefore, from Eq. (11), taking these eigenvalues as
m1 ¼ m2 ¼ m3 ¼ m4 ¼ jgj:
ð13Þ
Here the eigenvalues are repeated, and following the procedure of [1], we can not obtain the results directly. Ross [13] has given a procedure to tackle the problems with repeated eigenvalues, provided the governing vector differential equation is of the first order. Therefore, using this procedure, we obtain the displacement as: The equilibrium equations in transformed domain for an initially unstressed elastic medium are
kþl þ ig u v ; k þ 2l k þ 2l dx 2 d v kþl 2 k þ 2l u þ i ¼ g gv : 2 l l dx 2 d u 2
¼ g2
l
ð14Þ ð15Þ
Eqs. (14) and (15) can be unified in the following first order vector differential equation as
dM ¼ AM; dx
ð16Þ
where
2 3 u 6 v 7 6 7 M ¼ 6 du 7; 4 dx 5 dv dx
2
0 60 6 6 A ¼ 6 l g2 6 kþ2l 4 0
0
1
0
0
0
kþ2l
l
g2
0
3
7 7 kþl 7 7: 0 ig kþ2 l 7 5 i kþll g 0 1
ð17Þ
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Using the eigenvalue method to solve the first order vector differential Eq. (16), let the solution is of the form:
Mðx; kÞ ¼ EðkÞemk :
ð18Þ
We observe that eigenvalues m = m1 and m = m3 are both of multiplicity 2. Therefore, an eigenvector X1 corresponding to the eigenvalue m = m1 = jgj is found to be
2
ijgj 6g 6 X1 ¼ 6 2 4 ig
3 7 7 7; 5
ð19Þ
gjgj to find the other eigenvector X2 corresponding to the eigenvalue m = m2 = jgj, and follow the procedure of Ross [13]. Considering the matrix equation
ðA m1 IÞh ¼ X 1
ð20Þ
that is,
3 3 2 3 2 h ijgj 7 1 60 jgj 0 1 7 7 6 6 6 76 h2 7 6 g 7 7 6 kþl 76 7 ¼ 6 2 7: 6 g2 l g j i g 0 j 6 4 ig 5 kþ2l kþ2l 74 h3 5 5 4 h4 gjgj g2 kþ2l l ig kþ2l l jgj 0 2
jgj
0
1
0
ð21Þ
On solving the above, we have
h1 ¼
2iðk þ 2lÞ ; ðk þ lÞ
g
h2 ¼
; jgj ijgjðk þ 3lÞ h3 ¼ ; ðk þ lÞ h4 ¼ 0:
ð22Þ
Hence the eigenvector X2 (independent of X1) depending upon repeated eigenvalue m1 is
2 n o 3 l i jgjx 2 kþ2 kþ l 7 6 7 6 7 6g x 1 7 6 jgj : X 2 ¼ ðh þ xX 1 Þ ¼ 6 n o 7 7 6 6 ijgj jgjx kþ3l 7 kþl 5 4 gjgjx
ð23Þ
Also, the eigenvector X3 corresponding to the eigenvalue
m ¼ m3 ¼ jgj
ð24Þ
is obtained as
2
ijgj
6g 6 X3 ¼ 6 2 4 ig
3 7 7 7: 5
ð25Þ
gjgj Similarly, following the procedure adopted for X2, we obtain the eigenvector X4 corresponding to the repeated eigenvalue m4 as
n o 3 l i jgjx þ 2 kþ2 kþl 7 6 7 6 7 6g x þ 1 7 6 jgj : X4 ¼ 6 n o 7 7 6 kþ3 l 7 6 ijgj jgjx þ kþl 5 4 2
gjgjx
Therefore, the solution of first order vector differential Eq. (16) can be written as
ð26Þ
S. Chugh et al. / Applied Mathematics and Computation 217 (2011) 8683–8692
M ¼ ðD1 X 1 þ D2 X 2 Þejgjx þ ðD3 X 3 þ D4 X 4 Þejgjx ;
8687
ð27Þ
where D1, D2, D3 and D4 are the coefficients which may depend upon g and elastic constants k and l. Taking
2
3 M 21 6M 7 6 22 7 M¼6 7; 4 M 23 5
ð28Þ
M 24 where
M21 ¼
i ½fðk þ lÞD1 jgj þ D2 fðk þ lÞjgjx 2ðk þ 2lÞggejgjx fðk þ lÞD3 jgj þ D4 fðk þ lÞjgjx þ 2ðk þ 2lÞggejgjx ; kþl ð29Þ
M23 ¼
g ½fD1 jgj þ D2 fjgjx 1ggejgjx þ fD3 jgj þ D4 fjgjx þ 1ggejgjx ; jgj
M22 ¼
ð30Þ
i ½fðk þ lÞD1 g2 þ D2 jgjfðk þ lÞjgjx ðk þ 3lÞggejgjx þ fðk þ lÞD3 g2 þ D4 jgjfðk þ lÞjgjx þ ðk þ 3lÞggejgjx ; kþl ð31Þ
M24 ¼ gjgj½fD1 þ D2 xgejgjx fD3 þ D4 xgejgjx :
ð32Þ
On using Eqs. (28)–(30) in Eq. (17) we get the displacement field in the transformed domain as under:
¼ u
i ½fðk þ lÞD1 jgj þ D2 fðk þ lÞjgjx 2ðk þ 2lÞggejgjx fðk þ lÞD3 jgj þ D4 fðk þ lÞjgjx þ 2ðk þ 2lÞggejgjx ; kþl ð33Þ
v ¼
g jgj
½fD1 jgj þ D2 ðjgjx 1Þgejgjx þ fD3 jgj þ D4 ðjgjx þ 1Þgejgjx :
ð34Þ
Inverting these we obtained the displacement field due to inclined line load acting in an initially unstressed elastic medium
uðx; yÞ ¼
1 2p
Z
1
1
i ½fðk þ lÞD1 jgj þ D2 fðk þ lÞjgjx 2ðk þ 2lÞggejgjx fðk þ lÞD3 jgj þ D4 ððk þ lÞjgjx kþl
þ 2ðk þ 2lÞÞgejgjx eigy dg;
v ðx; yÞ ¼
1 2p
Z
1
1
ð35Þ
g ½fD1 jgj þ D2 ðjgjx 1Þgejgjx þ fD3 jgj þ D4 ðjgjx þ 1Þgejgjx eigy dg: jgj
ð36Þ
Further, the transformed stress–strain relations for plane strain problem for an initially unstressed medium are
du S11 ¼ ðk þ 2lÞ ikgv ; dx dv : S12 ¼ l igu dx Using Eqs. (28), (31) and (32) in (17), we obtain the values of
ð37Þ ð38Þ du dx
and ddxv as under:
du i ½fðk þ lÞD1 g2 þ D2 jgjfðk þ lÞjgjx ðk þ 3lÞggejgjx þ fðk þ lÞD3 g2 þ D4 jgjfðk þ lÞjgjx þ ðk þ 3lÞggejgjx ; ¼ dx kþl ð39Þ dv ¼ gjgj½fD1 þ D2 xgejgjx fD3 þ D4 xgejgjx : dx
ð40Þ
and v from Eqs. (35) and (36) and their derivatives from Eqs. (39) and (40) into Eqs. (37) and (38), we Using the values of u get the transformed stresses as
S11 ¼
2lð2k þ 3lÞ 2lð2k þ 3lÞ ejgjx þ 2lig2 D3 þ iD4 jgj 2ljgjx þ ejgjx ; 2lig2 D1 þ iD2 jgj 2ljgjx kþl kþl
ð41Þ
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ðk þ 2lÞ 1 jgjx ðk þ 2lÞ 1 jgjx e 2gjgj D3 þ D4 x þ e : S12 ¼ l 2gjgj D1 þ D2 x kþl jgj kþl jgj
ð42Þ
Inverting these we get the stresses in an initially unstressed elastic medium
Z 1 1 2lð2k þ 3lÞ ejgjx 2lig2 D1 þ iD2 jgj 2ljgjx 2p 1 kþl
2lð2k þ 3lÞ ejgjx eigy dg; þ 2lig2 D3 þ iD4 jgj 2ljgjx þ kþl
S11 ¼
S12 ¼
l 2p
ðk þ 2lÞ 1 jgjx ðk þ 2lÞ 1 jgjx igy e 2gjgj D3 þ D4 x þ e e dg: 2gjgj D1 þ D2 x kþl jgj kþl jgj 1
Z
ð43Þ
1
ð44Þ
2.3. Deformation due to line source When the infinite medium is free from the initial compressive stresses, we have the displacement and stress field due to a line-load for the medium I (x > 0), here (D1 = D2 = 0)
uI ðx; yÞ ¼
Z 1 i 1 fðk þ lÞD3 jgj þ D4 fðk þ lÞjgjx þ 2ðk þ 2lÞgg ejgjx eigy dg; 2p 1 k þ l
v I ðx; yÞ ¼
1 2p
SI11 ¼
1 2p
SI12 ¼
Z
1
ð45Þ
g fD3 jgj þ D4 ðjgjx þ 1Þg ejgjx eigy dg; jgj
ð46Þ
2lð2k þ 3lÞ ejgjx eigy dg; 2lig2 D3 þ iD4 jgj 2ljgjx þ kþl 1
ð47Þ
Z
l 2p
1
1
ðk þ 2lÞ 1 2gjgj D3 þ D4 x þ ejgjx eigy dg: kþl jgj 1
Z
1
ð48Þ
The deformation for the medium II (x < 0), here (D3 = D4 = 0) is
uII ðx; yÞ ¼
v II ðx; yÞ ¼ SII11 ¼
1 2p
SII12 ¼
i 2p
Z
1
1 fðk þ lÞD1 jgj þ D2 fðk þ lÞjgjx 2ðk þ 2lÞgg ejgjx eigy dg; kþl
1
Z
1
ð49Þ
g fD1 jgj þ D2 ðjgjx 1Þg ejgjx eigy dg; jgj
ð50Þ
2lð2k þ 3lÞ ejgjx eigy dg; 2lig2 D1 þ iD2 jgj 2ljgjx kþl 1
ð51Þ
1 2p
Z
l 2p
1
1
ðk þ 2lÞ 1 2gjgj D1 þ D2 x þ ejgjx eigy dg: kþl jgj 1
Z
1
ð52Þ
2.3.1. Normal line load Consider a normal line load F1 per unit length acting in the positive x-direction on the interface x = 0 along the z-axis. The boundary conditions [1, Eq. 28], are:
uI ðx; yÞ uII ðx; yÞ ¼ 0;
v I ðx; yÞ v II ðx; yÞ ¼ 0; ð53Þ
SI11 ðx; yÞ SII11 ðx; yÞ ¼ F 1 dðyÞ; SI12 ðx; yÞ SII12 ðx; yÞ ¼ 0; where
Z
1
1
dðyÞdy ¼ 1; dðyÞ ¼
1 2p
Z
1
eigy dg:
ð54Þ
1
Using these boundary conditions in Eqs. (45)–(52),we get
fjgjðk þ lÞðD3 þ D1 Þg þ 2fðk þ 2lÞðD2 D4 Þg ¼ 0;
ð55Þ
S. Chugh et al. / Applied Mathematics and Computation 217 (2011) 8683–8692
8689
fðD4 þ D2 Þ þ jgjðD3 D1 Þg ¼ 0;
ð56Þ
2lð2k þ 3lÞ F 1 2lg2 ðD3 D1 Þ þ jgjðD4 þ D2 Þ ¼ ; kþl i
ð57Þ
k þ 2l ðD4 D2 Þ ¼ 0: jgj k þ l
ð58Þ
ðD1 þ D3 Þ þ
1
Solving these equations for the following coefficients, we obtain
F1 kþl ; 2 i4lg k þ 2l F 1 kþl : D2 ¼ D4 ¼ i4ljgj k þ 2l D1 ¼ D3 ¼
ð59Þ ð60Þ
Substituting the values of these coefficients Di (i = 1, 2, 3, 4) in Eqs. (45)–(52) and solving, we obtain the following closed-form expressions for the displacements and stresses at any point in an initially unstressed medium as:
F1 1 k þ 3l kþl x2 log x2 þ y2 þ ; 2 k þ 2l 4pl k þ 2 l x2 þ y 2 F kþl xy ; vN ¼ 1 4pl k þ 2l x2 þ y2 " # ) ( 2 x x y2 F1 kþl x N 2 ; S11 ¼ 2 x þ y2 2p k þ 2l ðx2 þ y2 Þ uN ¼
SN12
" # F1 l y kþl yx2 : ¼ 2 2p k þ 2l x2 þ y2 k þ 2l ðx2 þ y2 Þ2
ð61Þ ð62Þ
ð63Þ
ð64Þ
2.3.2. Tangential line load Consider a line force F2 per unit length acting at the origin in the positive y-direction. The boundary conditions [1, Eq. 32], are:
uI ðx; yÞ uII ðx; yÞ ¼ 0;
v I ðx; yÞ v II ðx; yÞ ¼ 0; SI11 ðx; yÞ SII11 ðx; yÞ ¼ 0;
ð65Þ
SI12 ðx; yÞ SII12 ðx; yÞ ¼ F 2 dðyÞ: Using these boundary conditions in Eqs. (45)–(52), we get
fjgjðk þ lÞðD3 þ D1 Þg þ 2fðk þ 2lÞðD2 D4 Þg ¼ 0;
ð66Þ
jgjðD3 þ D1 Þ þ ðD4 þ D2 Þ ¼ 0;
ð67Þ
2lið2k þ 3lÞ 2lig2 ðD3 D1 Þ þ jgjðD4 þ D2 Þ ¼ 0; k þ 2l
ð68Þ
l 2gjgj D3 þ D4
k þ 2l 1 k þ 2l 1 2gjgj D1 D2 ¼ F 2 : k þ l jgj k þ l jgj
ð69Þ
Solving these equations for the following coefficients, we get
F2 ; 2gjgjl F2 kþl D2 ¼ D4 ¼ : 4lg k þ 2l
D1 ¼ D3 ¼
ð70Þ ð71Þ
Substituting the values of these coefficients Di (i = 1, 2, 3, 4) in Eqs. (45)–(52), we obtain the following closed-form expressions for the displacements and stresses at any point in an initially unstressed medium as:
F2 u ¼ 4pl T
"
# kþl xy2 ; k þ 2l ðx2 þ y2 Þ2
ð72Þ
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vT ¼
F2 1 k þ 3l kþl x2 log x2 þ y2 ; 2 k þ 2l 4pl k þ 2 l x2 þ y 2
ST11
F2 ¼ 2p
ST12
F2 ¼ 2p
"
kþl k þ 2l
"
kþl k þ 2l
)# ( y kþl yx2 ; 2 2 x2 þ y 2 k þ 2l ðx2 þ y2 Þ
ð73Þ
) ( 2 x x y2 ðx2 þ y2 Þ
2
# x 2 : x þ y2
ð74Þ
ð75Þ
The results obtained here agree with the corresponding results of Love [2] and Maruyama [3]. 2.3.3. Inclined line load For an inclined line load F0 per unit length, we have [15]
F 1 ¼ F 0 cos d;
F 2 ¼ F 0 sin d:
Fig. 1. Variation of (a) normal displacement (uN) and (b) tangential displacement (uT) against the horizontal distance (y) on the plane x = 1.0.
ð76Þ
S. Chugh et al. / Applied Mathematics and Computation 217 (2011) 8683–8692
8691
The stresses and displacements subjected to inclined line load can be obtained by superposition of the vertical and tangential cases. The deformation field is given by
uIN ðx; yÞ ¼ uN ðx; yÞ þ uT ðx; yÞ;
v
IN
ðx; yÞ ¼ v ðx; yÞ þ v ðx; yÞ; N
T
ð77Þ ð78Þ
N T SIN 12 ðx; yÞ ¼ S12 ðx; yÞ þ S12 ðx; yÞ;
ð79Þ
N T SIN 11 ðx; yÞ ¼ S11 ðx; yÞ þ S11 ðx; yÞ:
ð80Þ
These results i.e., for normal, tangential and inclined line load, cannot be obtained directly from Eqs. (1) and (2) by taking f = 0, as mentioned in the particular case [1]. 3. Numerical results In the present paper we have obtained the closed-form analytical expressions for the displacements and stresses at any point of an initially unstressed orthotropic elastic medium as a result of the inclined line load F0 per unit length acting on the z-axis with its inclination d with x-direction. For numerical computations, we use the values of elastic constants given by Kebeasy et al. [16] for Aswan crustal structure. In the present model, we take k = 2.22075 1011 dyne/cm2 and l = 1.90930 1011 dyne/cm2.
Fig. 2. Variation of (a) normal stress SN11 and (b) tangential stress ST11 against the horizontal distance (y) on the plane x = 1.0.
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We have plotted graphs in Fig. 1(a) to Fig. 2(b) for the variation of displacement and stresses with the horizontal distance y for a fixed value of x = 1.0 for four different values of d = 0°, 45°, 60° and 90°. The case d = 0° corresponds to a normal line load and d = 90° refers to a tangential line load. Fig. 1(a) exhibits the variation of normal displacement (uN) and Fig. 2(b) shows the tangential displacement (vT). These figures show that the displacements for d = 45°, 60° lie between the corresponding displacements for a normal line load and tangential line load. Fig. 2(a) shows the variation of normal stress SN12 and Fig. 2(b) shows tangential stress ST12 . In each figure, the displacement for d = 45°, 60° lie between the rest two curves. Acknowledgement The authors are thankful to the editor and reviewers for the improvement of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
M.M. Selim, M.K. Ahmed, Plane strain deformation of an initially stressed orthotropic elastic medium, Appl. Math. Comput. 175 (2006) 221–237. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. T. Mauryama, On two-dimensional elastic dislocations in an infinite and semi-infinite medium, Bull. Earthq. Res. Inst., University of Tokyo, 44, 1966. Y. Okada, Surface deformation due to inclined shear and tensile faults in a homogenous isotropic half-space, Bull. Seismol. Soc. Am. 75 (4) (1985) 1135– 1154. Y. Okada, Internal deformation due to inclined shear and tensile faults in a half-space, Bull. Seismol. Soc. Am. 82 (2) (1992) 1018–1040. R. Burridge, L. Knopoff, Body forces equivalent for seismic dislocations, Bull. Seismol. Soc. Am. 54 (6A) (1964) 1875–1888. K. Aki, P.G. Richards, Quantitative Seismology, Theory and Methods, Vol. I and II, W. H. Freeman, San Francisco, 1980. E. Pan, Static response of a transversely isotropic and layered half-space to general dislocation sources, Phys. Earth Planet. Int. 58 (1) (1989) 103–117. N.R. Garg, D.K. Madan, R.K. Sharma, Two-dimensional deformation of an orthotropic elastic medium due to seismic sources, Phys. Earth Planet. Int. 94 (1996) 43–62. R. Kumar, A. Miglani, N.R. Garg, Plane strain problem of poroelasticity using eigenvalue approach, Proc. Indian Acad. Sci. (Earth. Planet. Sci.) 109 (2000) 371–380. 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. N.R. Garg, R. Kumar, A. Goel, A. Miglani, Plane strain deformation of an orthotropic elastic medium using an eigenvalue approach, Earth Planet. Space 55 (2003) 3–8. S.L. Ross, Differential Equations, John Wiley & Sons Inc, New York, 1984. M.A. Biot, Mechanics of Incremental Deformation, John Wiley & Sons Inc, New York, 1965. A.S. Sadda, Elasticity-Theory and Applications, Pergamon Press Inc, New York, 1974. 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.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Plane strain deformation of an initially unstressed elastic medium Shamta Chugh a, Dinesh Kumar Madan b,⇑, Kuldip Singh a a b
Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar 125001, India Department of Mathematics, The Technological Institute of Textile and Sciences, Bhiwani 127021, India
a r t i c l e
i n f o
Keywords: Eigenvalue Plane strain Initial stress Line load Fourier transform
a b s t r a c t Selim and Ahmed [1] used the eigenvalue approach by assuming distinct eigenvalues to calculate the elastic deformation due to an inclined load at any point as a result of an inclined line load of initially stressed orthotropic elastic medium. They studied the plane strain problem and obtained the corresponding results for an unstressed orthotropic medium as a particular case. In the present paper, it is shown that all the eigenvalues do not remain distinct, but become repeated when the elastic medium is free from the initial compressive stresses. Further, the displacements and stresses for an unstressed elastic medium have been independently obtained. The variation of the displacements and stresses due to normal and tangential line load are also shown graphically. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The behavior of elastic materials due to loading is of great interest in engineering, soil mechanics and geophysics. When the source surface is very long in one direction in comparison to the others, the use of two-dimensional approximation is justified and consequently calculations are simplified to a great extent and one gets a closed-form analytical solution. A very long strip-source and a very long line source are examples of such two-dimensional sources. The deformation due to loading such as inclined line load, strip-load, continuous line load, etc., is useful in analyzing the field around mining tremors and drilling into the crust of the earth. It can also contribute to the theoretical consideration of the seismic and volcanic sources since it can account for the deformation fields in the entire volume surrounding the source region. Love [2] obtained expressions for the displacements due to line source in an isotropic elastic medium. Maruyama [3] obtained displacement and stress fields corresponding to long strike-slip faults in a homogenous isotropic half-space. Okada [4,5] provided compact analytical expressions for the surface deformation and internal deformation due to inclined shear and tensile faults in a homogenous isotropic half-space. Using the body-force equivalent of dislocation source, as discussed by Burridge and Knopoff [6] and Aki and Richards [7], Pan [8] obtained the response of transversely isotropic layered medium to general dislocation sources. Garg et al. [9] obtained the representation of seismic sources causing anti-plane strain deformation of an orthotropic medium. Kumar et al. [10] used eigenvalue approach to solve the plane strain problem of poroelasticity for an isotropic medium. The corresponding problem for a transversely isotropic medium has been discussed by Kumar et al. [11]. Garg et al. [12] has studied the general plane strain problem of an infinite orthotropic elastic medium, due to two-dimensional sources, without considering the effect of the initial stress present in the medium. By considering distinct eigenvalues, they have used eigenvalue approach to obtain the deformation due to inclined line load. Selim and Ahmed [1] used the same technique to obtain analytical expressions for displacements and stresses at any point as a result of an inclined line load of initially stressed orthotropic elastic medium. It has also been discussed there that the corresponding deformation for an ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (D.K. Madan). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.112
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unstressed medium can be obtained as a particular case. In the present paper we have shown that all the eigenvalues do not remain distinct, rather become repeated when the elastic medium is free from the initial compressive stresses. Therefore, this case cannot be considered as a particular case of the results of Selim and Ahmed [1]. Here, we have obtained the displacements and stresses for an unstressed elastic medium by using the novel analytical eigenvalue method given by Ross [13] for repeated eigenvalues. The variation of the displacements and stresses due to normal and tangential line load are also shown graphically. 2. Theory Selim and Ahmed [1] used an eigenvalue approach and obtained the analytical expressions for the displacement and stresses at any point due to a normal and tangential line loading at the origin of the xy-plane of an initially stressed orthotropic infinite elastic medium. They have considered the plane strain problem. For ready reference the expressions are reproduced hereunder: 2.1. Normal line load
uN ðx; yÞ ¼
F1 P1 log y2 þ n21 x2 P2 log y2 þ n22 x2 ; 4pB11 ðP2 n2 P1 n1 Þ
v N ðx; yÞ ¼ SN11 ðx; yÞ ¼
SN12 ðx; yÞ
F1 log y2 þ n21 x2 log y2 þ n22 x2 ; 4pB11 ðP2 n2 P 1 n1 Þ
! y2 P1 n21 P 2 n22 þ n21 n22 ðP1 P2 Þ B21 x2 yF 1 n21 n22 xF 1 ; 2pðP2 n2 P1 n1 Þ y2 þ n21 x2 y2 þ n22 x2 2pB11 ðP2 n2 P1 n1 Þ y2 þ n21 x2 y2 þ n22 x2
! y2 ðP1 P2 Þ xy n21 n22 þ x2 P 1 n22 P2 n21 yN 1 F 1 ¼ ; 2pB11 ðP 2 n2 P1 n1 Þ y2 þ n21 x2 y2 þ n22 x2
ð1Þ
where the upper sign is for medium I and the lower sign is for medium II and superscript (N) indicates the results due to the normal line load F1. 2.2. Tangential line load
uT ðx; yÞ ¼
v T ðx; yÞ ¼ ST11 ðx; yÞ
F 2 P1 P2 log y2 þ n21 x2 log y2 þ n22 x2 ; 4pN1 ðP2 n1 P1 n2 Þ
F 2 P2 log y2 þ n21 x2 P1 log y2 þ n22 x2 ; 4pN1 ðP 2 n1 P1 n2 Þ
! y2 ðP 1 P2 Þ þ x2 P1 n21 P 2 n22 B11 P1 P2 xy2 F 2 n21 n22 B21 yF 2 ¼ ; 2pN1 ðP2 n1 P1 n2 Þ y2 þ n21 x2 y2 þ n22 x2 2pN1 ðP2 n2 P 1 n1 Þ y2 þ n21 x2 y2 þ n22 x2
ST12 ðx; yÞ ¼
! x2 n21 n22 ðP1 P2 Þ þ y2 P 1 n22 P2 n21 P1 P2 n22 n21 x2 yF 2 xF 2 þ ; 2pðP2 n1 P1 n2 Þ y2 þ n21 x2 y2 þ n22 x2 y2 þ n21 x2 y2 þ n22 x2
ð2Þ
where (T) indicates the results due to tangential line load F2 . The stress–strain relations for initially unstressed orthotropic elastic medium for plane strain problem are Biot [14]
S11 ¼ B11 e11 þ B12 e22 ; S22 ¼ ðB12 PÞe11 þ B22 e22 ; S12 ¼ 2Q 3 e12 :
ð3Þ
Here Sij (i, j = 1, 2) are the incremental stress components, Bij and Q3 are the incremental elastic coefficients and shear modulus respectively. These incremental elastic coefficients are related to Lame’s coefficients k and l of the isotropic unstressed state and are given by
B11 ¼ k þ 2lð1 þ fÞ; B21 ¼ k;
B12 ¼ k þ 2lf;
B22 ¼ k þ 2l;
Q 3 ¼ l;
ð4Þ
S. Chugh et al. / Applied Mathematics and Computation 217 (2011) 8683–8692
N1 ¼ lð1 þ fÞ;
N2 ¼ k þ lð1 þ fÞ and N3 ¼ lð1 fÞ;
8685
ð5Þ
P 2l
where f ¼ is the initial stress parameter. Here, n1 and n2 are the eigenvalues corresponding to eigenvectors P1 and P2, respectively and are given by-
n21
¼
B0 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B20 4C 0
; 2 B2 þ N1 N3 N22 ; B0 ¼ 11 B11 N3
n21
¼
B0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B20 4C 0 2
;
ð6Þ
N1 C0 ¼ N3
and
Pk ¼ i
nk N2 n2k B11 N 1
n2k N1 B11 nk N2
¼ i
for k ¼ 1; 2:
ð7Þ
In Section (6) (p. 230) [1], a particular case has been considered to obtain the corresponding expressions of the displacements and stresses for an initially unstressed orthotropic elastic media, i.e., when f = 0, the medium is free from initial compressive stresses and the elastic coefficients reduce to
B11 ¼ B22 ¼ 2l þ k;
Q ¼l
B12 ¼ B21 ¼ k;
ð8Þ
and the values of N1, N2 and N3 are
N1 ¼ N 3 ¼ l and N2 ¼ k þ l:
ð9Þ
On using these values the characteristic equation [1, Eq. 16], for the initially unstressed orthotropic elastic medium becomes
m4 2g2 m2 þ g4 ¼ 0;
ð10Þ
provided that l – 0 and k + 2l – 0. It is observed that this characteristic equation is independent of elastic modulii and solving, the repeated eigenvalues are obtained, i.e.,
m2 ¼ g2 ; g2 :
ð11Þ
Also from [1], (on using Eqs. (36) and (37) in Eq. (17)), we obtain
B0 ¼ 2;
C 0 ¼ 1;
n21 ¼ n22 ¼ 1
and
m2 ¼ g2 ; g2 ;
ð12Þ
which coincides with the eigenvalues obtained by using Eq. (11). Therefore, from Eq. (11), taking these eigenvalues as
m1 ¼ m2 ¼ m3 ¼ m4 ¼ jgj:
ð13Þ
Here the eigenvalues are repeated, and following the procedure of [1], we can not obtain the results directly. Ross [13] has given a procedure to tackle the problems with repeated eigenvalues, provided the governing vector differential equation is of the first order. Therefore, using this procedure, we obtain the displacement as: The equilibrium equations in transformed domain for an initially unstressed elastic medium are
kþl þ ig u v ; k þ 2l k þ 2l dx 2 d v kþl 2 k þ 2l u þ i ¼ g gv : 2 l l dx 2 d u 2
¼ g2
l
ð14Þ ð15Þ
Eqs. (14) and (15) can be unified in the following first order vector differential equation as
dM ¼ AM; dx
ð16Þ
where
2 3 u 6 v 7 6 7 M ¼ 6 du 7; 4 dx 5 dv dx
2
0 60 6 6 A ¼ 6 l g2 6 kþ2l 4 0
0
1
0
0
0
kþ2l
l
g2
0
3
7 7 kþl 7 7: 0 ig kþ2 l 7 5 i kþll g 0 1
ð17Þ
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Using the eigenvalue method to solve the first order vector differential Eq. (16), let the solution is of the form:
Mðx; kÞ ¼ EðkÞemk :
ð18Þ
We observe that eigenvalues m = m1 and m = m3 are both of multiplicity 2. Therefore, an eigenvector X1 corresponding to the eigenvalue m = m1 = jgj is found to be
2
ijgj 6g 6 X1 ¼ 6 2 4 ig
3 7 7 7; 5
ð19Þ
gjgj to find the other eigenvector X2 corresponding to the eigenvalue m = m2 = jgj, and follow the procedure of Ross [13]. Considering the matrix equation
ðA m1 IÞh ¼ X 1
ð20Þ
that is,
3 3 2 3 2 h ijgj 7 1 60 jgj 0 1 7 7 6 6 6 76 h2 7 6 g 7 7 6 kþl 76 7 ¼ 6 2 7: 6 g2 l g j i g 0 j 6 4 ig 5 kþ2l kþ2l 74 h3 5 5 4 h4 gjgj g2 kþ2l l ig kþ2l l jgj 0 2
jgj
0
1
0
ð21Þ
On solving the above, we have
h1 ¼
2iðk þ 2lÞ ; ðk þ lÞ
g
h2 ¼
; jgj ijgjðk þ 3lÞ h3 ¼ ; ðk þ lÞ h4 ¼ 0:
ð22Þ
Hence the eigenvector X2 (independent of X1) depending upon repeated eigenvalue m1 is
2 n o 3 l i jgjx 2 kþ2 kþ l 7 6 7 6 7 6g x 1 7 6 jgj : X 2 ¼ ðh þ xX 1 Þ ¼ 6 n o 7 7 6 6 ijgj jgjx kþ3l 7 kþl 5 4 gjgjx
ð23Þ
Also, the eigenvector X3 corresponding to the eigenvalue
m ¼ m3 ¼ jgj
ð24Þ
is obtained as
2
ijgj
6g 6 X3 ¼ 6 2 4 ig
3 7 7 7: 5
ð25Þ
gjgj Similarly, following the procedure adopted for X2, we obtain the eigenvector X4 corresponding to the repeated eigenvalue m4 as
n o 3 l i jgjx þ 2 kþ2 kþl 7 6 7 6 7 6g x þ 1 7 6 jgj : X4 ¼ 6 n o 7 7 6 kþ3 l 7 6 ijgj jgjx þ kþl 5 4 2
gjgjx
Therefore, the solution of first order vector differential Eq. (16) can be written as
ð26Þ
S. Chugh et al. / Applied Mathematics and Computation 217 (2011) 8683–8692
M ¼ ðD1 X 1 þ D2 X 2 Þejgjx þ ðD3 X 3 þ D4 X 4 Þejgjx ;
8687
ð27Þ
where D1, D2, D3 and D4 are the coefficients which may depend upon g and elastic constants k and l. Taking
2
3 M 21 6M 7 6 22 7 M¼6 7; 4 M 23 5
ð28Þ
M 24 where
M21 ¼
i ½fðk þ lÞD1 jgj þ D2 fðk þ lÞjgjx 2ðk þ 2lÞggejgjx fðk þ lÞD3 jgj þ D4 fðk þ lÞjgjx þ 2ðk þ 2lÞggejgjx ; kþl ð29Þ
M23 ¼
g ½fD1 jgj þ D2 fjgjx 1ggejgjx þ fD3 jgj þ D4 fjgjx þ 1ggejgjx ; jgj
M22 ¼
ð30Þ
i ½fðk þ lÞD1 g2 þ D2 jgjfðk þ lÞjgjx ðk þ 3lÞggejgjx þ fðk þ lÞD3 g2 þ D4 jgjfðk þ lÞjgjx þ ðk þ 3lÞggejgjx ; kþl ð31Þ
M24 ¼ gjgj½fD1 þ D2 xgejgjx fD3 þ D4 xgejgjx :
ð32Þ
On using Eqs. (28)–(30) in Eq. (17) we get the displacement field in the transformed domain as under:
¼ u
i ½fðk þ lÞD1 jgj þ D2 fðk þ lÞjgjx 2ðk þ 2lÞggejgjx fðk þ lÞD3 jgj þ D4 fðk þ lÞjgjx þ 2ðk þ 2lÞggejgjx ; kþl ð33Þ
v ¼
g jgj
½fD1 jgj þ D2 ðjgjx 1Þgejgjx þ fD3 jgj þ D4 ðjgjx þ 1Þgejgjx :
ð34Þ
Inverting these we obtained the displacement field due to inclined line load acting in an initially unstressed elastic medium
uðx; yÞ ¼
1 2p
Z
1
1
i ½fðk þ lÞD1 jgj þ D2 fðk þ lÞjgjx 2ðk þ 2lÞggejgjx fðk þ lÞD3 jgj þ D4 ððk þ lÞjgjx kþl
þ 2ðk þ 2lÞÞgejgjx eigy dg;
v ðx; yÞ ¼
1 2p
Z
1
1
ð35Þ
g ½fD1 jgj þ D2 ðjgjx 1Þgejgjx þ fD3 jgj þ D4 ðjgjx þ 1Þgejgjx eigy dg: jgj
ð36Þ
Further, the transformed stress–strain relations for plane strain problem for an initially unstressed medium are
du S11 ¼ ðk þ 2lÞ ikgv ; dx dv : S12 ¼ l igu dx Using Eqs. (28), (31) and (32) in (17), we obtain the values of
ð37Þ ð38Þ du dx
and ddxv as under:
du i ½fðk þ lÞD1 g2 þ D2 jgjfðk þ lÞjgjx ðk þ 3lÞggejgjx þ fðk þ lÞD3 g2 þ D4 jgjfðk þ lÞjgjx þ ðk þ 3lÞggejgjx ; ¼ dx kþl ð39Þ dv ¼ gjgj½fD1 þ D2 xgejgjx fD3 þ D4 xgejgjx : dx
ð40Þ
and v from Eqs. (35) and (36) and their derivatives from Eqs. (39) and (40) into Eqs. (37) and (38), we Using the values of u get the transformed stresses as
S11 ¼
2lð2k þ 3lÞ 2lð2k þ 3lÞ ejgjx þ 2lig2 D3 þ iD4 jgj 2ljgjx þ ejgjx ; 2lig2 D1 þ iD2 jgj 2ljgjx kþl kþl
ð41Þ
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S. Chugh et al. / Applied Mathematics and Computation 217 (2011) 8683–8692
ðk þ 2lÞ 1 jgjx ðk þ 2lÞ 1 jgjx e 2gjgj D3 þ D4 x þ e : S12 ¼ l 2gjgj D1 þ D2 x kþl jgj kþl jgj
ð42Þ
Inverting these we get the stresses in an initially unstressed elastic medium
Z 1 1 2lð2k þ 3lÞ ejgjx 2lig2 D1 þ iD2 jgj 2ljgjx 2p 1 kþl
2lð2k þ 3lÞ ejgjx eigy dg; þ 2lig2 D3 þ iD4 jgj 2ljgjx þ kþl
S11 ¼
S12 ¼
l 2p
ðk þ 2lÞ 1 jgjx ðk þ 2lÞ 1 jgjx igy e 2gjgj D3 þ D4 x þ e e dg: 2gjgj D1 þ D2 x kþl jgj kþl jgj 1
Z
ð43Þ
1
ð44Þ
2.3. Deformation due to line source When the infinite medium is free from the initial compressive stresses, we have the displacement and stress field due to a line-load for the medium I (x > 0), here (D1 = D2 = 0)
uI ðx; yÞ ¼
Z 1 i 1 fðk þ lÞD3 jgj þ D4 fðk þ lÞjgjx þ 2ðk þ 2lÞgg ejgjx eigy dg; 2p 1 k þ l
v I ðx; yÞ ¼
1 2p
SI11 ¼
1 2p
SI12 ¼
Z
1
ð45Þ
g fD3 jgj þ D4 ðjgjx þ 1Þg ejgjx eigy dg; jgj
ð46Þ
2lð2k þ 3lÞ ejgjx eigy dg; 2lig2 D3 þ iD4 jgj 2ljgjx þ kþl 1
ð47Þ
Z
l 2p
1
1
ðk þ 2lÞ 1 2gjgj D3 þ D4 x þ ejgjx eigy dg: kþl jgj 1
Z
1
ð48Þ
The deformation for the medium II (x < 0), here (D3 = D4 = 0) is
uII ðx; yÞ ¼
v II ðx; yÞ ¼ SII11 ¼
1 2p
SII12 ¼
i 2p
Z
1
1 fðk þ lÞD1 jgj þ D2 fðk þ lÞjgjx 2ðk þ 2lÞgg ejgjx eigy dg; kþl
1
Z
1
ð49Þ
g fD1 jgj þ D2 ðjgjx 1Þg ejgjx eigy dg; jgj
ð50Þ
2lð2k þ 3lÞ ejgjx eigy dg; 2lig2 D1 þ iD2 jgj 2ljgjx kþl 1
ð51Þ
1 2p
Z
l 2p
1
1
ðk þ 2lÞ 1 2gjgj D1 þ D2 x þ ejgjx eigy dg: kþl jgj 1
Z
1
ð52Þ
2.3.1. Normal line load Consider a normal line load F1 per unit length acting in the positive x-direction on the interface x = 0 along the z-axis. The boundary conditions [1, Eq. 28], are:
uI ðx; yÞ uII ðx; yÞ ¼ 0;
v I ðx; yÞ v II ðx; yÞ ¼ 0; ð53Þ
SI11 ðx; yÞ SII11 ðx; yÞ ¼ F 1 dðyÞ; SI12 ðx; yÞ SII12 ðx; yÞ ¼ 0; where
Z
1
1
dðyÞdy ¼ 1; dðyÞ ¼
1 2p
Z
1
eigy dg:
ð54Þ
1
Using these boundary conditions in Eqs. (45)–(52),we get
fjgjðk þ lÞðD3 þ D1 Þg þ 2fðk þ 2lÞðD2 D4 Þg ¼ 0;
ð55Þ
S. Chugh et al. / Applied Mathematics and Computation 217 (2011) 8683–8692
8689
fðD4 þ D2 Þ þ jgjðD3 D1 Þg ¼ 0;
ð56Þ
2lð2k þ 3lÞ F 1 2lg2 ðD3 D1 Þ þ jgjðD4 þ D2 Þ ¼ ; kþl i
ð57Þ
k þ 2l ðD4 D2 Þ ¼ 0: jgj k þ l
ð58Þ
ðD1 þ D3 Þ þ
1
Solving these equations for the following coefficients, we obtain
F1 kþl ; 2 i4lg k þ 2l F 1 kþl : D2 ¼ D4 ¼ i4ljgj k þ 2l D1 ¼ D3 ¼
ð59Þ ð60Þ
Substituting the values of these coefficients Di (i = 1, 2, 3, 4) in Eqs. (45)–(52) and solving, we obtain the following closed-form expressions for the displacements and stresses at any point in an initially unstressed medium as:
F1 1 k þ 3l kþl x2 log x2 þ y2 þ ; 2 k þ 2l 4pl k þ 2 l x2 þ y 2 F kþl xy ; vN ¼ 1 4pl k þ 2l x2 þ y2 " # ) ( 2 x x y2 F1 kþl x N 2 ; S11 ¼ 2 x þ y2 2p k þ 2l ðx2 þ y2 Þ uN ¼
SN12
" # F1 l y kþl yx2 : ¼ 2 2p k þ 2l x2 þ y2 k þ 2l ðx2 þ y2 Þ2
ð61Þ ð62Þ
ð63Þ
ð64Þ
2.3.2. Tangential line load Consider a line force F2 per unit length acting at the origin in the positive y-direction. The boundary conditions [1, Eq. 32], are:
uI ðx; yÞ uII ðx; yÞ ¼ 0;
v I ðx; yÞ v II ðx; yÞ ¼ 0; SI11 ðx; yÞ SII11 ðx; yÞ ¼ 0;
ð65Þ
SI12 ðx; yÞ SII12 ðx; yÞ ¼ F 2 dðyÞ: Using these boundary conditions in Eqs. (45)–(52), we get
fjgjðk þ lÞðD3 þ D1 Þg þ 2fðk þ 2lÞðD2 D4 Þg ¼ 0;
ð66Þ
jgjðD3 þ D1 Þ þ ðD4 þ D2 Þ ¼ 0;
ð67Þ
2lið2k þ 3lÞ 2lig2 ðD3 D1 Þ þ jgjðD4 þ D2 Þ ¼ 0; k þ 2l
ð68Þ
l 2gjgj D3 þ D4
k þ 2l 1 k þ 2l 1 2gjgj D1 D2 ¼ F 2 : k þ l jgj k þ l jgj
ð69Þ
Solving these equations for the following coefficients, we get
F2 ; 2gjgjl F2 kþl D2 ¼ D4 ¼ : 4lg k þ 2l
D1 ¼ D3 ¼
ð70Þ ð71Þ
Substituting the values of these coefficients Di (i = 1, 2, 3, 4) in Eqs. (45)–(52), we obtain the following closed-form expressions for the displacements and stresses at any point in an initially unstressed medium as:
F2 u ¼ 4pl T
"
# kþl xy2 ; k þ 2l ðx2 þ y2 Þ2
ð72Þ
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vT ¼
F2 1 k þ 3l kþl x2 log x2 þ y2 ; 2 k þ 2l 4pl k þ 2 l x2 þ y 2
ST11
F2 ¼ 2p
ST12
F2 ¼ 2p
"
kþl k þ 2l
"
kþl k þ 2l
)# ( y kþl yx2 ; 2 2 x2 þ y 2 k þ 2l ðx2 þ y2 Þ
ð73Þ
) ( 2 x x y2 ðx2 þ y2 Þ
2
# x 2 : x þ y2
ð74Þ
ð75Þ
The results obtained here agree with the corresponding results of Love [2] and Maruyama [3]. 2.3.3. Inclined line load For an inclined line load F0 per unit length, we have [15]
F 1 ¼ F 0 cos d;
F 2 ¼ F 0 sin d:
Fig. 1. Variation of (a) normal displacement (uN) and (b) tangential displacement (uT) against the horizontal distance (y) on the plane x = 1.0.
ð76Þ
S. Chugh et al. / Applied Mathematics and Computation 217 (2011) 8683–8692
8691
The stresses and displacements subjected to inclined line load can be obtained by superposition of the vertical and tangential cases. The deformation field is given by
uIN ðx; yÞ ¼ uN ðx; yÞ þ uT ðx; yÞ;
v
IN
ðx; yÞ ¼ v ðx; yÞ þ v ðx; yÞ; N
T
ð77Þ ð78Þ
N T SIN 12 ðx; yÞ ¼ S12 ðx; yÞ þ S12 ðx; yÞ;
ð79Þ
N T SIN 11 ðx; yÞ ¼ S11 ðx; yÞ þ S11 ðx; yÞ:
ð80Þ
These results i.e., for normal, tangential and inclined line load, cannot be obtained directly from Eqs. (1) and (2) by taking f = 0, as mentioned in the particular case [1]. 3. Numerical results In the present paper we have obtained the closed-form analytical expressions for the displacements and stresses at any point of an initially unstressed orthotropic elastic medium as a result of the inclined line load F0 per unit length acting on the z-axis with its inclination d with x-direction. For numerical computations, we use the values of elastic constants given by Kebeasy et al. [16] for Aswan crustal structure. In the present model, we take k = 2.22075 1011 dyne/cm2 and l = 1.90930 1011 dyne/cm2.
Fig. 2. Variation of (a) normal stress SN11 and (b) tangential stress ST11 against the horizontal distance (y) on the plane x = 1.0.
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We have plotted graphs in Fig. 1(a) to Fig. 2(b) for the variation of displacement and stresses with the horizontal distance y for a fixed value of x = 1.0 for four different values of d = 0°, 45°, 60° and 90°. The case d = 0° corresponds to a normal line load and d = 90° refers to a tangential line load. Fig. 1(a) exhibits the variation of normal displacement (uN) and Fig. 2(b) shows the tangential displacement (vT). These figures show that the displacements for d = 45°, 60° lie between the corresponding displacements for a normal line load and tangential line load. Fig. 2(a) shows the variation of normal stress SN12 and Fig. 2(b) shows tangential stress ST12 . In each figure, the displacement for d = 45°, 60° lie between the rest two curves. Acknowledgement The authors are thankful to the editor and reviewers for the improvement of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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