Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2010, Article ID 817680, 11 pages
doi:10.1155/2010/817680
Research Article
Propagation of Elastic Waves in Prestressed Media
Inder Singh,1 Dinesh Kumar Madan,2 and Manish Gupta3
1
Department of Mathematics, J. V. M. G. R. R. (P.G) College, Charkhi Dadri, Haryana 127306, India
Department of Mathematics, The Technological Institute of Textile and Sciences, Bhiwani 127021, India
3
Department of Electronic & Instumentation, The Technological Institute of Textile and Sciences,
Bhiwani 127021, India
2
Correspondence should be addressed to Inder Singh, is gupta@yahoo.com
Received 13 June 2010; Accepted 29 October 2010
Academic Editor: George Jaiani
Copyright q 2010 Inder Singh et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
3D solutions of the dynamical equations in the presence of external forces are derived for a
homogeneous, prestressed medium. 2D plane waves solutions are obtained from general solutions
and show that there exist two types of plane waves, namely, quasi-P waves and quasi-SV waves.
Expressions for slowness surfaces and apparent velocities for these waves are derived analytically
as well as numerically and represented graphically.
1. Introduction
In fact, the Earth is prestressed medium, due to many physical causes, that is, gravity
variation, slow process of creep and variation of temperature, and so forth. Therefore, these
problems are of much interest to seismologists due to its application in mineral prospecting
and prediction of earthquakes.
For studying the propagation of elastic waves in prestressed solids of infinite extent,
Sidhu and Singh 1, 2, Norris 3, and Day et al. 4 used a much simpler form of equation
of motion of Biot 5. The medium considered by earlier investigators is a homogenous
prestressed with incremental elastic coefficients possessing orthotropic anisotropy due to
normal components of initial stresses.
In the present paper, 3D problem of propagation of P , SV and SH waves for a
homogeneous prestressed medium is discussed. The model of prestressed medium used here
is much more general than that used by earlier investigators.
The 2D plane waves solutions are obtained from general solutions for different
conditions of initial stresses with or without external forces. Graphs of slowness surfaces
are derived.
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2. Basic Equations
Consider a homogenous prestressed solid. The state of prestresses is, therefore, defined by six
components, that is, S11, S22, S33, S12 S21 , S31 S13 , and S23 S32 . Let all stress components
be functions of x, y, z. The state of initial stress introduces anisotropy so that even for an
initially isotropic solid defined by two Lame’s constants λ, μ, the number of the incremental
elastic coefficients Bij, Qi will be always larger than 2. Let X, Y, Z be components of body
forces along coordinate axes, respectively, where X, Y, Z are all constant. The general form of
dynamical equation for prestressed solids in the presence of external forces is given by Biot
5, page 52:
∂S11
∂S12
∂S31
∂s11 ∂s12 ∂s31
ρΔx ρ ωz Y − ωy Z − ρeX − exx
− eyy
− ezz
∂x
∂y
∂z
∂x
∂y
∂z
∂S31 ∂S12
∂S11 ∂S31
∂S12 ∂S11
∂Wz
− eyz
− ezx
− exy
S11 − S22 ∂y
∂z
∂z
∂x
∂x
∂y
∂y
∂Wy ∂W z
∂Wy
∂Wy
∂Wz
∂Wx
∂Wx
S12
S33 − S11 2S12
− S31
S23
−
− 2S12
∂x
∂z
∂z
∂x
∂y
∂y
∂z
ρ
∂2 u1
.
∂t2
2.1
The two other equations are obtained by cyclic permutation of x, y, z, 1, 2, 3 and X, Y, Z.
ρ is the density and u1 , u2 , u3 the displacement components along the axes. ΔXi are the
components of incremental body force, which are assumed to satisfy the equation
ΔXi uj Xi,j 0,
e exx eyy ezz .
2.2
The Sij are the components of prestress, which are assumed to satisfy the equilibrium
equation.
Sij,j ρXi 0,
2.3
and are related to the initial strain ij by Hooke’s law
Sij λii δij 2μij ,
2.4
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The incremental stresses sij are supposed to be linearly related to the incremental strains eij
through the incremental elastic coefficients Bij and Qi
s11 B11 exx B12 eyy B13 ezz,
s22 B21 exx B22 eyy B23 ezz,
s33 B31 exx B32 eyy B33 ezz,
2.5
s23 2Q1 eyz,
s31 2Q2 ezx,
s23 2Q3 exy.
We assume that
u1 , u2 , u3 u, v, w,
exx 1 ∂u ∂v
,
2 ∂y ∂x
1 ∂w ∂v
Wx −
,
2 ∂y ∂z
exy ∂u
,
∂x
eyy X1 , X2 , X3 X, Y, Z,
∂v
,
∂y
ezz 1 ∂v ∂w
,
2 ∂z ∂y
1 ∂u ∂w
−
Wy ,
2 ∂z ∂x
eyz ∂w
,
∂z
1 ∂w ∂u
,
2 ∂x ∂z
1 ∂v ∂u
−
Wz .
2 ∂x ∂y
ezx Substituting 2.2, 2.5, and 2.6 in 2.1, we have
B11
∂2 v 1
∂2 v
∂2 w
P1 ∂2 u
P3 ∂2 u
Q
−
Q
− S12 2 − S12 2 − S31 2
3
2
2
2
2
2 ∂y
2 ∂z
2
∂x
∂x
∂z
∂x
∂2 w
∂2 u
∂2 u
∂2 u
1
P1 ∂2 v
S23
S31
B12 Q3 − S31 2 S12
2
∂x∂y
∂y∂z
∂x∂z
2 ∂x∂y
∂y
∂2 u
1
1
P3 ∂2 w
∂2 v
∂2 v
S31
− S23
B13 Q2 −
2
∂y∂z 2
∂x∂z
2 ∂x∂z
1
∂S11
∂u
∂2 w
∂2 w
1
− S23
−
ρX
S12
2
∂y∂z 2
∂x∂y
∂x
∂x
∂u
1 ∂S12 ∂S11 ∂u 1 ∂S31 ∂S11
−
−
ρZ
2 ∂x
∂y ∂y 2 ∂x
∂z
∂z
2.6
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∂v
∂S12
∂v
1 ∂S12 ∂S11
− ρY
ρX
2 ∂x
∂y
∂x
∂y
∂y
1 ∂S31 ∂S12 ∂v 1 ∂S31 ∂S11
∂w
−
−
− ρZ
2 ∂y
∂z ∂z 2 ∂x
∂z
∂x
∂S31
∂w
1 ∂S31 ∂S12 ∂w
−
ρX
−
2 ∂y
∂z
∂y
∂z
∂z
−
ρ
∂2 u
∂t2
.
2.7
The two other equations are obtained from 2.7 by cycle permutation of x, y, z, 1, 2, 3, X, Y, Z
and u, v, w.
Here
P1 S11 − S22 ,
P2 S22 − S33 ,
2.8
P3 S33 − S11 .
3. Propagation of Waves
For elastic waves propagating in a direction specified by direction cosines l, m, n along the
axes, we take
v Vi expiP ,
u Ui expiP ,
w Wi expiP .
3.1
Ui , Vi , Wi are amplitude factors along the axes and P is phase factor:
P k ct − lx my nz ,
3.2
where c is phase velocity and k is wave number.
Putting 3.1 and 3.2 in 2.7, we get
Ω1 Ω1 ρc2 Ui 1 1 Vi K1 K1 Wi 0,
K2 K2 Ui Ω2 Ω2 ρc2 Vi 2 2 Wi 0,
3 3 Ui K3 K3 Vi Ω3 Ω3 ρc2 Wi 0,
3.3
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where
P1
P3
Ω1 − B11 l2 Q3 −
m2 Q2 n2 S12 lm S23 mn S31 nl ,
2
2
S31
S23
S12
P1
2
2
n − B12 Q3 lm −
mn ln,
1 S12 l 2
2
2
2
K1 S31 l2 Ω1
i
k
S12
S23
S31
P3
m2 − B13 Q2 −
ln −
mn ln,
2
2
2
2
∂S11
1 ∂S12 ∂S11
1 ∂S31 ∂S11
ρX l ρY m ρZ n ,
∂x
2 ∂x
∂y
2 ∂x
∂z
3.4
3.5
3.6
3.7
1 i
k
1 ∂S12 ∂S11
∂S12
1 ∂S31 ∂S12
− ρY l −
ρX m n ,
2 ∂x
∂y
∂y
2 ∂y
∂z
3.8
K1 i
k
1 ∂S31 ∂S11
1 ∂S31 ∂S12
1 ∂S31
− ρZ l ρX n .
m
2 ∂x
∂z
2 ∂y
∂z
2 ∂z
3.9
The explicit expressions for Ω2 , Ω3 , 2 , 3 , K2 , K3 , Ω2 , Ω3 , 2 , 3 , and K2 , K3 are obtained from 3.4–3.9, by cyclic permutation of x, y, z, 1, 2, 3, l, m, n and X, Y, Z,
respectively.
Setting the determinant of the coefficients of Ui , Vi and Wi of 3.3 equal to zero, on
simplification, we get the cubic equation in ρc2 :
ρ3 c6 ρ2 c4 A B D ρc2 AB BD DA − FJ − EI − HG
ABD EFG HIJ − AFJ − EID − HGB 0,
3.10
where
A Ω1 Ω1 ,
E 1 1 ,
H K1 K1 ,
B Ω2 Ω2 ,
F 2 2 ,
I K2 K2 ,
D Ω3 Ω3 ,
G 3 3 ,
J K3 K3 .
4. Special Cases
Two special cases may be dealt with immediately.
3.11
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4.1. Three Dimensional Prestressed Medium
4.1.1. Propagation of Waves along the Unique Axis
Putting n 1, l m 0 in 3.3–3.8 and in the expressions of Ω2 , Ω3 ,
2 , 3 ,K2 , K3 , Ω2 , Ω3 , 2 , 3 , and K2 , K3 , we get
P3
,
Ω11 − Q2 2
11 P2
Ω21 − Q1 −
,
2
S12
,
2
K11 0,
21 0,
K21 31 S31 ,
4.1
K31 S23 ,
∂S31 ∂S11
i ∂S23 ∂S22
i ∂S33
ρz ,
Ω21 ρz ,
Ω31 ρz ,
∂x
∂z
2k ∂y
∂z
k ∂x
i ∂S31 ∂S12
i ∂S23
i ∂S33 ∂S31
,
ρY ,
− ρX ,
11 21 −
31 2k ∂y
∂z
k ∂z
k ∂x
∂z
i ∂S31
i ∂S23 ∂S12
i ∂S33 ∂S23
K11 ρX ,
K21 ,
K31 ρY .
k ∂z
2k ∂x
∂z
2k ∂y
∂z
4.2
Ω11 i
2k
S12
,
2
Ω31 −B33 ,
Equation 3.10 takes the form
ρ3 c6 ρ2 c4 A1 B1 D1 ρc2 A1 B1 B1 D1 D1 A1 − F1 J1 − E1 I1 − H1 G1 A1 B1 D1 E1 F1 G1 H1 I1 J1 − A1 F 1 J1 − E1 I1 D1 − H1 G1 B1 0,
4.3
where
A1 Ω11 Ω11 ,
B1 Ω21 Ω21 ,
E1 11 11 ,
H1 K11 ,
F1 21 ,
I1 K21 K21 ,
D1 Ω31 Ω31 ,
G1 31 31 ,
4.4
J1 K31 K31 .
4.1.2. In the Absence of Body Forces
When X Y Z 0 and S11 , S22 , S33 ,S12 and so forth are all constant, then 4.3 becomes,
with the help of 4.2 and 4.4,
ρ3 c6 ρ2 c4 Ω11 Ω21 Ω31 ρc2 Ω11 Ω21 Ω21 Ω31 Ω31 Ω11 − 11 K21 Ω11 Ω21 Ω31 − 11 K21 Ω31 0.
4.5
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4.1.3. Prestressed Is Defined by Normal Components
There are only normal components of prestress present in the medium, that is putting S12 S31 S23 0 in 4.1, 4.5 takes the form
Ω11 ρc2
Ω21 ρc2
Ω31 ρc2 0;
4.6
on simplification, we get three values of c2 :
c2 Q2 P3 /2
,
ρ
c2 Q1 − P2 /2
,
ρ
c2 B33
.
ρ
4.7
4.2. Two Dimensional Prestressed Medium
4.2.1. Plane Waves Solution
Here, we consider the behaviour of plane waves in xy-plane perpendicular to the z-axis;
putting n 0 in 3.4–3.9 and in the expressions of Ω2 , Ω3 and so forth, we get the set of
equations from 3.3:
Ω111 Ω111 ρc2 Ui 111 111 Vi 0,
K211 K211 Ui Ω211 Ω211 ρc2 Vi 0,
4.8
where
P1
m2 S12 ml ,
Ω111 − B11 l2 Q3 −
2
P
111 S12 l2 − B12 Q3 1 ml ,
2
P1 2
l B22 m2 S12 ml ,
Ω211 − Q3 2
P1
nl ,
K211 S12 m2 − B12 Q3 −
2
∂S11
1 ∂S12 ∂S11
i
Ω111 ρX l ρY m ,
k
∂x
2 ∂x
∂y
∂S22
i 1 ∂S22 ∂S12
Ω211 ρX l ρY m ,
k 2 ∂x
∂y
∂y
∂S12
i 1 ∂S12 ∂S11
− ρY l −
ρX m ,
111 k 2 ∂x
∂y
∂y
∂S12
1 ∂S22 ∂S12
i
K211 ρY l ρX m .
k
∂x
2 ∂x
∂y
4.9
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Equation 4.8 has nontrivial solution when
ρ2 c4 Ω111 Ω211 Ω111 Ω211 ρc2 − 111 111 K211 K211 0.
4.10
It is quadratic equation in ρc2 , and it has two values of c2 corresponding to quasi-SV waves
and quasi-P waves.
In the absence of body forces and when S11 , S22 , and so forth are constants, then from
4.8 and 4.9, we get
Ω111 ρc2 Ui 111 Vi 0,
K211 Ui Ω211 ρc2 Vi 0.
4.11
If S12 S23 S31 0, then 4.11 takes the form
Ω1111 ρc2 Ui 1111 Vi 0,
K2111 Ui Ω2111 ρc2 Vi 0,
4.12
where
P1
Ω1111 − B11 l2 Q3 −
m2 ,
2
P
1111 − B12 Q3 1 ml,
2
2
P1
2
l B22 m ,
Ω2111 −
Q3 2
P1
lm.
K2111 − B21 Q3 −
2
4.13
The set of homogeneous 4.12 in Ui , Vi has a nontrivial solution when
Ω1111 ρc2
1111
0.
2 Ω2111 ρc
K2111
4.14
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This quadratic equation in ρc2 may be solved to obtain
2ρc2
P1 2
P1
l B21 Q3 −
m2
B11 Q3 2
2
2
P1 2
P1
P1
P1 2 2
l B22 − Q3 m2 4 B21 Q3 −
B12 Q3 l m.
±
B11 − Q3 −
2
2
2
2
4.15
Thus, in general, in this two-dimensional model of the prestressed medium, there exist two
types of plane waves, namely, quasi-P waves and quasi-SV waves whose phase velocities
correspond to upper and lower signs of 4.15.
4.2.2. Propagation of Plane Waves in Orthotropic Medium
Consider a homogenous prestressed elastic solid. The material is either isotropic in finite
strain or anisotropic with orthotropic symmetry. The principal directions of initial stress are
chosen to coincide with the directions of elastic symmetry and the coordinate axes. Let the
state of uniform initial stresses have principal stresses S11 , S22 , and S33 . We further assume
that S22 S33 and S11 and S22 are constant. The principal stress S33 does not enter explicitly
into the equations of motion. Its influence is, however, included indirectly in the values of the
incremental elastic coefficients. We put l sin θ and m cos θ; 4.15 can be written as
2ρc2 θ B11 Q3 ±
where
P1
P1
sin2 θ B21 Q3 −
cos2 θ
2
2
2
P1
P1
2
2
sin θ B22 − Q3 cos θ ,
B11 − Q3 −
2
2
4.16
4B12 Q3 P1 /2B12 Q3 P1 /2sin2 θcos2 θ, where
B21 − P1 B12 .
4.17
Let CP θ and CSV θ be the values of c associated with upper and lower signs in 4.16,
corresponding to the velocities for quasi-P waves and quasi-SV waves, respectively. Hence
these expressions coincide with the expressions obtained by Sidhu and Singh 1 for velocities
of quasi-P waves and quasi-SV waves.
5. Numerical Calculation and Discussions
Here we consider the model for an initially stressed medium:
B11 λ 2μ P1,
B12 λ P1 ,
B22 λ 2μ,
Q3 μ.
5.1
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Slowness of quasi-P wave
0.61
0.6
0.59
0.58
0.57
0.56
0.55
0.54
0.17
0.35
0.52 0.7
0.87 1.05 1.22
Angle of incidence θ
p −0.8
p −0.6
p0
p 0.2
1.4
1.57
1.4
1.57
p 0.4
p 0.6
p 0.8
Figure 1
Slowness of quasi-SV wave
1.2
1
0.8
0.6
0.4
0.2
0
0.17
0.35
0.52 0.7
0.87 1.05 1.22
Angle of incidence θ
p −0.8
p −0.6
p0
p 0.2
p 0.4
p 0.6
p 0.8
Figure 2
Using 5.1 in 4.17, we get a nondimensional form of velocity equations as
2 1/2
2
1 CP θ ,
δ 3 p δ 1 p δp δ 1 p sin θ
2
1/2 SV θ 1 δ 3 p − δ 1 p 2 δp δ 1 p sin2 θ
C
,
2
5.2
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11
where
δ
λ
,
μ
p
P1
,
2μ
β2 μ
.
ρ
5.3
The apparent velocities for quasi-P waves and quasi-SV waves can be obtained from 5.2 as
CP a θ P θ
C
,
sin θ
SV θ
C
CSVa θ .
sin θ
5.4
P θ, 1/C
SV θ have been
The numerical values of the dimensionless slowness 1/C
calculated from 5.2 assuming that δ 1 for different values of p vary from −0.8 to 0.8
and for different values of θ vary from 0◦ to 90◦ . Figures 1 and 2 show the variation of the
dimensionless slowness for the quasi-P and quasi-SV waves with the angle of incidence −0.8,
−0.6, 0.2, 0.4, 0.6, and 0.8.
6. Conclusion
The study shows that the phase velocities of quasi-P waves and quasi-SV waves are highly
affected by the initial stresses present in the medium and also the direction of propagation.
Acknowledgment
The authors are thankful to Dr. George Jaiani, Tbilisi State University, Georgia for his valuable
comments and encouragement to prepare the paper.
References
1 R. S. Sidhu and S. J. Singh, “Comments on “Reflection of elastic waves under initial stress at a free
surface: P and SV motion,” Journal of the Acoustical Society of America, vol. 74, pp. 1640–1644, 1983.
2 R. S. Sidhu and S. J. Singh, “Reflection of P and SV waves at the free surface of a prestressed elastic
half-space,” Journal of Acoustical Society of America, vol. 76, no. 2, pp. 594–598, 1984.
3 A. N. Norries, “Propagation of plane waves in a prestressed elastic medium,” Journal of the Acoustical
Society of America, vol. 47, pp. 1642–1643, 1983.
4 S. Day, N. Ray, and A. Datta, “Propagation of P and S waves and reflection of P in a medium under
normal initial stresses,” Geophysical Research Bulletin, vol. 23, pp. 1–28, 1985.
5 M. A. Biot, Mechanics of Incremental Deformations, John Wiley & Sons, New York, NY, USA, 1965.