I ntnat. J. Mh. Muh. Si.
Vol. 2 No. 4 (1979) 703-716
703
GENERATION OF SH-TYPE WAVES IN LAYERED ANISOTROPIC ELASTIC MEDIA
P.C. PAL
Department of Mathematics
Indian School of Mines
826004, India
Dhanbad
LOKENATH DEBNATH
Departments of Mathematics and Physics
East Carolina University
Greenville, N. C. 27832, U. S. A.
(Received ovember 8, 1978 and in revised form July 2, 1979)
ABSTRACT.
A study is made of the generation of SH-type wves at the free surface
of a layered anlsotroplc elastic medium due to an impulsive stress discontinuity
moving with uniform velocity along the interface of the layered medium.
The exact
solution for the displacement function is obtained by the Laplace and Fourier
transforms combined with the modified Cagnlard method.
The numerical results for
an important special case at two different distances are shown graphically.
The
results of the present study are found to be in excellent agreement with those
of isotropic elastic media.
KEY WORDS AN PHRASES. SH-type wav anisotropic lat av seismology and
sheuing strs discontinuity.
1980 Mathu Subject
Clasiation
Code:
13D15, 73D0, 73D30.
P. C. PAL AND L. DEBNATH
704
i.
INTRODUCTION.
In recent years an attention has been given to problems of generation and
propagation of waves in anlsotropic or inhomogeneous elastic solids.
The solutions
of these problems in various geometrical configurations, and in layered media are
important in geophysics, seismology, acoustics and electromagnetism.
Available
information strongly suggests that layered media, crystals, and various new materials
such as reinforced plastics, flbre-relnforced metals, and composite materials are
essentially anlsotropic and/or inhomogeneous in nature.
Naturally, there is a
growing need for elastodynamlcal analysis of anlsotroplc and/or inhomogeneous
problems in elastic materials.
Some recent works on the generation and propagation of waves in layered isotropic or anlsotroplc elastic media may be relevant to mention with a brief descrlp-
Anderson [i] made an interesting study of
tion in order to indicate our motivation.
elastic wave propagation in layered anlsotroplc media with applications.
He
discussed the period equations for waves of Raylelgh, Stoneley, and Love types.
It
was shown that anisotrophy can have a pronounced effect on both the range of
existence and the shape of the dispersion curves.
Finally, the single layer solu-
tlons in an anisotroplc medium were generalized to n-layer media by the use of
Haskell matrices.
Elastic properties are generally anlsotroplc (transversely isotroplc) in
sedimentary layers.
Studies of Uhrlng and Van Melle
[3] have indicated that anlsotrophy
[2], and Anderson and Harkrlder
is also present in the near surface layers,
and in the crust and upper mantle regions of the earth.
It has been observed that
the transition region between the crust and mantle is not a single layer but is
possibly formed by a set of thin layers.
generation of
Nag [4] made an investigation of the
SH-type waves at the free surface of an isotroplc elastic layered
medium due to an impulsive stress discontinuity moving with constant velocity after
creation along the interface of the medium.
In a subsequent paper Nag and Pal [5]
WAVES IN LAYERED ANISOTROPIC ELASTIC MEDIA
705
solved a problem similar to that of Nag with a shearing stress discontinuity at the
interface of two layers of finite thickness overlying a semi-inflnlte medium of
different elastic constants.
Mention may also be made of the work on forced
vibrations of an anlsotropic elastic spherical shell due to an uniformly distributed
internal and external pressure by Sheehan and Debnath [6].
The purpose of this paper is to study the generation of SH-type waves at the
free surface of a layered anlsotroplc elastic medium due to an impulsive stress
discontinuity moving with uniform velocity along the interface of the layered medium.
The displacement function is obtained for two different types of discontinuity in the
The numerical solution for one case at two different distances is
shearing stress.
Some special cases of physical interest are examined.
shown graphically.
2.
MATHEMATICAL FORMULATION OF THE PROBLEM.
We consider an anlsotroplc elstlc layer of thickness
LI,
N
and
1
Pl
h
over an anlsotropic half-space with constants
with elastic constants
L2,
N
2
and
P2"
The geometrical configuration of the anlsotroplc wave problem is depicted in Figure I.
With the
z-axls directed vertically downwards, the transversely Isotroplc layered
structure is referred to a rectangular Cartesian coordinate system with the origin at
the interface
(z=0)
of the layers I and II as shown in Figure i.
The wave gener-
atlng mechanlsn is a shearing stress discontinuity which occurs suddenly at the
interface and then moves with constant velocity
along the interface.
in the positive
x-dlrectlon
Since only the SH-type waves are considered, we can assume
that the displacement fields
and
V
u--w=0
and
v
is a function of space variable
x, z
t.
Neglecting body forces and assuming small deformations, the equations of motion
in the two layers can be written, with the usual notations, in the form
i
2vi
t
2
Ni
32vi2 + Li 32vi
3x
z2
t > 0,
(2.1ab)
P. C. PAL AND L. DEBNATH
706
where
1,2, v I
i
and
v
represent the displacement functions in the layers
2
I
and II respectively.
LI’NI’PI
Il -
+
o
L2 N202
GEOMETRY OF THE PROBLEM
FIG. I.
The boundary conditions are, in the usual notations,
(yz)l
v
0
v
I
(z) I
where
S(x,t)
of time
t,
(yz) i
2
-h,
z
at
z- 0,
at
(yr) 2
(2.2)
t > 0
(2.3)
t > 0
S(x,t) H(t)
is a function of
x
at
z
0
and
t;
H(t)
(2.4)
is the Heavislde unit function
and the shearing stress in an anisotroplc medium is given by
v
Li
---,i
i
(2.5ab)
i, 2
We complete the formulation of the problem by assuming the appropriate initial
conditions and the existence of the Laplace and Fourier transforms of the functions
vi(x,z;t)
with respect to time
t
and distance
x
respectively.
707
WAVES IN LA1.E AWIOTOP IC ELAS IC HEDLA
THE SOLUTIO OF THE PROBLEM.
3.
The above problem can readily be solved by using the Laplace and Fourier
transforms combined with the modified Cagnlard metho& The Laplace transform with
and the Fourier transform with respect to
t
respect to
I e-iXdx I
(,z;p)v
are defined by
x
e-pt v(x’z;t)dt’
where the tilda and the bar deote the lplace an4 t Fourier transforms respectively.
Application of the le tranBforms (3.i) to equations (2.1ab) give the
solutions in the upper
I
l(X’z;P)
(A cosh
v2(x,z;p)
sl
z
C exp (ix-
A, B
where the constants
(2.2)
loer layers with
and
+B
ns2
C
sinh
nsl
v
z)
2
/
0
as
z
/
in the form
eiXd,
(3.2)
(3.3)
z)d,
are to be determined from the boundary conditions
(2.4);
i
=(p2/
nsl
L
+
(3.4ab)
i- 1 2
1
82i
82i/ 811
@i
2/2)2
1
i
(i)2
81i
and
2
.Ni. 7
(3.5ab)
%pi
is the anisotropic parameter for the two layers
It follows fro the bom4ary conditions (2.2)
A
C, B cosh
(2.3) that
(3.6ab)
A slnh n h,
sl
slh
We now consider to different forms of the functions
Case (1):
S(x,t)
P, a < x < b + Yt
@,
where
P
I and II.
[
S(x,t).
(3.7)
elsewhere
is a constant.
This case implies that the stress discontinuity is created in the region
x
a
P. C. PAL AND L. DEBNATH
708
x
to
and then expands with constant velocity
b
particular, when
in the
x-dlrectlon.
In
the discontinuity is created at the origin and then
0,
b
a
V
expands with uniform velocity
V
When the disturbance expands
x-dlrectlon.
in the
in both directions after creation, this solution is to be added to the solution for
negative values of
x.
It follows from the transformed boundary condition (2.4) combined with (3.7)
that
[e -la e-ib + e-ib
+ p/V
p
2p
+
BLI sl CL2 s2
i
With the aid of (3.6ab)
and{3.8), we solve
J
(3.8)
for the three constants
A,B,C;
and then write down the integral solution for the displacement function at the free
surface
-h
z
h,p)
Vl(X,
in the form
P
=Tp
-
I (Llnslsinh
exp (ix)
nsl
e-iX_e-ib + e-ib
i+p/V
i
p
where
K(<I)
h
]
+ L2
ns2
cosh
nsl
h)
d
i (Llsl+L2s2) le-ia-e-ib
exp(ix-hnsl)
i
i+p/V
I
Ke
d,
(3.9)
represents the reflection parameter given by
K
Llsl-L2ns2
Llsl+L2s2
(3.10)
Using the inverse Laplace transform, we can rewrite (3.9) in a convenient form
where
Ii
-
Vl(X
L -I stands for the
P
(l-Ke
[-i(I 1
-h;p)
+
12
+
13),
(3.11)
inverse Laplace transform and
I
_2hslh_
,1
i (L Insl+L2ns2)
exp(fx
I
hnsl)d,
(3.12)
-
WAVES IN LAYERED ANISOTROPIC ElaSTIC MEDIA
(I-K
P
p
13
x
with
I
e
(i+p/V)
x
_2hnsl)_ I
exp(ix
(Llsl+L2s2)
and
a
x
2
hsl)d,
exp(ix 2
i (Llnsl+L2ns2)
709
(3.13)
(3.14)
hnsl)d,
2
b.
x
In order to evaluate the Laplace inversion integral, we shall use the modified
Cagniard method due to Garvin [7] who discussed the contour integration and mapping
It may be fair to avoid duplication of mathematical analysis similar to
in detail.
-
that of Garvin, and to quote some necessary results from that paper without proof.
We next substitute
p/811
(3.4ab) and 3.10) so that
in
1
nsl
L (i +
811
2
2@i)
i
and
K=
s2
+
2
@2 2)
(3.15ab)
i
+
2@22)2
2
(3.16)
i
+
LI(I+2012) + L2(811
S122
-
1
2
811 12 2
2
LI(I+2@12 )2 L2( 8.I 12
812
I
811
L
2
2
2
We thus obtain
2P
Ii
where
Im
[I exp’iXl-hnsl"
(Llnsl+L2ns2
(i
and
s2
-2hnsl
2
+K
I,
exp
2P
2
Im
1811
in
(3.17) by
i
-4hnsl
+ ...)d,
(3.17)
811/@2812.
Setting
we get
Iii,
p{_iXl+h(l+2@l 2)2} 81
2
d
/
1
2
@12)’" + L2 811
1
]
_+
i/@l
q (8122 + 2@22
The integrand of (3.18) has singularities at
+_
e
are given by (3.15ab).
Denoting the first term of
P
e
0
nsl
Iii
+ K
0,
(3.18)
and
710
P. C. PAL AND L. DEBNATH
i
[-ix I
t
+ h(l +
212) 21811
(3.19)
so that the inversion gives
1
811
(t)
[itx I + h{t
(x12+h212)
The singularities in the
in the
2
(Xl
2
+
h
2
i
2
)8112} 2--]
(3.20)
-plane are shown in Figure 2, and the mapping and contour
t-plane are depicted in Figure 3.
Making reference to these Figures and to
the paper [4], we find
t
i-i Illwhere
I(t
2P
’111
I ,1
[I, 1 (T)]dT
and
i
GI
Im[{l +
t[(t)]
(3.21)
0
i
t[tH(t)]
T)G
12I,I
2
(t)} + L2
2
q {811
812
i
+
2
2 (t)}2]
2
I,i
I
x
-i
d
i,i
d-
I,I (t)
H{t
(Xl
2
+
h2i 2)281
i
},
(3.22)
i
Since
h181
2
[{i +
i,I
I < t <
(x12 h212)281i the expression
1
2
1
2
2
L2
real.
(t)l } +q {811 + 22 i,I (t)}
+
is
8122
In general, it turns
out that
t
t-Ill ,n
I
2P
P1811
)GI ,n [i ,n (T)]dT
(t
(3.23)
0
where
I
Im[{l + 12 i 2, n(t) } + 52 { 811
G1 ,n [i .n (t)]
2
( t)}
,n
22i
I
i l,n(t)H{t (x12 + n2h212)TBll}
8122
d
x
I
2
+
1
(3.24)
n dt
and
{l,n(t)
Ell
(x12+n2h2l 2)
i
[ix,t + nh{t
2
2
(x I +
n2h212) 812} 2]
n
I, 3, 5
(3.25)
WAVES IN LAYERED ANISOTROPIC ELASTIC MEDIA
PLANE SHOWING THE SINGULARITIES
FIG.2.
PLANE SHOWING THE MAPPING AND THE CONTOUR
FIG.
711
P. C. PAL AND L. DEBNATH
712
.
so that
t-lI i
t-lIl, n,
(3.26)
n=1,3,5...
Sinilarly, we obtain
t-lI 2
Z
t-iI2
n=1,3,5...
(3.27)
n
A similar procedure gives that
L -11 3
L-II3,
[
n=1,3,5...
(3.28)
n
where
t
L-II3
x
f
2P.
PlBll
,n
(t
0
T)G
-
3 ,n
[2, n (T)]d
1
2
L
2,n(t)
is given by
2
1
2 (t)}2]
2,n (t)}2 + q {811
8122 + 2 2 2,n 1
2
G3 ,n [2 ,n (t)] Re[{I + i
n-i
d
"811 +
2,n(t)
]-I
(t)
K
I-V- i2,n
2,n dt
and
(3 29)
(3.25) with x
2
2
H{t
2
(x 2 +
in place of
x
2
n2h2l ) [i},
I.
Finally, a simple combination of the results (3.26)
value of the free surface displacement field
Case(ll):
where
P
S(x,t)
Ph (x
included in (3.31) so that
(3.28) gives the exact
h,t).
Vt),
(r)
is a constant,
Vl(X,
(3.30)
P
(3.31)
is the Dirac distribution and a factor
h
is
has the dimension of stress.
The boundary condition (2.4) given
BLlnsl + CL2ns2 "2
Solving
v
l(x,
-h;p)
Ph
A, B, and C
Ph
(3.32)
V(i+P/V)’
from (3.6 ab) and
(I+K e
+.
(Llsl+L2s2) (.
(3.32), we
exp(ix
obtain the solution as
hsl)d
(3.33)
WAVES IN LAYERED ANISOTROPIC ELASTIC MEDIA
713
A procedure similar to that of Case (i) gives the solution in the form
,t
v
l(x,
2Ph
-h,t)
I
rVSll
n=1,3,5
10 Gn[n(T)]dT,
(3.34)
where
Gn[n(t)]
1
2
Re[(1 +
811
t--+ in (t) ]-i
d
2
}
#12n2(t)} +i {811
8122 + .22n
n-i
n(t)
K
dt
2
L2
2
n2h2#12)l 8111,
H{t
(x +
(x 2 +
n2h2l2)
n(t)
I
2
2
(3.35)
and
1
811
n(t)
(x2+n2h212)
2
[itx + nh{t
If the stress discontinuity is taken as
(x
Vt),
H(x)
H(x
Vt)
in place of
the coresponding result on the right hand side of (3.33) differ only
by a constant factor from
4.
(3.36)
(with
13
a
b
0)
(3.12)
in
(3.14).
RICAL RESULTS
It is of interest to consider the initial behavior of the displacement field
vl(x,-
h,t)
for case (ll) numerically.
Following Nag and Pal [5], motion due to each of the pulses at the instant of
their arrival has been investigated.
0.9,
#I
2
I.I,
L2/L1
1.33,
The initial behavior of
(i)
When
KiVl(X,
x
5h, t
-h, t)
h
811
A(e3)cosh-I H(z
6.3) +
+
A(eT)cosh-I I-.i H(r
I0 I)
T
(811) csh-I
h,t)
sx
’ [A(01)cosh-1
+
+A
811/V
Vl(X
for
15. H(
We consider the following numerical values:
811/812
1.21,
at two different distances is examined.
initial values, we have
tt(:
5.2) +
A(e5)cosh-i 8--
T
+
15.2)],
1.02.
A(e9)cosh-i
H(T
H(,
8.2)
12.6) +
(4.1)
P. C. PAL AND L. DEBNATH
714
2.42 P
KI
where
Pl
(4.2)
-811
and
A(8 n
[(I-1.76cos28n )1/2-
Re
(i-i.
When
KlVl(X
76cos2e n i/2+
x
10h,
t
(1.2-
n
cose n
for six initial values, we have
811
[A(81)cosh-i 10.08 H(T- I0 08) +
+ A(83 )csh-I 10.71T H(T- i0.71) +
T
(85)cosh-I ii 94 H(T
+ A(89)cosh-i 53 H(T
T
T
A(8 ), n
n
Values of
.1cos2e n )i/2]
n-3
2
sin 8
(4.3)
Th
-h, t)
+ Ao
where
0. 95 (l 04_2
n-i
2
I, 3, 5,
n
(2)
2.1cos28n )1/2]
0.95(1.04
A(8
ii 94)
15 5)
i, 3, 5,
),
n
1, 3, 5,
+
+
A(87)cosh-I
T
13.64 H(T- 13.64)
A(811)cosh-i 17
T
62
+
(4.4)
17.62)],
H(T
are given by (4.3).
for
5h
x
and
10h
x
tabulated
are
below:
TABLE i.
A(Sn
A(81
A(e3
A(85
A(87
A(89
A(ell
x
5h
0.74
-0.301
0.052
-0.122
0.0009
-0.00012
x
10h
0.23
-0.02
0.0165
-0.0035
0.00055
-0.0002
5.
DISCUSSION AND CONCLUSIONS.
The exact form of the displacement field due to physically realistic shearing
stress discontinuity has been obtained.
and
x
10h
have been plotted against
The value of
T
1
T
T
0
KlVl(X,-h,t)
where
which the disturbances arrive at the point of observation.
T
0
for
x
5h
is the time at
The values of for
715
WAVES IN LAYERED ANISOTROPIC ELASTIC MEDIA
8O
SO
I0
0
4
3
2
)
7
0
9
$
I0
II
12
FIG.4.
x
5h
and
in Figure 4.
10h
x
5.2
are
and
respectively.
10.08
These results are shown
It is clear from Figure 4 that the curves start from the origin and
undergo sharp changes in their slopes with sudden Jerking as the dfferent pulses
arrive after undergoing reflections from the boundaries of the anlsotropc layers.
These changes occur for
for
x
10h
at
T
x
102 +
5h
at
0.81 n
T
2 1/2
(52
x
n
2 1/2
i, 5, 9,
n
the contribution due to the third pulse for
than for
+ 0.81
x
10h
n
i, 5, 9,....;
and
We also infer that
will arrive
n
a shorter tme
5h.
In addtlon to above two special disturbances, the present analyss ncorporates
other forms of physically realistic stress discontinuity for the generation of
SH
P. C. PAL AND L. DEBNATH
716
type waves in both isotropic and anisotropic elastic media.
In the case of the layered isotropic media,
i
i, i
1,2;
the results of
the present analysis reduce to those of Nag [4] for the corresponding isotropic
problem.
Finally the shearing stress discontinuity is always associated with the
propogation of cracks in earthquakes.
tions to geophysics and seismology.
Hence the present study has direct applica-
In addition, the present study is of interest
in connection with the propagation of cracks in sedimentary layers, and in particu-
far in elastic layers having anisotropic property in general.
ACKNOWLEDGEMENT.
The authors express their grateful thanks to Professor M. Bath,
University of Uppsala, Sweden and to Professor K. R. Nag for their help and
encouragement.
This work was partially supported by the East Carolina University
Research Committee.
REFERENCES
i.
Anderson, D. L., Wave Propagation in Anisotropic Media, J. Goephys. Res. 66
(1961) 2953-2963.
2.
Uhring, L. F. and Van Melle, E. M., Velocity Anisotrophy in the Stratified
Media, Geophysics 21 (1955) 774-779.
3.
Anderson, D. L. and Harkrider, D. G., The effect of anisotropy on Continental
and Oceanic Surface wave dispersion (abstract) J. Geophys. Res. 6_7 (1962)
1627.
4.
Nag, K. R., Generation of SH-type wave due to stress Discontimuity in a layered
elastic media, uart. J. Mech. and Appl. Math. 16 (1963) 295-303.
5.
Nag, K. R. and Pal, P. C., The Disturbance of SH-type due to shearing stress
Discontinuity at the Interface of two layers overlying a semi-infinite
medium,
Mec___h. 2__9 (1977) 821-827.
6.
Sheehan, J. and Debnath, L., Forced Vibrations of an Anisotropic Elastic
Sphere, Arch. Mech 24 (1972) 117-125.
7.
Garvin, W. W., Exact Transient Solution of the Buried Line Source Problem,
Proc. Roy. Soc. Lond. A234 (1956) 528-541.
.