Diffraction of torsional wave or plane harmonic
compressional wave by an annular rigid disc
RANJIT S. DHALIWAL and B. M. SINGH
Department of Mathematics and Statistics, University of Calgary, Alberta, Canada
J. VRBIK
Brock University, St. Catherines, Ontario, Canada
A. P. S. SELVADURAI
Department of Ovil Engineering, Carleton University, Ottawa, Ontario
In this paper we have considered the following two problems. Firstly the diffraction of normally
incident SH waves by a rigid annular disc situated at the interface of two elastic half-spaces is considered.
The solution of the problem is reduced into the solution of triple integral equations involving Bessel
functions. The solution of the triple integral equations is reduced into Fredholm integral equations of
th second kind. By finding the solution of the Fredholm integral equation, the numerical values for
the moment required to produce the rotation of disc are obtained. Secondly, the problem of diffraction of plane harmonic compressional wave by an annular circular disc embedded in an infinite
elastic space is considered. The annular disc is assumed to be perfectly welded with the infinite solid.
The solution of the problem is reduced into the solution of Fredholm integral equation of the second
kind. The Fredholm integral equation is solved numerically and the numerical values for the couple
applied on the disc are obtained.
INTRODUCTION
It is well known that the problems of cracks or inclusions
are of considerable interest in seismology and geophysics. If
the inclusions are located at the interface of layered media,
the study becomes more important. Loeber and Sih 1' 2 have
studied the scattering of torsional waves by line crack and
a penny-shaped crack laying on a bimaterial interface.
Diffraction of torsional waves or normal compressional
waves by a flat annular crack in an infinite elastic medium
has been studied by Shindo) '4 Diffraction of torsional
waves by a fiat annular crack at the interface of two bonded
dissimilar elastic solids has been discussed by Singh, Dhaliwal and Vrbik. s The problem of diffraction of torsional
waves by a circular rigid disc situated at the interface of
two dissimilar elastic half-spaces has been considered by
Singh, Rokne and Dhaliwal. 6
In this paper we have considered the problem of diffraction of torsional waves by an annular rigid disc situated at
the interface of two dissimilar elastic half-spaces. Using
Hankel transforms the problem has been reduced to the
solution of triple integral equations. The solution of triple
integral equations is reduced into the solution of the
Fredholm integral equation of the second kind. Numerical
solution of the Fredholm integral equation is used to find
Accepted January 1984. Discussion doses September1984.
150
the shear stress component on the disc. The solution of
such a type of problem may give some information of
building foundations in the event of earthquake waves. In
the second problem, a rigid annular disc embedded in an
infinite, isotropic elastic medium is assumed to be excited
by a normally incident plane, harmonic compressional
wave. The disc is assumed to be perfectly welded with the
surrounding solid. The solution of the problem is reduced
into the solution of the Fredholm integral equation of the
second kind. By finding numerical solution of the integral
equation the numerical values of the couple applied on the
disc are obtained.
FORMULATION AND SOLUTION OF PROBLEM (A)
Consider a cylindrical coordinate system (r,O,z) with
origin at the centre of the annular disc and let the disc
occupy the region a <~ r ~< b, z = 0 at the interface of two
half-spaces z > 0 and z < 0. The rigid disc is assumed to be
excited by a torsional wave originating at z = - oo. Now the
problem is to find the stress distribution subject to the
following boundary conditions:
uo(r, O+)=uo(r, O-)=uo(r)exp(--iwt),
uo(r,O+)=uo(r,O-),
Ooz(r, 0+) = Ooz(r, 0-),
Soil Dynamics and Earthquake Engineering, 1984, Vol. 3, No. 3
0<r<a,
r>b
0 < r < a, r > b
a<~r<<.b (1)
(2)
(3)
0261-7277/84/030150-07 $2.00
© 1984 CML Publications
Diffraction of torsional wave or plane harmonic compressional wave by annular rigid disc: R. S. Dhaliwal et al.
where w is circular frequency. In what follows, the factor
exp(--i¢ot) shall be suppressed.
The problem of determining the stress distribution
reduces to that of obtaining the solution of the displacement
equation:
~2uo
--
~r2
+
lau o
Uo
r ar
r2
~2Uo+ 2
+ --
~z 2
k uo
f
~Al(~)
0
(13)
(4)
[/al/~1 + U2/32] A 1(~) = (g 1 +/a2) B(~)
(14)
then the triple integral equations (12) and (13) can be
written in the following form:
f ~-IB(~)
~11(~) exp(--~lz) Jl(~r) d~, z >I 0
[1 + M ( ~ ) ]
Jl(~r) d~ = uo(r), a < r < b
(15)
0
=
(5)
f B(~) Jl(~r) d~ = O, 0 < r < a ,
r>b
(16)
0
~12(~j) exp([32z)Jl(~r ) d~, z <<.0
where
,0
Oj
r>b
Let
= 0
where k 2 = p/lz, # is Lame's constant, p is the density of the
elastic material and k is the wave number. The 'solution of
equation (4) can be written in the following form:
uo(r, z )
Jl(~r) d~ = O, 0 < r < a ,
[/at/31 +/a2/~21
!
(6)
g~ - PI~:
uj
, (/" = l, 2)
(7)
where suffixes 1 and 2 correspond to the half-spaces z > 0
and z < 0, respectively. In equations (5), AI(~) and A2(~)
are unknown functions which are to be determined by
using boundary conditions (1)-(3). With the help of the
following equations:
[ ~(/ax__+/~2)
1]
(17)
g ( ~ ) = L(/31ul+ ~2u2)
We can easily see that M(~) -->0 when ~ ~ oo. To find the
solution of triple integral equations (15) and (16) we
follow Cooke 7 and assume
f B(~) Jl(~r) d~ = g(r), a < r < b
0
Making use of the inversion theorem for Hankel transforms we get from equations (16) and (18) that
(8)
Ozo=la-~z ' OrO=la\ Or
(18)
b
B(~) = ~ f rg(r)Jx(~r) dr
we find that
(19)
a
oo
--121 f
~j/31Ax(g)exp(--[31z)Jl(~r) d~, z >~O
Substituting the value of B(~) from equation (19) into
(15) we get
0
ooz(r, z) =
b
(9)
f ug(u)L(u,r)du=F(r),
/a2 f ~¢2A2(~) exp(fl2z)Jl(~r) d~, z <~0
a<r<b
(20)
a
0
where
--~tl ; ~j2AI(/j) exp(--[Jl z) J2(~r) d~, z >t 0
7
F(r)=uo(r)-- I ~-lB(~)M(~)J1(~r)dlj,
0
Oro(r,z ) =,
--//'2 f ~2A2(~)exp(fl2z)J2(~r)
d~, z < 0
(21)
0
(10)
oo
a<r<b
2
L (u, r) nur
min (u, r)
r
J
s 2 ds
[(u2--s2)(r 2 _ s2)11/2
0
0
From the continuity conditions (1) and (2) of the displacement field uo(r, 0÷) = Uo(r, 0-) for all values of r we
find that:
A 1(~) = A2(~)
(1 1)
From conditions (1) and (3) we find that:
Cooke 7 has shown that the solution of integral equation
(20) can be written in the following form:
s2a(s) =
d f x2F(x) dx
(s2_x2)l~2
4s
~2~e=-~
a
b
;~,A:(~)Jl(~r) d~ =uo(r),
0
a<r<b
(12)
f tG(t) K(s, t) dt
x
~
, a<s<b
(22)
a
Soil Dynamics and Earthquake Engineering, 1984, Vot 3, No. 3
151
Diffraction of torsional wave or plane harmonic compressional wave by annular rigid disc." R. S. Dhaliwal et al.
where
a
K(s, t) = f
y2(a~--Y2) dy
(s 2 - y 2 ) ( t 2 - y : )
(23)
o
If a -+ 0, we find that Ka(s, t) -~ 0 and the kernel K2(s, t)
is the same as obtained by Singh, Rokne and Dhaliwal. 6 In
this way we can find solution of the problem of diffraction
of a torsional wave by a circular rigid disc at the interface
of two bonded dissimilar elastic solids.
We can easily write that:
b
= f g(r) dr
G(s)
J ( r 2 - - $2)1/2
sa
(24)
I(s, ~) = s~ex(s, ~) -~ x / c ~ J l ( ~ a )
(32a)
$
If u0(r) = uor, where Uo is constant, then we find that:
$
d (x2F(x) dx
where
oa
UoS(2S2 - a 2)
$
f rJo( r) dr
f ~-aB(~) M(~) I(s, ~) d~
a
ea(s, ~) = )(s-7~_r2)~7~
(32b)
a
o
(25)
where
Now we find that:
azo(r, O+)--Ozo(r, O-) = (pl + p2)g(r), a <r < b
$
(26)
I(s, ~) = ds
(33)
The moment required to produce the rotation of the disc is
given by:
a
Equation (24) is of the Abel type and hence its solution
can be written in the following form:
b
T = -- 2rr f r 2 [aoz(r, 0") -- aoz(r, 0-)] dr
a
b
2 df
sG(s) ds
g(r) = - - ~ - ~
(S2
r2)1/2
b
(27)
= -- 21r(px + P2) f r2g(r) dr
r
We can easily show that
;
(34)
a
Making use of equations (19) and (27) we find that
[-1B(~) M(~)I(s, ~) d~
b
2r
f. sG(s) ds
B(~) : ~ - [ - - ~ a J l ( O ) J ~ k
o
b
a
,o
= 2 f G(t)dt y M(~)l(t,~)l(s,~)d~
a
(28)
f sG(s) ea(s, ~)ds]
a
(35a)
Let
0
s=asec0,
Making use of the equations (25) and (28), we can write the
integral equation (22) in the following form:
s2a(s)
b
t=asec¢,
G(a sec ¢) sec2¢ = uoHl(~b)
(35b)
then the integral equation (29) can be written in the following form:
UoS(2S2 - a2)
-
sin 0 cos20HI(O)
b
see -1 ( b / a )
-- [ G(t) [Kl(s, t) + K2(s, t)] dt, a < s < b
= (1 + sin 20)-- sin 0 cos20
a
(29)
Y
o
where
x [Ka(O, c~)+ K2(O, ~b)]sin i~nl(~) dO,
4st
at. y2(a2_y2) dy
Kl(s, t) = n2 X/(s 2 _ a2)(t 2 _ a2) J (s 2 _ y2)(t2 _ y2)
0<0 <see-l(b)
(36)
0
2st
~r2x/(s 2-- a z) ( t 2 -- a 2) [t 2 -- s:]
KI(O, ¢) =
4 sec0 sec~b
zr2 tan0 tan ~ [sec2¢-- see20]
0
X [s(a2--s2)log s+a
x [sec ~btan Slog t a n ( 2 ) - - see 0 tan20
iotan(°)]
(3o)
Kds, t)=-); M(~)l(t,DI(s,t:)d~
(31)
(37)
K2(O, ~) = ~a
o
152
Soil Dynamics and Earthquake Engineering, 1984, Vol. 3, No. 3
M(~) I(a sec 0, ~) I(a sec 0, ~) d~
o
(38)
lh'ffraction of torsional wave or plane harmonic compressional wave by annular rigid disc: R. S. Dhaliwal et al.
Making use of equations (27), (35b) we find from equation
(34) that:
see- ~(b/a)
T
UoIt~
4aa(l+a)
f
[ l + 2 t a n 2 ~ ] H~(~b)d¢
(39)
o
We can easily find that:
k2 = k
where u0 is the amplitude, kl = w/c1 is the wave number in
the incident wave and cx is the P-wave velocity in the solid.
The time factor e x p ( - - i w t ) will be omitted in the subsequent analysis.
Since the problem is axisymmetric, the displacement
components in the scattered field ur(r, z), Uz(r, z) may be
conveniently expressed in terms of two scalar potential
functions ~b(r, z), ~O(r, z) by means of the equations:
(40)
P/-~I
1N PlIt2
The numerical values of lT/uoItal have been obtained
from (39) where numerical values of Hx(~) have been
obtained from (36). The numerical values of I T/UoItll have
been graphed in Figs. 1, 2 for OdO~ = 1 and for various
values of Itl/It2 and for Itl/It2 = 1 and for various values of
OdO2, respectively.
Ur =-~r
Jr
(41)
uz - az
~3z2
where k2 = ¢o/c2, c2 being the shear wave velocity in the
solid and ~, ~ satisfy Helmholtz equations
V2~b + k~q~ = 0
(42)
v2~ + k~ = 0
FORMULATION AND SOLUTION OF PROBLEM (B)
Consider a cylindrical polar coordinate system (r, 0, z) with
origin at the centre of the disc and z-axis perpendicular to
the plane of the annular disc. Let a plane harmonic compressional wave propagating along the z-axis be incident
normally on the disc. The displacement vector associated
with incident wave alone may be expressed as:
The stress components on the z = 0 plane are given by
(43)
exp(-- i¢ot) u°(r, z) = {0, O, Uo exp [ikl(z -- cx t)]}
where It is the shear modulus of the elastic solid.
Anticipating that the presence of the disc will introduce
discontinuities in the field quantities across z = 0, the solutions of (42) are postulated as follows following Mal, Ang
and Knopoff: 8
= 0.3
b=*.O
a
~=,.o
s.o
4.0 ~
~
~==oz
"
k
F= 04
3.0
vllzl)dk
0
/
2.0
--g7- Z.O
(44)
/
~
0 0
LO
2.0
3.0
4.0
~
5,0
-
-~t = '~.o
6.0
7.0
k
{-~2Q(k)-SQx(k)}Jo(kr)exp(--v21zl)dk
~(r,z) =
0
8.0
(45)
Figure 1. Variation o f Kx with l T/itl rio I for the annular
disc. [rio = (1 + a)2Uo; a = It2/Itl]
where
=[ ~ ,
k > k1
(46)
vj
t --iv~Z--~'
o=0.3
b = I.O
5.0
~-2
4o
7:"°
r---=
P, =0.~'
/
"
~
I~ °'~
'
2.0
1.0
- 5.0
1.0
2.0
3.0
(47)
8 = sgn(z)
/
2.0
0
k < k I, j = 1,2
4.0
5.0
6.0
7.0
8.0
K,
b3gure 2. Variation of K1 with [ T/Itluol for the annular
disc. [t~o = (I + a) 2 { 1 + (P2/Pl)} Uo; a = It2[Itl]
P(k), Px(k), Q(k), Ql(k) are unknown functions to be
determined from the boundary conditions. Since there is no
discontinuity in the physical properties of the medium
inside the region a < r < b, z = 0, which is occupied by the
annular disc, all the field quantities must be continuous
across z = 0 for r > b. The remaining boundary conditions
are as follows:
IAm ur(r , z) = O, a < r < b
(48)
Z ----~0
IAmuz(r,z)+uo=O , a < r < b
z.->O
(49)
The continuity of u r and uz across z = 0 for all r implies
that Pa(k) + Qa(k) = 0 and P(k) + Q(k) = 0. The remain-
Soil Dynamics and Earthquake Engineering, 1984, Vol. 3, No. 3
153
Diffraction of torsional wave or plane harmonic compressional wave by annular rigid disc: R. S. Dhaliwal et al.
ing boundary conditions can be shown to be satisfied
provided Pl(k) = 0 and P(k) is the solution of the following
triple integral equations:
Following Cooke 7 the solution of integral equation (58a)
can be written in the following form:
4s
2SUo
kP(k) So(kr) dk = O, 0 < r < a
(50)
H~(s)= ( k ] + k ~ ) ~
tHs(t)Kl(s,t) .
~ d t
a
o
k 91--
+ f G~(k)P(k)I~(s,k)dk,
P(k)Jo(lo')dk=--Uo, a < r < b (51)
o
(59)
a<s<b
o
where
8
kP(k)Jo(kr) dk = O, b <r
d f rJo(~r) dr
Ii(s, ~) = -~.] ( ~
(52)
(60)
o
rt
The normal traction on the annular disc is given by:
KI(s, t) = f
Ozz(r, O) = pk~ ; kP(k) Jo(kr) dk, a < r < b
(53/
1
2 (t s -- s s)
x [ (as -- ss) log s + a
The integral equation (51) can be written in the following
form:
o
_
o
0
2Uo
e(k) ]o(kr) dk = k~ + k----~+
(as _y2) dy
(s 2 _ y2) (t s _ y2)
S
s--a
(a s - t t2) l°g
t-all
t + a
(61)
b
aa(k) P(k) Jo(kr) dk,
Hs(s) =
o
a<r<b
f
h(r) dr
(rs $S)1/2' a < S < b
-
-
(62)
$
(54)
Equation (62) is of Abel type, hence its solution may be
written as:
where
G,(k) = 1 + k] + k---~ 9 x -
b
(551
2 d (sH2(s) ds
h (r) = -- rt dr J (sS--r-------~)
1--7s
It is to be noted that:
(63)
F
O(k-a) as k -~ ~¢
Gl(k: ) =
We can easily prove that:
Let us assume that
. ; kP(k) J0(kr) dk =h(r), a < r < b
ai(De(k)Ii(s,k)dk=
Hs(t)Ls(s,t)dt
(64)
(56)
0
o
a
where
Making use of the inversion theorem for Hankel transforms
we getfrom equations (50) and (56) that:
L 2 ( s , t ) = 2 f Ii(s,k)11(t,k)Gx(k)dk
b
P(k) = f rh (r) Jo(kr) dr
(65)
o
(57)
Hence the integral equation (59) can be written in the
following form:
a
Substituting the value of P(k) from equation (57) into (54)
we get:
2SUo
/-/s (st
b
f uh(u)M(u,r)du=F(r), a < r < b
~/sS _ a~(k~ +-~9
(58a)
b
+f I-Is(t)[L,(s,t)+ Ls(s,t)l dt,
a
where
a<s<b
(66)
a
where
F(r) - k] + k----~2F
Gl(k) P(k)Jo(kr) dk
(5Sb)
o
M(u, r) = ; Jo(kr) Jo(ku) dk
--2
L l(s,t)= n2X/(s2 a2)(t2 a2) [tS_sS )
(58c)
o
154
Soil Dynamics and Earthquake Engineering, 1984, Vol. 3, No. 3
x t(a2_s2)log s+a --s(a s --t s)log ~t + a [ ]J
t.
a--a
(67)
Dtffraction of torsional wave or plane harmonic compressional wave by annular rigid disc: R. S. Dhaliwal et al.
7.0
If a ~ 0, the integral equation (66) reduces to the corresponding equation of Mal. 9
We can write equation (60) in the following form:
$
/(s, k) = [ ~
s
,_,.o/ /
/t,
6.0
b-l.O
~"~z= °4
5.0
fk s,(
j ~ )dr
j l
(68)
e - 0.0
"
'~-Lo
P,
4.0
l&l
a
Let
s=asecO,
2.0
t=asecq~,
F.2
1.0
2uoR(O)
sin OH2(asec 0) = - (k~ + k~)
(69)
0
then we can write the integral equation (66) in the following
form:
f
I.O
2.0
3.0
4.0
5.0
6.0
7.0
8.0
K,
Figure3.
Variationof Kz with I T/lal fiol for the solid disc.
[~o = (1 + ~)2Uo; ~ =
s e e -x (b/a)
sin OR(O) = 1 + sin0
0
Is2/ud
sec2 CR (~b)
7.C
0
×{LI(*,O)+L2(*,O}d¢,
0=0.0
O<O<sec-'( b]
K¢
b =1.0
\a/
(70)
/,,z =LO
F',
5.C
where
4.C
4 cos 2 0 cos 2
LI(O,
~) = --
rr2 sin 0 sin ~ [cos 2~b-- cos2 0 ]
2.0 ~
l
--0"2
x [cos~ tan2~b log (tan ~- )
0 ['~
-- cosO tan20 log ( t a n O ) ]
L2(O, ¢) = 2 ;
G,(k)I(asec¢,k)I(asecO, k)dk
0
5-o ~
I.O
Z.O
3.0
4,0
(71)
(72)
5.0
6.0
7.0
8.0
KI
Figure4. Variationof Kl with I T/lal~ol for the solid disc.
[t~o= (1 + a) 2 { 1 + (P2/P2)}Uo; ~ = ~JPz]
o
1.5
?2,
I(a sec ~, k) = [sec ~ cot ¢Jo(ka) + k tan ~Jo(ka)
a
sec~
+ ga sec q~ f
I.O
e
1
x/a; sec~----~- - r ~ J
(73)
a
0.5
The couple on the disc is given by the equation:
b
M=
2.f
r2Ozz(r, O) dr
tO
2.0
3.0
a
4.0
5.0
6.0
7.0
8,0
K2
b-~gure 5.
disc
b
= 4/zk2 [ J
a
a
8puok~a3 set,b/a) sec20 (1 + 2 tan 20)
(k~ + k~)
sin 0
R(0) dO (74)
0
Variation of
K2
with IM/8puol for the annular
where g and X are Lame constants and c 1 is the P-wave
velocity in the solid and c2 is the velocity of the S-wave
associated with the displacement and v is the Poisson's ratio.
The numerical values of IM/8uola I have been graphed in
Fig. 3 against Ka for various values of a and v = 1/4.
We know that:
k,
(
NUMERICAL
,
k-~2 = ~'z = \ X---+-~/
- L ~ J
(75)
RESULTS
The numerical values for I TIpl?~ol and I TIpl~OI shown in
Figs. 1 and 2 have been obtained by making use of the
Soil Dynamics and Earthquake Engineering, 1984, Vol. 3, No. 3
155
Diffraction o f torsional wave or plane harmonic compressional wave by annular rigid disc: R. S. Dhaliwal et al.
the integral equation (70). The results given in Fig. 6 refer
to the limiting case of a solid penny-shaped inclusion. It
may be noted that kl and k2 have dimensions of (length)-1.
By taking K1 = bk: and K2 = bk2 we find that K1 and K2
are dimensionless. Since b in Figs. 1-6 have been set equal
to unity the values o f K t and K2 are dimensionless.
a-O
v=0.25
1.0
b=O.50
0.5
REFERENCES
~ 0 . 3 3
~ , . . r
,
~o
~.o
.~o
40
~.o
e'.o
r:o
~,=o.zo
8'o
K2
Figure 6.
inclusion
Variation o f K 2 with
Ig/a/~u01 for
the solid
result (39). The numerical values of the function Hl(q~)
required for the numerical evaluation of (39) have been
obtained from a numerical solution of the integral equation
(36). The results shown in Figs. 1 and 2 have been evaluated
for the cases when (i) P2/Pl = 1 and for various values of
/al//~2 and (ii)/al//a2 = 1 and for various values ofpl/P2. The
results given in Figs. 3 and 4 refer to the limiting case of a
solid penny-shaped inclusion. Similarly the numerical values
for ]M/8/~Uol shown in Figs. 5 and 6 have been obtained
by making use of the result (74). The numerical values of
the function R (s) required for the numerical evaluation
of (74) have been obtained from a numerical solution of
156
1 Loeber, J. F. and Sih, G. C. Transmission of anti-plane shear
waves past an interface crack in dissimilar media, Eng. Fracture
Mech. 1973, 5, 699
2 Loeber, J. F. and Sin, G. C. Torsional wave scattering about a
penny-shaped crack lying on a bimaterial interface, Dynamic
Crack Propagation, Sih, G. C., ed., Noordhoff International
Publishers, Leyden, pp. 513-528, 1973
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