Engineering, 2010, 2, 485-495
doi:10.4236/eng.2010.27064 Published Online July 2010 (http://www.SciRP.org/journal/eng)
485
Dynamic Stress Intensity Factors for Three Parallel Cracks
in an Infinite Plate Subject to Harmonic Stress Waves
Shouetsu Itou
Department of Mechanical Engineering, Kanagawa University, Yokohama, Japan
Received February 25, 2010; revised April 9, 2010; accepted April 22, 2010
Abstract
Dynamic stresses around three parallel cracks in an infinite elastic plate that is subjected to incident
time-harmonic stress waves normal to the cracks have been solved. Using the Fourier transform technique,
the boundary conditions are reduced to six simultaneous integral equations. To solve these equations, the
differences of displacements inside the cracks are expanded in a series. The unknown coefficients in those
series are solved using the Schmidt method such that the conditions inside the cracks are satisfied. Numerical
calculations are carried out for some crack configurations.
Keywords: Three Cracks, Time-Harmonic Problem, Stress Intensity Factor, Integral Equation, The Schmidt
Method
1. Introduction
A time-harmonic solution for stresses around a crack in
an infinite plate was reported by Loeber and Sih [1]. In
their study, they obtained the Mode III dynamic stress
intensity factor during the passage of a time-harmonic
anti-plane shear wave. Subsequently, they also solved the
crack problem for a compression wave and a vertically
polarized shear wave [2]. Adopting a somewhat different
approach, the same problem was studied independently
by Mal [3]. The corresponding three-dimensional solu-
tions for a penny-shaped crack have been obtained by
Sih and Loeber [4,5] and by Mal [6].
Materials are generally weakened by some cracks.
Therefore, it is of interest to reveal the mutual effect of
the cracks on the dynamic stress intensity factors. Itou
solved the dynamic stresses around two collinear cracks
in which a self-equilibrated system of pressure is varied
harmonically with time [7]. Later, the Mode III solution
was given for two collinear cracks by Itou [8]. As for
three collinear cracks, Mode I solution s were determined
under the condition that time-harmonic normal traction is
applied to the surfaces of the cracks [9].
Materials are occasionally weakened by some parallel
cracks. Takakuda solved the time-harmonic problem for
two parallel cracks in an infinite plane subjected to
waves that impinge perpendicular to the cracks [10]. So
and Huang analyzed the Mode III stress intensity factor
around two cracks located in arbitrary positions in an
infinite medium subjected to incident SH waves [11].
Meguid and Wang cleared the effect of the existence of
an arbitrarily located and oriented micro defect on the
dynamic stress intensity factors for a finite main crack
subjected to a plane incident wave [12]. Ayatollahi and
Fariborz provided the analysis of multiple curved cracks
in an infinite plane under in-plane time harmonic loads
[13]. Itou and Haliding assumed that two small collinear
cracks are situated symmetrically above a main crack in
an infinite plate and provided the dynamic stress inten-
sity factors dur ing passage of time-harmonic wave s [1 4 ] .
A peak value of the dynamic stress intensity factor
for collinear cracks in an infinite elastic plate is
generally about 1.20-1.60 times larger than that of the
corresponding static v alue
peak
i
K
s
tatic
i
K. However, in the paper
[10], it was found that a peak value of the dynamic stress
intensity factor for two parallel cracks is significantly
larger than those for the collinear cracks. For example,
for an infinite plate containing two parallel cracks of
length separated by a distance , the
2ahpeak
i
K
s
tatic
i
K ratio is 4.16 for 1.0ha [10]. It was also
shown that similar results appear for two parallel cracks
in an infinite orthotropic plate subjected to incident
time-harmonic stress waves [15]. From this fact, it is
expected that
p
eak static
ii
KK ratio will be very large for
three parallel cracks during passage of the time harmonic
stress waves.

S. ITOU
486
In investigating the
p
eak static
ii
KK
ratio for three par-
allel cracks, the mixed boundary value conditions are
reduced to six dual integral equations. It has been shown
that the integral equations can be converted into six sets
of an infinite series and that the unknown coefficients in
the series can be solved using the Schmidt method [16].
The author has developed a Fortran program to obtain
the unknowns in four dual infinite series [15,17]. How-
ever, it is very difficult to write a Fortran program that is
capable of solvin g the unknown s in six sets of an infinite
series, making it difficult to solve the time-harmonic
problem for three parallel cracks. The present author
decided to solve the time-harmonic dynamic crack prob-
lem for three parallel cracks in an infinite elastic plate
because it is of importance to provide the dynamic stress
intensity factors in fracture mechanics.
In this study, time-harmonic stresses are solved for
three parallel cracks in an infinite elastic plate during the
passage of time-harmonic stress waves propagating nor-
mal to the cracks. The boundary co nditions were redu ced
to dual integral equations with use of the Fourier trans-
form technique. In order to solve these equations, the
differences between the crack surface displacements are
expanded to a series of functions that are equal to zero
outside the cracks. The Schmidt method is modified so
as to solve for the unknown coefficients in six sets of an
infinite series.
A Fortran progr am has been developed to calculate the
stress intensity factors for several crack configurations
numerically.
2. Fundamental Equations
Consider a crack in an infinite plate located along the
x
axis from to at , with respect to the
rectangular coordinates
aa0y
)(,
x
y; an upper crack from b

to at ; a lower crack from to at
; and incident time-harmonic stress waves pro-
pagating normally to the cracks, as shown in Figure 1.
For convenience, is referred to as layer 1);
is referred to as layer 2); is re-
ferred to as the upper half-plane 3); and is re-
ferred to as the lower half-plane 4). Let and be
defined as the
b
y
2

1
hy
0
c
1
h
y
*
u
c
h
*
v
2
h
hy
1
h0y
y
2
x
and components of the displace-
ment, respectively. If the displacement components
and are expressed by two functions
y
*
u
*
v


*,,
x
yt 
and

,,

*
x
yt  such that
** ****
,uxyv x    y (1)
the equations of motion reduce to the following forms:
2*22*222 *2
2*2 2*222*2
1,
1
L
T
x
yc t
x
yc t
 
 
 

(2)
where is time. The dilatational wave velocity
t
L
c
and the shear wave velocity under plane stress con-
ditions can be given as follows:
T
c


2
21 ,
L
c2
L
c



  
 (3)
where

is the modulus of rigidity,

is Poisson’s
ratio, and

is the density of the material.
The stresses can be expressed by the equations




 


*2*2222*22*
*2*2222*22*
*2*2*2222*
22
22
22
yy L
xx L
xyL 2
x
ct
yct
xy
xy
x
yxct
 
 
 

 

 
 
(4)
with

2
221
LT
cc




(5)
The incident stress waves that propagate through the
infinite plate parallel to the –axis in the negative di-
rection can be expressed as follows:
y



*
*
exp exp
0
inc
yy L
LL
inc
xy
pitypiyct
cc





 
 






(6)
where is a constant and
p

is the circular fre-
quency.
Substituting the following relations


 
*
*
,, exp ,
,, exp


x
yti t
x
yti t




 (7)
into Equation (2) results in
22222
22222
0,
0
xy
xy



 

 
(8)
with

 
2
,
11
T
LTLT
LT T
c
ccc c
cc c


 







(9)
Hereafter, the time factor

exp it


is omitted from
the equations for convenience. Hence, displacements and
stresses are expressed, respectively, by the following:
,ux
vx
y
y






 (10)

S. ITOU487






22222 2
2*22 222
222222
22
22
22
yy L
xx L
xy L
x
cxy
y
cx
xy xc
 
 
 

 
 
y
(11)
The boundary conditions for this problem can be ex-
pressed as
1313 1
,at,
yyyy xyxyyh x
  (12)
1212
,at0,
yyyy xy xyyx
  (13)
2424 2
,at,
yyyy xyxyyhx
  (14)

1111
exp,0 at,
yyL xy
pihc yhx   b
(15)
31 311
0,0, at,uu vvyhbx (16)
11
,0at0,
yy xy
py   xa (17)
12 12
0,0, at0,uuvvy ax (18)

2222
exp,0at,
yyL xy
pihc yhx   c
(19)
24 242
0,0, at,uuvvy hcx  (20)
where the subscript indicates the layer , the
subscript 3 indicates the upper half-plane (3), and the
subscript 4 indicates the lower half-plane (4).

1, 2i

i
3. Analysis
To obtain a solution, the following Fourier transforms
are introduced:
 
  
exp ,
()1 2exp
ffxixdx
f
xif ix

d







 



(21)
Applying Equ aiton (18) to Equation (8), we obtain :


2222
2222
0,
0
ddy
ddy


 
 
(22)
In the Fourier transform domain, the displacements
and stresses are denoted, respectively, by the forms
,ui ddyvddyi
 
  (23)



222
22 22
222
212
212
212
yy
xx
xy
id dy
ddy idd
id dy
 
 
 
 
 
 
y

(24)
For the layer , the solutions of Equation (22)
have the following forms:

1, 2i



22 22
22 22
sinh cosh
sinh cosh
ii i
ii i
A
yB y
CyD
 
 

 y 
(25)
where ,,,
ii i
A
BD are unknown coefficients. For the
upper half-plane (3) and the lower half-plane (4), the
solutions of Equation (22) have the following forms in
terms of the unknown coefficients :
3344
,,,CDCD



22 22
33 3
22 22
33 3
exp exp
exp exp
CyCi
DyDi
 



y
y


(26)



22 22
44 4
22 22
44 4
exp exp
exp exp
CyCi
DyDi

 
 
 
y
y


(27)
The stresses and displacements can be expressed by
twelve unknowns: 111
,,
A
BC,
and . Using Equations (12), (13) and (14), which
are valid for
1222233
,,,,,,,DABCDCD
4
C4 D
x

, the twelve unknowns are reduced
to six unknowns, yielding the following relations:
21 111 121 131142 15216
21 211 221 23124225226
31 311 321 331342 35236
31 411 421 431 44245246
41511 521 531542 552
iCAfBfiCfi DfAfBf
iDAfBfiCfiDfAfBf
CAfBfiCfiDfAfBf
iDAfBfiCfi DfAfBf
CAfBfiCfiDfAfB
 
 
 
 
 
56
41 611621 631 64265266
f
iDAfBfiCfi DfAfBf 
(28)
where the expressions of the known functions


,1,2,,6
ij
fij
have been omitted.
To satisfy the boundary conditions (16), (18) and (20),
the differences of the displacements are expanded as
follows:


1
13 1sin 2sinfor
0for
bb n
n
uuanxbxb
bx









(29)
 
1
13 1cos21sinfor
0for
bb n
n
vv bnxbxb
bx









(30)
 
1
12 1sin 2sinfor
0for
aa n
n
uuc nxaxa
ax









(31)

S. ITOU
488
 
1
12 1cos21sinfor
0for
aan
n
vv dnxax
ax




 


a
(32)
 
1
24 1sin2sinfor
0for
ccn
n
uuenxcx
cx




 


c
(33)
 
1
24 1cos(21) sinfor
0for
cc n
n
vvf nxcx
cx




 


c
(34)
where and
,,,,
nnnnn
abcde n
f
are unknowns, and the
subscripts and indicate the values at ,ab c1,yh

and , respectively. The Fourier trans-
forms of Equations (29)-(34) are expressed by
0y2 yh
   
  
13 2
1
13 21
1
2
21
bb nn
n
bb nn
n
iuuanJ b
vvbnJ b







 
 


(35)
   
  
12 2
1
12 21
1
2
21
aa nn
n
aa nn
n
iuucnJ a
vvdnJ a








 


(36)
   
  
24 2
1
24 21
1
2
21
cc nn
n
cc nn
n
iuuenJc
vvfnJc







 
 


(37)
where

n
J

is the Bessel function.
However, the variables on the left-hand sides of Equa-
tions (35), (36) and (37) can be expressed in terms of the
unknowns 11 112 ,, ,,
A
BiCiDA and :
2
B
 

13111112 113 114215
216
131 211 221 231 24225226
bb
bb
iuuAh BhiCh iDhAh
Bh
vv AhBhiChiDhAhBh
 

  
(38)
 

121311 321 331342 35
236
121 411 421 431 44245246
aa
aa
iu uAh Bh iChiDhAh
Bh
vv AhBhiChiDhAhBh
 

 
(39)





24151152 153 154255
256
241 611621 631 64265266
cc
cc
iuuAh Bh iCh iDhAh
Bh
vv AhBhiChiDhAhBh


  
(40)
where the expressions of the known functions

,
ij
hij

have been omitted.

1, 2,, 6
Equating Equations (35), (36) and (37) with Equations.
(38), (39) and (40), the unknowns 1,
A
1,B1,iC1,iD 2
A
and can be indicated by the unknowns
and
2
B
n
e
,
n
a,
n
bc ,
n
,
n
dn
f
. For example, the unknown 1
A
is ex-
pressed as:
 
 
 
 
 
 
1112
11
21 2
31 2
11
41 2
51 2
1
61 2
1
2
21
2
21
2
21
nn
nn
n
nn
nn
n
nn
n
nn
n
n
n
A
anHJb b
nHJb
cnHJa d
nHJa
en HJc
fn HJac















 
 
 
 








(41)
where ij
H
is the cofactor of the element , and
ij
h

is
given by


,, 1,2,,6
ij
hij (42)
Consequently, stresses that satisfy the boundary con-
ditions (12), (13), (14), (16), (18) and (20) can be ex-
pressed in terms of the unknowns
and
,
n
a,
n
b,
n
c,
n
d,
n
d
n
f
. For example, stress 1
y
y

at is of the
form
1
yh

  
  
  
   
  
 
12
11 0
1
221
0
1
32
0
1
421
0
1
52
0
1
621
2c
(2 1)/cos
(2 )/cos
21 cos
2cos
(2 1)/co
nn
yy hn
nn
n
nn
n
nn
n
nn
n
nn
anQJb xd
bnQJbxd
cnQJa xd
dnQJ axd
enQJcxd
fnQJc
os























 

 

 







0
1s n

x
d





(43)
where the expressions of the known functions


i
Q


S. ITOU489

1,2,, 6i

have been omitted. Finally, the remaining
boundary conditions (15), (17) and (19), which must be
satisfied inside the cracks, reduce to the forms:
  
 
11
15
nn
nn
aFx
eF

1213 14
11 11
16 1
11 for
nnnn nn
nn nn
nn
nn
bFxcFx dFx
xfFxu xxb
 
 



 


(44)
 

21
25
nn
nn
aFx
eF


2223 24
11 11
26 2
11 for
nnnn nn
nn nn
nn
nn
bFxcFx dFx
xfFxu xxb
 
 



 


(45)
 
 
31
35
nn
nn
aF x
eF


32 33 34
11 11
36 3
11 for
nnnnnn
nn nn
nn
nn
bF xcF xdF x
xfFxu xxa
 
 



 


(46)
 
 
41
45
nn
nn
aFx
eF


4243 44
11 11
46 4
11 ()for
nnnn nn
nn nn
nn
nn
bFxcFx dFx
xfFxu xxa
 
 



 


(47)
 
 
51
55
nn
nn
aF x
eF


52 53 54
11 11
56 5
11 for
nnnnnn
nn nn
nn
nn
bF xcF xdF x
xfFxu xxc
 
 






(48)
  
 
61
65
nn
nn
aF x
eF

6263 64
11 11
66 6
11 for
nnnn nn
nn nn
nn
nn
bF xcF xdF x
xfFxu xxc
 
 



 


(49)
where the functions are known.
For example,

,1,2,,6
nij
Fxij

,
11n
F
x


12n
F
x66n and

F
x are ex-
pressed as
  
111 2
0
2cos
nn
F
xn QJbxd



 (50)
 


 

222 1
0
22 1
2
21
cos21sin
nLn
L
nQQJ b
12
cos
Fx
x
dQ bxnxb
 



 






(51)
  
66362 1
0
21 sin
nn
F
xnQJc xd



 

(52)
where the constant 2
L
Q is given by

22
L
L
QQ

 (53)
where
L

is a larger value of

. The functions


i
ux
are denoted by the equations:



 

1123
45 26
exp ,0,,
0,exp ,0
ux pihuxux p
uxuxpihux


 

  (54)
The unknowns and ,,,,
nnn nn
abcde n
f
in Equations
(44), (45), (46), (47), (48) and (49) can now be solved
using the Schmidt method described in Appendix A.
4. Stress Intensity Factors
Once the unknown coefficients and ,,,,
nnnnn
abcde n
f
have been solved, all the stresses and displacements can
likewise be solved. In fracture mechanics, it is important
to determine the stress intensity factors defined from the
stresses in the region near the crack ends. Using the rela-
tions
 
 

 

0
1/2 1/2
22 22
1/2 1/2
22 22
cos, sin
sin ,
cos
n
n
n
n
n
Jax xd
ax ax x an
ax axx anforax











 






 





(55)
the stress intensity factors can be expressed as




1
11(1
2
1
1
21(1
7
1
lim 2
21 1
lim 2
21
hyy h
xb
n
nL
n
hxy h
xb
n
nL
n
Kxb
bn Q
Kxb
anQ b

)
)
b















(56)



0
11(0)
16
1
0
21(0)
21
1
lim2 ()
21(1)
lim 2
21
yy
xa
n
nL
n
xy
xa
n
nL
n
Kxa
dn Q
Kxa
cnQ a

a















(57)

S. ITOU
490






2
142
30
1
2
242
35
1
lim 2
21 1
lim 2
21
hyy h
xc
n
nL
n
hxy h
xc
n
nL
n
Kxc
f
nQ
Kxc
enQ c

c



















(58)
where the constants are given
by expressions taking the similar form as Equation (53).

7,16,21,30,35
iL
Qi
5. Numerical Examples
The dynamic stress intensity factors were calculated nu-
merically with quadruplex precision using a Fortran pro-
gram, during the operation of which, overflow and un-
derflow do not occur within the range to .
Numerical calculations were performed for a Poisson’s
ratio
5500
105500 10
0.25

.
The semi-infinite integrals, which appear in the known
functions ()
nij
F
x must be evaluated
numerically. It can be verified that the numerical integra-
tions have been performed satisfactorily because the in-
tegrands decay rapidly as the integration variable
(,1,2, ,6)ij

increases. To solve the unknown coefficients
and
,
n
a,
n
b
,
n
c,
n
dn en
f
, the Schmidt method has been ap-
plied by truncating the infinite series in Equations (44),
(45), (46), (47), (48) and (49) by summing from 1n

to . It has been verified that the values for the
left-hand side of the equations coincide with those for the
right-hand side with acceptable accuracy.
8n
The absolute values of the stress intensity factors giv-
en in Equations (56), (57) and (58) are calculated for
1.0ba ca and 12 2.5ha ha; these are plot-
ted with respect to T ac

in Figure 2. The straight
dashed lines on the left-hand side of Figure 2 indicate
the corresponding static values given by Ishida [18].
Figure 3 shows the values for 12 1.0ha ha. The
dynamic stress intensity factors for two cracks in an infi-
nite plate have been solved by Takakuda [10]. In the
present study, the same problem has been reworked, and
the results are plotted in Figures 4 and 5 for 12.5ha

and 1.0, respectively, where 1.0ba is assumed, and
denotes the distance between the two parallel cracks.
For the case of two parallel cracks, the distance in
Figure 1 is considered to be infinite.
1
h
2
h
Figure 1. Geometry and coordinate system.
Figure 2. Stress intensity factors ,
1
1
h
K,
1
2
h
K,
0
1
K,
0
2
K
2
1h
K and 2
2h
K for12
2.5ha ha
and ba ca


. 1.0
6. Discussion
In the previous paper [14], time-harmonic stresses are
solved for three cracks in an infinite elastic plate during
the passage of time-harmonic stress waves. Two collin-
ear cracks are situated symmetrically on either side of the
main crack. The mixed boundary conditions with respect
to the three cracks are reduced to four sets of an infinite
series. The two sets of an infinite series are derived from

S. ITOU491
Figure 3. Stress intensity factors ,
1
1
h
K,
1
2
h
K,
0
1
K,
0
2
K
2
1h
K and 2
2h
K for12 1.0ha ha and ba ca


. 1.0
Figure 4. Stress intensity factors ,
1
1
h
K,
1
2
h
K0
1
K and
0
2
K for 12.5ha and 1.0ba

. (For the case of two
parallel cracks, the distance in Figure 1 is considered
to be infinite). 2
h
the boundary conditions inside the main crack, while the
other two sets are derived from those inside one of the
upper small cracks. The method to solve the unknown
coefficients in the infinite series has been already de-
scribed in [15,17].
In the present paper, time-harmonic stresses are solved
for an infinite elastic plate weakened by three parallel
cracks. As the three cracks are not collinear, the bound-
ary conditions with respect to the three cracks are re-
duced to six sets of an infinite series. Each of the two
Figure 5. Stress intensity factors ,
1
1
h
K,
1
2
h
K0
1
K and
0
2
K for 11.0ha and 1.0ba. (For the case of two
parallel cracks, the distance in Figure 1 is considered
to be infinite). 2
h
sets of an infinite series must satisfy the boundary condi-
tions inside the lower, middle and upper cracks, respec-
tively. Therefore, the Schmidt method is newly extended
in the present paper to determine the six sets of unknown
coefficients as described in section 1.
In this study, the minimum value of


12
ha ha
used during numerical calculations was 1.0. If the calcu-
lations were performed for lower values of
1/ha


2
ha, the resulting peak value of

0
1
K
pa would
be significantly larger than 12.8. There is scope for fur-
ther inquiry into whether or not the absolute values of the
stress intensity factors would increase in the case of four
parallel cracks in an infinite elastic plate. The author
considers that the peak value of

0
1
K
pa increases
as the number of parallel cracks increases. If a time-
harmonic load is applied to materials weakened by many
parallel cracks, we note that the dynamic stress intensity
factors would have a greater value than that for the cor-
responding static solution.
7. Conclusions
Based on the numerical calculations outlined above, and
with reference to Figures 2 through 5, the following
conclusions are reached:
1) There is a critical circular frequency in the three
cracks case near 0.9
T
ac


similar to that shown
by Takakuda in the two-cracks case [10]. The value

S. ITOU 492

0
1
K
pa is 12.8 for 12 1.0ha ha. For two
parallel cracks, the corresponding value is 5.87. It can be
seen that the presence of the third crack has a significant
effect upon the dynamic stress intensity factor around the
center crack between the upper and lower parallel cracks.
[7] S. Itou, “Dynamic Stress Concentration around Two Co-
planar Griffith Cracks in an Infinite Elastic Medium,”
ASME Journal of Applied Mechanics, Vol. 45, No. 12,
1978, pp. 803-806.
[8] S. Itou, “Diffraction of an Antiplane Shear Wave by Two
Coplanar Griffith Cracks in an Infinite Elastic Medium,”
International Journal of Solids and Structures, Vol. 16,
No. 12, 1980, pp. 1147- 1153.
2) For 12 2.5ha ha (Figure 2), the slope to the
peak value of

0
1
K
pa is comparatively gentle.
However, for 12 1.0ha ha (Figure 3), the curve
rises steeply to the maximum value of


0
1
K
pa ,
after which it declines equally steeply. The curve exhib-
its a sharp peak at the critical circular frequency
near 0.9
T
ac

.
[9] S. Itou, “Dynamic Stresses around Two Cracks Placed
Symmetrically to a Large Crack,” International Journal
of Fracture, Vol. 75, No. 3, 1996, pp. 261-271.
[10] K. Takakuda, “Scattering of Plane Harmonic Waves by
Cracks (in Japanese),” Transactions of Japan Society of
Mechanical Engineeris, Series A, Vol. 48, No. 432, 1982,
pp. 1014-1020.
[11] H. So and J. Y. Huang, “Determination of Dynamic
Stress Intensity Factors of Two Finite Cracks at Arbi-
trary Positions by Dislocation Model,” International
Journal of Engineering Science, Vol. 26, No. 2, 1988,
pp. 111-119.
3) In static solutions, the stress intensity factors for
three equal-length cracks decrease slightly as 1 ha

2
ha

decreases. However, this is accompanied by a
significant increase in the peak value of

0
1
K
pa. [12] S. A. Meguid and X. D. Wang, “On the Dynamic Interac-
tion between a Microdefect and a Main Crack,” Pro-
ceedings of the Royal Society of London, Series A, Vol.
448, No. 1934, 1995, pp. 449-464.
8. References [13] M. Ayatollahi and S. J. Fariborz, “Elastodynamic Analy-
sis of a Plane Weakened by Several Cracks,” Interna-
tional Journal of Solids and Structures, Vol. 46, No. 7-8,
2009, pp. 1743-1754.
[1] J. F. Loeber and G. C. Sih, “Diffraction of Antiplane
Shear by a Finite Crack,” Journal of the Acoustical Soci-
ety of America, Vol. 44, No. 1, 1968, pp. 90-98. [14] S. Itou and H. Haliding, “Dynamic Stress Intensity Fac-
tors around Three Cracks in an Infinite Elastic Plane
Subjected to Time-Hharmonic Stress Waves,” Interna-
tional Journal of Fracture, Vol. 83, No. 4, 1997, pp. 379-
391.
[2] G. C. Sih and J. F. Loeber, “Wave Propagation in an
Elastic Solid with a Line of Discontinuity or Finite
Crack,” Quarterly of Applied Mathematics, Vol. 27, No.
2, 1969, pp. 193-213.
[15] S. Itou and H. Haliding, “Dynamic Stress Intensity Fac-
tors around Two Parallel Cracks in an Infinite-Orth-
otropic Plane Subjected to Incident Harmonic Stress Wa v es, ”
International Journal of Solids and Structures, Vol. 34,
No. 9, 1997, pp. 1145-1165.
[3] A. K. Mal, “Interaction of Elastic Waves with a Griffith
Crack,” International Journal of Engineering Science,
Vol. 8, No. 9, 1970, pp. 763-776.
[4] G. C. Sih and J. F. Loeber, “Torsional Vibration of an
Elastic Solid Containing a Penny-Shaped Crack,” Journal
of the Acoustical Society of America, Vol. 44, No. 5, 1968,
pp. 1237-1245. [16] P. M. Morse and H. Feshbach, “Methods of Theoretical
Physics,” McGraw-Hill, New York, Vol. 1, 1958.
[17] S. Itou, “Axisymmetric Slipless Indentation of an Infinite
Elastic Hollow Cylinder,” Bulletin of the Calcutta Ma-
thematical Society, Vol. 68, No. 17, 1976, pp. 157-165.
[5] G. C. Sih and J. F. Loeber, “Normal Compression and
Radial Shear Waves Scattering at a Penny-Shaped Crack
in an Elastic Solid,” Journal of the Acoustical Society of
America, Vol. 46, No. 3B, 1969, pp. 711-721. [18] M. Ishida, “Elastic Analysis of Cracks and Stress In-
tensity Factors (in Japanese),” Fracture Mechanics
and Strength of Materials, Baifuukan Press, Tokyo,
Vol. 2, 1976.
[6] A. K. Mal, “Interaction of Elastic Waves with a Penny-
Shaped Crack,” International Journal of Engineering
Science, Vol. 8, No. 5, 1970, pp. 381-388.

S. ITOU 493
Appendix A
For convenience, Equations (42) through (47) can be
rewritten as
 
 
11 121314
11 11
15161
11 for
nn nnnnnn
nn nn
nn nn
nn
aFx bFxcFx dFx
eFxfFxu xxb
 
 



 


(A.1)
  
 
21 222324
11 11
2526 2
11 for
nn nnnnnn
nn nn
nn nn
nn
aFx bFxcFxdFx
eFxfFxu xxb
 
 



 


(A.2)
 
 
31 323334
11 11
3536 3
11 for
nn nnnnnn
nn nn
nn nn
nn
aF xbFxcF xdFx
eFxfFxu xxa
 
 



 


(A.3)
  
 
41 424344
11 11
4546 4
11 for
nn nnnnnn
nn nn
nn nn
nn
aFx bFxcFxdFx
eFxfFxu xxa
 
 



 


(A.4)
 
 
51 525354
11 11
5556 5
11 for
nn nnnnnn
nn nn
nn nn
nn
aF xbFxcF xdFx
eFxfFxu xxc
 
 



 


(A.5)
 
 
61 626364
11 11
6566 6
11 for
nn nnnnnn
nn nn
nn nn
nn
aF xbFxcFxdFx
eFxfFxu xxc
 
 



 


(A.6)
A set of functions that satisfy the orthogonal-
ity condition
()
n
Gx
 
2
00
,
bb
mnnmnn n
GxGx dxIIGx dx   

(A.7)
can be constructed from a given set of arbitrary function,
say 65()
n
F
x, such that
  
65
1
n
ninnni
i
GxPPFx


 (A.8)
where is the cofactor of the element of ,
which is defined as
in
Pin d'
n
D
 
11 121
21 65 65
0
1
',
n
c
nini
nnn
ddd
d
DdFx
dd






n
Fxdx
(A.9)
Representing the fifth series in Equation (A.6) by the
orthogonal series with coefficients ()
n
Gx n

, the fol-
lowing relationships are derivable:
 
 

65
11
66162
11 1
64 66
11
nnn n
nn
nn nnnn
nn n
nn nn
nn
eFxG x
uxaF xbF xcF x
dF xfF x



 
 



 




63
(A.10)
The second equality yields
 
  
 
66
00
1
62 63
00
11
64 66
00
11
1
for
cc
nn ini
i
n
cc
in iini
ii
cc
in iini
ii
Gxuxdxa GxFxdx
I
bGxFxdx cGxFxdx
dGxFxdx fGxFxdx
xc








 





1









(A.11)
and considering Equation (A.8), the first equality shows
that
0
11111
abc d
nniniiniiniinii ni
iiiii
eabcd f f


 
 
 


(A.12)
with
 
 
 
 
 
 
06
0
61
0
62
0
63
0
64
0
66
0
,
,
,
,
,
c
nj
nj
jn jj j
c
nj
a
nij i
jn jj j
c
nj
b
nij i
jn jj j
c
nj
c
nij i
jn jj j
c
nj
d
nij i
jn jj j
c
nj
f
nij i
jn jj j
PGxuxdx
PI
PGxF xdx
PI
PGxF xdx
PI
PGxF xdx
PI
PGxF xdx
PI
PGxF xdx
PI






























(A.13)

S. ITOU 494
Substituting Equation (A.12) into Equation (A.5), the
equality now becomes
 
 
**
51 5253
11 1
***
5456 5
11
for
nn nnnn
nn n
nn nn
nn
aF xbFxcF x
dF xfF xux
xc
 
 



 



*
(A.14)
with
 
 
 
 
 
 
*
51 5155
1
*
52 5255
1
*
53 5355
1
*
54 5455
1
*
56 5655
1
*0
55 55
1
,
,
,
,
,
a
nn ini
i
b
nn ini
i
c
nn ini
i
d
nn ini
i
f
nn ini
i
ii
i
F
xFx Fx
F
xFx Fx
F
xFx Fx
F
xFx Fx
F
xFx Fx
ux uxFx






























(A.15)
Using the same procedure, the orthogonal function
*()
n
H
x is constructed from *
56 () n
F
x as
  
**
56
1
n
ninnni
i
H
xQQF


x (A.16)
where is the cofactor of the element
in
Qin
g
of ,
which is defined as
''
n
D
 
11 121
21
1
**
56 56
0
'' ,
n
n
nn
c
in in
gg g
g
D
gg
n
g
FxFxdx






(A.17)
Using Equations (A.14) and (A.16), the coefficient
n
f
can be expressed by and as follows: ,,
nnn
abcn d
0
11
11
ab
nniniini
ii
cd
iniini
ii
fa
cd
b











(A.18)
with
 
 
 
 
 

0**
5
0
**
51
0
**
52
0
**
53
0
**
54
0
2
*
0
,
,
,
,
,
c
nj
nj
jn jj j
c
nj
a
nij i
jn jj j
c
nj
b
nij i
jn jj j
c
nj
c
nij i
jn jj j
c
nj
d
nij i
jn jj j
c
nn
Q
H
xu xdx
QJ
Q
H
xF xdx
QJ
Q
H
xF xdx
QJ
Q
H
xF xdx
QJ
Q
H
xF xdx
QJ
JHxdx




























(A.19)
Substituting Equation (A18) into Equation (A12), we
obtain the following relation:
0***
11
**
11
ab
nnini ini
ii
cd
iniini
ii
eab
cd







 



(A.20)
with
0* 00
1
*
1
*
1
*
1
*
1
,
,
,
,
f
nn jnj
j
aa af
ninijinj
j
bb bf
niniji nj
j
cc cf
niniji nj
j
dd d
niniji nj
j
f






























(A.21)
Replacing the coefficients and
n
en
f
in Equations
(A.1), (A.2), (A.3) and (A.4) with Equations (A.18) and
(A.20), the equality becomes
 
 
**
11 1213
11 1
**
14 1
1for
nn nnnn
nn n
nn
n
aF xbFxcF x
dFxuxxb
 
 






*
(A.22)
 
 
** *
21 2223
11 1
**
24 2
1for
nn nnnn
nnn
nn
n
aF xbF xcFx
dFxuxxb
 
 



 


(A.23)

S. ITOU 495
 
 
**
31 3233
11 1
**
34 3
1for
nn nnnn
nn n
nn
n
aF xbFxcF x
dFxuxxa
 
 



 


*
(A.24)
 
 
**
41 4243
111
**
44 4
1for
nn nnnn
nnn
nn
n
aF xbF xcFx
dFxu xxa
 
 



 


*
(A.25)
with
 
 
 
 
 
**
11 111516
1
**
12 151516
1
**
13 131516
1
**
14 141516
1
*0*
11 15
1
,
,
,
,
aa
nn iniini
i
bb
nn iniini
i
cc
nn iniini
i
dd
nn iniini
i
ii i
i
FxFxFx Fx
F
xFxFx Fx
FxFxFxFx
F
xFxFx Fx
uxuxF x
















 


 





 










016i
Fx


(A.26)
 

 
 
 
**
31 313536
1
**
32 353536
1
**
33 333536
1
**
34 343536
1
*0*
33 35
1
,
,
,
,
aa
nn iniini
i
bb
nn iniini
i
cc
nn iniini
i
dd
nn iniini
i
iii
i
F
xFxFx Fx
F
xFxFx Fx
F
xFxFx Fx
F
xFxFx Fx
ux uxF x

















 




 




 




 



 







036
i
Fx


(A.28)
 

 
 

**
21 212526
1
**
22 252526
1
**
23 232526
1
**
24 242526
1
*0*
22 25
1
,
,
,
,
aa
nn iniini
i
bb
nn iniini
i
cc
nn iniini
i
dd
nn iniini
i
iii
i
F
xFxFxFx
F
xFxFx Fx
F
xFxFx Fx
F
xFxFx Fx
ux uxF x



















 


 


 


 







026
i
Fx




(A.27)
 
 
 
 
 
**
41 414546
1
**
42 454546
1
**
43 434546
1
**
44 444546
1
*0*
44 45
1
,
,
,
,
aa
nn iniini
i
bb
nn iniini
i
cc
nn iniini
i
dd
nn iniini
i
iii
i
F
xFxFxFx
F
xFxFxFx
F
xFxFx Fx
F
xFxFxFx
ux uxF x






















 




 








 







046
i
Fx


(A.29)
Equations (A.2 2) , (A. 23 ), (A.24) and (A.25 ) have been
already solved for and in Ref. [17, 1 5] . ,,
nnn
abcn d