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Collinear periodical cracks with surface separation and frictional sliding
Theoretical and Applied Fracture Mechanics 8 (1987) 187-191 North-Holland
187
COLLINEAR PERIODICAL CRACKS W I T H SURFACE SEPARATION AND FRICTIONAL SLIDING
T.-H. HAO Department of Basic Science, China Textile University, Shanghai, China
Using the method of Muskhelishvili, an exact solution is obtained for the problem of collinear, periodical cracks whose surface undergo separation and frictional sliding.
The conditions at y = + oo are given by
1. Introduction When the crack surfaces slide in the presence of friction, the roughness gives rise to an upward separation as illustrated in Fig. 1. This model has been used by Kachanov [1] to account for the underestimate of the results in rock mechanics when frictional sliding forces are modelled in a classical fashion as flat contacting surfaces. As in [1], this paper considers the uplift of the crack surfaces when frictional sliding is present. The corresponding separation is assumed to be proportional to the slide. The formulation will utilize the complex potentials of Muskhelishvili.
oy = - o ,
~'xy= ~',
ox = 0,
where o and • are applied uniform stress. Following the work in [2], the two-dimensional stress field can be determined from two analytic functions ~ ( z ) and 12(z): oy -- i~'xy = q ~ ( z ) + ~ ( z )
+ 2iy~-r-~,
o,+Oy=2[ep(z)+~--~]=nReep(z). For this problem, the quantities
Ou 8v 3u av ox, Oy, r.y, Ox' Ox' Oy and Oy
f ( x + 2nby) = f ( x , y ) . Consider a series of collinear periodical cracks as shown in Fig. 2. They are placed at - a + 2nb <~x <~a + 2nb, y = 0 on the Oxy plane where n is an integer. On the surface of the cracks, there prevails the stress conditions:
(1)
with # being the frictional coefficient while oy and r,y are, respectively, the normal and shear stress. With y being a proportional coefficient, the displacement u and v are related as I v ] = ~/[u].
(2)
If the superscripts
+ and -
correspond to the
upper and lower crack surface, then [u] and [v] are defined as [v] = v + - v
-,
[u] = u + - u -.
(5)
are periodical functions of the form
2. Statement of problem and preliminaries
~0v+%=0, o~<0,
(4)
(3)
(6)
From the relation between Im ~ ( z ) and rotation 3 v / 3 x - 3u/Sy, there results
q~(z + 2nb) = ep(z),
~2(z + 2nb) = ~2(z). (7)
Making use of the function 2b z = - - a r c t g ~,
(8)
'IT
the region ABCD of into the f-plane with in Fig. 3. The points spond to those in Fig. f=tg
the z-plane in Fig. 2 maps the crack [4[ < tg(¢ra/2b) A B C D E F in Fig. 2 corre3. It is clear that on x = _+b
~-~ = tg ~ - ~ ( _ + b + i y )
= i cth Yl,
0167-8442/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
Yl = ' ~ y / 2 b -
(9)
7<.-H, Hao / Collinear periodical cracks with surface separation and frictional sliding
188
B(~)
y
:tg8
DN')
C(-:) K K'
]I ]
B{~} D{~}
!
Fig. 1.
Fig. 3,
Moreover, the following limits should be observed:
expressed as
[~,(t) + m,)] ~+ [e~(,) + ~(l il - 2p(t), Yl - - + "+-~ , cth Yl --' + 1: Yl--+0, c t h y l ~
(10)
Equations (9) and (10) imply that BC and BA in the ~-plane coincide with DC and DA, respectively, as indicated in Fig. 3. Because of the periodicity of q,(z) and I2(z), it is apparent that the values of q~(z) and I2(z) on any point K of A B C are equal to those on its corresponding point K ' of ADC. Hence, on both sides of BC and DC in the ~-plane (or BA and DA), ep(~) (or 82(~)) has the same value. This result implies that q,(~) and £2(~) are analytic on whole ~-plane except on the crack EF.
[4)(t)-/2(t)]+-[~(t)] =2q(t). -a<<.t..<
q(t)=-{[(o,: -a~-)-i('r~+-r~)]. There results from eq. (1) that
.o,, + ~,, : o.
(13)
(o, < o )
This leads to p ( t ) = 0, - i%v,
3. Analytic continuation
(12)
q(t)--0.
(14)
Hence, eq. (11) gives
The stress boundary conditions on the upper and lower crack surfaces in the z-plane can be
[~b(t) - ~](t)] + - [dp(t) - ~2(t)]
= O,
-a<~t<~a.
(15)
On I-plane, the above result is of the form
A(=)
2 E
B
K
=0.
A(--)
Fig. 2.
Z=X+iy
F D K'
IIIIIIII
I~l
'ira
~
r/=O.
(16)
As mentioned earlier, ~(£) and 82(~) are analytic on while [-plane except on the crack EF. In view of eq. (16), q~([) - I2([) has the same value on both sides of EF. The Liouville's theorem, therefore, yields
,~(~) - ~ ( ¢ ) = Co = ~([) =,~(¢)- c0.
cl + ic,.
189
T.-H. Hao / Collinear periodical cracks with surface separation and frictional sliding
Substituting eq. (25) into eq. (15) gives
The first equation of eqs. (11) becomes ~ ( ~ ) + + e p ( ~ ) - = o~, = ir~y + C~ +
'rra I~1 ~
l~{q~(~)+ + n(~)-+
iC2, (18)
~/=0.
"rra
: Im[~(~)++ ~ ( ~ ) - ] - C~, 'ITa
2--~(q,(~)++~(~)
]~l~tg~-~,
/~ Re[q~(~) + + 0 ( ~ ) - ] -/~C~
I~1 ~
1
=
Application of eq. (1) on E F gives
(19)
L ( ~ ) + } - ~C 1
-L(~)+-L(~)
},
T/=0,
(26)
E[q,(~)++,~(~) -] + L ( ~ ) + + L ( ~ ) • 2#C1 - 2 C 2 . =1
~/=0.
~(~)-+
1 +/~i
'
(27)
Similarly, the displacements in complex form are given by
E- /,i-1 /~i+l'
2G(u'+iv')=~q,(t)+g2(t),
Furthermore, eqs. (25) may be inserted into eq. (23) to yield
-a~
(20) where ~ = 3 - 4 o for plane strain and ~ = ( 3 o)/(1 + o) for plane stress with v being the Poisson's ratio. In eq. (20), u ' = d u / d t , o ' = d v / d t and G is the shear modulus. Application of eqs. (3) yields
L ( ~ ) + = 0, (28)
7=0.
It becomes apparent that -
+-
~@
-,
77=0.
D[q~(~)+- ~(~) ] + L ( ~ ) vi- 1 ~ra D- yi+l' I~1 ~
2G([u']+i[v']) +
rra I~]~
[~[~
-
= 0,
~=0.
(29)
(21)
- a <~ t <<.a,
Im{x[q~(t)+-O(t) -] + ~2(t)+-~2(t) - } 4. C o n d i t i o n s at infinity
=~, R e ( ~ [ ~ ( t ) + - ~ ( t ) -]
+ a ( t ) + - sa(,)- }, -a-%< t ~< a.
(22)
With reference to the ~'-plane, it can be shown that
De~(~) - L(~) - const = do,
Im{(~ + 1)[,(~j) + - (q~(~)-] } = Re[O(~) + - q'(~)- ] Y(~ + 1), qTa
I~l ~
(23)
=
(30)
-do
Putting eq. (30) into (27) it is found that
~1=0
It is convenient to introduce a new analytic function L(~)= ~(~) (= e?(~)) such that on the f-axis L ( ~ ) + = ~ ( ~ ) -,
Prior to the evaluation of qff~') and g2(~) from the stress conditions at infinity, the Liouville's theorem is again applied to obtain
L ( ~ ) - = O ( ~ ) +,
(E + D)[q~(~) + - q~(f)-] = 2d o + 2i ~a
141 ~
Im O(~) += 1 [ 0 ( ~ ) + - L ( ~ ) - ] ,
M = 2 d° + i(/~C, - 6"2)/(1 +/~i)
(E+D) (25)
C2
'
(31)
(24) Let M be defined such that
Im 4,(~)-= ~-~[0(~)-- L(~)+],
1 +#i
~/=o.
Re 4,(~5)+= ½[q,(~)+ + C ( ~ ) - ] ,
Re4,(~) = ½ [ q ~ ( ~ ) - + L ( t ) + ] ,
I~G -
(32)
From the definition of L(~'), there results lim L (~) = lim ~ - ~ . g'~oo ~--,oo
(33)
T.-H. Hao / Collinear periodical cracks with surface separation and frictional sliding
190
Introducing N as the following limit: lira L--~= N,
lim ~ ( ~ ) =
q,(z) and I2(z) take the forms (34) ~(z) =
~a
-
¼o
2(1 + y u ) cos~-~
which can be inserted into eq. (30) to give lim L ( ¢ ) = lim [Dq,(~)] - do, +
tg~-~ _ 1i [ 2 ~rz ~raV tg 2-b - tg2 2---b
(35)
N = D N - d o.
Therefore, eqs. (31) and (32) yield ~(~)+ + q~(~)- = M,
(¢-~,o)(v-0
X
,rra
[
"ffa
(4l)
2 cos-~-~ - (1 + 3'~t)
I~1 ~
l
(36)
s2(z) = ~ ( z ) -
Co = ~ ( z ) - ~o - ~ - .
Under these considerations, q~(z) and 12(z) are found: (N
1
qJ(z) =
.
5. Stress intensity factors
~rz
-~M)tg~-~
-
2 ,rrg
+ ½M,
,rra
tg ~-~ - t g 2 ~
(37)
(N
1
-
~(z)
=
.
qTZ
~M)tg~-~
2 'fig
tg ~ - ~ - tg
+ ½M-
2 ~a
C O.
It is useful to obtain the crack tip stress intensity factors k 1 and k 2. This requires only a knowledge of the q~(z) function in the first of eqs. (41), i.e., k s - ik 2 = 2~2 lira ~ - a q ~ ( z ) g~a
=
The conditions in eq. (4) can thus be invoked leading to lim
[O(z) + ~2(5)+ 2iyO-r~] = - o - i t ,
lira
{Re[~(z)] } = --~o.
y---, ± oc
y--, ± oo
N =
(38)
(39)
C1=½o,
Cz=r.
k s = ~,k2,
k2
(0"- #o) / 2b
l+y#
r ~ #o.
era
(43)
v~tg~-~'
k~ = 0 , . /-2-b ~a --~--tg~-~
k 2 = ('r --/.tO)V
It follows that Co, C 1 and C2 are given by Co=½O+i¢,
(42)
Note that k~ = 0 if y = 0. This means that k s or the uplifting of the crack surface is induced by the condition in eq. (2). Setting 3' = 0, k ~ - ik 2 becomes purely imaginary and hence
_ ¼o,
(1 - c o s T b ) ( ¢ - / x o ) ( 3 ' - 1) 'ha - ]o. 2(1 + Yt*) cos~--~
r ~ l~o.
Separating the real and imaginary parts, k s and k 2 are obtained;
Without going into details, the constants M and N can be obtained from
½M= - ( ¢ - / ~ 0 ) ( - / - i ) 2(1 + ~,)
( r - / ~ o ) ( y - i) /'-2b --~r-a 1 + y/~ V -~-tg-~-~ ,
(40)
Referring to the z-plane and loading at infinity,
r >/-/zo
(441)
,
This agrees with the classical results for a series of collinear cracks whose surfaces slide with friction. Some numerical results for # = 0.3 and 7 = 0, 0.1, etc. are listed in Table 1. Large values of 3' tend to increase [V]x= o and hence k 1.
T. -H. Hao / Collinear periodical cracks with surface separation and frictional sliding
191
Table 1
y=O:
kl=o,
k2=k~o,
[ulx~o = [U]o,
[V]x_o=0,
7=0.1: kl=0.097k2o,
k2=0.97k2 o,
[u]x=o=0-97[U]o,
[v]x=o=0.097[U]o,
y = 0.2: ka=0.188k2o ,
k2=0.94k2o,
[u]x,o=0.94[U]o,
[0]~=o=0.188[U]o,
~, = 0.3: kl=O.276k2o,
k2=0.92k2o,
[u]x_o=0.92[U]o,
[v]x=o=0.276[U]o .
References [1] L.M. Kachanov, Mechanics of Materials 1, p. 37 (1982). [2] N.I. Muskhelishvili, Some Basic Problems of Mathematical Theo~ of Elasticity, Noordhoff, Leyden (1975).
[3] H. Liebowitz, Fracture, Vol. 2, Academic Press, New York (1968). [4] T.-H. Hao, Internat. J. Fracture 30, 223-227 (1986). [5] T.-H. Hao, Engrg. Fracture Mech. 26(1L 59-63 (1987).
187
COLLINEAR PERIODICAL CRACKS W I T H SURFACE SEPARATION AND FRICTIONAL SLIDING
T.-H. HAO Department of Basic Science, China Textile University, Shanghai, China
Using the method of Muskhelishvili, an exact solution is obtained for the problem of collinear, periodical cracks whose surface undergo separation and frictional sliding.
The conditions at y = + oo are given by
1. Introduction When the crack surfaces slide in the presence of friction, the roughness gives rise to an upward separation as illustrated in Fig. 1. This model has been used by Kachanov [1] to account for the underestimate of the results in rock mechanics when frictional sliding forces are modelled in a classical fashion as flat contacting surfaces. As in [1], this paper considers the uplift of the crack surfaces when frictional sliding is present. The corresponding separation is assumed to be proportional to the slide. The formulation will utilize the complex potentials of Muskhelishvili.
oy = - o ,
~'xy= ~',
ox = 0,
where o and • are applied uniform stress. Following the work in [2], the two-dimensional stress field can be determined from two analytic functions ~ ( z ) and 12(z): oy -- i~'xy = q ~ ( z ) + ~ ( z )
+ 2iy~-r-~,
o,+Oy=2[ep(z)+~--~]=nReep(z). For this problem, the quantities
Ou 8v 3u av ox, Oy, r.y, Ox' Ox' Oy and Oy
f ( x + 2nby) = f ( x , y ) . Consider a series of collinear periodical cracks as shown in Fig. 2. They are placed at - a + 2nb <~x <~a + 2nb, y = 0 on the Oxy plane where n is an integer. On the surface of the cracks, there prevails the stress conditions:
(1)
with # being the frictional coefficient while oy and r,y are, respectively, the normal and shear stress. With y being a proportional coefficient, the displacement u and v are related as I v ] = ~/[u].
(2)
If the superscripts
+ and -
correspond to the
upper and lower crack surface, then [u] and [v] are defined as [v] = v + - v
-,
[u] = u + - u -.
(5)
are periodical functions of the form
2. Statement of problem and preliminaries
~0v+%=0, o~<0,
(4)
(3)
(6)
From the relation between Im ~ ( z ) and rotation 3 v / 3 x - 3u/Sy, there results
q~(z + 2nb) = ep(z),
~2(z + 2nb) = ~2(z). (7)
Making use of the function 2b z = - - a r c t g ~,
(8)
'IT
the region ABCD of into the f-plane with in Fig. 3. The points spond to those in Fig. f=tg
the z-plane in Fig. 2 maps the crack [4[ < tg(¢ra/2b) A B C D E F in Fig. 2 corre3. It is clear that on x = _+b
~-~ = tg ~ - ~ ( _ + b + i y )
= i cth Yl,
0167-8442/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
Yl = ' ~ y / 2 b -
(9)
7<.-H, Hao / Collinear periodical cracks with surface separation and frictional sliding
188
B(~)
y
:tg8
DN')
C(-:) K K'
]I ]
B{~} D{~}
!
Fig. 1.
Fig. 3,
Moreover, the following limits should be observed:
expressed as
[~,(t) + m,)] ~+ [e~(,) + ~(l il - 2p(t), Yl - - + "+-~ , cth Yl --' + 1: Yl--+0, c t h y l ~
(10)
Equations (9) and (10) imply that BC and BA in the ~-plane coincide with DC and DA, respectively, as indicated in Fig. 3. Because of the periodicity of q,(z) and I2(z), it is apparent that the values of q~(z) and I2(z) on any point K of A B C are equal to those on its corresponding point K ' of ADC. Hence, on both sides of BC and DC in the ~-plane (or BA and DA), ep(~) (or 82(~)) has the same value. This result implies that q,(~) and £2(~) are analytic on whole ~-plane except on the crack EF.
[4)(t)-/2(t)]+-[~(t)] =2q(t). -a<<.t..<
q(t)=-{[(o,: -a~-)-i('r~+-r~)]. There results from eq. (1) that
.o,, + ~,, : o.
(13)
(o, < o )
This leads to p ( t ) = 0, - i%v,
3. Analytic continuation
(12)
q(t)--0.
(14)
Hence, eq. (11) gives
The stress boundary conditions on the upper and lower crack surfaces in the z-plane can be
[~b(t) - ~](t)] + - [dp(t) - ~2(t)]
= O,
-a<~t<~a.
(15)
On I-plane, the above result is of the form
A(=)
2 E
B
K
=0.
A(--)
Fig. 2.
Z=X+iy
F D K'
IIIIIIII
I~l
'ira
~
r/=O.
(16)
As mentioned earlier, ~(£) and 82(~) are analytic on while [-plane except on the crack EF. In view of eq. (16), q~([) - I2([) has the same value on both sides of EF. The Liouville's theorem, therefore, yields
,~(~) - ~ ( ¢ ) = Co = ~([) =,~(¢)- c0.
cl + ic,.
189
T.-H. Hao / Collinear periodical cracks with surface separation and frictional sliding
Substituting eq. (25) into eq. (15) gives
The first equation of eqs. (11) becomes ~ ( ~ ) + + e p ( ~ ) - = o~, = ir~y + C~ +
'rra I~1 ~
l~{q~(~)+ + n(~)-+
iC2, (18)
~/=0.
"rra
: Im[~(~)++ ~ ( ~ ) - ] - C~, 'ITa
2--~(q,(~)++~(~)
]~l~tg~-~,
/~ Re[q~(~) + + 0 ( ~ ) - ] -/~C~
I~1 ~
1
=
Application of eq. (1) on E F gives
(19)
L ( ~ ) + } - ~C 1
-L(~)+-L(~)
},
T/=0,
(26)
E[q,(~)++,~(~) -] + L ( ~ ) + + L ( ~ ) • 2#C1 - 2 C 2 . =1
~/=0.
~(~)-+
1 +/~i
'
(27)
Similarly, the displacements in complex form are given by
E- /,i-1 /~i+l'
2G(u'+iv')=~q,(t)+g2(t),
Furthermore, eqs. (25) may be inserted into eq. (23) to yield
-a~
(20) where ~ = 3 - 4 o for plane strain and ~ = ( 3 o)/(1 + o) for plane stress with v being the Poisson's ratio. In eq. (20), u ' = d u / d t , o ' = d v / d t and G is the shear modulus. Application of eqs. (3) yields
L ( ~ ) + = 0, (28)
7=0.
It becomes apparent that -
+-
~@
-,
77=0.
D[q~(~)+- ~(~) ] + L ( ~ ) vi- 1 ~ra D- yi+l' I~1 ~
2G([u']+i[v']) +
rra I~]~
[~[~
-
= 0,
~=0.
(29)
(21)
- a <~ t <<.a,
Im{x[q~(t)+-O(t) -] + ~2(t)+-~2(t) - } 4. C o n d i t i o n s at infinity
=~, R e ( ~ [ ~ ( t ) + - ~ ( t ) -]
+ a ( t ) + - sa(,)- }, -a-%< t ~< a.
(22)
With reference to the ~'-plane, it can be shown that
De~(~) - L(~) - const = do,
Im{(~ + 1)[,(~j) + - (q~(~)-] } = Re[O(~) + - q'(~)- ] Y(~ + 1), qTa
I~l ~
(23)
=
(30)
-do
Putting eq. (30) into (27) it is found that
~1=0
It is convenient to introduce a new analytic function L(~)= ~(~) (= e?(~)) such that on the f-axis L ( ~ ) + = ~ ( ~ ) -,
Prior to the evaluation of qff~') and g2(~) from the stress conditions at infinity, the Liouville's theorem is again applied to obtain
L ( ~ ) - = O ( ~ ) +,
(E + D)[q~(~) + - q~(f)-] = 2d o + 2i ~a
141 ~
Im O(~) += 1 [ 0 ( ~ ) + - L ( ~ ) - ] ,
M = 2 d° + i(/~C, - 6"2)/(1 +/~i)
(E+D) (25)
C2
'
(31)
(24) Let M be defined such that
Im 4,(~)-= ~-~[0(~)-- L(~)+],
1 +#i
~/=o.
Re 4,(~5)+= ½[q,(~)+ + C ( ~ ) - ] ,
Re4,(~) = ½ [ q ~ ( ~ ) - + L ( t ) + ] ,
I~G -
(32)
From the definition of L(~'), there results lim L (~) = lim ~ - ~ . g'~oo ~--,oo
(33)
T.-H. Hao / Collinear periodical cracks with surface separation and frictional sliding
190
Introducing N as the following limit: lira L--~= N,
lim ~ ( ~ ) =
q,(z) and I2(z) take the forms (34) ~(z) =
~a
-
¼o
2(1 + y u ) cos~-~
which can be inserted into eq. (30) to give lim L ( ¢ ) = lim [Dq,(~)] - do, +
tg~-~ _ 1i [ 2 ~rz ~raV tg 2-b - tg2 2---b
(35)
N = D N - d o.
Therefore, eqs. (31) and (32) yield ~(~)+ + q~(~)- = M,
(¢-~,o)(v-0
X
,rra
[
"ffa
(4l)
2 cos-~-~ - (1 + 3'~t)
I~1 ~
l
(36)
s2(z) = ~ ( z ) -
Co = ~ ( z ) - ~o - ~ - .
Under these considerations, q~(z) and 12(z) are found: (N
1
qJ(z) =
.
5. Stress intensity factors
~rz
-~M)tg~-~
-
2 ,rrg
+ ½M,
,rra
tg ~-~ - t g 2 ~
(37)
(N
1
-
~(z)
=
.
qTZ
~M)tg~-~
2 'fig
tg ~ - ~ - tg
+ ½M-
2 ~a
C O.
It is useful to obtain the crack tip stress intensity factors k 1 and k 2. This requires only a knowledge of the q~(z) function in the first of eqs. (41), i.e., k s - ik 2 = 2~2 lira ~ - a q ~ ( z ) g~a
=
The conditions in eq. (4) can thus be invoked leading to lim
[O(z) + ~2(5)+ 2iyO-r~] = - o - i t ,
lira
{Re[~(z)] } = --~o.
y---, ± oc
y--, ± oo
N =
(38)
(39)
C1=½o,
Cz=r.
k s = ~,k2,
k2
(0"- #o) / 2b
l+y#
r ~ #o.
era
(43)
v~tg~-~'
k~ = 0 , . /-2-b ~a --~--tg~-~
k 2 = ('r --/.tO)V
It follows that Co, C 1 and C2 are given by Co=½O+i¢,
(42)
Note that k~ = 0 if y = 0. This means that k s or the uplifting of the crack surface is induced by the condition in eq. (2). Setting 3' = 0, k ~ - ik 2 becomes purely imaginary and hence
_ ¼o,
(1 - c o s T b ) ( ¢ - / x o ) ( 3 ' - 1) 'ha - ]o. 2(1 + Yt*) cos~--~
r ~ l~o.
Separating the real and imaginary parts, k s and k 2 are obtained;
Without going into details, the constants M and N can be obtained from
½M= - ( ¢ - / ~ 0 ) ( - / - i ) 2(1 + ~,)
( r - / ~ o ) ( y - i) /'-2b --~r-a 1 + y/~ V -~-tg-~-~ ,
(40)
Referring to the z-plane and loading at infinity,
r >/-/zo
(441)
,
This agrees with the classical results for a series of collinear cracks whose surfaces slide with friction. Some numerical results for # = 0.3 and 7 = 0, 0.1, etc. are listed in Table 1. Large values of 3' tend to increase [V]x= o and hence k 1.
T. -H. Hao / Collinear periodical cracks with surface separation and frictional sliding
191
Table 1
y=O:
kl=o,
k2=k~o,
[ulx~o = [U]o,
[V]x_o=0,
7=0.1: kl=0.097k2o,
k2=0.97k2 o,
[u]x=o=0-97[U]o,
[v]x=o=0.097[U]o,
y = 0.2: ka=0.188k2o ,
k2=0.94k2o,
[u]x,o=0.94[U]o,
[0]~=o=0.188[U]o,
~, = 0.3: kl=O.276k2o,
k2=0.92k2o,
[u]x_o=0.92[U]o,
[v]x=o=0.276[U]o .
References [1] L.M. Kachanov, Mechanics of Materials 1, p. 37 (1982). [2] N.I. Muskhelishvili, Some Basic Problems of Mathematical Theo~ of Elasticity, Noordhoff, Leyden (1975).
[3] H. Liebowitz, Fracture, Vol. 2, Academic Press, New York (1968). [4] T.-H. Hao, Internat. J. Fracture 30, 223-227 (1986). [5] T.-H. Hao, Engrg. Fracture Mech. 26(1L 59-63 (1987).