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Two finite length cracks moving between strips under plane extension
cm-7944/w $3.00+.00 0 1984 Pergamon Press Ltd.
Engineming Fracture Mechanics Vol. 19, No. 2. pp. 279-286. 1984 Printed in Great Britain.
TWO FINITE LENGTH CRACKS MOVING BETWEEN STRIPS UNDER PLANE EXTENSION R. S. DHALIWAL, Department
of Mathematics,
B. M. SINGH University
and JON ROKNE
of Calgary, Calgary, Alberta, Canada
Abstract-The problem of the two coplanar uniformly propagating finite cracks in the strip of elastic material is solved. Two specific conditions of loading on the strip with finite width are discussed. In the first case the rigidly clamped edges are pulled apart in opposite directions. In the second case equal and opposite tractions are applied to edges of the strip. By the use of Fourier transforms we reduce the first problem to solving a set of triple integral equations with cosine kernel and a weight function. These equations are solved by using finite Hilbert transform techniques for large values of h where 2h is the width of the strip. Finally, numerical values for dynamic stress intensity factors are presented graphically. It is shown that the solution of the second problem can be obtained in a manner similar to the first case.
1. INTRODUCTION years the study of elastodynamic problems of moving or extending cracks in linear elastic media has received much attention in view of its importance in the field of fracture mechanics. The problem of a strip of material being split by a semi-infinite crack propagating at constant velocity has been investigated by Sih. and Chen[ 11. Sih. and Chen[2] have also studied the finite length crack propagating with constant velocity in a strip when the anti-plane shear displacements and stresses are applied to the lateral boundaries of the strip. Sih. and Chen[3] have also studied the related problem of a uniformly propagating line crack in a strip of elastic material under plane extension. Recently Singh, Moodie and Haddow[4] obtained closed form solution for a finite length crack moving in a strip under anti-plane shear stress. Tait and Moodie[5] solved the problem of two coplanar moving Griffith cracks in a strip under anti-plane shear stress using complex variable method. Lowengrub and Srivastava[6] also considered the problem of determining the stress field in the vicinity of two stationary coplanar Griffith cracks in an infinitely long elastic strip. In this paper we have considered the problem of two coplanar moving cracks in the strip under plane extension. Two specific conditions of loading with finite width of the strip are discussed. In the problem (A), the rigidly clamped edges are pulled apart in opposite directions. In the problem (B) equal and opposite tractions are applied to edges of the strip. For problem (A) expression for dynamic stress intensity factor is obtained for values of h + 1. IN RECENT
2. ELASTODYNAMIC EQUATIONS AND FORMULATION OF THE PROBLEM Consider a strip of the elastic material --m < X < m, -h < Y < h with cracks moving in the interior of the material on the line Y = 0, c < IX/< 1 with velocity o. For the sake of completeness we shall write some results, which are given by Sih. and Chen[3]. In terms of two scalar functions 4 and I,$each of which is a function of X, Y and t, we find that [u
u 9
9
W]’
a24 a*+
a!+aJ, ?!?L_alC, ax aY’ax r3Y' 0
[
1
a*fja24
-+2=72,2+Z=~af2, ax aY c, at
ax
a2+ aY
1 2
I
a2’
(2)
(3) where c, is the compression (irrotational) wave speed, c2 the shear (equivoluminal) wave speed in an infinitely extended medium, A and CLare Lam6 coefficients and p is the mass density of the elastic medium. 279
R.SDHALIWAL et al.
280
For the constant velocity of the cracks, it is convenient to introduce a Galilean transformation x=X-r&
y=Y,
V=t.
(4)
In the transformed system, the wave equations are independent of time variable t’, i.e. s 2q+q4 ’ ax
ay
s 2q+q4 2 ax
3
ay
9
(5)
where
(6) The displacement and stress components can also be transformed into the new coordinate system [U
u
XT
p’
!%+!!!!~_~o
$$)I” z [
ax
ay' ay
ax'
and
I ’
(7)
hP’$+2p(~+-g), uy=hv20+2~(gf+$), ux =
a,,=p
( 8%a29,q 2-axay
ax
ay ) *
(8)
The two conditions of loading considered involve the specification of displacements on the strip boundaries and tractions on the crack surfaces. Now we will discuss the two problems. Problem (A) Let the edges of the strip be clamped rigidly and displaced by an amount SOin the direction normal to the propagating cracks and hence l&(x, rh) = 0,
--M < x
u&x, 0) = 0, 0 < 1x( 1, uy(x,-th) = &,, --m < x
(9) (10)
(11)
Because of symmetry, the following conditions on the stress components may be written down as uy(x,O)=O,
c
f&(x, 0) = 0, --co< x G @J.
(12)
In order to use the technique of integral transforms, it is necessary to solve an alternative but equivalent problem involving loading on the crack surfaces which can be obtained from the problem of a clamped strip (without cracks) subjected to a uniform strain. The dynamic stress intensity factors can be found by solving the equivalent problem -Et&J
a,(x,o)=h(l_V2)' uy(x,O)=O,
c
O<(x(
IxI>l,
frxxy(x,O)=O, O<(x(
(13) (14) (15)
TWO finite length cracks
moving
between
281
strips
and the displacement must satisfy r&.(x,+h) = u,(x, *h) = 0,
(16)
where v denotes Poisson’s ratio and E is the Young’s modulus. The solution to the original problem is then obtained by adding the solution of the equivalent problem and that of
ux=
vEao h(l-
3)7
UY=
Eao 4h’
(l-
ux, = 0. Solution of the eqn (5) can be written in the following form: d(x, Y)= ilrn [A(5) exp(s&) + B(5) ev(-s&)lco@W
d5
rL(x,Y)= i lo- [C(CZ)exp(s&y) + o(5) exp(-s&N sin(&)d5.
(17)
Where A(& B(t), etc. are the four unknowns in terms of the single variable 5; Substituting from (17) into eqns (7) and (8), it is found that displacements and stresses are given by: n, = f [
H-A(5) exp(s&)-
B(5) exp(-s&)
+ s~lC(5)
x exp(s&y) - D(T) exp(-dY)ll sin(txjr) 45
rx vy= $ o 5bM5) exph5y) - W exp(-s&)1 - W3 I x exp(s2&)- N5) d-&)I ~045~)45
(18)
(19)
~~=~~m52(-~(1-~~t2~12)[A(~)exp(s,5y)+8(51 x ew(-s&y)1 + sJC(t) eMs25y) - D(5) exd-s,5y)l} cW%) d5,
(20)
cr, = % ost2 it1 + s~M5) evbdy) + B(5) exp(-s15y)l. I ( -%[C(5) eM2W - D(t) evb2b)l} cos(W d5,
uxy
=
4EL ffi 2
-~
I
o
5
(
-s&W3
ew(s&)-
&t)
w-4-sdy)l
+!j (I+ %%W) exp(&) + D(5) exp(-s&)1} sin@x) d[.
3. SOLUTION OF PROBLEM (A) With the help of boundary conditions (15) and (16) we get A(5) = f,(@)B(5)
EFM Vol. 19, No. 2-F
(21)
(22)
R. S. DHALIWAL
282 C(t)
((1+ sdfdth)
=
exp(s&)
+ (I-
et al.
S,SZ) expt-s,~h)l(2sJ’
exp(-s&)B,(Z),
D(Z) = - I(1 - s,st)f,(@r) exp(s,@)+ (1 + s,sJ exp(-s,5h)l(2sJ-,
exp(s&h)B,(&
where f,(@) is defined by eqn (26) below. With the help of boundary conditions (13) and (14) and using the results (23) we get the following triple integral equations: cc
I
0
B,(t) cost~x) dt = 0,
I
I G,tSh)B,tI)
costtx) dt = -
0
G,t-$W =
Ml + szht5h)- (1+ w&W)
0 < x < c.
I < x,
(24)
TESo 4/Lh(1 - “2)’ c < x < 1,
+ Cl- s,sJr,(Zh)l} wzM5Wl ’
12s,sA(Zh) - (1+ s,sz)fMh) + (1-
a,(P)= l+f,b$h),
WO=f,t5h)--1,
8d@)= fdfh) exp[b, - s2Ml - exp[ - (s, - s,)th] v,(#) = fdY) expIs,+ sMl-
w[ - (s, + sM1
The eqn (25) can be written in the following form:
p&rB,(t) sin(&) d5+ Ir5MdZh)BMcost5x)d5 = 4ah~~~y2), c
0
0
where
M,(th) = - [P + Wh)l, P = P(s,, s*) = -
4s,s* t (1 t s$ c 2s,(l - s,3 I ’
(27)
Following Lowengrub and Srivastava[Sl, the solution of triple integral eqns (24) and (25) can be written in the following form:
where h(tp is the solution of the following Fredholm integral equation of the second kind: h(x’) t
’ h(?)K(x2, t) dt = F(x’),
IC
c
(2%
satisfying the condition
I’ c
h(t? dt = 0,
(30)
Two finite length cracks
moving
slrips
between
283
and
K(x', t) = -
(31)
M(y, t) = $ \Oa~~(~~) cos(uy) sin(&) du,
(32)
Rx3= ($$*+
(33)
[(xz_
&_
,2)]“2?
C being an arbitrary constant to be determined by condition (30). The stress along the line of the cracks is given by the formula: h(l-
v’)
’h(r2)M(x,t) dt.
(34
The dynamic stress intensity factor for inner and outer edges of the crack, defined by the following relations Ki’ = lim {[2(~ - x)]“*u~~(x, 0)}, .x-V-
(35)
Ko’= !ty {12(x- UliEujytx, 0th
(36)
are given by:
h(l- v?Ko’ =&l[(l-$)(I-$$+$) E&J
-$(3+C,tC3+O(h-‘)]
(38)
where F = F(;,
vm),
E = E(;,
Y/l-c’),
(39)
are complete elliptic integrals of the first and second kinds respectively and the constants Co, C1 and C2 are given by: Co= ltc*-2E/F,. C, .--_l_c*, _ (I- c3* 4G C2= c*~(E~F)C,. 10and f, denote the integrals given by the equations
s=fj-LrM,(u)du=-lu[l+$$)]du, 0
rr3M,(u) du = -~~Ul[l+~]d~, which are convergent and can be evaluated numerically.
R. S. DHALIWAL
284
et al.
We can write that
(&)“*= (-!&)‘i’
ko2p
s1=
d
2
2
l;$
(42)
-7
s*= J
(43)
1-z.
If we take u + 0, we get the results of the paper of Lowengrub and Srivastava[6]. We also notice that the eqn (4.8) of 161contains a misprint. The term 21,/a4(3 + C, t C,)should be -21,/S4(3 + C, t C2). The variations of the stress intensity factor for inner and outer edges of the cracks are shown graphically for different values of crack speed u/c2 and h in Figs. l-4. Problem (B)
If a uniform stretching load of intensity p. is applied to the upper and lower edges y = +-II of the strip, then the equivalent problem for this case involves the application of traction -p. on the crack surface. The boundary conditions of the problem can be written in the following form: u)!(x,O)=-p,
c<]x]
uy(x,O)=O, O<(x(
(44)
JxJ>l,
(45)
uxxy(x,O)=O, O
(46)
and the boundary conditions along the strip edges are q(x, kh) = crxxy(x,-+h) = 0.
(47)
c=a3 h=20 0.628 fh=15
0.623-
0.622-
0.621
’
0.1
I
0.2
I
0.3
I
0.4
I
0.5
I
0.6
I
0.7
1
0.8
Fig. 1. Variations of stress intensity factor for inner edge of crack with crack velocity for c = 0.3, h = 10.15, 20, Y = a.
Two finite length cracks
moving
between
285
strips cr0.5 Y.f tl=20 /h=l5
/
.T $ g =:w ’
0.506 0.505 0.504
t 1
iiL_&__;,. 0.1
0.4
0.3
0.2
L C2
Fig. 2. Variations
of stress intensity
factor
for inner edge of crack with crack velocity IS, 20, ” =a.
,h=
for c = 0.5. h = 10.
20
0.613 r 0.612 0.611 -
0.607-
;iij 0. I
,
,
0.2
0.3
,
,
,
0.4
0.5
0.6
u
d.7
d,*
.
CP
Fig. 3. Variations
of stress intensity
factor
for outer edge of crack with crack IS, 20, Y = a.
velocity
for c = 0.3, h = 10,
Making use of the boundary conditions (46) and (47) we get A(g) = fA5)BW 4~20 + s;)&(5)
B(5) = I
[~s,s# + ~22)S,([)- 8s,s2p2(5) cosh(&A) + 2(1+ ~3 siWs2hMQI c(t) = [4s1sz’2~),1: 1’ Lq2)n(*)] D(5) = 4W2PAl) ‘&(
(1+
~2%2(Q
1 + S23
exp(-@2h)B2(t)
eXP~SS2h~B2~t~
(48)
R. S. DHALIWAL
286
0.4 I Fig. 4. Variations
et al.
0.5
0.6
0.7
0.8
+c
c,
of stress intensity factor for outer edge of crack with crack velocity for c = 0.5, h = IO, IS, 20, Y = a.
With the help of the bounda~ conditions (44) and (45)we find that
I I
J
BA5) cos((x) dS = 0,
0
0 < x < c, 1< x,
(49)
m
0
G*(~h)E*(~)cos(&) d5 = $‘,
c < x < 1,
(50)
Wt 1+ s2?*a&) + Wd3&) sinht&) - W + s,3’r2(6> cosh(s&~)l~ G2(*h)= {4s,s2(1+ s?)&(5) - 8sIs2P2([)cosh(s2f‘h)+ 2(1+ s2”>‘~~(~> sinh(s&)) a2(Sh)
M,,hj
=
=
W&l - coW&) {k&t
- COShfS&)
1+f2@3),62(5h)= f2(5h)- 1
exp(-s&)1- (I+ s?)*sinh(s&) expt-s&h)1 eXp(S&)] + (I+ S23* sinh(s*~h)exp(s,~h)~*
(51)
The problem is now reduced to the solution of triple integral eqn (49)and (50)which are similarto the triple integral eqfis (24)and (25) and may be solved in a similar manner. REFEXENCES [l] [2] [3] (41
G. C. Sib. and E. P. Chen, Moving cracks in a finite strip under tearing action. J. Franklin Inst., 2!MJ,25 (1970). 0. C. Sib. and E. P. Chen, Mechanics of Fracture (Edited by G. C. Sih.), Vol. 4, Chap. 2, pp, 89-93, J,,eyden, Noordhoff (1977). 0. C. Sih. and E. P. Chen, Cracks propagating in a strip of material under plane extension. Inf. .f. Engng Sci. I& 537 (1972). 8. M. Singh, T. 8. Moodie and J. B. Haddow, Closed-form solutions for finite length crack moving in a strip under anti-plane shear stress. Acta Mechanica 38,99 (1981). [S] R. J. Tait and T. B. Moodie, Complex variable methods and the closed-form solutions to dynamic crack and punch problems in the classical theory of elasticity. Jnf. 1. Engng Sci. 19,221 (1981). [6] hi. Lowengrub and K. N. Srivastava, Two coplanar Griffith cracks in an infinitely long elastic strip. fat. J. Ehgng Sci. 6,425 (1968). (Receiued 31 January 1983; received for publication 29 March
1983)
Engineming Fracture Mechanics Vol. 19, No. 2. pp. 279-286. 1984 Printed in Great Britain.
TWO FINITE LENGTH CRACKS MOVING BETWEEN STRIPS UNDER PLANE EXTENSION R. S. DHALIWAL, Department
of Mathematics,
B. M. SINGH University
and JON ROKNE
of Calgary, Calgary, Alberta, Canada
Abstract-The problem of the two coplanar uniformly propagating finite cracks in the strip of elastic material is solved. Two specific conditions of loading on the strip with finite width are discussed. In the first case the rigidly clamped edges are pulled apart in opposite directions. In the second case equal and opposite tractions are applied to edges of the strip. By the use of Fourier transforms we reduce the first problem to solving a set of triple integral equations with cosine kernel and a weight function. These equations are solved by using finite Hilbert transform techniques for large values of h where 2h is the width of the strip. Finally, numerical values for dynamic stress intensity factors are presented graphically. It is shown that the solution of the second problem can be obtained in a manner similar to the first case.
1. INTRODUCTION years the study of elastodynamic problems of moving or extending cracks in linear elastic media has received much attention in view of its importance in the field of fracture mechanics. The problem of a strip of material being split by a semi-infinite crack propagating at constant velocity has been investigated by Sih. and Chen[ 11. Sih. and Chen[2] have also studied the finite length crack propagating with constant velocity in a strip when the anti-plane shear displacements and stresses are applied to the lateral boundaries of the strip. Sih. and Chen[3] have also studied the related problem of a uniformly propagating line crack in a strip of elastic material under plane extension. Recently Singh, Moodie and Haddow[4] obtained closed form solution for a finite length crack moving in a strip under anti-plane shear stress. Tait and Moodie[5] solved the problem of two coplanar moving Griffith cracks in a strip under anti-plane shear stress using complex variable method. Lowengrub and Srivastava[6] also considered the problem of determining the stress field in the vicinity of two stationary coplanar Griffith cracks in an infinitely long elastic strip. In this paper we have considered the problem of two coplanar moving cracks in the strip under plane extension. Two specific conditions of loading with finite width of the strip are discussed. In the problem (A), the rigidly clamped edges are pulled apart in opposite directions. In the problem (B) equal and opposite tractions are applied to edges of the strip. For problem (A) expression for dynamic stress intensity factor is obtained for values of h + 1. IN RECENT
2. ELASTODYNAMIC EQUATIONS AND FORMULATION OF THE PROBLEM Consider a strip of the elastic material --m < X < m, -h < Y < h with cracks moving in the interior of the material on the line Y = 0, c < IX/< 1 with velocity o. For the sake of completeness we shall write some results, which are given by Sih. and Chen[3]. In terms of two scalar functions 4 and I,$each of which is a function of X, Y and t, we find that [u
u 9
9
W]’
a24 a*+
a!+aJ, ?!?L_alC, ax aY’ax r3Y' 0
[
1
a*fja24
-+2=72,2+Z=~af2, ax aY c, at
ax
a2+ aY
1 2
I
a2’
(2)
(3) where c, is the compression (irrotational) wave speed, c2 the shear (equivoluminal) wave speed in an infinitely extended medium, A and CLare Lam6 coefficients and p is the mass density of the elastic medium. 279
R.SDHALIWAL et al.
280
For the constant velocity of the cracks, it is convenient to introduce a Galilean transformation x=X-r&
y=Y,
V=t.
(4)
In the transformed system, the wave equations are independent of time variable t’, i.e. s 2q+q4 ’ ax
ay
s 2q+q4 2 ax
3
ay
9
(5)
where
(6) The displacement and stress components can also be transformed into the new coordinate system [U
u
XT
p’
!%+!!!!~_~o
$$)I” z [
ax
ay' ay
ax'
and
I ’
(7)
hP’$+2p(~+-g), uy=hv20+2~(gf+$), ux =
a,,=p
( 8%a29,q 2-axay
ax
ay ) *
(8)
The two conditions of loading considered involve the specification of displacements on the strip boundaries and tractions on the crack surfaces. Now we will discuss the two problems. Problem (A) Let the edges of the strip be clamped rigidly and displaced by an amount SOin the direction normal to the propagating cracks and hence l&(x, rh) = 0,
--M < x
u&x, 0) = 0, 0 < 1x(
(9) (10)
(11)
Because of symmetry, the following conditions on the stress components may be written down as uy(x,O)=O,
c
f&(x, 0) = 0, --co< x G @J.
(12)
In order to use the technique of integral transforms, it is necessary to solve an alternative but equivalent problem involving loading on the crack surfaces which can be obtained from the problem of a clamped strip (without cracks) subjected to a uniform strain. The dynamic stress intensity factors can be found by solving the equivalent problem -Et&J
a,(x,o)=h(l_V2)' uy(x,O)=O,
c
O<(x(
IxI>l,
frxxy(x,O)=O, O<(x(
(13) (14) (15)
TWO finite length cracks
moving
between
281
strips
and the displacement must satisfy r&.(x,+h) = u,(x, *h) = 0,
(16)
where v denotes Poisson’s ratio and E is the Young’s modulus. The solution to the original problem is then obtained by adding the solution of the equivalent problem and that of
ux=
vEao h(l-
3)7
UY=
Eao 4h’
(l-
ux, = 0. Solution of the eqn (5) can be written in the following form: d(x, Y)= ilrn [A(5) exp(s&) + B(5) ev(-s&)lco@W
d5
rL(x,Y)= i lo- [C(CZ)exp(s&y) + o(5) exp(-s&N sin(&)d5.
(17)
Where A(& B(t), etc. are the four unknowns in terms of the single variable 5; Substituting from (17) into eqns (7) and (8), it is found that displacements and stresses are given by: n, = f [
H-A(5) exp(s&)-
B(5) exp(-s&)
+ s~lC(5)
x exp(s&y) - D(T) exp(-dY)ll sin(txjr) 45
rx vy= $ o 5bM5) exph5y) - W exp(-s&)1 - W3 I x exp(s2&)- N5) d-&)I ~045~)45
(18)
(19)
~~=~~m52(-~(1-~~t2~12)[A(~)exp(s,5y)+8(51 x ew(-s&y)1 + sJC(t) eMs25y) - D(5) exd-s,5y)l} cW%) d5,
(20)
cr, = % ost2 it1 + s~M5) evbdy) + B(5) exp(-s15y)l. I ( -%[C(5) eM2W - D(t) evb2b)l} cos(W d5,
uxy
=
4EL ffi 2
-~
I
o
5
(
-s&W3
ew(s&)-
&t)
w-4-sdy)l
+!j (I+ %%W) exp(&) + D(5) exp(-s&)1} sin@x) d[.
3. SOLUTION OF PROBLEM (A) With the help of boundary conditions (15) and (16) we get A(5) = f,(@)B(5)
EFM Vol. 19, No. 2-F
(21)
(22)
R. S. DHALIWAL
282 C(t)
((1+ sdfdth)
=
exp(s&)
+ (I-
et al.
S,SZ) expt-s,~h)l(2sJ’
exp(-s&)B,(Z),
D(Z) = - I(1 - s,st)f,(@r) exp(s,@)+ (1 + s,sJ exp(-s,5h)l(2sJ-,
exp(s&h)B,(&
where f,(@) is defined by eqn (26) below. With the help of boundary conditions (13) and (14) and using the results (23) we get the following triple integral equations: cc
I
0
B,(t) cost~x) dt = 0,
I
I G,tSh)B,tI)
costtx) dt = -
0
G,t-$W =
Ml + szht5h)- (1+ w&W)
0 < x < c.
I < x,
(24)
TESo 4/Lh(1 - “2)’ c < x < 1,
+ Cl- s,sJr,(Zh)l} wzM5Wl ’
12s,sA(Zh) - (1+ s,sz)fMh) + (1-
a,(P)= l+f,b$h),
WO=f,t5h)--1,
8d@)= fdfh) exp[b, - s2Ml - exp[ - (s, - s,)th] v,(#) = fdY) expIs,+ sMl-
w[ - (s, + sM1
The eqn (25) can be written in the following form:
p&rB,(t) sin(&) d5+ Ir5MdZh)BMcost5x)d5 = 4ah~~~y2), c
0
0
where
M,(th) = - [P + Wh)l, P = P(s,, s*) = -
4s,s* t (1 t s$ c 2s,(l - s,3 I ’
(27)
Following Lowengrub and Srivastava[Sl, the solution of triple integral eqns (24) and (25) can be written in the following form:
where h(tp is the solution of the following Fredholm integral equation of the second kind: h(x’) t
’ h(?)K(x2, t) dt = F(x’),
IC
c
(2%
satisfying the condition
I’ c
h(t? dt = 0,
(30)
Two finite length cracks
moving
slrips
between
283
and
K(x', t) = -
(31)
M(y, t) = $ \Oa~~(~~) cos(uy) sin(&) du,
(32)
Rx3= ($$*+
(33)
[(xz_
&_
,2)]“2?
C being an arbitrary constant to be determined by condition (30). The stress along the line of the cracks is given by the formula: h(l-
v’)
’h(r2)M(x,t) dt.
(34
The dynamic stress intensity factor for inner and outer edges of the crack, defined by the following relations Ki’ = lim {[2(~ - x)]“*u~~(x, 0)}, .x-V-
(35)
Ko’= !ty {12(x- UliEujytx, 0th
(36)
are given by:
h(l- v?Ko’ =&l[(l-$)(I-$$+$) E&J
-$(3+C,tC3+O(h-‘)]
(38)
where F = F(;,
vm),
E = E(;,
Y/l-c’),
(39)
are complete elliptic integrals of the first and second kinds respectively and the constants Co, C1 and C2 are given by: Co= ltc*-2E/F,. C, .--_l_c*, _ (I- c3* 4G C2= c*~(E~F)C,. 10and f, denote the integrals given by the equations
s=fj-LrM,(u)du=-lu[l+$$)]du, 0
rr3M,(u) du = -~~Ul[l+~]d~, which are convergent and can be evaluated numerically.
R. S. DHALIWAL
284
et al.
We can write that
(&)“*= (-!&)‘i’
ko2p
s1=
d
2
2
l;$
(42)
-7
s*= J
(43)
1-z.
If we take u + 0, we get the results of the paper of Lowengrub and Srivastava[6]. We also notice that the eqn (4.8) of 161contains a misprint. The term 21,/a4(3 + C, t C,)should be -21,/S4(3 + C, t C2). The variations of the stress intensity factor for inner and outer edges of the cracks are shown graphically for different values of crack speed u/c2 and h in Figs. l-4. Problem (B)
If a uniform stretching load of intensity p. is applied to the upper and lower edges y = +-II of the strip, then the equivalent problem for this case involves the application of traction -p. on the crack surface. The boundary conditions of the problem can be written in the following form: u)!(x,O)=-p,
c<]x]
uy(x,O)=O, O<(x(
(44)
JxJ>l,
(45)
uxxy(x,O)=O, O
(46)
and the boundary conditions along the strip edges are q(x, kh) = crxxy(x,-+h) = 0.
(47)
c=a3 h=20 0.628 fh=15
0.623-
0.622-
0.621
’
0.1
I
0.2
I
0.3
I
0.4
I
0.5
I
0.6
I
0.7
1
0.8
Fig. 1. Variations of stress intensity factor for inner edge of crack with crack velocity for c = 0.3, h = 10.15, 20, Y = a.
Two finite length cracks
moving
between
285
strips cr0.5 Y.f tl=20 /h=l5
/
.T $ g =:w ’
0.506 0.505 0.504
t 1
iiL_&__;,. 0.1
0.4
0.3
0.2
L C2
Fig. 2. Variations
of stress intensity
factor
for inner edge of crack with crack velocity IS, 20, ” =a.
,h=
for c = 0.5. h = 10.
20
0.613 r 0.612 0.611 -
0.607-
;iij 0. I
,
,
0.2
0.3
,
,
,
0.4
0.5
0.6
u
d.7
d,*
.
CP
Fig. 3. Variations
of stress intensity
factor
for outer edge of crack with crack IS, 20, Y = a.
velocity
for c = 0.3, h = 10,
Making use of the boundary conditions (46) and (47) we get A(g) = fA5)BW 4~20 + s;)&(5)
B(5) = I
[~s,s# + ~22)S,([)- 8s,s2p2(5) cosh(&A) + 2(1+ ~3 siWs2hMQI c(t) = [4s1sz’2~),1: 1’ Lq2)n(*)] D(5) = 4W2PAl) ‘&(
(1+
~2%2(Q
1 + S23
exp(-@2h)B2(t)
eXP~SS2h~B2~t~
(48)
R. S. DHALIWAL
286
0.4 I Fig. 4. Variations
et al.
0.5
0.6
0.7
0.8
+c
c,
of stress intensity factor for outer edge of crack with crack velocity for c = 0.5, h = IO, IS, 20, Y = a.
With the help of the bounda~ conditions (44) and (45)we find that
I I
J
BA5) cos((x) dS = 0,
0
0 < x < c, 1< x,
(49)
m
0
G*(~h)E*(~)cos(&) d5 = $‘,
c < x < 1,
(50)
Wt 1+ s2?*a&) + Wd3&) sinht&) - W + s,3’r2(6> cosh(s&~)l~ G2(*h)= {4s,s2(1+ s?)&(5) - 8sIs2P2([)cosh(s2f‘h)+ 2(1+ s2”>‘~~(~> sinh(s&)) a2(Sh)
M,,hj
=
=
W&l - coW&) {k&t
- COShfS&)
1+f2@3),62(5h)= f2(5h)- 1
exp(-s&)1- (I+ s?)*sinh(s&) expt-s&h)1 eXp(S&)] + (I+ S23* sinh(s*~h)exp(s,~h)~*
(51)
The problem is now reduced to the solution of triple integral eqn (49)and (50)which are similarto the triple integral eqfis (24)and (25) and may be solved in a similar manner. REFEXENCES [l] [2] [3] (41
G. C. Sib. and E. P. Chen, Moving cracks in a finite strip under tearing action. J. Franklin Inst., 2!MJ,25 (1970). 0. C. Sib. and E. P. Chen, Mechanics of Fracture (Edited by G. C. Sih.), Vol. 4, Chap. 2, pp, 89-93, J,,eyden, Noordhoff (1977). 0. C. Sih. and E. P. Chen, Cracks propagating in a strip of material under plane extension. Inf. .f. Engng Sci. I& 537 (1972). 8. M. Singh, T. 8. Moodie and J. B. Haddow, Closed-form solutions for finite length crack moving in a strip under anti-plane shear stress. Acta Mechanica 38,99 (1981). [S] R. J. Tait and T. B. Moodie, Complex variable methods and the closed-form solutions to dynamic crack and punch problems in the classical theory of elasticity. Jnf. 1. Engng Sci. 19,221 (1981). [6] hi. Lowengrub and K. N. Srivastava, Two coplanar Griffith cracks in an infinitely long elastic strip. fat. J. Ehgng Sci. 6,425 (1968). (Receiued 31 January 1983; received for publication 29 March
1983)