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Steady motion of a crack parallel to a bond plane—further results
hr. 1. Engng Sci. Vol. 19. pp, 805409. Printed in Great Britain.
0020-722S/8l/O6M545SO2.~/0 Pergaman Press Ltd.
Ml
STEADY MOTION OF A CRACK PARALLEL TO A BOND PLANE-FURTHER RESULTS S. H. CHEN Deere & Co., 3300River Drive, Moline, IL 61265,U.S.A. and L. M. KEER and J. D. A~HENBACH Department of Civil Engineering, Northwestern University, Evanston, IL 60201,U.S.A. (Communicated by I. N. SNEDDON)
Abstract-Thepresent analysis concerns the steady propagation of a crack of length 24 parallel to a bond plane between two half-planes having diierent material properties. The crack speed is less than the transverse wave speed of the half-plane in which it is located, but the material properties of the untracked material are such that the shear or dilatational wave speeds may be exceeded. A transition speed is seen to exist above which the untracked medium tends to act as a stiffened material.
INTRODUCTION
IN A previous paper[l], the near-tip stress fields were investigated for the case that a line crack
of length 2a moves at a constant speed c parallel to a bond-plane formed by two half planes occupied by dissimilar linearly elastic isotropic materials. The near tip stress fields were found to depend upon material combinations of the elastic constants, the distance between the crack and the bond-plane and the speed at which the crack moves. In [I], results were given only for the case when the crack moves at a speed less than the smaller transverse wave speed of the two materials, i.e. for the subsonic-subsonic case. In the present analysis, the cases of subsonic-transonic and subsonic-supersonic are studied for the same geometry (see Fig. 1). The crack speed c in material 1 is smaller than the transverse wave speed of material 1, and it is either between the transverse wave speed and the longitudinal wave speed (subsonic-transonic) or larger than the longitudinal wave speed of the untracked material 2 (subsonic-supersonic). FORMULATION
The geometry and coordinate system for the considered problem are shown in Fig. 1. The governing equations can be taken with minor modification from those in [l]. Two displacement potentials 4, and $ are introduced for material 2; for the subsonic-transonic case, they are governed by
(Ia, W
while for the subsonic-supersonic case, the governing equations are
ai4,xx- 4,YY = 0, Pk%- *,,yy =0 ,& =
(c*/c~- 1)“2, @T= (c’,‘+
1)“2,
tk b) (4%b)
where cL is the longitudinal wave speed, [(Af 2~)/pJ”‘, and cT is the transverse wave speed, and a comma refers to differentiation with respect to the indicated variable (i.e. WPSR, .x = ala,). To formulate either problem, the Fourier transform technique and the method of superposition can be employed as in 111 to reduce the governing equations and the boundary 805
806
S. H. CHEN et al.
h
k-20
--I
J_.-
x
-c
I
Fig. 1. Geometry and coordinate system.
conditions to a system of singular integral equations of the form
(6) where the constants kl and kt are defined as kl = - NP,,
~2)/27+3,(1-
W,
k2 = -
132VW32(1
~22)
Ua,b)
-a:,
while
f(x)=~[u,(y=Ol)-u,(y=o-)I, g(x) =
lX(
$ [U,(Y =0’) - u,(y = o-)1,
@a, b)
Ix/< a,
are dislocation densities. In eqns (7a, b) R&T
P2)
=
(I+
p:>’
-
(9)
‘@a2r
PI = (1 - c~/c~,)“~ and p2 = (1 - c2/4.,)1/2,
(loa, b)
where the subscripts 1 to cL and cT refer to the cracked material (see Fig. 1). The kernels kjk(X, t), j,k = 1,2, which are regular functions in x and t, are defined in the Appendix. The numerical method used here is the same as was employed in [l] to solve for the functions f(x) and g(x). RESULTS
AND DISCUSSION
The quantity K(8) is plotted vs 6 in Figs. 2 and 3. The near-tip circumferential stress field is related to K(B) by uee = u0V(a)K(O)/dC?r)
= (2rP[Kf11eete,
4 + K,dde,
41,
(11)
807
Steady motion of a crack parallel to a bond plane-further results
2.100 K
S-C,0 5,
0.600 I
-0.500
-
3; i
I .a00
-I80
-60
-120
0
‘b--J4 60
120
4%
0.2
--L
0.3
i
0.4
-+-
0.5
+
0.6
-e
,
0.8 i
180
8, deg Fig. 2. I@) vs 6 for I, = vz= 0.3, pl = pz, &p, = 0.5, and h/a = 0.4.
^.
I.700
-____-/ :
P i
1.300 0.900
f-\ !’
_...._
___ .___
._
5t /+?\ L
i
i!
:
1 *C “-
_-L
I__..
-180
-120
-60
0
60
120
.._-
-=o
51
0.2 0.3 0.4 0.5
* -
0.6
A
0.8
-J
I80
8, dw Fig. 3. K(e) vs 0 for Y,= v2= 0.3, pl = pz, g2/pr = 0.5, and /~/a= 5.
where K, and Krr are stress intensity factors in modes I and II, respectively, and & and r$ define near-tip stress fields arising from the contribution of oe to the symmetric (I) and anti-symmetric (II) cases, respectively. Figures 2 and 3 show results for the cases that Vi= y = 0.3, pI = p2, ,uJqr = 0.05, clcr, varying from 0 to 0.8, and h/a = 0.4 and 5 in Figs. 2 and 3, respectively. For this material combination, we have the subsonic-subsonic case for c < 0.223cr1, subsonic-transonic case for 0.223~~~< c < 0.418& and subsonic-supersonic case for c > 0.418cri. It is found that, for ~~ = 5, the cleavage angle increases as the crack speed increases. I-Iowever, for h/a = 0.4, it is found that the magnitude of the stress intensity factor & decreases rapidly when the speed of the moving crack exceeds a certain (transition) speed, which is in the subsonic-transonic range. The stress intensity factors K1 and KII are listed in Table 1 for various crack speeds in the subsonic-subsonic and subsonic-transonic range. For h/a = 0.4, the transition speed seems to be about 0.35~T1. For a smaller value of h/u, the transition speed will be fess. For large values of h/a, the transition speed may not be
S. H. CHEN et al. Table 1. Stress intensity factors for &CL, =
0
0.1 0.21 0.23 0.3 0.33 0.37 0.38 0.39 0.4 0.41
2.13 2.20 2.62 2.81 3.35 3.79 3Sl 2.66 1.59 0.772 0.428
0.569 0.617 0.915 1.09 1.81 2.56 3.% 3.69 2.85 1.78 0.907
observable, since for these cases the effect of the upper half plane will be small. When the crack speed is at or exceeds the transition speed, the upper medium tends to behave physically like a stiffened material.
Acknowledgement-This work was carried out in the course of research sponsored by the Air Force Office of Scientific Research (Grant AFOSR 78-3589). REFERENCE [l] S. H. CHEN, L. M. KEER and J. D. ACHENBACH, Inf. J. Engng Sci. 18,225 (1980).
(Received 5 June 1980)
APPENDIX The kernels in eqns (5) and (6) are defined as, Subsonic-supersonic
case
where 8, = 28,, 82 = PI f B2, j, = 282,
(A-2)
and
(A3a-c)
r121+ $21 = 2BliC11, r122f $22 = 2B,iC2, + (1 f BW12,
(A4a-c)
r123+ is123 = (1 + p!)C22, rzll + is2,, = (1 + SW3b r212+ is212 = (1 +/.G)C4~ -2B2iC32.
(A5a-c)
r213+ is213 = - 2B21C42, r221+ ih2, = (1 +BKh r222+ is222 = (1 f BW2, r223f is223 = - 2B2iC22,
where i = d/( - 1).
- 2B2iG2,
(A6a-c)
Steady motion of a crack parallel to a bond plane-further results
809
In eqns (A3-A6)
6% b) where p = 1,. . ,4, and u t iw = det (A). A is a 4 x 4 matrix defined by
A=
-28,;
-(l +B:)
-(1+/B
2P2i
2r?3c”’ Fl
(1-E); -284t42 _ Pi P4 1 I
(1-E); --1 B3
In eqn (A7) .$ = (u - iw).xik, j, k = 1,.
,4,
(A9)
where x, is the determinant of a 3 x 3 matrix obtained by deleting jth row and kth column of matrix A. The matrix F, which elements are A, jk = 1,. ,4, is defined by
-(1+/G)
F--l-
24’-s:,
(1 +/a* -281
‘+s: 28 2
28, -(l+B:)
1
(AI’3
-- l+P: 2 -- l+P:
1
-82
2P2 1+LG
l+s:
281 1+/-G -- 2
(1+B:)*
PI
282
_
In eqn (A8), j3 and j, are defined as ps = (c*/c;*- I)“*, Fd = (?I&
- 1)“2.
(Alla, b)
Subsonic-transonic case The equations for subsonic-transonic case are the same as those for subsonic-supersonic case except that the-elements Ai3,j = 1,. . ,4 in eqn (A8) are replaced by - 2i$i3kL2/p,, (1 - /&/~i, - I and - ij3, respectively. In this case, p3 has the definition of /J3= (1 - C*/C;*)“*.
UES Vol. 19. No. 6-E
(A12)
0020-722S/8l/O6M545SO2.~/0 Pergaman Press Ltd.
Ml
STEADY MOTION OF A CRACK PARALLEL TO A BOND PLANE-FURTHER RESULTS S. H. CHEN Deere & Co., 3300River Drive, Moline, IL 61265,U.S.A. and L. M. KEER and J. D. A~HENBACH Department of Civil Engineering, Northwestern University, Evanston, IL 60201,U.S.A. (Communicated by I. N. SNEDDON)
Abstract-Thepresent analysis concerns the steady propagation of a crack of length 24 parallel to a bond plane between two half-planes having diierent material properties. The crack speed is less than the transverse wave speed of the half-plane in which it is located, but the material properties of the untracked material are such that the shear or dilatational wave speeds may be exceeded. A transition speed is seen to exist above which the untracked medium tends to act as a stiffened material.
INTRODUCTION
IN A previous paper[l], the near-tip stress fields were investigated for the case that a line crack
of length 2a moves at a constant speed c parallel to a bond-plane formed by two half planes occupied by dissimilar linearly elastic isotropic materials. The near tip stress fields were found to depend upon material combinations of the elastic constants, the distance between the crack and the bond-plane and the speed at which the crack moves. In [I], results were given only for the case when the crack moves at a speed less than the smaller transverse wave speed of the two materials, i.e. for the subsonic-subsonic case. In the present analysis, the cases of subsonic-transonic and subsonic-supersonic are studied for the same geometry (see Fig. 1). The crack speed c in material 1 is smaller than the transverse wave speed of material 1, and it is either between the transverse wave speed and the longitudinal wave speed (subsonic-transonic) or larger than the longitudinal wave speed of the untracked material 2 (subsonic-supersonic). FORMULATION
The geometry and coordinate system for the considered problem are shown in Fig. 1. The governing equations can be taken with minor modification from those in [l]. Two displacement potentials 4, and $ are introduced for material 2; for the subsonic-transonic case, they are governed by
(Ia, W
while for the subsonic-supersonic case, the governing equations are
ai4,xx- 4,YY = 0, Pk%- *,,yy =0 ,& =
(c*/c~- 1)“2, @T= (c’,‘+
1)“2,
tk b) (4%b)
where cL is the longitudinal wave speed, [(Af 2~)/pJ”‘, and cT is the transverse wave speed, and a comma refers to differentiation with respect to the indicated variable (i.e. WPSR, .x = ala,). To formulate either problem, the Fourier transform technique and the method of superposition can be employed as in 111 to reduce the governing equations and the boundary 805
806
S. H. CHEN et al.
h
k-20
--I
J_.-
x
-c
I
Fig. 1. Geometry and coordinate system.
conditions to a system of singular integral equations of the form
(6) where the constants kl and kt are defined as kl = - NP,,
~2)/27+3,(1-
W,
k2 = -
132VW32(1
~22)
Ua,b)
-a:,
while
f(x)=~[u,(y=Ol)-u,(y=o-)I, g(x) =
lX(
$ [U,(Y =0’) - u,(y = o-)1,
@a, b)
Ix/< a,
are dislocation densities. In eqns (7a, b) R&T
P2)
=
(I+
p:>’
-
(9)
‘@a2r
PI = (1 - c~/c~,)“~ and p2 = (1 - c2/4.,)1/2,
(loa, b)
where the subscripts 1 to cL and cT refer to the cracked material (see Fig. 1). The kernels kjk(X, t), j,k = 1,2, which are regular functions in x and t, are defined in the Appendix. The numerical method used here is the same as was employed in [l] to solve for the functions f(x) and g(x). RESULTS
AND DISCUSSION
The quantity K(8) is plotted vs 6 in Figs. 2 and 3. The near-tip circumferential stress field is related to K(B) by uee = u0V(a)K(O)/dC?r)
= (2rP[Kf11eete,
4 + K,dde,
41,
(11)
807
Steady motion of a crack parallel to a bond plane-further results
2.100 K
S-C,0 5,
0.600 I
-0.500
-
3; i
I .a00
-I80
-60
-120
0
‘b--J4 60
120
4%
0.2
--L
0.3
i
0.4
-+-
0.5
+
0.6
-e
,
0.8 i
180
8, deg Fig. 2. I@) vs 6 for I, = vz= 0.3, pl = pz, &p, = 0.5, and h/a = 0.4.
^.
I.700
-____-/ :
P i
1.300 0.900
f-\ !’
_...._
___ .___
._
5t /+?\ L
i
i!
:
1 *C “-
_-L
I__..
-180
-120
-60
0
60
120
.._-
-=o
51
0.2 0.3 0.4 0.5
* -
0.6
A
0.8
-J
I80
8, dw Fig. 3. K(e) vs 0 for Y,= v2= 0.3, pl = pz, g2/pr = 0.5, and /~/a= 5.
where K, and Krr are stress intensity factors in modes I and II, respectively, and & and r$ define near-tip stress fields arising from the contribution of oe to the symmetric (I) and anti-symmetric (II) cases, respectively. Figures 2 and 3 show results for the cases that Vi= y = 0.3, pI = p2, ,uJqr = 0.05, clcr, varying from 0 to 0.8, and h/a = 0.4 and 5 in Figs. 2 and 3, respectively. For this material combination, we have the subsonic-subsonic case for c < 0.223cr1, subsonic-transonic case for 0.223~~~< c < 0.418& and subsonic-supersonic case for c > 0.418cri. It is found that, for ~~ = 5, the cleavage angle increases as the crack speed increases. I-Iowever, for h/a = 0.4, it is found that the magnitude of the stress intensity factor & decreases rapidly when the speed of the moving crack exceeds a certain (transition) speed, which is in the subsonic-transonic range. The stress intensity factors K1 and KII are listed in Table 1 for various crack speeds in the subsonic-subsonic and subsonic-transonic range. For h/a = 0.4, the transition speed seems to be about 0.35~T1. For a smaller value of h/u, the transition speed will be fess. For large values of h/a, the transition speed may not be
S. H. CHEN et al. Table 1. Stress intensity factors for &CL, =
0
0.1 0.21 0.23 0.3 0.33 0.37 0.38 0.39 0.4 0.41
2.13 2.20 2.62 2.81 3.35 3.79 3Sl 2.66 1.59 0.772 0.428
0.569 0.617 0.915 1.09 1.81 2.56 3.% 3.69 2.85 1.78 0.907
observable, since for these cases the effect of the upper half plane will be small. When the crack speed is at or exceeds the transition speed, the upper medium tends to behave physically like a stiffened material.
Acknowledgement-This work was carried out in the course of research sponsored by the Air Force Office of Scientific Research (Grant AFOSR 78-3589). REFERENCE [l] S. H. CHEN, L. M. KEER and J. D. ACHENBACH, Inf. J. Engng Sci. 18,225 (1980).
(Received 5 June 1980)
APPENDIX The kernels in eqns (5) and (6) are defined as, Subsonic-supersonic
case
where 8, = 28,, 82 = PI f B2, j, = 282,
(A-2)
and
(A3a-c)
r121+ $21 = 2BliC11, r122f $22 = 2B,iC2, + (1 f BW12,
(A4a-c)
r123+ is123 = (1 + p!)C22, rzll + is2,, = (1 + SW3b r212+ is212 = (1 +/.G)C4~ -2B2iC32.
(A5a-c)
r213+ is213 = - 2B21C42, r221+ ih2, = (1 +BKh r222+ is222 = (1 f BW2, r223f is223 = - 2B2iC22,
where i = d/( - 1).
- 2B2iG2,
(A6a-c)
Steady motion of a crack parallel to a bond plane-further results
809
In eqns (A3-A6)
6% b) where p = 1,. . ,4, and u t iw = det (A). A is a 4 x 4 matrix defined by
A=
-28,;
-(l +B:)
-(1+/B
2P2i
2r?3c”’ Fl
(1-E); -284t42 _ Pi P4 1 I
(1-E); --1 B3
In eqn (A7) .$ = (u - iw).xik, j, k = 1,.
,4,
(A9)
where x, is the determinant of a 3 x 3 matrix obtained by deleting jth row and kth column of matrix A. The matrix F, which elements are A, jk = 1,. ,4, is defined by
-(1+/G)
F--l-
24’-s:,
(1 +/a* -281
‘+s: 28 2
28, -(l+B:)
1
(AI’3
-- l+P: 2 -- l+P:
1
-82
2P2 1+LG
l+s:
281 1+/-G -- 2
(1+B:)*
PI
282
_
In eqn (A8), j3 and j, are defined as ps = (c*/c;*- I)“*, Fd = (?I&
- 1)“2.
(Alla, b)
Subsonic-transonic case The equations for subsonic-transonic case are the same as those for subsonic-supersonic case except that the-elements Ai3,j = 1,. . ,4 in eqn (A8) are replaced by - 2i$i3kL2/p,, (1 - /&/~i, - I and - ij3, respectively. In this case, p3 has the definition of /J3= (1 - C*/C;*)“*.
UES Vol. 19. No. 6-E
(A12)