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Kinked cracks in an anisotropic material
~i3*7~~1
at?i&mere Fraasre Medtmica Vol. 39, No. 5, pp. 927-930.1991 Printed in GreatBritain.
u.cio + 0.00
Pcrgamon Rem pk.
KINKED CRACKS ‘IN AN ANISOTROPIC MATERIAL c. R. CHIAN~ ment
of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 30043, Taiwan, R.O.C.
Abstract-A kinked semi-infinite,crack in an anisotropic mateeal is analyzd. The stress intensity factors of the kinked crack tip are expressed in terms of those associated with the main crack prior to kinking. The asymptotic stress tield around the crack tip is derived for the degenerate case in which the two complex parameters of the anisotropic material are equal. The general results for small kink angles are found to be. in agreament with tho$e derived directly by a perturbation analysis.
a semi-infinite crack loaded asymmetrically (i.e. both stress intensity factors K, and K,, are present). In general, the new (micro) crack would initiate at a certain angle to the main (macro) crack (Fig. 1). When the material is isotropic, several analyses and results have been reported[l-51. On the other hand, the corresponding anisotropic cases do not receive due attention. When this paper was being written two publi~tions appeared[6,7J. Because of their potential application to the study of fracture behavior in the composite materials, our primary objective is to determine the stress intensity factors at the branch crack tip in terms of those associated with the macro crack in an anisotropic material. The approximate solution method pioposed by Howard[8] is adopted here. The similar idea has also been used by Maiti[93in calc\ilating ~energy%&ase r$e of a kinked crack. The results are expected to be accurate for small kink angles. This is confirmed by comparing them to those obtained through a direct perturbation proocdure. As a byproduCt, we also derive the asymptotic stress field around a crack tip in an anisotropic material of which the complex parameters are identical. CONSIDER
2 FIELD EQUA~ON~ It is known that in an anisotropic material, the plane deformation and the antiplane deformation usually are unseparable; nevertheless, we shall assume that the symmetry of the material allows them to be decoupled. So the generalized Hooke’s law for the plane problem can be written as ~]=[ilzz]~~]
(1)
for plane stress. For plane strain problems, the coefficients aY should be replaced by &=a& - u~~~~~,~ (if = 1,2,6) and an additional stress component
should be included. Assume that no body forces are present, by in~~ucing Airy’s stress function F and using eq. (1), the compatibility condition b&mes
927
C.
928
R. CHIANG
Fig. 1 This equation Can be integrated ,by in~~ucing order[lO]
four linear differential operators of the first
where the operators are Dj=~
j = 1,2,3,4
-~j~
and p/ are the roots of the characteristic equation f a&p2 - 2az6fi+ a,, = 0. %*P4- 2a16p3+ (2~2,~ (5) It Can be shown that the roots must be complex and occur in conjugate pairs. It is also noted that if pl, p2 are known in particular coordinaiti (x, J) then for the Coordinates (x’, r’) rotated at an angle t3 with respect to (x, y), we have
fl; Z p,cos0 -sin0 COS8f@jSin0’ 3. ~Y~~~C
(6) THE CRACK TIP
STRESS FIELD ARO~
The asymptotic stress field around the Crack tip is given by[l1,12]
‘yy=&Re
i ~~~osBR+~~sinB)+~(~ssB+~~sinB)
(7) ‘xy~$$jRe I ~~~os~~~,inBj+~~~os~~~ninB) 1 where the origin of the polar coordinate (r, 0) is at the crack tip, and two coefficients A and B are reMed to the ~onv~tio~ stress intensity factors X, and X,, by A =i/42x1
+
41,
&
=
PI
4
+
4r (8)
c12 -h
PI
-P2
*
However, eq. (7) breaks down whenever ,u, = p2. In the following, we shall derive the result for ,u]= ju, which does not appear in any open literature. In the Case of equal Complex parameter, the Airy’s stress function has the following representation[ 101 F=2Re{G(z)+%Z(z)l
(9)
where z =x + py and d =x + fiy. Accordingly, the corresponding stress components have the fo~o~ng forms u xx
=
2 Re(CLAW+ W(~)l+ &#P(@)
uuu= 2 Re (V+(Z)+ Z&(Z) -I-2q(z)}
929
Kinked cracks in an anisotropic material
where JI (z) = G”(z), q(z) = H’(z), and primes denote the differentiation with respect to its functional argument. To derive the asymptotic stress field around the crack tip, we assume that and WI = NJ(271z)
cp(z) = B/,/(27rz)
(11)
where A and B are two complex constants to be determined from the boundary conditions. According to the definitions of the conventional stress intensity factors, we have, on 8 = 0 that Re {PA + (p/2 + p)B} = -Ku/2
Re {A + 3B/2} = K,/2,
(12)
and on 8 = II, the free surface requires that Im {PA + b/2 + fi)B} = 0.
Im (A + 3B/2} = 0,
(13)
It is concluded that for p = 6, + i &, we have
’ [(
‘Ose +iisine
2A-c0st?+psinf3
B)+2(1
+p)B]}
(14)
where A = K,/8 + i3(K,, + 6, K,)/8,
In particular,
B = K, /4 - i(K,, + 6, $)/46,.
for isotropic materials, with p = i the well known asymptotic field is recovered.
4. SOLUTION PROCEDURE From the asymptotic stress field, we can calculate the tractions cr,,y.and eiy, along the potential branch crack as
where o is the kink angle and the explicit forms of P and Q can be derived either from eq. (7) or from eq. (14) by suitable coordinate, transformations with 8 = w. Since it has been recognized that in an infinite domain for the collinear cracks, the tractions created on the line of the cracks are independent of the elastic constants provided that the loads acting on the crack surfaces have the resultant zero for each crack. Accordingly, we can estimate the stress intensity factors of the kinked crack tip as K:
=
JYK,,
K,,,
a)
=
Kf,
=
QUG, K,,,o) =
G,
(w)K,
C21(~)4
+
G,(~VG,
+
C22(~&
(16)
It is noted that (as expected) K: and Kf, are independent of the kinked (micro) crack length. They are dependent on the macro crack length contained implicitly in K, and K,,, however. When there exists a uniform stress T acting parallel to the macro crack, there will be an additional contribution from T which in fact is proportional to & i.e. the kinked micro length. There are some important implications for the stability of crack path due to the presence of T (see Cotterell and Rice[l] among others).
930
C. R. CHIANG
5. EXPRESSIONS FOR SMALL KINK ANGLES
If o is small, we expect that the result should coincide with the perturbation verify this assertion, note the following expansions 1
solution[l3]. To
x 1 -+Lo f(;+$*)o*--..*
J(cos 0 + p sin 0) cos 20 x 1 - 2w* + . . * sin20x20
-.+a.
(17)
It is concluded that for small w, we have c ,,= 1-~W2-_~2Re{~,~2} C 12 =
-3~
-im*Re{p,
+p2)
C21=0(l+fRe(~,~2})-~02Re(~,~2(~I+~2)} C,, = 1 + f w Re {p, + ~1~)- o*(i + i Re {,u?+ pI ,u2+ pi}).
(18)
It is noted that these expressions are valid even for p, = ,u2. In particular, when p, = p2 = i (the isotropic material), we have c,, = 1 -$o’ c,* = -;o c,, +lJ c**=
1 -fd.
(19)
These are found to be in agreement with previous results. 6. CONCLUDING REMARKS Due to the limitation of the present solution method, only the results of small kink angle are reliable. Nevertheless as far as the stability of crack path is concerned, the results of small kink angles are sufficient to serve the purpose. Unlike in the isotropic material, we have found that the maximum value of C,, does not necessarily occur when w = 0. Physically this is due to the fact that the maximum tangential stress does not occur directly ahead of the crack under the pure mode-1 loading. Therefore under pure mode-1 loading the crack may still kink in an anisotropic material despite the fact that pure tension perpendicular to the crack is applied. Of course, the strength anisotropy must also be taken into account in a real situation. REFERENCES [1] B. Cotterell and J. R. Rice, Slightly curved or kinked cracks. Inr. J. Fracture 16, 155-169 (1980). [2] B. L. Karihaloo, L. M. Keer, S. Nemat-Nasser and A. Oranratnachal, Approximate description of crack kinking and curving. J. appl. Mech. 48, 515-519 (1981). [3] M. Ichikawa and S. Tanaka, A critical analysis of the relationship between the energy release rate and the stress intensity factors for non-coplanar crack extension under combined mode loading. Int. J. Fracture 18, 19-28 (1982). [4] K. K. Lo, Analysis of branched cracks. J. appl. Mech. 45, 797-801 (1978). [S] P. S. Theocaris, Asymmetric branching of cracks. J. uppl. Mech. 44, 611618 (1977). [6] G. R. Miller and W. L. Stock, Analysis of branched interface cracks between dissimilar anisotropic media. J. appl. Mech. 56, 844849 (1989). [A M. Obata, S. Nemat-Nasser and Y. Goto, Branched cracks in anisotropic elastic solids. J. appt. Mech. 56,859864 (1989). [8] I. C. Howard, Simple approximate results for the stress intensity factors at the tip of a kinked crack. Inr. J. Fracture 14, R307-R310 (1978). [9] S. K. Maiti, An approximate method for calculating of strain energy release rate associated with kinking of a mode I crack loaded initially in an orthotropic direction. Inr. J. Fracture 32, R33-R36 (1986). [IO] S. G. Lekhnitskii, Anisotropic plates. Gorden & Breach, London (1968). [1I] G. C. Sih and H. Liebowitz, in Fracture (Edited by H. Liebowitz). Vol. 2. Academic Press, New York (1968). [12] S. Krenk, The stress distribution in an infinite anisotropic plate with colinear cracks. Znl. J. Soli& Strucf. l&449460 (1975). [13] C. R. Chiang, unpublished results (1988). (Received 6 August 1990)
at?i&mere Fraasre Medtmica Vol. 39, No. 5, pp. 927-930.1991 Printed in GreatBritain.
u.cio + 0.00
Pcrgamon Rem pk.
KINKED CRACKS ‘IN AN ANISOTROPIC MATERIAL c. R. CHIAN~ ment
of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 30043, Taiwan, R.O.C.
Abstract-A kinked semi-infinite,crack in an anisotropic mateeal is analyzd. The stress intensity factors of the kinked crack tip are expressed in terms of those associated with the main crack prior to kinking. The asymptotic stress tield around the crack tip is derived for the degenerate case in which the two complex parameters of the anisotropic material are equal. The general results for small kink angles are found to be. in agreament with tho$e derived directly by a perturbation analysis.
a semi-infinite crack loaded asymmetrically (i.e. both stress intensity factors K, and K,, are present). In general, the new (micro) crack would initiate at a certain angle to the main (macro) crack (Fig. 1). When the material is isotropic, several analyses and results have been reported[l-51. On the other hand, the corresponding anisotropic cases do not receive due attention. When this paper was being written two publi~tions appeared[6,7J. Because of their potential application to the study of fracture behavior in the composite materials, our primary objective is to determine the stress intensity factors at the branch crack tip in terms of those associated with the macro crack in an anisotropic material. The approximate solution method pioposed by Howard[8] is adopted here. The similar idea has also been used by Maiti[93in calc\ilating ~energy%&ase r$e of a kinked crack. The results are expected to be accurate for small kink angles. This is confirmed by comparing them to those obtained through a direct perturbation proocdure. As a byproduCt, we also derive the asymptotic stress field around a crack tip in an anisotropic material of which the complex parameters are identical. CONSIDER
2 FIELD EQUA~ON~ It is known that in an anisotropic material, the plane deformation and the antiplane deformation usually are unseparable; nevertheless, we shall assume that the symmetry of the material allows them to be decoupled. So the generalized Hooke’s law for the plane problem can be written as ~]=[ilzz]~~]
(1)
for plane stress. For plane strain problems, the coefficients aY should be replaced by &=a& - u~~~~~,~ (if = 1,2,6) and an additional stress component
should be included. Assume that no body forces are present, by in~~ucing Airy’s stress function F and using eq. (1), the compatibility condition b&mes
927
C.
928
R. CHIANG
Fig. 1 This equation Can be integrated ,by in~~ucing order[lO]
four linear differential operators of the first
where the operators are Dj=~
j = 1,2,3,4
-~j~
and p/ are the roots of the characteristic equation f a&p2 - 2az6fi+ a,, = 0. %*P4- 2a16p3+ (2~2,~ (5) It Can be shown that the roots must be complex and occur in conjugate pairs. It is also noted that if pl, p2 are known in particular coordinaiti (x, J) then for the Coordinates (x’, r’) rotated at an angle t3 with respect to (x, y), we have
fl; Z p,cos0 -sin0 COS8f@jSin0’ 3. ~Y~~~C
(6) THE CRACK TIP
STRESS FIELD ARO~
The asymptotic stress field around the Crack tip is given by[l1,12]
‘yy=&Re
i ~~~osBR+~~sinB)+~(~ssB+~~sinB)
(7) ‘xy~$$jRe I ~~~os~~~,inBj+~~~os~~~ninB) 1 where the origin of the polar coordinate (r, 0) is at the crack tip, and two coefficients A and B are reMed to the ~onv~tio~ stress intensity factors X, and X,, by A =i/42x1
+
41,
&
=
PI
4
+
4r (8)
c12 -h
PI
-P2
*
However, eq. (7) breaks down whenever ,u, = p2. In the following, we shall derive the result for ,u]= ju, which does not appear in any open literature. In the Case of equal Complex parameter, the Airy’s stress function has the following representation[ 101 F=2Re{G(z)+%Z(z)l
(9)
where z =x + py and d =x + fiy. Accordingly, the corresponding stress components have the fo~o~ng forms u xx
=
2 Re(CLAW+ W(~)l+ &#P(@)
uuu= 2 Re (V+(Z)+ Z&(Z) -I-2q(z)}
929
Kinked cracks in an anisotropic material
where JI (z) = G”(z), q(z) = H’(z), and primes denote the differentiation with respect to its functional argument. To derive the asymptotic stress field around the crack tip, we assume that and WI = NJ(271z)
cp(z) = B/,/(27rz)
(11)
where A and B are two complex constants to be determined from the boundary conditions. According to the definitions of the conventional stress intensity factors, we have, on 8 = 0 that Re {PA + (p/2 + p)B} = -Ku/2
Re {A + 3B/2} = K,/2,
(12)
and on 8 = II, the free surface requires that Im {PA + b/2 + fi)B} = 0.
Im (A + 3B/2} = 0,
(13)
It is concluded that for p = 6, + i &, we have
’ [(
‘Ose +iisine
2A-c0st?+psinf3
B)+2(1
+p)B]}
(14)
where A = K,/8 + i3(K,, + 6, K,)/8,
In particular,
B = K, /4 - i(K,, + 6, $)/46,.
for isotropic materials, with p = i the well known asymptotic field is recovered.
4. SOLUTION PROCEDURE From the asymptotic stress field, we can calculate the tractions cr,,y.and eiy, along the potential branch crack as
where o is the kink angle and the explicit forms of P and Q can be derived either from eq. (7) or from eq. (14) by suitable coordinate, transformations with 8 = w. Since it has been recognized that in an infinite domain for the collinear cracks, the tractions created on the line of the cracks are independent of the elastic constants provided that the loads acting on the crack surfaces have the resultant zero for each crack. Accordingly, we can estimate the stress intensity factors of the kinked crack tip as K:
=
JYK,,
K,,,
a)
=
Kf,
=
QUG, K,,,o) =
G,
(w)K,
C21(~)4
+
G,(~VG,
+
C22(~&
(16)
It is noted that (as expected) K: and Kf, are independent of the kinked (micro) crack length. They are dependent on the macro crack length contained implicitly in K, and K,,, however. When there exists a uniform stress T acting parallel to the macro crack, there will be an additional contribution from T which in fact is proportional to & i.e. the kinked micro length. There are some important implications for the stability of crack path due to the presence of T (see Cotterell and Rice[l] among others).
930
C. R. CHIANG
5. EXPRESSIONS FOR SMALL KINK ANGLES
If o is small, we expect that the result should coincide with the perturbation verify this assertion, note the following expansions 1
solution[l3]. To
x 1 -+Lo f(;+$*)o*--..*
J(cos 0 + p sin 0) cos 20 x 1 - 2w* + . . * sin20x20
-.+a.
(17)
It is concluded that for small w, we have c ,,= 1-~W2-_~2Re{~,~2} C 12 =
-3~
-im*Re{p,
+p2)
C21=0(l+fRe(~,~2})-~02Re(~,~2(~I+~2)} C,, = 1 + f w Re {p, + ~1~)- o*(i + i Re {,u?+ pI ,u2+ pi}).
(18)
It is noted that these expressions are valid even for p, = ,u2. In particular, when p, = p2 = i (the isotropic material), we have c,, = 1 -$o’ c,* = -;o c,, +lJ c**=
1 -fd.
(19)
These are found to be in agreement with previous results. 6. CONCLUDING REMARKS Due to the limitation of the present solution method, only the results of small kink angle are reliable. Nevertheless as far as the stability of crack path is concerned, the results of small kink angles are sufficient to serve the purpose. Unlike in the isotropic material, we have found that the maximum value of C,, does not necessarily occur when w = 0. Physically this is due to the fact that the maximum tangential stress does not occur directly ahead of the crack under the pure mode-1 loading. Therefore under pure mode-1 loading the crack may still kink in an anisotropic material despite the fact that pure tension perpendicular to the crack is applied. Of course, the strength anisotropy must also be taken into account in a real situation. REFERENCES [1] B. Cotterell and J. R. Rice, Slightly curved or kinked cracks. Inr. J. Fracture 16, 155-169 (1980). [2] B. L. Karihaloo, L. M. Keer, S. Nemat-Nasser and A. Oranratnachal, Approximate description of crack kinking and curving. J. appl. Mech. 48, 515-519 (1981). [3] M. Ichikawa and S. Tanaka, A critical analysis of the relationship between the energy release rate and the stress intensity factors for non-coplanar crack extension under combined mode loading. Int. J. Fracture 18, 19-28 (1982). [4] K. K. Lo, Analysis of branched cracks. J. appl. Mech. 45, 797-801 (1978). [S] P. S. Theocaris, Asymmetric branching of cracks. J. uppl. Mech. 44, 611618 (1977). [6] G. R. Miller and W. L. Stock, Analysis of branched interface cracks between dissimilar anisotropic media. J. appl. Mech. 56, 844849 (1989). [A M. Obata, S. Nemat-Nasser and Y. Goto, Branched cracks in anisotropic elastic solids. J. appt. Mech. 56,859864 (1989). [8] I. C. Howard, Simple approximate results for the stress intensity factors at the tip of a kinked crack. Inr. J. Fracture 14, R307-R310 (1978). [9] S. K. Maiti, An approximate method for calculating of strain energy release rate associated with kinking of a mode I crack loaded initially in an orthotropic direction. Inr. J. Fracture 32, R33-R36 (1986). [IO] S. G. Lekhnitskii, Anisotropic plates. Gorden & Breach, London (1968). [1I] G. C. Sih and H. Liebowitz, in Fracture (Edited by H. Liebowitz). Vol. 2. Academic Press, New York (1968). [12] S. Krenk, The stress distribution in an infinite anisotropic plate with colinear cracks. Znl. J. Soli& Strucf. l&449460 (1975). [13] C. R. Chiang, unpublished results (1988). (Received 6 August 1990)