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The early stage of fatigue crack propagation
EngineeringFracture ~ee~a~tes Vol. 42, No. 5, pp. 789-796, 1992
Printed in Great Britain.
THE EARLY STAGE OF FATIGUE PROPAGATION
Gill3-7944192 $5.00+ 0.00 Q 1992Pergamon Press Ltd.
CRACK
H. FAN and L. M. KEER Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, U.S.A. Abstract-It is observed that an early stage fatigue crack propagates along the maximum shear direction rather than ~r~ndi~~ar to the maximum tensile stress direction. In the present paper, an analytical consideration is given concerning the effects of localized plastic deformation on the early stage of fatigue crack propagation. it is seen that the localized plastic defo~ation shields the local mode II stress intensity factor and leads the micro-crack to propagate along the maximum shear stress direction. In this analytical modeling, the localized plastic zone is modeled as an elliptical inclusion with eigenstrains where the Eshelby inclusion solution [Pm-. R. Sot. A241, 376-396 (1957)] is employed.
NOMENCLATURE semi-axes of plastic zone elastic moduli yielding stress in shear Poisson’s ratio elastic shear modulus plastic shear modulus plastic strain eigenstrain domain of the PSB which is (.x,/u,)~ + (x2/a,)’ = 1 boundary of R (x,/c,)‘+ (x2/c2)* = 1, with q-+0 boundary of R
1. INTRODUCTION FATIGUE CRACK initiation and early stage micro-crack propagation have attracted many researchers in both experimental and theoretical fields of research. It has been observed experimentally (e.g. refs [l] and [2]) that there are some regions where plastic strain is highly concentrated while the neighborhood of these subdomains is still in the range of elasticity. These subdomains with localized high plastic strain are called persistent slip bands (PSB), which play an important role in the early stage of fatigue damage (crack initiation and early propagation). Some analytical efforts have been made to model the PSB (e.g. refs [3] and [4]) by using ellipsoid inclusions where the Eshelby solution [S] can be applied. Lin and Chen [6] treated the plastic zone as slipping layers in which micro-cracking is believed to be initiated. Following the crack initiation, there is the early stage of micro-crack propagation. A typical fatigue crack path is shown schematically in Fig. 1, where the crack propagates along the maximum shear direction, x,-axis, in the early stage, and gradually tends to the direction perpendicular to the maximum tensile stress, the X-axis. In the present paper, an analytical model for the above observation is given by using an eigenstrain formulation and calculation of local stress intensity factor. Based on the experimental observations, the as-initiated fatigue crack is embedded in a PSB where plastic deformation is highly concentrated. For the sake of simplicity, the PSB is modeled as an elliptic inclusion in an infinite elastic matrix (Fig. 2). The micro-crack surrounded by the plastic zone is also modeled as a smaller elliptical inclusion within the PSB. The eigenstrain in R, (crack) is determined such that the traction-free condition on the crack surfaces is satisfied. The stress intensity factor is then calculated by considering the stress jump across X&, which is different from conventional calculations in fracture mechanics. 789
790
H. FAN
and L. M. KEER
-X
Fig. I. A schematic
2. MODELING
of the fatigue
OF A MICRO-CRACK
crack
path.
EMBEDDED
IN PSB
Experimental observations show that plastic deformation under fatigue is localized in some slip bands (PSB), which in the following is modeled as a narrow elliptically shaped inclusion. Micro-cracks are believed to be initiated within the bands. In this band (inclusion), Hooke’s law is written as: cJij= /Gij(Qk - &) + 2&;, - ES). Since slip lines form in a favored direction, only shear deformation along these slip lines is plastic. By using the coordinates shown in Fig. 2, a pure shear case is considered hereafter (take T = @,z, L = t,2): z* = 2&c* - ck)
(1)
where E,, is the total shear strain and ci is the plastic part. From a hysteresis loop (Fig. 3) one can write that: TA
-
2&
TY
(2)
where b is the plastic modulus. In order to simplify the analysis, a stable loop with bi-linear shape is used in the present formulation. Cyclic hardening and softening are not considered. Although the power relation between D and c may be closer to reality, a qualitatively similar result to the present analysis is expected. Substituting eq. (2) into eq. (1)
By using the eigenstrain concept [7],
The early stage of fatigue
crack
791
propagation
i
x2
a2 -4
L
X1
8
?
23
J
L
1 Fig. 2. Model of a micro-crack
embedded
in PSB.
where t;” is the loading at the far field and TX is caused by the plastic strain. For an isotropic material, it is known from the Eshelby solution [5] (see Appendix) that: (5)
A
* Q2
B
Fig. 3. Hysteresis
loop of fatigue
792
H. FAN and L. M. KEER
where a, and a, are the geometrical parameters of this elliptical plastic zone. Equations (3), (4) and (5) lead to:
(6) The eigenstrain cpAis then solved from this equation as:
c; =
T;”
-5y
2P
2P ala2 (1 -vk+a2)2-
(7)
.
1 -P/Pp
Thus, 1 T,=TF----
T;c-Ty
ala2
(1 -v)(a,
+a,)’
1
ala2
1
(8)
(1 -v)(al+a2>2-~
and as another limit, ,u~= 0, zA = T,, . When pu,=p, we have z,=z;, Similarly, the stress in the plastic zone at point B (see Fig. 3) can be obtained as:
Then, t; +ty
'fY
tB=
-$-
1
211p
which in turn leads to: TB=
2P tg. 1 - P”lPp
-Ty-
(9)
The Eshelby solution also gives: r,=T,l--
2~
hb,
(1 -V)(6,+h2)ZLg
(10)
where ~7 is the applied stress at infinity at reversed loading and b, and b2 are the corresponding plastic zone semi-axes, which are not necessarily the same as a, and a,. As ca= T;” - z? and Ar = T: - T! are the far field and local loading amplitudes, respectively.
3. STRESS
INTENSITY
FACTOR ANALYSIS
During the early stage of crack propagation, the as-initiated micro-crack is still partly controlled by the PSB, where it has been initiated. At this stage, the crack size is much smaller than the PSB. Thus, the properties in the PSB, which have been formulated in the previous section, can be thought of as unchanged from the micro-crack in the PSB. Consider the configuration described in Fig. 2, where a crack is in the plastic zone. When the crack size is much smaller than plastic zone size, c, 4 a,, the crack is also modeled as an elliptical inclusion with eigenstrain cc. where cc is determined to satisfy the traction-free condition at the crack surfaces, i.e. in Q,, a$+rr?,+a&=O
in
f&
(lla)
a,“,+~:~+&~=0
in
Q
(1 lb)
where a; is the applied stress at infinity, 0: is the stress disturbance caused by the plastic zone, which can be obtained from the above known EP, and a:, is caused by the crack. Consider the pure shear loading at point A in Fig. 3, where cg is obtained in eq. (7). It is known that: *T2
=
__?k 1 -v(a,
a1a2 +a2)2E’
(12)
The early stage of fatigue crack propagation
aC2 = --
2P
793
Cl c2
(13)
1 - v (c, + c2)2tf2
where a, and a, are the two semi-axes of the plastic zone and c, and c2 are the two semi-axes of the crack; c2 tends to zero. For mode II, by substituting eqs (13) and (12) into eq. (1 la), with eq. (7), one may obtain:
2P
Cl
c2
1 -v(c,+c,)2
1 ala2 l---.--1 - v (a, + a2)’
cc cry” I2
1 - ry/rr 1 (1
-
1
ala2 VI
(4
+
(14)
a2)2-F&
where it is assumed that lim c2cC2= finite. C*-0
With these eigenstrains, the stresses at the crack tip may be calculated by considering the jump conditions across aR, [boundary of Q,, (x,/c,)’ + (x~/c~)~= l] as follows: [aij]nj = (ai,(out) - aij(in))nj = 0
(15)
A,n,
(16)
[Ui,j]=
Z4i,j(OUt) -
ui,j(ill)
=
where n is the unit normal vector of an,, and Ai are to be determined. Hooke’s law gives
[Qijlnj= cij/d([uk, 11- [ck:l)nj= 0
(17)
where [E:] = --E;, represent jumps of the eigenstrains. With eq. (16), eq. (17) can be taken in the form of C,,, Akn,n, = - C,,,.c ;,n, .
(18)
Solving eq. (18) for Ai, and then finding the stress jump will lead to the determination of the stress intensity factor for the crack. Because the SIF is a near-tip property, we may consider a point (x, , x2) on aQ which is near the crack tip. The normal vector of this can then be simplified as: CZXI
c2 (194
n’
=
n, =
&:x:
+
c-:x:/c:)
+
ClX2
JW
+l
tl9b)
as c2 + 0, x2 + 0 and x, + c,. By substituting these values into eq. (18) one has: A, (kn,n, + 1) + n,n,kA, = -2cZn,
(204
n, n,kA, + A,(&, n, + 1) = - 2cC,n,
(20b)
where k = (A + P)/L(. It is noticed that the only non-zero component of stress on the crack surface is a,, , whose jump condition is given as:
bI1l=O + &-4)b4,~1+4~,~1 = (A+ 2p)A, n, + hl,n,.
(21)
By substituting the solutions of A, and A,, which have been solved from eq. (20a, b), a,, = [a,,] = --
2P 2.5i2n,.
l-v
(22)
194
H. FAN
0.60
0.50
and L. M. KEER
:
0.00
I,lllill,“l’lll’r,l1111(111(11111111111 0.20 0.40 0.60 0.60 Ratio between Plestlc and Elastic Moduli
I.0 0
By noting the limit eq. (19a,b) and .$c2 = finite, with tiz in eq. (14) one may obtain: 0 II =
-2
“i cI
l----
r;”
2(c, -x)
1
1 - zu/r;
aI*2
1 -v(a,+az)’
1
(23) *Ia2
(1 - v) (a, + a2)2 - m The shielded stress intensity factor is found from the stress expression above as:
!+-_ II
1
1 - tu/z;
ala2
1 -v(a,+a,)’
1
ala2 (1 -v)(a,+a,)‘-1
1
(24)
where
The present solution has been checked as a limit of Wu and Chen’s [8] numerical calculation. It is worth noting that in the slip band model, the only non-zero plastic strain components are t,2 = 62,. Thus there is no shielding on K,.
4. DISCUSSION
AND CONCLUSION
Crack tip stresses shielded by local plastic deformation or damage have been discussed by many researchers. For example, Steif [9] considered a semi-infinite crack penetrating into a circular inclusion by using the complex variable approach. Further analysis on the structure of the damage zone shows that anisotropic damage can provide the maximal crack tip shielding (see Ortiz and Giannakopoulos [lo], for a review on this subject). The shielding effect considered in the present paper is different from those mentioned above. It should be pointed out that this plastic deformation is caused by fatigue loading rather than crack tip stress concentration. From Fig. 4 [eq. (24) drawn for v = 0.3 and a, /a, = lo], the stress intensity factor, K,, , is known to be decreased by a surrounding plastic zone, and the normalized crack tip stress intensity factor depends on the modulus of the ‘matrix’ and ‘inclusion’ (PSB). It has been seen that crack shielding in our configuration does not depend on the relative crack length and the position of the crack in the inclusion and is a result of the simplification used to evaluate eigenstrains [eqs (7) and (13)]. Eigenstrains were obtained for the plastic zone and crack separately without considering the interaction between them because c, =$a,, The analysis may not be close to reality for larger cracks where the relative size and position become more important. However, during the early stage of
795
The early stage of fatigue crack propagation
crack propagation, the crack is very small compared with the size of the damaged band, which is consistent with our assumption in the analysis in the previous section. With ali the above modeling and formulation, it is time to interpret our result as the observation of Fig. 1. Since the plastic modulus J+,in the PSI3 is very low [I], from Fig. 4 we know that the local stress intensity factor in model II (x, - x2 coordinate) is almost completely shielded. Recall the statement at the end of the previous section that K, is not shielded because of the absence of normal plastic strain in the PSB, then eq. (I 1) can be rewritten as:
K, can then be calculated by using the same procedure as in Section 3. It is clear that this unshielded mode I stress intensity factor is fully responsible for the early stage fatigue crack propagation direction, which coincides with the x,-axis, i.e. the maximum shear stress direction. The crack increases its length under cyclic loading along the x,-axis until it is so big that K,, cannot be shielded by the plastic deformation in the PSB. The crack is then out of the control of the PSB. The following stage of propagation is controlled by far field tensile loading. The crack path tends to the direction-perpendicular to the maximum tensile stress, the X-axis-in Fig. 1. Acknowledgments-This
Foundationthrough Grant MSM-8817869.
research was supported by the National Science
REFERENCES 111V. Essman, U. Gosele and H. Mughrabi, A model of extrusion and intrusion in fatigue metals. Phil. Mug. 44A, 405426 (1981).
121C. Laird, Mechanisms and theories of fatigue, in Fatigue
and Microstructure, pp. 149-204. ASM,Metals Park, Ohio (1978). [31 M. R. Lin, M. Fine and T. Mura, Experimental and theoretical study of fatigue crack initiation in metal. Metall. Trans. 34, 6 19W528 (1986). [41 Tanaka and Mura, A dislocation model for fatigue crack initiation. J. appl. Mech. 48, 97-103 (1981). [51J. D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and retated problems. Proc. R. Sot.
A241, 376-396 (1957).
H T. H. Lin and Q, Y. Chen, Introduction of slip bands in high-cycle fatigue crack initiation, in Micromechunics and ~~ho~~gefle~f~ (Edited by G. J. Weng, M. Taya and H. Abk),pp, 231-242. Springer-Verlag, New York (3989). [71T. Mura, ~ic~o~ee~a~jc~ of De&x% in Solid. Martinus NiihoK The Hague (19821. PI C. H. Wu and C. Chen, A d;ack”in a confocal elliptic inhomogeneity emgddeh in an infinite medium. 1. appl. Mech. 57, 91-96 (1990).
[91P. S. Steif, A semi-infinite crack penetrated in a circular inclusion. .f. uppI. Mech. 54, 87-92 (1987). DOIM. Ortiz and A E. Giannakopoulos, Maximal crack tip shielding by micro-crack. J. appt. Mech. 56, 279-283 (1989). APPENDIX For the sake of completeness. the Eshelby solution of an elliptical cylinder is listed here (let a,-+cg in Eshelby’s solution). The Eshelby tensor for isotropic materials is:
2va,
1
s 1133=----~
2(1-v)u,+a,’
S
2va,
I
**‘j= 2(1_Ij a, + (1*
s S2323 =
0, ~ 2(a,
a2 s,,,,
f
02)
’
=
____ 266
I +
~2)
The rest of the components are zero. The strain in the elliptical shaped inclusion caused by a unifo~ given by:
eigenstrain is
796
H. FAN and L. M. KEER
and the stress disturbance in the inclusion is: fJg= c,k,ta,&:”
- 4).
For the case in the previous sections, 34 ala2 CT,*=-----------_ 1-v(a,+a,)2
’
is the only component needed in the whole analysis. (Receioed 25 June 199 I)
Printed in Great Britain.
THE EARLY STAGE OF FATIGUE PROPAGATION
Gill3-7944192 $5.00+ 0.00 Q 1992Pergamon Press Ltd.
CRACK
H. FAN and L. M. KEER Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, U.S.A. Abstract-It is observed that an early stage fatigue crack propagates along the maximum shear direction rather than ~r~ndi~~ar to the maximum tensile stress direction. In the present paper, an analytical consideration is given concerning the effects of localized plastic deformation on the early stage of fatigue crack propagation. it is seen that the localized plastic defo~ation shields the local mode II stress intensity factor and leads the micro-crack to propagate along the maximum shear stress direction. In this analytical modeling, the localized plastic zone is modeled as an elliptical inclusion with eigenstrains where the Eshelby inclusion solution [Pm-. R. Sot. A241, 376-396 (1957)] is employed.
NOMENCLATURE semi-axes of plastic zone elastic moduli yielding stress in shear Poisson’s ratio elastic shear modulus plastic shear modulus plastic strain eigenstrain domain of the PSB which is (.x,/u,)~ + (x2/a,)’ = 1 boundary of R (x,/c,)‘+ (x2/c2)* = 1, with q-+0 boundary of R
1. INTRODUCTION FATIGUE CRACK initiation and early stage micro-crack propagation have attracted many researchers in both experimental and theoretical fields of research. It has been observed experimentally (e.g. refs [l] and [2]) that there are some regions where plastic strain is highly concentrated while the neighborhood of these subdomains is still in the range of elasticity. These subdomains with localized high plastic strain are called persistent slip bands (PSB), which play an important role in the early stage of fatigue damage (crack initiation and early propagation). Some analytical efforts have been made to model the PSB (e.g. refs [3] and [4]) by using ellipsoid inclusions where the Eshelby solution [S] can be applied. Lin and Chen [6] treated the plastic zone as slipping layers in which micro-cracking is believed to be initiated. Following the crack initiation, there is the early stage of micro-crack propagation. A typical fatigue crack path is shown schematically in Fig. 1, where the crack propagates along the maximum shear direction, x,-axis, in the early stage, and gradually tends to the direction perpendicular to the maximum tensile stress, the X-axis. In the present paper, an analytical model for the above observation is given by using an eigenstrain formulation and calculation of local stress intensity factor. Based on the experimental observations, the as-initiated fatigue crack is embedded in a PSB where plastic deformation is highly concentrated. For the sake of simplicity, the PSB is modeled as an elliptic inclusion in an infinite elastic matrix (Fig. 2). The micro-crack surrounded by the plastic zone is also modeled as a smaller elliptical inclusion within the PSB. The eigenstrain in R, (crack) is determined such that the traction-free condition on the crack surfaces is satisfied. The stress intensity factor is then calculated by considering the stress jump across X&, which is different from conventional calculations in fracture mechanics. 789
790
H. FAN
and L. M. KEER
-X
Fig. I. A schematic
2. MODELING
of the fatigue
OF A MICRO-CRACK
crack
path.
EMBEDDED
IN PSB
Experimental observations show that plastic deformation under fatigue is localized in some slip bands (PSB), which in the following is modeled as a narrow elliptically shaped inclusion. Micro-cracks are believed to be initiated within the bands. In this band (inclusion), Hooke’s law is written as: cJij= /Gij(Qk - &) + 2&;, - ES). Since slip lines form in a favored direction, only shear deformation along these slip lines is plastic. By using the coordinates shown in Fig. 2, a pure shear case is considered hereafter (take T = @,z, L = t,2): z* = 2&c* - ck)
(1)
where E,, is the total shear strain and ci is the plastic part. From a hysteresis loop (Fig. 3) one can write that: TA
-
2&
TY
(2)
where b is the plastic modulus. In order to simplify the analysis, a stable loop with bi-linear shape is used in the present formulation. Cyclic hardening and softening are not considered. Although the power relation between D and c may be closer to reality, a qualitatively similar result to the present analysis is expected. Substituting eq. (2) into eq. (1)
By using the eigenstrain concept [7],
The early stage of fatigue
crack
791
propagation
i
x2
a2 -4
L
X1
8
?
23
J
L
1 Fig. 2. Model of a micro-crack
embedded
in PSB.
where t;” is the loading at the far field and TX is caused by the plastic strain. For an isotropic material, it is known from the Eshelby solution [5] (see Appendix) that: (5)
A
* Q2
B
Fig. 3. Hysteresis
loop of fatigue
792
H. FAN and L. M. KEER
where a, and a, are the geometrical parameters of this elliptical plastic zone. Equations (3), (4) and (5) lead to:
(6) The eigenstrain cpAis then solved from this equation as:
c; =
T;”
-5y
2P
2P ala2 (1 -vk+a2)2-
(7)
.
1 -P/Pp
Thus, 1 T,=TF----
T;c-Ty
ala2
(1 -v)(a,
+a,)’
1
ala2
1
(8)
(1 -v)(al+a2>2-~
and as another limit, ,u~= 0, zA = T,, . When pu,=p, we have z,=z;, Similarly, the stress in the plastic zone at point B (see Fig. 3) can be obtained as:
Then, t; +ty
'fY
tB=
-$-
1
211p
which in turn leads to: TB=
2P tg. 1 - P”lPp
-Ty-
(9)
The Eshelby solution also gives: r,=T,l--
2~
hb,
(1 -V)(6,+h2)ZLg
(10)
where ~7 is the applied stress at infinity at reversed loading and b, and b2 are the corresponding plastic zone semi-axes, which are not necessarily the same as a, and a,. As ca= T;” - z? and Ar = T: - T! are the far field and local loading amplitudes, respectively.
3. STRESS
INTENSITY
FACTOR ANALYSIS
During the early stage of crack propagation, the as-initiated micro-crack is still partly controlled by the PSB, where it has been initiated. At this stage, the crack size is much smaller than the PSB. Thus, the properties in the PSB, which have been formulated in the previous section, can be thought of as unchanged from the micro-crack in the PSB. Consider the configuration described in Fig. 2, where a crack is in the plastic zone. When the crack size is much smaller than plastic zone size, c, 4 a,, the crack is also modeled as an elliptical inclusion with eigenstrain cc. where cc is determined to satisfy the traction-free condition at the crack surfaces, i.e. in Q,, a$+rr?,+a&=O
in
f&
(lla)
a,“,+~:~+&~=0
in
Q
(1 lb)
where a; is the applied stress at infinity, 0: is the stress disturbance caused by the plastic zone, which can be obtained from the above known EP, and a:, is caused by the crack. Consider the pure shear loading at point A in Fig. 3, where cg is obtained in eq. (7). It is known that: *T2
=
__?k 1 -v(a,
a1a2 +a2)2E’
(12)
The early stage of fatigue crack propagation
aC2 = --
2P
793
Cl c2
(13)
1 - v (c, + c2)2tf2
where a, and a, are the two semi-axes of the plastic zone and c, and c2 are the two semi-axes of the crack; c2 tends to zero. For mode II, by substituting eqs (13) and (12) into eq. (1 la), with eq. (7), one may obtain:
2P
Cl
c2
1 -v(c,+c,)2
1 ala2 l---.--1 - v (a, + a2)’
cc cry” I2
1 - ry/rr 1 (1
-
1
ala2 VI
(4
+
(14)
a2)2-F&
where it is assumed that lim c2cC2= finite. C*-0
With these eigenstrains, the stresses at the crack tip may be calculated by considering the jump conditions across aR, [boundary of Q,, (x,/c,)’ + (x~/c~)~= l] as follows: [aij]nj = (ai,(out) - aij(in))nj = 0
(15)
A,n,
(16)
[Ui,j]=
Z4i,j(OUt) -
ui,j(ill)
=
where n is the unit normal vector of an,, and Ai are to be determined. Hooke’s law gives
[Qijlnj= cij/d([uk, 11- [ck:l)nj= 0
(17)
where [E:] = --E;, represent jumps of the eigenstrains. With eq. (16), eq. (17) can be taken in the form of C,,, Akn,n, = - C,,,.c ;,n, .
(18)
Solving eq. (18) for Ai, and then finding the stress jump will lead to the determination of the stress intensity factor for the crack. Because the SIF is a near-tip property, we may consider a point (x, , x2) on aQ which is near the crack tip. The normal vector of this can then be simplified as: CZXI
c2 (194
n’
=
n, =
&:x:
+
c-:x:/c:)
+
ClX2
JW
+l
tl9b)
as c2 + 0, x2 + 0 and x, + c,. By substituting these values into eq. (18) one has: A, (kn,n, + 1) + n,n,kA, = -2cZn,
(204
n, n,kA, + A,(&, n, + 1) = - 2cC,n,
(20b)
where k = (A + P)/L(. It is noticed that the only non-zero component of stress on the crack surface is a,, , whose jump condition is given as:
bI1l=O + &-4)b4,~1+4~,~1 = (A+ 2p)A, n, + hl,n,.
(21)
By substituting the solutions of A, and A,, which have been solved from eq. (20a, b), a,, = [a,,] = --
2P 2.5i2n,.
l-v
(22)
194
H. FAN
0.60
0.50
and L. M. KEER
:
0.00
I,lllill,“l’lll’r,l1111(111(11111111111 0.20 0.40 0.60 0.60 Ratio between Plestlc and Elastic Moduli
I.0 0
By noting the limit eq. (19a,b) and .$c2 = finite, with tiz in eq. (14) one may obtain: 0 II =
-2
“i cI
l----
r;”
2(c, -x)
1
1 - zu/r;
aI*2
1 -v(a,+az)’
1
(23) *Ia2
(1 - v) (a, + a2)2 - m The shielded stress intensity factor is found from the stress expression above as:
!+-_ II
1
1 - tu/z;
ala2
1 -v(a,+a,)’
1
ala2 (1 -v)(a,+a,)‘-1
1
(24)
where
The present solution has been checked as a limit of Wu and Chen’s [8] numerical calculation. It is worth noting that in the slip band model, the only non-zero plastic strain components are t,2 = 62,. Thus there is no shielding on K,.
4. DISCUSSION
AND CONCLUSION
Crack tip stresses shielded by local plastic deformation or damage have been discussed by many researchers. For example, Steif [9] considered a semi-infinite crack penetrating into a circular inclusion by using the complex variable approach. Further analysis on the structure of the damage zone shows that anisotropic damage can provide the maximal crack tip shielding (see Ortiz and Giannakopoulos [lo], for a review on this subject). The shielding effect considered in the present paper is different from those mentioned above. It should be pointed out that this plastic deformation is caused by fatigue loading rather than crack tip stress concentration. From Fig. 4 [eq. (24) drawn for v = 0.3 and a, /a, = lo], the stress intensity factor, K,, , is known to be decreased by a surrounding plastic zone, and the normalized crack tip stress intensity factor depends on the modulus of the ‘matrix’ and ‘inclusion’ (PSB). It has been seen that crack shielding in our configuration does not depend on the relative crack length and the position of the crack in the inclusion and is a result of the simplification used to evaluate eigenstrains [eqs (7) and (13)]. Eigenstrains were obtained for the plastic zone and crack separately without considering the interaction between them because c, =$a,, The analysis may not be close to reality for larger cracks where the relative size and position become more important. However, during the early stage of
795
The early stage of fatigue crack propagation
crack propagation, the crack is very small compared with the size of the damaged band, which is consistent with our assumption in the analysis in the previous section. With ali the above modeling and formulation, it is time to interpret our result as the observation of Fig. 1. Since the plastic modulus J+,in the PSI3 is very low [I], from Fig. 4 we know that the local stress intensity factor in model II (x, - x2 coordinate) is almost completely shielded. Recall the statement at the end of the previous section that K, is not shielded because of the absence of normal plastic strain in the PSB, then eq. (I 1) can be rewritten as:
K, can then be calculated by using the same procedure as in Section 3. It is clear that this unshielded mode I stress intensity factor is fully responsible for the early stage fatigue crack propagation direction, which coincides with the x,-axis, i.e. the maximum shear stress direction. The crack increases its length under cyclic loading along the x,-axis until it is so big that K,, cannot be shielded by the plastic deformation in the PSB. The crack is then out of the control of the PSB. The following stage of propagation is controlled by far field tensile loading. The crack path tends to the direction-perpendicular to the maximum tensile stress, the X-axis-in Fig. 1. Acknowledgments-This
Foundationthrough Grant MSM-8817869.
research was supported by the National Science
REFERENCES 111V. Essman, U. Gosele and H. Mughrabi, A model of extrusion and intrusion in fatigue metals. Phil. Mug. 44A, 405426 (1981).
121C. Laird, Mechanisms and theories of fatigue, in Fatigue
and Microstructure, pp. 149-204. ASM,Metals Park, Ohio (1978). [31 M. R. Lin, M. Fine and T. Mura, Experimental and theoretical study of fatigue crack initiation in metal. Metall. Trans. 34, 6 19W528 (1986). [41 Tanaka and Mura, A dislocation model for fatigue crack initiation. J. appl. Mech. 48, 97-103 (1981). [51J. D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and retated problems. Proc. R. Sot.
A241, 376-396 (1957).
H T. H. Lin and Q, Y. Chen, Introduction of slip bands in high-cycle fatigue crack initiation, in Micromechunics and ~~ho~~gefle~f~ (Edited by G. J. Weng, M. Taya and H. Abk),pp, 231-242. Springer-Verlag, New York (3989). [71T. Mura, ~ic~o~ee~a~jc~ of De&x% in Solid. Martinus NiihoK The Hague (19821. PI C. H. Wu and C. Chen, A d;ack”in a confocal elliptic inhomogeneity emgddeh in an infinite medium. 1. appl. Mech. 57, 91-96 (1990).
[91P. S. Steif, A semi-infinite crack penetrated in a circular inclusion. .f. uppI. Mech. 54, 87-92 (1987). DOIM. Ortiz and A E. Giannakopoulos, Maximal crack tip shielding by micro-crack. J. appt. Mech. 56, 279-283 (1989). APPENDIX For the sake of completeness. the Eshelby solution of an elliptical cylinder is listed here (let a,-+cg in Eshelby’s solution). The Eshelby tensor for isotropic materials is:
2va,
1
s 1133=----~
2(1-v)u,+a,’
S
2va,
I
**‘j= 2(1_Ij a, + (1*
s S2323 =
0, ~ 2(a,
a2 s,,,,
f
02)
’
=
____ 266
I +
~2)
The rest of the components are zero. The strain in the elliptical shaped inclusion caused by a unifo~ given by:
eigenstrain is
796
H. FAN and L. M. KEER
and the stress disturbance in the inclusion is: fJg= c,k,ta,&:”
- 4).
For the case in the previous sections, 34 ala2 CT,*=-----------_ 1-v(a,+a,)2
’
is the only component needed in the whole analysis. (Receioed 25 June 199 I)