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Fatigue crack propagation for defects near a free surface
,5,,gmeermgFrircrure Me
FATIGUE
Institute
for Robotics,
I, pp. 23-32, 1987
4x13-7944/87 $3.00+ 00 Pergamon Journals Ltd.
CRACK PROPAGATION FOR NEAR A FREE SURFACE G. L. CHEN and R. ROBERTS Lehigh University, 200 West Packer Avenue,
DEFECTS
Bethlehem,
PA 18015. U.S.A
Abstract-The fatigue crack growth associated with an internal flaw as it approaches a free surface was studied. Previous researchers recommended when the plastic zone at the crack tip nearest the free surface reaches one-tenth of the ligament size which exists between the free surface and the crack tip that the flaw be treated as an edge crack. The results of this study show that the assumption of an edge crack can prove to be exceptionally conservative when employed for design purposes. Underestimates of fatigue life of as much as 1000% were observed. Improved methods for predicting the fatigue growth of near surface defects are offered in the paper.
INTRODUCTION THE DESIGN and analysis of engineering structures often require consideration of how cyclic loads and defect population affect total structural life. Examples of such evaluations can be found in the literature over the last lo-15 years. One such analysis for turbine wheel design was proposed by Brandt[ 11. Brandt considered the fatigue crack growth of internal flaws approaching a free surface. A defect of this type is shown schematically in Fig. 1. Gibbons et al.[2] considered the fracture behavior of similar internal defects in the vicinity of a free surface. Brandt in his analysis proposed that an internal defect be considered as an edge crack if the ligament between the crack tip nearest the free surface and the free surface is less than ten times the crack tip plastic zone size. This limiting condition can be written as
L 6 lOR,
(1)
where the plastic zone R, is defined by Irwin[3] as
(2) L is the ligament length, K the crack tip stress intensity factor and cY the material’s yield strength. Figure 2 shows the free surface, the adjacent crack tip and the plastic zones in relationship to each other. It also shows the proposed equivalent edge defect when the criterion given by eq. (1) is met. Gibbon’s results[2] suggest that the proportionality factor in eq. (I) should be approximately 11.4 as opposed to Brandt’s value of 10. The problem of assessing the interaction of an internal defect with a free surface is of great practical significance. As will be shown, the analysis procedures proposed by Brandt where an
Fig. 1. Internal
flaw approaching 23
a free surface.
G. L. CHEN
24
Fig. 2. Ligament
and
R. ROBERTS
and plastic zones
internal defect is modeled as an edge crack may at times prove to be conservative. To understand the importance of this, consider, for example, a turbine shaft containing an internal defect which is near a free surface and known to be actively growing due to fluctuations in turbine speed. For this application the decision to continue to operate the turbine given the active fatigue crack or to retire the shaft from service can have significant economic impact on the user. Thus it becomes mandatory to understand with a known degree of certainty how conservative a given calculation of structural life can be. Without this understanding, appropriate risk and economic impact analysis and decisions cannot be made. It is the intent of this paper to examine some of the proposals put forward by Brandt and Gibbons et al. with respect to the fatigue crack growth of internal defects near a free surface. Other possible analysis procedures will also be investigated. To this end the fatigue crack propagation in thin plate specimens of A36 steel containing internal defects with one crack tip approaching a free surface has been studied. These results and testing details are presented in the remaining portions of this paper. TEST PROGRAM For the purposes of this investigation thin plate type specimens were cut from a sheet of A36 steel material. Figure 3 shows the important dimensions of the specimens used. All specimens were 1.9 mm (0.075 in.) thick. Center crack specimens (CCS) and compact tensile specimens (CTS) were used to produce baseline fatigue crack propagation (FCP) data to which the off-center crack specimens (OCC) could be compared. Brandt suggested that the work of Zwicky[4] could be used to estimate the K level at the crack tip nearest the free surface for a body of infinite extent. This took the form KB = (0.7+0.035(2a/L))oJ2na.
(a) Fig. 3. Test specimen
configurations.
(b)
(c
)
(a) OCS, (b) CCS, (c) CTS. All dimensions
in mm
Fatigue
Fig. 4. Comparison
crack propagation
for defects near a free surface
between Isida’s and Brandt’s formulae for the stress intensity tip, K, = Isida’s Results, KB = Brandt’s Results.
2s
factor at the right crack
However, in this study, the stress intensity factors at both crack tips for the CCS and OCC specimens were calculated using Isida’s[S] results. This takes the form K= where rr is the applied
o&F(&it)
(3)
stress, 2a is the crack length shown in Fig. 3 and 6 and 1 are defined as
This was done because the specimens were of finite size and the Zwicky results are for an infinite body. A comparison of Ki and KB as a function of the specimen size is given in Fig. 4. As can be seen when the right ligament exceeds 250 mm the percentage error in KB is about 4%. For ligament sizes less than this the errors are too great to ignore. All fatigue testing was done on a 20 kip electrohydraulic closed loop test machine. For the plate specimens of the CCS and OCC variety a traveling microscope was used to measure the fatigue crack growth at each crack tip. All tests were constant load type tests where the load ratio R = ~,,JcJ~~~ was maintained at R = 0.1. The testing frequency was 20 Hz. For the CTS specimen, crack growth was measured with the use of a clip gage and compliance techniques[6]. The OCC specimens were run at three maximum load levels : 17.8 kN (4000 Ibs), 22.2 kN (5000 lbs) and 26.7 kN (6000 Ibs). The CCS specimen was tested at a load level of 26.7 kN (6000 lbs). TEST RESULTS The FCP results for the CCS and CTS specimens are shown in Figs 5 and 6, respectively, as plots of FCP rate vs AK, where AK = K,,,,, - K,i,. The results for both specimen types are very similar with the CCS-FCP data being slightly greater than the CTS-FCP data at comparable AK levels. This comparison is shown in Fig. 7. Here it can also be seen that for AK levels above 15 MPa,,/% these data can be represented by an equation of the form
where a is the crack length, N the number
of cycles and C and m material
constants
G. L. CHEN and R. ROBERTS
26 10-2t
I 4_ 6
7-
P
6
b
4
D 2
A36 steel eCTS352 4 CTS 353 + CTS 355
3
FCP rate
1
10-5 6 4
2
2 10-6
AK
(MPa M.5)
Fig. 5. FCP rate vs AK in CTS.
Figures 8 and 9 highlight the FCP rate data as a function of AK for the left and right crack tip respectively of the OCC specimens. It appears from these data that there is transition in the data at a AK level of about 20-22 MPa&. This corresponds to an R,/L ratio of about 0.14.15. Some of the more important details of the test data are summarized in Table 1. As a means of comparing the OCC-FCP data to the CCS-FCP data a new parameter A was introduced. This is defined as
(6)
A = (daldn)oc~l(daldn)~~s
‘“-ii/ A36
,,-I/ IO'
steel no. 361 (center crack)
4
2
AK
(MPa M.5)
Fig. 6. FCP rate vs AK in CCS
6
8
IO2
Fatigue
crack propagation
for defects near a free surface
2
4
AK Fig. 7. Comparison
(MPo
5
5
102
M.5)
of FCP rate in CTS and CCS.
where (da/dn)occ = FCP rate in OCC specimens for given AK level, (da/dn)co
= FCP rate in CCS specimens for given AK level.
lo-2
L
I 4_ 6
2.
,
A36 steel 0362 0363 n 364
FCP rate (left tip)
0
10-Z -
00 -0
6;; 6 N E
d % y’
4210-4
-
$8
z P
F
9”
4 *-
10-5
_ 64-
210-5
IO’
2
AK
4
(MPa
6
M.5)
Fig. 8. FCP rate vs AK for left tip
e
I02
27
28
G. L. CHEN
and
R. ROBERTS
A36 steel
FCP rate
(right
tip)
*t
lo+1 IO'
Ak‘(MPa
IO2
M.5)
Fig. 9. FCP rate vs AK for right tip.
Table 1. Partial
~________. .~~ WA. (da/d& Data No.
(MPafi
(10m4 mm/cyc)
summary
of test data for OCC specimens L
(WW~
----
(I&$& (10 -’ mm/cyc) ,.._.-.___.___-~_-.__-_-..__
AL
AR
(mm)
&IL
29 32 34 36 38 40 42 43 44 46 48
22.42 23.50 24.40 25.55 26.72 27.85 29.18 29.96 30.89 33.42 37.40
0.662 0.799 1.08 1.28 1.54 1.86 2.53 2.86 3.35 4.81 9.38
Specimen 21.96 22.90 23.66 24.62 25.60 26.44 27.43 28.00 28.66 30.30 32.69
362 (26.7 KN, R = 0.I) 0.592 1.09 0.761 1.10 0.943 1.27 1.18 1.25 1so 1.26 1.82 1.29 2.12 1.46 2.24 1.48 2.40 I .54 2.81 1.61 3.20 2.00
1.06 1.16 1.26 1.34 I .46 1.56 1.57 1.52 1.49 1.40 1.17
14.31 13.88 13.34 12.72 12.06 Il.46 10.83 IO.46 9.92 8.89 7.32
0.111 0.127 0.142 0.164 0.189 0.216 0.251 0.274 0.307 0.401 0.609
17 21 22 24 25 26 28 30 32 33 34
21.46 23.31 23.75 24.79 25.47 26.05 28.06 30.04 31.93 34.24 36.65
0.465 0.643 0.722 0.968 1.17 1.37 2.03 2.79 4.07 5.17 6.90
Specimen 20.65 22.10 22.42 23.18 23.67 24.10 25.35 26.55 27.68 28.72 29.85
363 (22.2 KN, R = 0.1) 0.421 0.91 0.572 0.91 0.620 1.00 0.774 1.07 0.899 1.16 I .03 I .25 I .44 1.37 1.83 I .4? 2.14 1.54 2.40 1.58 2.15 1.60
0.97 1.00 I .03 1.12 1.20 1.27 I .46 1.54 1.52 1.48 1.45
12.48 11.32 1I .06 10.45 10.02 9.66 8.73 7.78 6.93 6.29 5.56
0.118 0.153 0.163 0.187 0.206 0.224 0.287 0.370 0.470 0.594 0.770
30 34 36 38 39 40 41 42 44 45 46
19.58 20.88 21.41 22.35 22.85 23.39 24.01 24.61 26.87 28.56 30.80
0.370 0.496 0.573 0.694 0.807 0.943 1.17 1.42 2.15 2.19 4.25
Specimen 18.38 19.32 19.70 20.25 20.54 20.87 21.25 21.60 22.74 23.52 24.55
364 (17.8 KN, R = 0.I 0.295 1.05 0.358 1.09 0.398 1.14 0.450 1.16 0,495 I .24 0.555 1.32 0.63 1 1.47 0.738 1.62 0.95 1 1.73 1.12 1.80 1.39 1.97
1.08 I .07 1.10 1.12 1.16 I .22 1.29 1.42 1.49 I .53 1.59
10.76 9.76 9.35 8.87 8.56 8.24 7.88 7.49 6.51 4.83 4.96
0.114 0.142 0.156 0.180 0.194 0.212 0.233 0.258 0.354 0.446 0.610
Fatigue
I
crack propagation
l
for defects near a free surface
362
x 363
2
. 364
t
, '01
015
i, 02
I 025
I 03
I 05
Fig. 10. A vs RL for left tip shown on log-log
In addition
to A the quantity
29
RL is introduced
coordinates
where
R, is defined by eq. (2) and L is the ligament at the left crack tip. Figures 10 and 11 show the values of A plotted against RL on a log-log plot for both the right and left crack tips in the OCC specimens. In these figures it appears for values of RL greater than 0. I to 0.15 that A is proportional to RL.For levels greater than RL = 0.25, A remains constant at a level of approximately 1.5 for the right tip. For the left tip A remains proportional to RL but the constant of proportionality changes. In terms of the physical appearance of the specimen ligament, when RL exceeded 0.25 the surface of the ligament became very rough indicating, based on past experiences with similar tests, that the ligament had fully yielded.
DISCUSSION
OF RESULTS
Ultimately, the test of any model of crack tip-free surface interaction requires that the model be able to accurately predict the relationship between the crack length and the number of applied cycles. As already noted, Brandt proposed for R, > L/l0 that the internal defect be considered as an edge crack. This can be examined on a purely theoretical basis by considering the two configurations in Fig. 12. The K level for the equivalent edge crack in Fig. 12(b) is calculated[7] from KE = n&(Y)
(8)
0362 x. 363 . 364 2-
Fig. 11. A vs R, for right tip shown on log-log
coordinates.
30
G. L. CHEN
and
R. ROBERTS
_I --‘ L
(I
20
KE
KI
L
(a)
Fig. 12. Internal
20
(b)
defect (a) and equivalent
edge crack
(b).
where
(9) The K level for the corresponding right crack tip in Fig. 12(a) is calculated from Isida’s results ([4] eq. (3)). The ratios of these two K levels (KJK,) are plotted in Fig. 13 as a function of a/w and flw. For the range of specimen geometries studied in this investigation, this ratio fell between approximately 3.0 and 2.3. It should be clear if the left crack tip ligament does provide some support for the right crack tip, then estimates of FCP rate using the equivalent edge crack model could be significantly in error. FCP rate is a function of AK”’ and AK could be off by a maximum of 3.0 if Brandt’s proposal is used. Brandt also suggested when 2a
10
-< L
$ ted
and $ values
spscimsns
Fig. 13. Ratio of KEIKl
(10)
forcurrent
Fatigue
crack propagation
for defects near a free surface
Table 2. Calculation
data for various
362 Load (kN) 2a. (mm) L0 (mm) (WLh N, (cycle) N, NB NQ
NrIN, WNO NoIN,
formulae
Specimen No. 363
364
22.2 20.26 13.11 0.103 74,800 91,600 6,500 72,000 1.23 0.087 0.97
17.8 24.28 11.20 0.103 82.500 105,500 9,500 82,100 1.28 0.12 0.99
26.7 17.35 14.73 0.106 50,000 61,900 3,400 48,500 1.24 0.068 0.97 .____
31
~__~
Note : N, : cycles from experimental data ; NI : cycles using Isida’s results ; Na : cycles for Brandt’s proposal; NQ : modified Isida’s results.
that the crack be considered as an edge crack independent of any plastic zone considerations. This can also be examined theoretically. For the case where 2a/L = 10 the ratio of K, assuming an edge crack of length L + 2a and the K for the right tip, according to Isida is 1.376 if the body is of infinite extent. Clearly this will provide a conservative estimate of the K levels and the predicted FCP rate which is proportional to AK”. As already noted for finite width specimens such as used in this study, the ratio can be much greater. To further evaluate the usefulness of Brandt’s proposal, various schemes for calculating crack growth were used. As baseline data the FCP data from the CCS were used. At each point in the calculation the K levels at the right and left crack tips were calculated and the corresponding da/dn levels obtained from the FCP-CCS data. A value of da was then calculated for the left and right crack tips for a small dN. These crack extensions were added to the previous crack length. This new crack length was used to calculate a new AK and da/dn for the left and right crack tips. This cycle was continued until the desired final crack length was obtained. Calculations were made for RL levels starting at about 0.1 and were continued until the ligament fractured. Table 2 provides the results of three calculation procedures. Each of the three provide the number of cycles required to grow the crack from a length of RL = 0.1 to the point of ligament rupture. They are as follows : N, : this represents the number of cycles calculated using Isida’s K levels for the right and left crack tips and ignoring all plasticity effects in the calculations. NB : this represents the number of cycles required treating the crack solely as an edge crack. This is Brandt’s proposal. N, : this procedure used Isida’s K levels with a modification of FCP rates as calculated by eq. (11). In an attempt to improve the estimate of N,, produced by using just Isida’s results, the A values calculated were introduced. The curves shown in Figs 10 and 11 were simplified as shown in Fig. 14. The FCP rates were then calculated from da dn
=
CAKmQ
(11)
where
Q = UR; and for both left and right crack tip when
0.25 > RL > 0.15 u=
0.5:
v= 0.8
(12)
32
G. L. CHEN
-
For
left
---
For
right
and
R. ROBERTS
tip tip
I 01
015
025
06 RL
Fig. 14. A vs RL simplified
curves shown on log-log
coordinates.
and for
left tip
U = 2.39
V = 0.33
u=
v= 0.
1.5
These growth rates were used in conjunction with Isida’s estimate of K to calculate cycles of life. The results are given in Table 2. These results are clearly better than the results based solely on using Isida’s K levels, N,. As a result of the tests conducted in this study it is evident that using Brandt’s proposal for calculating fatigue life can produce results which are very conservative. For the three tests reported the results can be as much as 700% to 1000% too conservative. Also for the tests conducted, much better estimates of fatigue life were produced by ignoring plastic zone effects and just calculating da/dn based on AK levels for the internal defect. This produced results which overestimated the fatigue life by about 25-30%. Finally, a proposed modification of the growth rates based on A and RL provided the best prediction capability. However, actual use of such procedures should be based on more extensive testing. REFERENCES [l] [2] [3] [4] [5] [6]
[7]
D. E. Brandt, The development of a turbine wheel design criterion based upon fracture mechanics. ASME J. Engng Pwr93,p. 411 (1971). W. G. Gibbons, W. R. Andrews and G. A. Clarke, Fracture of defects approaching a free surface. Trans. ASME, 75. PVP-10 (1975). G. R. Irwin, Plastic zone near a crack and fracture toughness. Proc. 7th Sagamore Ordnance Materials Corzf. Syracuse University Research Institute (1960). E. E. Zwicky, unpublished work. M. Isida, Stress-intensity factor for the tension of eccentrically cracked strip. J. uppl. Mech. 33,674 (1966). R. S. Vecchio, D. A. Jablonski, B. H. Lee, R. W. Hertzberg, C. N. Newton, R. Roberts, G. Chen and G. Connelly, Development of an automated fatigue crack propagation test system. Automated Test Methods fr,r Fracture and Fatigue Crack Growth, ASTM STP 877, 44-66 (1985). Single-edge-crackedplate under tension. ASTM STP 410 (1969). (Received 3 January 1986)
FATIGUE
Institute
for Robotics,
I, pp. 23-32, 1987
4x13-7944/87 $3.00+ 00 Pergamon Journals Ltd.
CRACK PROPAGATION FOR NEAR A FREE SURFACE G. L. CHEN and R. ROBERTS Lehigh University, 200 West Packer Avenue,
DEFECTS
Bethlehem,
PA 18015. U.S.A
Abstract-The fatigue crack growth associated with an internal flaw as it approaches a free surface was studied. Previous researchers recommended when the plastic zone at the crack tip nearest the free surface reaches one-tenth of the ligament size which exists between the free surface and the crack tip that the flaw be treated as an edge crack. The results of this study show that the assumption of an edge crack can prove to be exceptionally conservative when employed for design purposes. Underestimates of fatigue life of as much as 1000% were observed. Improved methods for predicting the fatigue growth of near surface defects are offered in the paper.
INTRODUCTION THE DESIGN and analysis of engineering structures often require consideration of how cyclic loads and defect population affect total structural life. Examples of such evaluations can be found in the literature over the last lo-15 years. One such analysis for turbine wheel design was proposed by Brandt[ 11. Brandt considered the fatigue crack growth of internal flaws approaching a free surface. A defect of this type is shown schematically in Fig. 1. Gibbons et al.[2] considered the fracture behavior of similar internal defects in the vicinity of a free surface. Brandt in his analysis proposed that an internal defect be considered as an edge crack if the ligament between the crack tip nearest the free surface and the free surface is less than ten times the crack tip plastic zone size. This limiting condition can be written as
L 6 lOR,
(1)
where the plastic zone R, is defined by Irwin[3] as
(2) L is the ligament length, K the crack tip stress intensity factor and cY the material’s yield strength. Figure 2 shows the free surface, the adjacent crack tip and the plastic zones in relationship to each other. It also shows the proposed equivalent edge defect when the criterion given by eq. (1) is met. Gibbon’s results[2] suggest that the proportionality factor in eq. (I) should be approximately 11.4 as opposed to Brandt’s value of 10. The problem of assessing the interaction of an internal defect with a free surface is of great practical significance. As will be shown, the analysis procedures proposed by Brandt where an
Fig. 1. Internal
flaw approaching 23
a free surface.
G. L. CHEN
24
Fig. 2. Ligament
and
R. ROBERTS
and plastic zones
internal defect is modeled as an edge crack may at times prove to be conservative. To understand the importance of this, consider, for example, a turbine shaft containing an internal defect which is near a free surface and known to be actively growing due to fluctuations in turbine speed. For this application the decision to continue to operate the turbine given the active fatigue crack or to retire the shaft from service can have significant economic impact on the user. Thus it becomes mandatory to understand with a known degree of certainty how conservative a given calculation of structural life can be. Without this understanding, appropriate risk and economic impact analysis and decisions cannot be made. It is the intent of this paper to examine some of the proposals put forward by Brandt and Gibbons et al. with respect to the fatigue crack growth of internal defects near a free surface. Other possible analysis procedures will also be investigated. To this end the fatigue crack propagation in thin plate specimens of A36 steel containing internal defects with one crack tip approaching a free surface has been studied. These results and testing details are presented in the remaining portions of this paper. TEST PROGRAM For the purposes of this investigation thin plate type specimens were cut from a sheet of A36 steel material. Figure 3 shows the important dimensions of the specimens used. All specimens were 1.9 mm (0.075 in.) thick. Center crack specimens (CCS) and compact tensile specimens (CTS) were used to produce baseline fatigue crack propagation (FCP) data to which the off-center crack specimens (OCC) could be compared. Brandt suggested that the work of Zwicky[4] could be used to estimate the K level at the crack tip nearest the free surface for a body of infinite extent. This took the form KB = (0.7+0.035(2a/L))oJ2na.
(a) Fig. 3. Test specimen
configurations.
(b)
(c
)
(a) OCS, (b) CCS, (c) CTS. All dimensions
in mm
Fatigue
Fig. 4. Comparison
crack propagation
for defects near a free surface
between Isida’s and Brandt’s formulae for the stress intensity tip, K, = Isida’s Results, KB = Brandt’s Results.
2s
factor at the right crack
However, in this study, the stress intensity factors at both crack tips for the CCS and OCC specimens were calculated using Isida’s[S] results. This takes the form K= where rr is the applied
o&F(&it)
(3)
stress, 2a is the crack length shown in Fig. 3 and 6 and 1 are defined as
This was done because the specimens were of finite size and the Zwicky results are for an infinite body. A comparison of Ki and KB as a function of the specimen size is given in Fig. 4. As can be seen when the right ligament exceeds 250 mm the percentage error in KB is about 4%. For ligament sizes less than this the errors are too great to ignore. All fatigue testing was done on a 20 kip electrohydraulic closed loop test machine. For the plate specimens of the CCS and OCC variety a traveling microscope was used to measure the fatigue crack growth at each crack tip. All tests were constant load type tests where the load ratio R = ~,,JcJ~~~ was maintained at R = 0.1. The testing frequency was 20 Hz. For the CTS specimen, crack growth was measured with the use of a clip gage and compliance techniques[6]. The OCC specimens were run at three maximum load levels : 17.8 kN (4000 Ibs), 22.2 kN (5000 lbs) and 26.7 kN (6000 Ibs). The CCS specimen was tested at a load level of 26.7 kN (6000 lbs). TEST RESULTS The FCP results for the CCS and CTS specimens are shown in Figs 5 and 6, respectively, as plots of FCP rate vs AK, where AK = K,,,,, - K,i,. The results for both specimen types are very similar with the CCS-FCP data being slightly greater than the CTS-FCP data at comparable AK levels. This comparison is shown in Fig. 7. Here it can also be seen that for AK levels above 15 MPa,,/% these data can be represented by an equation of the form
where a is the crack length, N the number
of cycles and C and m material
constants
G. L. CHEN and R. ROBERTS
26 10-2t
I 4_ 6
7-
P
6
b
4
D 2
A36 steel eCTS352 4 CTS 353 + CTS 355
3
FCP rate
1
10-5 6 4
2
2 10-6
AK
(MPa M.5)
Fig. 5. FCP rate vs AK in CTS.
Figures 8 and 9 highlight the FCP rate data as a function of AK for the left and right crack tip respectively of the OCC specimens. It appears from these data that there is transition in the data at a AK level of about 20-22 MPa&. This corresponds to an R,/L ratio of about 0.14.15. Some of the more important details of the test data are summarized in Table 1. As a means of comparing the OCC-FCP data to the CCS-FCP data a new parameter A was introduced. This is defined as
(6)
A = (daldn)oc~l(daldn)~~s
‘“-ii/ A36
,,-I/ IO'
steel no. 361 (center crack)
4
2
AK
(MPa M.5)
Fig. 6. FCP rate vs AK in CCS
6
8
IO2
Fatigue
crack propagation
for defects near a free surface
2
4
AK Fig. 7. Comparison
(MPo
5
5
102
M.5)
of FCP rate in CTS and CCS.
where (da/dn)occ = FCP rate in OCC specimens for given AK level, (da/dn)co
= FCP rate in CCS specimens for given AK level.
lo-2
L
I 4_ 6
2.
,
A36 steel 0362 0363 n 364
FCP rate (left tip)
0
10-Z -
00 -0
6;; 6 N E
d % y’
4210-4
-
$8
z P
F
9”
4 *-
10-5
_ 64-
210-5
IO’
2
AK
4
(MPa
6
M.5)
Fig. 8. FCP rate vs AK for left tip
e
I02
27
28
G. L. CHEN
and
R. ROBERTS
A36 steel
FCP rate
(right
tip)
*t
lo+1 IO'
Ak‘(MPa
IO2
M.5)
Fig. 9. FCP rate vs AK for right tip.
Table 1. Partial
~________. .~~ WA. (da/d& Data No.
(MPafi
(10m4 mm/cyc)
summary
of test data for OCC specimens L
(WW~
----
(I&$& (10 -’ mm/cyc) ,.._.-.___.___-~_-.__-_-..__
AL
AR
(mm)
&IL
29 32 34 36 38 40 42 43 44 46 48
22.42 23.50 24.40 25.55 26.72 27.85 29.18 29.96 30.89 33.42 37.40
0.662 0.799 1.08 1.28 1.54 1.86 2.53 2.86 3.35 4.81 9.38
Specimen 21.96 22.90 23.66 24.62 25.60 26.44 27.43 28.00 28.66 30.30 32.69
362 (26.7 KN, R = 0.I) 0.592 1.09 0.761 1.10 0.943 1.27 1.18 1.25 1so 1.26 1.82 1.29 2.12 1.46 2.24 1.48 2.40 I .54 2.81 1.61 3.20 2.00
1.06 1.16 1.26 1.34 I .46 1.56 1.57 1.52 1.49 1.40 1.17
14.31 13.88 13.34 12.72 12.06 Il.46 10.83 IO.46 9.92 8.89 7.32
0.111 0.127 0.142 0.164 0.189 0.216 0.251 0.274 0.307 0.401 0.609
17 21 22 24 25 26 28 30 32 33 34
21.46 23.31 23.75 24.79 25.47 26.05 28.06 30.04 31.93 34.24 36.65
0.465 0.643 0.722 0.968 1.17 1.37 2.03 2.79 4.07 5.17 6.90
Specimen 20.65 22.10 22.42 23.18 23.67 24.10 25.35 26.55 27.68 28.72 29.85
363 (22.2 KN, R = 0.1) 0.421 0.91 0.572 0.91 0.620 1.00 0.774 1.07 0.899 1.16 I .03 I .25 I .44 1.37 1.83 I .4? 2.14 1.54 2.40 1.58 2.15 1.60
0.97 1.00 I .03 1.12 1.20 1.27 I .46 1.54 1.52 1.48 1.45
12.48 11.32 1I .06 10.45 10.02 9.66 8.73 7.78 6.93 6.29 5.56
0.118 0.153 0.163 0.187 0.206 0.224 0.287 0.370 0.470 0.594 0.770
30 34 36 38 39 40 41 42 44 45 46
19.58 20.88 21.41 22.35 22.85 23.39 24.01 24.61 26.87 28.56 30.80
0.370 0.496 0.573 0.694 0.807 0.943 1.17 1.42 2.15 2.19 4.25
Specimen 18.38 19.32 19.70 20.25 20.54 20.87 21.25 21.60 22.74 23.52 24.55
364 (17.8 KN, R = 0.I 0.295 1.05 0.358 1.09 0.398 1.14 0.450 1.16 0,495 I .24 0.555 1.32 0.63 1 1.47 0.738 1.62 0.95 1 1.73 1.12 1.80 1.39 1.97
1.08 I .07 1.10 1.12 1.16 I .22 1.29 1.42 1.49 I .53 1.59
10.76 9.76 9.35 8.87 8.56 8.24 7.88 7.49 6.51 4.83 4.96
0.114 0.142 0.156 0.180 0.194 0.212 0.233 0.258 0.354 0.446 0.610
Fatigue
I
crack propagation
l
for defects near a free surface
362
x 363
2
. 364
t
, '01
015
i, 02
I 025
I 03
I 05
Fig. 10. A vs RL for left tip shown on log-log
In addition
to A the quantity
29
RL is introduced
coordinates
where
R, is defined by eq. (2) and L is the ligament at the left crack tip. Figures 10 and 11 show the values of A plotted against RL on a log-log plot for both the right and left crack tips in the OCC specimens. In these figures it appears for values of RL greater than 0. I to 0.15 that A is proportional to RL.For levels greater than RL = 0.25, A remains constant at a level of approximately 1.5 for the right tip. For the left tip A remains proportional to RL but the constant of proportionality changes. In terms of the physical appearance of the specimen ligament, when RL exceeded 0.25 the surface of the ligament became very rough indicating, based on past experiences with similar tests, that the ligament had fully yielded.
DISCUSSION
OF RESULTS
Ultimately, the test of any model of crack tip-free surface interaction requires that the model be able to accurately predict the relationship between the crack length and the number of applied cycles. As already noted, Brandt proposed for R, > L/l0 that the internal defect be considered as an edge crack. This can be examined on a purely theoretical basis by considering the two configurations in Fig. 12. The K level for the equivalent edge crack in Fig. 12(b) is calculated[7] from KE = n&(Y)
(8)
0362 x. 363 . 364 2-
Fig. 11. A vs R, for right tip shown on log-log
coordinates.
30
G. L. CHEN
and
R. ROBERTS
_I --‘ L
(I
20
KE
KI
L
(a)
Fig. 12. Internal
20
(b)
defect (a) and equivalent
edge crack
(b).
where
(9) The K level for the corresponding right crack tip in Fig. 12(a) is calculated from Isida’s results ([4] eq. (3)). The ratios of these two K levels (KJK,) are plotted in Fig. 13 as a function of a/w and flw. For the range of specimen geometries studied in this investigation, this ratio fell between approximately 3.0 and 2.3. It should be clear if the left crack tip ligament does provide some support for the right crack tip, then estimates of FCP rate using the equivalent edge crack model could be significantly in error. FCP rate is a function of AK”’ and AK could be off by a maximum of 3.0 if Brandt’s proposal is used. Brandt also suggested when 2a
10
-< L
$ ted
and $ values
spscimsns
Fig. 13. Ratio of KEIKl
(10)
forcurrent
Fatigue
crack propagation
for defects near a free surface
Table 2. Calculation
data for various
362 Load (kN) 2a. (mm) L0 (mm) (WLh N, (cycle) N, NB NQ
NrIN, WNO NoIN,
formulae
Specimen No. 363
364
22.2 20.26 13.11 0.103 74,800 91,600 6,500 72,000 1.23 0.087 0.97
17.8 24.28 11.20 0.103 82.500 105,500 9,500 82,100 1.28 0.12 0.99
26.7 17.35 14.73 0.106 50,000 61,900 3,400 48,500 1.24 0.068 0.97 .____
31
~__~
Note : N, : cycles from experimental data ; NI : cycles using Isida’s results ; Na : cycles for Brandt’s proposal; NQ : modified Isida’s results.
that the crack be considered as an edge crack independent of any plastic zone considerations. This can also be examined theoretically. For the case where 2a/L = 10 the ratio of K, assuming an edge crack of length L + 2a and the K for the right tip, according to Isida is 1.376 if the body is of infinite extent. Clearly this will provide a conservative estimate of the K levels and the predicted FCP rate which is proportional to AK”. As already noted for finite width specimens such as used in this study, the ratio can be much greater. To further evaluate the usefulness of Brandt’s proposal, various schemes for calculating crack growth were used. As baseline data the FCP data from the CCS were used. At each point in the calculation the K levels at the right and left crack tips were calculated and the corresponding da/dn levels obtained from the FCP-CCS data. A value of da was then calculated for the left and right crack tips for a small dN. These crack extensions were added to the previous crack length. This new crack length was used to calculate a new AK and da/dn for the left and right crack tips. This cycle was continued until the desired final crack length was obtained. Calculations were made for RL levels starting at about 0.1 and were continued until the ligament fractured. Table 2 provides the results of three calculation procedures. Each of the three provide the number of cycles required to grow the crack from a length of RL = 0.1 to the point of ligament rupture. They are as follows : N, : this represents the number of cycles calculated using Isida’s K levels for the right and left crack tips and ignoring all plasticity effects in the calculations. NB : this represents the number of cycles required treating the crack solely as an edge crack. This is Brandt’s proposal. N, : this procedure used Isida’s K levels with a modification of FCP rates as calculated by eq. (11). In an attempt to improve the estimate of N,, produced by using just Isida’s results, the A values calculated were introduced. The curves shown in Figs 10 and 11 were simplified as shown in Fig. 14. The FCP rates were then calculated from da dn
=
CAKmQ
(11)
where
Q = UR; and for both left and right crack tip when
0.25 > RL > 0.15 u=
0.5:
v= 0.8
(12)
32
G. L. CHEN
-
For
left
---
For
right
and
R. ROBERTS
tip tip
I 01
015
025
06 RL
Fig. 14. A vs RL simplified
curves shown on log-log
coordinates.
and for
left tip
U = 2.39
V = 0.33
u=
v= 0.
1.5
These growth rates were used in conjunction with Isida’s estimate of K to calculate cycles of life. The results are given in Table 2. These results are clearly better than the results based solely on using Isida’s K levels, N,. As a result of the tests conducted in this study it is evident that using Brandt’s proposal for calculating fatigue life can produce results which are very conservative. For the three tests reported the results can be as much as 700% to 1000% too conservative. Also for the tests conducted, much better estimates of fatigue life were produced by ignoring plastic zone effects and just calculating da/dn based on AK levels for the internal defect. This produced results which overestimated the fatigue life by about 25-30%. Finally, a proposed modification of the growth rates based on A and RL provided the best prediction capability. However, actual use of such procedures should be based on more extensive testing. REFERENCES [l] [2] [3] [4] [5] [6]
[7]
D. E. Brandt, The development of a turbine wheel design criterion based upon fracture mechanics. ASME J. Engng Pwr93,p. 411 (1971). W. G. Gibbons, W. R. Andrews and G. A. Clarke, Fracture of defects approaching a free surface. Trans. ASME, 75. PVP-10 (1975). G. R. Irwin, Plastic zone near a crack and fracture toughness. Proc. 7th Sagamore Ordnance Materials Corzf. Syracuse University Research Institute (1960). E. E. Zwicky, unpublished work. M. Isida, Stress-intensity factor for the tension of eccentrically cracked strip. J. uppl. Mech. 33,674 (1966). R. S. Vecchio, D. A. Jablonski, B. H. Lee, R. W. Hertzberg, C. N. Newton, R. Roberts, G. Chen and G. Connelly, Development of an automated fatigue crack propagation test system. Automated Test Methods fr,r Fracture and Fatigue Crack Growth, ASTM STP 877, 44-66 (1985). Single-edge-crackedplate under tension. ASTM STP 410 (1969). (Received 3 January 1986)