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Role of plastic work in fatigue crack propagation in metals
Ealmte,,mf Fracttme M~/umlcs Vol ! I, L~. 791.404 Perllamou Prem Lid, 1979. Printed m Great BriUda
Q0131--WI-.0~IPJ~
R O L E O F P L A S T I C W O R K IN F A T I G U E C R A C K P R O P A G A T I O N IN M E T A L S t Y. IZUMI~ and M. E. FINE Department of Materials Science and Engineering and Materials Research Center, Northwestern Univarsity, Evanston, IL 60201, U.S.A. Almract--The plastic work required for a unit area of fatigue crack propagation U was measured by cementing tiny foil strain gages ahead of propagating fati~le cracks and recording the stress-strain curves as the crack approached. Measurements of U and plastic zone size in aluminumalloys 2024-T4,2219-T861, 2219 oversged, and AI-6.3 wt~ Cu--T4,and a binary Hi-base alloy with 7.2wt% AI are herein reported. The results are discussed along with previously reported measurements of U in three steels and 7050 aluminum alloy. When U is compared to the fatigue crack propngation rate at constant h ~ along with strength and modulus, the conclusion is drawn that U is one of the parameters which determines the rate of fatigue crack propagation. The relation of U to microstructure is also discussed. "Homolgeneous"plastic deformation in the plastic zone ahead of the crack seems desirable.
1. INTRODUCTION RECENTLYTHEintegrated plastic work expended in the plastic zone around a fatigue crack for a unit area of crack advance, U, was measured by cementing tiny foil strain 8ages ahead of fatigue cracks in center notched panel specimens[l]. Results for a high strength low alloy Nb steel and 7050 A1 alloy showed that the rate of fatigue crack propagation, IdcldN[~, depended approximately on 1/U. The relationship was further established by study of additional AI alloys and steels [2]. Recently, Davidson and Lankford [3] obtained the value of U in 0.05 wtg~ C steel by measuring the variation in subcell size vs distance from a fatigue crack tip using electron channeling. Using data of hysteresis loops vs cell size in uncracked iron, they integrated the hysteretic plastic work over the plastic zone. The U values obtained for San Antonio laboratory air and dry nitrogen correlated inversely with the smaller [dc/dN[aK for dry nitrogen. The U value determined for the Nb-doped 0.06 wt~ C steel in dry argon using the strain gage technique[l] was about 70~ smaller than U determined for the 0.05wt~ C steel using the electron channeling technique. The former steel, of course, has a higher yield strength and contains NbC precipitates. Theoretically, many equations for the fatigue crack propagation rate predict a Idc/dN[aK 1/U relationship. Weertman[4], Mura and Lin[5], Rice[6] and Cherepanov and Halmanov[7] derived simple functional relationships for the fatigue crack propagation rate of the form dc
(AK) 4
where A is a dimensionless constant, p is the shear modulus and o- is an appropriate strength parameter. Theoretical or empirical equations [2, 5] which predict [dc/dN[ ~ AK 2 also show an inverse dependence on U. In a previous paper[2], the authors proposed dc
(AK) 2 U/2y)
=8
(2)
where B is a dimensionless constant, E is Young's modulus, and 7 is the true surface energy. Much better agreement with experimental data[2], i.e. the dimensionless term was more nearly constant, was obtained with eqn (2) than with other (AK)2 type equations which did not contain U, such as dc/dN = B'(AK)Z/o'yE and dc/dN = B'(AK)Z/E ". With the set of data previously presented, the standard deviations for B' and B" were larger than their ave.rage values. tThis research was supported by the Air Force Once of ScientificResearch, O k e of Aerospace Research, Grant No. AF-AFOSR-76-2892A. ~Present address: Onada Cement Company, Tokyo, Japan. 791
792
Y IZUMI and M. E. FINE
Measurements of U for additional alloys using the foil strain gage technique are herein reported and sources of error in the measurement are examined. The results for U to date are discussed in terms of various equations which have been proposed for fatigue crack propagation vs AK. 2. EXPERIMENTAL PROC~JRES The chemical compositions of the alloys studied are given in Table 1. The commercial aluminum alloys, 7050, 2024 and 2219, were supplied by Alcoa Research Laboratory. The Al-6.3 w/o Cu alloy, which was studied previously in this laboratory[8], was provided by Kaiser Aluminum and Chemical Corp. The Ni-7.2 wlo AI alloy was prepared by International Nickel Company, Inc. Standard center notched panel specimens, 100 nun long, 20 mm wide, and 2.3-3.0 mm thick, were employed. The long direction of the specimens were cut parallel to the rolling direction. The surfaces were polished with 6 ~m diamond paste for the AI base alloys and 600 grit SiC for the Ni-AI alloy. A center notch approximately 3 mm long and 0.2 mm wide was introduced by electro-discharge machining using a flattened Cu wire tool. The heat treatments for the specimens are listed in Table 2. The plastic work associated with fatigue crack propagation was measured using the procedure previously described [1]. Tiny foil strain gages (Micro Measurements MA-13-008CL120 for the AI alloys and MA-06-008CL-120 for the Ni alloy), 210 ~m wide and 200 ~m long in the gage section were cemented with epoxy on the notched specimens astride and above the extended crack plane trace. The strain gages were normally mounted to measure only local strain in (Y) the tensile direction; however, local strains perpendicular to the tensile direction (defined as the X direction) were also measured in the 2024 Al alloy. The strain indicator used was readable to 10-4%. The fatigue crack propagation rates were first determined vs AK at 30 Hz using a 40X telemicroscope readable to 2 ~m. For all of the measurements the R-ratio was kept constant at 0.05 (pull-pull). The plastic work U was measured at constant AK in the range 7-16 MN/m 312. Table 1 Chemmal analysis of alloys studied Si
Fe
Cu
M~
C.r
NI
Z~
X__~_l
Z__~_r
7050 A1
01
.01
2.25
0
2.36
0
--
6 20
02
Ii
2024 A1
.14
.28
4.52
.59
1.41
.01
.00
.23
.04
2219 A1
.09
.23
6.19
.31
.000
.00
.0
.02
05
6.34
.000
000
.000
--
A1-6.3 w/0 Cu
004
009
N~-7 2 w/o Al
Chemlcal
~
006
V
--
Be
--
000
.09
000
14
0006
AI
C
P
Si
Fe
S
0
7 16
.001
0005
.013
.041
.001
0037
analyses were provided by the suppllers.
Table 2. Heat treatment of alloys Specimen
Condition T76
Solution treated 1/2 hr at 482JC, aged 4 days at rm 3 hrs at 121°C and 9 hrs at 163°C
T4
Solutlon
treated
i hr at 500°C and aged 7 days at rm
temp
2024 A1
T4
Solution
treated
1 hr at 495~C and aged 7 days at rm
temp,
2219 A1
T861
Solution treated l hr at 535~C, aged 7 days at rm. terap , elongated 27 at strain rate of 8.3 ~ lO-~ sec -x and aged 18 hrs at 177°C
overaged
Same as T861,
T4
Solution treated rm temp
1 hr at 510CC followed by aging 7 days at
Solutlon treated 625°C ~n argon
I hr at 900°C in argon and aged 2 days at
7050 A1
A1 - 6 3 w/o Cu
NI - 7 2 w/o A1
except the aging temperature
temp.,
was 202°C
Role of plastic work in fatigue crack propagation in metals
793
To keep AK constant, the maximum and minimum loads were decreased when the crack length increased 0.1--0.2 ram. During the tests, the frequency was periodically decreased from 30 to 0.3 Hz in order to graphically record the nominal stress vs strain gage output. All measurements were done in dried argon. The procedures for reducing the data to give U are given in the previous papers[l]. The minimum plastic strain at zero load was also measured in order to study the non-hysteretic plastic energy. For determining the cyclic stress-strain curve vs strain amplitude, which is required for the determination of U, a cfipon extensometer of 7.5 mm gage length was fastened to unnotched specimens and the strain amplitude was incrementally increased to -+1.25%, decreased, increased, etc. until a stable set of hysteresis loops were obtained. 3. EXPERIMENTAL RESULTS AND REDUCTION OF DATA Figures l(a) and (b) g i v e s o m e typical nominal stress-local strain curves for the 2024-T4 and 2219-T861. X is defined as the distance from the center of the gage to the crack tip along the horizontal axis; Y is the distance from the center of the gage above or below the crack plane. Non-closing curves (A and G) are due to permanent strain and were obtained only during the first cycle. The permanent strains during the first loading, elp, were 0.02%--0.03% for the AI alloys but only 0.005% for the Ni-AI alloy at AK = 15.5 MN/m 3/2. The rate of increase of permanent plastic strain, deP/dN, for all alloys studied was less than 104% per cycle after the first cycle and thus the loops appear closed on the scale used in the plots. The observation of a "closed" loop means that a residual compressive stress must be present at the minimum nominal stress. The cyclic stress-strain hysteresis loop obtained with an unnotched specimen of AI-6.3% Cu are shown in Fig. 2. Similar loops were obtained for the other alloys. A hysteresis loop for AK of 15.5 MN/m 3/2 and small X and Y is also shown. As previously[l], the local stress in the specimen was determined approximately from the cyclic stress-strain curves of unnotched specimens by equating the strains. The mean value of strain amplitude in the lower curve was shifted to coincide with zero strain of the upper curve, as shown in the figure, to take account of the residual compressive stresses near the crack tip. The exact amount of this shift in the origin has only a small effect on the amount of hysteretic work per cycle determined from the areas in the loops, the quantity of interest, since the slopes of the fines are parallel except at the large strain tips of the loops. 2 0 2 4 AI Z~K=15 5 M N / r n~2 Y : 150,u.m
E" E Z
:E
A
B
2OO
2 2 1 9 - T 861 AI A K : 15 5 M N / m 3~= Y= 320,u.m
c
LLI n." I'U3
D
E
F
4-"
E Z
G H 100
:E Z
u~
400 o
//
IZ co
z
Z
d2
d4
0'8
lb
4'2
STRAIN (%) notch crack
IIm.~foll gage
A B CDEF
(a)
E O Z
,'50
/
J
n ~ f O I I
/
GH
o2
o~
gage
I JK
o'~
o'8
STRAIN (%)
(b)
Fig. I. Typical nominal stress-local strain curves vs distance from crack tip determined using foil strain gages. The distances A - F and O-K are approximately 2 mum. Y is the distance from the gage center to the crack line in the vertical direction; (a) 2024; (b) 2219.
794
Y. IZUMI and M. E. FINE
-o~,
-o'2
6
o~
o'4
STRAIN C%)
Fq~ 2. Cyclic stnam-strain hysteresis loops obtained on an unnotched A1-6.3% Cu specimen. A hysteresis loop obama~ with a foil strain ~ on a specimen with a fatigue crack is also shown. This figure shows how the local stresses were determined in the present study.
The total plastic work to extend the fatigue crack surface by a unit area U is made up of two parts, hysteretic, Uh, and non-hysteretic U,. Since AK is kept constant during the measurement, the plastic zone size is essentially constant during the crack propagation. When most of the gage is out of the plastic zone the loop width is very small, as shown by the B and H curves in Fig. 1. The loops widen as the plastic zone moves into the gage area and are widest when the crack tip is closest to the gage (F). When the crack tip reaches or enters the gage, the gage fails. The hysteretic plastic work expended during extension of a fatigue crack by a unit area is obtained by equating it to the summation of the areas of the loops over the plastic zone (converted to local stress using the cyclic stress-sirain curves as described) divided by dc/dN. The hysteretic l~stic work per unit area for a belt (a unit length wide) parallel to the X axis and centered on Y, Uy, is given by the equation Ur = (S~)A + . . . . + (SXr)F = ~..( ).$xr., dc dN
(3)
where (Sxy)A is the area of the loop at A in Fig. l(a). When the increments in X are small, eqn (2) can be rewritten Uy = f Uxy dX.
Thus the total hysteretic plastic work, Uh, is simply
(4)
Role of plastic work in fatigue crack propagation in metals
ill
AKI -=6124 3 Cu A MN/m 3/a
NI - 72AI ZIK= 15 5MN/m 312
2/~1/-
E
F -)
Y: 190/~m
/
2
%
~K D
1
Y=12O~m
r,A / - Y = 5 ,'.'.'.'.O '.'.'.'.'~ m
D
,~ / - Y = 2 5 0 ~ . m
1
Y= 300#.rn ~. BG
,.o./-- Y = 23OF m
~'~.~',~S¥:4,~r~ ~ ll,~,c).. O
795
-
BG 1
2
0 X, DISTANCE FROM CRACK TIP(n7m)
X, DISTANCE FROM CRACK TIP(mm)
(b)
(a)
Fig. 3. Typical local hysteretic plastic work profiles along X for several values of Y. B.G. indicates background plastic work; (a) AI-6.3% Cu, A K = 12.4 MN/m3/2; (b) Ni-7.2% AI, A K = 15 5 MN/m 3/2.
It will be shown that the non-hysteretic plastic work per unit area of fatigue crack propagation is very small for the present measurements. Thus Uh - U. Examples of Uxy at constant Y plotted vs X are shown in Figs. 3(a) and (b). The Y values showware the nominal values. Since the crack did not move in a perfectly straight line, the actual Y values (these were used in the calculation) deviated from the nominal values. Some of the curves in Fig. 3 show tails indicating background plastic work, as observed previously[l]. Such backgound plastic work appears to result from experimental problems such as faulty bonding and was subtracted from the plastic work for determining Uy. One problem in the measurement of U is to determine the large value of plastic work near the crack tip, where the gages failed. As described and discussed in the previous paper[l], the hysteretic plastic work closer than 100/~m to the crack was estimated by extrapolating the strain amplitudes determined by the foil strain gage on a linear log-log plot to 1/~m. The plastic work per unit area so determined for closer than 100 ~tm to the crack tip is designated Uc. Typical extrapolations are shown in Fig. 4. Although U in eqn (4) was previously calculated using the local strain in the Y direction only, the plastic work due to other strain components must also be considered. The plastic work due to local strain in the X direction was measured for the 2024-T4 A1 alloy and found to be only 10% of the total U.
AI-63Cu
lo' bJ t7 D P
.
~"."
A
D
_ _ ( A 12.4
/~KtMN/m"'} l B 10.8 C93 2024 - T4 AK(MN/m~{ ED15785
<( _Z 16 2
x. DISTANCE FROM CRACK TIP (~.m)
Fig. 4. Strain amplitude measured with foil strain gase vs X extrapolated to X = 1 ltm assuming a linear relation on a log-log scale
796
Y. IZUMIand M. E. FINE ~I
"o
2024 - T4 AI o 15 5 MN/m 'JIz AK = o 78
10
o
AI - 6 3 Cu 'N
• 124 MN/rn :~'2 AK = .6. 1 0 8 o 93
8
~E "o x 0 20O
4OO
\
o \
5 4
1
, 8O0
6OO
\
7 6
0
\
9
1000 3
DISTANCE FROM C R A C K
LINE
Y
2
[,u.m]
(a)
t 2 ~30
0
' 400
' 600
800
1000
DISTANCE FROM CRACK LINE Y [ ~ m ]
(b)
Fig. 5. Localhyttereticplasticworkdueto err in a beltof unitwidthalongX (Uy) vs ¥ for 2024-T4. Uy for X of Y < 100 #m him beea mtlected ia ~
plots. Similar curves were obtained for all of the alloys,
treatments,andAg's measuredin this investigation.
The-hystet~k plastic work per unit area for further than 100 pm from the crack front, Uh was dotlgmiMd by plotthlg /Jr (eqn 4) vs Y and then measuring the area under the curve. Examples of such plots are shown in Figs. 5(a) and (b). The total hysteretic plastic work per unit area of fatigue crack ~ , Uh = Uc + UI, for the alloys and AK values of the present research are Oven in Table 3. The contribution to the plastic work from closer than 100 ~m to the crack front Uc varied from 5 to 40% of the total U. Since the minimum strain gradually increased with each cycle, i.e. the loops were not exactly closed, the non-hysterefic plastic work must be considered. The 6.3% Cu-A1 alloy was expected to have the h q m t non-hysteretic contribution (U,) of all the alloys studied. U, was detmrmia~ in ~ alloy by am analysis simitat to Uk and found to be approximately 2 x 103 J/m2 for AK = 12.4161Iram. This is a factor of 100 smaller ~ Uk. ThUS in the determination of U, U, may be nq0eeted. Plots of U,r (defined similarly to/Jr) vs Y for the three values of AK investigated are shown in Fig. 6. From the scale pierre_.__,it is obvious that the areas under the U, curves are much smaller than the corresponding values of Us, Fig. 5Co). In order to determine possible eR~,ts of specimen thickness on the U values, the fatigue cmzk I ~ rates were measun~ vs &K for specimens of 2219 which were 1, 3, and 6 mm thick. The results (shown in Fig. 7) illustrate a slower rate in the high range of AK for the 1 nun Table3. Hystereticplasticworkper unitareaof fatiguecrackextension AK
, ~J~ A16.3 w/o
Cu
2024 T4
2219 T861
2219 ~ e r a g ~
Ni7.2
wloAI
Uf
Uc
Ub
JJ¢
J/~
,T/~~
12.4
4.8xi0 b
l.OXl0s
5 8XlO s
10.8
5.7
0.4
6.1
9.3
I0.I
0.4
10.5
15.5
3.0
1.8
4.8
7.8
2.0
0.6
2.6
15.5
1.5
0.1
1.6
7.8
1.8
0.6
24
15.5
1.1 1.7
0.3
1.4
9.3
0.4
2.1
15.5
4.1
0.7
4.8
Role of plastic work in fatigue crack propagation in metals
w
AI - 6.3 Cu
~5
/xK=
z
mint/
E ~ 20
• 124 M N / m 3/z 108
30
40
50
coo
70 80
OZ
u 93
A
1 mm
-t
B 3 mm |
~3
c
x
~2
6n'rn
)
T861
ED 63mmmmI over oged
~
C
1 o o
i
2o0
4oo
6oo
8oo
10oo
DISTANCE FROM CRACK LINE Y[/~m]
16'
Fig. 6. DE
ld"
i
t
i
7
8
9 10
i
I
20 ~K /MN~ Fig. 7.
Fig. 6. Local non-hysteretic plastic work due to err in a belt of unit width along X(U.r) vs Y for AI-6.3% Cu. U.r when X or Y< 100/tin has been neglected. Fig. 7. Fatigue crack propagation rate vs AK of 2219 AI alloy for specimens of three different thicknesses.
thick specimen than for the 3 and 6 mm thick specimens. This is no doubt due to the fact that such thin specimens are more nearly in plain stress than the thicker specimens. Since the rate, Idc/dNlaK, at AK of 15.5 MN/m3~ for 3 and 6 mm thickness specimens was 1.3 times larger than that for 1 mm thick specimens, one expects only a 30% increase in U at this AK on reducing the specimen thickness to I mm. As an additional experiment to study the effect of specimen thickness, foil strain gages were cemented 200 and 550/zm from the crack tip on a 2024-T4 Al specimen 2.3 mm thick and hysteresis loops were measured under load control at AK of 12.4 MN/m 3~ and 7.8 MN/m 3~. The reverse side of the specimen was then carefully removed with a milling cutter to a thickness of 0.7 mm using sufficient oil to prevent the specimen from overheating. The hysteresis loops were remeasured. There was little change in the strain amplitudes and the loop widths between the two thicknesses at both AK values. One value appeared anamalous. This was attributed to gage damage during machining. The conclusion is that specimen thickness over the range of thickness studied has little affect on U except perhaps at the highest AK values. During preparation of the specimens, appreciable residual stresses may be introduced on and near the surface. The effects of such residual stresses were evaluated by cementing gages on notched and unnotched specimens and then shifting the lower grip by melting the Woods metal pot. Hysteresis loops were obtained on an mmotched specimen of 2024-T4 at a stress amplitude of ± 450 MN/m ~. No difference in loop width on changing the prestrain from -0.2 to 0.1 was observed. There was only a few percent change in strain amplitude. In a notched specimen of 7050-T76 with a fatigue crack and with strain gages cemented 100/zm ahead of the fatigue crack, there was little variation of the strain amplitude and the loop width on changing the prestrain from -0.03% to +0.03% (AK of 12.4 MN/m~). There was also little effect on the loop widths and strain amplitudes at AK of 6.2 MN/m3~ from overloading at 12.4 MN/m 3~. DISCUSSION OF F.~IIOMS IN THE I ~ [ f l ~ 4 l l N T OF U Neglected terms will first be discussed. In the present research, U was equated only to the hysterefic plastic work in the Y direction. As already mentioned for the present set of measurements, the hysteretic plastic work in the X direction was found to be only 10% of that EFgv~. li, go
798
Y. IZUMI and M. E. FINE
for the Y direction and the non-hysteretic plastic work for X or Y < 100 # m was found to be negligibly small. The effect of specimen thickness was discussed in the previous section. With respect to the effect of residual stress, prestrain of -0.2 to 0.1% changed the area of hysteresis loops to result in an estimated change of 20 to -10% in U. Residual stresses might arise from specimen preparation or in mounting of the specimen. The initial residual stresses are rapidly removed by cyclic loading and replaced by a macrostress profile characteristic of the local cyclic loading conditions[9, 10]. Since 104 cycles were usually required for crack initiation, initial residual effects are expected to be very small. Even if we neglect the effect of cycling on the residual stress, a prestrain of 0.2% which corresponds to a stress of 150 MN/m 2 for the 2024--T4 Al alloy would change the U value by only 20%. Negative values of the permanent plastic strain during the first load, which were occasionally observed, indicated some bending of the specimens. Such prestrains were less than _+0.03% and would affect the final values of U by no more than a few percent. Extrapolation of the local strain amplitude to closer than 100~zm from the crack tip, according to the technique described in the earlier paper[l] (Fig. 4), is the most uncertain part of the procedure used to determine U. Davidson and Lankford[ll] studied the plastic zone around the fatigue crack, using selected area electron channeling in 6061-'1'6 A! alloy and concluded that curves for strain amplitude vs distance from the crack tip on a log-log plot are not linear as assumed in this paper. Using their data for the strains vs X, the change in the total U calculated is only 10%. Kang and Liu[12] observed the strain amplitude around a fatigue crack from 100 ~m to I mm in the 2024 AI alloy using Moir6 fringes. Both sets of data for strain amplitude are approximately the same as those determined in the present study further than 100/,m from the crack tip. The electron channeling technique for measuring U developed by Davidson and Lankford[3] is more accurate than the foil strain gage technique close to the crack tip. The latter is more accurate for the contribution to U from > 100 ~m from the crack tip. The two techniques should be used on the same alloy to arrive at a more accurate value of U. The data in Table 3 shows U values for several different values of AK in the range 7.8-15.5 MN/m 3t2. The variation of the measured U with AK is less than 40% from the average value and may be within experimental error. A similar conclusion was reached for the Nb-doped 0.06 wt% C steel[I]. PLASTIC ZONE The foil strain gage technique also gives contour maps of the non-hysteretic and hysteretic plastic zones. The non-hysteretic plastic zone will be treated first. During cyclic loading, the permanent plastic strain measured at zero load ~ increased as the crack approaches the strain gage (decrease in X). Again the 6.3 wt% Cu-AI alloy was selected for study because it has the largest permanent strains of the alloys studied. The resuRs are shown in Fig. 8. The background ,P essentially occurred during the first cycle and may be associated with changes in the gage. In the lower AK experiment (Fig. 8a), new foil gages were cemented on the specimen after 8000 cycles. The total permanent strain recorded by both gages reached about 0.5% after 9000 cycles. At .the higher AK (Fig. 8b) rapid increases in permanent strains marked P were observed near the outer boundary of the plastic zone. Equi-permanent plastic strain lines around the fatigue crack tip were determined from the curves of ~P vs X for various Y's. These are shown in Figs. 9(a) and (b). The lines for Y larger than 600 tzm from the crack line were estimated by extrapolationand are indicated by dotted lines. Equi-hysteretic plastic work density lines around the fatigue crack tip for the various specimens were determined from the data of Uxy vs X for various Y's, examples of which are shown in Fig. 3. The hysteretic plastic work maps are shown in Figs. 10(a-d) for 2219-T861, 2219 overnged and Ni-7.3 wt% AI-Ni alloys, respectively. DISCUSSION
Although ~lcldN[Ag has been predicted to depend on l/~r or l/or2, Donahue et al. [ 13] presented some data to show that it was independent of o-. However, Burck and Weertman[14] utilizing the fact that the strength of iron and its alloys increase on cooling from room temperature to
799
Role of plastic work in ratine crack porpagafionin metals I
AI - 6.3 Cu /~K: 9 3 MN/m~'2
AI - 63Cu ~ K : 124 MNI m ~z
'\
10
;s
o°
z n..
""~ ~ ,,, H L ,,~/
1D
HL
P
" ~ '~~ K ~ " "~"" /
Z
ILl
:E .,.,,r.
O~
bJ Z
< 0.5 E cr
bJ n
LtJ O_
I
o
~
xY~o
~,~oo
X, DISTANCE FROM CRACK TiP (M,m) X,
(a)
DISTANCE FROM CRACK TIP (p.m)
(b)
Fig. 8. Permanent strain at zero stress ep for various values of T vs distance from crack tip (X) for AI-6.3~ Cu; (a) AE -- 9.3 MN/m3r~; (b) AK = 12.4 MN/m3r2. H.L. represents value of X where a hysteresis loop was first observed, p nutrks the value of X at A/( = 12.4 MN/m~ where a rapid increase in e P was observed. The baclr4pound e p occurred duri~ the first cycle and was due to changes in the gage, not the
material.
Y
( b ) Z~K = 12A M N / m 3'2
,,
.-:::;.:>;..:.--"
)'//;" ~oo
\p
[/J.m]
looo
i l
5oo
,
/
o
[p.~]
( o ) Z3K ,, g:3 M N / r n 3~2
IOOO
x
1 oo
o
[~.m] Fig. 9. Equi-permanontstrain (t~) ~ for AI-6.3~ Co alloy vs X and Y. Dotted lines are extrapolated. Numbers are in %. Points designated p correspond to those in Fig. 8. 770K showed a correlation of Idc/dNlax with l/
Y. IZUMI and M. E. FINE
800
Y(p.m) ((3) 2219 T861
~O00
(C) AI- 63Cu AK= 124 MN/m :112
Y (/~m)
1000
&K=78 MN / rr~ 2 4
,1 2 4 6 ~ ' X (/~rn)
I£~
500
: CRACK
(ffm) 2219 over aged AK= 93 MNI m31z
o
MNIm
Y (fire) 312
1000
~
1000
500
,1 2 4 6 X O~J.m) 1(~00
50o
(d) N, - 7 2 AI Z~K= 155
(b)
CRACK
i
X (/zm) 1(:;oo
0
500
500
500
CRACK 0
2, X(~m)
1(~00
4 6810
CRACK
500
Fig. 10. Contour maps of hysteretic plastic work Uxr vs X and Y for; (a) 2219-T861, ~K = 7.8 MN/m3/2; (b) 22219-overaged; (c) AI-6.3%Cu; and (d) Ni-7.2~ AI. The numbers are in units of 10'1J/m4. Table 4. Comparisonof products Idc/dN[t55o'2/~ for the five aluminumalloys and Nb-doped HSLA steel ldc/dN [15.5°~ ~
iT) 2 0 . 3 x 103
7050 T76 2024 T4
5.5
2219 T861
8.9
2219 overaged
5.6
Nl - 7 2 w/o A1
7.5
Nb-doped HSLA steel (0.6 w/o C)
1.5 3
Idc/dNlls.5 = fatxgue crack propagation rate at AK=I5.5 ~ / m 2 = 0.2% offset cyclic yield stress = shear modulus
proximately constant for all materials. Comparison of the AI alloys[8] and the Cu alloys[B] previously referred to shows that this is not observed. Table 4 compares the product Idc/dNll55 ~r2/~ for the five alloys studied in this research and the HSLA steel studied previously[16]. While the values for 2024-T4 and 2219 overased are nearly the same, this product is approximately three times larger for the 7050 AI alloy. For the Nb-doped HSLA steel, it is more than three times smaller than for 2025-T4. The variation of this product may be understood by introducing U as a variable. Decrease in U along with increase in o- also explains the observation of Donabue et aL The data for U of the present study and the previous studies in this laboratory are summarized in Table 5 along with the fatigue crack propalaXion rates, cyclic and moaotonic 0.2% offset yield stresses, the exponent m in the Paris eqeation, and the constant A in eqn (1). The fatigue crack propagation rates, IdcldN[ax, for the 2024-T4 and 6.3 w/o Cu-AI alloys in Table 4 are considerably faster than the rates for the Nb-doped HSLA steel; the cyclic yield stress, cry, for the HSLA steel is between those for the two alloys. These differences in IdcldNlAK cannot be explained by a difference in the shear modulus or or;. Introduction of the
Roleof plasticworkin fatiguecrackpropagationin metals
801
plastic work U rectifies this difference. The Cu-C steels[16] and the Ni-base alloy show a similar correlation between [dcldNIAg and the measured U. The value of U for the 2024 alloy is the highest of those for the commercial Ai alloys studied while its Idc/dNIAK is the lowest. The large differences in IdcldNhs.5 between the 7050-T76 AI alloy on the one hand and the Ni-base alloy and HSLA steel on the other correlate with the large differences in U. In view of these results, U is obviously playing a determinative role for the rate of fatigue crack propagation. Since the plastic work during cyclic loading results from the to and fro motion of dislocations, it is reasonable to consider that U, as well as the alloy's strength, c~, are implicitly related to the alloy's microstructure. The A1-6.3wt% Cu alloy aged at room temperature contains both GPI (Guiner-Preston zones) and the 0 phase[8]. The excess copper over the stability limit forms 0 particles which are 5-10/zm in diameter with a mean spacing of about 50 gm. The presence of GP zones tends to give inhomogeneous deformation with coarse planar slip bands. The presence of the large 0 precipitates, however, make the plastic deformation more uniform throughout the specimen, since the 0 particles are not penetrable by dislocations and thus dislocations must bend or cross slip around the particles. This results in a higher work hardening rate. We propose that this is the origin of the higher U for this alloy. Possibly because few dispersed particles of the required size are present in the 7050 AI alloy, the value for U is much smaller than that in the 6.3 wt% Cu-AI alloy. Unlike the binary A1 alloy, the 2024 AI alloy has S phase and the AbCu2Fe constitutent particles. Comparing the 6.3 wt% Cu and 2024 AI alloys, the cyclic yield stress of the latter is considerably larger, but its U is a factor 2 smaller. This may result from a non-uniform distribution of constituent particles. The value of U for 2024-T4 is twice that for 7050-T4. The GPI zones for the 7050-T4 alloy are spherical while those in 2024-T4 A! are platelets. The dispersoid precipitate in the 7050 alloy contains Zr while those in the A1 alloy contain Mn. The 7050 also contains fewer dispersoids. The higher U in the 2024-T4 alloy may possibly result from a greater difficulty for dislocations to avoid cutting platelets than spheres even though the spherical zones are a stronger barrier to dislocation motion (the strength for the 7050-T4 Ai alloy is larger). The 2219 AI alloy has high strength at elevated temperatures. The inclusions in this alloy[17] are similar to those in the 2024 alloy with 0 replacing the S-phase. The T861 and overaged (202°C for 18 hr) treatment give O', the particle size for the former being smaller[18]. Increasing O' size seems to have little effect on U even though it decreases the strength, Thus the observed decrease in IdcldNhs.s on overaging is related only to a decrease in strength. The large increase (50%) of c~ over ~ in the Ni-AI alloy is worthy of comment (Table 5). The aging treatment used gives aligned cube-shaped y' precipitates roughly 100 ~ on a side[19]. The flow stress increase due to cycling may be due to the disordering and creation of antiphase boundaries in the y' particles from activity on two intersecting slip systems, one for tension, one for compression. As stated in the Introduction, theoretical equations for the fatigue crack propagation rate may be classified into mainly two types: (a) IdcldN[Ag ~ (AK)4, Refs. [4-7] and (b) [dcldNIAK (AK) 2, Refs.[20-23]. Equation (2) followed from the suggestion of Donahue et al.[13] that IdcldNIAK is related to ductility which in turn is related to o~E. Thus introducing o/E into the equation dcldN = C(AK)21Ecr, which had been derived by several authors [20-23], removed the dependence on ~. Instead the present authors[2] introduced U as a measure of ductility and y to make the term dimensionless; 2y/(U+y) represents a "ductility factor". It is one for non-ductile materials. In metals, of course, U >>y so 2~//U is a good approximation giving from eqn (2). Experimentally, the exponent on AK, m, ranges from 2 to 8. For data for fatigue crack propagation compiled by Frost et a!.[24], the average value for 14 alloys is 3.94 with a standard deviation of __.0.86. With the 8 alloys and treatments studied in this laboratory (Table 5) the m values ranged from 3 to 4.5 giving m = 3.7__.0.5. Thus, eqn (1) seems to be the best theoretical equation for the present set of data. Table 5 gives the proportionality constants A for the 8 alloys and AK's studied assuming that o,~ the 0.2% offset cyclic yield stress, is the appropriate strength parameter. The average value
Y. IZUMI and M. E. FINE
802
Table 5. U values and mechanical properties U
Oy
~/.~
./crcza
j/~
~1,~
~/,~
,,
A
7050 AI T76"
15.5
30~I0 -s
0.63XI05
510
470
4.0
2.3XI0 -z
7050 AI T4*
12.4
30
0.54
410
320
3.5
2.8
AI-6.3 w/o Cu T4
12.4 10.8 9.3
4.7 2.0 1.2
5.8 6 1 10.5
230
130
4.0
1.6 1.2 2.3
2024 AI T4
15.5 7.8
14 1.6
3.2 2.6
390
300
3.0
3 1 4.4
2219 AI T861
15 5 7.8
25 1.5
1.6 2.4
370
350
4.0
2.5 3.5
2219 AI overaged
15.5 9.3
32 6.8
1.4 2.1
260
270
4.0
1.4 3.4
Nx-7.2 w/o AI Aged 2 days, 625°C
15.5
2.0
4.8
670
440
4 5
6.3
Nb-doped HSLA steel*
19.5 15.5 12.4 93
3.2xi0 -s 1.7 0.8 0.3
12 8 12 6
3~0
360
3 5
2.7 2.1 3.5 2 1
19.5
1.8
3.4
710
880
3.5
1.6
Steel - Aged 200 min, 19.5 500°C **
5.6
1.3
780
950
3.0
2.3
3
Steel - Aged 13 min, 500°C **
dc/dN
,
Ikeda~ Izumi and Fine (1) ~-~ Ikeda, Sakal and Fine (Is)
dc/dN = fatigue crack propagation rate = total hysterettc plastic work per unlt area 0.2% o f f s e t c y c h c yield stress Y o = 0.2v offset monoton£c y£eld stress m = Parls exponent, i.e., dc/dN = C(Ak) m A = constant in Eq (I) Speclmen thlcknesses were 2 3 to 3 w Dry argon atmosphere
of A is 2.7 × 10 -3 with a standard deviation of --+-1.2 x 10-3. /~ for all steels was taken to be 7.8 × 104 MNIm2; for all Al alloys, 2.6 × 104 MN/m 2, and for the Ni alloy, 8.4 × 104 MN/m 2. The strength parameter in eqn (1) needs further discussion. A cyclic mechanical property would seem to be more appropriate than a monotonic mechanical property since the material at the crack tip has been "conditioned" by the stress--strain cycling. Weertman [4] used the critical stress for fracture of the material at the crack tip, oro which is expected to be somewhat larger than or; or or. The actual flow stress of the material at the crack tip is, of course, not known. In the present study, only or; and or, for 0.2% offset, were considered to compute A. Slightly better statistics were obtained when or; was used rather than or; A ffi 2.73 (1 +_0.44) compared to 2.26 (1 +_0.47). More measurements are needed, of course, to establish the ranges of validity of the equation and the significance of the value of A. Kikukawa et all25] showed, following the suggestion of Elber[26], that if AK+~ (AK = K,ffi,-K~,m) is used in the Paris equation rather than AK, the exponent is near 2. Substituting AKd into eqn (2) is thus suggested. Following Kikukawa et aL, the closure stresses may be estimated from where the stress-strain curve bends on unloading, Fig. 3. The present measurements of U were made at constant AK with the load an~litude reduced to compensate for the crack growth. The values of AKd were 65-85% of AK and tended to decrease as the load was decreased. Substituting the data in Table 5 into eqn (2) and taking ~ = 2 J/m 2, the "constant" B showed only slightly larger relative standard deviation than A in eqn (1). Thus U, the plastic work of fatigue crack propagation, appears to be an important parameter for determining Ida/dNt~K whether the Paris exponent is 2 or 4. The plastic zones will next be discussed. The shapes of one-half of the permanent and hysteretic plastic zones are shown in Figs. 9 and 10, respectively. The half of the plastic zone below the crack is a mirror image of that above. The plastic zones are "butterfly-wing" shaped in keeping with their shapes being determined by the shear stresses. The definition of the size of the plastic zone requires discussion. Often gross plastic deformation only is considered (0.2% plastic strain, for example) and lesser amounts of plastic strain are neglected, that is, the 0.2% offset yield stress is often introduced into theoretical
Role of plasticworkin fatiguecrack propagationin metals
803
'and empirical equations for the size of the plastic zone. Since setting Ux,,, = 0 corresponds to a point where a hysteresis loop first appears on increasing the stress amplitude, the stress, crh, is defined as the maximum stress corresponding to Uxy = 0. This was determined simply by extrapolating the hysteretic plastic strain to zero using the cyclic stress--strain curve obtained with an unnotched specimen; o-h is the minimum stress to give to and from dislocation motion and it defines the boundary of the hysteretic plastic zone about a fatigue crack. The distance 2rh~'~ was defined as the maximum distance from the crack tip to the ~'h boundary. This was established by extrapolation of Uxy vs X, Y data along an appropriately selected line and rhm~ was fit to the equation r h~ = Ch(AK/o.h)2.
(6)
Values of 2rr, h x,
(7)
r m , = C'(A K/o"y) 2
and the results are also given in Table 6. Table6. Plasticzone sizes 2rhmax
2rP.
2rnmax
oh
p.m
cI
0.22 X 10"I
300
0.16
150
0.39 x i0-l
280
0.15
780 1050
I00 I00
0.45 x 10-I 0.34 x 10-I
630 820
0.19 0.14
7.8 15.5
680 910
150 150
1.2 x 10-I 0.43 x 10-I
170 650
0.15 0.21
2219-T861
7.8 15.5
630 1080
Ii0 110
0.63 x 10-I 0 . 2 7 X 10- 1
120 460
0.14 0.13
2219-0A
9.3 15.5
860 ii00
ii0 ii0
0.60 X 10-I 0.28 x 10-I
520 630
0.20 0.09
N1-7.2% A1
15.5
1280
190
0.96 X i0-l
160
0.15
~lm ~
~
ss/~
7050-T76
15.5
630
130
7050-T4
12.4
530
AI-6 3% Cu
9.3 12.4
2024-T4
ch
~m
II00 1800
C h ffi (5.25 • 3.1) x i0-a C i - (1.48 ± 0.55) x 10-I
A permanent strain plastic zone size may also be defined and determined from the data of e p vs X and Y for the AI-6.3% Cu alloy, Fig. 8. In this case, 2rPm~was defined as e p = background which essentially occurs during the first cycle and is due to changes in the gage. The permanent strain plastic zone size may be regarded as equal to the static plastic zone size. Theoretically, Mura and Lin[5] and Rice[6] derived that the constant in the equation relating the cyclic plastic zone to AK and yield stress is w/64 or 0.05 which is very close to C h. C , which corresponds to the 0.2% offset cycfic yield stress, is about three times larger. Rice suggested that the static plastic zone size is 4 times the cycfic zone size. Interestingly, rP~ffi,is only two times larger than rs~, and only 1.5 times larger than rhm~.Comparing the various alloys, the Ni-7.2% AI alloy has a surprising large r~m~but small r ~ in keeping with the large value of ~r~,, Table 5. CONCLUSIONS When all of the data for U, the plastic work per unit area of fatigue crack propagation is considered, the inescapable conclusion is reached that it is an important parameter which determines the rate of fatigue crack propagation irrespective of the observed exponent in the
804
Y. IZUMI and M. E. FINE
Paris equation. Enough measurements of U are now available so that a discussion of how it is related to the microstructure and plastic deformation modes can be begun. "Homogeneous" plastic deformation in the plastic zone ahead of the crack is desirable for resistance to fatigue crack propagation. Means of homogenizing the plastic deformation without introducing other deleterious effects (such as easy crack initiaition at dispersed phases) need to be developed. The measurement also yeilds a contour map of the plastic zone. Three plastic zone sizes are given: rma~,determined by the extent of permanent plastic deformation; r ~ , determined by the extent of cyclic plastic deformation bounded by the 0.2% cyclic yield strength; and r ~h , determined by the occurrence of hysteresis in the stress-strain curves. The coefficient for the latter in the equation r ~ = Ch(AK/trh) 2 is close to ~r/64 which is given by theory for the cyclic plastic zone. AcknowledgementwThe use of the Central Faciliues of Northwestern Umversity's Materials Research Center, supported under the NSF-MRL program grant DMR76-80847,facilitated this work.
REFERENCES [I] S. lkeda, Y. lzumi and M. E. Fine, Plastic work during fatigue crack propagation in a htgh strength low alloy steel and in 7050 Al-alloy. Engng Fracture Mech. 9, 123 (1977). [2] M. E. Fine and Y. Izumi, Role of ductility and yield stress in fatigue crack propagation in metals. Proc. 4th Int Conf. Strength of Metals and Alloys, Vol. II, p. 468, Nancy, France, Angust-Scptembcr (1976). [3] D. L. 1)avidson and J. Lankford, Jr., Environment sensitive fracture of engineering materials. TMS-AIME Proc. Symp. Chicago, Illinois 24-26 October (1977). [4] J. Wccrtman, Theory of fatigue crack growth based on a BCS crack theory with work hardening. Int. J. Fracture Mech. 9, 125 (1973). [5] T. Mura and C. T. Lin, Theory of fatigue crack for work hardening materials. Int. J. Fracture 10, 284 (1974). [6] J. R. Rice, Mechanics of crack tip deformation and extension by fatigue. ASTM STP 415, 247 (1967). [7] G. P. Cherepanov and H. Halmanov, On the theory of fatigue crack growth. Engng Fracture Mech. 4, 219 (1972). [8] J. S. Santner and M. E. Fine, Fatigue crack propagation in AI-bas¢copper alloys. Met. Trans. 7A, 583 (1976) [9] V. M. Radhakrishnan and C. R. Prasad, Relaxation of residual stress with fatigue loading. Engng Fracture Mech. 8, 593 (1976) [10] D. J. Quesoel, M. Meshii and J. B. Cohen, Residual stresses in high-strength low-alloy steel during low cycle fatigue. Mat. Sci. and Engng (in press). Ill] D. L. Davidson and J. Lankford, Jr., Plastic strain distribution at the tips of propagating fatigue cracks J. Engng Mat. and Tech. 95, 24 (1976). [12] T. S. Kang and H. W. Liu, Fatigue crack propagation and cyclic deformation at a crack tip. Int. J. Fracture 10, 201 (1974). [13] R. J. Donahue, H. M. Clark, P. Atanmo, R. Kumble and A. J. McEvily, Crack opening displacement and the rate of fatigue crack growth. Int. J. Fracture Mech. 8, 209 (1972) [14] L. H. Burck and J. Weertman, Fatigue crack propagation in iron and Fe-Mo sohd solution alloys (77-296°K). Met. Trans. 7A, 257 (1976). [15] A. Saxena and S. D. Antoiovlch, Low cycle fatigue, fatigue crack propagation and substructures m a series of polycrystalline Cu~AIalloys. Met. Trans. 6A, 1809(1975). [16] S. Ikeda, T. Sakai and M, E. Fine, Fatigue behavior of precipitation-hardeningmedmm C steels containing Cu J. Mat. Sci. 12, 675 (1977). [17] M Conserva, E. D. Russo and M. Burattl, Carattefistlche ed tmpieghi della lega AI-6.3wlo Cu+(Mn, Zr, V, Ti) (P-ACtMV2r). Aiuminio e nuova meta. $, 227 (1970). [18] J. H. Silcock, T J. Heal and H. K. Hardy, Structural aging characteristics of binary aluminum-copperalloys J. Inst. Metals $2, 239 (1953-54). [19] A J. Ardell and R. B. Nicholson, On the modulated structure of aged NI-AIalloys. Acta Met. 14, 1295 (1966). [20] F. A. McClintock, ASTM STP 415, 170 (1967). Discussion to article by C. Laird Discussion on in~uence of metallurgical structure. [21] R. W. Larduer, A dislocation model for fatigue crack growth in metals. Phil. Mag. 17, 71 (1968). [22] A. S. Kuo and H. W. Liu, An analysis of unzipping model for fatigue crack growth. Scripta Met. 10, 723 (1976). [23l N. E. Frost and J. R. Dixon, A theory of fatigue crack growth. Int. J. Fracture 3, 301 (1967). [24] N. E. Frost, K. J. Marsh and L. P. Pook, Metal Fatigue, p. 212. Clarendon Press, Oxford (1974). [25] M. Kikukawa, M. Jono and K. Tanaka, Fatigue crack closure behavior at low stress intensity level. Proc ICM-II, p 716, Boston, Mass. (1976). [26] W EIber,The Significanceof Fatigue Crack Closure. ASTM STP 486, 230 (1971). (Received 24 July 1978: recewed for publtcat#on 5 September 1978)
Q0131--WI-.0~IPJ~
R O L E O F P L A S T I C W O R K IN F A T I G U E C R A C K P R O P A G A T I O N IN M E T A L S t Y. IZUMI~ and M. E. FINE Department of Materials Science and Engineering and Materials Research Center, Northwestern Univarsity, Evanston, IL 60201, U.S.A. Almract--The plastic work required for a unit area of fatigue crack propagation U was measured by cementing tiny foil strain gages ahead of propagating fati~le cracks and recording the stress-strain curves as the crack approached. Measurements of U and plastic zone size in aluminumalloys 2024-T4,2219-T861, 2219 oversged, and AI-6.3 wt~ Cu--T4,and a binary Hi-base alloy with 7.2wt% AI are herein reported. The results are discussed along with previously reported measurements of U in three steels and 7050 aluminum alloy. When U is compared to the fatigue crack propngation rate at constant h ~ along with strength and modulus, the conclusion is drawn that U is one of the parameters which determines the rate of fatigue crack propagation. The relation of U to microstructure is also discussed. "Homolgeneous"plastic deformation in the plastic zone ahead of the crack seems desirable.
1. INTRODUCTION RECENTLYTHEintegrated plastic work expended in the plastic zone around a fatigue crack for a unit area of crack advance, U, was measured by cementing tiny foil strain 8ages ahead of fatigue cracks in center notched panel specimens[l]. Results for a high strength low alloy Nb steel and 7050 A1 alloy showed that the rate of fatigue crack propagation, IdcldN[~, depended approximately on 1/U. The relationship was further established by study of additional AI alloys and steels [2]. Recently, Davidson and Lankford [3] obtained the value of U in 0.05 wtg~ C steel by measuring the variation in subcell size vs distance from a fatigue crack tip using electron channeling. Using data of hysteresis loops vs cell size in uncracked iron, they integrated the hysteretic plastic work over the plastic zone. The U values obtained for San Antonio laboratory air and dry nitrogen correlated inversely with the smaller [dc/dN[aK for dry nitrogen. The U value determined for the Nb-doped 0.06 wt~ C steel in dry argon using the strain gage technique[l] was about 70~ smaller than U determined for the 0.05wt~ C steel using the electron channeling technique. The former steel, of course, has a higher yield strength and contains NbC precipitates. Theoretically, many equations for the fatigue crack propagation rate predict a Idc/dN[aK 1/U relationship. Weertman[4], Mura and Lin[5], Rice[6] and Cherepanov and Halmanov[7] derived simple functional relationships for the fatigue crack propagation rate of the form dc
(AK) 4
where A is a dimensionless constant, p is the shear modulus and o- is an appropriate strength parameter. Theoretical or empirical equations [2, 5] which predict [dc/dN[ ~ AK 2 also show an inverse dependence on U. In a previous paper[2], the authors proposed dc
(AK) 2 U/2y)
=8
(2)
where B is a dimensionless constant, E is Young's modulus, and 7 is the true surface energy. Much better agreement with experimental data[2], i.e. the dimensionless term was more nearly constant, was obtained with eqn (2) than with other (AK)2 type equations which did not contain U, such as dc/dN = B'(AK)Z/o'yE and dc/dN = B'(AK)Z/E ". With the set of data previously presented, the standard deviations for B' and B" were larger than their ave.rage values. tThis research was supported by the Air Force Once of ScientificResearch, O k e of Aerospace Research, Grant No. AF-AFOSR-76-2892A. ~Present address: Onada Cement Company, Tokyo, Japan. 791
792
Y IZUMI and M. E. FINE
Measurements of U for additional alloys using the foil strain gage technique are herein reported and sources of error in the measurement are examined. The results for U to date are discussed in terms of various equations which have been proposed for fatigue crack propagation vs AK. 2. EXPERIMENTAL PROC~JRES The chemical compositions of the alloys studied are given in Table 1. The commercial aluminum alloys, 7050, 2024 and 2219, were supplied by Alcoa Research Laboratory. The Al-6.3 w/o Cu alloy, which was studied previously in this laboratory[8], was provided by Kaiser Aluminum and Chemical Corp. The Ni-7.2 wlo AI alloy was prepared by International Nickel Company, Inc. Standard center notched panel specimens, 100 nun long, 20 mm wide, and 2.3-3.0 mm thick, were employed. The long direction of the specimens were cut parallel to the rolling direction. The surfaces were polished with 6 ~m diamond paste for the AI base alloys and 600 grit SiC for the Ni-AI alloy. A center notch approximately 3 mm long and 0.2 mm wide was introduced by electro-discharge machining using a flattened Cu wire tool. The heat treatments for the specimens are listed in Table 2. The plastic work associated with fatigue crack propagation was measured using the procedure previously described [1]. Tiny foil strain gages (Micro Measurements MA-13-008CL120 for the AI alloys and MA-06-008CL-120 for the Ni alloy), 210 ~m wide and 200 ~m long in the gage section were cemented with epoxy on the notched specimens astride and above the extended crack plane trace. The strain gages were normally mounted to measure only local strain in (Y) the tensile direction; however, local strains perpendicular to the tensile direction (defined as the X direction) were also measured in the 2024 Al alloy. The strain indicator used was readable to 10-4%. The fatigue crack propagation rates were first determined vs AK at 30 Hz using a 40X telemicroscope readable to 2 ~m. For all of the measurements the R-ratio was kept constant at 0.05 (pull-pull). The plastic work U was measured at constant AK in the range 7-16 MN/m 312. Table 1 Chemmal analysis of alloys studied Si
Fe
Cu
M~
C.r
NI
Z~
X__~_l
Z__~_r
7050 A1
01
.01
2.25
0
2.36
0
--
6 20
02
Ii
2024 A1
.14
.28
4.52
.59
1.41
.01
.00
.23
.04
2219 A1
.09
.23
6.19
.31
.000
.00
.0
.02
05
6.34
.000
000
.000
--
A1-6.3 w/0 Cu
004
009
N~-7 2 w/o Al
Chemlcal
~
006
V
--
Be
--
000
.09
000
14
0006
AI
C
P
Si
Fe
S
0
7 16
.001
0005
.013
.041
.001
0037
analyses were provided by the suppllers.
Table 2. Heat treatment of alloys Specimen
Condition T76
Solution treated 1/2 hr at 482JC, aged 4 days at rm 3 hrs at 121°C and 9 hrs at 163°C
T4
Solutlon
treated
i hr at 500°C and aged 7 days at rm
temp
2024 A1
T4
Solution
treated
1 hr at 495~C and aged 7 days at rm
temp,
2219 A1
T861
Solution treated l hr at 535~C, aged 7 days at rm. terap , elongated 27 at strain rate of 8.3 ~ lO-~ sec -x and aged 18 hrs at 177°C
overaged
Same as T861,
T4
Solution treated rm temp
1 hr at 510CC followed by aging 7 days at
Solutlon treated 625°C ~n argon
I hr at 900°C in argon and aged 2 days at
7050 A1
A1 - 6 3 w/o Cu
NI - 7 2 w/o A1
except the aging temperature
temp.,
was 202°C
Role of plastic work in fatigue crack propagation in metals
793
To keep AK constant, the maximum and minimum loads were decreased when the crack length increased 0.1--0.2 ram. During the tests, the frequency was periodically decreased from 30 to 0.3 Hz in order to graphically record the nominal stress vs strain gage output. All measurements were done in dried argon. The procedures for reducing the data to give U are given in the previous papers[l]. The minimum plastic strain at zero load was also measured in order to study the non-hysteretic plastic energy. For determining the cyclic stress-strain curve vs strain amplitude, which is required for the determination of U, a cfipon extensometer of 7.5 mm gage length was fastened to unnotched specimens and the strain amplitude was incrementally increased to -+1.25%, decreased, increased, etc. until a stable set of hysteresis loops were obtained. 3. EXPERIMENTAL RESULTS AND REDUCTION OF DATA Figures l(a) and (b) g i v e s o m e typical nominal stress-local strain curves for the 2024-T4 and 2219-T861. X is defined as the distance from the center of the gage to the crack tip along the horizontal axis; Y is the distance from the center of the gage above or below the crack plane. Non-closing curves (A and G) are due to permanent strain and were obtained only during the first cycle. The permanent strains during the first loading, elp, were 0.02%--0.03% for the AI alloys but only 0.005% for the Ni-AI alloy at AK = 15.5 MN/m 3/2. The rate of increase of permanent plastic strain, deP/dN, for all alloys studied was less than 104% per cycle after the first cycle and thus the loops appear closed on the scale used in the plots. The observation of a "closed" loop means that a residual compressive stress must be present at the minimum nominal stress. The cyclic stress-strain hysteresis loop obtained with an unnotched specimen of AI-6.3% Cu are shown in Fig. 2. Similar loops were obtained for the other alloys. A hysteresis loop for AK of 15.5 MN/m 3/2 and small X and Y is also shown. As previously[l], the local stress in the specimen was determined approximately from the cyclic stress-strain curves of unnotched specimens by equating the strains. The mean value of strain amplitude in the lower curve was shifted to coincide with zero strain of the upper curve, as shown in the figure, to take account of the residual compressive stresses near the crack tip. The exact amount of this shift in the origin has only a small effect on the amount of hysteretic work per cycle determined from the areas in the loops, the quantity of interest, since the slopes of the fines are parallel except at the large strain tips of the loops. 2 0 2 4 AI Z~K=15 5 M N / r n~2 Y : 150,u.m
E" E Z
:E
A
B
2OO
2 2 1 9 - T 861 AI A K : 15 5 M N / m 3~= Y= 320,u.m
c
LLI n." I'U3
D
E
F
4-"
E Z
G H 100
:E Z
u~
400 o
//
IZ co
z
Z
d2
d4
0'8
lb
4'2
STRAIN (%) notch crack
IIm.~foll gage
A B CDEF
(a)
E O Z
,'50
/
J
n ~ f O I I
/
GH
o2
o~
gage
I JK
o'~
o'8
STRAIN (%)
(b)
Fig. I. Typical nominal stress-local strain curves vs distance from crack tip determined using foil strain gages. The distances A - F and O-K are approximately 2 mum. Y is the distance from the gage center to the crack line in the vertical direction; (a) 2024; (b) 2219.
794
Y. IZUMI and M. E. FINE
-o~,
-o'2
6
o~
o'4
STRAIN C%)
Fq~ 2. Cyclic stnam-strain hysteresis loops obtained on an unnotched A1-6.3% Cu specimen. A hysteresis loop obama~ with a foil strain ~ on a specimen with a fatigue crack is also shown. This figure shows how the local stresses were determined in the present study.
The total plastic work to extend the fatigue crack surface by a unit area U is made up of two parts, hysteretic, Uh, and non-hysteretic U,. Since AK is kept constant during the measurement, the plastic zone size is essentially constant during the crack propagation. When most of the gage is out of the plastic zone the loop width is very small, as shown by the B and H curves in Fig. 1. The loops widen as the plastic zone moves into the gage area and are widest when the crack tip is closest to the gage (F). When the crack tip reaches or enters the gage, the gage fails. The hysteretic plastic work expended during extension of a fatigue crack by a unit area is obtained by equating it to the summation of the areas of the loops over the plastic zone (converted to local stress using the cyclic stress-sirain curves as described) divided by dc/dN. The hysteretic l~stic work per unit area for a belt (a unit length wide) parallel to the X axis and centered on Y, Uy, is given by the equation Ur = (S~)A + . . . . + (SXr)F = ~..( ).$xr., dc dN
(3)
where (Sxy)A is the area of the loop at A in Fig. l(a). When the increments in X are small, eqn (2) can be rewritten Uy = f Uxy dX.
Thus the total hysteretic plastic work, Uh, is simply
(4)
Role of plastic work in fatigue crack propagation in metals
ill
AKI -=6124 3 Cu A MN/m 3/a
NI - 72AI ZIK= 15 5MN/m 312
2/~1/-
E
F -)
Y: 190/~m
/
2
%
~K D
1
Y=12O~m
r,A / - Y = 5 ,'.'.'.'.O '.'.'.'.'~ m
D
,~ / - Y = 2 5 0 ~ . m
1
Y= 300#.rn ~. BG
,.o./-- Y = 23OF m
~'~.~',~S¥:4,~r~ ~ ll,~,c).. O
795
-
BG 1
2
0 X, DISTANCE FROM CRACK TIP(n7m)
X, DISTANCE FROM CRACK TIP(mm)
(b)
(a)
Fig. 3. Typical local hysteretic plastic work profiles along X for several values of Y. B.G. indicates background plastic work; (a) AI-6.3% Cu, A K = 12.4 MN/m3/2; (b) Ni-7.2% AI, A K = 15 5 MN/m 3/2.
It will be shown that the non-hysteretic plastic work per unit area of fatigue crack propagation is very small for the present measurements. Thus Uh - U. Examples of Uxy at constant Y plotted vs X are shown in Figs. 3(a) and (b). The Y values showware the nominal values. Since the crack did not move in a perfectly straight line, the actual Y values (these were used in the calculation) deviated from the nominal values. Some of the curves in Fig. 3 show tails indicating background plastic work, as observed previously[l]. Such backgound plastic work appears to result from experimental problems such as faulty bonding and was subtracted from the plastic work for determining Uy. One problem in the measurement of U is to determine the large value of plastic work near the crack tip, where the gages failed. As described and discussed in the previous paper[l], the hysteretic plastic work closer than 100/~m to the crack was estimated by extrapolating the strain amplitudes determined by the foil strain gage on a linear log-log plot to 1/~m. The plastic work per unit area so determined for closer than 100 ~tm to the crack tip is designated Uc. Typical extrapolations are shown in Fig. 4. Although U in eqn (4) was previously calculated using the local strain in the Y direction only, the plastic work due to other strain components must also be considered. The plastic work due to local strain in the X direction was measured for the 2024-T4 A1 alloy and found to be only 10% of the total U.
AI-63Cu
lo' bJ t7 D P
.
~"."
A
D
_ _ ( A 12.4
/~KtMN/m"'} l B 10.8 C93 2024 - T4 AK(MN/m~{ ED15785
<( _Z 16 2
x. DISTANCE FROM CRACK TIP (~.m)
Fig. 4. Strain amplitude measured with foil strain gase vs X extrapolated to X = 1 ltm assuming a linear relation on a log-log scale
796
Y. IZUMIand M. E. FINE ~I
"o
2024 - T4 AI o 15 5 MN/m 'JIz AK = o 78
10
o
AI - 6 3 Cu 'N
• 124 MN/rn :~'2 AK = .6. 1 0 8 o 93
8
~E "o x 0 20O
4OO
\
o \
5 4
1
, 8O0
6OO
\
7 6
0
\
9
1000 3
DISTANCE FROM C R A C K
LINE
Y
2
[,u.m]
(a)
t 2 ~30
0
' 400
' 600
800
1000
DISTANCE FROM CRACK LINE Y [ ~ m ]
(b)
Fig. 5. Localhyttereticplasticworkdueto err in a beltof unitwidthalongX (Uy) vs ¥ for 2024-T4. Uy for X of Y < 100 #m him beea mtlected ia ~
plots. Similar curves were obtained for all of the alloys,
treatments,andAg's measuredin this investigation.
The-hystet~k plastic work per unit area for further than 100 pm from the crack front, Uh was dotlgmiMd by plotthlg /Jr (eqn 4) vs Y and then measuring the area under the curve. Examples of such plots are shown in Figs. 5(a) and (b). The total hysteretic plastic work per unit area of fatigue crack ~ , Uh = Uc + UI, for the alloys and AK values of the present research are Oven in Table 3. The contribution to the plastic work from closer than 100 ~m to the crack front Uc varied from 5 to 40% of the total U. Since the minimum strain gradually increased with each cycle, i.e. the loops were not exactly closed, the non-hysterefic plastic work must be considered. The 6.3% Cu-A1 alloy was expected to have the h q m t non-hysteretic contribution (U,) of all the alloys studied. U, was detmrmia~ in ~ alloy by am analysis simitat to Uk and found to be approximately 2 x 103 J/m2 for AK = 12.4161Iram. This is a factor of 100 smaller ~ Uk. ThUS in the determination of U, U, may be nq0eeted. Plots of U,r (defined similarly to/Jr) vs Y for the three values of AK investigated are shown in Fig. 6. From the scale pierre_.__,it is obvious that the areas under the U, curves are much smaller than the corresponding values of Us, Fig. 5Co). In order to determine possible eR~,ts of specimen thickness on the U values, the fatigue cmzk I ~ rates were measun~ vs &K for specimens of 2219 which were 1, 3, and 6 mm thick. The results (shown in Fig. 7) illustrate a slower rate in the high range of AK for the 1 nun Table3. Hystereticplasticworkper unitareaof fatiguecrackextension AK
, ~J~ A16.3 w/o
Cu
2024 T4
2219 T861
2219 ~ e r a g ~
Ni7.2
wloAI
Uf
Uc
Ub
JJ¢
J/~
,T/~~
12.4
4.8xi0 b
l.OXl0s
5 8XlO s
10.8
5.7
0.4
6.1
9.3
I0.I
0.4
10.5
15.5
3.0
1.8
4.8
7.8
2.0
0.6
2.6
15.5
1.5
0.1
1.6
7.8
1.8
0.6
24
15.5
1.1 1.7
0.3
1.4
9.3
0.4
2.1
15.5
4.1
0.7
4.8
Role of plastic work in fatigue crack propagation in metals
w
AI - 6.3 Cu
~5
/xK=
z
mint/
E ~ 20
• 124 M N / m 3/z 108
30
40
50
coo
70 80
OZ
u 93
A
1 mm
-t
B 3 mm |
~3
c
x
~2
6n'rn
)
T861
ED 63mmmmI over oged
~
C
1 o o
i
2o0
4oo
6oo
8oo
10oo
DISTANCE FROM CRACK LINE Y[/~m]
16'
Fig. 6. DE
ld"
i
t
i
7
8
9 10
i
I
20 ~K /MN~ Fig. 7.
Fig. 6. Local non-hysteretic plastic work due to err in a belt of unit width along X(U.r) vs Y for AI-6.3% Cu. U.r when X or Y< 100/tin has been neglected. Fig. 7. Fatigue crack propagation rate vs AK of 2219 AI alloy for specimens of three different thicknesses.
thick specimen than for the 3 and 6 mm thick specimens. This is no doubt due to the fact that such thin specimens are more nearly in plain stress than the thicker specimens. Since the rate, Idc/dNlaK, at AK of 15.5 MN/m3~ for 3 and 6 mm thickness specimens was 1.3 times larger than that for 1 mm thick specimens, one expects only a 30% increase in U at this AK on reducing the specimen thickness to I mm. As an additional experiment to study the effect of specimen thickness, foil strain gages were cemented 200 and 550/zm from the crack tip on a 2024-T4 Al specimen 2.3 mm thick and hysteresis loops were measured under load control at AK of 12.4 MN/m 3~ and 7.8 MN/m 3~. The reverse side of the specimen was then carefully removed with a milling cutter to a thickness of 0.7 mm using sufficient oil to prevent the specimen from overheating. The hysteresis loops were remeasured. There was little change in the strain amplitudes and the loop widths between the two thicknesses at both AK values. One value appeared anamalous. This was attributed to gage damage during machining. The conclusion is that specimen thickness over the range of thickness studied has little affect on U except perhaps at the highest AK values. During preparation of the specimens, appreciable residual stresses may be introduced on and near the surface. The effects of such residual stresses were evaluated by cementing gages on notched and unnotched specimens and then shifting the lower grip by melting the Woods metal pot. Hysteresis loops were obtained on an mmotched specimen of 2024-T4 at a stress amplitude of ± 450 MN/m ~. No difference in loop width on changing the prestrain from -0.2 to 0.1 was observed. There was only a few percent change in strain amplitude. In a notched specimen of 7050-T76 with a fatigue crack and with strain gages cemented 100/zm ahead of the fatigue crack, there was little variation of the strain amplitude and the loop width on changing the prestrain from -0.03% to +0.03% (AK of 12.4 MN/m~). There was also little effect on the loop widths and strain amplitudes at AK of 6.2 MN/m3~ from overloading at 12.4 MN/m 3~. DISCUSSION OF F.~IIOMS IN THE I ~ [ f l ~ 4 l l N T OF U Neglected terms will first be discussed. In the present research, U was equated only to the hysterefic plastic work in the Y direction. As already mentioned for the present set of measurements, the hysteretic plastic work in the X direction was found to be only 10% of that EFgv~. li, go
798
Y. IZUMI and M. E. FINE
for the Y direction and the non-hysteretic plastic work for X or Y < 100 # m was found to be negligibly small. The effect of specimen thickness was discussed in the previous section. With respect to the effect of residual stress, prestrain of -0.2 to 0.1% changed the area of hysteresis loops to result in an estimated change of 20 to -10% in U. Residual stresses might arise from specimen preparation or in mounting of the specimen. The initial residual stresses are rapidly removed by cyclic loading and replaced by a macrostress profile characteristic of the local cyclic loading conditions[9, 10]. Since 104 cycles were usually required for crack initiation, initial residual effects are expected to be very small. Even if we neglect the effect of cycling on the residual stress, a prestrain of 0.2% which corresponds to a stress of 150 MN/m 2 for the 2024--T4 Al alloy would change the U value by only 20%. Negative values of the permanent plastic strain during the first load, which were occasionally observed, indicated some bending of the specimens. Such prestrains were less than _+0.03% and would affect the final values of U by no more than a few percent. Extrapolation of the local strain amplitude to closer than 100~zm from the crack tip, according to the technique described in the earlier paper[l] (Fig. 4), is the most uncertain part of the procedure used to determine U. Davidson and Lankford[ll] studied the plastic zone around the fatigue crack, using selected area electron channeling in 6061-'1'6 A! alloy and concluded that curves for strain amplitude vs distance from the crack tip on a log-log plot are not linear as assumed in this paper. Using their data for the strains vs X, the change in the total U calculated is only 10%. Kang and Liu[12] observed the strain amplitude around a fatigue crack from 100 ~m to I mm in the 2024 AI alloy using Moir6 fringes. Both sets of data for strain amplitude are approximately the same as those determined in the present study further than 100/,m from the crack tip. The electron channeling technique for measuring U developed by Davidson and Lankford[3] is more accurate than the foil strain gage technique close to the crack tip. The latter is more accurate for the contribution to U from > 100 ~m from the crack tip. The two techniques should be used on the same alloy to arrive at a more accurate value of U. The data in Table 3 shows U values for several different values of AK in the range 7.8-15.5 MN/m 3t2. The variation of the measured U with AK is less than 40% from the average value and may be within experimental error. A similar conclusion was reached for the Nb-doped 0.06 wt% C steel[I]. PLASTIC ZONE The foil strain gage technique also gives contour maps of the non-hysteretic and hysteretic plastic zones. The non-hysteretic plastic zone will be treated first. During cyclic loading, the permanent plastic strain measured at zero load ~ increased as the crack approaches the strain gage (decrease in X). Again the 6.3 wt% Cu-AI alloy was selected for study because it has the largest permanent strains of the alloys studied. The resuRs are shown in Fig. 8. The background ,P essentially occurred during the first cycle and may be associated with changes in the gage. In the lower AK experiment (Fig. 8a), new foil gages were cemented on the specimen after 8000 cycles. The total permanent strain recorded by both gages reached about 0.5% after 9000 cycles. At .the higher AK (Fig. 8b) rapid increases in permanent strains marked P were observed near the outer boundary of the plastic zone. Equi-permanent plastic strain lines around the fatigue crack tip were determined from the curves of ~P vs X for various Y's. These are shown in Figs. 9(a) and (b). The lines for Y larger than 600 tzm from the crack line were estimated by extrapolationand are indicated by dotted lines. Equi-hysteretic plastic work density lines around the fatigue crack tip for the various specimens were determined from the data of Uxy vs X for various Y's, examples of which are shown in Fig. 3. The hysteretic plastic work maps are shown in Figs. 10(a-d) for 2219-T861, 2219 overnged and Ni-7.3 wt% AI-Ni alloys, respectively. DISCUSSION
Although ~lcldN[Ag has been predicted to depend on l/~r or l/or2, Donahue et al. [ 13] presented some data to show that it was independent of o-. However, Burck and Weertman[14] utilizing the fact that the strength of iron and its alloys increase on cooling from room temperature to
799
Role of plastic work in ratine crack porpagafionin metals I
AI - 6.3 Cu /~K: 9 3 MN/m~'2
AI - 63Cu ~ K : 124 MNI m ~z
'\
10
;s
o°
z n..
""~ ~ ,,, H L ,,~/
1D
HL
P
" ~ '~~ K ~ " "~"" /
Z
ILl
:E .,.,,r.
O~
bJ Z
< 0.5 E cr
bJ n
LtJ O_
I
o
~
xY~o
~,~oo
X, DISTANCE FROM CRACK TiP (M,m) X,
(a)
DISTANCE FROM CRACK TIP (p.m)
(b)
Fig. 8. Permanent strain at zero stress ep for various values of T vs distance from crack tip (X) for AI-6.3~ Cu; (a) AE -- 9.3 MN/m3r~; (b) AK = 12.4 MN/m3r2. H.L. represents value of X where a hysteresis loop was first observed, p nutrks the value of X at A/( = 12.4 MN/m~ where a rapid increase in e P was observed. The baclr4pound e p occurred duri~ the first cycle and was due to changes in the gage, not the
material.
Y
( b ) Z~K = 12A M N / m 3'2
,,
.-:::;.:>;..:.--"
)'//;" ~oo
\p
[/J.m]
looo
i l
5oo
,
/
o
[p.~]
( o ) Z3K ,, g:3 M N / r n 3~2
IOOO
x
1 oo
o
[~.m] Fig. 9. Equi-permanontstrain (t~) ~ for AI-6.3~ Co alloy vs X and Y. Dotted lines are extrapolated. Numbers are in %. Points designated p correspond to those in Fig. 8. 770K showed a correlation of Idc/dNlax with l/
Y. IZUMI and M. E. FINE
800
Y(p.m) ((3) 2219 T861
~O00
(C) AI- 63Cu AK= 124 MN/m :112
Y (/~m)
1000
&K=78 MN / rr~ 2 4
,1 2 4 6 ~ ' X (/~rn)
I£~
500
: CRACK
(ffm) 2219 over aged AK= 93 MNI m31z
o
MNIm
Y (fire) 312
1000
~
1000
500
,1 2 4 6 X O~J.m) 1(~00
50o
(d) N, - 7 2 AI Z~K= 155
(b)
CRACK
i
X (/zm) 1(:;oo
0
500
500
500
CRACK 0
2, X(~m)
1(~00
4 6810
CRACK
500
Fig. 10. Contour maps of hysteretic plastic work Uxr vs X and Y for; (a) 2219-T861, ~K = 7.8 MN/m3/2; (b) 22219-overaged; (c) AI-6.3%Cu; and (d) Ni-7.2~ AI. The numbers are in units of 10'1J/m4. Table 4. Comparisonof products Idc/dN[t55o'2/~ for the five aluminumalloys and Nb-doped HSLA steel ldc/dN [15.5°~ ~
iT) 2 0 . 3 x 103
7050 T76 2024 T4
5.5
2219 T861
8.9
2219 overaged
5.6
Nl - 7 2 w/o A1
7.5
Nb-doped HSLA steel (0.6 w/o C)
1.5 3
Idc/dNlls.5 = fatxgue crack propagation rate at AK=I5.5 ~ / m 2 = 0.2% offset cyclic yield stress = shear modulus
proximately constant for all materials. Comparison of the AI alloys[8] and the Cu alloys[B] previously referred to shows that this is not observed. Table 4 compares the product Idc/dNll55 ~r2/~ for the five alloys studied in this research and the HSLA steel studied previously[16]. While the values for 2024-T4 and 2219 overased are nearly the same, this product is approximately three times larger for the 7050 AI alloy. For the Nb-doped HSLA steel, it is more than three times smaller than for 2025-T4. The variation of this product may be understood by introducing U as a variable. Decrease in U along with increase in o- also explains the observation of Donabue et aL The data for U of the present study and the previous studies in this laboratory are summarized in Table 5 along with the fatigue crack propalaXion rates, cyclic and moaotonic 0.2% offset yield stresses, the exponent m in the Paris eqeation, and the constant A in eqn (1). The fatigue crack propagation rates, IdcldN[ax, for the 2024-T4 and 6.3 w/o Cu-AI alloys in Table 4 are considerably faster than the rates for the Nb-doped HSLA steel; the cyclic yield stress, cry, for the HSLA steel is between those for the two alloys. These differences in IdcldNlAK cannot be explained by a difference in the shear modulus or or;. Introduction of the
Roleof plasticworkin fatiguecrackpropagationin metals
801
plastic work U rectifies this difference. The Cu-C steels[16] and the Ni-base alloy show a similar correlation between [dcldNIAg and the measured U. The value of U for the 2024 alloy is the highest of those for the commercial Ai alloys studied while its Idc/dNIAK is the lowest. The large differences in IdcldNhs.5 between the 7050-T76 AI alloy on the one hand and the Ni-base alloy and HSLA steel on the other correlate with the large differences in U. In view of these results, U is obviously playing a determinative role for the rate of fatigue crack propagation. Since the plastic work during cyclic loading results from the to and fro motion of dislocations, it is reasonable to consider that U, as well as the alloy's strength, c~, are implicitly related to the alloy's microstructure. The A1-6.3wt% Cu alloy aged at room temperature contains both GPI (Guiner-Preston zones) and the 0 phase[8]. The excess copper over the stability limit forms 0 particles which are 5-10/zm in diameter with a mean spacing of about 50 gm. The presence of GP zones tends to give inhomogeneous deformation with coarse planar slip bands. The presence of the large 0 precipitates, however, make the plastic deformation more uniform throughout the specimen, since the 0 particles are not penetrable by dislocations and thus dislocations must bend or cross slip around the particles. This results in a higher work hardening rate. We propose that this is the origin of the higher U for this alloy. Possibly because few dispersed particles of the required size are present in the 7050 AI alloy, the value for U is much smaller than that in the 6.3 wt% Cu-AI alloy. Unlike the binary A1 alloy, the 2024 AI alloy has S phase and the AbCu2Fe constitutent particles. Comparing the 6.3 wt% Cu and 2024 AI alloys, the cyclic yield stress of the latter is considerably larger, but its U is a factor 2 smaller. This may result from a non-uniform distribution of constituent particles. The value of U for 2024-T4 is twice that for 7050-T4. The GPI zones for the 7050-T4 alloy are spherical while those in 2024-T4 A! are platelets. The dispersoid precipitate in the 7050 alloy contains Zr while those in the A1 alloy contain Mn. The 7050 also contains fewer dispersoids. The higher U in the 2024-T4 alloy may possibly result from a greater difficulty for dislocations to avoid cutting platelets than spheres even though the spherical zones are a stronger barrier to dislocation motion (the strength for the 7050-T4 Ai alloy is larger). The 2219 AI alloy has high strength at elevated temperatures. The inclusions in this alloy[17] are similar to those in the 2024 alloy with 0 replacing the S-phase. The T861 and overaged (202°C for 18 hr) treatment give O', the particle size for the former being smaller[18]. Increasing O' size seems to have little effect on U even though it decreases the strength, Thus the observed decrease in IdcldNhs.s on overaging is related only to a decrease in strength. The large increase (50%) of c~ over ~ in the Ni-AI alloy is worthy of comment (Table 5). The aging treatment used gives aligned cube-shaped y' precipitates roughly 100 ~ on a side[19]. The flow stress increase due to cycling may be due to the disordering and creation of antiphase boundaries in the y' particles from activity on two intersecting slip systems, one for tension, one for compression. As stated in the Introduction, theoretical equations for the fatigue crack propagation rate may be classified into mainly two types: (a) IdcldN[Ag ~ (AK)4, Refs. [4-7] and (b) [dcldNIAK (AK) 2, Refs.[20-23]. Equation (2) followed from the suggestion of Donahue et al.[13] that IdcldNIAK is related to ductility which in turn is related to o~E. Thus introducing o/E into the equation dcldN = C(AK)21Ecr, which had been derived by several authors [20-23], removed the dependence on ~. Instead the present authors[2] introduced U as a measure of ductility and y to make the term dimensionless; 2y/(U+y) represents a "ductility factor". It is one for non-ductile materials. In metals, of course, U >>y so 2~//U is a good approximation giving from eqn (2). Experimentally, the exponent on AK, m, ranges from 2 to 8. For data for fatigue crack propagation compiled by Frost et a!.[24], the average value for 14 alloys is 3.94 with a standard deviation of __.0.86. With the 8 alloys and treatments studied in this laboratory (Table 5) the m values ranged from 3 to 4.5 giving m = 3.7__.0.5. Thus, eqn (1) seems to be the best theoretical equation for the present set of data. Table 5 gives the proportionality constants A for the 8 alloys and AK's studied assuming that o,~ the 0.2% offset cyclic yield stress, is the appropriate strength parameter. The average value
Y. IZUMI and M. E. FINE
802
Table 5. U values and mechanical properties U
Oy
~/.~
./crcza
j/~
~1,~
~/,~
,,
A
7050 AI T76"
15.5
30~I0 -s
0.63XI05
510
470
4.0
2.3XI0 -z
7050 AI T4*
12.4
30
0.54
410
320
3.5
2.8
AI-6.3 w/o Cu T4
12.4 10.8 9.3
4.7 2.0 1.2
5.8 6 1 10.5
230
130
4.0
1.6 1.2 2.3
2024 AI T4
15.5 7.8
14 1.6
3.2 2.6
390
300
3.0
3 1 4.4
2219 AI T861
15 5 7.8
25 1.5
1.6 2.4
370
350
4.0
2.5 3.5
2219 AI overaged
15.5 9.3
32 6.8
1.4 2.1
260
270
4.0
1.4 3.4
Nx-7.2 w/o AI Aged 2 days, 625°C
15.5
2.0
4.8
670
440
4 5
6.3
Nb-doped HSLA steel*
19.5 15.5 12.4 93
3.2xi0 -s 1.7 0.8 0.3
12 8 12 6
3~0
360
3 5
2.7 2.1 3.5 2 1
19.5
1.8
3.4
710
880
3.5
1.6
Steel - Aged 200 min, 19.5 500°C **
5.6
1.3
780
950
3.0
2.3
3
Steel - Aged 13 min, 500°C **
dc/dN
,
Ikeda~ Izumi and Fine (1) ~-~ Ikeda, Sakal and Fine (Is)
dc/dN = fatigue crack propagation rate = total hysterettc plastic work per unlt area 0.2% o f f s e t c y c h c yield stress Y o = 0.2v offset monoton£c y£eld stress m = Parls exponent, i.e., dc/dN = C(Ak) m A = constant in Eq (I) Speclmen thlcknesses were 2 3 to 3 w Dry argon atmosphere
of A is 2.7 × 10 -3 with a standard deviation of --+-1.2 x 10-3. /~ for all steels was taken to be 7.8 × 104 MNIm2; for all Al alloys, 2.6 × 104 MN/m 2, and for the Ni alloy, 8.4 × 104 MN/m 2. The strength parameter in eqn (1) needs further discussion. A cyclic mechanical property would seem to be more appropriate than a monotonic mechanical property since the material at the crack tip has been "conditioned" by the stress--strain cycling. Weertman [4] used the critical stress for fracture of the material at the crack tip, oro which is expected to be somewhat larger than or; or or. The actual flow stress of the material at the crack tip is, of course, not known. In the present study, only or; and or, for 0.2% offset, were considered to compute A. Slightly better statistics were obtained when or; was used rather than or; A ffi 2.73 (1 +_0.44) compared to 2.26 (1 +_0.47). More measurements are needed, of course, to establish the ranges of validity of the equation and the significance of the value of A. Kikukawa et all25] showed, following the suggestion of Elber[26], that if AK+~ (AK = K,ffi,-K~,m) is used in the Paris equation rather than AK, the exponent is near 2. Substituting AKd into eqn (2) is thus suggested. Following Kikukawa et aL, the closure stresses may be estimated from where the stress-strain curve bends on unloading, Fig. 3. The present measurements of U were made at constant AK with the load an~litude reduced to compensate for the crack growth. The values of AKd were 65-85% of AK and tended to decrease as the load was decreased. Substituting the data in Table 5 into eqn (2) and taking ~ = 2 J/m 2, the "constant" B showed only slightly larger relative standard deviation than A in eqn (1). Thus U, the plastic work of fatigue crack propagation, appears to be an important parameter for determining Ida/dNt~K whether the Paris exponent is 2 or 4. The plastic zones will next be discussed. The shapes of one-half of the permanent and hysteretic plastic zones are shown in Figs. 9 and 10, respectively. The half of the plastic zone below the crack is a mirror image of that above. The plastic zones are "butterfly-wing" shaped in keeping with their shapes being determined by the shear stresses. The definition of the size of the plastic zone requires discussion. Often gross plastic deformation only is considered (0.2% plastic strain, for example) and lesser amounts of plastic strain are neglected, that is, the 0.2% offset yield stress is often introduced into theoretical
Role of plasticworkin fatiguecrack propagationin metals
803
'and empirical equations for the size of the plastic zone. Since setting Ux,,, = 0 corresponds to a point where a hysteresis loop first appears on increasing the stress amplitude, the stress, crh, is defined as the maximum stress corresponding to Uxy = 0. This was determined simply by extrapolating the hysteretic plastic strain to zero using the cyclic stress--strain curve obtained with an unnotched specimen; o-h is the minimum stress to give to and from dislocation motion and it defines the boundary of the hysteretic plastic zone about a fatigue crack. The distance 2rh~'~ was defined as the maximum distance from the crack tip to the ~'h boundary. This was established by extrapolation of Uxy vs X, Y data along an appropriately selected line and rhm~ was fit to the equation r h~ = Ch(AK/o.h)2.
(6)
Values of 2rr, h x,
(7)
r m , = C'(A K/o"y) 2
and the results are also given in Table 6. Table6. Plasticzone sizes 2rhmax
2rP.
2rnmax
oh
p.m
cI
0.22 X 10"I
300
0.16
150
0.39 x i0-l
280
0.15
780 1050
I00 I00
0.45 x 10-I 0.34 x 10-I
630 820
0.19 0.14
7.8 15.5
680 910
150 150
1.2 x 10-I 0.43 x 10-I
170 650
0.15 0.21
2219-T861
7.8 15.5
630 1080
Ii0 110
0.63 x 10-I 0 . 2 7 X 10- 1
120 460
0.14 0.13
2219-0A
9.3 15.5
860 ii00
ii0 ii0
0.60 X 10-I 0.28 x 10-I
520 630
0.20 0.09
N1-7.2% A1
15.5
1280
190
0.96 X i0-l
160
0.15
~lm ~
~
ss/~
7050-T76
15.5
630
130
7050-T4
12.4
530
AI-6 3% Cu
9.3 12.4
2024-T4
ch
~m
II00 1800
C h ffi (5.25 • 3.1) x i0-a C i - (1.48 ± 0.55) x 10-I
A permanent strain plastic zone size may also be defined and determined from the data of e p vs X and Y for the AI-6.3% Cu alloy, Fig. 8. In this case, 2rPm~was defined as e p = background which essentially occurs during the first cycle and is due to changes in the gage. The permanent strain plastic zone size may be regarded as equal to the static plastic zone size. Theoretically, Mura and Lin[5] and Rice[6] derived that the constant in the equation relating the cyclic plastic zone to AK and yield stress is w/64 or 0.05 which is very close to C h. C , which corresponds to the 0.2% offset cycfic yield stress, is about three times larger. Rice suggested that the static plastic zone size is 4 times the cycfic zone size. Interestingly, rP~ffi,is only two times larger than rs~, and only 1.5 times larger than rhm~.Comparing the various alloys, the Ni-7.2% AI alloy has a surprising large r~m~but small r ~ in keeping with the large value of ~r~,, Table 5. CONCLUSIONS When all of the data for U, the plastic work per unit area of fatigue crack propagation is considered, the inescapable conclusion is reached that it is an important parameter which determines the rate of fatigue crack propagation irrespective of the observed exponent in the
804
Y. IZUMI and M. E. FINE
Paris equation. Enough measurements of U are now available so that a discussion of how it is related to the microstructure and plastic deformation modes can be begun. "Homogeneous" plastic deformation in the plastic zone ahead of the crack is desirable for resistance to fatigue crack propagation. Means of homogenizing the plastic deformation without introducing other deleterious effects (such as easy crack initiaition at dispersed phases) need to be developed. The measurement also yeilds a contour map of the plastic zone. Three plastic zone sizes are given: rma~,determined by the extent of permanent plastic deformation; r ~ , determined by the extent of cyclic plastic deformation bounded by the 0.2% cyclic yield strength; and r ~h , determined by the occurrence of hysteresis in the stress-strain curves. The coefficient for the latter in the equation r ~ = Ch(AK/trh) 2 is close to ~r/64 which is given by theory for the cyclic plastic zone. AcknowledgementwThe use of the Central Faciliues of Northwestern Umversity's Materials Research Center, supported under the NSF-MRL program grant DMR76-80847,facilitated this work.
REFERENCES [I] S. lkeda, Y. lzumi and M. E. Fine, Plastic work during fatigue crack propagation in a htgh strength low alloy steel and in 7050 Al-alloy. Engng Fracture Mech. 9, 123 (1977). [2] M. E. Fine and Y. Izumi, Role of ductility and yield stress in fatigue crack propagation in metals. Proc. 4th Int Conf. Strength of Metals and Alloys, Vol. II, p. 468, Nancy, France, Angust-Scptembcr (1976). [3] D. L. 1)avidson and J. Lankford, Jr., Environment sensitive fracture of engineering materials. TMS-AIME Proc. Symp. Chicago, Illinois 24-26 October (1977). [4] J. Wccrtman, Theory of fatigue crack growth based on a BCS crack theory with work hardening. Int. J. Fracture Mech. 9, 125 (1973). [5] T. Mura and C. T. Lin, Theory of fatigue crack for work hardening materials. Int. J. Fracture 10, 284 (1974). [6] J. R. Rice, Mechanics of crack tip deformation and extension by fatigue. ASTM STP 415, 247 (1967). [7] G. P. Cherepanov and H. Halmanov, On the theory of fatigue crack growth. Engng Fracture Mech. 4, 219 (1972). [8] J. S. Santner and M. E. Fine, Fatigue crack propagation in AI-bas¢copper alloys. Met. Trans. 7A, 583 (1976) [9] V. M. Radhakrishnan and C. R. Prasad, Relaxation of residual stress with fatigue loading. Engng Fracture Mech. 8, 593 (1976) [10] D. J. Quesoel, M. Meshii and J. B. Cohen, Residual stresses in high-strength low-alloy steel during low cycle fatigue. Mat. Sci. and Engng (in press). Ill] D. L. Davidson and J. Lankford, Jr., Plastic strain distribution at the tips of propagating fatigue cracks J. Engng Mat. and Tech. 95, 24 (1976). [12] T. S. Kang and H. W. Liu, Fatigue crack propagation and cyclic deformation at a crack tip. Int. J. Fracture 10, 201 (1974). [13] R. J. Donahue, H. M. Clark, P. Atanmo, R. Kumble and A. J. McEvily, Crack opening displacement and the rate of fatigue crack growth. Int. J. Fracture Mech. 8, 209 (1972) [14] L. H. Burck and J. Weertman, Fatigue crack propagation in iron and Fe-Mo sohd solution alloys (77-296°K). Met. Trans. 7A, 257 (1976). [15] A. Saxena and S. D. Antoiovlch, Low cycle fatigue, fatigue crack propagation and substructures m a series of polycrystalline Cu~AIalloys. Met. Trans. 6A, 1809(1975). [16] S. Ikeda, T. Sakai and M, E. Fine, Fatigue behavior of precipitation-hardeningmedmm C steels containing Cu J. Mat. Sci. 12, 675 (1977). [17] M Conserva, E. D. Russo and M. Burattl, Carattefistlche ed tmpieghi della lega AI-6.3wlo Cu+(Mn, Zr, V, Ti) (P-ACtMV2r). Aiuminio e nuova meta. $, 227 (1970). [18] J. H. Silcock, T J. Heal and H. K. Hardy, Structural aging characteristics of binary aluminum-copperalloys J. Inst. Metals $2, 239 (1953-54). [19] A J. Ardell and R. B. Nicholson, On the modulated structure of aged NI-AIalloys. Acta Met. 14, 1295 (1966). [20] F. A. McClintock, ASTM STP 415, 170 (1967). Discussion to article by C. Laird Discussion on in~uence of metallurgical structure. [21] R. W. Larduer, A dislocation model for fatigue crack growth in metals. Phil. Mag. 17, 71 (1968). [22] A. S. Kuo and H. W. Liu, An analysis of unzipping model for fatigue crack growth. Scripta Met. 10, 723 (1976). [23l N. E. Frost and J. R. Dixon, A theory of fatigue crack growth. Int. J. Fracture 3, 301 (1967). [24] N. E. Frost, K. J. Marsh and L. P. Pook, Metal Fatigue, p. 212. Clarendon Press, Oxford (1974). [25] M. Kikukawa, M. Jono and K. Tanaka, Fatigue crack closure behavior at low stress intensity level. Proc ICM-II, p 716, Boston, Mass. (1976). [26] W EIber,The Significanceof Fatigue Crack Closure. ASTM STP 486, 230 (1971). (Received 24 July 1978: recewed for publtcat#on 5 September 1978)