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A small-crack growth law and its related phenomena
0013.7944/92 $5.00+ 0.00 @ 1992PergamonPresspk.
fh&tvr& Fracture ,W&adca Vol. 41, No. 4, pp. 499-513, 1992 P&W in Great Britain.
A SMALL-CRACK GROWTH LAW AND ITS RELATED PHENOMENA Department of Magi
HIRONOBU NISITANI and Strength of Solids, Kyushu W~~ity,
Department of Mad
Fukuoka, 812, Japan
MASAHIRO GOT0 En~n~~S, Oita Unive~ity, Oita, 870-l 1, Japan
NORIO KAWAGOISHI Faculty of Engineering, Kagoshima University, Kagos~m~ 890, Japan Ahatraet-In modern design, it is an important task to estimate fatigue life. The growth law of a small crack must be known in order to estimate the fatigue life of machines and structurea, because the fatigue life of members is controlled mainly by the behavior of a small crack. The growth rate of a small crackcannot usually be predicted by linear elastic fracture mechanics, but is determined uniquely by the term u:l. In this paper, two fatigue crack growth laws, dlfdN = C AK’and dildiV = C,o:l, which hold in large and small cracks, mapectively, are taken as the ~m~~tive onea expressing the crack growth rate of ductile materials. Many experimental results indicate that the relation dtfd.NccdP holds under low nominal stresses and the relation dt/dNoca:l holds under high nominal stresses. A u&@&g explanation for two growth laws is made based on an assumption that the crack growth rate is proportional to the reversible plastic zone size. Moreover, an effective and convenient method based on a smalLcrack growth law, dl/dN = C, (a,,/o,)V, in which the e&t of mechanical properties is partly considered, is proposed for predicting the fatigue life of plain members, where a, is the ultimate tensile strength. The validity of this method is confirmed through its a~li~tio~ to other researchers’ data.
IN MODERNdesign,
it is an important task to estimate fatigue life precisely. The growth law of a small crack must be known in order to estimate the fatigue life, because the fatigue life of members is controlled mainly by the propagation life of a small crack. For example, in the fatigue tests of annealed carbon steels, about 70% of fatigue life of a plain specimen is occupied by the life in which a crack propagates from an initial size up to 1 mm[i]. Until now, many researchers have reported that the growth rate of fatigue cracks can be determined in terms of the function of stress intensity factor range AK. AK is the effective parameter for the cracks when the condition of small scale yielding is satisfied. However, dl/dlV of a small crack cannot be determined uniquely by AK* because the condition of small scale yielding is not usually satisfied in a small crack. For an in&&e plate with a crack, the value of K is given by the equation K = g&. This equation indicates that the stress range has to be high enough in a small crack in order to get the same growth rate as in a large crack. Therefore, when a sufficiently small crack propagates at a finite growth rate (for example, 1W6-1O-3mm/cycle), the unction of small scale yielding is not satisfied in general. Accordingly, a basic problem in estimating the fatigue life of plain members is to clarify the growth law of a small surface crack. Recently, ~n~de~ng the silent of small cracks in the eviction of fatigue damage, many studies on the behavior of a small crack have been made[l-7]. The present authors have carried out a series of researches concerning the propagation of a small crack, and indicated that dl/dN of a small surface crack is dete~ned uniquely by the term a:l[l, 41,where a,, is the nominal stress amplitude, I is the crack length and n is a constant. In this paper, the new experimental data and results shown in the previous reports are ~o~idc~ s~~eti~lly, and the physical background and effectiveness of the small-crack growth law are discussed. For estimating the fatigue life of plain members, an effective and convenient method based on the small-crack growth law and the ultimate tensile strength is proposed. The validity of this method is checked through its application to the results obtained by the other researchers. 499
HIRONOBU
500
2. PHYSICAL
NISITANI
BACKGROUND
et al.
ON CRACK
GROWTH
2.1. Relation between the crack growth rate and the cyclic plastic zone size Many investigations of fatigue crack propagation have been carried out and many growth laws have been reported (for example, there are 54 crack growth equations in the book written by Kocanda[8]). All of these growth laws have succeeded in describing crack growth behaviors over a limited experimental program, though some of them formally contradict each other. The propagation of a fatigue crack occurs by the accumulation of irreversible deformation at the crack tip. Therefore, the cyclic crack tip displacement CTOD, seems to be appropriate as a measure of crack propagation. However, it is more advantageous to use the cyclic plastic zone size 5, which is closely related to CTOD,, because the definition of CTOD, is not clear, and is somewhat obscure and difficult to measure in polycrystalline materials. In this section, it is shown that the following relation[9] is reasonable for crack propagation.
The experimental results which support this assumption are shown in the following. The plastic zone around a fatigue crack depends on the crack length and the stress level. Figure 1 shows an example of the plastic zone around a crack tip of an Fe-3% Si alloy specimen revealed by etching. Figures l(a) and (b) show the plastic zone on the specimen surface and in the interior, respectively. In the figure, the plastic zone can be divided into two parts; a heavily strained, light etched zone close to the fatigue crack and a lightly strained, dark etched zone formed around the heavily strained zone[lO]. It is not clear how these two types of plastic zone correspond to the monotonic and cyclic loadings. In several studies[l&l2], however, it is suggested that a light etched zone is identified as a cyclically deformed zone and a dark etched zone as a monotonically deformed zone. In this paper, therefore, we assume that the size of the reversible plastic zone corresponds to the size of the light etched zone. As seen from the figure, the closer to the crack tip, the larger becomes the size of plastic zone measured as the extent of the highly deformed zone perpendicular to the direction of crack growth. Figure 2 shows the relation between the crack growth rate and the reversible plastic zone size at a crack tip. The zone sizes on the surface and in the interior are defined by the mean width of the heavily strained zone at both crack tips and the one at a crack tip below a hole bottom, respectively. The crack growth rate on the surface is derived from the relation dl’/dN = (dl/dN)/2 and that in the interior from dl’/dN = db/dN, b = 0.361. For all the stress levels studied, the fatigue crack growth rate is nearly proportional to the reversible plastic zone size both on the surface and in the interior, namely eq. (1) holds regardless of the scale of yielding. 2.2. Crack opening ratio The other phase of the fatigue crack growth is crack closure phenomena[l3]. The crack tip does not open always during the stress cycling. Kikukawa et al.1141 have systematically measured the value of the crack opening ratio, U = Aa,e/Aa (=&,/AK), in many materials. Their results show that U is nearly proportional to AK in the limited range of AK as shown in Fig. 3. Figures 4 and 5 show the results of low cycle bending fatigue tests of annealed 0.45% C steels in SEM[l5]. Figure 4 is the 6/1-x/1 relation. The shape of crack opening is geometrically similar under the constant strain (stress). The value of U is shown in Fig. 5. The value of U is nearly constant independent of both the crack length and strain range, where U is the value measured at the point 10 pm apart from the crack tip and is defined by the following equation.
The results of Figs 4 and 5 suggest that dl/dN under high stress range is proportional to crack length 1. Two relations, UaAK for low stress level with the exception of AK near threshold level and U = constant for high stress level, are available when examining the crack growth laws shown in the following section.
A small-crack growth law and its related phenomena
(a)
(b)
On the
In
the
surtaoe
interior
Fig. 1. The example of a plastic zone around a propagating crack (Aa = 640 MPa, I= 2.6 mm).
501
503
A small-crack growth law and its related phenomena
,.,..‘...ei**l
pkisticztSsizekjE? Fig. 2. The crack growth rate against the reversible plastic zone size.
Fig. 3. Effect of AK on opening ratio U (example taken from Kikukawa er aZ.‘sresults[l4], in-plane bending fatigue of plate specimens).
3. A UNIFYING TREATMENT FOR LARGE CRACK GROWTH AND SMALL CRACK GROWTH
The present authors have reported that dr/dN of a small crack is determined uniquely by the term a:l. In this section, the following two equations are taken as the representative ones expressing crack growth rates of ductile materials, and a unifying treatment of fatigue crack growth is made:
The first equation holds for large cracks (low nominal stress) and the second one for small cracks (high nominal stress). These two equations, however, contradict each other except when m = n = 2 (usually m is about 4 and n is much larger than 2). If we make an assumption that dl/dN is proportional to the reversible plastic zone size $,, the two equations will not contradict each other. The formal contradiction comes from the difference in the ranges where the two equations hold. The ranges are mainly controlled by the stress level. 3.1. A crack growth law
forlarge cracks
In general, eq. (3) gives good results in the case when a large crack propagates by a small nominal stress. In this case, U CCAK holds. This indicates that the effective stress intensity factor l.t 1
0 A 0 *
(5.!!$ l bx 0.4% i 0.8% 0.576 al 1.014 0.729 x 1.198 dE, =0.5%
=0.1
I-
0 jj0.S
P
I
i-
“,A=2ACt 0.4
A&t
Y
0
2
6
4
8 x10-z
x/t Fig. 4. Relation between 611 and x/l mar the crack tip.
0
*r
0 0.50 A 0.66 a 0.9s
30.2 u 0
,
I
0‘5
1.0 1
‘
1,s mm
1
2.0
Fig. 5. Crack opening ratio U of low cycle bending fatigue test.
HIRONOBU NISITANI et al.
504 1
Large cr~ks(~QS4,opproxi~t~ty) UaAK
.’ &K.fr &AKaAK2----4a)
rpr Q: U$ (I : Const).-4 b)
Fig. 6. Schematic explanation of fatigue crack growth laws.
range AK& is proportional to AK*. On the other hand, Ypris considered to be proportional to AK& From these considerations, we can obtain the crack growth law given by eq. (3), namely dl]dN = C AK4 (see Fig. 6). 3.2. A crack growth law for small cracks A small crack does not propagate unless the nominal stress is high enough. Accordingly, when a sufficiently small crack propagates with a finite growth rate (for example, 10-6-10-3 mm/cycle), the condition of small scale yielding is not satisfied. In this case, eq. (4) holds. It is natural to assume that U is nearly constant when the nominal stress is high and the condition of small scale yielding does not hold (Figs 4 and 5). In this case, the dependency of Ypon G and i can be estimated from the dependency of 5, the monotonic plastic zone size, on d and f under u~~~tion~ loading. In the Dugdale model[l(i] for unidirectional loading, the relation ~/Zoc(a/a,,>” holds. The index n is 2 for ct/r.rY 4 1 and as a/a,, tends to unity, the value of n becomes much larger than 2. The plastic zone size 5 is proportional to crack length under constant stress. In this case, eq. (4) holds and is explained on the assumption based on eq, (1) (see Fig. 6). As mentioned above, although eqs (3) and (4) formally contradict each other, these two growth laws are explained consistently from the same physical background, based on an assumption that the crack growth rate is proportional to rp. 3.3. ~~e~tigatio~ of two growth laws based on the exFeri~nta~ results Figure 7 shows the crack growth data for an annealed 0.45% C steel plain specimen under axial directional loading (R = - l)[l]. Figure 7(a) shows the relation between crack length and the relative number of cycles N/N,.. It is found that a crack initiates at N/N,z 0.2 and the fatigue life of a plain specimen is controlled mainly by the time in which a crack propagates from an initial size (10 pm in this material) to about 1 mm; this life time is about 0.7N,. This means that we must know the growth law of small cracks if we want to predict the fatigue life of plain members. Figure 7(b) shows the relation between the crack growth rate and the stress intensity factor range AK. Here AK is the effective parameter for the propogation of large cracks in which the condition of small scale yielding is satisfied. However, dt/dN of a small crack under high stress is not determined uniqueiy by AK, as shown in this figure. Figure 7(c) shows the relation between dZ/dN and the term ail. The growth rate of a small crack is determined uniquely by ail (in this material, n is about 8) and this will be seen more clearly in later figures. In order to measure the crack growth rate at stress levels below the fatigue limit (225 MPa = 0.63a,, uY:yield stress), fatigue tests on drilled specimens (both the diameter and depth are 0.1 mm) and 0.5 mm pre-cracked specimens were also carried out. The dl/dN vs AK relation
A small-crack growth law and its reIated phenomena
so5
(c)
(b)
Fig. 7. Crack growth data for a 0.45% C steel plain specimen: (a) a crack initiates at N/N,1 0.2. The vaha of I is mainly controlled by N/N/ alone; (b) dl/dN is not determined by AK in the case of u, > 0.6u,; (c) d//dN is determined by a:l in the case of a, > 0.6q.
is shown in Fig. 8. dl/d‘dNof a smalf crack in which the condition of small scale yielding is satisfied is dete~n~ ~iq~ly by AK Figure 9 shows the growth data of Fe-3% Si alloy[l’lf which are shown in Figs 1 and 2. It is found that eq. (3) holds for low nominal stress and eq. (4) holds for high nominal stress. Figure 10 shows the growth data of prestrained 0.45% C steel plain specimens ((I; = 550 MPa) under axial directional loading (R = - 1)[18]. Equation (3) holds for the range of u,, < OSa, (Aa < 570 MPa) and eq. (4) for the range of a, > 0.60~ (do > 630 MPa). Although dl/dN for the range of O.&r,< a, < 0,6a, cannot be determined by tbe convenient parameter, this range is relatively narrow and almost all crack growth rates can be evaluated approximately by eqs (3) and (4).
tdl-
3
II AK
10
W
30
Fig. 8. Crack grow& data in 0.45% C steel plain qecimene with a small bfind bofa or 0.5 mm pm-crack, dr/dN Is determined by Aly alone in the case of a, < Ma,. E3T.i 4114-D
HIRONOBU
506
70*
I
50
10
I
NISITANI
er al.
I
AK M?%%T
(a)
(b)
Fig. 9. Crack growth data for an Fe-3% Si aiioy.
The experimental results shown in Figs 7, 9 and 10 indicate that we must use eq. (4) as the growth law of a crack when evaluating the fatigue life of plain members, because the fatigue limit of plain specimens is relatively large (a, > O&r,).
4. APPLICATION OF SMALL-CRACK GROWTH LAW TO THE VARIOUS KXNDS OF FATIGUE DATA Until now, only eq. (3) has been studied for a wide range of conditions. In the present section, the relations between eq. (4) and some factors affecting fatigue damage are investigated. 4.1. The efect
ofloading
pattern on the small-crack growth law
The rotating-bending (R-B) fatigue tests of annealed 0.45% C steel specimens with the same cross-section as the tension-compression (T-C) specimens whose growth data are shown in Fig. 7, were carried out. The results show that no significant difference in the fatigue behavior between R-B loading and T-C loading is observed and the fatigue life of R-B loading is also occupied mainly by the life in which a crack propagates from an initial size up to 1 mm. The growth rate of such a small crack is determined uniquely by eq. (4) with n = 7.5[19]. Figure 11 shows the relation between the aspect ratio and the crack length. Although the aspect ratio of R-B loading decreases with increase in crack length, its value can be regarded as approximately constant in the range smaller than 1 mm and is nearly equal to the value of T-C loading. Figure 12 shows that dl/dN vs ail relation of R-B loading. It is found that eq. (4) is valid for the determination of dZ/dN of small cracks. The dotted straight line is the result of T-C loading. The growth rate of R-B loading is smaller than that of T-C loading, because in the case of R-B loading there is the stress gradient towards the center of the cross-section. Moreover, in order to study the effect of stress ratio R on eq. (4), the fatigue tests of T-C loading with several values of R (R = - 1.5, -0.5 and 0) were carried out. The results show that eq. (4) holds for each R. The effect of R on eq. (4) is represented by the different values of C,[20]. 4.2. The effect of material properties on the small-crack growth law Equation (4) can express the growth rate of a small crack in a given material, but it is not convenient for the comparison of crack growth behaviors among different materials. Therefore,
A small-crack growth law and its related phenomena
507
lo-’
10” -
Tanoion-ceqWee*ion 2-0.3mn-l.Om
VY I a
-
yo-
-
8 ‘I
510-' 9
G
s 8 alo-’ pi
-
“r &
d-O.lm D=5mm
I
10-G ’
I
I
10" =a
Cal
G..t
I
I
I
10'6
10" 9lAa)~'hl
(b)
Fig. 10. Crack growth data for a prestrained 0.45% C steel.
we proposed the following crack growth law in which the effect of material partly considered[ 11,
properties
is
(5) where Q,, is the monotonic yield stress. By using eq. (5), the effect of material properties can be estimated approximately. In comparison of fatigue behavior among different materials, it is more reasonable to use the cyclic yield stress uF instead of the monotonic one uy, although a,, was used in eq. (5) instead of bvc for the sake of simplicity. Tanaka et a1.[21] have inestigated the low-cycle fatigue properties of several different carbon steels. They showed that the excellent correlation between the cyclic yield stress and the ultimate
o
Tanslon-campnsslon(R=-1)
Cr&
length
I2
mm
3
Fig. 11. Relation between aspect ratio and crack length.
10-6 10'7
1
I
I 10'8 6.6
1
I
I
,
1020 10'9 (rP~l4lM
Fig. 12. dl/dN vs a:1 relation (n = 7.5).
508
HIRONOBU NISITANI et al. Table I. Values of C, and C, and n in eqs (4) and (7)
No.
Range of 0, lay
n
c, x 10-2’
c, x lo-)
8.5 9.0 1.5 8.0 7.3 7.5 7.5
6.4 x 10-3 5.4 x lo-’ 1.2 5.1 x lo-* 6.7 1.1 1.1
4.6 x 10-l 8.8 x 10-l 1.3
215-287 230-315 235-358 235-372 279-373
0.87-1.07 0.89-l .07 0.78-1.04 0.72-099 0.71-1.08 0.65-1.02 0.861.09
1:+: 9:9 17.4
0.3-1.0 0.5-2.0 0.05-1.5 0.05-1.5 0.05-1.5 0.05-1.5 O-5-2.5
of normalimd3y;bon 8 8 353 8 433
stcci (8-10) 265-320 568 290-350 721 340430
0.824.99 0.82-0.99 0.78-0.99
7.5 8.5 9.5
2.3 3.6 x lO-3 1.2 x 10-b
3.9 9.3 17.0
0.05-l .5 0.3-2.0 0.3-2.0
of heat-treaW4;bon 8 8 750 8 821 8 975 8 1068 5 1161 5 1376 8 1353
steels (1 l-18) 673 380-480 833 455-700 966 500-650 1112 L%o-800 1224 700-850 1243 700-1000 1510 800-900 1571 85WWl
0.78-0.98 0.61-0.93 0.61-0.80 0.62-0.82 0.66-0.80 0.60-0.94 0.58-0.65 0.63-0.67
10 9.3 7.2 6.0 5.3 5.0 3.0 2.0
4.6 x lo-’ 3.4 x IO-’
8.8 5.0 4.3 2.3
0.05-2.0 0.05-2.0 0.2-2.0 0.2-2.0 0.2-2.0 0.3-l .o OS-l.5 0.2-2.0
Material
(a) For th;lz of annealed carbon steels (I-7) 1 SlOC 85 224 206 380 372 22iL240 190-220 : 5 6 I (b) p 9 10
SZOC s35c s35c s45c s5oc thi2a; s35c S45c
(c) For the we 11 s35c 12 s45c 13 s45c 14 s45c 15 s45c 16 s5oc 17 s45c 18 SW
5 : 1:
317 276 331 361 347
469 512 592 631 674
7:3 t*; :: x 5.4 x 3.5 x 4.9 x
z-’ 103 104 .^ 10’” IO’”
::: ::5
Range of 1 (mm)
d: Diameter of the specimen.
tensile strength a, holds independent of the ma&id properties, and the cyclic yield stress is related to the ultimate tensile strength as follows: hue= 0.608a,.
(6)
Accordingly, it is regarded that but should be considered in the follo~ng law[22]: (7) Here, a reciprocal of constant C, represents the resistance for crack propagation in each material. Table 1 compares the constants C1 and C, for various carbon steels under rotating-bending fatigue. In the comparison of ex~~rnen~l results under the different conditions, it is necessary to know the effects of size and shape of a specimen on the growth law. However, it has been found that these effects are negligible[lg]. As seen in Table 1, C, is a stable constant for materials with similar mechanical properties and therefore is suitable for the comparison of different materials. 4.3. Method for inferring the constants C3 and n in eq. (7) Figures 13 and 14 show the C, vs a, and the n vs a, relations for rotating-bending fatigue, respectively[23,24]. For both &ures, good correlation is obtained in each heat treatment condition, namely C,=Aaf n=Eu,+F.
(8) (9)
The constants A, D, E and F are as follows: A = 5.8 x lo-“, D = 6.2, E = -4.7 x iOm3 and F = 10.5 for annealed steels; A = 5.3 x lo-“, D = 3.7, E = -7.9 x IO-” and F = 13.1 foi normalized steels; and A = 9.8 x lO’, D = -25 E = - 9.8 x 10m3and F = 16.5 for heat-treated steels. Therefore, we can infer the constants C, and n from the vale of a,, in carbon steels. Figure 15 shows the a, vs H,, Vickers hardness, relation, The following relation holds between a,, and H,: a” = G&l,
(10)
A small-crack growth law and its related phcnotncna
509
r
Fig. 14. n vs u, relation for rotating-bending fatigue.
Fig. 13. C, vs u, relation for rotating-bending fatigue.
where C, = 3.53 for annealed and normalized steels and C, = 3.08 for heat-treated steels. This means that we can infer the constants C, and n by using the hardness which can be measured easily. 5. METHOD FOR PREDICTING FATIGUE LIFE OF CARBON STEEL PLAIN MEMBERS 5.1. Fatigue lge prediction based on the small-crack growth law
As shown in the previous sections, the growth rate of a small crack is determined uniquely by eq. (7). Accordingly, when we want to predict the fatigue life of plain specimens, eq. (7) can be used as a means for prediction. The present authors have carried out the fatigue tests of various kinds of carbon steel plain specimens having a diameter of 5 mm, and obtained the results that: (i) fatigue life is controlled mainly by the propagation life of a crack smaller than 1 mm; and (ii) dZ/dPJ of a crack in the range of 0.05 mm c f < 1 mm is decent uniquely by eq. (7). A~o~in~y, the small-crack propagation life of specimens of 5 mm in diameter can be written as follows, based on eq. (7):
(11)
0
loo
Vickers
200
300 hardness
Fig. 15. a, vs H, relation.
400 Hv
!m
HIRONOBU NISITANI et al.
510
Here, putting lo= 0.05 mm and I, = 1 mm:
The fatigue life of plain specimens of 5 mm in diameter Nf, can be represented by the next relation: 03) where No,oS is the initiation life of a crack of 0.05 mm and N,,is the life in which a crack propagates from 1 mm up to failure. On the other hand, the ratio of total fatigue life to small-crack propa~tion life 0 = 0.05+ 1 mm), N0,05-r1 /N,, was measured for the various kinds of steels, and the next relation nearly holds for each stee1[23,24]: (14) From eqs (12) and (14), the fatigue life of a plain specimen of 5 mm in diameter N,, can be predicted approximately by eq. (15).
The second phase of the investigation in this section deals with the fatigue life prediction of specimens having an arbitrary diameter d (d = a$, a & 1,4 = 5 mm). Figure 16 shows the crack growth curves of the geometrically similar specimens (ratio of similarity is 1: 2). The material used was an annealed 0.39% C steel. The load was rotating-bending and the specimens had a small blind hole (both the diameter and dgrth of hole were 0.3 mm) in the central part. The ordinate is the logarithm of relative crack length l/d and the abscissa is the number of cycles counted from the time when l/d = 0.08, that is, I = 0.5 and 1.0 mm for d = 6 and 12 mm, respectively. The following is apparent from Fig. 16: dl/dN in the geometrically similar specimens subjected to the same stress is proportional to the d&meter of the specimen. This also supports the assumption of eq. (1). In order to evaluate the effect of an increase in diameter on the fatigue life, it is advantageous to divide the fatigue life into the following four stages: (1) (2) (3) (4)
crack initiation life; propagation life of a microcrack {propagation behavior is affected by the ~~os~ct~e); propa~tion period in which eq. (4) holds; unstable propagation life up to failure [eq. (4) does not hold in this period].
When the specimen diameter increases, the life in stages (1) and (2) decreases because of a decrease in stress gradient. However, we neglect the decrease in lives of these stages by taking account of the fact that the lives of stages (1) and (2) are relatively small compared with the total fatigue life. In stages (3) and (4), paying attention to the results of Fig. 16, it is concluded that the propagation life from &,up to failure for a smail specimen (4 in diameter) is equal to the life
Fig. 16. Crack growth curves of 0.39% C steel similar spwimtns having a small blind hole.
511
A small-crack growth law and its reiated phenomena
from al0 up to failure for a large specimen (ad in diameter). Therefore, the propagation life from & up to al,, is the increase in life due to the increasing diameter. Putting the life of each stage as follows: (1) + (2) = N0.0S,(3) = lvo.DS_cr, and (4) = N,,, the fatigue life of a specimen of d = ctd,,in diameter Nf, is expressed by N,, = No.05 + Noas-, + K-r (16)
= No.05 + ~0.0S~0.0k + No,os+0: + L/.
Here, Fig. 16 indicates that the relations of No,o5 _ t = NOoh_* and Nt _,,z N, + hold. From eqs (13) and (14), No.oS + N1+ is nearly equal to N,,,,, . That is, eq. (16) is rewritten as follows: (17)
+# = No.05 -@ 0.0% + 2No.os +1 *
By using eqs (11) and (12), the fatigue life of a plain specimen of d = 5a mm in diameter, I$, can be predicted by the following equation: N,a= & : 3 0
a (ln a + 6).
(18)
11
For rotating-bending fatigue, the constants C3 and n can be determined by eqs (8) and (9), respectively, for the given value of 6,. 5.2. App~ic~t~n of fatigue life prediction method In order to confirm the validity of the fatigue life prediction method of eq. (18), its application to the rotating-bending fatigue data published by the National Research Institute for Metals (NRIM)[25] and the data of Cazaud et aZ.[26]was carried out. Figures 17 and 18 show the application of eq. (18) to the NRIM data and the experimental results of Caxaud et al., respectively. The prediction is in good agreement with the ex~~mental results over a wide range, with the exception of stresses very near the fatigue limit (a, < 1.OSa,, 0, : fatigue limit). Here, fatigue limits of steels are approximated by the relation G, r aJ2, except for extremely hard steel whose tensile strength is larger than 1500MPa. Accordingly, the lower limiting
Annralcd,S?SMPa
e200 Gi
100
I ’
‘*
I
I
Numt&
cycles
loo-
I t%
llurr
Nt
”
Fig. 17. Eat&nation of fatigue __--__ life: _ appfication of eq. (18) to NRIM
data.
Rotating bending of 0.36% C strrl -:Estimation bawd on Eq.(16) 0,&0:6xperim~nt by R.Cazaud -*-:cf.~l.OSu” H-t : Heat-heated
pzooc 5 _
-:Eatimation bawd on Eq.(l8) O,A,A:Expwlmmt by NRtM ----:apl.o5d, H-T :Heat -Trca ted N :Normalired
I
I O
10‘
I
5
Number ‘%t cycles
1
I
a
tdOtailurc
I Nt
10’
Fig. 18. Estimation of._fatigue .life: application _ _ of aq. (18) to caajutt’s
expennmtai
mutts.
HIRONOBU NISITANI et al.
512
Fig. 19. Schematic explanation
of fatigue life prediction method for the rotating-bending carbon steel plain specimens.
fatigue of
stress of the predicting method of eq. (18) is given approximately by the relation 6. > 0.53 0, (= l.O5a,), and this method is valid as long as o, is smaller than the yield stress. The schematic explanation of a fatigue life predicting method is shown in Fig. 19. It can be concluded that eq. (18) is effective and convenient for predicting the fatigue life of carbon steel plain members, because the evaluation of fatigue life is easily attained by using only the ultimate tensile strength with the type of heat treatment neglecting the effects of microstructure, chemical composition, a slight difference in temperature for heat treatments, etc. 6. CONCLUSIONS The follo~ng conclusions were reached with regard to a small-crack growth law of carbon steel plain specimens under constant stress amplitudes. (1) The crack growth of a plain specimen is not determined by the stress intensity factor range AK. The growth rate increases with an increase in the applied stress range for a given value of AK and the growth rate of a plain specimen is determined uniquely by the term ail, where 6, is nominal stress amplitude, 1 is crack length, and n is a constant. (2) The growth rate of fatigue cracks under a wide range of o;, can be determined by the following two growth laws. (a) When a crack is propagating under low stress,
(b) When a crack is propagating under high nominal stress,
(3) Both these crack growth laws can be explained consistently from the same physical background, based on the assumption that crack growth rate is proportional to the reversible plastic zone size Y, . The background of this assumption was investigated by using the experimental results of F+3% Si alloy specimens.
A small-crack growth law and its related phenomena
513
(4) At high stress ranges, the growth rate of a small crackin one material cannot compare with that of a different material. Therefore the following small-crack growth law is proposed: ’ where r~,is the ultimate tensile strength. In this equation, the effect of material properties is partly considered. The constant l/C, represents the resistance to crack propagation in each material and is a stable constant for those materials with similar rn~ha~~al properties. (5) An effective and convenient method based on the small-crack growth law and ~tirna~ tensile strength was proposed for predicting the fatigue life of plain members. The validity of this method was confirmed from its application to the data reported by other researchers. REFERENCES PI H. Nisitani and M. Goto, A small crack growth law and its application to the evaluation of fatigue life, in The lkhcuicnr
ofShorr Fatigue Cracks (Edited by K. J. Miller and E. R. de 10s Rios), pp. 461478.
Mech. Eng. Publiiers (1987).
PI H. Kitagawa, S. Takahashi, C. M. Suh and S. Miyashita, Quantitative analysts of fatigue pmcus-microcracks and
slip lines under cyclic strains. ASTM STP 675,4204t9 (1979). I31 M. H. El Haddad, N. E. Dowling, T. H. Topper and K. N. Smith, I integral applications for short fatigue cracks at notches. fnr. $. Fractm 14, 15-30 (1980). cracks, in Mec!ro&es [41 H. Nisitani, Unif~~ treatment of fat&m crack growth laws in small, large and nonpro~~ti~ o~~u~i~-~~~ (Edited by T. Mura), pp. 151-166. ASME (1981). S. J. Hudak, Jr, Small crack behavior and the prediction of fatigue life. Trcx.r. ASME H&26-35 (1981). ;I K. J. Miller, The short crack probiem. Fatigue ZGtgngMuter. Structwe~ 5, 223-232 (1982). VI M. Goto, Statistical investigation of the behaviour of microcracks in carbon steels. Fu!igue Brgrtg Muter. Srruetuns 14, 833-845 (1991). &I S. Kocanda, Fufigue Failure of Metals. Sijthoff & Noordhoff, Amsterdam (1978). energy approach. Trans. ASME, J. bar. &gng 191H. W. Liu, Fatigue crack propagation and applied stress ran-an 85, 116-122 (1963). [lOI G. T. Hahn, R. G. Hoagland and A. R. Rosenfield, Local yielding attending fatigue crack growth. Metal/. Trans. 3, 1189-1202 (1972). [ill K. Ando, N. Ogura and T. Nishioka, Effect of grain size on the fatigue crack propagation rate and plastic behavior near a fatigue crack. Truns. Jpn Sot. Mech. Engng 44,40144023 (1978). I121 K. Tanaka, M. Hojo and Y. Nakai, Fatigue crack initiation and early propagation in 3% silicon iron. ASTM STP 811, 207-232 (1983). W. E&r, The ~~ifi~~ of fatigue crack closure, ASTM STP rsS, 230-242 (1971). f:$ M. Kikukawa, M. Jono and K. Tanaka, Fatigue crack cfosure behavior at low stress intensity kvel. Proc. znd Znf. Co& Me&. Behaviot of ~o~er~~ (pp. 716720) (1976). I151 H. Nisitani, Y. Gda and T. Yakus~ji~~onside~tio~s of the smallcrack growth law based on CGD-low-cycle bending fatigue test of an annealed 0.45% steel in the SEM. JSME Int. 1. 33. 243-248 f1990). D. 8. Dugdale, Yielding of steel sheets containing slits. J. Mech. Phyk Soli& & 10ollOS (1960). ;I; H. Nisitani and N. Kawagoishi, Relation between fatigue crack growth law and reversible plastic zone size in Fe-3% Si alloy. Proc. 6rh Inr. Cong. Experimenral Mechanics (pp. 795-800) (1988). VI H. Nisitani and M. Goto, Relation between the shape of S-N curve and the crack growth law in prestrained 0.45% C steel specimens with a small hole. Trans. Jpn Sot. Mech. Engng 51, 1430-1435 (1985). WI H. Nisitani and M. Goto, Relation between small crack growth law and fatigue life of machines and structures, Trans. Jpn Sot. Mech. Engng 51, 332-341 (1985). 1201H. Nisitani and M. Goto, Effect of stress ratio on the propagation of small crack of plain specimens under high and low stress amplitudes-fatigue under axial loading of annealed 0.45% C steel. Trans. Jpn Sot. Mech. Engttg SO, 1090-1096 (1984). 1211K. Tanaka, S. Nis~ji~, S. Matsuoka, T. Abe and F. Kouzu, Low and high-cycle fatigue properties of various steels specified in JIS for machine structural use. Fczigne Engng Mater. Strnclures 4, 97-108 (1981). 1221M. Goto and H. Nisitani, Inference of a small-crack growth law based on the S-N curves. Truns. J&nSot. &fee/t.Engng 56, 1938-1944 (1990). ~231H, Nisitani and M. Goto and H. Miyagawa, A method for predicting fatigue life of carbon steel plain members, Trans. Jpn Sot. Mech. Engng 53, 2238-2247 (1987). 1241M. Goto, H. Nisitani, H. Miyagawa and T. Abe, A method for predicting fatigue life of heat-treated low alloy steels. Truns. Jpn Sot. Me& Engng 56, 1011-1019 (1990). National Research Institute for Metals, NRIM Fatigue Data Sheet, Nos 1, 2, 3 and 4 (1978). t;:; R. Caxaud, G. Pomey, P. Rabbe and Ch. Janssen, La Farigue des Metaux. Dun& Paris (1969) (Recebed 5 February 1991)
fh&tvr& Fracture ,W&adca Vol. 41, No. 4, pp. 499-513, 1992 P&W in Great Britain.
A SMALL-CRACK GROWTH LAW AND ITS RELATED PHENOMENA Department of Magi
HIRONOBU NISITANI and Strength of Solids, Kyushu W~~ity,
Department of Mad
Fukuoka, 812, Japan
MASAHIRO GOT0 En~n~~S, Oita Unive~ity, Oita, 870-l 1, Japan
NORIO KAWAGOISHI Faculty of Engineering, Kagoshima University, Kagos~m~ 890, Japan Ahatraet-In modern design, it is an important task to estimate fatigue life. The growth law of a small crack must be known in order to estimate the fatigue life of machines and structurea, because the fatigue life of members is controlled mainly by the behavior of a small crack. The growth rate of a small crackcannot usually be predicted by linear elastic fracture mechanics, but is determined uniquely by the term u:l. In this paper, two fatigue crack growth laws, dlfdN = C AK’and dildiV = C,o:l, which hold in large and small cracks, mapectively, are taken as the ~m~~tive onea expressing the crack growth rate of ductile materials. Many experimental results indicate that the relation dtfd.NccdP holds under low nominal stresses and the relation dt/dNoca:l holds under high nominal stresses. A u&@&g explanation for two growth laws is made based on an assumption that the crack growth rate is proportional to the reversible plastic zone size. Moreover, an effective and convenient method based on a smalLcrack growth law, dl/dN = C, (a,,/o,)V, in which the e&t of mechanical properties is partly considered, is proposed for predicting the fatigue life of plain members, where a, is the ultimate tensile strength. The validity of this method is confirmed through its a~li~tio~ to other researchers’ data.
IN MODERNdesign,
it is an important task to estimate fatigue life precisely. The growth law of a small crack must be known in order to estimate the fatigue life, because the fatigue life of members is controlled mainly by the propagation life of a small crack. For example, in the fatigue tests of annealed carbon steels, about 70% of fatigue life of a plain specimen is occupied by the life in which a crack propagates from an initial size up to 1 mm[i]. Until now, many researchers have reported that the growth rate of fatigue cracks can be determined in terms of the function of stress intensity factor range AK. AK is the effective parameter for the cracks when the condition of small scale yielding is satisfied. However, dl/dlV of a small crack cannot be determined uniquely by AK* because the condition of small scale yielding is not usually satisfied in a small crack. For an in&&e plate with a crack, the value of K is given by the equation K = g&. This equation indicates that the stress range has to be high enough in a small crack in order to get the same growth rate as in a large crack. Therefore, when a sufficiently small crack propagates at a finite growth rate (for example, 1W6-1O-3mm/cycle), the unction of small scale yielding is not satisfied in general. Accordingly, a basic problem in estimating the fatigue life of plain members is to clarify the growth law of a small surface crack. Recently, ~n~de~ng the silent of small cracks in the eviction of fatigue damage, many studies on the behavior of a small crack have been made[l-7]. The present authors have carried out a series of researches concerning the propagation of a small crack, and indicated that dl/dN of a small surface crack is dete~ned uniquely by the term a:l[l, 41,where a,, is the nominal stress amplitude, I is the crack length and n is a constant. In this paper, the new experimental data and results shown in the previous reports are ~o~idc~ s~~eti~lly, and the physical background and effectiveness of the small-crack growth law are discussed. For estimating the fatigue life of plain members, an effective and convenient method based on the small-crack growth law and the ultimate tensile strength is proposed. The validity of this method is checked through its application to the results obtained by the other researchers. 499
HIRONOBU
500
2. PHYSICAL
NISITANI
BACKGROUND
et al.
ON CRACK
GROWTH
2.1. Relation between the crack growth rate and the cyclic plastic zone size Many investigations of fatigue crack propagation have been carried out and many growth laws have been reported (for example, there are 54 crack growth equations in the book written by Kocanda[8]). All of these growth laws have succeeded in describing crack growth behaviors over a limited experimental program, though some of them formally contradict each other. The propagation of a fatigue crack occurs by the accumulation of irreversible deformation at the crack tip. Therefore, the cyclic crack tip displacement CTOD, seems to be appropriate as a measure of crack propagation. However, it is more advantageous to use the cyclic plastic zone size 5, which is closely related to CTOD,, because the definition of CTOD, is not clear, and is somewhat obscure and difficult to measure in polycrystalline materials. In this section, it is shown that the following relation[9] is reasonable for crack propagation.
The experimental results which support this assumption are shown in the following. The plastic zone around a fatigue crack depends on the crack length and the stress level. Figure 1 shows an example of the plastic zone around a crack tip of an Fe-3% Si alloy specimen revealed by etching. Figures l(a) and (b) show the plastic zone on the specimen surface and in the interior, respectively. In the figure, the plastic zone can be divided into two parts; a heavily strained, light etched zone close to the fatigue crack and a lightly strained, dark etched zone formed around the heavily strained zone[lO]. It is not clear how these two types of plastic zone correspond to the monotonic and cyclic loadings. In several studies[l&l2], however, it is suggested that a light etched zone is identified as a cyclically deformed zone and a dark etched zone as a monotonically deformed zone. In this paper, therefore, we assume that the size of the reversible plastic zone corresponds to the size of the light etched zone. As seen from the figure, the closer to the crack tip, the larger becomes the size of plastic zone measured as the extent of the highly deformed zone perpendicular to the direction of crack growth. Figure 2 shows the relation between the crack growth rate and the reversible plastic zone size at a crack tip. The zone sizes on the surface and in the interior are defined by the mean width of the heavily strained zone at both crack tips and the one at a crack tip below a hole bottom, respectively. The crack growth rate on the surface is derived from the relation dl’/dN = (dl/dN)/2 and that in the interior from dl’/dN = db/dN, b = 0.361. For all the stress levels studied, the fatigue crack growth rate is nearly proportional to the reversible plastic zone size both on the surface and in the interior, namely eq. (1) holds regardless of the scale of yielding. 2.2. Crack opening ratio The other phase of the fatigue crack growth is crack closure phenomena[l3]. The crack tip does not open always during the stress cycling. Kikukawa et al.1141 have systematically measured the value of the crack opening ratio, U = Aa,e/Aa (=&,/AK), in many materials. Their results show that U is nearly proportional to AK in the limited range of AK as shown in Fig. 3. Figures 4 and 5 show the results of low cycle bending fatigue tests of annealed 0.45% C steels in SEM[l5]. Figure 4 is the 6/1-x/1 relation. The shape of crack opening is geometrically similar under the constant strain (stress). The value of U is shown in Fig. 5. The value of U is nearly constant independent of both the crack length and strain range, where U is the value measured at the point 10 pm apart from the crack tip and is defined by the following equation.
The results of Figs 4 and 5 suggest that dl/dN under high stress range is proportional to crack length 1. Two relations, UaAK for low stress level with the exception of AK near threshold level and U = constant for high stress level, are available when examining the crack growth laws shown in the following section.
A small-crack growth law and its related phenomena
(a)
(b)
On the
In
the
surtaoe
interior
Fig. 1. The example of a plastic zone around a propagating crack (Aa = 640 MPa, I= 2.6 mm).
501
503
A small-crack growth law and its related phenomena
,.,..‘...ei**l
pkisticztSsizekjE? Fig. 2. The crack growth rate against the reversible plastic zone size.
Fig. 3. Effect of AK on opening ratio U (example taken from Kikukawa er aZ.‘sresults[l4], in-plane bending fatigue of plate specimens).
3. A UNIFYING TREATMENT FOR LARGE CRACK GROWTH AND SMALL CRACK GROWTH
The present authors have reported that dr/dN of a small crack is determined uniquely by the term a:l. In this section, the following two equations are taken as the representative ones expressing crack growth rates of ductile materials, and a unifying treatment of fatigue crack growth is made:
The first equation holds for large cracks (low nominal stress) and the second one for small cracks (high nominal stress). These two equations, however, contradict each other except when m = n = 2 (usually m is about 4 and n is much larger than 2). If we make an assumption that dl/dN is proportional to the reversible plastic zone size $,, the two equations will not contradict each other. The formal contradiction comes from the difference in the ranges where the two equations hold. The ranges are mainly controlled by the stress level. 3.1. A crack growth law
forlarge cracks
In general, eq. (3) gives good results in the case when a large crack propagates by a small nominal stress. In this case, U CCAK holds. This indicates that the effective stress intensity factor l.t 1
0 A 0 *
(5.!!$ l bx 0.4% i 0.8% 0.576 al 1.014 0.729 x 1.198 dE, =0.5%
=0.1
I-
0 jj0.S
P
I
i-
“,A=2ACt 0.4
A&t
Y
0
2
6
4
8 x10-z
x/t Fig. 4. Relation between 611 and x/l mar the crack tip.
0
*r
0 0.50 A 0.66 a 0.9s
30.2 u 0
,
I
0‘5
1.0 1
‘
1,s mm
1
2.0
Fig. 5. Crack opening ratio U of low cycle bending fatigue test.
HIRONOBU NISITANI et al.
504 1
Large cr~ks(~QS4,opproxi~t~ty) UaAK
.’ &K.fr &AKaAK2----4a)
rpr Q: U$ (I : Const).-4 b)
Fig. 6. Schematic explanation of fatigue crack growth laws.
range AK& is proportional to AK*. On the other hand, Ypris considered to be proportional to AK& From these considerations, we can obtain the crack growth law given by eq. (3), namely dl]dN = C AK4 (see Fig. 6). 3.2. A crack growth law for small cracks A small crack does not propagate unless the nominal stress is high enough. Accordingly, when a sufficiently small crack propagates with a finite growth rate (for example, 10-6-10-3 mm/cycle), the condition of small scale yielding is not satisfied. In this case, eq. (4) holds. It is natural to assume that U is nearly constant when the nominal stress is high and the condition of small scale yielding does not hold (Figs 4 and 5). In this case, the dependency of Ypon G and i can be estimated from the dependency of 5, the monotonic plastic zone size, on d and f under u~~~tion~ loading. In the Dugdale model[l(i] for unidirectional loading, the relation ~/Zoc(a/a,,>” holds. The index n is 2 for ct/r.rY 4 1 and as a/a,, tends to unity, the value of n becomes much larger than 2. The plastic zone size 5 is proportional to crack length under constant stress. In this case, eq. (4) holds and is explained on the assumption based on eq, (1) (see Fig. 6). As mentioned above, although eqs (3) and (4) formally contradict each other, these two growth laws are explained consistently from the same physical background, based on an assumption that the crack growth rate is proportional to rp. 3.3. ~~e~tigatio~ of two growth laws based on the exFeri~nta~ results Figure 7 shows the crack growth data for an annealed 0.45% C steel plain specimen under axial directional loading (R = - l)[l]. Figure 7(a) shows the relation between crack length and the relative number of cycles N/N,.. It is found that a crack initiates at N/N,z 0.2 and the fatigue life of a plain specimen is controlled mainly by the time in which a crack propagates from an initial size (10 pm in this material) to about 1 mm; this life time is about 0.7N,. This means that we must know the growth law of small cracks if we want to predict the fatigue life of plain members. Figure 7(b) shows the relation between the crack growth rate and the stress intensity factor range AK. Here AK is the effective parameter for the propogation of large cracks in which the condition of small scale yielding is satisfied. However, dt/dN of a small crack under high stress is not determined uniqueiy by AK, as shown in this figure. Figure 7(c) shows the relation between dZ/dN and the term ail. The growth rate of a small crack is determined uniquely by ail (in this material, n is about 8) and this will be seen more clearly in later figures. In order to measure the crack growth rate at stress levels below the fatigue limit (225 MPa = 0.63a,, uY:yield stress), fatigue tests on drilled specimens (both the diameter and depth are 0.1 mm) and 0.5 mm pre-cracked specimens were also carried out. The dl/dN vs AK relation
A small-crack growth law and its reIated phenomena
so5
(c)
(b)
Fig. 7. Crack growth data for a 0.45% C steel plain specimen: (a) a crack initiates at N/N,1 0.2. The vaha of I is mainly controlled by N/N/ alone; (b) dl/dN is not determined by AK in the case of u, > 0.6u,; (c) d//dN is determined by a:l in the case of a, > 0.6q.
is shown in Fig. 8. dl/d‘dNof a smalf crack in which the condition of small scale yielding is satisfied is dete~n~ ~iq~ly by AK Figure 9 shows the growth data of Fe-3% Si alloy[l’lf which are shown in Figs 1 and 2. It is found that eq. (3) holds for low nominal stress and eq. (4) holds for high nominal stress. Figure 10 shows the growth data of prestrained 0.45% C steel plain specimens ((I; = 550 MPa) under axial directional loading (R = - 1)[18]. Equation (3) holds for the range of u,, < OSa, (Aa < 570 MPa) and eq. (4) for the range of a, > 0.60~ (do > 630 MPa). Although dl/dN for the range of O.&r,< a, < 0,6a, cannot be determined by tbe convenient parameter, this range is relatively narrow and almost all crack growth rates can be evaluated approximately by eqs (3) and (4).
tdl-
3
II AK
10
W
30
Fig. 8. Crack grow& data in 0.45% C steel plain qecimene with a small bfind bofa or 0.5 mm pm-crack, dr/dN Is determined by Aly alone in the case of a, < Ma,. E3T.i 4114-D
HIRONOBU
506
70*
I
50
10
I
NISITANI
er al.
I
AK M?%%T
(a)
(b)
Fig. 9. Crack growth data for an Fe-3% Si aiioy.
The experimental results shown in Figs 7, 9 and 10 indicate that we must use eq. (4) as the growth law of a crack when evaluating the fatigue life of plain members, because the fatigue limit of plain specimens is relatively large (a, > O&r,).
4. APPLICATION OF SMALL-CRACK GROWTH LAW TO THE VARIOUS KXNDS OF FATIGUE DATA Until now, only eq. (3) has been studied for a wide range of conditions. In the present section, the relations between eq. (4) and some factors affecting fatigue damage are investigated. 4.1. The efect
ofloading
pattern on the small-crack growth law
The rotating-bending (R-B) fatigue tests of annealed 0.45% C steel specimens with the same cross-section as the tension-compression (T-C) specimens whose growth data are shown in Fig. 7, were carried out. The results show that no significant difference in the fatigue behavior between R-B loading and T-C loading is observed and the fatigue life of R-B loading is also occupied mainly by the life in which a crack propagates from an initial size up to 1 mm. The growth rate of such a small crack is determined uniquely by eq. (4) with n = 7.5[19]. Figure 11 shows the relation between the aspect ratio and the crack length. Although the aspect ratio of R-B loading decreases with increase in crack length, its value can be regarded as approximately constant in the range smaller than 1 mm and is nearly equal to the value of T-C loading. Figure 12 shows that dl/dN vs ail relation of R-B loading. It is found that eq. (4) is valid for the determination of dZ/dN of small cracks. The dotted straight line is the result of T-C loading. The growth rate of R-B loading is smaller than that of T-C loading, because in the case of R-B loading there is the stress gradient towards the center of the cross-section. Moreover, in order to study the effect of stress ratio R on eq. (4), the fatigue tests of T-C loading with several values of R (R = - 1.5, -0.5 and 0) were carried out. The results show that eq. (4) holds for each R. The effect of R on eq. (4) is represented by the different values of C,[20]. 4.2. The effect of material properties on the small-crack growth law Equation (4) can express the growth rate of a small crack in a given material, but it is not convenient for the comparison of crack growth behaviors among different materials. Therefore,
A small-crack growth law and its related phenomena
507
lo-’
10” -
Tanoion-ceqWee*ion 2-0.3mn-l.Om
VY I a
-
yo-
-
8 ‘I
510-' 9
G
s 8 alo-’ pi
-
“r &
d-O.lm D=5mm
I
10-G ’
I
I
10" =a
Cal
G..t
I
I
I
10'6
10" 9lAa)~'hl
(b)
Fig. 10. Crack growth data for a prestrained 0.45% C steel.
we proposed the following crack growth law in which the effect of material partly considered[ 11,
properties
is
(5) where Q,, is the monotonic yield stress. By using eq. (5), the effect of material properties can be estimated approximately. In comparison of fatigue behavior among different materials, it is more reasonable to use the cyclic yield stress uF instead of the monotonic one uy, although a,, was used in eq. (5) instead of bvc for the sake of simplicity. Tanaka et a1.[21] have inestigated the low-cycle fatigue properties of several different carbon steels. They showed that the excellent correlation between the cyclic yield stress and the ultimate
o
Tanslon-campnsslon(R=-1)
Cr&
length
I2
mm
3
Fig. 11. Relation between aspect ratio and crack length.
10-6 10'7
1
I
I 10'8 6.6
1
I
I
,
1020 10'9 (rP~l4lM
Fig. 12. dl/dN vs a:1 relation (n = 7.5).
508
HIRONOBU NISITANI et al. Table I. Values of C, and C, and n in eqs (4) and (7)
No.
Range of 0, lay
n
c, x 10-2’
c, x lo-)
8.5 9.0 1.5 8.0 7.3 7.5 7.5
6.4 x 10-3 5.4 x lo-’ 1.2 5.1 x lo-* 6.7 1.1 1.1
4.6 x 10-l 8.8 x 10-l 1.3
215-287 230-315 235-358 235-372 279-373
0.87-1.07 0.89-l .07 0.78-1.04 0.72-099 0.71-1.08 0.65-1.02 0.861.09
1:+: 9:9 17.4
0.3-1.0 0.5-2.0 0.05-1.5 0.05-1.5 0.05-1.5 0.05-1.5 O-5-2.5
of normalimd3y;bon 8 8 353 8 433
stcci (8-10) 265-320 568 290-350 721 340430
0.824.99 0.82-0.99 0.78-0.99
7.5 8.5 9.5
2.3 3.6 x lO-3 1.2 x 10-b
3.9 9.3 17.0
0.05-l .5 0.3-2.0 0.3-2.0
of heat-treaW4;bon 8 8 750 8 821 8 975 8 1068 5 1161 5 1376 8 1353
steels (1 l-18) 673 380-480 833 455-700 966 500-650 1112 L%o-800 1224 700-850 1243 700-1000 1510 800-900 1571 85WWl
0.78-0.98 0.61-0.93 0.61-0.80 0.62-0.82 0.66-0.80 0.60-0.94 0.58-0.65 0.63-0.67
10 9.3 7.2 6.0 5.3 5.0 3.0 2.0
4.6 x lo-’ 3.4 x IO-’
8.8 5.0 4.3 2.3
0.05-2.0 0.05-2.0 0.2-2.0 0.2-2.0 0.2-2.0 0.3-l .o OS-l.5 0.2-2.0
Material
(a) For th;lz of annealed carbon steels (I-7) 1 SlOC 85 224 206 380 372 22iL240 190-220 : 5 6 I (b) p 9 10
SZOC s35c s35c s45c s5oc thi2a; s35c S45c
(c) For the we 11 s35c 12 s45c 13 s45c 14 s45c 15 s45c 16 s5oc 17 s45c 18 SW
5 : 1:
317 276 331 361 347
469 512 592 631 674
7:3 t*; :: x 5.4 x 3.5 x 4.9 x
z-’ 103 104 .^ 10’” IO’”
::: ::5
Range of 1 (mm)
d: Diameter of the specimen.
tensile strength a, holds independent of the ma&id properties, and the cyclic yield stress is related to the ultimate tensile strength as follows: hue= 0.608a,.
(6)
Accordingly, it is regarded that but should be considered in the follo~ng law[22]: (7) Here, a reciprocal of constant C, represents the resistance for crack propagation in each material. Table 1 compares the constants C1 and C, for various carbon steels under rotating-bending fatigue. In the comparison of ex~~rnen~l results under the different conditions, it is necessary to know the effects of size and shape of a specimen on the growth law. However, it has been found that these effects are negligible[lg]. As seen in Table 1, C, is a stable constant for materials with similar mechanical properties and therefore is suitable for the comparison of different materials. 4.3. Method for inferring the constants C3 and n in eq. (7) Figures 13 and 14 show the C, vs a, and the n vs a, relations for rotating-bending fatigue, respectively[23,24]. For both &ures, good correlation is obtained in each heat treatment condition, namely C,=Aaf n=Eu,+F.
(8) (9)
The constants A, D, E and F are as follows: A = 5.8 x lo-“, D = 6.2, E = -4.7 x iOm3 and F = 10.5 for annealed steels; A = 5.3 x lo-“, D = 3.7, E = -7.9 x IO-” and F = 13.1 foi normalized steels; and A = 9.8 x lO’, D = -25 E = - 9.8 x 10m3and F = 16.5 for heat-treated steels. Therefore, we can infer the constants C, and n from the vale of a,, in carbon steels. Figure 15 shows the a, vs H,, Vickers hardness, relation, The following relation holds between a,, and H,: a” = G&l,
(10)
A small-crack growth law and its related phcnotncna
509
r
Fig. 14. n vs u, relation for rotating-bending fatigue.
Fig. 13. C, vs u, relation for rotating-bending fatigue.
where C, = 3.53 for annealed and normalized steels and C, = 3.08 for heat-treated steels. This means that we can infer the constants C, and n by using the hardness which can be measured easily. 5. METHOD FOR PREDICTING FATIGUE LIFE OF CARBON STEEL PLAIN MEMBERS 5.1. Fatigue lge prediction based on the small-crack growth law
As shown in the previous sections, the growth rate of a small crack is determined uniquely by eq. (7). Accordingly, when we want to predict the fatigue life of plain specimens, eq. (7) can be used as a means for prediction. The present authors have carried out the fatigue tests of various kinds of carbon steel plain specimens having a diameter of 5 mm, and obtained the results that: (i) fatigue life is controlled mainly by the propagation life of a crack smaller than 1 mm; and (ii) dZ/dPJ of a crack in the range of 0.05 mm c f < 1 mm is decent uniquely by eq. (7). A~o~in~y, the small-crack propagation life of specimens of 5 mm in diameter can be written as follows, based on eq. (7):
(11)
0
loo
Vickers
200
300 hardness
Fig. 15. a, vs H, relation.
400 Hv
!m
HIRONOBU NISITANI et al.
510
Here, putting lo= 0.05 mm and I, = 1 mm:
The fatigue life of plain specimens of 5 mm in diameter Nf, can be represented by the next relation: 03) where No,oS is the initiation life of a crack of 0.05 mm and N,,is the life in which a crack propagates from 1 mm up to failure. On the other hand, the ratio of total fatigue life to small-crack propa~tion life 0 = 0.05+ 1 mm), N0,05-r1 /N,, was measured for the various kinds of steels, and the next relation nearly holds for each stee1[23,24]: (14) From eqs (12) and (14), the fatigue life of a plain specimen of 5 mm in diameter N,, can be predicted approximately by eq. (15).
The second phase of the investigation in this section deals with the fatigue life prediction of specimens having an arbitrary diameter d (d = a$, a & 1,4 = 5 mm). Figure 16 shows the crack growth curves of the geometrically similar specimens (ratio of similarity is 1: 2). The material used was an annealed 0.39% C steel. The load was rotating-bending and the specimens had a small blind hole (both the diameter and dgrth of hole were 0.3 mm) in the central part. The ordinate is the logarithm of relative crack length l/d and the abscissa is the number of cycles counted from the time when l/d = 0.08, that is, I = 0.5 and 1.0 mm for d = 6 and 12 mm, respectively. The following is apparent from Fig. 16: dl/dN in the geometrically similar specimens subjected to the same stress is proportional to the d&meter of the specimen. This also supports the assumption of eq. (1). In order to evaluate the effect of an increase in diameter on the fatigue life, it is advantageous to divide the fatigue life into the following four stages: (1) (2) (3) (4)
crack initiation life; propagation life of a microcrack {propagation behavior is affected by the ~~os~ct~e); propa~tion period in which eq. (4) holds; unstable propagation life up to failure [eq. (4) does not hold in this period].
When the specimen diameter increases, the life in stages (1) and (2) decreases because of a decrease in stress gradient. However, we neglect the decrease in lives of these stages by taking account of the fact that the lives of stages (1) and (2) are relatively small compared with the total fatigue life. In stages (3) and (4), paying attention to the results of Fig. 16, it is concluded that the propagation life from &,up to failure for a smail specimen (4 in diameter) is equal to the life
Fig. 16. Crack growth curves of 0.39% C steel similar spwimtns having a small blind hole.
511
A small-crack growth law and its reiated phenomena
from al0 up to failure for a large specimen (ad in diameter). Therefore, the propagation life from & up to al,, is the increase in life due to the increasing diameter. Putting the life of each stage as follows: (1) + (2) = N0.0S,(3) = lvo.DS_cr, and (4) = N,,, the fatigue life of a specimen of d = ctd,,in diameter Nf, is expressed by N,, = No.05 + Noas-, + K-r (16)
= No.05 + ~0.0S~0.0k + No,os+0: + L/.
Here, Fig. 16 indicates that the relations of No,o5 _ t = NOoh_* and Nt _,,z N, + hold. From eqs (13) and (14), No.oS + N1+ is nearly equal to N,,,,, . That is, eq. (16) is rewritten as follows: (17)
+# = No.05 -@ 0.0% + 2No.os +1 *
By using eqs (11) and (12), the fatigue life of a plain specimen of d = 5a mm in diameter, I$, can be predicted by the following equation: N,a= & : 3 0
a (ln a + 6).
(18)
11
For rotating-bending fatigue, the constants C3 and n can be determined by eqs (8) and (9), respectively, for the given value of 6,. 5.2. App~ic~t~n of fatigue life prediction method In order to confirm the validity of the fatigue life prediction method of eq. (18), its application to the rotating-bending fatigue data published by the National Research Institute for Metals (NRIM)[25] and the data of Cazaud et aZ.[26]was carried out. Figures 17 and 18 show the application of eq. (18) to the NRIM data and the experimental results of Caxaud et al., respectively. The prediction is in good agreement with the ex~~mental results over a wide range, with the exception of stresses very near the fatigue limit (a, < 1.OSa,, 0, : fatigue limit). Here, fatigue limits of steels are approximated by the relation G, r aJ2, except for extremely hard steel whose tensile strength is larger than 1500MPa. Accordingly, the lower limiting
Annralcd,S?SMPa
e200 Gi
100
I ’
‘*
I
I
Numt&
cycles
loo-
I t%
llurr
Nt
”
Fig. 17. Eat&nation of fatigue __--__ life: _ appfication of eq. (18) to NRIM
data.
Rotating bending of 0.36% C strrl -:Estimation bawd on Eq.(16) 0,&0:6xperim~nt by R.Cazaud -*-:cf.~l.OSu” H-t : Heat-heated
pzooc 5 _
-:Eatimation bawd on Eq.(l8) O,A,A:Expwlmmt by NRtM ----:apl.o5d, H-T :Heat -Trca ted N :Normalired
I
I O
10‘
I
5
Number ‘%t cycles
1
I
a
tdOtailurc
I Nt
10’
Fig. 18. Estimation of._fatigue .life: application _ _ of aq. (18) to caajutt’s
expennmtai
mutts.
HIRONOBU NISITANI et al.
512
Fig. 19. Schematic explanation
of fatigue life prediction method for the rotating-bending carbon steel plain specimens.
fatigue of
stress of the predicting method of eq. (18) is given approximately by the relation 6. > 0.53 0, (= l.O5a,), and this method is valid as long as o, is smaller than the yield stress. The schematic explanation of a fatigue life predicting method is shown in Fig. 19. It can be concluded that eq. (18) is effective and convenient for predicting the fatigue life of carbon steel plain members, because the evaluation of fatigue life is easily attained by using only the ultimate tensile strength with the type of heat treatment neglecting the effects of microstructure, chemical composition, a slight difference in temperature for heat treatments, etc. 6. CONCLUSIONS The follo~ng conclusions were reached with regard to a small-crack growth law of carbon steel plain specimens under constant stress amplitudes. (1) The crack growth of a plain specimen is not determined by the stress intensity factor range AK. The growth rate increases with an increase in the applied stress range for a given value of AK and the growth rate of a plain specimen is determined uniquely by the term ail, where 6, is nominal stress amplitude, 1 is crack length, and n is a constant. (2) The growth rate of fatigue cracks under a wide range of o;, can be determined by the following two growth laws. (a) When a crack is propagating under low stress,
(b) When a crack is propagating under high nominal stress,
(3) Both these crack growth laws can be explained consistently from the same physical background, based on the assumption that crack growth rate is proportional to the reversible plastic zone size Y, . The background of this assumption was investigated by using the experimental results of F+3% Si alloy specimens.
A small-crack growth law and its related phenomena
513
(4) At high stress ranges, the growth rate of a small crackin one material cannot compare with that of a different material. Therefore the following small-crack growth law is proposed: ’ where r~,is the ultimate tensile strength. In this equation, the effect of material properties is partly considered. The constant l/C, represents the resistance to crack propagation in each material and is a stable constant for those materials with similar rn~ha~~al properties. (5) An effective and convenient method based on the small-crack growth law and ~tirna~ tensile strength was proposed for predicting the fatigue life of plain members. The validity of this method was confirmed from its application to the data reported by other researchers. REFERENCES PI H. Nisitani and M. Goto, A small crack growth law and its application to the evaluation of fatigue life, in The lkhcuicnr
ofShorr Fatigue Cracks (Edited by K. J. Miller and E. R. de 10s Rios), pp. 461478.
Mech. Eng. Publiiers (1987).
PI H. Kitagawa, S. Takahashi, C. M. Suh and S. Miyashita, Quantitative analysts of fatigue pmcus-microcracks and
slip lines under cyclic strains. ASTM STP 675,4204t9 (1979). I31 M. H. El Haddad, N. E. Dowling, T. H. Topper and K. N. Smith, I integral applications for short fatigue cracks at notches. fnr. $. Fractm 14, 15-30 (1980). cracks, in Mec!ro&es [41 H. Nisitani, Unif~~ treatment of fat&m crack growth laws in small, large and nonpro~~ti~ o~~u~i~-~~~ (Edited by T. Mura), pp. 151-166. ASME (1981). S. J. Hudak, Jr, Small crack behavior and the prediction of fatigue life. Trcx.r. ASME H&26-35 (1981). ;I K. J. Miller, The short crack probiem. Fatigue ZGtgngMuter. Structwe~ 5, 223-232 (1982). VI M. Goto, Statistical investigation of the behaviour of microcracks in carbon steels. Fu!igue Brgrtg Muter. Srruetuns 14, 833-845 (1991). &I S. Kocanda, Fufigue Failure of Metals. Sijthoff & Noordhoff, Amsterdam (1978). energy approach. Trans. ASME, J. bar. &gng 191H. W. Liu, Fatigue crack propagation and applied stress ran-an 85, 116-122 (1963). [lOI G. T. Hahn, R. G. Hoagland and A. R. Rosenfield, Local yielding attending fatigue crack growth. Metal/. Trans. 3, 1189-1202 (1972). [ill K. Ando, N. Ogura and T. Nishioka, Effect of grain size on the fatigue crack propagation rate and plastic behavior near a fatigue crack. Truns. Jpn Sot. Mech. Engng 44,40144023 (1978). I121 K. Tanaka, M. Hojo and Y. Nakai, Fatigue crack initiation and early propagation in 3% silicon iron. ASTM STP 811, 207-232 (1983). W. E&r, The ~~ifi~~ of fatigue crack closure, ASTM STP rsS, 230-242 (1971). f:$ M. Kikukawa, M. Jono and K. Tanaka, Fatigue crack cfosure behavior at low stress intensity kvel. Proc. znd Znf. Co& Me&. Behaviot of ~o~er~~ (pp. 716720) (1976). I151 H. Nisitani, Y. Gda and T. Yakus~ji~~onside~tio~s of the smallcrack growth law based on CGD-low-cycle bending fatigue test of an annealed 0.45% steel in the SEM. JSME Int. 1. 33. 243-248 f1990). D. 8. Dugdale, Yielding of steel sheets containing slits. J. Mech. Phyk Soli& & 10ollOS (1960). ;I; H. Nisitani and N. Kawagoishi, Relation between fatigue crack growth law and reversible plastic zone size in Fe-3% Si alloy. Proc. 6rh Inr. Cong. Experimenral Mechanics (pp. 795-800) (1988). VI H. Nisitani and M. Goto, Relation between the shape of S-N curve and the crack growth law in prestrained 0.45% C steel specimens with a small hole. Trans. Jpn Sot. Mech. Engng 51, 1430-1435 (1985). WI H. Nisitani and M. Goto, Relation between small crack growth law and fatigue life of machines and structures, Trans. Jpn Sot. Mech. Engng 51, 332-341 (1985). 1201H. Nisitani and M. Goto, Effect of stress ratio on the propagation of small crack of plain specimens under high and low stress amplitudes-fatigue under axial loading of annealed 0.45% C steel. Trans. Jpn Sot. Mech. Engttg SO, 1090-1096 (1984). 1211K. Tanaka, S. Nis~ji~, S. Matsuoka, T. Abe and F. Kouzu, Low and high-cycle fatigue properties of various steels specified in JIS for machine structural use. Fczigne Engng Mater. Strnclures 4, 97-108 (1981). 1221M. Goto and H. Nisitani, Inference of a small-crack growth law based on the S-N curves. Truns. J&nSot. &fee/t.Engng 56, 1938-1944 (1990). ~231H, Nisitani and M. Goto and H. Miyagawa, A method for predicting fatigue life of carbon steel plain members, Trans. Jpn Sot. Mech. Engng 53, 2238-2247 (1987). 1241M. Goto, H. Nisitani, H. Miyagawa and T. Abe, A method for predicting fatigue life of heat-treated low alloy steels. Truns. Jpn Sot. Me& Engng 56, 1011-1019 (1990). National Research Institute for Metals, NRIM Fatigue Data Sheet, Nos 1, 2, 3 and 4 (1978). t;:; R. Caxaud, G. Pomey, P. Rabbe and Ch. Janssen, La Farigue des Metaux. Dun& Paris (1969) (Recebed 5 February 1991)