Minimum load Crack opening l o a d Maximum load Plasticily
! -- I
Oxide
IC ,ncrement crock Iosed I l % I
'- :1
Crock length, O v! a _ - I o _ v |
Roughness
iCIosed crock i
I n c r e m e n l I
Crock lenglh, o,. _J o ]
FIG. 4--Three mechanisms of fatigue crack closure.
of a/to*. For a given value of a/o~*, the values of ~~ and A~r~,/~rv are determined from the figure. When a is small as compared with to*, ~rop is nearly zero and U is about unity.
As the crack gets longer, U decreases and approaches a constant value of 0.443.
The finite-element method has been used to calculate the plasticity-induced closure under both plane stress and plane strain [16,17]. Several mechanical models have been proposed for both oxide-induced and roughness-induced crack closures [18,19]. Roughness-induced closure increases with increasing dimension of the material microstructure.
Elastic-Plastic Parameters
Under large-scale yielding, gross and general yielding conditions, CTOD and J-integral are usable as a characterizing parameter of the crack-tip filed. Under Masing's assumption
TANAKA ON FATIGUE CRACK PROPAGATION 157
i I I I I I I
,: 1.0-
0.5
| 0.2 ~ ,
=. o , / o r .
~ 0.1 ---- aOttilO'~'
n," \
i I i I I I ~ . a
0.1 1 10 1OO 1000
R e L a t i v e c r a c k L e n g t h , a / w "
FIG. 5--Changes of stress amplitude and effective stress with crack length relative to reversed plastic zone size.
[20] that either branch of the stress-strain hysteresis is geometrically similar to the monotonic stress strain curve with a scale factor of two, the range of J-integral AJ is proved to be path- independent and characterize the range of change in stress, strain and displacement during one loading cycle [21,22].
The Similitude Concept
A unique correlation between the fatigue crack propagation rate and the stress intensity factor is based on the concept of similitude. Equal stress intensity (or J) will have equal consequences. This similitude requirement is not always satisfied, especially when we deal with small cracks.
Surface
Crack mlhahon
Stage 1 growth t - ~
/ / "~,\\
_ ~ ~ _ _ _ S!oge_ll growth ! /(/
/ N (J)- slngulanty field
(3.) Intense
shear bands
Environment ~ )
~ ~ /Grain
( b ) - - , \ 2 / boundary
Crack wake . ~
FIG. 6--A fatigue crack nucleated from the surface," (a) Stage I and Stage lI growth; (b) Details of fracture-related zone at the crack tip.
158 FRACTURE MECHANICS: TWENTIETH SYMPOSIUM
Figure 6a illustrates a fatigue crack nucleated from the specimen surface. The spread of the K-singularity field near the crack tip where the singular term is predominant is about one-tenth of the crack length [23]. When the crack is large and when the physical (chemical) process of fatigue cracking occurs within the K-singularity field as shown in Fig. 6a, the rate of crack growth will be unique. On the other hand, when the crack length is small, the K similitude breaks down; the crack growth rate is no longer a unique function of K-values.
For elastic-plastic cracks, the J-integral is to be used instead of K as a near-tip field parameter.
Referring to the fracture-related zone for the fatigue crack shown in Fig. 6b, the break- down of the K similitude due to the shortness of the crack is possibly induced through several processes:
1. Breakdown of the microstructural similitude. When the crack length is on the order of the material microstructure, for example, the grain size, plastic deformation near the crack tip is very much influenced by the material microstructure, and the grain boundaries will block crack growth. The assumption of a macroscopic continuum in fracture mechanics is violated; the similitude is broken down because of microstructural inhomogeneity. These short cracks are microstructurally short cracks [2].
2. Breakdown of the mechanical similitude. The AK value loses its meaning as a crack driving force for the following two cases. The plastic zone size is large compared with the crack length, that is, the crack-tip yielding is of large scale. The crack closure is not fully developed because of the short crack wake. These cracks are mechanically short cracks [3].
3. Breakdown of the environmental similitude. The crack-tip chemical environment con- trois the crack growth rate and is often different from the bulk environment. Short cracks may have a different crack-tip environment from long cracks, so that the propagation rate will be different. Such cracks are chemically short cracks [24].
Table 1 presents the classification of the crack size according to mechanical and micro- structural influences [25]. When the crack length is large compared with the microstructural dimension (microstructurally large), the material resistance is expected to be homogeneous.
The subject is to find out an appropriate driving force. For a small crack whose length is comparable to the microstructural dimension (microstructurally small), the material resis- tance is influenced by crystal orientations and grain boundaries. Therefore, even when the mechanical driving force is properly expressed in terms of SIF (like in the case of Type 3 crack), a crack shows irregular, anomalous growth behavior.
The limitation of K for determining crack-tip plasticity due to large-scale yielding can be estimated on the basis of Dugdale model analysis [7]. Ten percent deviation of the plastic
TABLE 1--Classification of crack size according to mechanical and chemical influences.
Microstructural Size
Mechanical Size
Large: Small:
a/to > 4 - 20 (SSY) a/o~ < 4 - 20 (LSY)
Large Crack Wake Small Crack Wake
Large:
a/M > 5 - 10 to/M ~> 1
Small:
a/M < 5 - 10 co/M ~ 1
Type 1: Mechanically and microstructurally large (LEFM valid)
Type 3: Mechanically large/
microstructurally small
Type 2: Mechanically small/
microstructurally large Type 4: Mechanically and
microstructurally small
TANAKA ON FATIGUE CRACK PROPAGATION 159 zone size from Eq 5 occurs at the stress level Grmax/13t Y = 0.30, and the corresponding plastic zone size relative to crack length is 0.12. Ten percent deviation of C T O D from Eq 6 occurs at Crma,/trV = 0.45, and the corresponding to/a is 0.32. When a similar 10% deviation criterion for the SSY limit is applied to the cyclic components, the SSY limit based on the reversed plastic zone size is at Acr/2trv = 0.30 and to*/a = 0.12, and that based on ACTOD is at the Acr/2cry = 0.45 and to*/a = 0.32, provided that there is no crack closure. Tanaka and Nakai [15] showed that the AK~tf-approach was applicable to even larger stress levels than the AK-approach. While crack closure is reduced due to the small amount of plasticity, to*
and ACTOD are determined as a unique function of AKe~, but not of AK.
Even for a long crack, the similitude will break down when the load is suddenly changed.
Fatigue crack propagation is temporarily retarded immediately after overloading or load reduction. This violation comes from the history effect; the crack-tip deformation is different from that under constant-amplitude loading. Crack closure consideration will show that AKef f is a proper local parameter.
Propagation of Long Fatigue Cracks
Relation between Crack Propagation Rate and Stress Intensity Factor
For a long crack, the steady-state fatigue crack growth rate is a function of the stress intensity factor. Paris and Erdogan [5] first obtained the fourth-power law between da/dN and A K
da/dN = C ( A K ) ' (12)
A large number of works done in the following two decades have shown that the exponent of the power relation between da/dN and AK is not necessarily four, but varies between two to seven. Thus, a generalized Paris law is
da/dN = C(AK)" (13)
where C is a weak function of R. A wide range of data of da/dN versus A K shows a sigmoidal variation, as shown in Fig. 7. The power law only represents the relation in the intermediate- rate regime, Regime B, from 10 -9 to 10 6 m/cycle. In Regime A , there is a threshold stress intensity range AK, h below which the rate is practically zero. A t high AK in Regime C, the acceleration of crack growth takes place, and unstable fracture starts at the maximum stress- intensity factor equal to the fracture toughness. The characteristics of each regime are described in Fig. 7 [26].
Many formulae have been proposed for expressing a sigmoidal variation in the relation between da/dN and AK. Several equations, Eqs 14 to 18, are listed in Table 2 [27-32].
Equations 14, 16, and 17 are empirical, and others are derived on the basis of some physical consideration. Even today, we do not have universal equations, and the values of C and m in the above equations need to be determined experimentally.
Mean-Stress and Variable-Amplitude Effects
The mean stress superposed on the fluctuating stress will influence the amount of static- mode growth and crack closure. The former contribution in Regime C is taken into account as a denominator in Eqs 14, 15, and 18. For some cases, the mean stress has a large influence on the growth rate even in Regime B. Several equations, Eqs 19 to 22, which include the
160 FRACTURE MECHANICS: TWENTIETH SYMPOSIUM - 2
I0
16 4
Regime A
NON-CONTINUUM MECHANISMS I large influence of:
1 . microstructure
li. mean stress
~ ) ill. environment.
t.) E
a
Z
" o o
10 -10
d o ' ~
1
I
~IRESIIOLD A Zll II h
Primary Mechanisms R._eegime B
CONTINUUM MECHANISM (striation growth) little influence of:
1 m i c r o s t r u c t u r e a i , mean s t r e s s i i i . d i l u t e e n v i r o n m e n t i v . t h i c k n e s s 9
B
8 !
! FINAL t t FAILURE
= C ( 6 K ) m u
R ~ i m e C
"STATIC MODE" MECHANISMS (cleavage, i n t e r g r a n u l a r
fibrous) - -
large influence of:
i . m i c r o s t r u e t u r e l i . mean s t r e s s i l l . t h i c k n e s s .
l i t t l e i n f l u e n c e o f :
! i v . e n v i r o n m e n t . i
Log AK
FIG. 7--Relation between crack propagation rate and stress intensity range.
mean-stress effect are given in Table 2 [33-37]. AG is the range of the energy release rate and has been used for cracking of adhesive joints. AS is the range of the strain energy density. Under Mode I loading, AG and AS are equivalent to Km,x 2 - Kmm 2 except for coefficients.
For brittle materials with low toughness, such as PMMA, epoxy, and graphite, the static- mode crack growth takes place even in the intermediate-rate regime. The value of Km,, as well as AK contribute the total amount of crack growth. In Eq 19, 3' = q / ( P + q) indicates the relative contribution of the maximum stress.
For ductile materials, the effect of the mean stress on the crack propagation rate in low- and intermediate-rate regimes has been ascribed to crack closure. The effect of R diminishes in the relation between cla/dN and AKoff, as shown in Fig. 8. The crack propagation rate is expressed by Elber equation [11] for any R value.
da/dN = C ( A K e f f ) m ( 2 3 )
The mean-stress effect is pronounced near the threshold. The AK, h value normally de- creases with increasing R. Phenomenological relations between AKth and R proposed by Klesnil and Lukas [34] are
AK, h = AK,ho(1 - R) ~ (24)
TANAKA ON FATIGUE CRACK PROPAGATION TABLE 2--Empirical and semi-empirical equations for fatigue crack propagation.
161
Equation
Number Equation Author Year
da AKm
(14) d--N = C (1 - R)Kc - AK Forman [27] 1967
da AK 4
(15) dN C Kc 2 _ Kmax 2 Weertman [28,29] 1966, 1969
(16) d--N = da C ( A K " - AKth") Klesnil [30] 1972
1 C~ C2 C2
(17) da/dN - _ _(AK)"---~ + " - ~ 2 _ (AK) _ [K,(1 - R)] ~2 Saxena [31] 1979 (18) dNd--'aa = C ( A K - AK,h) 2 1 + K~ -~Km~ McEvily [32] 1983
Roberts [33] 1965
(19) d__aa = C AKP Kmax q
dN Klesnil [34] 1972
(20) d-N da = C ( K m a x 2 - Kmm2)m Arad [35] 1971
(21) d'-'N = da C(AG)m Mostovoy [36] 1975
(22) d'--N da = C(AS)m Badaliance [37] 1980
where AK, ho is the AKth value at R = 0. Schmidt and Paris [38] proposed AKth = AKtho(1 -- KR)
AKth = AKtho(1 -- KRc) = const
for R < Rc for R > Rc
(25)
Figure 9 shows the relation between AKth and R for steels [39]. The contribution of crack closure has been found to be large near the threshold and the effective threshold stress intensity range has been found to be constant. The AKth value consists of
A K t h : AK.f~h + AKa,h (26)
where the first term is the intrinsic material resistance and AKc,h (=Kop,h -- Kin,,) is the extrinsic one. AK~,,h equals to AKth at stress ratios larger than Rc. It is between 2 to 3 MPaV'-mm for steels. It is now known that crack closure also accounts for the major part of the microstructural effect on the threshold.
Crack closure plays a significant role in the fatigue crack growth behavior under variable- amplitude spectrum loading. Once the value of AKoff is evaluated experimentally or theo- retically, the subsequent crack growth rate can be predicted from the d a / d N - AK~ff relation obtained in the test under constant-amplitude loading. Therefore, the prediction of crack closure is the topics of current researches. For variable-amplitude loading including the stress level below the threshold, AKe,,h may disappear and the power relation between d a / d N and AK.ff should be extended below AK~,,h for predicting crack growth rate [40].
Although crack closure is primarily significant in crack growth under single or multiple
162 FRACTURE MECHANICS: "TWENTIETH SYMPOSIUM 10 -6
10 -7
~g
U u
E
v
10 -8 z
d
~ 10 -9
e.-
r o
~. 10 -~o
SM41B I WOL
R 0.!
0.3 0.5 0.7 0.85 C T
R 0.1
9
9 9
03
Solid marks: AKet
~> • Half-open marks:
A K = A K e f f
~0-" I 9 ~?l ~ ? l t
2 5 10 20 50
Stress intensity range. AK, Ai~ef f (MPam ~12)
FIG. 8--Relation between crack propagation rate and stress intensity range for mild steel under various R values.
over loads, there may be other factors such as crack branching contributing crack growth retardation [41].
Mixed-Mode and Biaxial Loading Effects
The growth direction of fatigue a crack under mixed-mode loading is not coplanar in most cases because of crack-tip stress asymmetry, and is much influenced by material anisotropy [42-44]. The threshold condition for fatigue crack growth under a mixed mode of I and II was examined for a mild steel [42]. The shear-mode growth took place at low stress intensity, and the tensile-mode growth took over at higher stress intensities. The growth direction of tensile crack growth is roughly along the direction of the maximum tangential stress or the minimum strain energy density. A crack grows as to reduce the KH component in isotropic materials. The AS parameter was used to predict the mixed-mode crack growth [45].
Fatigue cracks in anisotropic materials such as carbon-fiber-reinforced plastics (CFRP) grow coplanarly even under mixed-mode loading [44]. Figure 10 shows the rate of delam-
C" -E tl} 13. 3; c- <3 {71 t- .t,-, c c ul i/1 o 1D l- I/) o r 0
I Solid marks :'Martensitic steel'"'-"-L':'-.~ ",, Half-open marks: Austenitic steels'"-"i~'.'.'~':.. I I I I E i i i I "-~,1; 0.2 0.4 0.6 0.8 Stress ratio, R
5" -E r 13. :E t- d ol c- E o c-
14 , , , , (Mip), , i , Ov a Ref. :3 En 3A 303 Jerram(1973) S] Mild steel 366 Present study <] $35C 372 Kikukawa(1981) 9 En 24 1275 Cook (1975) SMZ,1B Present study 9 D6c 1640 Mautz(1976) 0 1.0 Stress o O 2 Open marks : Ferritic-pearlitic steels Solid marks : Martensitic steels I I I I 1 I I I I 0 0.2 0.4 0.6 0.8 1.0 9 atio, R FIG. 9--Effect of stress ratio on threshold stress intensity range.
7_ i> I> 3 7_ "11 5 !1 ?) rl I> l) D 5 7_ ...1. o)
164 FRACTURE MECHANICS: TWENTIETH SYMPOSIUM
,..I
U
I I I ~ I ' I
[ ' . 'I'. 9 9
m ~/__~ ~1~ _o ~'~ ~,
0
I , I , I , I , I ,
(apA:)/uJ) Np/op 'aloJ uo!|~odoJd )poJ D
I ' I ' I ' I ' I '
-
~ v <>'%~. ~'-- o o o
~>"#-<>o_. oo o B o
"" <~ O 0 @ 0
~o% %
o ~ <I 0
I , I , I , I ~ I
~ "o % % %
( a l ~ l W ) NPlOP ' aloJ uo!Io~odoJd )l::)oJ3
8 z
,9
, - - ~
=
o,
~z
r-) .~(}'--"
~ ~
~j
TANAKA ON FATIGUE CRACK PROPAGATION 165 ination fatigue cracks under Mode I and a mixed mode of I and II at various R-values.
Mixed mode data were obtained by using cracked lap shear specimens (CLS) where the ratio of Mode II component of the strain energy release rate AGH to the total value AG is 0.70. The crack growth rate is much higher at higher R ratios and under mixed mode loading when it is correlated to z~K~. When da/dN is correlated to AG ( = z~G~ + A G n ) , the effect of R disappears. However, z~G is not enough to account for the contribution of Mode II component, and a new correlating parameter is necessary. The effect of anisotropy on crack growth under mixed-mode loading will be different depending on the types of composites.
Further study is necessary.
Nonsingular stresses applied parallel to the crack plane often influence the fatigue crack growth rate in the biaxial stress field [46-48]. The plastic zone ahead of the crack tip tends to increase under pure shear (lateral compression) loading, while decreasing in biaxial tensile loading. Thus, a fatigue crack grows faster in pure shear than in biaxial tension in the large- scale yielding situation. There is no nonsingular stress effect in the SSY situation [48].
Elastic-Plastic Loading Effect
The rate of fatigue crack growth under elastic-plastic and gross plasticity conditions is higher than that predicted from the relation between da/dN and AK. Among various elastic- plastic parameters, the J-integral range seems to be most successful. Dowling and Begley [49] first correlated the rate to AJ and obtained the power-law relation
da/dN = C(AJ)" (27)
Under SSY conditions, Eq 27 becomes equivalent to Eq 23 because AKr = ~/EAJ. Figure 11 shows the growth rate plotted against AKaf and AJ [50]. The data obtained under strain- controlled condition show the acceleration in Fig. l l a , while all the data fall on the single line in Fig. l l b .
Other parameters which have been proposed are ACTOD [51,52] and the plastic zone size (PZS) [53,54]. The crack growth rate is given as a power function as
da/dN = C(ACTOD)" (28)
da/dN = C(PZS)" (29)
Mechanisms of Fatigue Crack Propagation Crack-Tip Blunting Model
Previous models for fatigue crack propagation are classified into two categories, one based on crack-tip blunting and the other based on damage accumulation [32]. In the intermediate- rate regime where no static mode fracture is involved, the crack-tip blunting model has been supported by various experimental evidences.
Crack-tip blunting model was first proposed by Laird and Smith [55] on the basis of the direct observation of crack opening profile. This type of model has been advanced by several investigators. Neumann [56] and Kikukawa et al. [57] made quantitative observation of the opening and closing of a crack and showed that the amount of crack growth per cycle was half the ACTOD, as derived from a simple geometrical relation. This geometrical relation is not satisfied for a wide range of fatigue crack growth data [58-60].
10 4 \ E z ~IC~ ~
t- O -7 ~,0
I I 8M41B (I) Deflection controlled tests CCP (2W=16mm ,B=6mm)
I I 1 I I I I A((%) R, 9 0.15~0.5 -I 9 02 -I 9 0.5 -I 9 0.6 - I Loed controlled tests CT (W-4Omm, B=lOmm) I~=x (kN) Rot A 1.96 0.1 0 2.94 0.1 , /
"~ 10-5 Z . ~I0 6
t- O
7 ~,o- lif e _o
3.92 0.1 ~ ~q~ [] 5.88 0.1 _. <> 7.36 o.i "'~. _~ CT (W=2OOmm. B-lOmm) ~/ Pma(kN) Ro" .r-/ ,4.7 0.05 ,~' j~/ @ 29.4 0.05 v ~" /\ ~
~/~ /
4 { Kmox~ z All CT data satisfy W-a _~ ~---~/ . I I I I I 5 I0 Effective stress intensity range , AKeff
(a) I I I I i0-9 I0-9 50 I00 ( MPa.,/~" ) I ' I ' I ' SM41B (]) Deflection controlled tests CCP (2W=16mm, B=6mm)
9 /
A((%) R. .~ -- 9 0.15~0.5 -I 9 0.2 - I ~" 9 0.5 -I v~ 9 0.6 - I / / Load controlled tests CT ( W =40mm, B =lOmm) /,~ P..=(kN) R. /~ A 1.96 0.1 /~'~ 0 2.94 0.1 / 34, 3.92 0.1 ~/~ " o 588 o, ~ 7.36 0.1 ..~-v CT (W-20Omm, O=lOmm) Pmox (kN) Ro" _~i~/ ~1 14.7 0.05 vo~7//r~ q~ , @ 29.4 0.05 9~;'- \ 02 \ ~g o ~/- do/dN=4.61xl(~lalAd)l 57 4 (Km~ f All CT data satisfy W-a ;= ?r~ cry / 9 I , I , I I0 z 103 104 d integral range 6J (N/m)
(b) i 105 FIG. ll--Crack propagation rate plotted against effective stress intensity range and J-integral range for mild steel: (a) da/dN versus AKelr; (b) da/dN versus AJ.
"n 0 c m E m o 7- z or) m z .-I N -I .-r .< E -.Q 0 E
TANAKA ON FATIGUE CRACK PROPAGATION 167
Y
CQ). (c)
Y
(b) (d)
FIG. 12--Crack-tip blunting by alternate shear.
Figure 12 shows crack-tip blunting model depicted by Kuo and Liu [58]. The amount of crack extension per cycle Aa ( = da/dN) is proportional to ACTOD as
da/dN = ~ACTOD (30)
where ~ is
= 0.5 cot ~J (31)
and 0 is half the crack-tip opening angle. For the small-scale yielding case, ACTOD is given by
ACTOD = ~'AK2/(E~rr' ) (32)
where err' is the cyclic yield strength. From Eqs 30 and 32, we have the crack growth law
da/dN = hAK2/(E~rr' ) (33)
where h = 13'8. In the above derivation, AK is replaced by AKeff when there is crack closure.
Under elastic-plastic loading, AJ is used for AK, and the above equation becomes
d a / d N = hA JAr r' (34)
The crack opening angle 0 is usually assumed to be 45 deg, which gives ~ = 0.5. According to the elastic-plastic analysis of a stationary crack, t3' is between 0.15 and 0.73 [61]. There- fore, k is between 0.08 and 0.37. Kuo and Liu [58] calculated h = 0.019 by using the unzipping model, in which only a fraction of ACTOD is assumed to be effective for crack advance. Figure 13a shows the da/dN-AJ relation for three metals, and in Fig. 13b the rate is plotted against AJ/crr' (~r' = the cyclic yield stress) [60]. Lines of several h values are drawn in the figure. Electron fractography indicated complete covering of the fracture surface by striations at rates from 2 x 10 -7 to 5 x 10 -5 m/cycle. The value of h decreases with decreasing rate. Careful observation of crack-tip opening profiles showed that the opening profile was not geometrically similar, as shown in Fig. 14. This gives rise to the exponent in Paris law larger than two even if striation formation is an operating mode. For several
0-4 10 -5 u E ~ I:1 o 8 &,o- o [ ~ iO -~ 0-;
' I [ ' I -7 ;( O OR"IC copper i~J/~ -- -- [] 0.04%C steel ... ~ SUS ~)4 / ^/z~,,. ~~ da/dN = Cj (t'.J) mJ C,) 1.52x10 ~ 693x10~, 1.47x10 'e ma 2.29 1.55 1.37 /, oy I I i I I I 0 2 10 a 10 4 IO s d integr01 range , AJ (N/m)
0-4 10-5 E I0 "~ 2 5 o Q. 2 I i I i I I0" t 10-s 10 -5 10 -4 AO/o" 4 (m) FIG. 13--Relation between crack propagation rate and J-integral: (left) da/dN versus A J; (right) da/dN versus AJ/(rv'. 1
~0 "11 0 c- "11 rn rn 0 "I- Z O0 .~ 11 E -t 5 .-I 1" /) < El 3 n
TANAKA ON FATIGUE CRACK PROPAGATION 169
- 6 0
tx~/2 (t,m)
J-eo 6,~ (*~)
| 0 1.0
- - ~ 4
I I I I f ~ I I I Id I .
-50 -40 "30 "20 -I0 0 I0 20 30 40 50
x' (tan) FIG. 14--Crack-tip opening profile in pure copper.
metals, geometrical similarity is satisfied and the striation spacing is proportional to A K 2 or to the crack length [62].
d a / d N = A a
(35)
Crack Propagation Through Multiple Mechanisms
When several mechanisms for crack growth are operating at the same time, the resultant crack growth rate will be a sum of the rate due to each mechanism. Consider the case in which only two mechanisms are operating. When two mechanisms operate in one cycle, the rate will be
d a / d N = (da/dN)~ + (da/dN)2 (36)
where (da/dN)~ is the rate of crack growth due to Mechanism 1 and (da/dN)2 is that due to Mechanism 2. When each mechanism occurs in only a certain area fraction of the fracture surface, the fracture surface consists of the area made by Mechanism 1 and that by Mechanism 2. The resultant rate will be
d a / d N = fl(da/dN)2 + fz(da/dN)z (37)
where fl and f2 are area fractions. In sequential model for crack growth, we have
d a / d N = 1/[fl/(da/dN)l + f2/(da/dN)2] (38)
The resultant growth law is obtained by substituting the d a / d N - A K relation for each mech- anism.
Mechanisms o f Crack Propagation Threshold
The threshold stress-intensity factor AK,, can be decomposed into the intrinsic (effective) component and the extrinsic (closure) component as given by Eq 26. Several models have been proposed for the threshold effective stress-intensity range AKoff, h. The lower limiting