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An analysis of collective damage for short fatigue cracks based on equilibrium of crack numerical density
PII:
Engineering Fracture Mechanics Vol. 59, No. 2, pp. 151±163, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0013-7944/98 $19.00 + 0.00 S0013-7944(97)00098-2
AN ANALYSIS OF COLLECTIVE DAMAGE FOR SHORT FATIGUE CRACKS BASED ON EQUILIBRIUM OF CRACK NUMERICAL DENSITY YU QIAO and YOUSHI HONG{ Laboratory for Nonlinear Mechanics of Continuous Media, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China AbstractÐCollective damage of short fatigue cracks was analyzed in the light of equilibrium of crack numerical density. With the estimation of crack growth rate and crack nucleation rate, the solution of the equilibrium equation was studied to reveal the distinct feature of saturation distribution for crack numerical density. The critical time that characterized the transition of short and long-crack regimes was estimated, in which the in¯uences of grain size and grain-boundary obstacle eect were investigated. Furthermore, the total number of cracks and the ®rst order of damage moment were discussed. # 1998 Elsevier Science Ltd KeywordsÐshort fatigue cracks, crack numerical density, collective damage, fatigue life, damage moment.
1. INTRODUCTION IN THE past two decades, dierent characteristics of fatigue damage have been observed in the primary and the ®nal stages of fatigue failure. In the primary stage of fatigue failure, the length of most cracks is comparable with the grain size. One distinct growth behaviour of short cracks has been reported as the crack growth rates with deceleration and acceleration patterns, which has been attributed to the interaction of crack tip plastic zone with microstructural barriers to plastic ¯ow [1±4]. In the recent years, another distinct phenomenon of short cracks has been revealed in the literature[5±10], namely that in short-crack regime fatigue process presents collective damage characteristics with the gradual evolution of crack numerical density, showing the number of short fatigue cracks increases with increasing number of fatigue cycles. The method based on the equilibrium of crack numerical density [11] is a possible approach to describe and analyze the collective evolution process of dispersed short cracks [12]. The basic consideration of the method is that the total number of short cracks is composed of cracks due to crack nucleation and crack growth. It is known that the short-crack regime takes up a large portion of total fatigue life and the approach to life prediction of this regime is still a problem remained for solution. One essential aspect of this problem is to de®ne the transition point between short and long-crack regimes, and then to determine the critical values of loading cycles and crack length. This is an important step toward the prediction of total fatigue life. The main objective of this paper is to present a new method to evaluate the collective behaviour of dispersive short-fatigue-cracks by using the concept of the equilibrium of crack numerical density, in which the crack numerical density denotes the number of cracks within unit area and unit crack length. After setting the expressions for crack growth rate and crack nucleation rate, the solution of the equilibrium equation and the evolution characteristics of crack population are studied. Three cases of transition criteria are discussed, with each showing the variation of critical time with grain size and microstructural obstacle eect. The total number of cracks and the ®rst order of damage moment (i.e. the total length of cracks) are also investigated, which are two factors available for the description of the extent of short-crack damage. {Author to whom correspondence should be addressed. 151
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Yu Qiao and Youshi Hong
2. BASIC EQUATIONS In order to evaluate the collective damage of short fatigue cracks, the equilibrium equation of crack numerical density [11] is used in the following analysis. Its non-dimensional form is [12]: @ @ n c; t A c n c; t Ng nN c @t @c
1
where n(c, t) is the crack numerical density, with n(c, t)dc being the number of cracks with length between c and c + dc at time t; A(c) is crack growth rate; and nN(c) is crack nucleation rate. All of these physical quantities are dimensionless, and the non-dimensional coecient Ng
nN d ; n A
with n*N being the characteristic crack nucleation rate, A* the characteristic crack growth rate, n* the characteristic crack numerical density, and d the characteristic dimension of the material concerned (e.g. the grain diameter). Note that d A is equivalent to the characteristic time. Equation (1) describes the equilibrium of crack numerical density in phase space. The second term at the left side describes the ¯ow of crack numerical density in phase space, which is attributed to crack growth; and the term at the right side describes the contribution to crack numerical density made by crack nucleation. If n(c, 0) = 0 and the threshold value of crack length for growth is zero, the theoretical solution of eq. (1) is [11]: Zc 1 Ng nN c0 dc0 2 n c; t A c Z c;t The lower integral boundary Z(c, t) is of such a de®nition that for a crack with an initial length of Z(c, t) at t = 0, its length will advance to c at time t under the growth rate of A(c). Figure 1 is a schematic illustration showing the deceleration-acceleration pattern of shortcrack growth [1]. The curves imply that before the crack tip reaches the ®rst grain boundary, crack growth rate decreases with the increasing of crack length. If crack length is relatively large, the crack growth rate will tend to LEFM regime. Thus, in general, we may construct the
Fig. 1. Schematic of crack growth rate versus crack length in short crack regime. (Grain 1 < Grain 2 < Grain 3<
An analysis of collective damage for short fatigue cracks
following expression for A(c):
A c
1 ÿ 1 ÿ Ad c c 1 d c c > 1
153
3
where Ad is the crack growth rate when the normalized crack length c = 1, d is the ratio of nondimensional average grain size d to non-dimensional initial crack growth rate A0, i.e. the normalized grain size. If c is normalized by grain diameter, Ad is the crack growth rate at the time the crack tip reaches the ®rst grain boundary. Equation (3) shows the eects of crack length c, grain size d, and grain-boundary obstacle eect Ad on crack growth rate. The value of Ad is within the range from 0 to 1, which re¯ects a number of experimental measurements [1±4]. If Ad=0, eq. (3) has the meaning that short cracks stop propagating when crack tips reach grain boundaries. This is the extreme condition that the largest grain-boundary obstacle eect prevails. In experiments [7, 13], it was shown that the length of most cracks at initiation is less than the average grain size, and there was almost not a crack at nucleation with length larger than 2 d. Note that the crack length at initiation also depends on the time interval used for observation. Thus, the crack nucleation rate nN is constructed as follows: 1 ÿ c=2 c 2 nN 4 0 c > 2 After setting above estimations for A(c) and nN(c), the distribution of n(c, t) versus c is readily calculated from eq. (1). Figure 2 shows the variation of n(c, t) with c for dierent time stages. Figure 2 also shows that there is a saturation distribution in the evolution process of collective short cracks (dash line), i.e. with the progress of fatigue process, the distribution of crack numerical density gradually tends to a saturation curve from small to large value of crack size. It is obvious that the saturation curve presents the stable distribution of crack numerical density. Referring to eq. (2), we may write the saturation curve as: Zc 1 Ng nN c0 dc0 5 n0 c A c 0 For the explicitness in the following, we write: Z1 Ng nN c dc a 0
and
Z D0
1 0
n c; t dc
Fig. 2. Variation of n(c, t) with c for dierent time stages, the dash line representing the saturation curve. (Ad=0, d=1).
6
7
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Yu Qiao and Youshi Hong
It is clear that a is the number of cracks produced within unit time duration, and D0 is the number of total cracks at time t. Regarding the equilibrium of crack numerical density, we have Z1 n c; t dc a t 8 D0 0
Assuming that at time t, the distribution of crack numerical density in the area of c < c*(t) would converge to the saturation curve, as such, the number of cracks in non-saturation area is de®ned as D*0, where c* is the length of the crack (at time t) whose initial length is zero at t = 0., Thus from eq. (3), the evolution of c* can be expressed as: 1 ÿ 1 ÿ Ad c c 1 c_ A c 9 c > 1 d c The solution of eq. (9) is:
(
c
1 1ÿAd
f1 ÿ exp ÿ 1 ÿ Ad tg b exp d t
t t0 t > t0
10
where b =Add= 1ÿAd , and t0
1 1 ln ; b d
which is the time when c* = 1. 3. TOTAL NUMBER OF CRACKS, MAXIMUM CRACK LENGTH AND CRITICAL TIME From eq. (2), it is possible to derive the evolutionary features of crack numerical density. However, the crack numerical density n(c, t) cannot be used directly to determine the termination of short-crack regime. The boundary of short-crack regime and the start of long-crack regime should be indicated by the appearance of a crack with critical length. Thus we need to know the maximum crack length cmax(t) as a boundary value of the evolution system. It is stipulated that the numerical-density-equilibrium analysis of short crack is not valid when cmax(t)e ccr, where ccr is the critical crack length, beyond which the fatigue damage enters into long-crack regime. In relation to the stipulation of criteria from dierent viewpoints, the corresponding value of cmax(t) can be identi®ed. The following three cases are discussed. 3.0.1. Case I. Assume that the upper boundary of the saturation area c* be the maximum crack length cmax(t) (i.e. cmax=c*). The important premise of Case I is that only the crack numerical density at the region of c < c* may distribute stably. The ful®ll of the criterion c* = cmax=ccr is equivalent to the statement that a crack with its initial length of zero develops to a crack with critical length ccr. The evolution of cmax can be shown as: ( 1 1ÿAd f1 ÿ exp ÿ 1 ÿ Ad tg t t0 11 cmax c b exp d t t > t0 Substituting eq. (8) into (11), one obtains: 8 1 ÿ 1ÿA < 1ÿAd 1 ÿ exp ÿ a d D0 cmax 0 : b exp dD a
t t0 t > t0
12
Because it is certain that ccr>1, then the eq. (12) shows the variation of cmax with Ad and d. critical time tcr of the termination of short crack regime can be derived from eq. (11): 1 ccr 13 tcr t jcmax ccr ln d b Fig. 3 shows the relationship between tcr and the parameters of Ad and d for this case. It is seen
An analysis of collective damage for short fatigue cracks
155
Fig. 3. tcr as a function of Ad and d for case I.
suggesting that the that the value of tcr increases with the decrease in the values of Ad and d, tolerance of collective short-crack damage is enhanced if the material is of small grain size and large obstacle eect of grain boundary. 3.0.2. Case II. The criterion of cmax is set as Z1 1 n c dc ; n cmax which implies that there must be at least one crack with the length larger than cmax. Here n* is the characteristic crack numerical density and the situation of n* = 1 is considered. Then the criterion can be rewritten as Z1 n c dc 1: cmax
Let t* be the time when c* = cmaxeccr. If ccre2, then at time t*, the number of cracks in nonsaturation area is: Zc Z ccr 1 Ng nN ^c d^c dc D0 a t ÿ A c 0 0 Z 1 Z 2 Z ccr Zc 1 a t ÿ Ng nN ^c d^c dc A c 0 2 0 1 24 Ad ÿ 15 A2d 4 4Ad ÿ 3 ln Ad ÿ 10 5 1 ccr 14 ln a t ÿ Ng 2 8d d 16 1 ÿ Ad 3 In the following, the circumstances of D*0<1, and D*0>1 are separately discussed. (a) If D*0<1, from the equilibrium of crack numerical density, tcr can be derived: 1 tcr t 1 ÿ D0 a where z
Ng ccr ln z d a 2
24Ad ÿ 15A2d 4 4Ad ÿ 3 ln Ad ÿ 10 5Ng 1 1 Ng a 16 1 ÿ Ad 3 8d
15
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Yu Qiao and Youshi Hong
From eq. (15), the relationship between cmax and D0 is therefore obtained: " # d cmax 2 exp D0 ÿ z Ng
16
Fig. 4 shows the variation of tcr with Ad and d for ccr=10. It is observed that the short which respectively re¯ects crack regime may last longer with smaller values of Ad and d, stronger grain-boundary obstacle eect and smaller grain size. (b) If D*0>1 at time t*, then tcr has another form: 1 t~cr t ÿ D0 ÿ 1 D~ 0 a where D~ 0
Z
ccr c jtt~cr
1 A c
Z
c 0
17
Ng nN c0 dc0 dc ÿ D0 jtt~cr ÿ1
18
However, in the present numerical calculation, we examine the trend of the response for D*0 and do not observe the occurrence of D*0>1. 3.0.3. Case III. Assume that the criterion of cmax be to satisfy n(cmax, t)E ncr, where Zc 1 0 0 ncr Ng nN c dc A c 0 cccr The ful®ll of the criterion means that the crack numerical density at c = ccr reaches a pre-set critical value. Note that in general, ccr is always several times greater than the grain size. Consequently for ccre2, referring to the de®nition of Z, we have: 8 > t 1 ln ccr ccr exp ÿd t < d Z ccr ; t 19 > 1 : 1 f1 b~ exp 1 ÿ Ad tg t > ln c cr 1ÿAd d
where ÿ 1ÿAd =d : b~ ÿAd ccr
Fig. 4. tcr as a function of Ad and d for case II.
An analysis of collective damage for short fatigue cracks
157
eq. (19) indicates that if t<
1 ÿ1 ; ln 1 ÿ Ad b~
then the crack numerical density would not be in saturation condition at c = ccr. Thus, from eq. (2), the evolution of crack numerical density at ccr is therefore obtained: Zc 1 0 0 n ccr ; t Ng nN c dc A c Z cccr ( Ng
ccr
2ÿZ 4 0
2
ÿ ÿ
t 1 ln c2cr d
t < 1 ln c2cr
20
d
If 1 ccr t > ln ; 2 d according to eq. (20), we may derive: Zcr Z ccr ; t jnncr 2 1 ÿ
p ccr ncr =Ng
By substituting the above expression into eq. (19), tcr can be solved as: 8 > ccr 1 1 > p ln 2 ccr ncr Ng =4 > < d 1ÿ ccr ncr =Ng tcr p > 2 1ÿAd 1ÿ ccr ncr =Ng ÿ1 > 1 > ln ccr ncr > Ng =4: : 1ÿA d b~
21
Figure 5 shows tcr as a function of Ad and d for this case. By comparison with Figs 3 and 4, It is seen that the extent of in¯uFig. 5 illustrates the same trend of tcr varying with Ad and d. ence of grain-boundary obstacle eect on tcr is greater than that of grain size, and if Ad and d are small, tcr is more sensitive to them. In fact, since grain-boundary obstacle eect Ad is a fac tor relatively dicult to control, people may pay more attention to the in¯uence of grain size d.
Fig. 5. tcr as a function of Ad and d for case III.
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Yu Qiao and Youshi Hong
Considering that Equation (15 and 21) both show that tcr is linear proportional to 1/d. short-crack regime takes up a large part of fatigue life, we assume that fatigue life is also linear namely proportional to 1/d, Nf f1
1 f2 d
22
where Nf is fatigue life, d is grain size, f1 and f2 are parameters related to fatigue stress range. Figure 6 shows the agreement between the theoretical estimation obtained from eq. (22) and experimental data from ref. [14]. It is worth pointing out that eq. (22) is proposed for the circumstance that short-crack regime takes up a large part of fatigue life. From eq. (21), the relationship between cmax and D0 is obtained: 8 > cmax a 1 > p ln cmax ncr Ng =4 > 2 1ÿ cmax ncr =Ng < d 23 D0 p > 2 1ÿAd 1ÿ cmax ncr =Ng ÿ1 > a > cmax ncr > Ng =4 : 1ÿAd ln b~ 1
where d =d b~ Ad c 1ÿA max
and cmax>2. Figure 7 shows the comparison of the relationship between D0 and cmax for the three cases. Evidently, the three curves almost increase exponentially with the enlargement of D0. There are two implications in Fig. 7. First, when the value of cmax is between 7 and 10, the resultant values of D0 for the three cases are within a narrow range. Secondly, if cmax is large, D0 is less sensitive to cmax (for example, D0vcmax=9 and D0vcmax=10 are nearly the same). Noting that ccr=cmax and D0=at, thus we may state that, when ccr is around 10, the estimation of critical time tcr is less dependent on the adoption of anyone of the criteria discussed. Here tcr describes the critical time that a few long cracks start to dominate the fatigue damage. 4. THE FIRST ORDER OF DAMAGE MOMENT The ®rst order of damage moment is de®ned as: Z1 n c; t c dc D1 g 0
Fig. 6. Fatigue life versus 1/d for two stress levels.
24
An analysis of collective damage for short fatigue cracks
159
Fig. 7. Relation between D0 and cmax for the three cases.
where g is a non-dimensional coecient. D1 is used to describe the extent of fatigue damage, which corresponds to the expectation of total crack length. From eq. (1), the evolution equation of c n(c, t) can be written as: @c n c; t @A c n c; t c Ng c nN A c n c; t @t @c
25
eq. (25) has the same form with eq. (1) if c n(c, t) is regarded as one variable. Referring to the equilibrium of evolution system, one may obtain: Zt Z1 dt0 Ng c nN A c n c; t0 dc D1 g 0
0
g a~ t g D1 where
Z a~
and D1
Z
0
t
0
dt
Z
1 0
D1 t ÿ d
Z
t 0
Z Y t
Ng c nN c dc
0
Z d t 0 A c ÿ dc n c; t dc D1 t0 dt0 g 0
eq. (26) can also be expressed as:
where
1
26
t
0
D1 t0 dt0 g a~ t g Y t
dt0
Let Ad=1 ÿ Ad, w=Ad+d,
Z
1 0
Z a1
A c ÿ d c n c; t0 dc
1 0
Ng nN c dc;
27
28
29
30
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Yu Qiao and Youshi Hong
and
Z a2
0
1
Ng c nN dc:
Then, substituting eq. (2 and3) into (30), one gains: Zt Z1 0 Y t dt 1 ÿ w c n c; t0 dc 0
Z
t
0
dt0
Z
0
1 0
n c; t0 dc ÿ w
Z
t 0
dt0
Z
1 0
c n c; t0 dc
31
Considering the equilibrium of crack numerical density for cracks with length between 0 and 1, one may write the total number of cracks as: Z t0 Z1 n c; t0 dc a1 t0 ÿ Ad n 1; t dt 32 0
0
Using the same method, one ®nds: Z1 Z t0 Z1 c n c; t0 dc a2 t0 A c n c; t0 dc ÿ Ad c n c; tc1 dt 0
0
0
33
Substituting eq. (32 and33) into eq. (31), one gets:
Z t0 Zt 1 ÿ w a2 1 ÿ da t2 ÿ Ad 1 ÿ w ÿ d dt0 n 1; t dt Y t ÿA d Y t 2 0 0
34
Note that from eq. (20), Zt Z t0 Z t0 Z 1 Zt 1 0 0 dt n 1; t dt dt Ng nN c dc dt 0 0 Ad Z 1;t 0 0
x x2 t x3 t2 x4 eAd t x5 e2Ad t
t t0
3Ng 2 8Ad t
t > t0
35
where x1
8 2A d ÿ 1 Ad Ng ; 16 A 4d
x3
3 A 2d ÿ 4 A d 1 Ng ; 8 A 2d Ad
t0
x2 ÿ x4
4 2A d ÿ 1 Ad Ng 8 A 3d
2 A d ÿ 1 Ng ; 2 A 4
x5
d
Ad Ng 16 A 4d
1 1 ln Ad Ad
By substituting eq. (35) into (34), Y(t) can be ®nally solved: 2 A d t x5 e2Ad t Y t x1 x2 t x3 t x24 e x6 t
t t0 t > t0
where x1 ÿ Ad q x1 ;
x2 ÿAd q x2 ;
x3 x ÿ Ad q x3 ;
x4 ÿ Ad q x4 ;
x5 ÿAd q x5 ;
x
q
1 ÿ g ÿ d ; 1 A d
3 x6 x ÿ q N g 8
1 ÿ g a2 1 ÿ da ; 2 1 A d
36
An analysis of collective damage for short fatigue cracks
161
If the initial condition is D1(0) = 0, the solution of eq. (29) will be: Z t Y 0 0 dt 0 0 ÿdt 0 D1 t g d e Y t a~ t e dt ÿ g~a t Y t d 0 Therefore D1(t) is obtained by substituting eq. (36) into eq. (37), 8 < x~ 1 x~ 2 t x~ 3 eA d t x~ 4 e2A d t x~ 5 edt g D1 t 2gx 2x ~ a d : ÿ d 6 t 6d2 g edt ÿ 1 where
t t0 t > t0
37
38
~x1 ÿ x2 a~ ÿ 2 x3 ; x2 ÿ 2 x3 ; x~ 3 x 1 Ad ; 4 d d2 d A d ÿ d ! x2 a~ 2 x3 d A d A d ~ ~x4 x 1 ; x5 ÿ x4 x5 5 d 2 A d ÿ d 2 A d ÿ d d2 A d ÿ d
eq. (38) is the theoretical solution to the evolution equation of the ®rst order of damage moment, and the relation between D1 g and t is illustrated in Fig. 8. Since D1 is often used to assess damage extent but hard to survey directly, and D0 is relatively easy to determine, Fig. 8 may provide a way to link together exper iments and theoretical analysis. Figure 9 shows the in¯uence of material parameters (Ad and d) on the ®rst order of damage moment, which indicates that the in¯uence of Ad is relatively weak Since x6 is considerably small in comparison with a d, it is easy to compared with that of d. ®nd that, when t>t0, eq. (38) reduced to : a~ g exp d t ÿ 1 D1 t d
39
The above equation corresponds to the resultant expression for D1 from a numerical simulation in ref. [12], in which
Fig. 8. Relation between D1/g and t at Ad=0 and d=1.
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Yu Qiao and Youshi Hong
Fig. 9. Variation of D1/g with Ad and d at t = 3.0.
D1 t
o exp k t ÿ 1 k
40
where o and k are material parameters. Equation (39 and40) suggest that D_ 1 is linearly proportional to D1: D_ 1 g1 D1 g2
41
where g1 and g2 are material parameters. Equations (39)±(41) express the relationship between damage evolution rate and damage extent, thus re¯ect the tendency of damage evolution, which is a result of equilibrium equation of crack numerical density and is independent of material properties and experimental environments. From eq. (39), it also can be seen that fatigue damage may develop faster with larger grain size, and the impact of grain size d on D1 has the form of [exp(d) ÿ 1]/d. Note that in eqs (39)±(41), the in¯uence of Ad is omitted, which suggests that D1 is independent of grain-boundary obstacle eect. 5. CONCLUSIONS The concept of the equilibrium of crack numerical density is applied to describe the collective evolution of short fatigue cracks, and the following conclusions are drawn: 1. There exists a saturation distribution of crack numerical density during the development of fatigue process. On the basis of this, the critical time that characterizes the termination of short-crack regime can be obtained upon the stipulation of a criterion. 2. The value of the critical time tcr is in¯uenced by the normalized grain size d and the obstacle eect of grain boundary Ad. tcr becomes larger with the decrease of d and increases with the reduction in Ad. The total fatigue life Nf is linearly proportional to 1/d. 3. The total number of cracks D0 increases linearly with time t (relevant to the number of fatigue cycles), whereas the ®rst order of damage moment D1 increases exponentially with t. When t>t0, D_ 1 is linearly proportional to D1. The in¯uence of grain-boundary obstacle eect on D1 can be omitted, and the in¯uence of grain size d has the form of [exp(d) ÿ 1]/d.
AcknowledgementsÐThis paper was supported by the National Outstanding Youth Scienti®c Award of China, the National Natural Science Foundation of China, and the Chinese Academy of Sciences.
An analysis of collective damage for short fatigue cracks
163
REFERENCES 1. Lankford, J. The growth of small fatigue cracks in 7076-T6 aluminum. Fatigue of Engineering Materials and Structures, 1982, 5, 233±248. 2. Zurek, A. K., James, M. R. and Morris, W. L. The eect of grain size on fatigue crack growth of short crack. Metallurgical Transactions A, 1983, 14, 1697±1705. 3. Fathulla, A., Weiss, B. and Stickler, R., Short fatigue cracks in technical Pm-Mo alloys. In The Behaviour of Short Fatigue Cracks, ed. Miller, K. J. and de los Rios, E. R.. Mechanical Engng Publisher, London, 1996, pp. 115±132. 4. Navarro, A. and de los Rios, E. R. A model for short fatigue crack propagation with an interpretation of the shortlong crack transition. Fatigue of Engineering Materials and Structures, 1987, 10, 169±186. 5. Goto, M. Statistical investigation of the behaviour of small cracks and fatigue life in carbon steel with dierent ferrite grain size. Fatigue and Fracture of Engineering Materials and Structures, 1994, 17, 635±649. 6. Weiss, J. and Pineau, A., Continuous and sequential multiaxial low-cycle fatigue damage in 316 stainless steel. In Advance in Multiaxial Fatigue, ASTM STP 1191, ed. McDowell, D. L. and Ellis, R.. American Society for Testing and Materials, Philadelphia, 1993, pp. 183±203. 7. Hong, Y., Gu, Z., Fang, B. and Bai, Y., Collective evolution characteristics and computer simulation of short fatigue cracks. Philosophical Magazine A, 1997, 75, 1517±1531. 8. Hong., Y., Lu, Y. and Zheng, Z. Initiation and propagation of short fatigue cracks in a weld metal. Fatigue and Fracture of Engineering Materials and Structures, 1989, 12, 323±331. 9. Hong, Y., Lu, Y. and Zheng, Z. Orientation preference and fractal character of short fatigue cracks in a weld metal. Journal of Materials Science, 1991, 26, 1821±1826. 10. Suh, C. M., Lee, J. J., Kang, Y. G., Ahn, H. J. and Woo, B. C. A simulation of the fatigue crack process in type 304 stainless steel at 5388C. Fatigue and Fracture of Engineering Materials and Structures, 1992, 15, 671±684. 11. Ke, F., Bai, Y. and Xia, M. Evolution of ideal micro-crack system (in Chinese). Science in China A, 1990, no. 6, 621±631. 12. Fang, B., Hong, Y. and Bai, Y. Experimental and theoretical study on numerical density evolution of short fatigue cracks. ACTA Mechanica Sinica (English Edition), 1995, 11, 144±152. 13. Kendall, J. M. and King, J. E. Short fatigue crack growth behaviour: data analysis eects. International Journal of Fracture, 1988, 10, 163±170. 14. Turnbull, A. and de los Rios, E. R. The eect of grain size on fatigue crack growth in an aluminium magnesium alloy. Fatigue and Fracture of Engineering Materials and Structures, 1995, 18, 1355±1366. (Received 28 January 1997, in final form 27 July 1997, accepted 27 July 1997)
Engineering Fracture Mechanics Vol. 59, No. 2, pp. 151±163, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0013-7944/98 $19.00 + 0.00 S0013-7944(97)00098-2
AN ANALYSIS OF COLLECTIVE DAMAGE FOR SHORT FATIGUE CRACKS BASED ON EQUILIBRIUM OF CRACK NUMERICAL DENSITY YU QIAO and YOUSHI HONG{ Laboratory for Nonlinear Mechanics of Continuous Media, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China AbstractÐCollective damage of short fatigue cracks was analyzed in the light of equilibrium of crack numerical density. With the estimation of crack growth rate and crack nucleation rate, the solution of the equilibrium equation was studied to reveal the distinct feature of saturation distribution for crack numerical density. The critical time that characterized the transition of short and long-crack regimes was estimated, in which the in¯uences of grain size and grain-boundary obstacle eect were investigated. Furthermore, the total number of cracks and the ®rst order of damage moment were discussed. # 1998 Elsevier Science Ltd KeywordsÐshort fatigue cracks, crack numerical density, collective damage, fatigue life, damage moment.
1. INTRODUCTION IN THE past two decades, dierent characteristics of fatigue damage have been observed in the primary and the ®nal stages of fatigue failure. In the primary stage of fatigue failure, the length of most cracks is comparable with the grain size. One distinct growth behaviour of short cracks has been reported as the crack growth rates with deceleration and acceleration patterns, which has been attributed to the interaction of crack tip plastic zone with microstructural barriers to plastic ¯ow [1±4]. In the recent years, another distinct phenomenon of short cracks has been revealed in the literature[5±10], namely that in short-crack regime fatigue process presents collective damage characteristics with the gradual evolution of crack numerical density, showing the number of short fatigue cracks increases with increasing number of fatigue cycles. The method based on the equilibrium of crack numerical density [11] is a possible approach to describe and analyze the collective evolution process of dispersed short cracks [12]. The basic consideration of the method is that the total number of short cracks is composed of cracks due to crack nucleation and crack growth. It is known that the short-crack regime takes up a large portion of total fatigue life and the approach to life prediction of this regime is still a problem remained for solution. One essential aspect of this problem is to de®ne the transition point between short and long-crack regimes, and then to determine the critical values of loading cycles and crack length. This is an important step toward the prediction of total fatigue life. The main objective of this paper is to present a new method to evaluate the collective behaviour of dispersive short-fatigue-cracks by using the concept of the equilibrium of crack numerical density, in which the crack numerical density denotes the number of cracks within unit area and unit crack length. After setting the expressions for crack growth rate and crack nucleation rate, the solution of the equilibrium equation and the evolution characteristics of crack population are studied. Three cases of transition criteria are discussed, with each showing the variation of critical time with grain size and microstructural obstacle eect. The total number of cracks and the ®rst order of damage moment (i.e. the total length of cracks) are also investigated, which are two factors available for the description of the extent of short-crack damage. {Author to whom correspondence should be addressed. 151
152
Yu Qiao and Youshi Hong
2. BASIC EQUATIONS In order to evaluate the collective damage of short fatigue cracks, the equilibrium equation of crack numerical density [11] is used in the following analysis. Its non-dimensional form is [12]: @ @ n c; t A c n c; t Ng nN c @t @c
1
where n(c, t) is the crack numerical density, with n(c, t)dc being the number of cracks with length between c and c + dc at time t; A(c) is crack growth rate; and nN(c) is crack nucleation rate. All of these physical quantities are dimensionless, and the non-dimensional coecient Ng
nN d ; n A
with n*N being the characteristic crack nucleation rate, A* the characteristic crack growth rate, n* the characteristic crack numerical density, and d the characteristic dimension of the material concerned (e.g. the grain diameter). Note that d A is equivalent to the characteristic time. Equation (1) describes the equilibrium of crack numerical density in phase space. The second term at the left side describes the ¯ow of crack numerical density in phase space, which is attributed to crack growth; and the term at the right side describes the contribution to crack numerical density made by crack nucleation. If n(c, 0) = 0 and the threshold value of crack length for growth is zero, the theoretical solution of eq. (1) is [11]: Zc 1 Ng nN c0 dc0 2 n c; t A c Z c;t The lower integral boundary Z(c, t) is of such a de®nition that for a crack with an initial length of Z(c, t) at t = 0, its length will advance to c at time t under the growth rate of A(c). Figure 1 is a schematic illustration showing the deceleration-acceleration pattern of shortcrack growth [1]. The curves imply that before the crack tip reaches the ®rst grain boundary, crack growth rate decreases with the increasing of crack length. If crack length is relatively large, the crack growth rate will tend to LEFM regime. Thus, in general, we may construct the
Fig. 1. Schematic of crack growth rate versus crack length in short crack regime. (Grain 1 < Grain 2 < Grain 3<
An analysis of collective damage for short fatigue cracks
following expression for A(c):
A c
1 ÿ 1 ÿ Ad c c 1 d c c > 1
153
3
where Ad is the crack growth rate when the normalized crack length c = 1, d is the ratio of nondimensional average grain size d to non-dimensional initial crack growth rate A0, i.e. the normalized grain size. If c is normalized by grain diameter, Ad is the crack growth rate at the time the crack tip reaches the ®rst grain boundary. Equation (3) shows the eects of crack length c, grain size d, and grain-boundary obstacle eect Ad on crack growth rate. The value of Ad is within the range from 0 to 1, which re¯ects a number of experimental measurements [1±4]. If Ad=0, eq. (3) has the meaning that short cracks stop propagating when crack tips reach grain boundaries. This is the extreme condition that the largest grain-boundary obstacle eect prevails. In experiments [7, 13], it was shown that the length of most cracks at initiation is less than the average grain size, and there was almost not a crack at nucleation with length larger than 2 d. Note that the crack length at initiation also depends on the time interval used for observation. Thus, the crack nucleation rate nN is constructed as follows: 1 ÿ c=2 c 2 nN 4 0 c > 2 After setting above estimations for A(c) and nN(c), the distribution of n(c, t) versus c is readily calculated from eq. (1). Figure 2 shows the variation of n(c, t) with c for dierent time stages. Figure 2 also shows that there is a saturation distribution in the evolution process of collective short cracks (dash line), i.e. with the progress of fatigue process, the distribution of crack numerical density gradually tends to a saturation curve from small to large value of crack size. It is obvious that the saturation curve presents the stable distribution of crack numerical density. Referring to eq. (2), we may write the saturation curve as: Zc 1 Ng nN c0 dc0 5 n0 c A c 0 For the explicitness in the following, we write: Z1 Ng nN c dc a 0
and
Z D0
1 0
n c; t dc
Fig. 2. Variation of n(c, t) with c for dierent time stages, the dash line representing the saturation curve. (Ad=0, d=1).
6
7
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Yu Qiao and Youshi Hong
It is clear that a is the number of cracks produced within unit time duration, and D0 is the number of total cracks at time t. Regarding the equilibrium of crack numerical density, we have Z1 n c; t dc a t 8 D0 0
Assuming that at time t, the distribution of crack numerical density in the area of c < c*(t) would converge to the saturation curve, as such, the number of cracks in non-saturation area is de®ned as D*0, where c* is the length of the crack (at time t) whose initial length is zero at t = 0., Thus from eq. (3), the evolution of c* can be expressed as: 1 ÿ 1 ÿ Ad c c 1 c_ A c 9 c > 1 d c The solution of eq. (9) is:
(
c
1 1ÿAd
f1 ÿ exp ÿ 1 ÿ Ad tg b exp d t
t t0 t > t0
10
where b =Add= 1ÿAd , and t0
1 1 ln ; b d
which is the time when c* = 1. 3. TOTAL NUMBER OF CRACKS, MAXIMUM CRACK LENGTH AND CRITICAL TIME From eq. (2), it is possible to derive the evolutionary features of crack numerical density. However, the crack numerical density n(c, t) cannot be used directly to determine the termination of short-crack regime. The boundary of short-crack regime and the start of long-crack regime should be indicated by the appearance of a crack with critical length. Thus we need to know the maximum crack length cmax(t) as a boundary value of the evolution system. It is stipulated that the numerical-density-equilibrium analysis of short crack is not valid when cmax(t)e ccr, where ccr is the critical crack length, beyond which the fatigue damage enters into long-crack regime. In relation to the stipulation of criteria from dierent viewpoints, the corresponding value of cmax(t) can be identi®ed. The following three cases are discussed. 3.0.1. Case I. Assume that the upper boundary of the saturation area c* be the maximum crack length cmax(t) (i.e. cmax=c*). The important premise of Case I is that only the crack numerical density at the region of c < c* may distribute stably. The ful®ll of the criterion c* = cmax=ccr is equivalent to the statement that a crack with its initial length of zero develops to a crack with critical length ccr. The evolution of cmax can be shown as: ( 1 1ÿAd f1 ÿ exp ÿ 1 ÿ Ad tg t t0 11 cmax c b exp d t t > t0 Substituting eq. (8) into (11), one obtains: 8 1 ÿ 1ÿA < 1ÿAd 1 ÿ exp ÿ a d D0 cmax 0 : b exp dD a
t t0 t > t0
12
Because it is certain that ccr>1, then the eq. (12) shows the variation of cmax with Ad and d. critical time tcr of the termination of short crack regime can be derived from eq. (11): 1 ccr 13 tcr t jcmax ccr ln d b Fig. 3 shows the relationship between tcr and the parameters of Ad and d for this case. It is seen
An analysis of collective damage for short fatigue cracks
155
Fig. 3. tcr as a function of Ad and d for case I.
suggesting that the that the value of tcr increases with the decrease in the values of Ad and d, tolerance of collective short-crack damage is enhanced if the material is of small grain size and large obstacle eect of grain boundary. 3.0.2. Case II. The criterion of cmax is set as Z1 1 n c dc ; n cmax which implies that there must be at least one crack with the length larger than cmax. Here n* is the characteristic crack numerical density and the situation of n* = 1 is considered. Then the criterion can be rewritten as Z1 n c dc 1: cmax
Let t* be the time when c* = cmaxeccr. If ccre2, then at time t*, the number of cracks in nonsaturation area is: Zc Z ccr 1 Ng nN ^c d^c dc D0 a t ÿ A c 0 0 Z 1 Z 2 Z ccr Zc 1 a t ÿ Ng nN ^c d^c dc A c 0 2 0 1 24 Ad ÿ 15 A2d 4 4Ad ÿ 3 ln Ad ÿ 10 5 1 ccr 14 ln a t ÿ Ng 2 8d d 16 1 ÿ Ad 3 In the following, the circumstances of D*0<1, and D*0>1 are separately discussed. (a) If D*0<1, from the equilibrium of crack numerical density, tcr can be derived: 1 tcr t 1 ÿ D0 a where z
Ng ccr ln z d a 2
24Ad ÿ 15A2d 4 4Ad ÿ 3 ln Ad ÿ 10 5Ng 1 1 Ng a 16 1 ÿ Ad 3 8d
15
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Yu Qiao and Youshi Hong
From eq. (15), the relationship between cmax and D0 is therefore obtained: " # d cmax 2 exp D0 ÿ z Ng
16
Fig. 4 shows the variation of tcr with Ad and d for ccr=10. It is observed that the short which respectively re¯ects crack regime may last longer with smaller values of Ad and d, stronger grain-boundary obstacle eect and smaller grain size. (b) If D*0>1 at time t*, then tcr has another form: 1 t~cr t ÿ D0 ÿ 1 D~ 0 a where D~ 0
Z
ccr c jtt~cr
1 A c
Z
c 0
17
Ng nN c0 dc0 dc ÿ D0 jtt~cr ÿ1
18
However, in the present numerical calculation, we examine the trend of the response for D*0 and do not observe the occurrence of D*0>1. 3.0.3. Case III. Assume that the criterion of cmax be to satisfy n(cmax, t)E ncr, where Zc 1 0 0 ncr Ng nN c dc A c 0 cccr The ful®ll of the criterion means that the crack numerical density at c = ccr reaches a pre-set critical value. Note that in general, ccr is always several times greater than the grain size. Consequently for ccre2, referring to the de®nition of Z, we have: 8 > t 1 ln ccr ccr exp ÿd t < d Z ccr ; t 19 > 1 : 1 f1 b~ exp 1 ÿ Ad tg t > ln c cr 1ÿAd d
where ÿ 1ÿAd =d : b~ ÿAd ccr
Fig. 4. tcr as a function of Ad and d for case II.
An analysis of collective damage for short fatigue cracks
157
eq. (19) indicates that if t<
1 ÿ1 ; ln 1 ÿ Ad b~
then the crack numerical density would not be in saturation condition at c = ccr. Thus, from eq. (2), the evolution of crack numerical density at ccr is therefore obtained: Zc 1 0 0 n ccr ; t Ng nN c dc A c Z cccr ( Ng
ccr
2ÿZ 4 0
2
ÿ ÿ
t 1 ln c2cr d
t < 1 ln c2cr
20
d
If 1 ccr t > ln ; 2 d according to eq. (20), we may derive: Zcr Z ccr ; t jnncr 2 1 ÿ
p ccr ncr =Ng
By substituting the above expression into eq. (19), tcr can be solved as: 8 > ccr 1 1 > p ln 2 ccr ncr Ng =4 > < d 1ÿ ccr ncr =Ng tcr p > 2 1ÿAd 1ÿ ccr ncr =Ng ÿ1 > 1 > ln ccr ncr > Ng =4: : 1ÿA d b~
21
Figure 5 shows tcr as a function of Ad and d for this case. By comparison with Figs 3 and 4, It is seen that the extent of in¯uFig. 5 illustrates the same trend of tcr varying with Ad and d. ence of grain-boundary obstacle eect on tcr is greater than that of grain size, and if Ad and d are small, tcr is more sensitive to them. In fact, since grain-boundary obstacle eect Ad is a fac tor relatively dicult to control, people may pay more attention to the in¯uence of grain size d.
Fig. 5. tcr as a function of Ad and d for case III.
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Yu Qiao and Youshi Hong
Considering that Equation (15 and 21) both show that tcr is linear proportional to 1/d. short-crack regime takes up a large part of fatigue life, we assume that fatigue life is also linear namely proportional to 1/d, Nf f1
1 f2 d
22
where Nf is fatigue life, d is grain size, f1 and f2 are parameters related to fatigue stress range. Figure 6 shows the agreement between the theoretical estimation obtained from eq. (22) and experimental data from ref. [14]. It is worth pointing out that eq. (22) is proposed for the circumstance that short-crack regime takes up a large part of fatigue life. From eq. (21), the relationship between cmax and D0 is obtained: 8 > cmax a 1 > p ln cmax ncr Ng =4 > 2 1ÿ cmax ncr =Ng < d 23 D0 p > 2 1ÿAd 1ÿ cmax ncr =Ng ÿ1 > a > cmax ncr > Ng =4 : 1ÿAd ln b~ 1
where d =d b~ Ad c 1ÿA max
and cmax>2. Figure 7 shows the comparison of the relationship between D0 and cmax for the three cases. Evidently, the three curves almost increase exponentially with the enlargement of D0. There are two implications in Fig. 7. First, when the value of cmax is between 7 and 10, the resultant values of D0 for the three cases are within a narrow range. Secondly, if cmax is large, D0 is less sensitive to cmax (for example, D0vcmax=9 and D0vcmax=10 are nearly the same). Noting that ccr=cmax and D0=at, thus we may state that, when ccr is around 10, the estimation of critical time tcr is less dependent on the adoption of anyone of the criteria discussed. Here tcr describes the critical time that a few long cracks start to dominate the fatigue damage. 4. THE FIRST ORDER OF DAMAGE MOMENT The ®rst order of damage moment is de®ned as: Z1 n c; t c dc D1 g 0
Fig. 6. Fatigue life versus 1/d for two stress levels.
24
An analysis of collective damage for short fatigue cracks
159
Fig. 7. Relation between D0 and cmax for the three cases.
where g is a non-dimensional coecient. D1 is used to describe the extent of fatigue damage, which corresponds to the expectation of total crack length. From eq. (1), the evolution equation of c n(c, t) can be written as: @c n c; t @A c n c; t c Ng c nN A c n c; t @t @c
25
eq. (25) has the same form with eq. (1) if c n(c, t) is regarded as one variable. Referring to the equilibrium of evolution system, one may obtain: Zt Z1 dt0 Ng c nN A c n c; t0 dc D1 g 0
0
g a~ t g D1 where
Z a~
and D1
Z
0
t
0
dt
Z
1 0
D1 t ÿ d
Z
t 0
Z Y t
Ng c nN c dc
0
Z d t 0 A c ÿ dc n c; t dc D1 t0 dt0 g 0
eq. (26) can also be expressed as:
where
1
26
t
0
D1 t0 dt0 g a~ t g Y t
dt0
Let Ad=1 ÿ Ad, w=Ad+d,
Z
1 0
Z a1
A c ÿ d c n c; t0 dc
1 0
Ng nN c dc;
27
28
29
30
160
Yu Qiao and Youshi Hong
and
Z a2
0
1
Ng c nN dc:
Then, substituting eq. (2 and3) into (30), one gains: Zt Z1 0 Y t dt 1 ÿ w c n c; t0 dc 0
Z
t
0
dt0
Z
0
1 0
n c; t0 dc ÿ w
Z
t 0
dt0
Z
1 0
c n c; t0 dc
31
Considering the equilibrium of crack numerical density for cracks with length between 0 and 1, one may write the total number of cracks as: Z t0 Z1 n c; t0 dc a1 t0 ÿ Ad n 1; t dt 32 0
0
Using the same method, one ®nds: Z1 Z t0 Z1 c n c; t0 dc a2 t0 A c n c; t0 dc ÿ Ad c n c; tc1 dt 0
0
0
33
Substituting eq. (32 and33) into eq. (31), one gets:
Z t0 Zt 1 ÿ w a2 1 ÿ da t2 ÿ Ad 1 ÿ w ÿ d dt0 n 1; t dt Y t ÿA d Y t 2 0 0
34
Note that from eq. (20), Zt Z t0 Z t0 Z 1 Zt 1 0 0 dt n 1; t dt dt Ng nN c dc dt 0 0 Ad Z 1;t 0 0
x x2 t x3 t2 x4 eAd t x5 e2Ad t
t t0
3Ng 2 8Ad t
t > t0
35
where x1
8 2A d ÿ 1 Ad Ng ; 16 A 4d
x3
3 A 2d ÿ 4 A d 1 Ng ; 8 A 2d Ad
t0
x2 ÿ x4
4 2A d ÿ 1 Ad Ng 8 A 3d
2 A d ÿ 1 Ng ; 2 A 4
x5
d
Ad Ng 16 A 4d
1 1 ln Ad Ad
By substituting eq. (35) into (34), Y(t) can be ®nally solved: 2 A d t x5 e2Ad t Y t x1 x2 t x3 t x24 e x6 t
t t0 t > t0
where x1 ÿ Ad q x1 ;
x2 ÿAd q x2 ;
x3 x ÿ Ad q x3 ;
x4 ÿ Ad q x4 ;
x5 ÿAd q x5 ;
x
q
1 ÿ g ÿ d ; 1 A d
3 x6 x ÿ q N g 8
1 ÿ g a2 1 ÿ da ; 2 1 A d
36
An analysis of collective damage for short fatigue cracks
161
If the initial condition is D1(0) = 0, the solution of eq. (29) will be: Z t Y 0 0 dt 0 0 ÿdt 0 D1 t g d e Y t a~ t e dt ÿ g~a t Y t d 0 Therefore D1(t) is obtained by substituting eq. (36) into eq. (37), 8 < x~ 1 x~ 2 t x~ 3 eA d t x~ 4 e2A d t x~ 5 edt g D1 t 2gx 2x ~ a d : ÿ d 6 t 6d2 g edt ÿ 1 where
t t0 t > t0
37
38
~x1 ÿ x2 a~ ÿ 2 x3 ; x2 ÿ 2 x3 ; x~ 3 x 1 Ad ; 4 d d2 d A d ÿ d ! x2 a~ 2 x3 d A d A d ~ ~x4 x 1 ; x5 ÿ x4 x5 5 d 2 A d ÿ d 2 A d ÿ d d2 A d ÿ d
eq. (38) is the theoretical solution to the evolution equation of the ®rst order of damage moment, and the relation between D1 g and t is illustrated in Fig. 8. Since D1 is often used to assess damage extent but hard to survey directly, and D0 is relatively easy to determine, Fig. 8 may provide a way to link together exper iments and theoretical analysis. Figure 9 shows the in¯uence of material parameters (Ad and d) on the ®rst order of damage moment, which indicates that the in¯uence of Ad is relatively weak Since x6 is considerably small in comparison with a d, it is easy to compared with that of d. ®nd that, when t>t0, eq. (38) reduced to : a~ g exp d t ÿ 1 D1 t d
39
The above equation corresponds to the resultant expression for D1 from a numerical simulation in ref. [12], in which
Fig. 8. Relation between D1/g and t at Ad=0 and d=1.
162
Yu Qiao and Youshi Hong
Fig. 9. Variation of D1/g with Ad and d at t = 3.0.
D1 t
o exp k t ÿ 1 k
40
where o and k are material parameters. Equation (39 and40) suggest that D_ 1 is linearly proportional to D1: D_ 1 g1 D1 g2
41
where g1 and g2 are material parameters. Equations (39)±(41) express the relationship between damage evolution rate and damage extent, thus re¯ect the tendency of damage evolution, which is a result of equilibrium equation of crack numerical density and is independent of material properties and experimental environments. From eq. (39), it also can be seen that fatigue damage may develop faster with larger grain size, and the impact of grain size d on D1 has the form of [exp(d) ÿ 1]/d. Note that in eqs (39)±(41), the in¯uence of Ad is omitted, which suggests that D1 is independent of grain-boundary obstacle eect. 5. CONCLUSIONS The concept of the equilibrium of crack numerical density is applied to describe the collective evolution of short fatigue cracks, and the following conclusions are drawn: 1. There exists a saturation distribution of crack numerical density during the development of fatigue process. On the basis of this, the critical time that characterizes the termination of short-crack regime can be obtained upon the stipulation of a criterion. 2. The value of the critical time tcr is in¯uenced by the normalized grain size d and the obstacle eect of grain boundary Ad. tcr becomes larger with the decrease of d and increases with the reduction in Ad. The total fatigue life Nf is linearly proportional to 1/d. 3. The total number of cracks D0 increases linearly with time t (relevant to the number of fatigue cycles), whereas the ®rst order of damage moment D1 increases exponentially with t. When t>t0, D_ 1 is linearly proportional to D1. The in¯uence of grain-boundary obstacle eect on D1 can be omitted, and the in¯uence of grain size d has the form of [exp(d) ÿ 1]/d.
AcknowledgementsÐThis paper was supported by the National Outstanding Youth Scienti®c Award of China, the National Natural Science Foundation of China, and the Chinese Academy of Sciences.
An analysis of collective damage for short fatigue cracks
163
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