Local fatigue damage accumulation around notch attending crack initiation

Local Fatigue Damage Accumulation around Notch Attending Crack Initiation

Y. IINO

The subsequent recrystallization technique was used to study the process of local damage accumulation around a notch under conditions of low-cycle fatigue. A 0.8-in. compact tension specimen of 304 stainless steel with a notch radius of 1 mm was used. The accumulated plastic zone around notch increases with the number of cycles N. The accumulated plastic strain within the zone also increases with N, producing the strain gradient (damage gradient). A fatigue crack initiates when the accumulated plastic strain at the notch root reaches a critical value equal to the fracture strain of the material; that is, when the accumulated plastic work at the crack initiation site becomes critical. The fatigue crack emanating from a notch root grows through the pre-existing damaged zone. It is shown that this local damage accumulation approach can explain the fast growth of a short crack from a notch.

I. I N T R O D U C T I O N

BECAUSE fatigue cracks emanating from stress rais- ers such as notches, holes, and fillets are one of the pri- mary causes of failure in many structural and machine components, the initiation of these cracks has been a subject of active interest. Prediction of the crack initiation (that is, of the number of stress cycles before crack initiation, N~) has been examined mainly by two methods.

A. Fracture Mechanics Approach

The relation between Ni and a parameter AK/~v/p, where AK is the stress-intensity-factor range and p is the notch radius, is plotted on a log-log scale. Lz-6J A linear relation is obtained when the notch tip radius is constant, but the corresponding A K / V p levels decrease as p increases. Iz,71

B. Local Strain Approach

The basis of this approach is that even though nominal stress of a component is elastic, the local stress and strain at a notch root are inelastic and that the notch-root fatigue behavior could be related to low cycle fatigue laws. The Ni is determined by use of the quantity ~ , where E is Young's modulus, Air is the local cyclic stress, and Ae is the local cyclic strain. The relation between log ~ and log N, is linear, t7-~~ In this analysis, however, the strain gradient ahead of the notch is not considered. If the component is very large and the notch radius is also large--that is, where the strain gradient is not s teep-- the previous analysis is realistic. When the component is small or the notch radius is small, the analysis would be inadequate, because the strain gradient is significant when the plastic zone is small, llJ,J2j and the strain distribution would vary with N because of damage accumulation and work hardening. The local strain approach has been slightly modified by taking account of the surface-strain distribution measured with a strain gage t~3J and of the calculated strain distribution, t~4,~SJ

Y. IINO, Professor, is with the Toyota Technological Institute, Hisakata, Tempaku, 468 Nagoya, Japan.

Manuscript submitted June 11, 1993.

Crack initiation has also been treated in terms of damage accumulation: (1) a crack initiates when the summation of the plastic shear deformation due to notch bulk plasticity exceeds a threshold value, t161 and (2) a crack initiates when the summation of the plastic strain energy dissipated by one cycle reaches a critical value, l~TJ The plastic strain energy is the total energy around the notch; how much of the energy is accumulated at the notch root where crack initiates is not known.

The process of damage accumulation before crack initiation, however, is not taken into consideration in these approaches and treatments; i.e., whether or not the plastic zone formed during the first cycle increases with N, how extensive the damaged zone is at crack initiation, how steep the damage gradient is, and how much the damage is at crack initiation. Fundamental studies ad- dressing these questions seem not to have been carried out, probably because of the difficulty of measuring the very localized plastic strain and the very localized damage zone.

Metallographic techniques such as etching t~2'~Bj and recrystallization t19-23] could help answer these questions, because (1) they are applicable to the inner section of the notch part, (2) microscopic or semimicroscopic measurement is possible, and (3) the recrystallization is caused by plastic energy intimately related to the damage accumulation.

The accumulated plastic zone and the accumulated plastic strain around a growing long fatigue crack have been investigated using the recrystallization technique [19"2~ and the microstructures ahead of the crack tip have been investigated using the etching technique, t~sl X-ray microbeams, t24~ and transmission elec- tron microscopy, tzSj Some results relevant to the study on notch damage accumulation are the following.

(1) An accumulated plastic zone is formed around a fatigue crack, and the crack growth rate is proportional to the square of the accumulated plastic zone size. (2) Accumulated plastic strain e,, increases significantly very near the tip of a fatigue crack, and the crack propagates when this becomes equal to the fracture strain ef. (3) For a given growth rate, the accumulated plastic zone size increases with increasing strain-hardening coefficient n.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 26A, JUNE 1995-- 1419

(4) Plastic work per unit length crack growth increases with n. (5) The fatigue-crack-tip region is significantly strained and damaged; dislocation density is high.

Extending these findings to fatigue-crack initiation, we would expect the following.

(1) A fatigue crack initiates when the accumulated plastic strain at a notch root reaches a critical value. (2) The accumulated plastic zone increases with the increasing number of cycles until a crack initiates. (3) The accumulated plastic zone size at crack initiation increases with increasing n.

The present work analyzed the local damage accumulation process around a notch before crack initiation in the low-cycle fatigue region. This process was analyzed metallographically using a newly developed subsequent recrystallizations technique p6~ that measures zones with various amounts of accumulated plastic strain. The results of this analysis are used to explain the fatigue- crack initiation process and the growth behavior of short cracks emanating from notches.

II. SUBSEQUENT R E C R Y S T A L L I Z A T I O N S T E C H N I Q U E

This metallographic method observes and measures the plastic zone and strain by use of the critical strain for recrystallization to occur ecR, and by using the relation between plastic strain e and the recrystallized grain size D, where ecR and e are equivalent plastic strains (in the case of monotonic tension, they are true plastic strain), p~ The technique can therefore be applied to a three- dimensional strain state, such as that of a notch and crack tip. Thermal etching was found to be effective for vis- ualizing the plastic zone. [231 Usually only one anneal is carried out for one specimen, by which one plastic zone with strain above ~cR is observed.

To observe plastic zones with various amounts of strain, the subsequent recrystallizations technique was developed. [26] The recrystallization temperature is increased stepwise to first observe the plastic zone with high plastic strain, then a zone with lower plastic strain, and so on. The ecR of the 304 stainless steel used in this work were, as shown in Figure 1,1261 0.5 for the anneal at 1023 K for 86.4 ks (anneal A), 0.12 for the subsequent anneal at 1173 K for 86.4 ks (anneal B), and 0.02 for the subsequent anneal at 1223 K for 86.4 ks (anneal C). The relation between e and D, determined by linear analysis, is plotted in Figure 2. The sample microstructure after anneal A was too poor for the grain size to be determined, so the relation is shown by a dotted line. The microstructures after anneals B and C were clear, and in both cases, D decreased with increasing e from ecR, and then increased and decreased again. The increase in D is due to the secondary recrystallization. Since the recrystallization is caused by the stored energy given by the external work, the technique is suitable for measuring the amount of the fatigue damage. The degree of the damage is expressed by the accumulated plastic strain eac. The amount of strain e~c is that for which the recrystallized microstructure after the anneal in the fatigue- damaged specimen is the same as in the specimen

1.0

tO

0.5

0.1

~176 I o

ECR o 0.5 O

�9 O

�9 O �9 n

0 0 0

0 0 0

0 0 0

o 0.12 z~ 0 0

O O

I I I

I023K I023K 1173K 86.4ks 86.4ks 86Aks

+ 11?3K 86~ks

A B annea I

o o 0.02

I I

1023K 1223K 86.41~ 86Aks

+

1173K 86Aks

-4.- 1223K 86.4ks

C

Fig. 1--Crit ical strain for recrystallization ecR for three anneals of 304 stainless steel. 126~

i I i I i

1.5 ~-~-~ I' i ~ - I onneal I A C

t 1.0 ~ -

t I

\ \

0.5 EcR =0.5 ~ ,, c

0 0.0 2 ~ -

i t , I~,~,I I i i Iltitl I i i

I Illl

10 0 101 102 10 3 D , lain

Fig. 2--Rela t ion between recrystallized grain size D and true stain e for three anneals.

1420--VOLUME 26A, JUNE 1995 METALLURGICAL AND MATERIALS TRANSACTIONS A

monotonically strained to the same amount of true tensile strain e. The strain eac is equivalent strain because the strain state around notch is triaxial.

IlL EXPERIMENTAL PROCEDURE

The material used in this study was a commercially available type 304 stainless steel plate 5.8-mm thick. The 0.8-in. compact tension specimens (Figure 3) were machined so that the loading direction was parallel with the rolling direction. The notch (radius p = 1 mm) was machined by electric-discharge wire cutting to avoid surface deformation during machining. Specimens with p :- 0.25 and 4 mm were also machined for comparison of the damage accumulation at crack initiation. Most tests, however, were carried out on samples with p = 1 mm. The notch root was carefully mechanically polished using emery paper numbers 320, 800, and 1500.

The specimen was low-cycle fatigued at room temperature (20 ~ to 23 ~ using an MTS testing machine under fully reversed load control of the maximum apparent stress intensity factor Kpmax = 34.4 M N . m -3/2 and the minimum Kpm~, = - 3 4 . 4 MN" m -3/2. The frequency was 0.1 to 0.3 Hz. During the fatigue the notch root was observed using a stereoscope. The number of cycles to fatigue-crack initiation Ni here was defined as the number of cycles after which a 1-mm-long crack on the notch-root surface was visible at 20-fold magnifi- cation. To observe the damage accumulation process, specimens were fatigued to various numbers of cycles: for p = 1 mm, N = 1/2 (monotonic tension), 5, 10, 20, 275, 550 = Ni; for p = 0.25 mm, N = 100 = Ni; and for p = 4 mm, N = 3280 = - N i (N was a little more than Ni).

All the tests were stopped after unloading from the tensile part of the last cycle. After the fatigue test, specimens were sectioned at midthickness into two parts by electric-discharge wire cutting.

t =5.8

~ ~ 16

, . m

50

",,1"

Fig. 3 - -Tes t specimen.

METALLURGICAL AND MATERIALS TRANSACTIONS A

The cut surfaces were polished using emery papers of grade numbers up to 1500 under water cooling. One of the cut specimens was annealed in a vacuum furnace at 1023 K for 86.4 ks (anneal A) to observe the accumulated plastic zone with eac ~ 0.5 (APZo.5). The other cut specimen was treated with anneal B to observe the accumulated plastic zone with Eac ~ 0.12 (APZo.~2), fol- lowed by anneal C to observe the accumulated plastic zone with eac ~ 0.02 (APZo.02). Microstructure after anneal A was observed by chemical etching in solution of C2HsOH: HC1 :HNO3 (100: 30: 15). Microstructures after anneals B and C were well thermally etched during the anneal and were observed directly after the annealings. A micro-Vickers hardness test (with 0.295 N) was also done on midthickness surface of the as-fatigued specimens.

IV. RESULTS

A. Notch Tip Radius of 1 mm

Figure 4 shows the microstructures around the notch after anneal C. The recrystallized zone is APZo.02. At N = 1/2 (monotonic tension), no recrystallized grain is observed. The plastic strain within about 0.1 mm of the notch tip is said to be below 0.02, since the gage length of the recrystallization technique to measure plastic strain after anneal C is about 0.1 mm. With increasing N, the APZo.o2 increases not only ahead of the notch tip (in the X direction) but also in the Y direction and along the notch root. The increase of APZ0.02 with N is shown in Figure 5. From Figure 5(a) it is seen that APZo.o2 first increases much in the X direction and then in the Y direction. Figure 5(b) shows the increase of R,o.o2, the distance from the notch tip to the APZo.02 front in the crack-growth direction (X direction) and of Rot.02, the distance from the notch tip along the notch root to the APZo.02 front. The angle from the notch tip to the APZo.02 front, 00.02, is also plotted. It is seen that both R~o.o2 and R00.02, and 00.02, increase rapidly during the initial 20 cycles, and then slowly and linearly. The term 00.02 at N = --400 is estimated to be - 9 0 deg, which means that all the notch root is covered by the accumulated plastic strain above 0.02. It increases further until 00.02 = 100 deg. The large increase of R~o.02 near N = 500 to 550 is due to the crack formation, as can be seen in Figure 4(d). Careful observation showed the formation of many microcracks of 20 to 30 /zm along the notch root at N = 275, which were not detected at the notch root by the stereoscope.

Figure 6 shows the microstructures after anneal B. The accumulated plastic zone with APZo.~2 is not observed at N = 1/2 and 10, but at N = 20 (Figure 6(b)), and it increases with N. Since the specimens are the same as those shown in Figure 4, one can compare the zone sizes directly. The APZo.12 is smaller than APZ0.o2 but the tendency of its increase with increasing N is similar to that of APZo o2.

Figure 7 shows the microstructures after anneal A. The APZo5 is not seen at N = 20, but is formed at N = 275 along the notch root with only 20 to 25-/xm zone width and very near the microcracks (Figure 7(e)). The zone width does not increase so much even at N = 550; 80

VOLUME 26A, JUNE 1995-- 1421

(a) (b)

(c) (d)

Fig. 4--Micros t ructure after anneal C; recrystallized zone showing the accumulated plastic zone with accumulated plastic strain ~c => 0.02 (APZ0.02). Number of cycles N equals (a) 1/2 (monotonic tension), (b) 20, (c) 275, and (d) 550.

to 100/xm (Figures 7(c), (g)). The APZ0.5 is also formed around a somewhat long fatigue crack (Figures 7(c), (f)).

The accumulated plastic strain distribution inferred from these observations is shown in Figure 8. The development of three APZs is clear. There is, of course, also an accumulated plastic zone with Eac ~ Ey (yield strain), APZ~,. around APZ0.02, but it is not shown in Figure 8. It should be noted that APZ0.5 does not become large,

but is limited to the vicinity of the notch root even at N = 550 and that as the fatigue crack grows it leaves a small wake of APZ05 through the fatigue-damaged zone. The accumulated plastic strain gradient--that is, the damage gradient--is thus significant.

The hardness distribution near midthickness ahead of the notch tip is plotted in Figure 9. The hardened zone increases with N and the value of hardness at a given


I I I I I - z2o 2.o

E E 1.5 - ,t~z~,z 90

o.oz w

g.

6 0 . 1.C -

- - ~ R x o.oz t"9 30

~T 0.5 ~

0 , , , , i . . . . I , I , t , [ I [ I 0 0 100 200 300 400 5 0 0

~ - ~ N , cycle

(a) (b)

Fig. 5 - -Re la t ion between the number of cycles N and (a) accumulated plastic zone with ear --> 0.02 (APT~.~) around notch and (b) the zone size in the crack-growth direction (R~o.o2) and along the notch root (Roo.o2) and the increase of angle (80.02).

point increases with N. Thus, fatigue hardening of this material is remarkable.

B. Notch Tip Radius of 0.25 and 4 mm

Figures 10(a) and (b) show the recrystallized microstructure after anneal C at N = -Ni. The recrystallized zone is APZ002. The shape of the zone of p = 0.25 mm is similar to that of p = 1 mm but the zone size is smaller (compare with Figure 4(d)). For p = 4 mm, however, the shape is different: the upper and lower parts of the notch are not recrystallized. The zone size is, however, greater than that for p = 0.25 mm and p = 1 mm. Figures 10(c) and (d) show the recrystallized microstructure after anneal B. The recrystallized zone is APZ0 ~2, and the tendency of their sizes is similar to that for APZo 02. It is seen from Figure 10(d) that the crack leaves a wake of eac => 0.12 as it propagates.

The microstructure near the notch tip after anneal A is shown in Figure 11. The recrystallized zone is APZ0.s. The zone size in crack-growth direction, R~os, is 50/zm for p -- 0.25 mm, 80 to 100/zm for p = 1 ram.

Figure 12 shows the effect of notch tip radius p on the accumulated plastic zone size, R~o.~2 and 2Rro.02 at Ni. In the range 0.25 =< p < 4 mm, the zone size increases with p.

V. DISCUSSION

A. Development of the Accumulated Plastic Zone around the Notch

Both the accumulated plastic zone and the accumulated plastic strain within the zone increase with N until

METALLURGICAL AND MATERIALS TRANSACTIONS A

fatigue-crack initiation. In the present constant nominal stress range of the specimen with p = 1 mm, Kp = ---34.4 MN. m -3/2, the notch opening displacement is constant during the fatigue and the notch tip plastic strain during the first cycle, N = 1/2, is less 0.02 (no recrystallization). Nevertheless, the recrystaUized zone developed and increases with N. This could be explained as follows. The plastic zone formed at N = 1/2 is cy- clically deformed during the following cycles and work hardens. Yield stress of the zone is increased and plastic deformation decreases. Since notch opening displacement is constant, the region surrounding the work- hardened zone is plastically deformed to compensate the decrease of deformation in the hardened zone. If the surrounding zone work hardens, the fn'st work-hardened zone begins to deform plastically to increase the accumulated plastic strain and at the same time parts beyond the surrounding region are plastically deformed. Thus, APZ increases and e~ within APZ also increases.

B. Damage Accumulation Process

From the accumulated plastic strain distribution ahead of the notch tip (p = 1 mm) (Figure 13), it is seen that damage accumulation is the highest at the notch tips and the notch root, and becomes small with increasing distance from the notch tip and the notch root. From Figure 13, the damage accumulation r a t e - - t h e amount of increase of eac per cycle, deac/dN--is roughly estimated. The rough deaffdN values at five points, whose positions can be designated using X-Y coordinates Pj(0, 0), P2(0.2, 0), P3(0.5, 0), P4(-0 .25 , 0.75), and Ps(1.0, 1.0) in Figure 13, are listed in Table I and plotted in Figure 14. The deac/dN values at points P , , / 2 ,

VOLUME 26A, JUNE 1995--1423

(a) (b)

(c) (d)

Fig. 6--Microstructure after anneal B: recrystallized zone showing the accumulated plastic zone with eoc => 0.12 (APZo ~2). Number of cycles N equals (a) 1/2, (b) 20, (c) 275, and (d) 550.

P3, and P4 decrease with N until N = 275 because of work hardening. The increase of de,JdN at P2, P3, and P4 in the range N = 275 to 550 is due to the lengthening of the crack. It should be noted that de,JdN at point P5 is 0.00025 in the range N = 275 to 550, whereas it is- almost zero (elastic deformation) at N below 275.

If linear accumulation of the cyclic plastic strain is assumed, the half value of deac/dN is cyclic plastic strain amplitude. Notch tip (P0 strain amplitude is estimated from Figure 14 to be about 0.01 at initial fatigue cycles (N = 1 to 5), and decreases to about 0.001 at N from 20 to 100. Thus, cyclic strain amplitude at a given point


(a) (b) (c)

Fig. 7- -Micros t ruc ture after anneal A. The fine recrystallized grains show the accumulated plastic zone with eoc => 0.5 (APZ0.5). Number of cycles N equals (a) and (d) 20, (b) and (e) 275, and (c), ( f ) , and (g) 550. (d), (e) and (f), and (g) are enlarged views of the part indicated by " [ " in (a), (b), and (c), respectively.

varies during fatigue even if the remote nominal stress is held constant. This is, of course, due to stress and strain gradients and to work hardening.

The critical value of 6.ac at crack initiation is difficult to determine experimentally. But given the observations that (1) the accumulated plastic zone APZ0.5 is formed for all three p values (Figures 7 and 11) and yet the accumulated plastic strain increases very steeply toward the notch tip (Figure 13), (2) eac at the tip of a long growing crack is the fracture s t r a in f 9'2~ and (3) the dislocation density at the tip of the fatigue crack is very high, t24,251 it can be said a fatigue crack initiates when the local accumulated plastic strain at the crack initiation site (notch root) is equal to the fracture strain e s. In terms of energy, this means that crack initiates when the local accumulated plastic work at the initiation site reaches a critical value.

It is obvious from the previous consideration that the

strain-hardening coefficient n has significant effect of damage accumulation. If n is zero or very small, the plastic zone formed during the first cycle will no t increase or will increase little, since cyclic plastic deformation corresponding to notch opening displacement can occur within the first plastic zone. Damage will accu- mulate within the zone until finally a crack initiates. The present material, 304 stainless steel, has very high value of n (0.5) at room temperature. So, the accumulated plastic zone size increases significantly during fatigue, as shown in the results.

Theoretically, the maximum stress around a notch takes place at the notch tip, and this implies that the crack initiates at the notch tip. But many microcracks are formed along the notch root. On the micrographs in Figures 7(b) and (c), a number of microcracks were 15 at N = 275 and 17 at N = 550. The number of the crack with length a is shown in Figure 15. The range 20 of angles in which

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 26A, JUNE 1995--1425

(d) ( f )

(e) (g)

Fig. 7 Cont . - -Micros t ruc ture after anneal A. The fine recrystallized grains show the accumulated plastic zone with eac -> 0.5 (APZ0.5). Number of cycles N equals (a) and (d) 20, (b) and (e) 275, and (c), ( f ) , and (g) 550. (d), (e) and (f), and (g) are enlarged views of the part indicated by " [ " in (a), (b), and (c), respectively.


cracks were observed was 70 deg at N = 275 and 106 deg at N = 550. The number of microcracks at crack initiation for p --- 0.25 and 4 m m is 13 and 30, respectively, and 20 is 135 and 70 deg, respectively. This means that an equal damage-accumulation region is formed along the notch root. The free-surface effect, which causes a lower yield stress in a surface layer than in bulk material in monotonic loading t27'281 and a higher dislocation density in a surface layer than in bulk material in cyclic loading, t291 might contribute to this phenomenon. This possibly should be examined.

C. Local Damage Accumulation Approach for Cracks Emanating from Notches

One of the microcracks becomes a main crack. The longest crack at N = 550 indicated by the arrow in Figure 15 is such a crack. It grows through the damage- accumulated zone around the notch which was formed

before crack initiation. In other words, the crack grows through the damage-accumulation mountain (whose height is the amount of the accumulated plastic strain), forming its own accumulated plastic zone on the mountain. The situation in the case of low-cycle fatigue regions is shown schematically in Figure 16. Figure 16(a) is drawn ac- cording to Figures 7(c) and (f), where APZ0.5 is formed around the crack. Since the line o f eac = 0.5 near the crack is the sum of the pre-existing accumulated plastic

Fig. 9 - -Mic ro -Vicke r s hardness Hv distribution ahead of notch tip for various numbers of cycles N. d: distance from the notch tip in the X direction.

(a) (c)

Fig. 8 - - A c c u m u l a t e d plastic strain distributions around notch at various numbers of cycles N.

(b) (d)

Fig. lO--Microstructure at N = -N~ (a) and (b) after anneal C and (c) and (d) after anneal B. The recrystallized zone of (a) and (b) is APZo.o2 and that of (c) and (d) is APZo 12. (a) and (c) p = 0.25 mm, N = 100; (b) and (d) p = 4 mm, N = 3280.


(a)

Fig. 13--Accumulated plastic strain e.,. distribution ahead of the notch tip and ahead of the notch root at 0 = 45 deg.

(b)

Fig. l l - -Micros t ruc ture after anneal A at N = ~Ni. The recrystallized zone is APZ0.5. (a) p = 0.25 mm, N --- 100; (b) p = 4 ram, N = 3280.

Fig. 12--Effect of notch tip radius p on the accumulated plastic zone size R~o.o2 and 2R~oo: at crack initiation.

Fig. 14--Average accumulated plastic strain increase rate de~c/dN vs number of cycles at five points in Fig. 13.


I I N I

0 275 A 55O

o IOO 200 O j p m

2e 70~

106"(N-550) -

= I I I ,,/A,

300 1400

Fig. 15 - -Number of microcracks at notch root. a: length of microcracks, f: number of microcracks.

strain of the notch, ~'acnt, and that of the fatigue crack, ~'acfc ( i .e . , Eactota I = ~'acnt ql_ ~'acfc), ~'acfc can be determined from e~r At deflection points a and a' of the e,~ = 0.5 line, e~:~ is the yield strain, e~ = ey. Accordingly, the accumulated yield strain zone of crack APZ~f~ can be drawn by the dotted line. It is obvious that APZ~f~ is very small near the notch tip. Half of the distance aa' is the accumulated plastic zone size in the Y direction. In the case shown in Figure 7(f), the value is about 50/.~m. The growth rate of the crack can be estimated to be ~0.8 ~m/cycle , since the crack grew about 0.2 mm in 275 cycles. The accumulated plastic zone size in the Y direction at d a / d N = 0.8 /zm/cycle of long crack is 1000 to 1500 ~ m , [22l although the stress ratio is 0.05. Thus, the fatigue crack in the highly damaged zone can grow with significantly small accumulated plastic

zone. The term eacfc within the dotted line can be estimated as follows. The value of eac:c at point a" = 0.3 (e,,ctotal = 0.5 and eac,, = 0.2); e,~:~ at a] = 1.5 - 0.5 = 1.0; and eacf: at a2 = 1.5 - 0.2 = 1.3 (1.5 is the fracture strain). Thus, eacfc of the crack tip increases with increasing crack length.

Figure 16(b) is a schematic drawing of eac after the crack passed through the pre-existing damaged zone. The concept of a damage-accumulation mountain will again be helpful. As in Figure 16(a), ea~yr at points a, a ' , b, b' , c, c ' , d, and d' is yield strain. The dotted line shows the APZ~f~. The size increases with crack length until the crack tip reaches the line of ear = ey. This means that a fatigue crack in the notch-damage zone can grow with less damage accumulation; i .e . , with less plastic work. Accordingly, when externally given plastic work is constant, the growth rate in the damaged zone is greater than that in a zone with no damage. The more the damage, the higher the acceleration. Thus, not only the damage but also the accumulated plastic strain gradient (damage gradient) plays an important role in this acceleration. The real situation is more complex, i .e . , the effect of the notch still exists, because of the shortness of the fatigue crack, and notch-damage accumulation would continue to increase. Part of the plastic work (energy) goes into this increased accumulation and this acts to decrease the crack-growth rate. This requires further study.

In the region very near the notch root, where many microcracks are initiated, the plastic work is distributed to these cracks. So, the acceleration would also be re- duced. It is when main crack begins to grow that the aforementioned acceleration becomes significant.

It has been shown that abnormal behavior (fast crack- growth rate) of a small crack from a notch is due to crack

Ey Ey

I" . . . . . . .

(a) (b)

Fig. 16--Schemat ic drawing of the accumulated plastic strain state of a fatigue crack emanating from a notch in low-cycle fatigue. The crack propagates through the pre-existing damage-accumulated zone. (a) The crack tip is within APZ0 t2. The dotted line shows the accumulated plastic zone with ear -> ey (yield strain), APZ~y due to the fatigue crack. (b) The crack tip beyond APZ~y. Accumulated plastic strain due to the crack at a, a' , b, b', c, c' , d, and d' is yield strain e r The dotted line shows the APZ~y of the fatigue crack. APZs beyond the line dd' (beyond the pre- existing notch APZ,~) are all due to the growing fatigue crack. The number inside the dotted line is the e,c value of the fatigue crack: eoc:c.


Table I. Accumulated Plastic Strain Increase Rate de~JdN at Five Points around Notch Shown in Figure 13"

N

PI P2 P3 P4 P5 deac de~c deac deac deac

AN e~c Aea,, dN e~ Ae~ dN e~ Ae~ dN eac Ae~ dN e~ Ae~ dN

0 0

- 0 0.01 0.0005 0

-0 .02 0 0 4 0.08 0.02 0.02 0.005

-0 .1 0.02 0 0.02 0.004

15 0.15 0.01 0.05 0.0003 0.02** 0.03 0.0003 20 --0.25 0.07 - 0 . 0 1 --0.05 0

255 0.5 0.002 0.04 0.00016 0.02 0.00008 0.06 0.0002 275 0.7 0.11 0.03 0.11 --0.01

275 0.14 0.0005 0.10 0.00036 0.17 0.0006 0.07 0.00025 550 0.14 0.13 0.28 0.08

*N: number of cycles, AN: increment of N, eo,,: accumulated plastic strain, Aeac: increment of e,,c during AN, and deo~/dN: increase rate of ea,, per one cycle.

**N = 10.

closure (more properly, absence of closure) or notch plasticity control. Without these ideas, the present local damage-accumulation approach could explain the abnormal growth of the crack emanating from notch.

VI. CONCLUSIONS

1. A recrystallization technique was used to study the local fatigue damage accumulation in 304 stainless steel. This technique enables accumulated plastic zones with various amounts of accumulated plastic strain to be observed clearly.

2. With increasing numbers of cycles, fatigue damage accumulates around a notch and the size of the accumulated plastic zone increases.

3. The area of the accumulated plastic zone at crack initiation increases with increasing notch tip radius from 0.25 to 4 mm.

4. Damage-accumulation rate, the rate at which accumulated plastic strain increases per cycle, is high during the initial stage of fatigue and then decreases. This is because of work hardening and the spreading of the plastically deformed zone.

5. A fatigue crack initiates at notch when the local accumulated plastic strain at the notch root reaches a critical value, which would be equal to fracture strain of the material--that is, a crack initiates when the local accumulated plastic work at the crack initiation site reaches a critical value.

6. The local damage-accumulation approach gives an explanation of the abnormal crack growth behavior of a fatigue crack emanating from a notch. It grows through the pre-existing damage-accumulated zone around the notch with less plastic work, resulting in faster crack growth.

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Local fatigue damage accumulation around notch attending crack initiation