Fatigue Crack Propagation in Trip Steels G. R. CHANANI, STEPHEN D. ANTOLOVICH, AND W. W. GERBERICH

The fat igue c r a c k propagat ion behav io r of a c l a s s of me tas t ab le aus teni t ic s t e e l s ca l l ed TRIP s t ee l s has been inves t iga ted . The al loy compos i t ion was chosen to have the Ms wel l below r o o m t e m p e r a t u r e and the M D above r o o m t e m p e r a t u r e af ter t h e r m o m e c h a n i c a l p r o c e s s i n g . A s imple t h e o r e t i c a l mode l of fat igue c r a c k p r o p a g a - tion (FCP) based on f r a c t u r e mechan ic s was developed. Fa t igue c r a c k propagat ion t e s t s on SEN s p e c i m e n s at va r ious s t r e s s in tensi ty r anges (AK) w e r e c a r r i e d out, and two s tage p l a s t i c - c a r b o n r e p l i c a w e r e used to obse rve the f r a c t u r e su r f ace of the FCP s p e c i m e n s . To a f i r s t approx imat ion , both the e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s fol lowed the usua l r e l a t ionsh ip be tween AK and FCP r a t e s ; i .e. da/dn oc (LxK). 4 The fat igue f r a c t u r e su r f ace contained fat igue s t r i a t i o n s , quas i c l eavage and elongated d i m - p ies ; a r e f l ec t i on of the complex s t r u c t u r e of TRIP s t e e l s . A benef ic ia l e f fec t of s t r a i n induced m a r t e n s i t e t r a n s f o r m a t i o n with r e g a r d s to fat igue c r ack propagat ion was found. TRIP s t ee l s showed be t t e r F C P p r o p e r t i e s than a number of alloy s t ee l s of s i m i l a r s t reng th l eve l s and c o m p a r e d favorably with marag ing s t ee l s in the low AK range .

AFTER t h e r m o m e c h a n i c a l p r o c e s s i n g , ce r t a in me ta s t ab l e aus teni t ic s t ee l s have been found to y ie ld unusual combina t ions of s t reng th , duc t i l i ty , and tough- nes s 1-5 Th i s p a r t i c u l a r c l a s s of s t ee l s has been given the des ignat ion " T R I P " , which is an ac ronym for T R a n s f o r m a t i o n Induced _Plasticity. It has been shown that the high toughness and duct i l i ty a s soc i a t ed with these s t e e l s a r e due to the t r a n s f o r m a t i o n of m e t a s t a b l e aus teni te to m a r t e n s i t e dur ing the t ens i l e tes t ing at t e m p e r a t u r e s below M D .1-6

At a t e s t t e m p e r a t u r e below M D t he re is a c r i t i c a l s t r a in or s t r e s s * beyond which the pa ren t aus teni te

*Whether the transformation is strata-induced or stress-induced from a funda- mental point of view is as yet unanswered. It is common to refer to the transfor- mation as strain-induced when it occurs after yielding of the austemte and stress- induced when it occurs before yielding of the austemte.

p a r t i a l l y t r a n s f o r m s to m a r t e n s i t e . The t r a n s f o r m a - tion to m a r t e n s i t e causes s t rengthening of the d e - f o r m e d reg ion . Because of this , p las t i c ins tabi l i ty is p reven ted and the un i fo rm elongat ion is i n c r e a s e d . The behav ior of these s t e e l s under monotonic loading has been studied ex tens ive ly as a function of t e s t t e m p e r a t u r e ,2,s,7 and t h e r m o m e e h a n i c a l p r o e e s - s ing. l '2 's

In the p r e s e n t inves t iga t ion , the fat igue c r ack p r o - pagat ion c h a r a c t e r i s t i c s of s e v e r a l TRIP s t e e l s w e r e studied as a function of the aus teni te s tabi l i ty by c o r r e l a t i n g the r a t e of fat igue c r ack propagat ion to the s t r e s s in tensi ty p a r a m e t e r . F o r this pu rpose , the alloy compos i t ion was chosen as to have the Ms t e m p e r a t u r e wel l below r o o m t e m p e r a t u r e and the M D t e m p e r a t u r e above r o o m t e m p e r a t u r e af ter t h e r m o m e c h a n i c a l p r o c e s s i n g . 1

The f r a c t u r e mechan ic s approach has been widely used to analyze fat igue c r a c k propagat ion 8-x4 and it is gene ra l ly found that the c r a c k propagat ion r a t e and s t r e s s in tens i ty fac to r range a re r e l a t ed by a power re la t ionsh ip of the f o r m

G. R. CHANANI and STEPHEN D. ANTOLOVICH are Post-Doc- torial Fellow and Associate Professor of Materials Science, respectively, Department of Materials Science and Metallurgical Engineering, Uni- versity of Cincinnati, Cincinnati, Ohio. W.W. GERBER1CH is As- sociate Professor, Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapohs, Minn.

Manuscript submitted February 21, 1972.

da/dn ~ (AK) m [1]

where da/dn = fat igue c r a c k propagat ion r a t e

AK = m a x i m u m minus m i n i m u m s t r e s s in tens i ty m = n u m e r i c a l constant which is often about 4.

I) E X P E R I M E N T A L PROCEDURE

The alloy compos i t ion was chosen based upon p r e - vious work with TRIP s t ee l s 1-5 and was such that the aus ten i te had a high work hardening r a t e , high s t a b i - l i ty and ex tens ive p rec ip i t a t ion hardening with p r i o r de fo rmat ion at a sui table e leva ted t e m p e r a t u r e . The nomina l compos i t ion of the al loy was 9 C r , 8 Ni, 3 Mn, 3 Si, 4 Mo, 0.25 C, and balance Fe . The actual compos i t ions a re shown in Tab le I.

The s t ee l s w e r e p r e p a r e d by induction mel t ing of high pur i ty e l e m e n t s under an ine r t he l ium a t m o s - phe re . Af te r homogeniz ing at l l00~ for 3 d, the

s in. th ickness ingots w e r e hot forged at l l00~ to -~ (for subsequent 80 pct reduct ion) and ~o in. th ickness (for subsequent 20 pct reduct ion) f lat p i e c e s . Af te r c leaning the su r face by sand blas t ing and acid p ickl ing , these b a r s w e r e aus ten i t i zed at 1200~ for 3 h under an a tm osphe re of 4 pct H in he l ium (forming gas) and then b r ine quenched. The m a t e r ) a l was then f la t r o l l ed at 250 ~ and 450~ to 0.075 in. thick p la t e s . In o rde r to main ta in c lose t e m p e r a t u r e con t ro l dur ing ro l l ing , the r o l l s w e r e p rehea ted and the m a t e r i a l was r e -

Table I. Chemical Composition of the Alloys Used for Fatigue Crack Propagation Tests

Alloy Identification

No. Cr Ni Mn Si Mo C Fe

a 9 7.4 2.8 2.8 4.0 0 24 balance a~ 9 7.4 2.8 2.8 4.0 0 25 balance b 9 7.5 2.9 2.8 4.0 0.25 balance b~ 9 7.5 2.9 2.8 4 0 0.25 balance c 9 7.5 2 9 2.8 4 0 0.24 balance el 9 7.5 2.9 2.8 4.0 0 25 balance d 9 7.4 2 9 2.8 4.0 0.24 balance dl 9 7.4 2 9 2.8 4.0 0.24 balance d2 9 7 4 2.8 2.8 4 0 0.24 balance

METALLURGICAL TRANSACTIONS VOLUME 3, OCTOBER 1972- 2661

heated in an e lec t r i c furnace between pa s se s . The total t ime taken in the rol l ing p rocess was approxi - mate ly 1 h.

T e n s i l e , F ig . 1, and Single Edge Notched (SEN) fatigue c rack propagat ion spec imens , Fig. 2, were machined f rom the rol led plates so that the tens i le axis was pa ra l l e l to the ro i l ing d i rec t ion . The notch in the SEN spec imens was perpend icu la r to the ro l l ing d i rec t ion . These spec imen designs were chosen for their good a l ignment and ease of p r e p a - ra t ion . Tens i l e tes ts were ca r r i ed out us ing an Ins t ron at a s t r a i n ra te of 0.04 per minute , while FCP tes ts were pe r fo rmed us ing a 300 KIP MTS un ive r s a l tes t ing sys t em. The amount of m a r t e n - s i t ic t r ans fo rma t ion dur ing tens i le tes t ing was measu red with a specia l ly designed p e r m e a m e t e r desc r ibed e l sewhere . 1~

I/4" O

~ [ ~ I/8" R

z3/,~'

0125" - - --

GAGE LENGTH: I"

THICKNESS = 0.075" Fig. 1--Tensile sheet specimen.

0.008"

0.05" max j

-<3-

~- 2"

0.85"

_1_

0.85"

THICKNESS = 0.075" Fig. 2--Single edge notched specimen.

2662-VOLUME 3, OCTOBER 1972

6"

F r a c t u r e tes ts were pe r fo rmed on SEN spec imens at an extens ion rate of 0.6 ipm. P r i o r to all t e s t s , the SEN spec imens were fatigue p rec racked per ASTM recommended practice.16 All the FCP tes ts were c a r r i e d under t ens ion - t ens ion loading at 4 Hz in the load control mode. Load and s t roke length were cont inuously moni tored dur ing the tes t us ing a high speed s t r ip char t r e c o r d e r . Both the m i n i m u m and the max imum load dur ing each run were noted. They usual ly r e m a i n e d cons tant throughout the d u r a - t ion of the tes t except for a few runs in the higher s t r e s s in tens i ty range where a g radua l but sma l l change in the ma x i mum value of load occu r r ed . In these cases the values of the load at the beginning and at the end of the run were noted. The ra t io P m a x / Pmin, /3 , was usual ly main ta ined at about 8. As a r e s u l t the effect of/3, if any, was negl ig ib le .

The c rack was grown (approximate ly by 30 mi ls ) dur ing each run and the c rack length before and after the run was m e a s u r e d with a t rave l ing m i c r o - scope. The FCP ra te was ca lcula ted f rom:

da/dn = A a / A N = (aen d - a s tar t) /AN [2]

where

as tar t = crack length at s t a r t of run aend = c rack length at end of run

AN = number of cycles

Since the c rack grew by a s m a l l amount dur ing a run , the value of AK inc reased only s l ight ly. Hence the values of AK both at the s t a r t and at the end of each run were calcula ted and the average of these two AK values was taken as the AK giving r i s e to the da/dn for that pa r t i cu l a r run .

Samples were cut f rom both tens i le and s ingle edge notched spec imens for meta l lographie examina t ion . After mechanica l pol ishing, the spec imens were e lec t ropol i shed using a solut ion of 90 pct acet ic and 10 pet pe rch lo r ic acid at 0~ A f r e sh solut ion of 5 g cupr ic ch lor ide , 100 ml hydrochlor ic acid, 100 ml ethyl alcohol, and 100 ml d i s t i l l ed water was used for etching each t ime.

A " two s t age" p l a s t i c - c a r b o n technique was used for f rac tographic s tudies , iv

II) THEORETICAL CONSIDERATIONS

Fat igue c rack propagat ion tes ts are no rma l ly con- ducted under t ens ion - t ens ion loading. 1~ During these tes ts e i ther a cons tant cyclic s t roke ampli tude or a cons tant cyclic load ampli tude is main ta ined . In the p r e s e n t inves t iga t ion the applied cyclic load range was main ta ined constant throughout the dura t ion of one run .

The fatigue crack growth ra te , da/dn, for a given m a t e r i a l and env i ronmen t depends upon load- t ime h i s to ry , the configurat ion of the s t r uc t u r e containing the c rack , and the crack geomet ry . It has been shown that FCP ra tes co r re l a t e ve ry wel l with A/~ -14 a p a r a m e t e r which includes l o a d / c r a c k geomet ry .

In order to es tab l i sh a quant i ta t ive re la t ionsh ip between fatigue c rack growth ra te and s t r e s s in tens i ty p a r a m e t e r for TRIP s tee l s , it can be a s sumed that the growth is governed by the accumula t ion of damage due to cyclic p las t ic s t r a in as has been done in the past . ls,19

METALLURGICAL TRANSACTIONS

F o r this p u r p o s e a quant i ta t ive f r a c t u r e c r i t e r i o n app l i cab le to cyc l i c loading i s n e c e s s a r y , i . e . a c r i - t e r ion which wi l l d e t e r m i n e how much damage wi l l have to take p l ace be fo re the c r a c k p r o p a g a t e s . I t has been shown that for cons tan t s t r a i n ampl i tude t e s t s on TRIP s t e e l s , ~5'2~ the Cof f in -Manson law holds :

i ~ F <e : v ~s [3]

where

N/= number of cycles to failure s = cyclic plastic strain range Ef = monotonic fracture strain

From Eq. [3], it follows

2 N / n ( e p i c / ) : 1 [4]

i . e . ,

4Yf[ep/ef] 2= 1 [5]

F r o m Eq. [5] a g e n e r a l equation for f r a c t u r e c r i - t e r ion under fa t igue as shown below has been obta ined . The g e n e r a l equat ion i s :

4 [ ~ / , f ] 2 A Y : 1 [6]

w h e r e

~ = a v e r a g e p l a s t i c s t r a i n over the r eg ion in which the f r a c t u r e c r i t e r i o n is to be s a t i s f i e d

AN = number of c y c l e s to p ropaga t e through this r eg ion

- a In o r d e r to d e t e r m i n e c p , an e l a s t i c - p l a s t i c ana -

l y s i s in f ron t of the c r a c k i s needed . As ye t no such a n a l y s i s i s ava i l ab le for mode I loading . However , a t ens i l e analogy of Hult and McCl in tock ' s s t r a i n d i s t r i b u t i o n ls'2~'22 for longi tudinal s h e a r has been used to d e s c r i b e e l a s t i c - p l a s t i c s t r a i n d i s t r i bu t i on , z3-e5 F r o m this ana logy , the s t r a i n at a d i s t ance r f r o m the c r a c k - t i p i s :

c = a y s / E ( R p / r )

w h e r e

ays = yie ld s t r e s s E = Young 's modulus

Rp = p l a s t i c zone r = d i s t ance ahead of c r a c k

In order to determine g~, a small region in the plastic zone just ahead of the crack-tip as shown in Fig. 3 has been considered by McClintock and others. 23 In Eq. [6] g~ can be obtained by integrating the strain over the region l shown in Fig. 3.

1 ~ ~- 1/(12A0/2) f e p r A O d r [8]

0

In the r eg ion l the e l a s t i c s t r a i n is s m a l l and hence Eq. [7] can be used to d e s c r i b e c p . Subst i tut ing Eq. [8] into [6] and so lv ing y i e ld s :

I / A N = da /dn = J(E y / c f ) 2 R p 2 / l [9 ]

where J = n u m e r i c a l cons tan t

~y = un iax ia l t ens i l e y ie ld s t r a i n

In a p r ev ious study of TRIP s t e e l s , G e r b e r i c h 3 o b s e r v e d Dugda le - type p l a s t i c zones and s i m i l a r zones we re a l so obse rved in th is s tudy. The Dugdale 2~ p l a s t i c zone s ize for monotonic loading i s given by:

CRACK

~- R -~- P

Fig. 3--Structural size l and plastic zone at the crack-tip.

R~ : a[sec(Trcr/2~rys)- 1] [10]

w h e r e

= r e m o t e app l ied s t r e s s

F o r cyc l i c loading , 2~ Eq. [10] can be modi f ied to y ie ld :

Rp : a [ s e c ( n A ~ / 4 ~ y s ) - 1] [11]

Subst i tu t ing this in the fa t igue c r a c k r a t e , Eq. [9] y i e l d s :

da /dn = J / l (~y /~ f )~a2[sec (~At l / 4 t l y s ) - 1] 2 [12]

Dugda l e ' s model is va l id only for an inf ini te p l a t e . A f in i t e -wid th c o r r e c t i o n mus t be app l ied . The r e s u l t i s :

da /dn = J / l ( c y / E f ) 2

a2[ sec (~Ae /4~ys ) - 1] 2 �9 Y ' ( a / w ) / ~ z [13]

whe re

Y(a /u ) = c a l i b r a t i o n function which depends on s p e c i m e n g e o m e t r y

Expans ion and s imp l i f i c a t i on of Eq. [13] g ives :

d a / d n = AZaZ(A(~/(Xys)4 + 2ASa3(A(r/(~ys)~

+ BZa4(Acz/(Xys) 8 [14]

where

A = g(a , w) B = h(a, w)

F o r the s ingle edge notched s p e c i m e n s used in th is investigationZa:

AK = YAff(a) I12 [15]

where

Y = 1.99 - 0 .41(a/w) + 18 .70(a /w) 2 - 38.48(a/w) 3

+ 53.85(a/w) 4

Subst i tu t ing Eq. [15] into Eq. [14] y ie lds :*

*The form of Eq. [16] does not seem to depend on the character of the plas- tic enclave. We have verified that use of a modified Irwin zone yields essentially the same result as was obtained using a Dugdale zone.

VOLUME 3, OCTOBER 1 9 7 2 - 2663 METALLURGICAL TRANSACTIONS

da/dn = C a ( A K / a y s ) 4 + 2 C D ( A K / a y s ) 6 + D 2

(Ag/~y~) ~ [t6] w h e r e

C : A / ~ D = B / Y 4

Thus, it can be s een that d a / d n may be dependent upon the higher p o w e r s of AK as we l l as (AK) 4 depend- ing on the va lues of C and D.

III) RESULTS AND DISCUSSION

A) T e n s i l e T e s t Resu l t s

Room temperature tens i l e and fracture proper t i e s are l i s ted in Table II. As ment ioned e a r l i e r , the m a r - tens i t i c transformat ion was fol lowed during tens ion test ing by a " p e r m e a m e t e r " . The amount of m a r t e n - s i te as a function of percent s train for four r e p r e s e n - tat ive c a s e s i s shown in F i g s . 4 and 5 (broken l ines ) . T h e s e f igures a lso show the engineer ing s t r e s s - s t r a i n behavior for these c a s e s (full l ines ) . The tens i l e p r o - p e r t i e s of TRIP s t e e l s have been d i s c u s s e d in detai l

280

~.zoo O O O

td ~o

W

i r

W ,oo Z

W

I I I I

/ / 2 0 %

- - . . . . 8 0 % / /

/ /

/

/ /

/ /

/ / / ] / / PRIOR [~FORMATION TEMP 450QC

/ i / / / TEST TEMP aS+C

/ / /

) - - - - ~ - " ~ I I I I I 0 O I 0 2 0 3 0 4 0 5

S T R A I N

I00~

so

so ~

40 ~

ZO

0 0 6

Fig . 4 - - E n g i n e e r i n g s t r e s s - s t r a i n (cont inuous cu rve ) a n d v o l u m e p e r c e n t m a r t e n s i t e - s t r a i n c u r v e (broken cu rve ) f o r the a l l o y s a and d, d e f o r m e d a t 450~

by s e v e r a l inves t iga tors ~'2'5-7 and wi l l be only br ie f ly cons idered here . As s een in Table II, these s t e e l s have both high s trength and elongation at r o o m t e m - pera ture .

A l loys that w e r e de formed 80 pet exhibit a yie ld point and the Liiders s tra in was rather large . Another c h a r a c t e r i s t i c observed in the r o o m temperature t ens i l e t e s t s of these a l loys was that fa i lures occurred just after reaching the m a x i m u m load. For a l loys de formed 20 pet , no y ie ld point was observed and both the final amount of m a r t e n s i t e and the vo lume percent of mar tens i t e per unit s tra in m e a s u r e d in a r o o m temperature t ens i l e t e s t w e r e l e s s than for those that w e r e deformed 80 pet . This indicates that 80 pet pr ior deformat ion r e d u c e s the austenite s tabi l i ty r e l a - t ive to mar tens i t e format ion . The r e a s o n for this has been attributed to the grea ter de format ion at the e l e - vated t emperature causing m o r e al loy carb ides to f o r m , thereby deplet ing the m a t r i x of austenite s tab i - l i z ing e lements .~ -7

B) Metal lography

F i g s . 6 and 7 show typical m i c r o s t r u c t u r e s of the a l loys used in presen t invest igat ion after var ious t h e r m o - m e c h a n i c a l t r e a t m e n t s . De format ion markings

2 8 0

~ 200

g

uJ bOO z

w

i i i [

~ 8 0 %

/ 1

" ~ ao'~

_ -- - -- 8 0 %

/ /

/

/

/

/ -- -- -- - 2 0 ~

/ I j r / /

i i / / /

/ / PRIOR DEFORMATION TEMP 2SO~ / / ~ ~ J / TEST T E M P 2S~

~ -- -- - - I - - I I I I O O O O I 0 2 0 3 0 4 0 5 0 6

S T R A I N

w

80

o=

so

=, 40 o

2 0

F i g . 5---Same a s Fig. 4 f o r a l l o y s b and c , d e f o r m e d a t 250~

Table II. Summary of Stress-Intensity and Tensile Data at 25~ for 0.075 In. Thick TRIP Steel Plate After Various Thermomechanical Processing

Yield TensLle , Strength, Strength, Elongatton, Crittcal Stress-Intensity Magnetic Charactensttcs

Processing ksi ksi Pct Factor, K c , ksHn. �89

Alloy Identification

No, Before Test After Test

a 20 pct at 450~ 137 183 46 A M a~ 20 pct at 450~ 138 187 42 331t A M b 20 pct at 250~ 126 174 55 A M b~ 20 pct at 250~ 119 160 49 279 A M c 80 pct at 250~ 237 237 36 A M Cl 80 pct at 250~ 237 258 38 296 A M d 80 pct at 450~ 233 248 34 A M d~ 80 pct at 450~ 231 243 32 334t A M d2 80 pct at 450~ 238 260 36 A M

*A = Nonmagnetic, M = Magnetic "}'Tested at a cross-head speed of 0.2 lps. At 0.02 xps crack grew slowly gLving no instability.

2664 -VOLUME 3, OCTOBER 1972 METALLURGICAL T R A N S A C T I O N S

Fig. 6--Microstructures of (a) Alloy d, deformed 80 pct at 450~ (b) Alloyc, deformed 80 pct at 250~

Fig. 7--Mierostructures of (a) Alloy a, deformed 20 pct at 450~ (b) Alloy b, deformed 20 pct at 250~

in the fo rm of sl ip t r a c e s , deformat ion twins, and elongated g ra ins can be seen in these f igures . All the s t r u c t u r e s are e s sen t i a l l y aus ten i t i c . The effect of i nc r e a se d deformat ion is also apparent . The aus teni te g ra ins a re highly elongated in the ro l l ing d i rec t ion for those al loys deformed 80 pct.

C) Fat igue Crack Propaga t ion Resu l t s

F igs . 8 through 11 show log-log plots of LxK vs da/dn for the four t he rmomechan ica l t r e a tmen t s inves t iga ted . Except for those at high LxK al l the tes t s were run at 4 Hz. A somewhat lower cycl ic speed was used for the tes t s at high LxK. A line of slope of 4 has been drawn through the data points and it can be seen that a r e la t ionsh ip of the type da/dn cc (LxK) 4 provides a good approximat ion over the lower ranges of data for the four TRIP s tee l s .

Eq. [16] developed for TRIP s tee l s indica tes that bes ides the fourth power of AK, the higher powers of LXK can affect the ra te of c rack propagat ion. How- ever , F igs . 8 through 11 show that for mos t of the AK range tes ted , no s igni f icant effect of the higher powers of LxK on c rack propagat ion ra te was observed .

METALLURGICAL TRANSACTIONS VOLUME 3, OCTOBER 1972- 2665

o~

10 -2

151-3

l0 -4

10 -5

10 -6

10 -7

w

u

m

I I i lliil I I I I liIIl i v

20% Deformed at 450"C

Alloy a and a I

I I tllllii I I lllllll I I 2 5 i0 20 50 i00 200

AK, Stress Intens• Range, Ksi-ln I/2

i l llil

I l i l i l 500

Fig. 8 - -Fa t igue c r a c k p r o p a g a t i o n r a t e vs s t r e s s i n t e n s i t y f a c t o r r a n g e for the a l l oys d e f o r m e d 20 pc t a t 450~

This can be explained as follows for mos t of the t e s t s in the p r e sen t inves t iga t ion where A(z/(~y s was usual ly below 0.2 and a/w was between 0.34 and 0.65.

Eq. [16] can be wr i t t en as:

da/dn = C2(AK/(Yys)4[1 + 2D/C(AK/ays) 2

+ D2/Ce(AK/Crys) 4 ] [17]

F o r a/w = 0.34 and Aa/ay s = 0.2 the above equation af ter solution and s impl i f iea t ion r educes to

an~an = r [1 + 2 • 10 -2 + 8 • 10-5] [18]

while for a/w = 0.65 and Aa/ay s = 0.2, Eq. [17] r educes t o :

da/dn : C2aK4/gys[1 + 3 x 10 -2 + 2 x 10 -4] [19]

In both Eqs . [18] and [19] the second and the third t e r m s in the b racke t s r e p r e s e n t the r e l a t i ve c o n t r i - butions of the sixth and eight powers of AK to the c r ack propagat ion r a t e . It can be seen that this con- t r ibut ion is not s igni f icant . However , at h igher va lues of A(z/rZy s some contr ibut ion due to higher o rde r t e r m s is expected . Indicat ions of this a re seen in F ig s . 8 and 11, where at lower AK va lues , the data a re m o r e or l e s s un i fo rmly above or below the FCP* cu rve while

*FCP curve henceforth refers to the log-tog plot of stress-intensity range, AK, as a function of crack propagatmn rate da/dn.

r~

D

q

10 -2

10 -3

10 -4

10 -5

10 -6

l0 -7

I I I I I I I I I I I I I I I I I lai~ I I I I

o/ _ 20% Deformed at 250~

Alloy b and b I ~ --

! Z

T I I I I I I I [ l l i l i t i I I U I I I

lO i00 1000 AK, Stress Intenslty Range, Ksl-ln I/2

Fig. 9 - -Fa t igue c r a c k p r o p a g a t i o n r a t e vs s t r e s s i n t e n s i t y f a c t o r r a n g e for the a l l o y s d e f o r m e d 20 pc t a t 250~

at higher AK va lu es , they are usua l ly above the FCP curve . This deviat ion at higher va lues of A<r/ays may be a lso due to the change in nature of crack propaga- tion b es id es the inf luence of higher power t e r m s . At this s tage , it can be tentat ive ly concluded that there i s a genera l tendency towards f lattening.

Since for most of the inves t igat ion the higher order t e r m s in Eq. [16] did not have any ef fect , they can be neglec ted and Eq. [16] can be s impl i f i ed to yield:

d a / d n = 8 / l ( r y / C f ) 2 ( A K / ( r y s ) 4 [201

The va lues of s El, and ay s w e r e obtained f rom the t ens i l e data shown in Tab le IIo The c r a c k growth r a t e , da/dn was de t e rmined at &K = 50 ks i4"~ , for each e a s e f r o m the co r r e spond ing F C P c u r v e . The va lues of l ca lcu la ted f r o m these data a r e l i s ted in Table III and a re on the o rde r of 15 • 10 -4 in.

As was ment ioned e a r l i e r , l is the r eg ion in which the f r a c t u r e c r i t e r i o n , Eq. [6], is sa t i s f i ed . The va lues of l for the four e a s e s inves t iga ted a re within a fac to r of th ree and a re of the s a m e o r d e r of m a g - nitude as a typical ce l l or subgra in s i z e Y '3~ Hence, l is apparent ly r e l a t ed to the d i s loca t ion subs t ruc tu re and appears to be a m a t e r i a l p r o p e r t y . However , the

2666 VOLUME 3, OCTOBER 1972 METALLURGICAL TRANSACTIONS

va lues of l for the four treatments do not exhibit any obvious corre la t ion with the rate of crack pro- pagation, da/dn.

Eq. [20] can be writ ten as

da/dn = R(AK) 4 [21]

where

R i s a rate constant

The magnitude of R i s a m e a s u r e of the ease of fatigue crack propagation in a mater ia l . The magnitude of R indicates the s ens i t i v i ty of the rate of fatigue crack propagation to the dif ferent p r o c e s s i n g t rea tments . Values of R determined from the FCP curve are shown in Table IV.

It appears that the treatment with the lowes t value of KC (20 pet deformat ion at 250~ had the highest value of R, i.e., m a x i m u m ease of fatigue crack pro- pagation. Since in the present case the plane strain fracture toughness parameter KIC was not determined , nothing could be inferred as to whether there is any corre la t ion between the KIC and da/dn as suggested by Krafft. 3~ However , the present re su l t s did not indicate any obvious corre lat ion with KC. Wei et al. 32 have a l so found no corre lat ion of FCP with KC for e i ther m e d i u m - c a r b o n low a l loy , ultrahigh strength s t e e l s or maraging s t e e l s in an inert env ironment .

F ig . 12 shows a compar i son of FCP behavior of a 80 pct deformed TRIP s t ee l with that of a group of high strength a l loys having a s i m i l a r UTS (250 grade maraging s t e e l , 300 M s tee l , AM 355 CRT s t a i n l e s s s t ee l and PH 15-7 Mo s tee l ) . The sou rces of these data as we l l as UTS of the mater ia l s are a l so indi - cated in the f igure . It can be s e e n that the s t e e l deformed 80 pct at 450~ is super ior to the 300 M, AM 355 CRT and PH 15-7 Mo s t e e l s and also c o m - pares favorably with maraging s t e e l s in the low s t r e s s intens i ty range.

D) Fractography

Typica l fatigue fractographs are shown in F i g s . 13 through 16. They show s tr ia t ions , quas ic l eavage and elongated d i m p l e s . Although fatigue s tr ia t ions were

Table I I I . Structural S i z e / f o r Tr ip Steels

da/dn at 50 ks], Processing m 1A X 10 s in/cycle 1 X 104 , m.

20 pct deformation at 450~ 6 40 26.2 20 pet deformation at 250~ 15.80 9 5 80 pct deformation at 250~ 6.25 9 7 80 pct deformation at 450~ 6.50 10.3

2

10 -2

i0 -3

10 -4

10 -5

i0 -6

I I I I I I I 1 I I I I l I I I I I I I I I I I I

80% Deformed at 250~ i

Alloy c and

D

~llti

I

u

,4

o

~9

J

/ 1o-7 I I I l l l i I I I I I I I I I I I I I

i0 i00 i000 AK, Stress Intenslty Range, Ksl-ln I/2

Fig. 10- -Fa t igue c r a c k propagat ion ra te v s s t r e s s in tens i ty f a c t o r range for the a l l o y s d e f o r m e d 80 pc t a t 250~

10 -2

10 -3

10 -4

10 -5

10 -6

l0 -7

- ' '"'"Jl ' '"'",]o' ",,,- j - -- z

- - 80% Deformed at 450~

Alloy d, dl, and d 2

i

I JJL,IJll I I IIJIJl] J I 2 5 i0 20 50 i00 200

~K, Stress Intensity Range, Ksi-in I/2

l l l l l 500

Fig. 11 - -Fa t igue c r a c k propagat ion rate v s s t r e s s i n t e n s i t y f a c t o r r a n g e for the a l l o y s d e f o r m e d 80 pet at 450~

METALLURGICAL TRANSACTIONS VOLUME 3, OCTOBER 1972 2667

observed in both 20 and 80 pct deformed a l loys , F igs . 13 and 15, they were not always p resen t or wel l - developed. The absence of numerous wel l -developed s t r ia t ions is quite common in high strength s tee ls 35 It has been shown by var ious inves t igators 36 that one s t r ia t ion is formed during each load cycle . Hence, the s t r ia t ion spacing can be used as a measure of the c rack propagation ra te . Considerable effort has been devoted to the study of the re la t ionship between the s t r ia t ions and the fatigue c rack growth ra te for high strength aluminum alloys probably because the s t r ia t ions are numerous and ex t remely well defined. Reasonable agreement has been repor ted between s t r ia t ion spacing and c rack growth ra te for these al loys. 3v However, not much work has been repor ted on such re la t ionships in high strength s tee l s , because the s t r ia t ions are difficult to observe in these ma te - r i a l s .

Typical s t r ia t ion spacings along with the c o r r e s - ponding macroscopic c rack propagation ra t e s for TRIP s tee ls are l is ted in Table V. This table shows that as da/dn i n c r ea se s , the spacing also i nc rea se s . Since da/dn cor responds to the s t r ia t ion spacing, the s t r ia t ion spacing should also vary in the same manner as da/dn does with AK. The resu l t s of the p resen t investigation were broadly consis tent with such a dependence as shown in F igs . 8 and 11, where the r a t e s de termined from s t r ia t ion spacings are indi- cated by fi l led c i r c l e s .

In the la te r s tages of c rack propagation, i.e at high s elongated dimples were observed, F ig . 15(c). Elongated dimples a re associa ted with the change in the f rac ture mode from flat to shear type. 37 The dimples were or iented with the long axis pe rpen- dicular to the c rack growth direct ion. The occurrence of these dimples at high c rack propagation r a t e s has been observed e a r l i e r by Gerber ich et al. 3a Somewhat s imi l a r but more uniformly elongated dimples have also been observed in low-cycle fatigue f rac tu re . ~5'2~

Fig. 14 (obtained from the fatigue sur faces of the alloy deformed 20 pet at 450~ and Fig. 16(b) (ob- tained from the fatigue surface of the alloy deformed 80 pet at 450~ exhibit fea tures of br i t t le fa i lure . This was expected, since in addition to austenite, the f rac ture surface contained mar tens i te as was

Table IV. Values of Crack Propagation Constant R for the Four Processing Treatments

Cntical Stress- R X I02a Intensity Factor, Kc,

Processing in. 7 lb. "4 cycle'lKc ksi-m. �89

20 pct deformation at 450~ 1.02 331 20 pct deformation at 250~ 2.55 276 80 pct deformation at 250~ 1.00 296 80 pct deformation at 450~ 1.2 334

Table V. Comparison of Observed Striation Spacing and the Corresponding Crack Propagation Rate

Striation Spacing Crack Propagataon Processing S X l0 s m. Rate* da/dn X 10Sin.

1.48 1.00 20 pct at 450~ 1.68 1.00

19.70 24.03

0.785 0.62 0.79 0.62 1.48 1.78

80 pct at 450~ 1.73 1.78 1,83 1.78 1.97 3.94 2.10 3.94

*The da/dn values listed are the average of the da/dn values In the fracture sur- face from which the replica was obtained

shown by low cycle fatigue r e su l t s . 15'2~ In Fig. 14 quasicleavage a r eas can be observed along with a region resembl ing fatigue s t r i a t ions . Since the spacing of these "fat igue s t r i a t i o n s " is approximately one hundredth of the c rack propagat ion ra te (spacing = 5.9 • 10 -6 in . /cyc le and da/dn = 6.57 • 10 -4 i n / cycle) , these " s t r i a t i o n s " are apparent ly rubbing marks s imi l a r to the " t i r e t r a c k s " observed in low cycle fatigue f rac ture sur face . 15'2~ F e a t u r e s r e s e m - bling r i ve r markings can also be seen in F ig . 14 and these are always associa ted with cleavage f rac tu re . The fea ture less f rac tograph in F ig . 16(b) (from the fatigue f rac ture surface of the alloy deformed 80 pct at 450~ is believed to be due to f rac tur ing of m a r -

2 0 0 i i t i I i i i i i i i i I i i i i i i i i I i ! A t - L I i i i i i I , i i

-~ I O O L - ~ - - 2 5 0 K S I ULT IMATE TRIP S T E E L O ~ ~ / { ~ / ~

,, <> %0 ~ '

LLI f - __Z

KSI ULTIMATE

-", Z 5 0 MAR 2 5 0 , REF 32 O Z 5 0 MAR 250, REF 34 V 3 O O M 275, REF 33 O AM 355 CRT 250, REF 3 3

P H I S - T M o 240 , REF 33

, i l l ] I I i I I I I i i i i i i I I I I i i i | i i 1 [ 10-5 10-4 10-5

do/dN, CRACK GROWTH RATE, IN./CYCLE

I I 1

Fig. 1 2 - - C o m p a r i s o n of c r a c k - p r o p a g a t i o n r a t e s of 250 k s i u l t i m a t e T R I P s t e e l s (80 pc t d e f o r m e d a t 250~ wi th o t h e r h igh s t r e n g t h a l l o y s in the s a m e t e n s i l e s t r e n g t h r ange .

2668 -VOLUME 3, OCTOBER 1972 METALLURGICAL TRANSACTIONS

(a)

(57 Fig. 13--Fractographs from the SEN specimen of the alloy deformed 20 pct at 450~ (a) and (b) show striations.

t ens i t e induced in the p a r e n t aus t en i t e dur ing load cyc l ing . In F i g . 16(a) a s t r u c t u r e r e s e m b l i n g " b r i t t l e s t r i a t i o n s " can be seen . T h e s e u sua l ly occur in high s t r e n g t h a l loys 17 and in the p r e s e n t c a s e , could be due to the p r e s e n c e of m a r t e n s i t e on the f r a c t u r e s u r f a c e . The ex i s t ence of b r i t t l e s t r i a t i o n s can a l so be in fe r r ed f r o m F i g . 17 where s m a l l c r a c k s b ranch ing f r o m the ma in c r a c k can be seen . F o r s y t h a~ and L a i r d 36 have d i s c u s s e d c r a c k - b r a n c h i n g in de t a i l . Acco rd ing to them, these c r a c k s a r e f o r m e d by the c l eavage f r a c -

Fig. 14--Fractographs from the SEN specimen of the alloy deformed 20 pct at 450~ showing " t i re t racks" in the bottom left and cleavage features.

t u re which subsequen t ly and in the c o u r s e of the t e n - s ion p a r t of the load cyc le changes to duc t i l e s h e a r f r a c t u r e . Based on the i r mode l , the spac ing be tween s u c c e s s i v e b ranched c r a c k s can be a s s o c i a t e d with each cyc l e . Hence , the spac ing be tween the two b r a n c h e s should be a p p r o x i m a t e l y equal to the c r a c k p ropaga t i on r a t e if only c l eavage s t r i a t i o n s a r e f o r m - ing. In the p r e s e n t c a s e , the spac ing was roughly of the o r d e r of 10 (da/dn) which s e e m s to d i sa l low the p o s s i b i l i t y of c l eavage s t r i a t i o n s . However , in TRIP s t e e l s both magne t ic m e a s u r e m e n t s 2~ and f r a c t o g r a p h i c r e s u l t s ind ica ted that the f r a c t u r e a r e a u sua l l y con- ta ined m o r e aus t en i t e than m a r t e n s i t e . As a r e s u l t , the m a j o r p a r t of the c r a c k p ropaga t i on i s by duc t i le f r a c t u r e while the r e m a i n d e r is by c l eavage f r a c t u r e . Hence the o b s e r v e d b r a n c h - s p a c i n g is u n d e r s t a n d a b l e .

E) Ef fec t Of T h e r m o m e c h a n i c a l P r o c e s s i n g On The F C P Behav io r Of TRIP S tee l s

T h e r m o m e c h a n i c a l p r o c e s s i n g can change the t h e r m o d y n a m i c s t ab i l i t y of the a l loys with r e s p e c t to the s t r a i n - i n d u c e d m a r t e n s i t i c t r a n s f o r m a t i o n . I n c r e a s i n g e i t he r the amount of p r i o r d e f o r m a t i o n o r d e f o r m a t i o n t e m p e r a t u r e p r o m o t e s p r e c i p i t a t i o n of complex c a r b i d e s , which contain aus t en i t e s t a b i - l i z e r s , t h e r e b y , r e n d e r i n g the m a t r i x uns t ab le with r e s p e c t to m a r t e n s i t e f o rma t ion . 1'7 The p r i o r t h e r - m a l - m e c h a n i c a l t r e a t m e n t h i s t o r y has been found to

METALLURGICAL TRANSACTIONS VOLUME 3, OCTOBER 1972- 2669

have a considerable effect on tensile and fracture properties of TRIP steels, where the austenite to martensite transformation plays a beneficial role. 1'3'4~

(c)

However , for fat igue c r a c k p ropaga t i on , i t can be seen in F i g . 18 that th ree of the four p r o c e s s i n g t r e a t m e n t s showed a p p r o x i m a t e l y the s a m e behav io r while the four th one, i .e . , the m a t e r i a l d e f o r m e d 20 pc t at 250~ showed somewhat p o o r e r F C P p r o p e r t i e s . Here i t should be noted that the a l loy d e f o r m e d 20 pc t a t 250~

(a)

~)

(b) Fig. 15--Fractographs from the SEN specimen of the alloy deformed 80 pet at 450~ (a) and (b) show striations while (c) shows dimples elongated perpendicular to the direction of crack growth.

2670 -VOLUME 3, OCTOBER Iq72

(b) Fig. 16--Fractographs from the SEN specimen of the alloy deformed 80 pct at 450~ (a) shows brittle striations while (b) shows flat fracture. Arrow indicates fractured marten- site plate.

METALLURGICAL TRANSACTIONS

has the h ighes t s t ab i l i t y because of the l e a s t d e f o r m a - tion and lowes t p r o c e s s i n g t e m p e r a t u r e . No m a r t e n - s i t e ahead of c r a c k - t i p was d e t e c t e d e i t he r by a hand- magne t o r by s u r f a c e r e l i e f e f fec t and, hence , l i t t l e if any m a r t e n s i t e f o r m e d dur ing fa t igue . Thus , t he re

Fig. 17--Fatigue crack path in the SEN path for the alloy deformed 20 pct at 450~ The arrows indicate branching cracks.

~~ I I l illll I I I I IIIII I I I |11__.

i _

10-3 ~

i i :od

i0-6 -------'- / ~ !

lo-7 I , i l l , i l l I I l l l l l l l I I I I I I I 2 s ~o 20 50 ~oo 200 soo

AK, Stress Intenslty Range, Ksl-in I/2 Fig. 18--The fatigue crack propagation rate vs s t ress m- tensity factor range trend for all processing treatments.

Fig. 19--Fatigue crack path in the SEN specimen for the alloy deformed 80 pct at 450~ Martensite formed during cycling distinctly outlines the crack.

a p p e a r s to be a d i s t i n c t bene f i c i a l e f fec t of the s t r a i n - induced t r a n s f o r m a t i o n in F C P .

The i m p r o v e m e n t in F C P for the uns t ab le a l loys can be expla ined in the fol lowing way. Fa t i gue c r a c k p ropaga t ion is the r e s u l t of the cumula t ive damage caused by s t r a i n cyc l ing of the m a t e r i a l at the c r a c k t ip . The m a t e r i a l at some poin t ahead of the c r a c k tip e x p e r i e n c e s i n c r e a s i n g s t r a i n ampl i t udes as the c r a c k p r o p a g a t e s toward that poin t and the c u m u l a - t ive s t r a i n could be v e r y high. The high cyc l i c p l a s t i c s t r a i n c a u s e s m a r t e n s i t e to f o r m in an aus ten i t e m a t r i x which s u r r o u n d s the p ropaga t ing c r a c k as shown in F ig . 19. M a r t e n s i t e f o r m a t i o n tends to i n c r e a s e the ef fec t ive s t r a i n harden ing r a t e . Cotterel141 has shown that a high s t r a i n ha rden ing r a t e g ives r i s e to a high c r a c k p ropaga t i on r a t e . F u r t h e r m o r e , the f o r m a t i o n of m a r t e n s i t e in duc t i le aus ten i t e d e c r e a s e s the amount of s t r a i n a c c u m u l a - t ion r e q u i r e d to cause f r ac tu re - -< f in Eq. [8]. How- e v e r , these nega t ive f a c t o r s appea r to be m o r e than offset by the l a r g e bene f i c i a l e n e r g y - a b s o r b i n g ef fec ts of the s t r a i n - i n d u c e d phase t r a n s f o r m a t i o n that a r e u sua l ly obtained in t ens i l e and f r a c t u r e toughness t e s t ing . 1'3'7'4~

IV) SUMMARY AND CONCLUSION

The results may be summarized as follows: I) For low values of AK, the experimental results

were in agreement with the theoretical model which predicted a relation of the form.

da/dn = R(AK) 4

2) The fat igue c r a c k p ropaga t i on r a t e s for th ree of the four p r o c e s s i n g t r e a t m e n t s i nves t iga t ed we re s i m i l a r and r e l a t i v e l y i n sens i t i ve to s t r eng th , d u c t i - l i ty and toughness . The four th p r o c e s s i n g t r e a t m e n t ( i .e . 20 pc t a t 250~ showed somewha t p o o r e r fat igue c r a c k p ropaga t ion p r o p e r t i e s .

METALLURGICAL TRANSACTIONS VOLUME 3, OCTOBER 1972 2671

3) A distinct beneficial effect of the strain induced transformation with regards to fatigue crack propa- gation was found.

4) In fatigue crack propagation tests small cracks branching from the main cracks were observed. These branches were associated with cleavage fracture.

5) The fracture surfaces of fatigue-crack specimens of TRIP steels , contained fatigue striations, quasi- cleavage and elongated dimples reflecting the ex- tremely complex structure of TRIP steels .

6) The crack growth rates computed using striation spacings were in agreement with the corresponding microscopic crack growth rates.

7) Three of the four alloys investigated showed somewhat similar fatigue-crack propagation pro- perties, and their FCP property was superior to a number of high alloy steels of similar tensile strength levels.

ACKNOWLEDGMENTS

Part of this work was carried out under the aus- pices of the United States Atomic Energy Commission. We are grateful to Professors E. R. Parker and V. F. Zackay of the University of California, Berkeley, for their continued interest and encouragement. One of us, Stephen D. Antolovich, also wishes to acknowl- edge the American Iron and Steel Institute for a research grant (Grant No. 63-272) supporting this work.

REFERENCES

1. V. F. Zackay, E. R. Parker, D. Fahr, and R. Busch: Trans. ASM, 1967, vol 60, p. 252.

2. G R. Chanam: M. S. thesis, University of Califorma, Berkeley, Sept. 1967. 3. W. W. Gerberich, P. L. Hemmings, V. F. Zackay, and E. R. Parker Fracture

1969, Proc. of the 2nd lnt. Conf. on Fracture, Brighton, April, 1969, p. 288. 4. S D. Antolovich: Trans. TMS-AIME, 1968, vol 242, p. 2371. 5. D Fahr: Ph.D. thesis, Unwerslty of California, Berkeley, Sept. 1969. 6. J. A. Hall, V. F. Zackay, and E R. Parker: Trans. ASM, 1969, vol. 62, p. 965.

7. G. R. Chanam, V. F. Zackay, and E. R. Parker: Met. Trans., 1971, vol. 2, p. 133.

8. P. C. Pans and F. Erdogen: J. BasicEng., TransASME, 1963, vol. 85, p. 528. 9. P. C. Paris: Fatigue-An Interdisciplinary Approach, Syracuse University Press,

Syracuse, N. Y., 1964, vol. 107 10. J. R. Rice: Fatigue Craek Propagation, Amer. Soc. Test. Mater., Special Tech.

Publ., No. 415, 1967, p. 247. 11. J. Weertman. lnt. J. Fract. Mech., 1966, vol. 2, p. 460. 12. B. Tomkms: Phil. Mag., 1968, vol. 18, p. 1042. 13. J. R. Rice: Lehigh University Inst. Res. Rep., Dec. 1962. 14. K. R. Lehr and H. W. Liu: lnt. Z Fract. Mech., 1969, vol. 5, p. 45. 15. G. R. Chanani Ph.D. eng. thesis, Unwersity of Califorma, July, 1970. 16. Amer. Soc. Test. Mater., Annual Book of Standards, Part 31, E 399-70 T, 1970. 17. A. Phillips, V. Kenlins, and B. V. Whiteson: Tech. Rep. ML-TDR-64-416, Air

Force Mat. Lab. 1965. 18. B. A. Bllby, A. H. CottreU, and K. H. Swmden: Proc. Roy. Soc., 1963, vol.

272A, p. 304. 19. F. A. McChntock and G. R Irwin: Amer. Soc. Test. Mater., Special Tech. Publ.

381, 1965, p. 84. 20. G. R. Chanani, W. W. Gerberich, and V. F. Zackay: Unwersity of California,

Berkeley, unpublished research, 1971. 21. J. A. H. Hult and F. A. McClintock: 9thlnt. Congr. o f Appl. Mech., University

of Brussels, 1957, vol 8, p. 51. 22. F. A. McChntock: J. Appl. Mech., 1958, vol 25, p. 382. 23. F. A. McClintock: Fracture of Solids, p. 65, Gordon and Breach, 1963, vol.

25. 24. F. A. McClintock: Mater Res. Stand, 1961, rot. 1, p. 277. 25. W. W. Gerbench: Exp. Mech., 1964, p. 335 26. D. S. Dugdale J. Mech. Phys. Solids, 1960, vol. 8, p. 10. 27. H. H. Johnson and P. C. Pans: Eng. Fract. Mech., 1968, vol. 1, p. 3, 28. W. F. Brown and J. E. Srawley: Amer. Soc. Test. Mater., Special Tech. Publ.

410, 1966, p. 12. 29. J. C. Grosskreutz: J. Appl. Phys., 1962, vol. 33, p. 1787. 30. K. D. Challenger and J. Moteff: ScriptaMet., 1972, vol. 6, p. 155. 31. J. M. Krafft: Trans. ASM, 1965, vol. 58, p. 691. 32. R. P. Wei, P. M. Talda, and Che-Yu Li: Amer. Soc. Test. Mater., Special Tech.

Publ. 415, 1967, p. 460. 33. D. R. Donaldson and W. E. Anderson: Proc. Crack Propagation Syrup. Crane-

field, England, Sept. 1961, vol. 2, p. 375. 34. C. M. Carman and J. M. Kathn: 66-MET-3, ASME, April 1966. 35. G. A. Miller: Trans. ASM, 1969, vol. 62, p. 651. 36. C. Laird: Arner. Soc. Test. Mater., Special Tech. Publ. 415, 1967, p. 131. 37. R. W. Hertzberg and P. C. Paris: Proc. oflnt. Conf. on Frac., Sendal, Japan,

1965, p. 459 38. W. W. Gerberich and C. E. Hartbower. Trans. ASM, 1968, vol. 61, p. 184. 39. =. J. E. Forsyth: Acta Met. 1963, vol. 11, p. 703. 40. S. D. Antolovich and B. Singh. Met. Trans., 1971, vol. 2, p. 2135. 41. B. Cotterell Trans. ASME, J. Basic Eng., 1965, vol. 87, p. 230.

2 6 7 2 - V O L U M E 3, OCTOBER 1972 M E T A L L U R G I C A L TRANSACTIONS