The Effect of Loading Mode on Hydrogen Embrittlement C. ST. J O H N

AND

W. W. GERBERICH

Hydrogen e m b r i t t l e m e n t is shown to occur v e r y e a s i l y in n o t c h e d - r o u n d b a r s under opening mode I (tension) but not u n d e r a n t i p l a n e s h e a r mode III (torsion). The s t r e s s t e n s o r i n v a r i ants u n d e r mode I, II, and III loadings and how these affect i n t e r s t i t i a l diffusion a r e d i s c u s s e d . It is suggested that long r a n g e diffusion of hydrogen down o r t h o g o n a l t r a j e c t o r i e s to the v i c i n i t y of the c r a c k tip, which can o c c u r u n d e r mode I but not mode III, is a key p a r t of any hydrogen e m b r i t t l e m e n t m e c h a n i s m . This p r e m i s e was e v a l u a t e d with AISI 4340 s t e e l heat t r e a t e d to u l t r a h i g h s t r e n g t h l e v e l s . It was found that an i n i t i a l mode I s t r e s s i n t e n s i t y level of 17,000 p s i - i n , l/a p r o d u c e d f a i l u r e in s e v e r a l m i n u t e s . Mode III s t r e s s i n t e n s i t y l e v e l s t h r e e t i m e s this p r o d u c e d no c r a c k i n i t i a t i o n in 300 m i n . F u r t h e r a n a l y s i s of the t i m e - d e p e n d e n t h y d r o g e n c o n c e n t r a t i n g effect utilized a s t r e s s wave e m i s s i o n t e c h nique. This p r o d u c e d p l a u s i b l e c r i t i c a l hydrogen c o n c e n t r a t i o n s even though the p r e s e n t e l a s t i c a n a l y s i s is a f i r s t o r d e r a p p r o x i m a t i o n of the s t r e s s field. T H E p r o b l e m of h y d r o g e n e m b r i t t l e m e n t has g e n e r ated c o n s i d e r a b l e d i s c u s s i o n r e g a r d i n g the exact m e c h a n i s m ( s ) involved, l'a T h e r e does appear to be some a g r e e m e n t however on the r o l e that the l o c a l i z e d s t r e s s fields in the c r a c k tip r e g i o n p l a y in all t h r e e c u r r e n t l y held m e c h a n i s m s ; which is, to c r e a t e h y d r o g e n c o n c e n t r a t i o n s via " u p - h i l l " diffusion that can be c o n s i d e r ably higher than those away f r o m the c r a c k tip. The concept of an e l a s t i c i n t e r a c t i o n b e t w e e n the v o l u m e t r i c c o m p o n e n t of a c r a c k tip s t r e s s t e n s o r and the d i l a tion a s s o c i a t e d with an i n t e r s t i t i a l h y d r o g e n atom has b e e n d i s c u s s e d b e f o r e . 3 More r e c e n t l y , 4 an a n a l y s i s of the h y d r o s t a t i c p r e s s u r e field of a c r a c k under mode I loading coupled with F i c k ' s f i r s t law has yielded steady state solute c o n c e n t r a t i o n g r a d i e n t s s i m i l a r to those obtained d i r e c t l y f r o m t h e r m o d y n a m i c s , s

C = CoeU/kT/(1 -- C O + CoeU/kT) ~- CoeU/kT for C o s m a l l

[1]

where u is the i n t e r a c t i o n e n e r g y b e t w e e n the c r a c k tip p r e s s u r e field and hydrogen atom dilation, and C o is the h o m o g e n e o u s h y d r o g e n solid s o lution c o n c e n t r a t i o n f a r f r o m the c r a c k tip. The s t r e s s t e n s o r a s s o c i a t e d with the tip of a m a t h e m a t i c a l l y s h a r p c r a c k in an i s o t r o p i c c o n t i n u u m is of the following f o r m in p o l a r c o o r d i n a t e s :

ai.i = ~ r

Yiik(O/2)

[21

The h y d r o s t a t i c p r e s s u r e is defined in t e r m s of the s t r e s s t e n s o r i n v a r i a n t , tYii (where the usual s u m m a tion c o n v e n t i o n is a s s u m e d ) , =

[3]

On e x a m i n i n g the f i j ( O / 2 ) for each of the t h r e e i n d e C. ST. JOHN and W. W. GERBERICH, formerly Graduate Student and Lecturer, respectively, University of California, Berkeley, are now Research Scientist, Centre des Materiaux de l'Ecole des Mines, CorbeilEssone, France and Associate Professor, University of Minnesota, Minneapolis, Minn. 55455. Manuscript submitted January 24, 1972. METALLURGICAL TRANSACTIONS

*See Appendix. -P -

= A Cos (0/2)

mode I

P = - A Sin (0/2)

[4]

mode llI

- P =0 where A -

mode II

2Kk for the p l a n e s t r e s s condition 3 q 21rr 2/~

(1 +v) for the plane s t r a i n condition.

R e a r r a n g i n g the above r e l a t i o n s h i p s , the i s o p r e s s u r e c u r v e s for each c r a c k loading mode a r e of the f o r m , r = B Cos 2 (0/2)

mode I

r = -B sina(0/2)

mode II

[5]

No v o l u m e t r i c t e r m for mode III. The orthogonal t r a j e c t o r i e s or g r a d i e n t s to the i s o p r e s s u r e c u r v e s can be d e r i v e d d i r e c t l y and r e p r e s e n t the h y d r o g e n diffusion paths, in the a b s e n c e of any p l a s t i c i t y , t o w a r d s r e g i o n s of i n c r e a s i n g negative p r e s s u r e . In p o l a r c o o r d i n a t e s , the p r e s s u r e g r a d i e n t c u r v e s for each loading mode a r e r = C Sin a (0/2)

mode I

r = - C Cos 2 (0/2)

mode II

[6]

No v o l u m e t r i c t e r m for mode IIL

where K k is the s t r e s s i n t e n s i t y factor for a p a r t i t u l a r loading mode, k.

-v

p e n d e n t loading m o d e s it can be e a s i l y shown that,*

The above two s e t s of r e l a t i o n s show a s y m m e t r y b e tween mode I and mode II in that the i s o p r e s s u r e c u r v e s of one mode a r e the g r a d i e n t c u r v e s of the other mode and vice v e r s a , as is shown in Fig. 1. The m o s t i m p o r t a n t point to be d e r i v e d f r o m the h y d r o s t a t i c p r e s s u r e a n a l y s i s of the c r a c k tip r e g i o n is the hydrogen c o n c e n t r a t i n g effect of the g r a d i e n t l i n e s in m o d e s I and II and the a b s e n c e of this effect in mode III. One would expect t h e r e f o r e that h y d r o g e n c o n t a i n i n g s t e e l s would be much m o r e s u s c e p t i b l e to hydrogen e m b r i t t l e m e n t while loaded in e i t h e r modes I or / / than in mode Ill. The object of the e x p e r i m e n tal p o r t i o n of this work was to c o m p a r e , in a c u r s o r y fashion, the effect of loading mode on the s u s c e p t i b i l i t y of a high s t r e n g t h s t e e l to hydrogen e m b r i t t l e m e n t . VOLUME 4, FEBRUARY 1973-589

/

Y

/",,

/

Table I. Critical Loads, Stress Intensities, and Hold Times for Mode / and Mode I I I Testing

Mode Ill Hydrogen Content

Mode I

PIC, lb.

Loading History

without All uncharged specimens 57,000 hydrogen loaded directly to failure (two tests) chafing

CRACK

X

R=BCOS2(~2)

R = C 51N2(~'2)

Fig. 1--Isopressure contours and gradients for a crack loaded in modes I or II, 0 < 0/2 < 90 deg. MATERIALS AND PROCEDURE A c o m m e r c i a l high s t r e n g t h steel, SAE-AISI 4340, was e m p l o y e d in t h i s study a f t e r b e i n g a u s t e n i t i z e d for 1 h at 1550~ oil quenched, and t e m p e r e d for 4 h at 550~ The h y d r o g e n a t e d s p e c i m e n s w e r e c a t h o d i c a l l y c h a r g e d in a 4 pct s u l f u r i c acid solution poisoned with yellow p h o s p h o r u s which, a c c o r d i n g to a n o t h e r study, 8 yields a predictable relationship between hydrogen content and applied c u r r e n t d e n s i t y . In this c a s e , a c u r r e n t d e n s i t y of 0.0003 a m p / s q in. for 24 h should have r e s u l t e d in 3.8 ppm of h y d r o g e n in solid solution. I m m e d i a t e l y a f t e r cathodic c h a r g i n g , the s p e c i m e n s w e r e t r a n s p o r t e d while i m m e r s e d in alcohol and d r y ice to the f r a c t u r e t e s t i n g e q u i p m e n t and loaded. All t e s t i n g was at r o o m t e m p e r a t u r e . The m o s t c o n v e n i e n t mode III loading a r r a n g e m e n t is the t o r s i o n of notched r o u n d b a r s . F o r c o m p a r i s o n p u r p o s e s both mode I and mode III t e s t s w e r e c a r r i e d out on 60 deg V - n o t c h e d r o u n d s p e c i m e n s with a 0.750 in. u n i f o r m d i a m e t e r and a 0.500 in. d i a m at the Vnotch root. The notch root r a d i i w e r e d e t e r m i n e d to be close to 0.003 in. except for one s p e c i m e n which had a r a d i u s of 0.004 in. The s p e c i m e n s were not p r e c r a c k e d v i a the s t a n d a r d fatigue p r o c e d u r e b e c a u s e of the difficulty in o b t a i n i n g a u n i f o r m c r a c k p e n e t r a t i o n in notched r o u n d c o n f i g u r a t i o n s . The mode I f r a c t u r e t e s t i n g of h y d r o g e n a t e d s p e c i m e n s c o n s i s t e d of loading to p r e s e l e c t e d t e n s i l e s t r e s s e s and holding for v a r i o u s t i m e s while s i m u l t a n e o u s l y m o n i t o r i n g for s t r e s s wave e m i s s i o n s (SWE). ? The mode III h y d r o g e n a t e d s p e c i m e n s were t e s t e d in b a s i c a l l y the s a m e m a n n e r as the mode I c o u n t e r p a r t s except that the loading was in t o r sion and the s t r e s s wave e m i s s i o n m o n i t o r i n g was not available. It would have b e e n b e s t to e v a l u a t e all t h r e e loading modes. However, it was d e s i r a b l e to use the s a m e s p e c i m e n c o n f i g u r a t i o n so that c h a r g i n g c o n d i t i o n s and the c o r r e s p o n d i n g h y d r o g e n d i s t r i b u t i o n s w e r e i d e n t i cal. Since it is difficult if not i m p o s s i b l e to achieve a mode H t e s t with a c i r c u m f e r e n t i a l notched round, we confined the p r e s e n t study to mode I and III t e s t s . RESULTS AND DISCUSSION The c r i t i c a l loads and s t r e s s i n t e n s i t i e s for f a i l u r e along with loading h i s t o r i e s a r e s u m m a r i z e d in Table I. 5 9 0 - V O L U M E 4, F E B R U A R Y 1973

with (a) 20 min at 8,240 lb. hydrogen with no SWE*; failure char~ng after 2 min at 9250 lb. with many ~3100 SWE (b) 20 min at 2000 in.lb., rising torque to failure (c) 330 min at 3600 in.lb., rising torque to failure

9250

TlllC,

..K~ttC, K[C, psi-in. ~A in.-lb, psi-in.89 115,000

5075

63,500

4200

52,500

4380

54,700

26,200

+Except for specimen(a) whichfailedfrom a sharpcrack, the KIc and KIIIC valuesare apparentonesin the sensethat failurewas initiatedfroma relatively bluntnotch.Nevertheless,the qualitativecomparisonsof crackingunderthese widelyvaryingdegreesof stressintensificationis illustrative. *Stress waveemission.

The r e s u l t s a r e a r r a n g e d a c c o r d i n g to h y d r o g e n content and loading mode. Specimen (a), h y d r o g e n - c h a r g e d and loaded in mode I, exhibited slow c r a c k p r o p a g a t i o n to e v e n t u a l c a t a strophic f a i l u r e at a c r i t i c a l s t r e s s i n t e n s i t y that is typical of u l t r a h i g h s t r e n g t h steel. 8 The slow c r a c k growth was followed d u r i n g the t e s t b y m o n i t o r i n g the s t r e s s wave e m i s s i o n s (SWE). In Fig. 2(a), the f r a c t u r e s u r f a c e shows the i n i t i a l slow c r a c k growth r e gion and the s u b s e q u e n t r a p i d growth a c r o s s the r e m a i n i n g s e g m e n t after KIC had b e e n attained. The KIC was e s t i m a t e d f r o m the following r e l a t i o n s h i p , ~

.72

-1.2

[7]

where P is the applied load, D is the d i a m e t e r of u n i f o r m c r o s s - s e c t i o n , d is the d i a m e t e r of r e d u c e d c r o s s - s e c t i o n , which is this c a s e was e s t i m a t e d by the a v e r a g e d i a m e t e r of the u n c r a c k e d s e g m e n t at the o n s e t of r a p i d c r a c k growth. Two static torque loads were e x a m i n e d in the mode III, h y d r o g e n a t e d s p e c i m e n s . The f i r s t t r i a l , s p e c i m e n (b), utilized an e q u i v a l e n t s t r e s s i n t e n s i t y that p r o v e d to be fatal for s p e c i m e n (a). The mode III s t r e s s i n t e n s i t y f o r m u l a employed was obtained f r o m a finite e l e m e n t a n a l y s i s ; t~ YT g i i I - (d/2)2.~ where

[8]

Y is a c o n s t a n t for any p a r t i c u l a r (d/D) and T is the applied t o r q u e .

Specimen (b) showed no e v i d e n c e of slow c r a c k growth after 20 m i n at an e q u i v a l e n t s t r e s s i n t e n s i t y that p r o vided f a i l u r e in 2 m i n under mode I loading. The torque was then slowly i n c r e a s e d u n t i l a slow ductile f a i l u r e e n s u e d , the r e s u l t i n g ductile f r a c t u r e s u r f a c e is shown in Fig. 2(c). Specimen (c) was loaded at 80pct of the torque that induced p l a s t i c s h e a r f a i l u r e in s p e M E T A L L U R G I C A L TRANSACTIONS

(a)

(b)

(a) (c) Fig. 2--Mode I fracture surfaces: (a) hydrogen charged, (b) control; Mode Ili fracture surfaces: (c) hydrogen charged, (d) control. c i m e n (b). This was held for 330 m i n with no e v i d e n c e of c r a c k growth. The t o r q u e was then s l o w l y r a i s e d to s p e c i m e n f a i l u r e that o c c u r r e d in the s a m e m a n n e r and at a p p r o x i m a t e l y the s a m e load a s f o r s p e c i m e n (b). The f r a c t u r e s u r f a c e for s p e c i m e n (c) was i n d i s t i n g u i s h a b l e f r o m s p e c i m e n (b). T h r e e u n c h a r g e d s p e c i m e n s w e r e l o a d e d d i r e c t l y to f a i l u r e , two in mode I and one in m o d e llI. No e v i d e n c e of slow c r a c k growth at low s t r e s s i n t e n s i t i e s was o b METALLURGICAL TRANSACTIONS

s e r v e d on the f r a c t u r e s u r f a c e s of any of t h e s e c o n t r o l s p e c i m e n s a s i s shown in F i g s . 2(b) and 2(d). The load n e c e s s a r y to induce c r a c k i n i t i a t i o n on mode I was a p p r o x i m a t e l y 450 p c t g r e a t e r than that for the h y d r o g e n a t e d m a t e r i a l . In mode III, h o w e v e r , the c r i t i c a l t o r q u e was only 17 p c t g r e a t e r than that o b t a i n e d f r o m the c h a r g e d s p e c i m e n s . T h i s d i f f e r e n c e b e t w e e n the c h a r g e d and u n c h a r g e d s p e c i m e n s l o a d e d in mode III m a y be e x p l a i n e d by the a n o m a l o u s l y l a r g e notch r o o t VOLUME 4, FEBRUARY 1973-591

Fig. 3--Fractography of mode I: (a) replica of hydrogen charged sample; (b) scanning electron microscopy of hydrogen charged sample: Fractography of mode III: (c) scanning electron microscopy typical of hydrogen charged and control samples.

(a)

(b)

592-VOLUME 4, FEBRUARY 1973

r a d i u s of the c o n t r o l s p e c i m e n which would d e c r e a s e the s t r e s s c o n c e n t r a t i o n . The f r a c t u r e s u r f a c e s for c h a r g e d and u n c h a r g e d mode I I I s p e c i m e n s w e r e i d e n t i c a l . The d i f f e r e n c e between mode I and I I I f r a c t u r e s u r f a c e s , for the s p e c i m e n with p r i o r h y d r o g e n a t i o n , a r e m o r e c l e a r l y i l l u s t r a t e d by s c a n n i n g and r e p l i c a e l e c t r o n f r a c t o g r a p h y of the f r a c t u r e s u r f a c e s in Fig. 3. Here it is s e e n that the slow c r a c k growth p r o c e s s in mode I is d e f i n i t e l y an i n t e r g r a n u l a r f r a c t u r e p r o c e s s a s m a y be a s s o c i a t e d with hydrogen e m b r i t t l e m e n t . On the other hand, s c a n n i n g of the o u t e r p e r i p h e r y of the c r a c k s u r f a c e s f r o m the mode I I I s p e c i m e n only r e v e a l e d the morphology shown in Fig. 3(c) for both c h a r g e d and u n c h a r g e d s p e c i m e n s . This would be c h a r a c t e r i s t i c of a ductile f r a c t u r e w h e r e the s u r f a c e s had r u b b e d t o g e t h e r . Thus, one m a y conclude that both m e c h a n i c a l t e s t data and f r a c t u r e s u r f a c e o b s e r v a t i o n s v e r i f y the h y drogen e m b r i t t l e m e n t m e c h a n i s m under mode I loading but not under mode III. The m a i n r e a s o n would a p p e a r to be the d i f f e r e n c e in the s t r e s s field i n t e r a c t i o n with hydrogen. The fact that mode I I I did not p r o d u c e hydrogen e m b r i t t l e m e n t should be put in p e r s p e c t i v e s i n c e some i n t e r a c t i o n of the hydrogen with the s t r e s s field is p o s s i b l e . For example, if h y d r o g e n in solution p r o d u c e s a t e t r a g o n a l d i s t o r t i o n , the n o n i s o t r o p i c s t r e s s field can i n t e r a c t with the e l a s t i c s h e a r s t r e s s as pointed out by Li, O r i a n i , and Darken, 11 for c a r b o n in a - F e . However, this would p r o b a b l y be a s e c o n d o r d e r effect as c o m p a r e d to the i n t e r a c t i o n with the d i l i t a n t field of mode I. C o n s i d e r i n g the SWE data f r o m s p e c i m e n (a) t e s t e d in mode I, 3100 SWE were detected d u r i n g the slow c r a c k growth p e r i o d . A s m a l l p o r t i o n of o s c i l l o g r a p h playback in Fig. 4 i n d i c a t e s the d i s c o n t i n u o u s n a t u r e of the e m b r i t t l e m e n t p r o c e s s . These w e r e a s s o c i a t e d with a total c r a c k growth a r e a of 0.061 in. 2 so that on the a v e r a g e , each SWE is a s s o c i a t e d with an a r e a of 2 • 10 -5 infl or a l i n e a r d i m e n s i o n of about 0.0045 in. METALLURGICALTRANSACTIONS

f r a c t u r e m e c h a n i c s a n a l y s e s . In p a r t i c u l a r , the s t r e s s t e n s o r i n v a r i a n t a s s o c i a t e d with the d r i v i n g f o r c e for long r a n g e h y d r o g e n diffusion at the tip of a c r a c k is d e t e r m i n e d . Two m a t h e m a t i c a l e x t r e m e s a r e c o n s i d e r e d , p l a n e s t r e s s w h e r e a z = 0 and p l a n e s t r a i n w h e r e e z = 0 and hence 0-z = v(0-x + cry). F o r the opening Mode I : In t e r m s of p o l a r c o o r d i n a t e s , r , 0, the s t r e s s e s a r e a function of p o s i t i o n and the magnitude of the s t r e s s i n t e n s i t y f a c t o r , K I.

I TIMING LINE = 0'5 SECONDS

Fig. 4--A few of the SWE recorded during the slow crack growth stage of specimen (a) under mode I loading.

If the d i s t a n c e o v e r which the h y d r o g e n c o n c e n t r a t e s and n u c l e a t e s f r a c t u r e i s on the s a m e o r d e r of m a g n i tude a s t h i s f r a c t u r e r e g i o n , then the p r e c e d i n g e q u a tions m a y be u t i l i z e d to e s t i m a t e the a m o u n t of h y d r o gen c o n c e n t r a t i n g in t h i s r e g i o n . Using the t h e r m o d y n a m i c e q u i v a l e n t to Eq. [19] in Ref. 4, the e q u i l i b r i u m c o n c e n t r a t i o n i s given by

~-#_~__~:~-~-

C = Co exp { 2 ( 1 + V)VHK }

[9]

a x - (2rrr)X-/~ c o s -~-

-

sin -~- sin - -

0-y - (27rr)i-]y c o s ~-

+ sin ~- sin - -

g 9 0 38 ~'xy - (2~r)I/z s i n - ~ c o s ~- c o s 2

[A-l]

a z = 0 for p l a n e s t r e s s a z = v(a x + ay) for p l a n e s t r a i n T): z = T y z = 0

Using the a v e r a g e s t r e s s i n t e n s i t y d u r i n g slow c r a c k growth of 21,400 p s i - i n . / (2.38 • 109 dyne-cm-3/2), the l i n e a r d i m e n s i o n of 0.0045 in. (0.0115 cm), and the v a l u e of 2.0 c m 3 / g - a t o m for the p a r t i a l m o l a l v o l u m e of h y d r o g e n in i r o n for VH,3 one c a l c u l a t e s at r o o m t e m p e r a t u r e t h a t C = 1.9C o. If the p r e v i o u s l y e s t i m a t e d v a l u e of 3.8 p p m H i s a s s u m e d , then C -~ 7.2 p p m H which i s v e r y c l o s e to o t h e r e s t i m a t e s 1~ for the c r i t i c a l h y d r o g e n c o n c e n t r a t i o n . Of c o u r s e , one m u s t t a k e into c o n s i d e r a t i o n that a) with the s h o r t t i m e involved e q u i l i b r i u m p r o b a b l y was not a c h i e v e d and b) this is a p u r e l y e l a s t i c s o l u t i o n and the g r a d i e n t due to the p l a s t i c zone w a s not c o n s i d e r e d . N e v e r t h e l e s s , s i n c e the v a l u e p r e d i c t e d i s s u f f i c i e n t l y c l o s e to what one might e x p e c t , t h i s a p p r o a c h m e r i t s f u r t h e r c o n s i d e r a tion. SUMMARY The c o m p a r a t i v e t e s t s p e r f o r m e d on h y d r o g e n c h a r g e d and c o n t r o l s p e c i m e n s give s t r o n g e v i d e n c e that high s t r e n g t h s t e e l l o a d e d in the mode III c o n f i g u r a t i o n is i n s e n s i t i v e to h y d r o g e n e m b r i t t l e m e n t when c o m p a r e d to mode I loading. An i n t e r p r e t a t i o n of t h e s e r e s u l t s is that the h y d r o s t a t i c p r e s s u r e f i e l d s a s s o c i a t e d with mode I s t r e s s c o n c e n t r a t i o n s r a i s e the h y d r o g e n s o l i d s o l u t i o n c o n c e n t r a t i o n s to c r i t i c a l v a l u e s that a r e n e c e s s a r y to i n i t i a t e slow c r a c k growth b y any one of the c u r r e n t l y held m e c h a n i s m s . ACKNOWLEDGMENTS T h i s w o r k w a s s u p p o r t e d by the United S t a t e s A t o m i c E n e r g y C o m m i s s i o n . The a u t h o r s would like to thank Y. T. Chen and J. Cohen for a s s i s t a n c e with e l e c t r o n f r a c t o g r a p h y and s c a n n i n g e l e c t r o n m i c r o s c o p y .

D e t e r m i n a t i o n of the p r e s s u r e is s i m p l i f i e d b y r e c o g n i z i n g that ax-

ax~v~

+ T max

0-a = % +2 0-Y - Tm~x 0-3 = "(0-x +

[A-2]

%).

Thus, for p l a n e s t r e s s , w h e r e 0-3 = 0,

-p=

0-, + 0"2 + 0-a

0-x + fly

3

3

and for p l a n e s t r a i n , (1 + v)(0-x + ay) 3

-p_-

[A-3]

Combining Eqs. [ A - I ] and [A-3] give -P

= 3(27rr)l/z c o s

for p l a n e s t r e s s

-P

2(1 + v ) K I /0\ = 3(27rr)1/2 c o s ~-~-) for p l a n e s t r a i n

[A-4]

a s given in Eq. [4]. F o r the i n - p l a n e s h e a r Mode H: git

ay = ~

KH

sin02

2 +cos ~ cos --

0 0 30 sin ~- c o s ~- c o s

KH 0[1 0 ~'xy - (2n~r)lZ c o s ~- - sin ~- sin

_~__]

[A-5]

0-z = 0 for p l a n e s t r e s s APPENDIX

0-z = v(0-x + ay) for p l a n e s t r a i n

The s t r e s s f i e l d s about c r a c k s under v a r i o u s t y p e s of loading have b e e n d i s c u s s e d at g r e a t length,* but

Combining Eqs. [ A - l ] and [A-5] give

*For example, see Paris and Sih ~3 for a discussion of both stress analysis and a description of the various types of loading modes.

will be r e i t e r a t e d h e r e to a s s i s t t h o s e u n f a m i l i a r with METALLURGICAL TRANSACTIONS

rxz = "ryz = 0

-P

=-

2KH /0\ 3(27rr)l/z sin [-4--) for p l a n e s t r e s s [A-6] VOLUME 4, F E B R U A R Y 1 9 7 3 - 5 9 3

-p

=-

2(1 + u)K H sin(O) for plane strain 3(21rr)1/2

also given in Eq. [4]. For the antiplane shear Mode ax = a ~ = a

rxy -

z =zxy

-KtII

(2~r)~-~

III"

=0

sin 0

T;

ry~

-

KIII

(2~r)t- ~- c o s

0

T

-P=O

REFERENCES 1. D. P. Williams and H. G. Nelson: Met. Trans., 1971, vol. 2, p. 1987. 2. C. F. Barth and E. A. Steigerwald: Met. Trans., 1971, vol. 2, p. 1988. 3. R. A. Oriani: FundamentalAspects of Stress Corrosion Cracking, p. 32, National Association of Corrosion Engineers, Houston, 1969.

5 0 4 - V O L U M E 4, F E B R U A R Y 1973

4. H. W. Liu: J. Basic Eng., ASME, 1970, vol. 92, p. 633. 5. D. McLean: Grain Boundaries in Metals, p. 118, Oxford at the Clarendon Press, 1957. 6. C. F. Barth and E. A. Steigerwald: Met. Trans., 1970, vol. 1, p. 3451. 7. C. E. Hartbower, W. W. Gerbedch, and P. P. Crimmins: WeMingZ Res. Supp., 1968, vol. 47, no. 1,p. Is. 8. A. J. Baker, F. J. Lauta, and R. P. Wei: Amer. Soc. Test. Mater. Special Tech. Publ. 370, 1965, p. 3. 9. W. F. Brown and J. E. Srawley: Amer. Soc. Test. Mater. Special Tech. Publ. 410, 1966. 10. W. K. Wilson, W. G. Clark, Jr., and E. T. Wessel: Fracture Mechanics Technology for Combined Loading and Low-To.Intermediate Strength Metals, Westinghouse Research Lab., Contract DAAE 07-67-C-4021, November 1968. 11. J. C. M. Li, R. A. Oriani, and L. S. Darken: Z. Physik. Chem., 1966, vol. 49, p. 271. 12. R. A. Oriani: FundamentalAspects of Stress Corrosion Cracking, p. 49, National Assoc. of Corrosion Engineers, Houston, 1969; see also W. W. Gerbedch and C. E. Hartbower, p. 426;M. Smialowski, p. 463. 13. P. C. Paris and G. C. Sih: Amer. Soc. Test. Mater. Special Tech. Publ. 381, 1965, p. 30.

METALLURGICAL TRANSACTIONS