N.K. Sinha, J. Materials Sci., 23 (1988), p. 4415

8/6/2019 N.K. Sinha, J. Materials Sci., 23 (1988), p. 4415

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JOURNAL OF MATERIALS SCIENCE 23 (1988) 4415-4428

C r a c k - e n h a n c e d c r e e p in p o l y c r y s t a ll in em a t e r i a l : s t r a i n - r a t e s e n s i t i v e s t r e n g t h a n dd e f o r m a t i o n o f i c eN I R M A L K . S I N H AIns t itu te fo r Research in Const ruct ion , N at iona l Research Co unc i l o f Canada,Ot tawa K IA ORC, Canada

A n o n - l i n e a r v i s c o e l a s t i c c r e e p e q u a t i o n f o r p o l y c r y s t a l l in e m a t e r ia l is p r e s e n t e d . I t i n c o r -p o r a t e s t h e e f f e c t o f c r a c k i n g a n d i s c a p a b l e o f d e s c r i b i n g p r i m a r y , s e c o n d a r y a n d t e r t i a r yb e h a v i o u r . T h e m o d e l p r e d i c t s t h e f o r m a t i o n o f m i c r o c r a c k s a n d t h u s t h e d a m a g e s t a t e d u e t ot h e h i g h - t e m p e r a t u r e g r a i n - b o u n d a r y e m b r i t t le m e n t p r o c e ss . T h i s p a p e r d e s c r ib e s it s a p p l i c a -t i o n i n f o r m u l a t i n g c r a c k - e n h a n c e d c r e e p a n d m a t e r i a l r e s p o n s e u n d e r c o n s t a n t s t r a in - r a t el o a d i n g c o n d i t i o n s ( t h e o r e t i c a l l y t h e s i m p l e s t c a s e b u t a c t u a l l y t h e m o s t d i f f i c u l t t o m a i n t a i n ) .T h e f o r m u l a t i o n m a k e s i t p o s s i b l e t o d e f i n e t h e r a t e e f f e c t o n s t r e s s - s t r a i n r e s p o n s e a n d t h era te sens i t i v i t y o f s t reng th , f a i l u re t ime , fa i l u re s t ra i n , damage and damage ra te , s t ra i n recovery ,e tc . N u m e r i c a l c o r r e s p o n d e n c e b e t w e e n t h e o r y a n d e x p e r i m e n t w a s o b s e rv e d w h e n p r e d i c -t i o n s w e r e c o m p a r e d w i t h a v a i l a b l e c l o s e d - l o o p , c o n t r o l l e d , c o n s t a n t s t r a i n - r a t e s t r e n g t h a n dd e f o r m a t i o n d a t a o n p u r e i c e . C a l c u l a t i o n s m a d e u s e o f m a t e r i a l c o n s t a n t s d e t e r m i n e d f r o mi n d e p e n d e n t c o n s t a n t - l o a d c r e e p te sts .

1 . I n t r o d u c t i o nT h e m e c h a n i c a l r e s p o n s e o f p o l y c r y s t a l l i n e ic e [1 , 2] ,f o r e x a m p l e t h e d e p e n d e n c e o f s t r e n g t h o n s t r a i n r a t e ,t h e d e p e n d e n c e o f f ai l u re t im e o n l o a d a n d t h e n e a rc o n s t a n c y i n f a i l u r e s t r a i n , i s v e r y s i m i l a r , p h e n o m -e n o l o g i c a l l y , t o t h e m e c h a n i c a l r e s p o n s e o f p o l y c r y s -t a l l i ne ma te r i a l s i n gene ra l [3 -7 ] . I c e i s one o f t hes t r o n g es t m a t e r ia l s i n t e r m s o f w o r k i n g t e m p e r a t u r e ,T , e x p r e s s e d a s h o m o l o g o u s t e m p e r a t u r e , T/Trn,w h e r e Tm i s t h e m e l t i n g p o i n t a n d b o t h T a n d Tm a r ein t he abso lu t e sca l e (Ke lv in ) . Seve ra l phys i ca l andm i c r o s t r u c t u r a l c h a r a c t e r i s t i c s i n f l u e n c e i t s s t r e n g t h .L o w l a t ti c e ( a n d g r a i n - b o u n d a r y ) d i f f u s i v i ty , l e a d i n gt o r e l a t i v e l y s l o w e r m a t r i x r e l a x a t i o n , a n d l a r g e rg r a i n s ( u s u a l l y > 1 m m ) , l e a d i n g t o s l o w e r g r a i n -b o u n d a r y d i f f u s i o n , a r e p r i m a r i l y r e s p o n s i b l e [ 8 ] .

A t t h e s a m e h o m o l o g o u s t e m p e r a t u r e t h e r a t e o fd i f f u s i o n i n ic e i s t w o o r t h r e e o r d e r s o f m a g n i t u d el o w e r t h a n t h a t o f m o s t o t h e r p o l y c r y s t a l li n e m a t e r i a ls[ 9 ] . A d i r e c t c o n s e q u e n c e o f t h i s p h e n o m e n o n i s ap r o p e n s i t y f o r c r a c k i n g a c t i v i t y . G r a i n - b o u n d a r ye m b r i t t l e m e n t p r o c e ss e s p l a y a d o m i n a n t r o l e i n d e te r -m i n i n g s t r e n g t h a n d d e f o r m a t i o n i n ic e. F o r t h i s r e a s o na n d i n s p i t e o f i ts r a t e - c o n t r o l l i n g e f fe c t, i n t r a g r a n u l a rp l a s t i c i t y c o n t r i b u t e s l e s s t o t h e t o t a l d e f o r m a t i o n a tf a i l u r e i n i c e t h a n i t d o e s i n m a n y o t h e r m a t e r i a l s a tst ress levels (a > 1 x 10 4E~ w h e r e E i s Y o u n g ' smo dulu s) a nd s t r a in r a t e s (4 > 1 x 10 -7 sec -1 ) o fp r a c t i c a l i n t e r e s t i n m o s t e n g i n e e r i n g s i t u a t i o n s . I nc o n j u n c t i o n w i t h i t s o p t i c a l t r a n s p a r e n c y , t h i s m a k e sp u r e i c e a n i d e a l m a t e r i a l f o r s t u d i e s o f m i c r o m e c h a n -i c s a n d , t h e r e b y , h i g h - t e m p e r a t u r e m a t e r i a l s c i e n c e .

T h e p r e s e n t p a p e r d e s c r i b es h o w a h i g h - t e m p e r a t u r e0022-2461/88 $03.00 + .12 9 1988 Chapman and Hall Ltd.

r h e o l o g i c a l m o d e l w i t h g ra i n - si z e d e p e n d e n t t r a n s i e n tc r e e p [ 1 0] c a n b e u s e d , w i t h t h e i n t r o d u c t i o n o f c r a c ke n h a n c e m e n t o f c re e p , f o r p r e d i c t i n g s t r e n g t h a n ds t r e s s - s t r a i n b e h a v i o u r u n d e r c o n s t a n t s t r a i n - r a t el o a d i n g c o n d i t i o n s . T h e t h e o r y i s t e s t e d w i t h d a t a o ni ce . T h e m o d e l p e r m i t s c o m m o n l y o b s e r v e d p h e n o m -e n a a n d e m p i r i c a l l y d e v e l o p e d e q u a t i o n s t o b e d e s -c r i b e d q u a n t i t a t i v e l y .2 . Pre l imin ary an a lys isCo nsid e r t he t e s t r e su l t s i n F ig . 1 fo r pu re S-2 i ce :o p t i c a l l y t r a n s p a r e n t , b u b b l e - f r e e , t r a n s v e r s e l y i s o -t r o p i c , c o l u m n a r - g r a i n e d . A t - 1 0 ___ 0 . 1 ~ a n du n d e r a c l o s e d - l o o p c o n t r o l l e d c o n s t a n t c o m p r e s s i v es t r a in r a t e o f 3 10 -5 sec -1 , a m ax im um s t re ss o f4 . 6 M N m - 2 w a s r e a c h e d . E s s e n t i a ll y , t h i s te s t w a sc o n d u c t e d f o l l o w i n g t h e m e t h o d d e s c ri b e d b y S i n h a[ 2 ] , a l t h o u g h a l a r g e r c o n t r o l l i n g g a u g e l e n g t h o f2 00 m m a n d a n i m p r o v e d t e c h n i q u e f o r m o u n t i n g t h eg a u g e o n t h e s p e c i m e n w a s u s e d [ 1 1 ]. T h e l o a d w a sr e m o v e d q u i c k l y ( in ~ 0 .1 s ec ) s o o n a f t e r t h e m a x i -m u m o r u p p e r y i e l d fa i l u re s tr e s s w a s r e a c h e d . S t r a i nr e c o v e r y h i s t o r y s h o w s a n i n s t a n t a n e o u s e l a st i c s t r a in ,ee (me asur ed a f t e r fu l l un lo ad in g , ~ 0 .1 sec ), fo l l ow edb y a d e l a y e d e l a s t ic r e c o v e r y , e d, a n d a p e r m a n e n t o rv i scous s t r a in , ev . Thus the ax i a l s t r a in , e , c an bed e s c r i b e d p h e n o m e n o l o g i c a l l y a s

e = ~e + ed + ev (1)I n t h i s c a se t h e t o t a l s t r a i n a t t h e p o i n t o f u n l o a d i n gc o n s i s t e d o f a b o u t 2 3 % e l a st ic s tr a i n , 2 8 % d e l a y e d -e l a s t i c s t r a i n , a n d 4 9 % v i s c o u s o r p e r m a n e n t s t r a i n .E l a s t ic a s w e l l a s d e l a y e d - e l a s ti c s t r a i n p l a y i m p o r t a n t441 5


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rIEZ=Eb113

e~

020

',=, 15,.-iku

I 0Z

~- 5

I I If

e-O- = 3.5 MN

I I

I I

-2rn4 =3 x 10 5 sec" I

250ram

50ramI ' I

_ r 1 T r 1 4

200 30 60 90 120 150 1200T I M E , t ( s e c )

Figure 1 Stress and strain histories forcolumnar-grained S-2 ice, average graindiameter of 5.0mm at -1 0 ~ (0.96Tin)under closed-loop controlled constantstrain rate of 3 x 10-Ssec-L

r o l e s i n d e t e r m i n i n g t o t a l s t r a i n. T h e o r e t i c a l d e v e l o p -m e n t m u s t t h e r e f o r e c o n s i d e r a l l t h e m e c h a n i s m s r e s-p o n s i b l e f o r t h e v a r i o u s s t r a i n c o m p o n e n t s . M o s ta n a l y s e s i n t h e l i t e r a t u r e o n p o l y c r y s t a l l i n e m a t e r i a l sh a v e i n g e n e r a l b e e n b a s e d p r i m a r i l y o n a s t e a d y - s t a t e( o f t e n c a l l e d p l a s t ic ) f l o w m e c h a n i s m .

F i g . 2 s h o w s t h e a b o v e - m e n t i o n e d s p e c i m e n a f t e rt e s t in g a n d a n o t h e r t e s t e d a t a lo w e r s t r a i n r a t e

( 4 1 0 - 6 s e c - 1 ). T h e s e c o n d s p e c i m e n w a s u n l o a d e ds o o n a f t e r r e a c h i n g i ts m a x i m u m l o a d o f 2. 4 M N m - 2 .T h e s i g n i f i c a n t d i ff e r e n c e s in t h e c r a c k d e n s i t i e s o f t h et w o s p e c i m e n s a s w e l l a s s e v e r a l e a r l y e x a m p l e s [ 1 , 2 ]m a k e i t c l e a r t h a t c r a c k i n g a c t i v i t y is h i g h l y r a t e -s e n s i t i v e .

T h e c r a c k s w e r e l o n g a n d n a r r o w a n d p a r a l l e l t o t h el e n g t h o f t h e c o l u m n a r g r a i n s ( F i g s 2 a a n d b ) . F i g . 2 c

Figure 2 Specimens (50 mm x 100 mm x 250 mm) after testing at - 10~C under strain rate of (a) 4 x 10 -6 sec i and (b) 3 x 10 5 sec-1of Fig. I, an d (c) a 5 mm thick section from mid-plane of specimen in (b).4 4 1 6


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Figure 3 A 0.8 mm double-micro-tomed th in section, made fromthick sec tion of Fig. 2c, und er(a) cross-polarized light, (b) com-bined polarized ight and scatteredlight, exhibiting predominantlygrain-boundary cracks.

shows that the p lane of the cracks t ends to be para l l e lto the axis of the impos ed com pressive s t ress . Similarobservat ions under cons tan t load t es t have beenrepor ted by Gold [12 , 13] fo r co lumnar-gra ined iceand by Co le [14] for isotropic gr anu lar ice. Fig. 2cshows that the cracks t end to p ropagate a long gra inboundar ies o r ien ted wi th in 20 ~ of the ax is o f theappl ied load . More incl ined gra in boundar ies arecracked at high s train rates (Figs 3 and 4) during laterdeformat ion when the crack dens i ty i s h igher , o raroun d shear p lanes in compress ion t es ts in which thewedging act ion of the grains leads to t r iaxial s t resss ta tes . Simi lar observat ions have been made formetals a nd al loys [15, 16], for exam ple, on s tainlesssteel of Types 304 and 316 [17, 18]. In m etals an dceramics , fo r which t es t s are usual ly performed intens ion , cracks t end to p ropagate a long gra in bound-aries oriented nor mal to the axis of the appl ied tensi lestress.

These cracks are kno wn as wedge- type cracks. I t i sno t a s imple mat ter to separa te a wedge- type crackfrom an opening created by the coalescence of smal lrounded cavi t ies , as emphasized 30 years ago byMcL ean [19] . The developme nt o f l arge vo ids af fec tsthe defo rma tion processes in a s ignificant ma nn er [20].Analys i s o f s t reng th and deformat ion mus t thereforeinclude crack ing ac t iv i ty and ra te dependence.

3. Microcracking mechanismsMicrocracks, comparable in s ize to a grain facet , seemto occur under a co ns tan t - load creep condi t ion wherestress exceeds a cri t ical v alue (> 10 -5 E, where E isY o u n g ' s m o d u l u s ) co m m o n l y k n o w n as t h e S t ro h -McL ean t rans i t ion . S t roh [21] descr ibed th i s min im umstress in terms o f the length o f a s l iding interface, thesurface-f ree energy and the shear modulus . McLean[22] used St roh 's re la t ion for g ra in-b oundar y crack ingby iden t i fy ing the in ter face wi th the gra in b oun daryand surface energy with the effect ive fractur e surfaceenergy. Sinha [23] has s how n the general ap pl icabi l i tyof the S t roh -M cL ean equat ion to po lycrys ta l line i ce .Calcu la t ions conf i rm that the pred ic ted min imumstress is about twice the value actual ly observed[12, 24]. I t has been po inted out , how ever, th at thisclassical t reatment as well as that proposed by Smithand Barn by [25] do n o t g ive any in form at ion on thes t ress - and temperature-dependent incubat ion t imeusually required und er mode rate stress ( > 10 4E) beforethe ini t iat ion of cracks a nd the s ignificant depen denceof crack ing ac t iv i ty wi th fur ther load ing .

By invoking Wi l l i ams ' [26] t rea tme nt o f Cot t re l l ' sequ at ion [27] for the s tabi l i ty of a void form ed bydislocat ions on two intersec t ing planes, and by replac-ing the wedge heigh t wi th gra in-boundary d i sp lace-ment , Sinha [23] showed that a cr i ti ca l g ra in-bounda ry

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s l i d i ng ( g . b . s . ) d i s p l a c e m e n t , a nd he nc e a c r i t i c a l g r a i n -b o u n d a r y s li d in g s t r ai n , m i g h t b e r e q u i re d f o r a v o ida t t h e i n t e r f a c e t o g r o w i n a n u n s t a b l e m a n n e r t h a tl e a d s t o a w e d g e - t y p e c r a c k .3 .1 . Cr i t i ca l g .b .s , d i sp l acement fo r c racki n i t i a t i onT h e h y p o t h e s i s o f a c r it i c a l g .b . s , d i s p l a c e m e n t , 2 o , f o rt h e f o r m a t i o n o f c ra c k s c o m p a r a b l e i n si ze to a g r a i nf a c e t w a s i n it ia l ly a p p l i e d t o o b s e r v a t i o n s o f p o l y c r y s -t a l li n e i c e [2 8 ]. D e t a i l e d s t u d i e s [ 23 , 2 8] s h o w e d t h a t 2 ci n t h e l im i t e d u n i a x i a l s t r e s s r a n g e o f 5 x 1 0 - S E t o2 x 1 0 - 4 E a n d t e m p e r a t u r e r a n g e o f 0 . 8 8 Tm t o0 . 9 8 T i n i n p r e v i o u s l y u n d e f o r m e d c l e a r i c e d e p e n d so n l y o n te m p e r a t u r e . T h u s

dl2 c = ( M , - m , T ) ~ = ( M , - m , T ) ( 2)w h e r e d ~ i s t h e u n i t d i m e n s i o n ( = 1 m ) ; M r , m ~ a r ec o n s t a n t s , a n d K ( a s s u m e d t o b e u n i ty ) i s a n o t h e rc o n s t a n t a r i si n g f r o m G i f k i n s ' [1 5] e q u a t i o n o f p r o -p o r t i o n a l i t y b e t w e e n g . b . s, s t r a i n , S gb s, a n d a v e r a g eg . b . s , d i s p l a c e m e n t , 2 ,

/~gbs = d ( 3 )w h e r e d i s t h e a v e r a g e g r a i n d i a m e t e r e x p r e s s e d i nun i t s o f d~ ( m ) .

E q u a t i o n 2 i s q u a l i t a ti v e l y s u p p o r t e d b y d i re c te x p e r i m e n t a l o b s e r v a t i o n s i n o t h e r m a t e ri a l s. T h ed e p e n d e n c e o f c a v i ty f o r m a t i o n o n t h e a m o u n t o fg r a i n - b o u n d a r y s li d in g w a s r e p o r t e d b y I n t r a t e r a n dM a c h l i n [ 2 9 ] i n c o p p e r b i c r y s t a l s , a n d s i m i l a r o b s e r v a -t i o n s h a v e b e e n r e p o r t e d b y F l e c k e t a I . [30] in ac o p p e r a l l o y . E x p e r i m e n t a l s u p p o r t f o r a c r i t i c a lg r a i n - b o u n d a r y s li d in g d i s p l a c e m e n t f o r c r a c k i n i ti -a t io n h a s b e e n p r o v i d e d b y W a t a n a b e [3 1] o n c o p p e rb i c r y s t a l s a n d M o r r i s a n d H a r r i e s [ 18 ] i n s t ee l .

T h e k e y h y p o t h e s is i n t h e d e v e l o p m e n t o f E q u a t i o n2 is t h a t d e l a y e d e l a s t ic s t r a i n , e a , i s a s s o c i a t e d w i t h ag r a i n - b o u n d a r y s l id i n g m e c h a n i s m [ 10 ] g i v e n , a s a fi r s ta p p r o x i m a t i o n , b y

= K _ ~ed = ~;gbs d (4 )E q u a t i o n s 2 a n d 4 g i v e t h e c r i t i c a l d e l a y e d e l a s t i cs t r a i n f o r t h e o n s e t o f c r a c k i n g a c t i v i t y a s

dle~ = ( M , - m , T ) - ~ ( 5 )

3 . 2 . O n s e t o f c r e e p c r a c k i n gE q u a t i o n 2 p r o v e d t o b e u s e f u l i n p r e d i c t in g t h e s tr e s sa n d t e m p e r a t u r e d e p e n d e n c e o f t im e , t f~ , f o r t h e o n s e to f c r a c k i n g a c t i v i t y i n p o l y c r y s t a l l i n e m a t e r i a l s [ 23 ]:{ Ifc = r r - In 1 - ~ (6)o r

w h e r e X = K 2 ~ / c l d ~ i s a c o n s t a n t a t a c o n s t a n t t e rn -4 4 1 8

p e r a t u r e a n d r r i s t h e t e m p e r a t u r e - d e p e n d e n t g r a in -b o u n d a r y r e l a x a t i o n t i m e , h a v i n g t h e s a m e a c t i v a t i o ne n e r g y a s th a t o f m a t r i x d e f o r m a t i o n ; c l , s a n d ba r e m a t e r i a l c o n s t a n t s . T h e c o n s t a n t s i s t h e s t re s se x p o n e n t f o r g r a i n - b o u n d a r y s l id i n g [1 0 ], a n d b i s t h et i m e e x p o n e n t f o r t r a n s i e n t , o r s t r i c t l y s p e a k i n g ,de l a ye d e l a s t i c s t r a i n [ 32 ] .

T h e v a l u e f o r s m a y b e b e t w e e n 1 a n d 4 f o r p o l y -c r y s t a l l in e m a t e r i a l s . T h e i n i t i al r a t e o f s l id i n g h a sb e e n f o u n d t o b e l in e a r (s = 1 ) f o r b o t h c o p p e r [3 3]a nd t i n [ 34 ] . S t r u t t e t a l . [ 35 ] f o u n d a v a l u e o f s = 2 . 8f o r l e a d b i cr y s t a ls , a n d L a n g d o n [3 6] f o u n d o n e o f,~ 2 . 4 f o r a lu m i n i u m a n d m a g n e s i u m a l l o y . ' A v a l u e o f4 . 7 f o r a l o w - c a r b o n s t e e l a n d o f 3 . 8 f o r 3 1 6 s t a i n l e s ss t e e l w a s r e p o r t e d b y H o r t o n [ 37 ]. G a t e s [ 3 8 ] a l s of o u n d a v a l u e o f 3. 8 f o r 3 1 6 s ta i n l e s s s te e l . F o r i c e ,h o w e v e r , t h e r e a r e n o e x p e r i m e n t a l d a t a o n g r a i n -b o u n d a r y s l i di n g. T h e a s s u m p t i o n g i v e n i n E q u a t i o n4 a n d e x p e r i m e n t a l o b s e r v a t i o n s o n c r e e p [ 3 2] l e dS i n h a [ 1 0] t o c o n c l u d e t h a t s ~ 1 f o r ic e ( s e e T a b l e I ) .

S i n c e A n d r a d e ' s p r o p o s a l i n 1 9 1 0 , l it t le a t t e n t i o nh a s b e e n p a i d t o t r a n s i e n t c r e e p i n m e t a l s a n d a l l o y s .V e r y l it t le i s k n o w n a b o u t d e l a y e d e l a s t i c it y i n g e n e r al ,a l t h o u g h p h e n o m e n o l o g i c a l t i m e - h a r d e n i n g a n d S tr a in -h a r d e n i n g e q u a t i o n s h a v e b e e n p r o p o s e d [ 3 ]. S h e a r o rs l i d i n g i n t h e g r a i n - b o u n d a r y r e g i o n s d u r i n g l o a d i n g ,o r i g i n a l l y s u g g e s t e d b y Z e n e r [ 4 0 ] , w a s h y p o t h e s i z e db y S i n h a [ J0 ] a s t h e p r i m a r y m i c r o m e c h a n i s m t h a tg i v e s ri s e t o a d e l a y e d e l a s t ic e f f e c t i n a p o l y c r y s t a l l i n em a t e r i a l . T h i s i d e a , t o g e t h e r w i t h t h e a d d i t i o n a lh y p o t h e s i s d i s c u s s e d i n E q u a t i o n 4 a n d t h e a s s u m p -t i o n t h a t i n t r a g r a n u l a r d i s l o c a t io n c r e e p is i n d e p e n -d e n t o f g ra i n s i z e, le d t o m o d i f i c a t io n o f t h e p h e n o m -e n o l o g i c a l e q u a t i o n p r o p o s e d f o r tr a n s i e n t c r e e p [ 32 ]a n d t h e i n t r o d u c t i o n o f a h y b r i d s o l u t i o n t h a t i n t ro -duc e s a g r a i n s i z e e f f e c t [ 10 ] ,

c , d ~ ( E ) S { 1 - e x p [ - ( a r t ) b ] } ( 8)~d - dw h e r e t i s t i m e a n d a r = l / r ? ; o t h e r s y m b o l s h a v eb e e n d e f i n e d e a r l i e r .

E q u a t i o n s 8 a n d 4 g iv e t h e s t r e s s - t i m e - t e m p e r a t u r ed e p e n d e n c e o f 2 ,

8dd C l d l ( E ) S { 1 - e x p [ - - ( a r t ) b ] } (9 )2 = K - K

T A B L E | M a t e r i a l c o n s t a n t s f o r c r e e p a n d c r a c k s f o r p u r e ic eo b t a i n e d f r o m e a r l i e r c r e e p e x p e r i m e n t s a n d a n a l y s e sR e f . C o n s t a n t[ 3 2] E = 9 . 5 G N m 2

C l = 9 x 1 0 3 d l = 1 m ( c h o s e n u n i t )n = 3b = 0 . 3 4 ~ 1In

a r = l / z r = 2 . 5 x 1 0 4 s e c i ( T = 2 6 3 K )~v0 = 1 . 7 6 x 1 0 7 s e c - I ( T = 2 6 3 K , o 0 = I M N m 2 )

[101 X = 1s = l

[ 2 3 ] M I = 1 . 6 7 x 10 - 6m I = 4 . 5 5 x 1 0 9 K - I

[ 3 9 ] N c = 5 5 0 m - 2~ b = 1 . 3 3 x 1 0 7 m - t


5/14

E q u a t i o n s 8 a n d 9 c l a r if y th e s i g n if i c an c e o f s a n d bu s e d e a r l i e r , b u t t h e r e a p p e a r s t o b e n o a v a i l a b l ei n f o r m a t i o n o n t h e v a l u e s o f b f o r o t h e r m a t e r i a l s .S i n h a [ 1 0 ] f o u n d t h a t b , , ~ 1 / n f o r p o l y c r y s t a l l i n e i cew h e r e n i s t h e s t r e s s e x p o n e n t u s e d i n p o w e r - l a we x p r e s s i o n s f o r t h e v i s c o u s f l o w r a te , o f t e n i n c o r r e c t l yr e f e r r e d t o a s t h e s t e a d y - s t a t e o r m i n i m u m c r e e p r a t e .T h e e x p o n e n t n i s a s s o c i a t e d w i t h a n i n t r a g r a n u l a rd i s l o c a t i o n m e c h a n i s m w i t h o u t a n y i n f l u e n c e f r o mc r a c k s . W h e t h e r t h i s s i m p l i f i c a t i o n ( b , , ~ l / n ) c a n b ee x t e n d e d t o o t h e r m a t e r i a l s r e m a i n s t o b e s e e n .

For i c e , Equa t ion 7 s impl i f i e s t o,re = Z r { - - l n l l - z ( E ) I } " ( l O )

w h i c h p r o v i d e s a t f ~ - a r e l a t i o n v e r y s i m i l a r t o t h es t re s s d e p e n d e n c e o f r u p t u r e l if e , c o m m o n l y o b s e r v e d ,and f i t t ed by e ssen t i a l l y em pi r i ca l equ a t ion s [41 , 42 ]. I tis m i c r o m e c h a n i c a l ly ba s e d a n d h a s b e e n s h o w n t o b eb e t t e r [ 23 ] t h a n t h e w e l l - k n o w n Z h u r k o v [ 43 ] e q u a t i o nu s e d b y G o l d [ 1 2 , 4 4 ] f o r i c e a n d f o r d e s c r i b i n g t h ed e p e n d e n c e o f t e n s il e r u p t u r e t i m e o n s t r e s s f o r m e t a l sa n d a l l o y s [ 4 5 ]. I n a d d i t i o n , E q u a t i o n 1 0 d o e s n o ts u ff e r f r o m t h e i m p o s s i b il i ty t h a t Z h u r k o v ' s e q u a t i o npred i c t s , i . e . a f i n i t e rup tu re t ime even fo r van i sh ings m a l l s tr e s s. T h i s i n c o n g r u i t y e m p h a s i z e s t h e e m p i r i -c a l c h a r a c t e r is t i c s o f t h e Z h u r k o v e q u a t i o n , a l t h o u g he f f o r t s h a v e b e e n m a d e t o d e s c r i b e i t a s a k i n e t i ct h e o r y o f f r a c t u r e [ 46 ].3 .3 . M i n i m u m s t r es s f o r c r e e p c r a c k i n gS u b s t i t u t i o n o f tfo = o o i n E q u a t i o n 6 a n d r e a r r a n g e -m en t g ive s t he m in i m um s t r e ss , O 'min, fo r t he o nse t o fc r a c k i n g a c t i v it y , ( K 2 c ' ] '/ "

O 'm in ~ - E \ c - - ~ I J = E Z l / s (11)U s u a l l y E d e c r e a s e s w i t h i n c r e a s e i n t e m p e r a t u r e .E q u a t i o n 2 s h o w s t h a t 2~ a l s o d e c r ea s e s w i t h i n c r e a s ei n t e m p e r a t u r e . T h u s E q u a t i o n l 1 s h o w s t h a t O-mi i s ad e c r e a si n g f u n c t i o n o f t e m p e r a t u r e a n d p r o v i d e s r e s u lt s[ 2 3 ] t h a t a g r e e e x t r e m e l y w e l l w i t h t h e e x p e r i m e n t a lo b s e r v a t io n s o f G o l d [ 12 ] a n d Z a r e t s k y e t a l . [24].3.4. Creep damageW h e n t h e a p p l i e d s t re s s i s g r e a t e r t h a n ~ min a n d t h el o a d d u r a t i o n i s g r e a t e r t h a n tr o, t h e n w i t h c o n t i n u e dd e f o r m a t i o n m o r e a n d m o r e s it es o f s t re s s c o n c e n tr a -t i o n r e a c h a c r i ti c a l p o i n t a n d t h e n u m b e r o f c ra c k si n c r e a s e s . T h i s c o n t i n u e d d a m a g e m e c h a n i s m h a sb e e n i n v e s t i g a te d b y S i n h a [ 3 9] , w h o n o t e d t h a t G o l d ' sa v a i l a b l e e x p e r i m e n t a l o b s e r v a t i o n s [ 1 2 , 13 ] o n t h es t re s s a n d t i m e d e p e n d e n c e o f c r a c k d e n s i t y , N , i n ic ec a n b e e x p r e ss e d a s a f u n c t i o n o f g r a i n - b o u n d a r ys l i d i n g d i s p l a c e m e n t . T h e s i m p l e s t e q u a t i o n w o u l d b eo f t h e f o r m

N = Nc exp [t) (2 -- 2~)] (12)w h e r e 2 i s g i v e n b y E q u a t i o n 9 a n d 2 c b y E q u a t i o n 2 .Arc i s t he c rack den s i t y co r re sp ond ing t o 2~ on f i r s tc r a c k s , a n d ~ i s a c o n s t a n t . A b e t t e r p r e s e n t a t i o n o fE q u a t i o n 12 w o u l d b e

N = Arc {ex p tO (2 - X c)] - 1} (13)

T h i s a v o i d s t h e p r o b l e m i n h e r e n t i n E q u a t i o n 1 2, i .e .t h a t t h e c r a c k d e n s i t y , e v e n a t t h e b e g i n n i n g o f t h et e s t, h a s a p o s i t i v e , t h o u g h v e r y s m a l l , v a l u e . E q u a t i o n13 a l so show s tha t c rack s fo rm a t trc wh en 2 = 2c .

E q u a t i o n 1 3 , o n t h e o t h e r h a n d , p r e d i c t s t h a t t h ec r a c k d e n s i t y is " n e g a t i v e " f o r 2 < 2 c. T h i s a b s u r dc o n d i t i o n c a n b e a v o i d e d m a t h e m a t i c a l l y b y r e c o g n iz -i n g t h a t a n e g a t i v e v a l u e f o r N i s e q u i v a l e n t t o n oc racks .

A n e x p l ic i t f o r m o f N i n t e r m s o f d e l a y e d e l a s ti c i tyo r s t r e ss and t ime i s g iven by i nse r t i ng 2 f romE q u a t i o n 4 i n E q u a t i o n 1 3 , g i v i n g

- ' tW i t h E q u a t i o n 8 t h i s g i v e sN = N o ( e x p { f f l f - ~ - Z ( E ) , {1 - exp t - - (ar t )b]}

4 . C r e e p w i t h o u t d a m a g eF o r stre sses less th an 0"rain, th at is a < O'min, he thr eet e r m s i n E q u a t i o n 1 f o r c o n s t a n t - s t re s s l o a d i n g c a n b edesc r ibed a s [10 ]

= 7 ~ + - y -4- ~vot ~ (16)

w h e r e t h e s e c o n d t e r m ( ~d ) i s d e s c r i b e d b y E q u a t i o n 8a n d t h e t h i r d t e r m (S v) r e p r e s e n ts t h e u s u a l p o w e r - l a wvi scous c reep ; t hu s ~v0 i s t he v i scou s s t r a in r a t e co r re -s p o n d i n g t o t h e c h o s e n u n i t o f s tr es s , a0 ( = 1 M N m - 2) .T h e r e c o v e r y c u r v e i s g i v e n b y t h e m i r r o r i m a g e o f a a.5 . C r e e p w i t h d a m a g eF o r cr < O'mi a n d t > 0 or fo r c~ > O'mi a n d t < trc,E q u a t i o n 1 6 i s a p p l i c a b le , b u t f o r l o a d i n g c o n d i t i o n s ,o- > a~in an d t > t r c , t h e d e f o r m a t i o n b e h a v i o u r i sc o n t i n u a l l y m o d i f i e d b y t h e i n c r e a si n g n u m b e r o fm i c r o c r a c k s t h a t d e v e l o p d u r i n g l o a d i n g . T h e s e a f f e c te l a st i c a s w e l l a s o t h e r c o m p o n e n t s o f s t r a in .5 .1 . E l a s t i c c r e e pB e c a u s e o f i n t e r n a l s t re s s c o n c e n t r a t i o n s , t h e n u c l e a -t i o n a n d g r o w t h o f c r a c k s o r c a v it ie s in d u c e a d d i t i o n a le l a s t ic s t ra in , l e ad ing t o e l a s ti c creep o r t ime -d epen den te l as ti c m o d u l u s . V e n k a t e s w a r a n a n d H a s s e l m a n [ 47 ]c o n c l u d e d , h o w e v e r , t h a t t h e t o t a l s t r a i n o f e l a st i cc r e e p b y c r a c k g r o w t h i s o f th e o r d e r o f a s m a l lmu l t i p l e (2 t o 3 ) o f the i n i t i a l e l a s ti c s t r a in t o w h ich t hem a t e r i a l i s s u b j e c t ed d u r i n g i n i ti a l lo a d i n g . T h i s p o i n ti s s u p p o r t e d b y d a t a c o m p i l e d b y M i l l er a n d L a n g d o n[ 48 ], s u g g e s ti n g t h a t f o r m a n y m e t a l s t h e t o t a l c a v i t yv o l u m e f r a c t i o n r a r e l y e x c e e d s 1 % . A l t h o u g h e l a s t i cc reep con t r i bu t e s l i t t l e t o t he t o t a l s t r a in , t h i s mechan-i s m s h o u l d b e g i v e n c o n s i d e r a t i o n i f t h e c a v i t y v o l u m ef rac t i on i s l a rge o r i f e la s t i c s t r a in con t r i bu t e s s i gn if i -can t ly t o t o t a l s t r a in .

441 9


6/14

C r a c k - e n h a n c e d e l a s t i c c r e e p c a n b e e s t i m a t e dr e a d i l y e n o u g h f r o m e x p e r i m e n t a l r e c o v e r y c u r v e s( F i g . 1 ). T h e i n s t a n t a n e o u s l y r e c o v e r e d s t r a i n o f4 .5 x 10 4 , i n t h i s c a se co r re spo nd in g t o a s tr e ss o f3 . 5 M N m - 2 a t t h e t i m e o f u n l o a d i n g , p r o v i d e d a ne f fe c ti v e Y o u n g ' s m o d u l u s v a l u e o f 7 . 8 M N m - z , T h i si s a b o u t 1 8 % l o w e r t h a n t h e v a l u e f o r Y o u n g ' sm o d u l u s , E , o f 9 . 5 M N m -2 d e t e r m i ne d f o r a nu n c r a c k e d s p e c i m e n [ 32 ] a n d g i v e n in T a b l e I . E l a s ti cc reep s t r a in i n excess o f t he i n i t i a l e l a s t i c s t r a int h e r ef o r e c o n t r i b u t e d o n l y a b o u t 4 % t o t h e t o t a ls t r a i n i n s p it e o f t h e e x t e n s iv e d a m a g e s h o w n i n F i g . 2 .U n l e s s t h e d e f o r m a t i o n i s e x t e n d e d t o l a r g e s t r a i n s o rs t r a in r a t e s , i t wo u ld b e s imp le r t o neg l ec t t he e f fec t o fe l a s t i c c reep . The same app l i e s t o de l ayed e l a s t i cs t r a in . Th i s po in t wi l l be d i scussed l a t e r .

d i a m e t e r , d . A s s u m i n g t h e a r e a o f a h e x a g o n o f s i d e s2 a t o b e e q u a l t o t h e a r e a o f a c i rc l e o f d i a m e t e r d , itc a n b e s h o w n t h a t

a 2 _ _ 7 r d2 (19)24 x 3 1 /2o r t h a t c r a c k s i z es a r e a b o u t h a l f t h e g r a i n d i a m e t e r s( i .e . 2a ~ 0 .55d) .

S u b s t i t u t i o n o f a 2 f r o m E q u a t i o n 1 9 i n E q u a t i o n 1 8gives

= i0 ~v0 1 -~- Nd 2Fl 1/2 a t (20)~v cV i s c o u s s t r a i n i n v o l v i n g c r a c k i n g t h e r e f o r e d e p e n d sn o t o n l y o n N , w h i c h d e p e n d s o n a a n d t a t a c o n s t a n tt e m p e r a t u r e , b u t a l s o o n g r a i n s i z e .

5 .2 . C r a c k - e n h a n c e d v i s c o u s c r ee pC o n s i d e r i n g a c r a c k a s a n a r r a y o f d is l o c a t io n s ,W e e r t m a n [ 49 ] d e r i v e d t h e e f f e c t o f c r a c k s o n v i s c o u sc r e e p r a te . F o r d i l u t e c o n c e n t r a t i o n s o f n o n - i n t e r a c t -i n g l o n g n a r r o w c r a c k s o r i e n t e d w i t h t h e i r m a j o rp l a n e s p e r p e n d i c u l a r t o a u n i a x i a l l y a p p l i e d t e n s i l el o a d , a n d f o r m a t e r i a l s o b e y i n g a p o w e r - l a w v i s c o u sc r e e p r a t e s u c h a s t h e t h i r d t e r m i n E q u a t i o n 1 6 , t h eenh anc ed c reep ra t e , Svc , i s g iven by

ivc = Sv (1 + 2 7 z Na Z n1/2) (17)w h e r e s v i s th e c r e e p r a t e f o r m a t e r i a l w i t h o u t c r a c k sa n d N i s t h e n u m b e r o f cr a c k s p e r u n i t a r e a . A l l t h ec r a c k s a r e o f u n i f o r m s i ze , a n d 2 a i s t h e c r a c k w i d t h .H e r e N i n d i c a te s t h e n u m b e r o f c r a c k s t h a t c r o s s au n i t c r o s s - s e c ti o n a l a r e a o r i e n t e d p e r p e n d i c u l a r t o t h ep l a n e o f t h e cr a c k s . T h e c o n d i t i o n f o r n o n - i n t e r a c t i o no f c r a c k s i s d e f i n e d b y a 2N ,~ 1.

A d o p t i n g E q u a t i o n 1 7, t h e n o n - l i n e a r v is c o u s s t r a i nu n d e r c o n s t a n t - s t r e s s c r e e p c o n d i t i o n s , i n v o l v i n gc rack ing , i s g iven by

= f0e vca t = f0iv0 (1 + 2 r c Na 2 nI /2) d t~ v ( 1 8 )S i n c e N = N ( a , t ) a t c o n s t a n t t e m p e r a t u r e , i t i s

i m p o s s i b l e t o u s e E q u a t i o n 1 8 w i t h o u t k n o w l e d g e o ft h e d e p e n d e n c e o f N o n s t r e s s a n d t i m e . T h i s e x p l a in s ,p e r h a p s, w h y W e e r t m a n ' s t h e o r e ti c a l d e v e l o p m e n th a s n e v e r , t o t h e a u t h o r ' s k n o w l e d g e , b e e n u s e d i n t h eq u a n t i t a t i v e a n a l y s i s o f c r e e p p r o b l e m s i n v o l v i n gc r a c k s , a l t h o u g h i t h a s b e e n d i s c u s s e d b y H a s s e l m a na n d V e n k a t e s w a r a n [ 5 0 ] f o r q u a l i t a t i v e d e s c r ip t i o n so f t h e e f f e c ts o f c r a c k s o n c r e e p i n p o l y c r y s t a l l i n ece ramics .E q u a t i o n 1 8, w i t h i ts u n d e r l y i n g t h e o r e t i ca l a s s u m p -t i o n s a n d l i m i t a t io n s , i s p a r t i c u l a r l y a p p l i c a b le t o t h et y p e o f d a m a g e t h a t o c c u r s i n u n i d i r e c t i o n a l l y s o l id i -f i e d c o l u m n a r - g r a i n e d s o l i d s s u c h a s t h o s e d i s c u s s e dea r l i e r (F igs 1 and 2 ) . I t c an r ead i ly be app l i ed becau seN i s d e s c r i b e d b y E q u a t i o n 1 5 . S o m e a s s u m p t i o n sm u s t o f c o u r s e b e m a d e a s t o t h e s iz e, 2 a , o f t h e c r a c k s .

I f th e c r a c k s a r e a s s u m e d t o b e t h e s a m e s i z e a st h o s e o f t h e g r a i n f a c e ts , a s s u g g e s te d b y F i g . 3 b a n dF i g . 4 , a n d i f t h e c r o s s - s e c ti o n a l g e o m e t r y n o r m a l t ot h e l e n g t h s o f th e g r a i n s i s a s s u m e d t o b e h e x a g o n a l ,t h e n 2 a c a n b e e x p r e s s e d i n t e r m s o f t h e g r a i n

5 . 3 . C o n s t i t u t i v e e q u a t i o nR e p l a c in g t h e th i r d t e r m i n E q u a t i o n 1 6 b y E q u a t i o n20 and neg l ec t i ng e l a s ti c creep , t he rheo log i ca l equa t io ni s ob t a ined :

,S = ge + ~d + ~v

= E+-d-a cl d I {1 -- exp [ - - ( a r t ) b ] }

+ ;o ~vo I + N d2 n 1/2 d t(21 )

w h e r e N i s f r o m t h e s e c o n d ( d e l a y e d e l a s t i c ) t e r mt h r o u g h E q u a t i o n s 1 4 o r 1 5 a n d a 2 N ,~ 1 or n d Z N /24 x 3 I/2 ~ 1.

E q u a t i o n 2 1 r e d u c e s t o E q u a t i o n 1 6 f o r a n y s t r e s san d t < trc or for an y t ime an d o- < ami .. In the f i rs tc a s e, t h e c r e e p p e r i o d w o u l d b e i n t h e t r a n s i e n t r a n g e( o f t e n c a l le d p r i m a r y c r e e p ) a n d t h e s e c o n d t e r m c o u l dd o m i n a t e t h e d e f o r m a t i o n p r o c e s s , re s u l t i n g i n a p r o -no un ced g ra in s i ze e f fec t. In t he s econd ca se , t he g ra in -s iz e d e p e n d e n t c r e e p c u r v e w o u l d e v e n t u a l l y l e ad t o ac o n s t a n t s t r a i n r a t e ( o f t e n c a l l e d a s t e a d y s t a t e ) t h a td o e s n o t d e p e n d o n g r a i n s iz e. F o r c o n d i t i o n s a >amen, i nvo lv ing c racks , t he c reep cu rve w ou ld passt h r o u g h a m i n i m u m c r e e p r a t e t o a n a c c e le r a t in g c re e pr a t e k n o w n a s t h e t e r t i a r y s t a t e . T h u s E q u a t i o n 2 1d e s c r ib e s t h e c o m p l e t e c r ee p c u r v e s n o r m a l l y o b s e r v e d( f u r t h e r i m p l i c a t i o n s w i l l b e d i s c u s s e d i n a n o t h e rp a p e r ) . I t s a p p l i c a t i o n i n p r e d i c t i n g s t r e s s - s t r a in , a n dh e n c e s t r e n g t h , r e s p o n s e f o r c o n s t a n t s t r a i n - r a t e t e s tswi l l be deve loped because ( i ) i t i s t heore t i ca l l y t hesimplest case , ( i i ) i t i s chal lenging, and ( i i i ) exper i -m e n t a l d a t a h a v e b e c o m e a v a i la b l e t h r o u g h t h e u s e o ft h e n e w g e n e r a t i o n o f c l o s e d - l o o p s t r a i n - c o n t r o l l e dt e s t m a c h i n e s .6 . C o n s t a n t s t r a in r a t eA t h e o r y f o r p r e d i c t i n g s t r a i n r e s p o n s e a n d h e n c es t r e s s - s t r a i n d i a g r a m s f o r n o n - l i n e a r m a t e r i a l s a tv a r i o u s t e m p e r a tu r e s c o r r e s p o n d i n g t o m o n o t o n i c a l l yi n c r e a s in g s t r e s s h i s to r i e s h a s a l r e a d y b e e n d e v e l o p e d[ 5 1 ] . I t i s b a s e d o n E q u a t i o n 1 6 a n d c o n s e q u e n t l y i sa p p l i c a b l e s t r i c t l y f o r c o n d i t i o n s i n w h i c h n o c r a c k -i n g a c ti v i t y i s i n v o l v e d . A l t h o u g h g o o d a g r e e m e n tb e t w e e n t h e o r y a n d e x p e r i m e n t w a s o b t a i n e d e v e n f o r

4420


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c o n d i t i o n s o f a f a i r d e g r e e o f c r a c k i n g , t h e t h e o r ys h o u l d n o t b e e x t r a p o l a t e d f a r b e y o n d i t s r a n g e o fapp l i ca t i on . The ana lys i s p rov ided , how eve r , a ba s is fo rapp ly ing a cons t an t - s t r e ss c reep equ a t ion t o a va r i ab l e -s t re s s l o a d i n g c o n d i t i o n . I t al s o p r o v i d e d s u p p o r t f o rt h e t w o b a s i c a s s u m p t i o n s u s e d a t c o n s t a n t t e m p e r a -tu re : ( i) t ha t t he de l ayed-e l a s t i c s t r a in r a t e dep ends ong r a i n s i z e a n d o n t h e e n t i r e l o a d i n g h i s t o r y , w h e r e a s( ii ) t h a t t h e v i s c o u s s t r a in r a t e d e p e n d s o n l y o n t h ec u r r e n t s t r e s s le ve l. I t m a y b e s e e n f r o m E q u a t i o n 1 6t h a t a t c o n s t a n t s t r a i n r a t e t h e s t r e s s w o u l d i n c r e a s ea n d a s y m p t o t i c a l l y a p p r o a c h a v a l u e d e t e r m i n e d b yt h e v i s c o u s t e r m . W h i l e t h i s p r e d i c t i o n a g r e e s w i t he x p e r i m e n t a l r e s u l t s u n d e r l o w s t r a i n r a t e s , u p p e ry i e ld - typ e f a i lu re occurs a t l a rge r r a t e s , i .e . i n i c e a tg rea t e r t ha n abou t 1 10 -7sec -1 [1] .

F o r u n i a x i a l c o n s t a n t s t r a i n r a t e s ( ~ ), e q u a l i n t e r v a l so f t ime (At ) r e su l t i n equa l s t r a in s t eps

A s = ~A t ( 2 2 )so t ha t t he t o t a l s t r a in e a f t e r S in t e rva l s o f t ime ,w o u l d b e

e = S ~ A t (23)C a l c u l a t i o n s f o r a c o n s t a n t s t r a i n r a t e a r e r e d u c e d t ot h e q u e s t i o n o f f i n d i n g s t re s s s t e p s , A a s , c o r r e s p o n d -ing t o each i n t e rva l such t ha t Equa t ion 22 i s sa t i s f i eda t a l l t imes .

The s t r e ss s tep ACrl app l i ed a t t = 0 wi ll p ro du ce a ni n s t a n t a n e o u s e l a s ti c s t ra i n o f A a n / E a n d c o n t r i b u -t i o n s f r o m d e l a y e d e l a s t i c a n d v i s c o u s f l o w w i l l b eneg l ig ib l e . At t he end o f t he f i r s t i n t e rva l o f t ime ,t = At , Eq ua t ion 21 g ives

A ~ ) s= tA t - E + A(A~r {1 - exp [ - - ( a rA t )b ] }

+ g v 0 ( A 6 - - ~ l " ( 1 + BNI) At ( 2 4 )k a 0 /w h e r e A = C l d l / ( d E ~ ) , B = r c Z d 2 n l / 2 / ( 1 2 3 I/2) an dN~ i s t he c rack dens i t y a t t = 0 . F or p rev io us lyu n d e f o r m e d m a t e r i a l N~ = 0 . F o r a g i v e n ~ a n d A t ,A a ~ c a n b e d e t e r m i n e d b y a n i t e r a t i o n m e t h o d .

A t t h e e n d o f t h e s e c o n d i n t e r v a l, t = 2 A t , t h e t o t a lst ress is cr = Aa~ + A~2, wh ere Aa2 is the new st ressa p p l i e d a t t h e b e g i n n i n g o f th i s i n te r v a l . T h e t o t a ls t r a in , fo l l ow ing t he p r inc ip l e de sc r ibed by S inh a {51],i s g iven bye = 2~At

Acq + A o " [- E + A L(Acr')S { 1 -- exp [-- ( a r 2 A t ) e ] }+ (Aa2) ~ {1 - exp [- - (awA t)b]}l

J

L \ o -j /

w h e r e N 2 i s t h e c r a c k d e n s i t y a t t h e b e g i n n i n g o f t h es e c o n d i n t e r v a l a n d i s d e t e r m i n e d b y t h e m a g n i t u d e o fthe de l ay ed e l a s t ic s t r a in a t t ha t t ime , i. e . a t t he end o f

the p rev iou s pe r iod . I t is ob t a in ed by rep l ac ing e~ i nE q u a t i o n 1 4 b y t h e s e c o n d o r t h e r e q u i r e d d e l a y e de l a s t i c s t r a i n t e r m o f E q u a t i o n 2 4 :

-~ - ( a c t , ) '

{ l - - e x p [ - - ( a rA t )h ] } - 2 c ) ] - - I t(25b)I t s h o u l d b e n o t e d i n E q u a t i o n 2 5 a t h a t t h e t o t a l

de l ay ed e l a s t ic s t r a in i s g iven by t he su m of t hea m o u n t s p r o d u c e d b y A a ~ a p p l ie d f o r 2 A t a n d A a 2a p p l i e d f o r A t , t h u s i n c o r p o r a t i n g t h e m e m o r y e f f e c tp r o p o s e d b y S i n h a [ 5 1 ]. T h e t o t a l v i s c o u s s tr a i n , o nt h e o t h e r h a n d , i s g i v en b y t h e s u m o f th e a m o u n tp r o d u c e d b y A a , a t t h e e n d o f t h e f i rs t p e r i o d , a t a r a t et h a t d e p e n d s o n A a j , a n d t h e a m o u n t p r o d u c e d b yAo-j + Aa2 dur ing t he ne x t pe r io d ( a t a ne w ra t ed e p e n d i n g o n t h e n e w t o t a l s t re s s ). S i n c e A a~ is k n o w nf rom the f i rs t i t e ra t i on , c a l cu l a t i on s s impl i fy to t hes o l u t i o n f o r A a 2.A s t h e n u m b e r o f s t e ps i n c r ea s e s a n d t h e s t r a in -s o f t e n i n g d u e t o c r a c k - e n h a n c e d c r e e p in c r e a se s , a ni n t e r v a l i s r e a c h e d b e y o n d w h i c h t h e t o t a l s t r a in , w i t h -o u t a n y f u r t h e r c h a n g e i n s t r e s s , w i l l b e g r e a t e r t h a nt h a t i m p o s e d b y t h e s t r a i n r a t e . S u p p o s e t h i s l i m i t i n gs t r e ss i s r eache d a t t he (L - 1 ) th i n t e rva l . The s t r e ssl ev e l m u s t b e r e d u c e d b y Ao- L a t t h e b e g i n n i n g o f t h eL t h p e r i o d , s o t h a t t h e c o m p u t e d s t r a i n a t t h e e n d o ft h i s p e r i o d ( g i v i n g d u e c o n s i d e r a t i o n t o t h e s t r a i nr e c o v e r y p r o c e s s i n t h e d e l a y e d e l a s t ic i t y ) e q u a l s t h a tr e q u i r e d b y E q u a t i o n 2 3 . T h i s i s r e p r e s e n t e d b ys = L ~ A t

A o " + A O " -'F . . . . A ~LE

+ A ((Aa , ) s { 1 - exp [ - ( a r L A t ) b ] }+ ( A ~2 )' {1 - e x p [ - ( a r ( L - 1 ) A t )q }+ . . . . ( A a L ) ' e x p [ - - ( a r A t ) q )

L \ a , /( A ~ + A o 2 ) "+ (1 + B N 2 ) + . . .o -

(26)w h e r e N L is o b t a i n e d f r o m t h e t o t a l d e l a y e d e l a s ti cs t r a in a t t h e e n d o f t h e ( L - 1 ) th p e r i o d a n d E q u a t i o n1 4. W i t h t h e i n c r e as e i n l o a d i n g t i m e , m o r e a n d m o r er e c o v e r y te r m s w i l l o f c o u r s e b e a d d e d i n E q u a t i o n 2 6f o r t h e s u b s e q u e n t r e d u c t i o n s i n s t r e s s .

I n t h i s e q u a t i o n , t h e r e d u c t i o n i n e l a st i c s t r a i n a n dt h e r e d u c t i o n i n v i s c o u s s t r a i n r a t e d u e t o r e d u c e dt o t a l s t re s s a r e s t r a i g h t f o r w a r d . T h e p r i n c i p l e o f m i r -r o r i m a g e f o r r e c o v e r y is f o l l o w e d h e r e in f o r m u l a t i n gt h e r e d u c t i o n i n t h e d e l a y e d e l a s t ic s t r a in . T h i s m e c h -a n i s m i s s t r i c t l y v a l i d f o r p r e v i o u s l y u n d e f o r m e d o r

4 4 2 1


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bL /3L~ Jr ~I - -

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~ , ~ 3 x 1 0 - 6~ - -~1 .7x 10 -6

\ 1 x 1 0 - 6~ x l 0 - 7 s e e - 1

~ - - 2 x 1 0 - 5

l x l 0 - 5~ 5 x 1 0 - 6

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16

i c e [ 8 , 5 2 ] , a s m a y b e s e e n i n d a t a c o m p i l e d b y G i t t u s( [5 3 , T a b l e 4 . 7 A ]) . T h e v a l u e o f E , d e t e r m i n e d e x p e r i -m e n t a l l y [ 1 0 ] , a l s o a g r e e s w e l l w i t h t h a t c a l c u l a t e df r o m s i n g l e - c r y s ta l e l a s t i c c o n s t a n t s [5 4 ].

S t r e s s - s t ra i n d i a g r a m s f o r p u r e c o l u m n a r - g r a i n e dS- 2 ice a t 0 .96Tin ( - 10~ C ) a r e r e p r o d u c e d i n F i g . 5[ 2 ]. T h e o r e t i c a l p r e d i c t i o n s f o r t h e s e t e s t c o n d i t i o n sa r e p r e s e n t e d i n F i g . 6 . A g r e e m e n t , q u a l i t a t i v e a s w e l la s q u a n t i t a t i v e , b e t w e e n t h e o r y a n d e x p e r i m e n t o n t h es t r e s s - s t r a i n ( a n d h e n c e t i m e ) d i a g r a m m a y b e c o n -s i d e re d t o b e e x c e l le n t , a n d t h e r e f o r e e n c o u r a g e se x a m i n a t i o n o f t h e d et a il s . A s t h e m a x i m u m s t re s s o rs t r e n g t h i s o f g e n e r a l i n t e r e s t , t h i s a s p e c t w i l l b e d i s -

cussed f i r s t . F ig . 7 i l lus t r a tes the s t r a in- r a te de pen -d e n c e o f u p p e r y i e l d f a i l u r e s t r es s , o - r, a n d f a i l u r es t r a in , e f . I t a l so inc lud es new tes t r esu l t s ( F ig . 1 ). Th et h e o r y s e e m s t o j u s t i f y , e v e n i n n u m e r i c a l t e r m s , t h ee m p i r i c a l l y o b t a i n e d p o w e r l a w b e t w e e n ~rf a n dp r o p o s e d e a r l i e r [ 2 ]

err = P (27 )if 0

w h e r e P ( = 2 1 2 ) a n d p ( = 0 . 3 45 ) a re c o n s t a n t s ; to i s t h eu n i t s t r a i n r a t e ( = 1 s e c - ~ ) .

B o t h t h e o r y a n d e x p e r i m e n t s h o w t h a t f a i l u r es t r a i n s a r e s m a l l ( < 0 . 1 5 % ) a n d t h a t t h e s t r a i n - r a t e

c -JiEz 4

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F i g u r e 8 D e p e n d e n c e o f u p p e r -y i e l d f a i l u r e t i m e o n s t r a i n r a t e .T e m p e r a t u r e - 1 0 ~ S - 2 i ce ,d = 4 . 5 m m + 0 . 5 r a m . E x p e r i -m e n t a l p o i n t s ( O ) f r o m S i n h a [ 2] ,( e ) n e w t e s ts . ( ) T h e o r y ,( - - - ) t f = 7 .3 x 10-34 -0 .82 [2 ].

sens i t iv i ty of s f , g iven b y the ex po ne nt o f k , i s sign i fi -c a n t l y l e s s t h a n t h a t o f o -r. T h e t h e o r y s h o w s , h o w e v e r ,t h a t t h e e m p i r i c a l p o w e r l a w b e t w e e n e r a n d ~ [2 ]s h o u l d n o t b e a p p l i e d t o l o w e r r a te s . T h i s p o w e r l a ww a s d e r i v e d f r o m t h e e x p e r i m e n t a l l y o b s e r v e d d e p e n -d e n c e o f f a i l u r e t i m e t r o n ~ ,

, !to Q (28)wh ere t o = 1 sec, 4o = I sec i . S in ce er = t r Y ~ t o ' o , tg ives

( ~ l - q (29)E f = Q \ ' S 0 J

E x p e r i m e n t a l v a l u e s f o r Q a n d q w e r e , r e s p e c t i v e l y ,7.3 10 -3 an d 0.82 (Fig. 8) .

E q u a t i o n s 2 7 a n d 2 8 g i v et f Q p q / p ( f f f ~ - q / P

- = - - ( 3 0 )tl \ al /

T h i s e q u a t i o n b e a r s a r e m a r k a b l e s i m i l a r i t y t o t h ed e p e n d e n c e o f te n s il e c r e e p r u p t u r e t i m e o n s t re s s f o rm e t a l s a n d a l l o y s a t h i g h t e m p e r a t u r e s [ 4 , 5 , 7 ] . T h el i m i t a t i o n s o f t h e e m p i r i c a l l y fi t t e d E q u a t i o n s 2 8 a n d3 0 c a n b e s e e n in F i g s 8 a n d 9 ; c l e a r ly t h e y c a n n o t b ea p p l i e d t o l o w e r r a t e s .

T h e t h e o r y i n d i c a t e s a n i n c r e a s i n g l y l o n g e r f a i l u r et i m e , a s t h e s t r a i n r a t e d e c r e a s e s , t h a n t h a t g i v e n b yt h e e m p i r i c a l r e l a t i o n . A s a c o n s e q u e n c e , a r e v e r s a l o ft h e e r - ~ r e l a t i o n i s p r e d i c t e d i n F i g . 7 ( i n s t e a d o f t h ed e c r e a s i n g e r g i v e n b y E q u a t i o n 2 9 ) . T h i s c h a n g e i n t h ed u c t i l i t y o f th e m a t e r i a l i s, in f a c t , u n d e r s t a n d a b l e .M i x e d - m o d e f a i l u re , in v o l v i n g w e d g e c r a c k i n g , s h o u l de v e n t u a l l y (a t s o m e l o w r a te s ) g o t h r o u g h a t r a n s i t i o nw h e r e t h e g r o w t h o f r o u n d e d c a v i ti e s w o u l d p l a y a ni m p o r t a n t r o l e [ 5 5 ] , u n t i l a t s o m e e v e n l o w e r r a t e i tw o u l d f l o w w i t h o u t a n y c r a c k i n g o r v o i d f o r m a t i o n .I n t h i s c a se E q u a t i o n 1 6 w i ll a p p l y a n d t h e s t r e ss w i lla s y m p t o t i c a l l y r e a c h a v a l u e t h a t d e p e n d s o n t h es t r a i n r a t e ( i n r e a l i ty , d i f f u s i o n a l fl o w w o u l d c o m -p l i c a t e t h e f l o w s t r e s s a n d i t s d e p e n d e n c e o n s t r a i n

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F i g u r e 9 D e p e n d e n c e o f f a i l u r et i m e o n s t r e s s . T e m p e r a t u r e- 1 0 ~ S - 2 i ce , d = 4 . 5 m m 4 -0 . 5 r a m . E x p e r i m e n t a l p o i n t s ( o )f r o m S i n h a [ 2 ] , ( e ) n e w t e s t s .( ) T h e o r y , ( - - ) t r = 2 .4 103 ar 237 [2].


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r a t e ) . C o m p r e s s i o n f a i l u r e , a t t h e s e l o w r a t e s , i s i l ld e f i n e d a n d r e q u i r e s a n e w , m o r e p r e c i s e d e f i n i t i o n ,p e r h a p s l i m i te d b y a l l o w a b l e s t r a in . T h i s s i t u a t i o n h a sa c t u a l l y b e e n o b s e r v e d a n d w a s d e s c r i b e d a s " v i s c o u sy i e l d " b y S i n h a [ 1 ] w h i l e a n a l y s i n g c o n s t a n t c r o s s -h e a d - r a t e s t r e n g t h t e st s. A n a n a l o g o u s s i t u a t i o n a r i s esi n c o n s t a n t - s t r e s s l o a d i n g a s t h e s t r e s s d e c r e a s e s a n dt h e t r - a r e l a t i o n i s g i v e n a d i f f e r e n t s l o p e f o r l o w e rs t r e s s e s t h a n t h a t f o r h i g h e r s t r e s s e s [ 7 ] .

A b e t t e r a p p r e c i a t i o n o f t h i s as p e c t o f th e d e f o r m a -t i o n p r o c e s s i s i n d i c a t e d i n F i g . 1 0 , w h i c h s h o w s t h es t r a i n -r a t e d e p e n d e n c e o f c r a c k d e n s i t y a t f a i lu r e N r,o b t a i n e d f r o m d e t a i l e d c a l c u la t i o n s . E s t i m a t i o n o f t h em i n i m u m s t r a i n r a t e f o r t h e t r a n s i t i o n f r o m p u r ev i s c o u s f l o w t o f a i l u r e w i t h c r a c k i n g c a n b e r e a d i l yo b t a i n e d w i t h o u t r i g o r o u s c a l c u l a t i o n s . E q u a t i o n 1 1i n c o n j u n c t i o n w i t h E q u a t i o n 2 a n d T a b l e I g i v e sO'mi ~ - - - 0 . 5 M N m - 2 a t - 1 0 ~ T h e t h i r d t e r m i nE q u a t i o n 1 6 g i v e s t h e v i s c o u s s t r a i n r a t e a t t h i s s t r e s sas 2.2 x 10 8 s e c - L T h u s , t h e m i n i m u m d u c t i l it y in /3 fa t a b o u t 5 x 1 0 - T s e c - 1 i n F i g . 7 d o e s n o t i n d i c a t ea n y t r a n s i t i o n i n m i c r o m e c h a n i c s , m e r e l y i n d i c a t in gt h e s t r a i n r a t e b e l o w w h i c h v i s c o u s s t r a i n d o m i n a t e st h e t o t a l d e f o r m a t i o n . Q u a n t i t a t i v e c l a r i f i c a t i o n o ft h i s s t a t e m e n t c a n b e s e e n i n F i g . 1 1, w h i c h i l l u s t r a t e s

t h e s t r a i n - r a t e d e p e n d e n c e o f e l a s t i c st r a i n , ~ ef , d e l a y e de las t ic s t r a in , sdr, an d v i sco us s t r a in , evf, a t u pp er y ie ld .I t s h o w s h o w v i s c o u s f l o w d o m i n a t e s t h e d e f o r m a t i o na t l o w e r s t r a i n r a t e s ( < 1 x 1 0 - 6 s e c - l ) a n d e l a s t i cs t ra i n d o m i n a t e s it a t h i g h e r r a t e s ( > 1 x 1 0 - 5 s e c - l ) .T h e r e i s a t e n d e n c y f o r v i s c o u s f l o w t o i n c r e a se a g a i na t s t i l l h i g h e r r a t e s ( > 1 x 1 0 0 4 ) , d u e t o t h e c r a c ke n h a n c e m e n t , b u t t h e o c c u r r e n c e o f p r e m a t u r e s p li t-t i n g - ty p e f r a c t u r e s l i m it s e x t e n s i o n o f t h e t h e o r y t oh i g h e r s t r a i n r a t e s . T h e t i m e - d e p e n d e n t r e c o v e r a b l es t r a i n a t f a i l u re , ed f h o w e v e r , i n c r ea s e s m o n o t o n i c a l l yw i t h s t r a in r a t e . T h e a m o u n t i s s m a l l b u t m e a s u r a b l ei n c o m p a r i s o n w i t h o t h e r c o m p o n e n t s a t h i g h e r ra t e s.A t l o w e r r at e s i ts c o n t r i b u t i o n t o t o t a l s t r a i n c o u l d b el a r g e r t h a n t h a t o f t h e e la s ti c c o m p o n e n t . E x t e n s i o n t oh i g h e r r a t e s , n o t d i s c u s s e d h e r e , m a y a l s o b e l i m i t e d .8 . D i s c u s s i o nT h e r a t e - c o n t r o l l i n g e f f e c t s o f t h e m i c r o m e c h a n i s m sc o n s i d e r e d c a n b e s e e n i n F i g . 1 2, w h i c h i l l u s t r a t e s t h eh i s t o r y o f t h e t h r e e s t r a in c o m p o n e n t s a s w e l l a s t h ec r a c k i n g a c t i v i t y . A l t h o u g h e l a s t i c s t r a i n m a y d o m i -n a t e t h e t o t a l d e f o r m a t i o n b e y o n d 1 x 1 0 - S s ec -1( F i g . 11 ), i ts r a t e o f c h a n g e i s z e r o a t u p p e r y i e l db e c a u s e ~ = 0 a t c r = a t . C o n s e q u e n t l y , t h e t o t a l

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s t r a i n ra t e a t t h i s p e a k s t r e s s m u s t e q u a l t h e s u m o f t h ev i s c o u s s t r a i n r a t e a n d d e l a y e d e l a s t i c s t r a i n r a t e . A st h e l a t t e r q u a n t i t y i s s m a l l , t h e r a t e - c o n t r o l l i n g m e c h -a n i s m a t f a i l u r e a n d d u r i n g t h e p o s t - f a i l u r e r e g i m e i sd o m i n a t e d b y t h e v i s c o u s s t r a in r a te .

N o t e t h e a s y m p t o t i c a p p r o a c h o f t h e v i s c o u s s t ra i nr a t e s t o v a l u e s c l o s e t o t h e i m p o s e d r a t e s i n F i g . 1 2 .C l e a r ly , i t i s t h e r e a s o n f o r s t r o n g , a l m o s t o n e - t o - o n e

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n u m e r i c a l c o r r e s p o n d e n c e ( i . e . l i p ~ n a n d p - U p , , ~i v0 ) e s t a b l is h e d b e t w e e n t h e e m p i r i c a ll y o b t a i n e ds t ra i n -r a t e d e p e n d e n c e o f s t re n g t h ( E q u a t i o n 2 7 ) a n dt h e p o w e r - l a w d e p e n d e n c e o f v i s c o u s f lo w r a t e o ns t re s s ( t h i r d te r m i n E q u a t i o n 1 6 ) f o r c o n d i t i o n s i nw h i c h t h e c r a c k d e n s i t y i s s t il l r e a s o n a b l y l o w [ 2] . S u c ha n u m e r i c a l c o rr e s p o n d e n c e , h o w e v e r , c a n n o t b e e s t a b -l i s h e d i f c o n v e n t i o n a l t e s t s y s t e m s w i t h l o w s t i ff n e s s e s


13/14

a r e u s e d . I n t h i s c a s e t h e i m p o s e d s p e c i m e n s t r a i n r a t ew o u l d v a r y d u r i n g l o a d i n g . T h i s t h e m e w i l l b e p r e -s e n t e d e l s e w h e r e . I t i s a p p r o p r i a t e , h o w e v e r , t o m e n -t i o n h e r e t h a t t h e t h e o r e t i c a l l y p r e d i c t e d d e p e n d e n c eo f c r a c k i n g a c t i v i t y d u r i n g l o a d i n g , s h o w n i n F i g . 1 2,b e a r s a c l o s e r e s e m b l a n c e t o t h o s e a c t u a l l y o b s e r v e d i npure S-2 i ce [56] us ing acous t i c emi ss ions and conven-t i o n a l t e s t m a c h i n e s .

E s t i m a t i o n o f t h e n u m b e r o f g r ai n s i n a p l an e c a nb e m a d e b y Ng = 1/[~(d/2)2]. I n c o l u m n a r - g r a i n e dm a t e r i a l , N g g iv e s t h e g r a i n d e n s i t y i f t h e p l a n e i sn o r m a l t o l e n g th o f th e c o lu m n s . F o r d = 4 . 5 r a m ,Ng = 6 .3 x 10- 4m -2 and i s show n in F ig . 10 . A tf a i l u r e , o n e c r a c k d e v e l o p s f o r e v e r y 5 0 g r a i n s a t1 x 10-7sec -1. Th i s r a t i o i nc rea se s t o ab ou t o nec r a c k f o r e v e r y th r e e g r a i n s a t 1 x 1 0 - 4 s e c - J . T h ec r a c k d e n s i t y e v e n a t f a i l u r e i s t h e r e f o r e r e a s o n a b l yl o w fo r l o a d i n g c o n d i t i o n s ~ < 1 1 0 - 4 se c - t .

T h e p r e s e n t t h e o r y i s v a l i d f o r c o n d i t i o n s o f n o n -i n t e r a c t i n g c r a c k s , s o t h a t a2N ,~ 1 o r , a cco rd ing t oEqua t ion 19 , r~d2N/24 3 ~/2 ~ 1. Th e valu e of the left-h a n d s i d e , c o r r e s p o n d i n g t o t h e f a i lu r e c r a c k d e n s i t ya t 1 x 10-4sec l , i s 0 .035 . Thu s , fo r s t r a in ra t e s le sst h a n l x 1 0 4, c r a c k s c a n b e a s s u m e d t o b e n o n -i n t e r a c t i n g e v e n t o f a i lu r e .

S ince t he de l ayed e l a s t i c s t r a in i nc rea se s wi thd e c r e a s e i n g r a i n s i z e f o r t h e s a m e v a l u e o f t h e g . b .s .d i s p l a c e m e n t , w h e r e a s t h e c r a c k s i z e d e c r e a s e s w i t hdec rea se i n g ra in d i am e te r , t he g ra in s i ze i n f luences t hes t r ai n c o m p o n e n t s i n a c o m p l e x m a n n e r . T h e e f f ec t o fg r a i n s i z e o n t h e r a t e s e n s i t i v it y o f t h e s t r e n g t h a n ds t r e s s - s t r a i n d i a g r a m s h a s b e e n e x a m i n e d a n d w i l l b ep r e s e n t e d e l s e w h e r e b e c a u s e o f s p a c e l i m i t a t io n s .B r i ef l y, i t h a s b e e n f o u n d t h a t t h e r a t e s e n s i t i v i ty o fs t r eng th i s no t a f f ec t ed by g ra in s i ze , bu t t he f a i l u res t r a in an d c rac k den s i t y a t f a i l u re i nc rea se s i gn i f ican t lywi th dec rea se i n g ra in s i ze , and t he s t r a in r eve r sa lp o i n t s h i f t s t o h i g h e r s t r a i n r a t e s . T h u s , f i n e - g r a i n e dm a t e r i a l b e h a v e s i n a m o r e d u c t i l e m a n n e r t h a nc o a r s e - g r a i n e d m a t e r i a l .9 . S u m m a r y a n d c o n c l u s i o nA n o n - l i n e a r c o n s t i t u t i v e e q u a t i o n f o r h i g h t e m p e r a -tu re s ha s been p re sen t ed . I t cons i s t s o f e la s t ic , de l ayed-e l a s t i c a n d v i s c o u s c o m p o n e n t s c o r r e s p o n d i n g t ot h r e e m i c r o m e c h a n i s m s : l a t t i c e d e f o r m a t i o n , i n t e r -g r a n u l a r s l id i n g a n d i n t r a g r a n u l a r d i s l o c at i o n m o t i o n .

T h e m o d e l i n c o r p o r a t e s t h e p r e d i c t a b i l i t y o f t h eo n s e t o f c r a c k in g a c t i v it y a n d d a m a g e a c c u m u l a t i o nd u e t o t h e m e c h a n i s m o f h i g h - te m p e r a t u r e g r a i n -b o u n d a r y e m b r i t t l e m e n t . G r a i n - f a c e t l o n g c r a c k sd e v e l o p w h e n a c r i t i c al g r a i n - b o u n d a r y s l id i n g ( g .b . s. )d i s p l a c e m e n t o r a n e q u i v a l e n t d e l a y e d e l a s t ic s t r a in i sr e a c h e d . F u r t h e r d a m a g e i s g i v e n i n t e rm s o f t h eexcess g .b . s , d i sp l acement ove r i t s c r i t i c a l va lue . Asc r a c k s f o r m , t h e y e n h a n c e th e m a t r i x d e f o r m a t i o na f f e c ti n g t h e o v e r a l l c re e p r a t e , l e a d i n g t o a m i n i m u mr a t e a n d t h e n t e r t i a r y c r e e p .

F o r m u l a t i o n s h a v e b e e n d e v e l o p e d , u s i n g t h i sm o d e l , f o r p r e d i c t i n g t h e d e f o r m a t i o n a n d c r a c k i n ga c t i v i ty f o r c o n d i t i o n s o f c o n s t a n t s t r a i n - r a t e s t r e n g t ht e s t s . T h e t h e o r y w a s t e s t e d w i t h p u b l i s h e d e x p e r i -m e n t a l d a t a o n t h e s t r a i n - r a t e s e n s i t iv i t y o f t h e

c o m p r e s s i v e s t r e n g t h o f t r a n s v e r s e l y i s o t r o p i c , c o l -u m n a r - g r a i n e d , p u r e p o l y c r y s t a l l i n e i ce w i t h a l o a da p p l i e d i n t h e p l a n e o f i s o t r o p y . C a l c u l a t i o n s u s i n gm a t e r i a l c o n s t a n t s w e r e o b t a i n e d f r o m c o n s t a n t - s t r e s sc r e e p e x p e r im e n t s t o t a l l y i n d e p e n d e n t o f t h e s t r e n g t ht es t s. O n e - t o - o n e c o r r e s p o n d e n c e o f t h e o r y a n d e x p er i-m e n t s w a s n o t e d f o r t h e d e p e n d e n c e o f s t r e n g t h , f a il -u r e s t r a i n a n d f a i l u r e t i m e o n s t r a i n r a t e . T h e t h e o r yt h e r e f o r e p r e d i c t s t h e e m p i r i c a l l y o b t a i n e d r e l a t i o n sb e t w e e n t h e s e q u a n t i t i e s . I t a l s o p o i n t s o u t t h e l i m i t a -t i o n s o f t h e e m p i r i c a l r e l a t io n s , f o r e x a m p l e t h e n o n -a p p l i c a b i l i t y o f t h e e m p i r i c a l l y o b t a i n e d p o w e r - l a wre l a t i on be tw een t f and crf o r be tw een s r and ~ a t r a t e sl o w e r t h a n t h e e x p e r i m e n t a l r a n g e .

The t heory p red i c t s a c r i t i c a l s t r a in r a t e , ~c ,be low w hich no wed ge c racks fo rm. Fo r i c e, ~c -~2 . 2 x 1 0 - S s e c -~ a t 0 . 96 T i n ( - 1 0 ~ A t ~ < ~ t h ef l o w s t r e s s i n c r e a s e s w i t h d e f o r m a t i o n a n d a s y m p -t o t i c a l ly a p p r o a c h e s a s t e a d y - s t a t e v a l u e d e p e n d i n gon 8 . W i th i nc rea se i n s t r a in r a t e , d > ~c, m ic ro c rack sd e v e l o p d u r i n g l o a d i n g a n d s t r e s s - s t r a i n d i a g r a m se x h i b i t a d i s t i n c t " u p p e r y i e l d t y p e " f a i l u r e i n w h i c hthe s t r e ss i nc rea se s t o a peak va lue and t hen dec rea se sw i t h f u r t h e r s t r a in . T h i s m a x i m u m s t re s s , o r s t r e n g t h ,s h o w s a p o w e r - l a w r a t e s e n s i t i v i t y s i m i l a r t o t h a t o ft h e m a t r i x o r d i s l o c a t i o n c r e ep ( v i s c o us ) r a t e , w i t h o u ta n y c r a c k e n h a n c e m e n t . C o n t i n u i t y i n t h e ~ r - ~ r e l a -t i o n i s m a i n t a i n e d , t h e r e f o r e , o v e r a w i d e d r a n g e . T h efa i l u re s t r a in e l, howe ve r , dec rea se s a s ~ i nc rea se s f ro mdc a n d i n d i c a t e s a p o i n t o f m i n i m u m d u c t i l i ty a t s o m es t ra in r a t e . I t i nc rea se s aga in wi th fu r the r i nc rea se i n4 . Th i s r eve r sa l p o in t i s a r e su l t o f t he r a t e de pen den ceo f t h e r e l a ti v e c o n t r i b u t i o n s o f t h e t h r e e s t r a i n c o m -p o n e n t s t o t o t a l s t ra i n . I t o c c u r s p r i m a r i l y a s a r e s u lto f t h e o p p o s i n g e f f e c t s o f v i s c o u s s t r a i n a n d e l a s t i cs t r a in . F or ~ > go, s t r e ss an d t he t ime to on se t o fc rack ing ac t i v i t y depend on 8 , a s does t he l eve l o fd a m a g e o r t h e c r a c k d e n s i t y a t af . A t t h e s e s t r a in r a t e st h e p r e d i c t e d c r a c k i n g r a t e i n c r e a s e s m o n o t o n i c a l l yw i t h t i m e a n d a p p r o a c h e s a n e a r l y c o n s t a n t v a l u ed u r i n g t h e p o s t - u p p e r y i e ld p e r io d , t h u s e x p l a i n i n g th ep h e n o m e n o l o g i c a l o b s e r v a t i o n o f d i r e c t d e p e n d e n c eo f c r a c k i n g r a t e o n s t e a d y - s t a t e f lo w r a t e .A c k n o w l e d g e m e n t sT h e a u t h o r i s i n d e b t e d t o R . J e r o m e f o r d e v e l o p m e n to f th e c o m p u t e r p r o g r a m a n d a s s i s ta n c e a t al l s t a g eso f th i s w o r k . T h i s p u b l i c a t i o n is a c o n t r i b u t i o n f r o mt h e I n s t i t u t e f o r R e s e a r c h i n C o n s t r u c t i o n , N a t i o n a lR e s e a r c h C o u n c i l o f C a n a d a .R e f e r e n c e s

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R e c e i ve d 3 N ov e m be r 1987and accepted 25 February 1988

N.K. Sinha, J. Materials Sci., 23 (1988), p. 4415

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