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T o c i t e t h i s p a p e r : I n t . J . R o c k M e c h . & M i n . S c i . 3 4 : 3 - 4 , p a p e r N o . 2 8 3 . C o p y r i g h t 1 9 9 7 E l s e v i e r S c i e n c e L t d

C o p y r i g h t 1 9 9 7 E l s e v i e r S c i e n c e L t dI n t . J . R o c k M e c h . & M i n . S c i . V o l. 3 4 , N o . 3 - 4 , 1 9 9 7 I S S N 0 1 4 8 - 9 0 6 2T o c i t e t h i s p a p e r : Int. J. RockMech. &Min. Sci. 3 4 : 3 - 4 , P a p e r N o . 2 8 3

A N E L A S T O - P L A S T I C C O N S T I T U T I V E M O D E L F O RB R I T T L E - D U C T I L E T R A N S I T I O N I N P O R O U S R O C K S

K e t a n R . S h a hCorne l l F ra c tu re Group a nd Corne l l T he ory Ce n te r , Corne l l Unive r s i ty , I tha c a , Ne w York 14853 , US A

A B S T R A C TA uni f ie d e la s to -p la s t ic c ons t i tu t ive mod e l ba se d o n the obse rve d be ha v ior o f porous roc ks i s p ropose d .T he mo de l u t i l i z e s the c on c e p t o f c r i t ic a l s ta te (CS) , use d fo r so i l s in the C a m c la y mode ls , to s e pa ra tebr i t t l e d i l a ta n t de form a t ions f ro m the c om pa c t ive one s . T he CS re pre se n ts u l t ima te ( r e s idua l ) s t a te o fs t re s s a t whic h la rge she a r de form a t ions oc c urs wi th s te a dy le ve l o f s tr e s se s a nd f ixe d poros i ty . T heC a m - c l a y p h e n o m e n o l o g y i s c o m b i n e d w i t h t h e p o r o e l a s t ic i t y t h e o r y to d e v e l o p a c o m p r e h e n s i v ee la s to -p la s ti c mo de l fo r porous roc ks . T he y ie ld su r fa c e i s a l lowe d to e vo lve wi th bo th p la s t i c v o lum e t r ica nd de v ia to r ic s t r a in suc h tha t d i l a ta n t ha rde n ing /so f te n ing a nd c omp a c t ive y ie ld ing c a n be p re d ic te d .T he po te n t ia l func t ion i s a s sume d to ha ve the s a m e fo rm a s the y ie ld func t ion , bu t the c ons ta n t s ha ved i f fe re n t va lue s to p re d ic t non-a s soc ia t ive d i la ta nc y . T he m ode l pa ra me te r s a re e va lua te d f rom thee xpe r ime n ta l da ta o f s a nds tone s a nd the mode l c a pa b i l i t i e s a re de mons t ra te d th rough the c a lc u la te dme c ha n ic a l r e spo nse in t r i a x ia l t e s ts unde r a wid e r a nge o f c onf in ing p re s sure s .

C o p y r ig h t @ 1 9 9 7 E l se v ie r S c i e n c e L tdK E Y W O R D SC o n s t i t u t i v e R e l a t i o n s E l a s t o - p l a s t i c M o d e l C r i t i c a l S ta t e B r i t t l e - d u c t i l e T r a n s i t i o n C a mC l a y M o d e l P o r o e l a s t ic i t y D i la t a n c y N o n - a s s o c i a t i v i ty I n e l a st i c C o m p a c t i o n P o r o u s R o c k s

I N T R O D U C T I O NSt re s s -s t r a in c urve o f porous roc ks und e r hydro s ta t i c loa d ing o r h ig h p re s sure ha s a n in f le c t io n po in twh ic h i s fo l lowe d by a l a rge poros i ty r e duc t ion (e .g . Br a c e , R i le y 1972). T he fa i lu re is th roughdis t r ibu te d ine la st i c me c ha n ism s suc h a s pore c o l la pse a nd g ra in c rush ing re su l t ing in to c a ta c la s ti c f lowa n d p e r m a n e n t c o m p a c t i o n ( C u r r a n , C a r r o ll 1 97 9; Z h a n g e t a l . 1990). The requ ired dif fe rent ia l st ressfo r the onse t o f ine la s t i c f low de c re a se s wi th the p re s sure a nd the y ie ld su r fa c e i s we l l r e p re se n te d by a ne l l ip t i c a l c a p in me a n-de v ia to r ic s t re s s (p - q ) spa c e (W ong e t a l . 1997). T he low-pre s sure de forma t ion sa re b r i tt l e a nd d i la ta n t wi th f a i lu re t a k ing p la c e in the fo rm o f loc a l iz a t ion in to she a r f a u l ts . T he she a rs t r e ng th ha s pos i t ive p re s sure -de pe nde n c y wi th f r i c t iona l y ie ld su r fa c e suc h as Mo hr -Co ulom b onef i t t ing the da ta we l l ( J a e ge r , Co ok 1979). A t the in te rme dia te r a nge o f p re s sure s , the s ta te o f de forma t ioni s m o r e c o m p l e x a n d m a y b e a c c o m p a n i e d b y m u l t ip l e c o n j u g a t e s h ea r b a n d s ( k a g e s o n - L o e e t a l . 1993)g r o w i n g w i t h c o n s t a n t v o l u m e o r c o m p a c t i o n .T he re a re e ng ine e r ing p rob le m s in whic h m e a n s t re s s va r ies so muc h tha t bo th b r i t tl e a nd duc t i l e mo de sof de form a t ion a re p re se n t . An inde n te r p re s s ing a ga ins t the roc k su r fa c e induc e s l a rge c omp re s s ives t r es se s in i ts v ic in i ty l e a d ing to c ompa c t ion (M i l le r , C he a tha m 1972; Sua re z -R ive ra e t a l . 1990).

I S S N 0 1 4 8 - 9 0 6 2


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St re sse s ou ts ide o f th i s c omp a c t ion z one a re low e noug h so tha t the b r i t t le f r a c tu r ing m a y be induc e d .Dy na m ic loa d ing a s in b la s t ing a nd pe r fo ra t ing o i l we l l s induc e s a wide r a nge o f s t re s ses a nd d i f f e re n tde form a t ion mode s a t a s ing le loc a t ion . S imula t ion o f the a bove p rob le m s re qu i re s a un i f ie d mo de le nc om pa ss ing bo th pos i t ive a nd ne ga t ive p re s sure -de pe nde n t ine la s t i c de forma t ion .A c o nve n t iona l a pproa c h ha s be e n to a ugm e nt a b r i tt l e y ie ld su r fa c e wi th a n e nc los in g c a p inme a n-de v ia to r ic s t re s s spa c e (D iM a gg io , S a nd le r 1971 ; L uba rda e t a l . 1996) . Such cap models requiretwo d i f f e re n t func t iona l fo rms o f the y ie ld su r fa c e s a nd the i r s e pa ra te e vo lu t ion la ws a s we l l a s thec om ple x ma n ipu la t ions to t r a c k the c om e r a t the junc t io n . T he ne e d fo r a smooth s ing le y ie ld su r fa c e ha sb e e n r e c o g n i z e d i n t h e w o r k o f D e s a i e t a l . 1988 and Desa i 1989. Desa i e t a l . 1988 in t rod uc e d a f a mi lyo f h i e r a rc h i c a l s in g l e s u r fa c e m o d e l s w h i c h r e q u i r e a m i n i m u m n u m b e r o f p a r a m e te r s.Ca m c la y mode ls use a s ing le e ll ip t i c a l o r s imi la r y ie ld su r fa c e inp - q spa c e a nd ha ve b e e n suc c e s sfu l inpre d ic t ing the re sponse o f d i ff e re n t so i l s (Sc h of ie ld , W ro th 1968). T he se mod e ls a re ba se d o n thec onc e p t o f c r i t ic a l s t ate (CS) w hic h re pre se n ts a l im i t ing s te a dy s tate o f s t r e ss . T he c r i t i c a l s t ate fo r roc ksm a y re pre se n t r e s idua l s tate o f s t re s s (Ge org ia nn opo ulos , Brow n 1978). W ork ha rde n in g /sof te n in g inCa m c la y i s a s sume d to be a func t ion o f the p la s t ic vo lum e t r ic s tr a in wh ic h be c om e s z e ro on the c r i t i c a ls ta te l ine (CSL ) . T he de form a t ions a re d i l a ta n t a nd c o mpa c t ive , r e spe c t ive ly , to the l e f t a nd f igh t o f CSL .Ca m c la y mode ls a long wi th the c onc e p t o f CS prov ide s a ge ne ra l f r a me w ork to s imula te bo th b r i t t l e andc a ta c la st i c de form a t ions fo r roc ks .C a m c l a y m o d e l s u t i l iz e e x p e r i m e n t a l l y o b s e rv e d l i n e a r r e la t i o n s h ip b e t w e e n t h e v o i d r a t io a n d l o g a r i t h mof e f fe c t ive me a n s t r e s s fo r bo th loa d ing a nd un loa d ing . T he e la s t i c i ty be c om e s non l ine a r a ndpre s sure -de pe n de n t a nd th i s i s the ma in w a y in wh ic h C a m c la y d i f fe r s f rom the c la s sic a l p la s t i c i tyfo rmu la t ion wh ic h inc lude s s ing le - sur fa c e mo de ls o f De sa i 1989 . Ca m c la y mode ls a re no t d i r e c t lys u i ta b l e f o r th e m e c h a n i c a l s i m u l a ti o n o f p o r o u s r o c k s a n d w i l l n e e d t o b e m o d i f i e d t o a c c o u n t f o r t h e i rqua l i t a t ive d i f f e re nc e s wi th so i ls .O n e o f t h e m a j o r w e a k n e s s o f C a m c l a y i s t h at d i l a t a n c y al w a y s i n d u c e s s o f t e n i n g . T h e i n t er c e p t o f aCa m c la y y ie ld su r fac e wi th the p -a x is i s use d a s a ha rde n ing /so f te n ing pa ra m e te r a nd i t r e duc e s wi thd i la ta nc y (Sc ho f ie ld , Wro th 1968). T he f r i c t iona l ha rde n ing wi th she a r s t ra in m a y be supe r im pose d suc ht h a t t h e d i l a ta n t h a r d e n i n g m a y b e p r e d ic t ed . T h e d i l a t an c y i s k n o w n t o b e n o n - a s s o c ia t i v e w h i c h m a y b ea c c o u n t e d f o r b y u s i n g a p o t e n t i a l f u n c ti o n d e r i v e d f r o m t h e s i m p l e m o d i f i c a t i o n o f t h e y i e l d f u n c t i o n(Dre sc he r e t a l . 1995). Co he s ion a nd te ns i le s t re ng th o f roc ks i s ge ne ra l ly no t ne g l ig ib le a nd c a n a l so bei n c l u d e d i n t h e m o d e l b y s i m p l y s h i f ti n g th e y i e l d s u r f ac e a l o n g t h e p a x is .Ca m c la y d i s re ga rds the c om pre s s ib i l i ty o f so l id ma t r ix a n d pore f lu id . T h is m a y no t be r e a son a b le fo rporous roc ks a n d the c ompre s s ib le c ons t i tue n ts m a y c on t r ibu te more tha n 10% of the vo lum e t r icde form a t ions fo r porous s a nds tone s . B io t ' s 194 1 c oup le d the ory fo r f lu id - sa tu ra te d me dium prov ide s ara t iona l ba s i s to a c c oun t fo r so l id a nd f lu id c om pre s s ib i li ty . T he c ons t i tu t ive e qua t ions o f Bio t ' sp o r o e l a s t i c it y w i l l b e w r i tt e n i n i n c r e m e n t a l f o r m t o i n t r o d u c e n o n l i n e a r i t y ( D e t o u m a y , C h e n g 1 99 3) .T he C a m c la y p la s t i c i ty wi l l be a dd e d to i t to fo rm ula te a c ompre h e ns ive c ons t i tu t ive mo de l fo r roc ks .T he mod e l de ve lo pe d he re wi l l be ve r i f i e d by c om pa r ing the r e su l t s wi th the e xpe r ime nta l da ta o fs a n d s to n e s g i v e n b y W o n g e t a l . 1997.E X P E R I M E N T A L D A T A A N D C R I T I C A L S T A T E F O R R O C K SPos i t ive p re s sure -de pe nde n t she a r s t r e ng th a nd d i la ta nc y o f roc ks in the b r i t t l e r e g ime h a ve be e n w e l ld o c u m e n t e d ( e .g . J a e g e r , C o o k 1 97 9) a n d n u m e r o u s e f f o r ts t h ro u g h M o h r - C o u l o m b o r D r u c k e r - P r ag e r

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p l a s t i c it y h a v e b e e n m a d e t o m o d e l t h e m . T h e o n s e t o f d i l a t a n c y an d p l a s t i c i ty is f o l l o w e d b y h a r d e n i n g ,pe a k s t r eng th , a nd so f te n ing . Loc a l iz a t ion in to she a r f a u l t s a c c ompa nie s the so f te n ing a nd u l t ima te ly thele ve l o f s t r e s ses r e a c he s the l im i t ing re s idua l s t r e ng th . Th is s te a dy s ta te a l so ha s p os i t ivepre s sure -de p e nde nc y a n d i s p r ima r i ly de te rmine d by f r i c t ion a long the f a u lt s .Br a c e , R i le y 1972 c omp re s se d d i f f e re n t roc ks unde r un ia x ia l s t ra in c ond i t ion up to the p re s sure o f 30k i loba r s (3 GPa ) a n d obse rve d tha t a ppre c ia b le pe rm a ne n t c om pa c t ion ta ke s p la c e . Mos t o f theh i g h - p o r o s i t y ro c k s s h o w e d a n i n f l e c t i o n p o i n t i n t h e i r s t re s s- s tr a in r e s p o n s e w h i c h w a s f o l l o w e d b yra p id ine la s t i c c om pa c t ion a nd ha rde n ing . T he i r m ic ros t ruc tu ra l obse rva t ions fo r Ind ia na l ime s toneind ic a te d tha t pore s a re e l im ina te d th rough the p la s t i c de form a t ion o f c a lc i t e g ra ins . Ca ta c la s t i c f low inporous s i l i ca te roc ks i s due to pe rva s ive in t r a gra nu la r c ra c k ing (Ed m on d , Pa te r son 1972 ; Hi r th , Tu l l i s1989; Zhang e t a l . 1 99 0). T h e o n s e t o f c o m p a c t i o n y i e ld i n g h a s n e g a t i v e p r es s u r e - d e p e n d e n c y ( H i r t h ,Tull is 1989; Wong e t a l . 1997). The y ie ld is fo l low e d by the ha rde n in g a t a de c re a s ing ra te a nd thes t r es se s a pproa c h the l im i t ing re s idua l va lue s . A t l a rge s t r a ins , d i l a ta nc y ma y in i t i a te a nd c a use thefa u l t ing (H i r th , Tu l l i s 1989 ; Wong e t a l . 1992). The re s idua l s t r e ng th in suc h a c a se i s gove rn e d by thef r ic t ion s imi la r to the b r i t t l e de forma t ions . I t c a n be c on c lude d f rom the se e xpe r im e nta l da ta tha t in bo thbr i t t l e a nd duc t i l e r e g ime s , a l im i t ing , pos i t ive p re s sure -de pe n de n t r e s idua l s tate o f s t r e ss c a n be de f ine d .The c o nc e p t o f c r it i c a l st a te (CS) ha s be e n ba se d on the ob se rve d c r i t i c a l vo id r a t io fo r so i l s a t wh ic hla rge she a r de form a t ions in i t i a te a t pe a k loa ds wi th ou t fu r the r c ha nge in the vo lum e (Sc h of ie l d , W ro th1968). The lev e l of s t resses do no t a l te r and so the c r i t ica l s ta te represents the s tead y res idua l s ta te ofs tr es s. S inc e the e xpe r im e nta l da ta fo r porous ro c ks show a n e x i s te nc e o f r e s idua l s t r eng th , the y ma y beuse d to de f ine the c r i t ic a l s ta te. A l tho ugh the e x pe r ime nta l da ta a re no t c onc lus ive , i t wo uld be r e qu i re dtha t pore vo lum e t r ic s tr a ins a re z e ro in the r e s idua l s ta te to de f ine i t a s CS . The v o lum e c ha nge in thec a ta c la s ti c re g im e i s ne ga t ive w he re a s fo r the b r i t t le de form a t ion i t i s pos it ive . The re g e ne ra l ly e x i s t s a nin te rme dia te s ta te o f s tr e ss in wh ic h la rge she a r de form a t ions pe r s is t a t c ons ta n t vo lum e (Wo ng e t a l .1997). The CS se pa rate s d i la ta n t de form a t ions f rom the c omp a c t ive one s a nd so r e pre se n ts thebr i t t l e -duc t i l e t r a ns i t ion .W o n g e t a l . 1 99 7 p r o v i d e d a c o m p r e h e n s i v e d a ta b a s e f o r t h e m e c h a n i c a l b e h a v i o r o f a v a r i e t y o fsa nds tone s . The i r da ta fo r the onse t o f d i l a ta nc y a nd pe a k s t r e ng th in b r i t t l e r e g ime a nd fo r the in i t i a t ionof ine la s ti c c om pa c t ion in duc t i l e r e g ime fo r Ada m swi l le r s a nds tone a re show n in F igure 1 . The y show e dtha t the d i f f e re n t ia l s t re s s a t the onse t o f c a ta cla s t i c flow ha s a ne ga t ive p re s sure -de pe n de nc y ( shown byf i l l e d squa re s in F igure 1 ) a nd a n e l l ip t i c a l c a p f i t s the da ta we l l . The o nse t o f d i l a ta nc y ( shown b y )ta ke s p la c e a t l e s s tha n 60% of the f a i lu re s t re s s ( in F igure 1 ) a nd c o ns ide ra b le ha rde n in g i s p re se n t .C A M C L A Y M O D E L F O R R O C K SCon s ide r a r e pre se n ta t ive e le me nt o f vo lume V in a porous roc k w i th m e a n s t re s s t e nsor (yO.a nd pore f lu idpre s sure P a s s ta te va r ia b le s. B io t ' s 1941 fo rm ula t ion unc ou p le s the bu lk she a r de form a t ion f rom thef lu id f low fo r a n i so t rop ic ma te r ia l a nd so d e v ia to r ic s t ra in i s de pe nde n t on the d e v ia to r ic s t re s s on ly .W r i t ing th i s in a n inc re me n ta l fo rm:

w h e r e s i j = ( Yi j - P S i j is the dev ia tor ic s t ress tenso r wi th p be in g the m ean s tress (yk#/3 and 8~. be in g the

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K r o n e c k e r d e l ta t e n s o r , e e i j i s th e e l a s t i c p a r t o f d e v i a t o r i c s t r a i n t e n s o r d e f i n e d a s De i j - De k / f l i j / 3 w h e r eD i j i s t h e s t r a i n t e n s o r a n d t h e s u p e r s c r i p t e is u s e d t o d e n o t e t h e e l a s t ic p a rt . C o m p r e s s i o n w i l l b ea s s u me d t o b e p o s i t i v e f o r s t r e ss e s , p r e s s u r e a n d s t r a in s . T h e b u l k s h e a r mo d u l u s ~ t i s i n g e n e r a l afun c t ion o f s t a t e var i a b les (y~).a n d P. T h e b u l k v o l u m e t r i c s t r ai n a n d t h e c h a n g e i n s p e c i f ic v o l u m e v ( b u lkv o l u m e p e r u n i t v o l u m e o f s o l i d s ) ar e l i n e a r l y r e l a t e d to t h e e f f e c t i v e p r e s s u r e p ' = p - P a n d p o r ep r e s s u r e P ( B i o t 19 4 1 ; R i c e , C l e a r y 1 9 7 6; D e t o u r n a y , C h e n g 1 9 9 3 ) :

de~ k = d t d PK -t K ~ ( 2 )1

- - V( 3 )

w h e r e K i s t h e d r a in e d b u lk m o d u l u s , = ( v - 1 ) / v i s t h e p o r o s i t y , a n d K s i s th e s o l i d b u l k m o d u l u s . T h e s ee q u a t i o n s a r e s i m p l i f i e d f r o m t h a t o f B i o t 1 9 4 1 r e l a t i o n s h i p s u n d e r t h e a s s u m p t i o n o f i s o tr o p i c a n d i n e r ts o l i d a n d c o n t i n u o u s p o r e s p a c e ( R i c e , C l e a r y 1 9 7 6) . A l t h o u g h n o t m e n t i o n e d e x p li c it ly , b u l k m o d u l i Ka n d K s a r e i n g e n e r a l t h e f u n c t i o n o f s tr e s s te n s o r a n d p o r e p r e s su r e . I t w o u l d b e a s s u m e d h e r e t h a t t h es o l i d is e l a s t ic a n d b u l k m o d u l u s K s r e ma i n s a c o n s t a n t .I t c a n b e s e e n f r o m E q n . 3 t h a t e l a st i c in c r e m e n t a l c h a n g e i n s p e c i f i c v o l u m e i s d i r e c t l y p r o p o r t i o n a l t ot h e i n c r e m e n t a l e f f e c t iv e p r e s s u re . T h e v o i d r a ti o e a n d s o t h e s p e c i f ic v o l u m e ( v = 1 + e ) a r e a s s u m e d t ob e a n e x p l i c i t f u n c t i o n o f e f f e c t i v e p r e s s u r e f o r s o i ls i n Ca m c l a y mo d e l s . Sp e c i f i c a l l y , v f o r e l a s t icl o a d i n g o r u n l o a d i n g i s l i n e a r i n th e l n p ' . T h i s r e la t i o n s h i p b r e a k s d o w n w h e n p ' i s z e r o o r n e g a t i v e a n ds i n c e ro c k s m a y h a v e f i n it e t e n s il e s t re n g t h , w e a s s u m e v t o b e a l i n e a r f u n c t io n o f In ( p ' + P o ) w i t h p ob e i n g t h e i s o t r o p i c t e n s i l e s t r e n g t h .

= - I n ( p ' + p o ) ( 4 )w h e r e ~ : i s a s lo p e o f v v e r s u s I n ( p ' + P o ) l i n e a n d v o i s t h e s p e c if i c v o l u m e a t p ' = -P o + 1 in e l as t i cr e g i m e . D i f f e r e n t ia t i n g th e a b o v e e q u a t i o n w i t h r e s p e c t to p ' a n d c o m p a r i n g w i t h E q n . 3 g i v e s

1 1 ~;- - = + ( 5 )K(p ' ) ( 1 - ) K ~ (1: /+ o)vI t w o u l d b e a s s u m e d t h a t th e s h e a r m o d u l u s r e m a i n s c o n s t a n t a n d h e n c e t h e P o i s s o n ' s r a t io v i s a f u n c t i o no f e f f e c t i v e p r e s s u r e g i v e n b y

3 K ( p ' ) - 2 t , ( 6 )" ( P ' ) = 2 ( 3 K ( p ' ) + u )

T h e P o i s s o n ' s r a ti o i s p r e s s u r e - d e p e n d e n t a n d m a y b e c o m e n e g a t i v e f o r l o w v a l u e s o f c o n f i n i n g p r e s su r ew h i c h i s c o n s i s t e n t w i t h t h e e x p e r i m e n t a l d a ta o f W o n g e t a l . 1 9 97 . T h e a l t e r n a ti v e f o r m u l a t i o n i n w h i c hv i s a s s u m e d c o n s t a n t m a y b e u s e d t o a l le v i a t e t h e n e g a t i v e v .I t c a n b e s h o w n f r o m t h e e l e m e n t a r y r e la t i o n s h i p V = v V s w h e r e V i s t h e s o l id v o l u m e t h a t th e b u l k

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volum e t r ic st r a in de c om pose s in to the fo l lowin g fo rm:A V A v A V ,= + ( 7 )V - - k k = v V ,

Sinc e i t i s a s sume d tha t so l id ma t r ix be h a ve s e la s ti c a l ly , bu lk p la s t i c vo lum e t r ic s t r a in i s ne ga t ive o f theAvP/v , i.e.,

Av / 'g k = ( s )vwhe re the su pe r sc r ip tp de no te s the p la s ti c par t . Th is r e la t ionsh ip (Eqn . 8 ) i s use d subse que n t ly in thede ve lo pm e nt o f the e la s top la s ti c mod e l .C a m c l a y p l a s t ic i t yThe m odi f ie d C a m c la y mode l use s a n e l l ip t i c a l y ie ld su r fa c e in me a n-de v ia to r ic s tr e ss spa c e suc h tha t i tin te r se c ts the me a n s t r e ss a x i s a t 0 a ndpc . The c o nso l ida t io n p re s su re pc i s the pa s t ma xim um e f fe c t ivepre s sure unde r hyd ros ta t i c loa d ing a n d i s u t il i z e d to de f ine ha rde n ing a nd so f te n ing . Dur ing a n i so t rop ice la s top la s t i c loa d ing , spe c i f ic vo lum e i s l ine a r ly r e la te d to In (p ' + P o ) s im ila r to the e las t ic par t :

v = v l - A I n ( p ' + Po) ( 9 )whe re )~ i s a s lope o fv ve r sus In (p ' + P o ) l ine a nd Yl i s the spe c i f ic vo lum e a t p ' = -P o + 1. The plas t icvo lum e t r ic s tr a in f rom Eqns . 8 , 4, a nd 9 c a n be wr i t t e n a s :

d ~ k k = _ ( d V v dve)v = ( ~ - + P o ) (10)F o r h y d r o s t a t i c lo a d i n g , p ' = P c a n d s o t h e e v o l u t i o n la w f o r p c b e c o m e s :

d v~ = v ( p ' + p o ) d ~ k

A l t h o u g h C a m c l a y m o d e l s h a v e b e e n d e v e l o p e d w i t h o n l y l o g a r i t h m i c ( o r ig i n a l) a n d e l l i p ti c a l(modi f ie d) y ie ld su r fac e s (Sc h of ie ld , W ro th 1968), a pos s ib le f a m i ly o f su r fa c es ma y be de ve lop e d byre qu i r ing tha t e a c h su r fa c e in te r se c t s the e f fe c t ive p re s sure a x i s e x a c t ly twic e. Th e se two in te r se c t ionsm u s t b e -P o andp~, withp~. > -Po . This ge ne ra l y ie ld su r fa c e F de pe nds upon the e f fe c tive p re s su re p ' a nda pa ra me te r r e la te d to the s e c on d inva r ia n t o f de v ia to r ic s tr e ss t e nsor sij den oted as q . I t is def ine d to be~ / 3 s q s q suc h tha t i t e qua l s the d i f f e re n t ia l s t re s s fo r t r i a x ia l t e st s . As soc ia t iv i ty i s ge ne ra l ly a s sume d2with Ca m c la y mode ls , bu t th i s r e qu i re m e nt i s r e la xe d he re a nd a d i f f e re n t po te n t ia l func t ion G(p ', q ) = 0s imi la r to D re sc he r et a l . 1995 w i l l be de f ine d . The p la s t i c s t r ains a re g ive n by

I S S N 0 1 4 8 - 9 0 6 2


6/13


OGThe c r i t i ca l s t a te i s t ha t limi t ing s t a t e in wh ich ince s s an t de fo rma t ions t ake p lace wi th in f in i t e s ima lchan ge in the s t resses . Ma them at ica l ly , wi t h in the con text of e las toplas t ic m odel in g , th is requires do~). = 0= P ; d F = 0 and dD~). 0 . I fp~ . i s the on ly hard enin g para me ter then th is imp l ies dG. = 0 = d D p k k (Eqn.11). The nece ssary co ndi t ion for th is f rom the f low ru le (Eqn. 12) i s tha t

OC/Op'=O (13)U n i f i ed m o d e l f o r p o r o u s r o c k sThe above fo rm u la t ion improved the C am c lay mod e l s by accoun t ing fo r cohes ion ( th rough thepa ram e te rpo ) and s o l id ma t r ix com pres s ib i l i ty . The f a i lu re s u r f ace fo r a va r i e ty o f r ocks may be w e l lapp rox ima ted by an e l l ip s e (F igu re 1 ). The f i rs t y i e ld ing in the b r it t le r eg ime o r the d i l a t ancy in i t i a t e s a t as ign i f i can t lower l eve l o f s tr e s ses . A gene ra l y i e ld func t ion wh ich m ay be s kewed i s needed to be t te r f i tt he expe r ime n ta l da ta . A func t ion wh ich a l lows to ad ju s t t he va lue o f e f f ec t ive p r e s s u re a t t he peak o f thefunct ion and as a specia l case is reduced to the e l l ipse is g iven by:

F = q 2 M ~pc (P ' +---P~"(Pc-P')= 0 ( 1 4 )\ P c ]w h e r e M 1 and n are the param eters . For n = 1 , F reduce s to be an e l l ipse and M 1 becomes the s lope o fc r i ti ca l s ta t e l ine fo r the a s s oc ia t ed f low ru le . The s kewed s u r face tha t was u s ed b y Dres ch e r e t a l . 1995

ti s ob ta ined by s ubs t i tu t ing n = 2 . The va lue o f e f f ec tive p r e ss u re p peak a t w h i c h q i s m a x i m u m i sn p c - P o ( 1 5 )P p e a k - - n + 1

S ince the equa t ion fo r the s lope o f the l ine connec t ing the po in t (-Po, 0) to the peak is not t r iv ia l , as l igh t ly d if f e r en t f o rm o f the func t ion i s wr i t t en s uch tha t t he pa r am e te r M equa l s tha t s lope .

F = q 2 _ M 2 ( n + l ) n - 1 ( P t + P o ~ r 'n n _ ( P c + P o ) - P c T ~ o o , I ( P c - P ' ) = 0 ( 1 6 )D r e s c h e r e t a l . 1995 fo rmu la ted a non -as s oc ia t ive C am c lay mode l to p r ed ic t t he peak and pos t -peakres pons e o f a no rm a l ly cons o l ida ted s o i l i n und ra ined cond i t ion . The non -a s s oc ia t iv i ty was in t roduced byus ing an o the r pa r ame te r N r ep lac ing the M in the y ie ld func t ion to de f ine the po ten t ia l f unc t ion . Thepa rame te r N w as r equ i r ed to be h ighe r than M. The s ame idea is u s ed he re to ob ta in G f rom Eqn . 16

G = q 2 _ N 2 ( n + l ) " - I ( P ' + P o ) '~( g + P ) (t7)S ince G i s de f ined a s a s u r f ace pas s ing th rough a g ive n (p ' , q ) po in t f o l low ing the above Eqn . 17, i t does

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no t in t e r s ec t the ho r i zon ta l ax i s a tp~. , bu t a tpg c . I t c an a l so be s hown tha t wh en N > M, I 3G/3p t

an elasto-plastic constitutive model for

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