1972_Brittle Failure in Rocks

7/29/2019 1972_Brittle Failure in Rocks

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Int. J. Rock Mech. Min. Sci. V o l . 9, p p . 8 7 - 1 0 2 . P e r g a m o n P r e s s 1 97 2. P r i n t e d i n G r e a t B r i t a i n

A M E T H O D F O R T H E A N A L Y S I S O F ST R ES S INB R IT T L E R O C K

G. BARLA*Henry Krumb School of Mines, Columbia University, New York

(Received 25 August 1970)Abstraet--A numerical method for the determination of the stresses associated with thefracture processes of a brittle rock material is proposed. It is shown that the various zonesof fracture around an underground opening can be predicted and a stability analysis of therock structure can be performed, while taking the complete physical behavior of the rockmaterial into account. The method is based upon the finite-element analysis and a process ofsuccessive approximations. An example of application for a norite rock mass containing acircular opening, under the conditions of plane strain and a uniaxial stress field, is given forillustration.

INTRODUCTIONFIELD observations reveal that excavations in rock which exhibits a brittle fracture behavi orare usually surrounded by a fracture zone. This empirical evidence is supported by theexperimental measurement of the velocity of sound in the region around the opening. Adefinite increase of this velocity with relation to the distance from the opening contour isgenerally revealed. This indicates that a corre sponding increase of the elastic modulus of therock mass is to be expected.

Based on such observations, the rock surrounding the opening may be given differentelastic moduli and the boundary of the fracture zone may be located at varying distancesfi'om the opening contour. Different assumptions for the geometry and the mechanicalproperties of the fracture zone may be made [1, 2].

An ot he r approach has been proposed by ZIEN~UEWlCZ, VALLIAPPANand KING [3], whotreated the rock as a 'no-tension' material and determined the distribution of stress inseveral structures. Also in this case, the assumptions introduced are unlikely to representthe nature of the real problem, particularly if structures in rock at great depth are considered.

The determination of rock fracture around u ndergro und openings is of great importanceto the rock mechanics engineer, as safety depends upon the ability to control fracture. Theunderstanding of the re-distribution of stress as fracture progresses is a necessary step inachieving such control. A method, which accounts for the complete physical behavior of arock material under different loading conditions, contributes to this understanding. It isthe purpose of the present paper to propose such a method, which is based upon the use ofthe finite-element technique and a process of successive approx imations.

MECHANISM OF BRITTLE FRACTUREThe mechanism of brittle fracture of rock as a result of an applied load has been investi-

gated by BIENIAWSKI 4]. Some of his results are used in the course o f this pape r and are herebriefly reviewed.* Present address: Istituto di Arte Mineraria del Politecnico, Torino, Italy.

87


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8 8 G . B A R L ABrittle fracture in multi-axial compression comprises five distinct processes: (1) crackclosure, (2) fracture initiation (i.e. stable fracture propagation), (3) critical energy release

(i.e. unstable fracture propagation), (4) strength failure and (5) rupture. The same processesoccur also in tension. However, crack closure is absent and the processes of stable andunstable fracture propagations virtually non-existent.It is important to notice that the initiation of fracture in compression does not mean that arock material has lost its load-carrying characteristic. However, in general, a change in thestate of the material occurs. Strength failure is a characteristic property of the rock material,while rupture is found to be a fundamental property of the rock structure.

~ ~ / / ~ ~ U N S T A B L E. . . . . . . . . . . 7- - " - ~A~T~RE~ ~ PROPAGATION

l E l ~FRACTURE NITIATION

CRACK LOSURESTRAIN

F IG . 1 . D i a g r a m m a t i c a l r e p r e s e n t a t i o n o f t h e c o m p l e t e p h y s i c a l b e h a v i o r o f a b r i t t le r o c k m a t e r i a l(BIENIAWSKI [4]).

The characteristic changes shown in Fig. 1 occur in the mechanical response of a rockmaterial as the applied load is increased. Fracture initiation is manifested by a departurefrom linearity of the stress-lateral strain curve, with a consequent increase in the value ofPoisson's ratio for the rock material. The stress-axial strain curve, however, remains linear,i.e. the modulus o f elasticity is constant. The onset of unstable fracture propagation, whichis found to coincide with the critical energy release, corresponds to the value o f the long-termstrength of a rock material. If the load is increased further, the modulus of elasticity decreaseswhile the Poisson's ratio increases due to a considerable cracking which is taking place inthe rock. The fracture process is concluded when strength failure is reached.GRIFFITH'S original theory [5] for open cracks and the modification of this theory, dueto MCCLI~,rrOCKand WALSH [6] (modified Gritfith criterion), to account for crack closure,have been used for predicting fracture initia tion in rock [7]. These studies cannot be usedto determine the ultimate stability of a rock structure. However, they are found to be veryuseful in comparative analyses where the choice is to be made for the best shape and layoutof mining excavations [8].


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A M ETH O D F O R TH E A N A LY S IS O F S TRES S I N BRI TTLE RO CK 8 9F o r p r a c t i c a l p u r p o s e s , i t is p o s s i b le t o a s s u m e t h a t t h e G r i f f i t h ' s t h e o r y i s v a l i d a l s o t o

p r e d i c t t h e s t r e n g t h f a i l u r e i n t e n s i o n , a s f r a c t u r e i n i t i a t i o n a n d s t r e n g t h f a i l u r e i n t e n s i o no c c u r s i m u l t a n e o u s l y . H o w e v e r , a p p l i c a t i o n o f t h e m o d i f i e d G r i f f i t h c r i t e r i o n t o s t r e n g t hf a i l u r e i n c o m p r e s s i o n i s n o t j u s t if i e d , a s h a s b e e n s h o w n b y BIEN IA W SK I [ 9 ] and BRAC~!,PAULDING an d SCHOL Z [10].

M E T H O D O F A N A L Y S IS

A r o c k m a t e r i a l u n d e r t e n s i o n r e v e a ls a p e c u l i a r n o n - l i n e a r b e h a v i o r p a t t e r n o f t h e s t r e s s -s t r a i n c u r v e , i. e. n o n - s y m m e t r y o f th e c u r v e w i t h r e s p e c t t o t h e o r i g i n o f th e c o o r d i n a t ea x e s ( in t h e f o l l o w i n g th i s n o n - l i n e a r i t y w i ll b e r e f e r r e d t o a s t e n s i l e - c o m p r e s s i v e n o n -l i n e a r it y ) . F o r s o m e r o c k t y p e s , t h is n o n - l i n e a r b e h a v i o r i s q u i t e p r o n o u n c e d , f o r o t h e r s i t i sv i r t u a l ly n o n - e x i s t e n t . A s d e s c r i b e d i n t h e p r e v i o u s a r t i c l e, n o n - l i n e a r e f f e c ts a r e a l s oe x h i b i t e d b y r o c k d u r i n g t h e m e c h a n i c a l r e s p o n s e u n d e r i n c r e a s i n g c o m p r e s s i v e l o a d .T h e i n f l u e n c e o f th e s e d i f f e re n t a n o m a l i e s i n t h e p h y s i c a l b e h a v i o r o f a r o c k m a t e r i a l i sn e g l e c t e d i n t h e a n a l y s e s p r e d i c a t e d u n d e r t h e a s s u m p t i o n o f l in e a r e l a s t ic i ty . T h e e a s i es tt h e o r e t i c a l p a t h b y w h i c h t h i s s i m p l i f y i n g a s s u m p t i o n m a y b e r e m o v e d i n a n y t h e o r e t i c a lt r e a t m e n t i s to v a r y t h e c o n s t i tu t i v e e q u a t io n s o f th e r o c k m a t e r i al . T h e d e v e l o p m e n t o fc o n s t i t u t i v e e q u a t i o n s f o r t h e d e s c r i p t i o n o f t h e v a r i o u s m e c h a n i c a l b e h a v i o r p a t t e r n se x h i b i t e d b y a r o c k is a s u b j e c t o f i n t e n se r e s e a r c h . T h e p r o b l e m i s t o e s t a b l is h a d e q u a t ea n d p r a c t ic a l p r o c e d u r e s f o r t h e d e t e r m i n a t i o n o f t he m a t e r i a l f u n c t io n s w h i c h e n t e r t h ev a r i o u s c o n s t i t u t i v e e q u a t i o n s [1 1].

I n t h e s o l u t i o n o f a b o u n d a r y v a l u e p r o b l e m w i t h n o n - l i n e a r m a t e r i a l p r o p e r ti e s , t h e u s eo f n u m e r i c a l m e t h o d s o f a n a l y si s e v e n f o r s im p l e g e o m e t r i e s, is im p e r a t i v e . I n p a r t i c u l a r , t h ea d v a n t a g e o f th e f i n i t e -e l e m e n t m e t h o d i n t h e s o l u t i o n o f p r o b l e m s o f t h is t y p e is f o u n d i ni ts o w n n a t u r e o f f o r m u l a t i o n , a s o n e c a n e a s il y t r e a t d i f f e r e n t c o n s t i t u t i v e b e h a v i o r s [1 2] .

T h e m e t h o d o f a n a ly s i s p r e s e n t e d i n th i s p a p e r c o n s i s t s o f s o lv i n g in a s e q u e n c e t h ef o l l o w i n g s e p a r a t e p r o b l e m s b y a p r o c e s s o f i n c r e m e n t a l l o a d i n g :

( a ) T o d e t e r m i n e t h e d i s t r i b u t i o n o f s t re s s in a r o c k s t r u c t u r e w h e r e t h e t e n s i l e - c o m p r e s s i v en o n - l i n e a r i t y o f t h e r o c k m a t e r i a l is ta k e n i n t o a c c o u n t .

( b ) T o e s t i m a t e t h e z o n e s i n w h i c h t h e p r o c e s s o f f r a c t u r e i n i t i a t i o n ( s t a b le f r a c t u r ep r o p a g a t i o n ) t a k e s p l a c e a n d t o d e t e r m i n e t h e n e w d i s t r i b u t i o n o f s t re s s w h i c h f o l l o w s t h ei n a b i l it y o f t h e r o c k t o s u s t a i n a n y l o a d i n t h e d i r e c t i o n o f th e t e n s il e st r es s a n d t h e d e p a r t u r ef r o m l i n e a r it y o f t h e s t r e s s -l a t e r a l s t r a i n c u r v e i n c o m p r e s s i o n .

( c) T o e s t a b li s h t h e o n s e t o f u n s ta b l e f r a c t u r e p r o p a g a t i o n i n t h e r o c k s t r u c t u r e a n d t oc o m p u t e t h e r e - d i s t r i b u t i o n o f s t re s s w h i l e a c c o u n t i n g f o r t h e n o n - l i n e a r r e s p o n s e i n t h el o n g i t u d i n a l a n d l a t e r a l d e f o r m a t i o n s o f t h e r o c k m a t e r i a l i n c o m p r e s s i o n .

( d ) T o p r o v i d e a m e a n s f o r a s c e r t a i n i n g t h e o n s e t o f s t r e n g t h f a il u r e in t h e r o c k s t r u c t u r ea n d t o d e t e r m i n e t h e r e s u l t i n g s tr e ss d i s t r i b u t i o n .

A s i m p l e m e t h o d o f s o l u t io n w h i c h w i l l b e u s e d i n t h is p a p e r i s t h a t o f s u c ce s s iv e a p p r o x i -m a t i o n s [1 3] . S in c e e a c h a p p r o x i m a t i o n i n v o l v e s a s o l u t i o n o f a l in e a r p r o b l e m , i t i s o n l yn e c e s s a r y t o p r o v i d e a m e a n s f o r t h e c o m p u t a t i o n o f th e s t if fn e s s m a t r i x b e f o r e e a c h a p p r o x i -m a t i o n .

( a ) A f t e r t h e e l a s t i c s o l u t i o n f o r t h e s t r u c t u r e h a s b e e n r e a c h e d b y t h e f i n i t e - e l e m e n tm e t h o d , t h e m i n i m u m p r i n c i p a l s t r e s s i n s o m e e l e m e n t s m a y b e t e n s i l e a n d t h e m a x i m u mp r i n c i p a l s t r es s c o m p r e s s i v e . T h e c o n s t i t u t i v e e q u a t i o n o f th e e l e m e n t s i n w h i c h t h i s o c c u r s


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9 0 G . B A R L Ai s t h a t o f a t ra n s v e r s e l y i s o t r o p i c m e d i u m . W i t h r e f e r en c e t o t h e c o o r d i n a t e s y s t e m o f th ee n t i r e s t r u c t u r e t h i s e q u a t i o n w r i t e s a f t e r t r a n s f o r m a t i o n

{ ~-} = [ A V [ c ] [ a ] ( , } ( 1 )w h e r e

c o s 2 f l s i n 2 f l - - 2 s in f l c o s f l ][A ]---- / s i n u /3 c o s 2 f l 2 s i n f c o s 3 J ( 2)

/Ls i n 13 c o s f l - s i n 3 c o s 3 c o s z 3 - s i n 2 3f o r p l a n e s t re s s c o n d i t i o n s :

[C ] = (1 - - mr2 2) a z 1 0 (3 )0 m ( 1 - - n c r 2 2 )

f o r p l a n e s t r a i n c o n d i t i o n s :I n (1 - - m r 2 2 ) n ~ 2 ( l + ~ i ) 0 ]E 2 / n ~ 2 ( l + or1) (1 - - ol 2) 0 ] (4)

[C] = (1 -k ( rx) (1 - - (r~ - - 2no,22) m(1 -k o r , )[ o 0 (1 - - t r 1 - - 2 n t x 2 2 ) _ ]f l i s e q u a l t o t h e a n g l e b e t w e e n t h e d i r e c t i o n o f th e t e n s i le s t re s s a n d t h e h o r i z o n t a l , /7 1 a n dE 2 a r e t h e e l a s ti c m o d u l i i n t h e d i r e c t i o n o f t h e t e n s il e a n d c o m p r e s s i v e s t r e s s e s r e s p e c t iv e l y .~ x a n d or2 a r e t h e a s s o c i a t e d P o i s s o n ' s r a t i o s . G 2 is t h e i n d e p e n d e n t s h e a r m o d u l u s , m a n d na r e t a k e n a s

E ~ G 2n : - - m : - - ( 5)E 2 ' E 2a n d t h e in d e x T i n d i ca t e s m a t r i x t r a n s p o s e .T h e s t if f n e ss o f th e r o c k s t r u c t u r e i s r e f o r m e d a n d a n e w e l a s ti c s o l u t i o n i s a t t a i n e d . T h ep r o c e s s i s r e p e a t e d u n t i l t h e d i f f e r e n c e s i n s t r e s s e s a n d d i s p l a c e m e n t s o b t a i n e d i n s u c c e s s i v ea p p r o x i m a t i o n s i s n e g l ig i b le .( b ) T h e z o n e i n t h e r o c k s t r u c t u r e w h e r e t h e p r o c e s s o f fr a c t u r e i s i n it ia t e d is d e t e rm i n e db y a p p l y i n g t h e G r i f f it h a n d m o d i f i e d G r i f f i th c r i t e r io n s o f f r a c t u r e i n i t i a t i o n [ 7] .

A f t e r s o l v i n g t h e s t r u c t u r e w i t h t e n s i l e - - c o m p r e s s i v e n o n - l i n e a r i t y f o r t h e r o c k m a t e r i a l ,t h e s t r e s s e s i n e a c h e l e m e n t ( i ) a r e :

{ r } , = ( r ~ } , ( 6 )( i = I , . . . , N )


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A M E T H O D F O R T H E A N A L Y S IS O F S TR E SS I N B R I T T L E R O C K 91w h e r e N s t a n d s f o r t h e n u m b e r o f e le m e n t s i n t h e s t ru c t u r e . T h e r e f o r e , t h e G r i f f it h a n dm o d i f i e d G r i f f i th c r i t e r i o n s o f f r a c t u r e i n i t i a t io n c a n b e a p p l i e d a s f o l l o w s :

( i ) F o r

~ ( ' ] ~< - - 0 . 3 3 ( 7 )\ ~ 1 / if r a c t u r e i n i t ia t e s w h e n

(~) l > / To(; = l . . . . , N ) .

( i i ) F o r

(8 )

- - 0 " 3 3 ~< ~< - - ( 9 ), [ 1 + 2 tL - - 2 V ( (1 + /z2 )) ]f r a c t u r e i n i t i a t e s w h e n

2 C o [ ( r 2 ) ] [ , v / ( ( l - ~ - / x 2 ) ) - - / z1 +1t(~,) , >/ 0 o )

( i i i ) F o r( i : 1 . . . , N ) .

1 ( ~ ' 2 ) [ x / ( ( l q - / x 2 ) ) - U ] ( 1 1 )[1 + 2~ + 2 V( (1 + ~ 2))] ~< ~ , ~< IV((1 + ~2)) + t~]

f r a c t u r e i n i t i a t e s w h e n

( i v ) F o r

[v '( (1 + ~ ) ) - n ] C o

( i = 1 . . . , N ) .

( 1 2 )

~ - ~ ) [ V ( ( 1 + t ~ ) ) - t~ ]/>, [ V ~ ( ( 1 - I- ~ 2 ) ) ~ _ t t ]( i = 1 , . . . , N )

n o f r a c t u r e c a n i n i t ia t e .

(13)

C o i s t h e l o n g i t u d i n a l s t r e s s a t f r a c t u r e i n i t i a t i o n u n d e r u n i a x i a l l o a d i n g . T o i s t h e t e n s i les t r e n g t h , a n d t~ t h e c o e f f i c i e n t o f i n t e r n a l f r i c t i o n o f t h e r o c k m a t e r i a l .

T h e f in i te e l e m e n t s ( L ) w h i c h i n i ti a t e f r a c t u r e a c c o r d i n g t o ( i) a r e a s s u m e d t o o b e y b y t h ec o n s t i t u t i v e e q u a t i o n ( 1 ) . H o w e v e r , i n t h i s c a s e th e m o d u l u s o f e l a s t i c i ty ( E 1 )j ( j = 1 . . . , L )i s s e t e q u a l t o z e ro . A c c o r d i n g l y , t h is c o n d i t i o n i s e q u i v a l e n t t o a s s u m e t h a t f r a c t u r e i n i t ia -t i o n a n d s t r e n g t h f a i l u re i n t e n s i o n o c c u r s i m u l t a n e o u s l y , w h i c h i s k n o w n t o b e s a t i s f a c t o r yf o r a n y p r a c t i c a l p u r p o s e .


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92 G. BARLAT h e f i n it e e l em e n t s , w h i c h i n i t i a te f r a c t u r e a c c o r d i n g t o e i t h e r ( i i) o r ( i ii ) a n d f o r w h i c h t h e

m i n i m u m p r i n c i p a l s t r e s s is te n s il e , a r e t r e a t e d a s i n ( a) . H o w e v e r , t h e f i n i te e le m e n t s ( M )w h i c h i n i t i a t e f r a c t u r e a c c o r d i n g t o ( i i i ) a n d f o r w h i c h t h e m i n i m u m p r i n c i p a l s t r e s s i sc o m p r e s s i v e , a r e t r e a t e d a s a n i s o t r o p i c a n d l i n e a r l y e l a s ti c m e d i u m w i t h t h e P o i s s o n ' sr a t i o ( c rl )j ( = 1 . . . . M ) p r o p e r l y i n c r e as e d , in o r d e r t o a c c o u n t f o r t h e a l re a d y m e n t i o n e dd e p a r t u r e f r o m l i n e a r i t y o f t h e s t r e s s - l a t e r a l s t ra i n c u r v e . F o r t h e f in i te e l e m e n t s w h i c h o b e y( iv ) , t h e m e c h a n i c a l p r o p e r t i e s a r e n a t u r a l l y l e f t u n c h a n g e d .

T h e s o l u t i o n i s a g a i n a t t a i n e d b y a p r o c e s s o f su c c e ss i ve a p p r o x i m a t i o n s . H o w e v e r , int h e p r e s e n t c a s e t h e s o l u t i o n i s a c c e p t e d w h e n t h e r e a r e n o m o r e e l e m e n t s f a i l i n g a c c o r d i n gt o ( i ) .

( c ) T h e p r o b l e m is n o w t h a t o f d e t e r m i n i n g f o r t h e r o c k s t r u c t u r e t h e t r a n s i t i o n f r o ms t a b l e t o u n s t a b l e f r a c t u r e p r o p a g a t i o n u n d e r c o m p r e s s i v e l o a d . A c r i t e r i o n e x i s t s w h i c ha l l o w s o n e t o a s c e r t a i n t h e o n s e t o f u n s t a b l e f r a c t u r e p r o p a g a t i o n [ 14 ].

E s s e nt ia l ly , f r a c t u r e p r o p a g a t i o n b e c o m e s u n s t a b l e w h e n t h e e n e r g y r el e a se d p e r u n i tc r a c k s u r f a c e a t t a i n s a k n o w n c r i t i c a l v a l u e . E x p e r i m e n t s h a v e b e e n c a r r i e d o u t b yBII?N IA W SK I [ 4] t o d e t e r m i n e s u c h a c r i t i c a l v a l u e a l s o i n s o m e r o c k t y p e s .

S o m e a t t e m p t s h a v e b e e n m a d e b y t h e a u t h o r t o a p p l y t h i s c r i t e r i o n i n c o n j u n c t i o nw i t h t h e f i n i t e- e l em e n t m e t h o d . H o w e v e r , f o r t h e p u r p o s e o f t h e p r e s e n t i n v e s t ig a t io n , u s e i sm a d e o n l y o f a c r i te r i o n e s ta b l is h e d o n p u r e l y p h e n o m e n o l o g i c a l g r o u n d s . T h e r e f o r e , i tw i ll b e a s s u m e d t h a t t h e o n s e t o f u n s t a b l e f r a c t u r e p r o p a g a t i o n t a k e s p la c e w h e n e v e r

[ ( ~ , ) , - ( ~ , ) , ] / > S o J F + A F ( ~ ' ) ' + ( ~ 2) , "2 ( ] 4 )w h e r e S o J r , A t , a n d m r a r e c h a ra c t e r is t i c c o n s t a n t s f o r t h e r o c k m a t e r i a l . T h e s e c o n s t a n t sc a n b e e a s i ly d e t e r m i n e d b y p l o t t i n g o n l o g a r i th m i c s c al es t h e r e s u lt s o f c o n v e n t i o n a lu n i a x i a l a n d t r i a x i a l c o m p r e s s i o n t e s t s a n d b y f i t t i n g t h e b e s t s t r a i g h t l i n e s t o t h e e x p e r i -m e n t a l p o i n t s b y t h e m e t h o d o f le a s t s q u a r e s [1 5]. C l e a r ly , in t h i s c a s e t h e r e s u lt s a r e t h o s ef o r t h e o n s e t o f u n s t a b le f r a c t u r e p r o p a g a t i o n .

T h e f in i te e le m e n t s ( P ) w h i c h r e s u l t to b e i n a c o n d i t i o n o f u n s t a b le f r a c t u r e p r o p a g a t i o na c c o r d i n g t o t h e p h e n o m e n o l o g i c a t c r i t e r i o n ( 1 4 ) w i l l b e g i v e n v a l u e s o f ( E k ) l and (ek)~( k = 1 , 2 ; i = 1, P ) , w h i c h a c c o u n t f o r t h e n o n - l i n e a r r e s p o n s e o f b o t h t h e a x i a l a n d t r a n s -v e r s a l d e f o r m a t i o n s . A p r o c e s s o f s u c c es s iv e a p p r o x i m a t i o n s i s u s e d t o a t t a i n a p i e c e - w i s en o n - l i n e a r s o l u t io n .

( d ) T h e c r i t e r i o n u s e d i n o r d e r t o a s c e r t a i n t h e s t r e n g t h f a i l u r e i s a g a i n e s t a b l i s h e d o n ap u r e l y p h e n o m e n o l o g i c a l b a s is . I t i s a ss u m e d f o r t h e o n s e t o f s t re n g t h f a i l u re

1 [ ] 0 5 )2w h e r e S o J s , A s a n d m s a r e t he s a m e p a r a m e t e r s d e f i n ed i n ( c ) b u t w i t h r e f e r e n c e t o t h es t r e n g t h f a i l u r e e n v e l o p e , a s i n d i c a t e d b y t h e s u b s c r i p t S .

T h e f i n it e e l e m e n t s ( Q ) w h i c h f ai l a c c o r d i n g t o ( 1 5 ) a r e a s s u m e d t o h a v e c o m p l e t e l y l o s tt h e i r l o a d - c a r r y i n g c h a r a c t e r i s t i c a n d a r e g i v e n z e r o v a l u e s o f (Ek )~ ( k --= 1 , 2 ; i - = 1, Q ) .A g a i n , t h e p r o c e s s o f s u cc e ss iv e a p p r o x i m a t i o n s c a n b e u s e d t o a t t a i n t h e c o r r e s p o n d i n gs o l u t i o n .

I t s h o u l d b e o b s e r v e d a t t hi s p o i n t t h a t t h e a b o v e a s s u m p t i o n d o e s n o t h o l d t r u e i n r e a lc a s e s . S o m e r o c k m a t e r i a l s w i t h b r i t t l e f r a c t u r e b e h a v i o r a r e k n o w n t o e x h i b i t , a l s o a f t e r


7/16

A M E T H O D F O R T H E A N A L Y S I S O F S T R E SS I N B R I T T L E R O C K 9 3

t R E A ~ I N P U T O A T A IKJ=0 ~ JK=0l ,_PRINT ELEMENTS MECHANCALPROPERTESI~I FORMELEMENT STIFFNESSES

I O~TAIN LINEARLY ELASTICSOLUmION II PRINT ANOP'OT OSPLACEMENmSANOSTRESSESI

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d~N~TK ~ YESNO

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t APPLYCRITERIONS OFFRACTUREINITIATION IPRINT ELEMENTSANDRANGESIN WHCHFRACTUREIS INITIATEDI ASSIGN MATERIAL PROPERTES TOEACHELEMENT IAS IN (b)

APPLY CRITERION FORUNSTABLEFRACTUREPROPAGATONPRINT ELEMENTSIN UNSTABLEFRACTUREPROPAGATON

CHANGEMECHANCAL PROPERTESAS IN (c)

CHECKFORELEMENTSIN TENSIONANDASSIGN MATERIALPROPERTESACCORDNGTO ( a )CHECKFORELEMENTSIN TENSION -NOT FAILING-ANDASSIGN MATERIALPROPERTESACCORDNGTO (a)

I PPLYCRITERION FORSTRENGTHFAILURE IPRINT ELEMENTS IN STRENGTHFAILURE ~ ~oCHANGEMECHANCAL PROPERTESAS IN (d)NITER ~ = NUMBEROFITERATIONS NEEDEDFORCONVERGENCEOF

NON-LINEAR SOLUTONNITER 2 = NUMBEROFITERATIONS NEEDEDFORCOMPLETE

SOLUTONF IG . 2 . S i m p l i f ie d f l o w d i a g r a m o f t h e c o m p u t e r p r o g r a m f o r t h e n o n - l in e a r s o l u t i o n s .

v


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94 G. BAR LAstrength failure, a carrying load characteristic, which depends upon the stiffness of thecomplete rock mass where the opening is located.

A computer program was written in order to perform the above analysis. A simplifiedflow scheme of this program is shown in Fig. 2. It is apparent that each problem from (a)to (d) can be solved either separately or in a sequence as shown in the flow scheme. In the lastcase, a solution is attained which accounts for the complete physical behavior of the rockmaterial up to strength failure.

Some comments on the convergence of the proposed method of solution are appropriateat this point. One is generally not assured of convergence. However, as shown in thefollowing article, an acceptable solution can be at tained after a sufficient number of succes-sive l inear analyses. This number is increased as the non-linearity increases.

Furthermore, it should be noticed that the final solution is valid only for elastic materials,i.e. it is assumed that the same constitutive equations hold true for either the loading or theunloading processes.

NUMERICAL EXAMPLEIn order to illustrate the foregoing theoretical approach, a numerical example for acircular opening located in norite is presented in the following. This particular rock type

has been chosen for the analysis as experimental results on the complete physical behaviorof norite--up to strength failure--were available [4]. These results for norite tested incompression are reported in Table 1. Then, the diagrams of Figs 3 and 4 are easily derivedin order to obtain, as reported in Table 2, the data needed for the mathematical descriptionof the various processes of brittle fracture. It should be observed that for the purpose ofthe following analysis the coefficients mp and ms are assumed to be equal to unity so that astraight line is fitted to the experimental data as shown in Fig. 4. The material properties

TABLE 1. DATAFOR NORITE TESTED IN COMPRESSION BIENIAWSKI[4])Longitudinalstress Longitudinal stress (psi) atfracture unstable strengthLateral initiation fracture failurestress propagation

oo 12,000 34,000 44,50038" 2 13,350 39,530 69,50013" 4 17,280 54,400 111,8009" 2 21,900 59,000 143,990

TABLE 2. DATA DERIVED FOR THE MATHEMATICALDESCRIPTION OFTHE VARIOUSPROCESSES OF BR1TTLE FRACTUREFracture initiation -- 0.750Uns tab le fracture propagation 6831" 63 0" 608Stre ngth failure 7108.37 0" 725

SoJl, , SoJs tL , A t , A spsi


9/16

A M E T H O D F O R T H E A N A L Y S I S O F S T R E S S I N B R I T T L E R O C K 9 5TABLE 3. MATERIAL PROPERTIES ASSUMED FOR THE NORITE IN

UNI AXI AL COMPRESSION AND TENSION

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T h e n e x t s t e p i s, a s a l w a y s i n t h e f i n it e - e l e m e n t a n a l y s is , t o c o n s t r u c t a n a p p r o p r i a t ef i n it e - e le m e n t m o d e l . D u e t o t h e g eo m e t r i c a l s y m m e t r y o f t h e s tr u c tu r e a n d t h e n a t u r e o ft h e s t r e s s f i e l d h e r e c o n s i d e r e d ( g r a v i t y e f f e c t s a r e n e g l e c t e d , p l a n e - s t r a i n c o n d i t i o n s a r ea s s u m e d , a n d a u n i f o r m u n i a x i a l s t re s s fi el d i s a p p l i e d ) , t h e m o d e l c o n s i s t s o f a q u a r t e r - p l a t e

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R e g i o n 1TTF1G. 6. Circular opening in a norite ro ck m ass under a uniaxial stress field ( '~v ~ 5000 psi )--F rac ture zonesand principal stresses.

o n l y . P o i n t s a l o n g t h e h o r i z o n t a l a x i s a r e g i v e n z e r o v e r t ic a l d i s p l a c e m e n t s . C o n v e r s e l y ,p o i n t s a l o n g t h e v e r t i c a l a x i s a r e g i v e n z e r o h o r i z o n t a l d i s p l a c e m e n t s . T h e u n i a x i a l s t r e s sf ie ld is o b t a i n e d w i t h a l o a d u n i f o r m l y d i s t r i b u te d o n t h e h o r i z o n t a l o u t e r b o u n d a r y a n d b yl e a v i n g th e p o i n t s a l o n g t h e v e r t i c a l o u t e r b o u n d a r y f r e e t o d is p l a c e in a n y d i r e c t i o n .

T h e s t r e s s i n c r e m e n t s i n t h e a n a l y s i s a r e s e t to b e e q u a l t o 2 5 0 0 p s i , s t a r t i n g w i t h a v a l u e o fr v = 5 0 0 0 p s i. A s t h e t e n s i l e - c o m p r e s s i v e n o n - l i n e a r i t y i s f o r n o r i t e v i r t u a l l y n o n - e x i s t e n t ,o n l y f o u r i t e r a t i o n s w e r e n e e d e d i n o r d e r t o o b t a i n t h e f ir s t n o n - l i n e a r s o l u t i o n (i .e . N ]T ER1 = 4 i n th e d i a g r a m o f F ig . 2 ). E i g h t c o m p l e t e c y c l es w e r e i n s t e a d m a d e f o r e a c h s t r e s si n c r e m e n t ( i. e. N IT E R 2 = 8 i n t h e d i a g r a m o f F i g . 2) .

T h e n u m e r i c a l r e s u l t s a r e i l l u s t ra t e d i n F i g s 5 - 1 0 . E a c h f ig u r e r e f e rs t o a c o m p l e t e s o l u t i o n .R e p o r t e d a r e ( in a q u a r t e r - p l a t e o n l y ) t h e z o n e s i n w h i c h t h e v a r i o u s p r o c e s s e s o f f r a c t u r ea r e e x p e c t e d t o o c c u r . A l s o , t h e p r i n c i p a l s t r es s e s , n o r m a l i z e d t o t h e a p p l i e d s t r e s s T o, a r ep l o t t e d i n e a c h fi g u r e a l o n g t h e h o r i z o n t a l a n d v e r t i c a l a x e s .

F i g u r e 5 s h o w s t h e s o l u t i o n o b t a i n e d f o r T o ---- 3 0 0 0 p s i . F o r t h i s a p p l i e d s t r e s s f r a c t u r e i nr e g i o n ( I ) i s p r e s e n t a t t h e b a c k a n d f l o o r o f t h e o p e n i n g o n l y . T h e r e s u l t i n g s t r e s se s a r es i m i l a r t o t h o s e d e r i v e d t h r o u g h t h e l i n e a r l y e l a s ti c s o l u t io n .

A s i l l u s t r a te d i n F i g . 6 , f o r a n a p p l i e d s t r e ss % = 5 0 00 p s i t h e f r a c t u r e z o n e s a t t h e b a c ka n d f l o o r a r e e x t e n d e d a n d f r a c t u r e i n i t i a t e s , i n r e g i o n ( I I ) , a t f o u r p o i n t s i n t h e s t r u c t u r e ,R O C K 9 1 1 - - O


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Region rrrF IG . 7 . C i r c u l a r o p e n in g i n a n o r i t e r o c k m a s s u n d e r a u n i a x i a l st re ss f ie l d (T o = 7 5 0 0 p s i ) - -F ra c tu re z o n e s

and principal stresses.a n d i n r e g i o n ( I I I ) , a t t h e r i bs . A c h a n g e i n t h e d i s t r i b u t i o n o f s t re s s a l o n g t h e v e r t i c a l a x iso c c u r s . T h e s t re s s e s a l o n g t h e h o r i z o n t a l a x is , h o we v e r , a r e n o t a f f e c te d .

I f t h e a p p l i e d s t r e s s ~'o i s f u r t h e r i n c r e a s e d , t h e t h r e e z o n e s i n w h i c h f r a c t u r e o c c u r s e x t e n d .F i g u r e 7 r e f e r s t o ro = 7 5 0 0 p s i . I t s h o w s , w h e n c o m p a r e d w i t h F i g . 6 , t h a t f r a c t u r e i nr e g i o n ( I ) p r o p a g a t e s a l o n g t h e v e r t i c a l a x i s a n d f r o m t h e o p e n i n g o u t w a r d s . N o t a b l ez o n e s i n wh i c h f r a c t u r e i s i n i t i a t e d i n e i t h e r r e g i o n ( I I ) o r ( I I I ) a r e n o w p r e s e n t . Wh i l e t h ed i s t r i b u t i o n o f s t re s s a l o n g t h e v e r t i c a l ax i s i s a g a i n c h a n g e d , t h e s t r e ss e s a l o n g t h e h o r i z o n t a la r e p r a c t i c a l l y e q u a l t o t h o s e o b t a i n e d w i t h t h e l i n e a r ly e la s t ic s o l u t i o n .

F i g u r e s 8 a n d 9 s h o w t h a t t h e f r a c t u r e i n r e g i o n ( I ) s t a b i l i z e s , wh e n t h e a p p l i e d s t r e s s i si n c r ea s e d . H o w e v e r , f r a c t u r e i n r e g i o n ( I I ) m o v e s t o w a r d t h e v e r ti c a l a x is a n d f r a c t u r e i nr e g i o n ( I I I ) e x te n d s t o a m a j o r p o r t i o n o f t h e s t r u c tu r e . T h e d i s t r i b u t i o n o f s t re s s is in b o t hc a s e s e q u a l t o t h a t o f t h e p r e v i o u s f i g u re .

A c o n s i d e r a b l e c h a n g e i n t h e f r a c t u r e p a t t e r n o c c u r s f o r T o = 1 5,0 00 p s i ( F i g . 1 0 ) . T h ec o m p l e t e s o l u t i o n o b t a i n e d i n t h is c a s e s h o w s t w o z o n e s i n w h i c h t h e p r o c e s s e s o f u n s t a b l ef r a c t u r e p r o p a g a t i o n a n d s t r e n g t h fa i l u r e o c c u r i n t h e p r o x i m i t y o f t h e o p e n i n g r i bs . T h ed i s t r i b u t i o n o f s t r e s s a l o n g t h e h o r i z o n t a l a x i s i s c h a n g e d c o n s i d e r a b l y a n d a t r a n s f e r o fs t re s s f r o m t h e d e s t r o y e d z o n e , i n t h e n e a r v i c i n i t y o f t h e o p e n i n g , t o t h e r o c k u n d e r f r a c t u r ei n i t i a t i o n t a k e s p l a c e . T h e s t re s s e s a l o n g t h e v e r t i c a l a x i s r e m a i n t h e s a m e .

A f u r t h e r i n c r e m e n t o f 25 0 0 ps i i n t h e a p p l i e d s t re s s le a d s t o a n u n s t a b l e s o l u t i o n . T h e r e -f o r e , i t i s c o n c l u d e d t h a t , a c c o r d i n g t o t h e p r e s e n t m e t h o d o f a n a l y si s , s t r e n g t h f a i lu r e i n a


13/16

A M E T H O D F O R T H E A N A L Y S I S O F S TR ES S I N B R I T T L E R O C K 9 9n o r i t e r o c k m a s s c o n t a i n i n g a c i r c u l a r o p e n i n g w o u l d o c c u r , u n d e r u n i a x i a l s t r es s fi el d,f o r % = 1 5 ,0 0 0 p s i, w h e r e a s t h e c o m p r e s s i v e s t r e n g t h f o r t h e n o r i t e i s 4 4 , 5 0 0 ps i . A l s o , af u r t h e r i n c r e a s e i n t h e a p p l i e d s t r e ss o f 2 5 0 0 p s i w o u l d r e s u l t i n t h e c o m p l e t e f a i l u r e o f t h er o c k s t r u c t u r e .

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R e g i o n f fFIG. 8. Circular opening in a norite ro ck m ass und er a uniaxial stress field (~-~ = 10,000 ps i)--F rac tur e zonesand principal stresses.

C O N C L U D I N G R E M A R K ST h e t h e o r e t i c a l m e t h o d p r o p o s e d i n t h is p a p e r a l l ow s o n e t o e f fe c t a s t a b il i ty a n a l y s i s

o f a r o c k s t r u c t u r e . T h e c o m p l e t e p h y s i c a l b e h a v i o r o f a r o c k m a t e r i a l w h i c h e x h i b it sb r i t t l e f r a c t u r e i s c o n s i d e r e d .

T h e z o n e s i n t h e r o c k s t r u c t u r e w h i c h u n d e r g o t h e v a r i o u s p r o c e s s e s o f f r a c t u r e (i .e .f r a c t u r e i n i ti a t i o n , u n s t a b l e f r a c t u r e p r o p a g a t i o n a n d s t r e n g t h f a i lu r e ) c a n b e d e t e r m i n e da n d t h e d i s t ri b u t i o n o f s t re s s c o m p u t e d . T h e G r i ff i th a n d m o d i f i e d G r i f fi t h c r i t e r io n s a r eu s e d f o r p r e d i c t i n g f r a c t u r e in i t ia t i o n . T w o c r it e r io n s , f o r m u l a t e d o n a p u r e l y p h e n o m e n o -l o g i c al b a s is , a r e i n t r o d u c e d i n o r d e r t o d e t e r m i n e t h e o n s e t o f u n s t a b l e f r a c t u r e p r o p a g a t i o na n d s t r e n g t h f a i lu r e . I t is a s s u m e d t h a t f r a c t u r e i n i ti a t i o n a n d s t r e n g t h f a i l u re i n t e n s i o no c c u r s i m u l t a n e o u s l y .


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I00 G. BARLA

T h e m e t h o d o f s o l u t io n i s b a s e d u p o n t h e a p p l i c a t i o n o f t h e fi n i te - e le m e n t a n a ly s is a n d ap r o c e s s o f s u c c e ss i v e a p p r o x i m a t i o n s . T h e a p p l i e d s t re s s i s i n c r e a s e d b y s t e p s u n t i l ac o m p l e t e f a i l u r e o f t h e s t r u c t u r e i s r e a c h e d . I t i s i m p l i e d t h a t t h e s t r e n g t h f a i l u r e o f t h e r o c ks t r u c t u r e i s ac h i e v e d w h e n e v e r i t i s i m p o s s i b le t o m a k e t h e p r o c e d u r e c o n v e r g e n t ,

L.

T 2 / T v

-I 0

+ 2r / o

TI /Tv

T 2 / T V--.r-.-.-.-

3

iEE~Lq] R e g i o n IF rac t i on [T IT T ~ ] R e g i o n 1"[ U n i o x i a l s t r e s s ( r v )i n i t ia t i o n 1 2 , 5 O 0 psi

I - ' - - - - ] R e g i o n T I -[FIG. 9. Circular opening in a nor ite rock mass under a uniaxial stress field (zv = 12,500psi)--Fracture zonesand principal stresses.

A n u m e r i c a l e x a m p l e f o r a n o r i t e r o c k m a s s c o n t a i n i n g a c i r c u la r o p e n i n g s h o w s t h a t ac a r e f u l d e s c r ip t i o n o f th e f r a c t u r e p r o c e s s e s w h i c h a r e e x p e c t e d t o o c c u r i n t h e r o c k s t r u c tu r e ,u n d e r i n c r e a s in g c o m p r e s s i v e l o a d , c a n b e e f fe c te d . I n p a r t i c u l a r , i t h a s b e e n s h o w n t h a tu n d e r a u n i a x i a l s t re s s fi el d t h e f r a c t u r e w o u l d i n i t i a te a t t h e b a c k a n d f l o o r o f t h e o p e n i n ga n d i t w o u l d p r o p a g a t e a l o n g t h e v e r ti c a l a x is , f r o m t h e o p e n i n g o u t w a r d s . A s t h e a p p l i e ds t r es s is i n c r e a s e d , a r e g i o n i n t h e n e a r v i c i n i t y o f t h e o p e n i n g r i b s u n d e r g o e s , i n s e q u e n c e ,t h e p r o c e s s e s o f f r a c t u r e i n i t ia t i o n , u n s t a b l e f r a c t u r e p r o p a g a t i o n a n d s t r e n g t h f a i l u re .

I f t h e r e s u lt i n g d i s tr i b u t i o n o f st re s s i s c o m p a r e d w i t h t h e o n e o b t a i n e d b y t h e l i n e a r lye l a s t ic s o l u t i o n , i t is s e e n t h a t , a f t e r t h e i n i t i a t i o n o f f r a c t u r e i n t h e b a c k a n d f l o o r r e g i o n s ,t h e s t r e s se s a l o n g t h e v e r t i c a l ax i s a r e c h a n g e d c o n s i d e r a b l y . A l s o , t h e i n i t i a t i o n o f f r a c t u r ei n o t h e r r e g i o n s o f th e s t r u c t u r e d o e s n o t m o d i f y t h e s t re s se s a l o n g t h e h o r i z o n t a l a x is . T h es a m e s t r e ss e s a re g r e a t l y a f f ec t e d a f t e r th e o n s e t o f u n st a b l e f r a c t u r e p ~ p a g a t i o n a n ds t r e n g t h f a i l u r e a t t h e r i b s.


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A METH OD FOR THE ANALYSIS OF STRESS IN BRITTLE ROCK 101

~ i ~ L . : ~

] ' -2 / "v

I- I

+ o I 2

" 1" / - r v. , . . . . . _ _ . . . _

"1"2 / -r v

3r / a

F r a c t u r e I - ~i n i t i o t i o n

U n s t o b l ef r a c t u r ep r o p a g a t i o nS t r e n g t hf a i l u r e

R e g i o n IR e g i o n T [ U n i a x i o l s t r e s s ( 'i v)1 5, 0 0 0 p s iR e g i o n ] I ] [

FIG. 10. Circular opening in a norite rock mass under a uniaxial stress field (~'v = 15,000 psi )--Fra ctu re zonesand principal stresses.

REFERENCESI. BLAKE W. Application o f the finite-element met hod in solving boundar y values prob lems in rockmechanics. Int. J. Rock Mech. Min. Sci. 3, 169-180 (1966).2. BARLA G. The Distributio n of Stress around Under groun d Openings --Effect s of some Geologic andMechanical Features of the Rock Mass, Proe. il primo convegno lnternazionale sui Problemi Tecnicinella Costruzione di Gallerie, Torino (1969).3. ZmNKIEWICZ O, C., VALLIAPPAN S. and KINO I. P. Stress analysis of rock as a 'no ten sion' ma teria l.G~oteehnique 18, 56-66 (1968).4. BIENIAWSKIZ. T. Mechanism of Brittle Fracture of Rock, Report CSIR (S. Aft) No. 580 (1967).5. GRIFFITH A. A. Theory of Rupture, Proceedings of the International Congress of Applied Mechanics, pp.55-63, J. Waltman Jr, Delft (1925).6. McCLl~rrocK F. A. and WALSH J. B. Friction on Griffith Cracks in Rocks unde r Pressure, Proceedingsof the Fourth U.S. National Congress of Applied Mechanics, pp. 1015-1021 (1963).7. HOEK E. Rock Fracture a round Mining Excavations, Proceedings of the Fourth International Conferenceon Strata Control in Rock Mechanics, pp. 334~348, New York (1964).8. BIENIAWSKIZ. T. and VAN TONDER C. P. G. A photoe lastic model study of stress dist ribution and rockfracture around mining excavations. Expl Mech. 9, 75-81 (1969).9. BIENIAWSKIZ. Y. Mechanism of Rock Fracture in Compression, Report CSIR (S. Afr.) No. 459 (1966).10. BRACEW. F., PAULDING B. W. and SCHOLZ C. Di latancy in the fractu re of crys talline rocks. J. geophys.Res. 71, 3939-3953 (1966).11. BARLA G. Some Constituti ve Equations for Rock Materials, Proceedings of the Eleventh Symposiumon Rock Mechanics, Berkeley (1969).


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102 G. BARLA12. Z~ENK~wI~z ~. C. and CHEUN~Y.K. The Finite E~ement Meth~d in Structural and C~ntmuum`~echanics~McGraw-Hill, London (1967).13. WILSONE. L. Finite Element Analysis of Two-dimensional Structure, Doctoral Dissertation, Universityof California, Berkeley (1963).14, IRWIN G. R. Fracture Mechanics, in Structural Mechanics (J. N. Goodier and N. J. Hoff, Eds)pp. 557-592, Pergamon Press (1960).15. HOEK E. Brittle Fracture of Rock, in Rock Mechanics in Engineering Practice, pp. 99-124, Wiley, NewYork (1968).

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1972_Brittle Failure in Rocks