2. MARCO EPISTEMOLÓGICO 50
2.4. EL PROFESOR INVESTIGADOR DE SU PROPIA ENSEÑANZA 55
As d isc u s s e d at th e en d o f C h a p te r 3, a slip -d ep e n d e n t c o n stitu tiv e relatio n fo r sh ear failu re has b e e n c o n stru c te d from ad v a n ce s in fractu re m ec h a n ic s th eories o f failure. T he c o n s titu tiv e relatio n can b e ex p ressed m ath e m a tic a lly as (O h n ak a, 1996):
x = f ( D , D , a f , T , e , > . , , C E ) (6.5) w here x a n d are th e sh e ar and effe c tiv e no rm al stresses a c tin g on a fault, D is the slip d isp la c e m e n t across th e fault, D is th e slip velocity, T is th e tem p eratu re, é is the strain rate , Xc is th e c h a ra c te ristic len g th o f th e g eo m etrical irreg u larities on th e fault p lan e a n d C E d en otes th e ch e m ica l e ffe c t o f th e p o re fluid. It has b een a ssu m e d th a t t d ep en d s p rim a rily on D , w ith th e d e p e n d en c e o f o th er p aram eters b e in g o f seco ndary sig n ific a n ce . T o d escrib e fu n ctio n (6.5), all th e exp erim en tal d a ta (typically show n as
c a lc u lated in S ectio n 5 .5 .4 from the biaxial stress e q u a tio n s. T he slip along the fault m ust now be ca lc u la te d from the follow ing equation:
A1
(6.5)
Dapp is know n as a p p a r e n t slip sin ce it in clu d es the lin ear H ookean elastic d eform ation o f the sam ple, w h ich is seen as the line A in F igure 6.3a. L in e A is represented by:
x = - a + bD^i (6.6)
A ssu m in g elastic d efo rm atio n , this elastic part, Dei, m u st be rem oved in o rd er to d eterm in e the true slip d isp lace m e n t, D, along the fault.
( t + a)
(6.7)
T h is resu lts in a rep resen tatio n o f a se lf-c o n siste n t co n stitu tiv e relation for sh ear rupture; the sh ea r stress, i, versus slip d isp lace m e n t, D for in tact T su k u b a g ranite (F ig u re 6.3b). T h e sh ad e d part rep resents the fracture en e rg y and is d efined by (R ice
1980): = p [ i ( D ) - x J d D (6.8) Ji>f, (a) 400 5 300 m 200 « 100 K110595 0 .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 app 500 K110595 4 00 K AXb
IlkrtniTT^ i
Residual friction stress level 300 S V) V)0) V) 200 s CO 100 '5W 0.6 0.8 1.0 1.2 0.0 0.2 0.4 S lip d i s p l a c e m e n t (m m)F ig u re 6.3 Deriving the slip-dependent constitutive relation for shear rupture from experimental data from this study, x, is the critical (initial) shear stress at which stain-hardening occurs, Xp is the peak shear stress. Xr is the residual shear (frictional sliding) stress. AXh is the difference between Xp and Xr. Da is the slip displacement at which peak stress is attained. D«w is the slip displacement over which slip-weakening occurs, and Dc is the critical slip displacement, which is the amount of displacement required to degrade the stress to a residual level (from Ohnaka et.al,1997)
T he m ain differen ce b e tw e en this analy sis a n d p rev io u sly p u b lish e d w ork (R ice, 1980; W ong 1982b) is th at by o nly rem o v in g th e lin e a r p a rt o f th e lo a d in g curve th ere still rem ain s a sm all p re-failu re p o rtio n o f the curve. P rev io u s stu d ies only rec a st th e sh ear stress d a ta from th e p e a k sh e ar stress. T h is p re-failu re p o rtio n is u su ally term ed the stra in -h a rd e n in g p h a se an d is a m ea su re o f the re sista n c e to the fra c tu re pro p ag atio n . W hen p lo tted as a stress-slip d isp la c e m e n t cu rv e th is m ay lead to co n fu sio n th a t the stra in -h a rd e n in g is th e sam e as slip -h ard e n in g observ ed in fric tio n al ex p erim en ts. T he v alid ity o f th is w ill b e d isc u sse d h ere as it co u ld be arg u ed th a t th ere is no real slip o c c u rrin g in the stra in h a rd e n in g p hase.
If th e fracture energy, Gc, is rep re sen te d as th e are a b e n e ath th e stress-slip d isp la c em e n t curve, a p o rtio n o f it sh o u ld b e a ttrib u te d to processes asso c ia te d w ith m icro crack in g . W ong (1 9 8 2 b ) used S E M to e x a m in e m ic ro cra ck s o f sam ples close to failu re an d p o st-failu re. H e e stim a ted the en e rg y in p u t for m ic ro c ra c k in g by m u ltip ly in g the stress-in d u ced c ra c k su rface area p e r u n it v olum e a n d th e ten sile single crystal specific su rface energy. H e sh ow ed th at the energ y a sso c ia te d w ith m ic ro cra ck in g should, as expected, b e m u ch sm aller th a n th e to tal en ergy in p u t fo r pre-failu re an d p o st-failu re d eform ation. H e also show ed th a t th e ratio o f m ic ro c ra c k in g rela te d en erg y to total en erg y is g reater for p re-failu re th a n p o st-failu re defo rm atio n . O n e reaso n fo r this m ay b e d ue to d iffe re n c e in th e en erg y b u d g e ts for th e pre- a n d p o st-failu re defo rm atio n ; fo r ex am ple, h ig h am o u n ts o f en erg y w ill b e d issip a te d d u e to frictio n al h eatin g an d aco u stic em issio n s for p o st-fa ilu re defo rm atio n . T he m ec h a n ism s for energy d issip a tio n d u rin g p re-failu re d e fo rm atio n are d o m in a n tly m ic ro cra ck related, as m ajo r en ergy sinks do n o t ex ist in th e in ta c t ro ck m ass.
In co m p ressiv e lo ad in g o f in itia lly in ta c t rock, ela stic d e fo rm atio n is usu ally lim ited to th e first 4 0 -5 0 % (ij, in F ig u re 6 .3 b ) o f th e p e a k stre n g th (Xp, in F ig u re 6.3b). A bove Tj, m ic ro cra ck in g develops in th e ro c k as it deform s u n d e r load. A t h ig h e r stresses, th e cracks w ill co alesce a n d in te rac t to form a p la n a r zone o f d a m a g e th a t is later m an ifest as the fau lt p lan e (as d isc u sse d in sectio n 3.7). T herefore, b rittle failu re o f in itially in ta c t ro ck in th is reg im e w ill alw ays h a v e som e n o n -ela stic d e fo rm atio n in th e zone w h ere a fau lt w ill occur, and this ra te o f d e fo rm atio n (strain h ard en in g ) w ill decrease as p e a k stren g th is ap p ro ach ed .
C o n tin u u m d a m a g e theo ries (A shby an d H allam , 1986; H orii a n d N em at- N asser, 1986) are b a sed on m odel ex p e rim e n ts w here w in g crack s d evelop fro m an
in itial flaw . T hese w in g crack s are h eld o p en by slip along th e flaw , and p ro p ag a te by v irtu e o f the slip p in g flaw . T his w ill n e c essa rily o c c u r p rio r to p e a k strength, since these crack s w ill n eed to in te ra c t close to p e a k stren g th values in o rd er to cause m acro sco p ic rupture. F on sek a, M urrell a n d B arn es (1 9 8 5 ) h av e seen w in g cra c k p ro p ag a tio n in rock. T he m ic ro sc o p ic slips acco m m o d a te d along th e m icro sco p ic cracks w ill add u p to th e slip p rio r to p e a k strength.
N o n -elastic p re-failu re d e fo rm atio n can b e seen in frictio n al slip ex p erim en ts too (O h n a k a and Y a m a sh ita, 1989). Is it rea so n a b le to c o m p a re p re-failu re strain h ard e n in g along a p re-e x istin g fau lt w ith th a t in a b u lk m e d iu m th at has n o t yet failed? C o n sid e r a sliding fault, as a no rm al stress is ap p lied , th e fau lt surfaces are p u sh e d to g eth e r and d ue to su rface ro u g h n ess, p ro tru d in g a sp erities w ill im p in g e on th e surfaces. A t very h igh no rm al stresses, the stre n g th o f th ese asp erity ju n c tio n s m ay even reach th e sh ear stren g th o f th e rock. A s a sh e arin g stress is ap plied, th e asp erity ju n c tio n s w ill shear, c a u sin g slip a m o n g st th e w eak est a sp erities a lth o u g h th e m ain fractu re is still intact. S lip is th ere fo re p ro ce e d in g even th o u g h sh e ar stress is in cre asin g in o rd er to b reak th e stro n g est asperities. P e a k stren g th is rea c h e d w hen th e largest asperity stren g th can b e overcom e. In th e in ta c t m ed iu m , I a rg u e th at p re-p e a k slip m ay be reg a rd e d as rep re sen tin g in te g rate d a m o u n ts o f slips c a u se d by m ic ro cra ck in g th at n ecessarily occurs as a p rep a ra to ry p h a se o f th e im m in e n t m ac ro sco p ic failu re b efore p eak stress is attained. S ince b u lk fractu re a n d frictio n al in sta b ility do share sim ilar m ec h a n ic a l p ro perties an d th e m ath e m a tic s d e scrib in g th e m (C o u lo m b ’s L aw and B y e rle e ’s R ule) are a lm o st fu n ctio n ally id e n tic a l, I su g g est th a t it is rea so n a b le to com pare p re-failu re strain h a rd e n in g alo n g a p re-e x istin g fau lt w ith th a t in a b u lk m ed iu m th a t has n o t yet failed.
A fter strain lo ca lisa tio n an d m ic ro cra ck m ass lin k ag e, th e in ta c t ro c k w ill fail m acro sco p ically an d slip on th e m ac ro sco p ic failu re surfaces occu rs w ith a d ecreasin g sh ear stress. T his is called slip -w e a k en in g in th e co n tex t o f th e c o n stitu tiv e relatio n (R ice, 1980).
A p p en d ix 6 c o n tain s th e resolved sh e a r stress versus slip d isp la c em e n t curves for th e e n tire p rocess o f sh ear ru p tu re o f in ta c t T su k u b a g ran ite, d erived from th e axial stress-strain curves show in sectio n 5.2. It is im p o rta n t to n o te th at som e ex p erim en ts involved a ja c k e t ru p tu re d u rin g th e fractu re p ro cess and th e curves d id n o t rea c h a c o n sta n t resid u al slid in g level. T herefore, reliab le p o st failu re p a ram eters su ch as Tr, Dc, Dsw, AXb an d Gc could n o t be o b tain e d from th ese curves. O nly reliab le d a ta are p lo tted
on the graphs to investigate the behaviour o f con stitu tive relation parameters. T he d atapoints used in graphs are tabulated in the low er tables in A p p en dix 5.