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Creep in rock as a stochastic process
Engineering Geology, 20 (1984) 301--310 Elsevier Science Publishers B V., Amsterdam -- Printed in The Netherlands
301
C R E E P I N R O C K AS A S T O C H A S T I C P R O C E S S
ROLAND P u s c H Department of Engineering Geology, Lund University of Technology and Natural Sciences; Swedish Geological, Lund (Sweden)
(Received March 16, 1984; accepted after revision July 30, 1984)
ABSTRACT Pusch, R., 1984. Creep in rock as a stochastic process. Eng. Geol., 20: 301--310. The heterogeneous character of rock suggests the use of statistical mechanics for the formulation of models of plasticity and creep. A stochastic creep model which allows for large variations in strength and size of the involved rock mass is outlined. The model, which is analogous to a previously developed concept for clastic materials, is shown to account rather well for typical creep behaviour of rock at moderate stresses and temperatures. Thus, it yields the often observed linear relation between the creep strain and the logarithm of time. INTRODUCTION
M a n y s e d i m e n t a r y and m e t a m o r p h i c r o c k s e x h i b i t significant c r e e p t h a t a f f e c t s the stability o f r o c k s t r u c t u r e s . H a r d igneous r o c k w i t h o u t clay s e a m s a n d g r a p h i t e inclusions usually shows insignificant s h o r t - t e r m creep o n loading or u n l o a d i n g . Still, t i m e - d e p e n d e n t strain a c c u m u l a t e s in the r o o f and walls o f tunnels and caverns even in such r o c k and m a y u l t i m a t e l y yield r o c k fall a n d collapse. T h e m a t t e r is o f p a r t i c u l a r i m p o r t a n c e f o r r o c k caverns used for oil a n d gas storage a n d f o r d e p o s i t i o n o f h a z a r d o u s w a s t e p r o d u c t s . F o r m u l a t i o n o f design criteria a n d c h o i c e o f suitable stabilization t e c h n i q u e s f o r such s t r u c t u r e s require u n d e r s t a n d i n g o f t h e involved m e c h a n i s m s f o r w h i c h physical m o d e l s i m p l y i n g s t r u c t u r a l h e t e r o g e n e i t y should f o r m the basis. Such a m o d e l is o u t l i n e d here for the pre-failure stage, yielding the o f t e n o b s e r v e d log-time e x p r e s s i o n o f the c r e e p f o r m o d e r a t e stress and t e m p e r a t u r e . T h e involved m e c h a n i s m s are, in principle, similar f o r small crystal m a t r i x e l e m e n t s and large r o c k masses, w h i c h explains t h e size-invariant c r e e p b e h a v i o u r as long as t h e n u m b e r o f i n t e r a c t i n g s t r u c t u r a l c o m p o n e n t s o f each c o n s i d e r e d s y s t e m is c o n s t a n t . This c o n d i t i o n is r e a s o n a b l y : w e l l fulfilled in uniaxial ( c o m p r e s s i o n ) c r e e p tests a n d in field p l a t e l o a d tests as well as in the early p h a s e o f t u n n e l c o n v e r g e n c e w h e n strain is still small.
0013-7952/84/$03.00
© 1984 Elsevier Science Publishers B.V.
302 PHYSICAL MECHANISMS As f o r m o s t crystalline materials, t r a n s i e n t creep in r o c k can be e x p l a i n e u in t e r m s o f m i g r a t i o n and i n t e r a c t i o n o f dislocations. This is the case for individual crystals o f various silicates, for instance, and for small crystal aggregates, b u t f o r e l e m e n t s c o m p o s e d o f d i f f e r e n t minerals w i t h varying stress/strain and t h e r m a l p r o p e r t i e s and w i t h a m o r e or less h e t e r o g e n e o u s s t r u c t u r e , o t h e r m e c h a n i s m s b e c o m e decisive in t h e creep b e h a v i o u r . T h u s , a m a i n s o u r c e o f c r e e p o f f r a c t u r e - p o o r r o c k is shear d i s p l a c e m e n t along crystal lattice planes and a l o n g i n t e r f a c e s b e t w e e n c o n t a c t i n g , b o n d e d as well as n o n - b o n d e d crystals. T h e latter are r e p r e s e n t e d b y d i f f e r e n t n e i g h b o u r i n g minerals w h i c h o f t e n have i n c o m p l e t e c o m m o n b o u n d a r i e s ( F i g . l ) , a f e a t u r e w h i c h is largely responsible f o r the p o r o s i t y o f crystalline igneous and m e t a m o r p h i c r o c k and f o r t h e w e l l - k n o w n influence o f the cell pressure on the s t r e n g t h o f r o c k s a m p l e s t e s t e d u n d e r triaxial stress c o n d i t i o n s . T h e s e f e a t u r e s i m p l y a c o n s i d e r a b l e variation in b o n d strength, a p r o p e r t y which is especially o b v i o u s for larger v o l u m e s w h e r e the n u m b e r o f crystals with
Imm
a-a B
X
~/
",
~
r-.
I t
oz b
/
y
A
"E
Fig. 1. Graphical interpretation of a typical granite section. (a~a) represents a cross-section through the crystal matrix. A = polished surface of the 30 um thick section for microscopy B = actual irregular surface exposed in a joint; C = incomplete intercrystalline contacts; D = fissure through crystal; E = tightly contacting crystals.
303 different properties, orientation and spacing yields a strongly heterogeneous matrix. In even larger rock masses, fissures and joints represent the major weaknesses with a corresponding strength span and, in the case of metamorphic rock, also a marked anisotropy. Each discontinuity of this sort is characterized by asperities and a varying a m o u n t of contact "islands". The variation in strength over cross-sections or volumes of any size suggests the use of statistical mechanics for the derivation of a plausible model for creep in rock. Considering the various scale-dependent creep-producing features, such a model should account for the following. (1) The application of a sufficiently high shear stress initiates slip, i.e. local plasticity, in the form of translatory shear displacements at overstressed crystal contacts and along certain atom planes in individual crystals, as well as slip at the interface between contacting asperities in fissures and joints. (2) Redistribution of microstresses is facilitated by slip, by which higher local stresses are produced where previously they were relatively low.. This causes local stress increase and additional microstructural plasticity by which the creep rate is enhanced. (3) Deformation will induce local displacements such that stronger units will make contact and help to strengthen the structure. This involves microdilatancy and mechanical interlocking, which are the main "healing" processes by which the creep rate is retarded. In principle, creep is produced by the continuation of the slip process initiated by the activation and formation of slip units. The net effect of the various mechanisms is governed by the magnitude of the deviat)r stress and the total strain. Sufficiently high stresses produce accumulated structural breakdown by which the damaging processes outweigh the healing ones and the strain rate accelerates, leading to failure. At lower stresses, short-term creep is retarded but the geometrical conditions become decisive in the macroscopic behaviour of the system. If expansion is prevented or very moderate, such as when a heavily loaded foundation rests on a free surface, the maximum number of interacting units is constant and the creep strain rate decelerates; but if expansion can take place, disintegration eventually occurs and the number of such units diminishes. The latter case, which is represented by tunnel, shaft and slope excavation activities, ultimately leads to progressive failure. We will consider here the condition of intermediate shear stresses with a non-decreasing number of activated slip units, i.e. the early, pre-failure creep stage. Creep of rock, as well as of soils, is temperature-sensitive, which suggests that it is basically a thermally activated process. It is therefore logical to apply the concept of molecular jump activation and a rate theory according to which displacements within the crystal matrix are treated as thermally assisted passages over energy barriers aided by the external application of stress. This approach has been made by other investigators as well but without applying statistical mechanics, which is called for by the heterogeneous structure and the corresponding large spectrum of activation energies (cf. Langer, 1979).
304 MATHEMATICAL FORM
T h e following general features o f the creep m o d e l are suggested as a conseq u e n c e o f the described p h e n o m e n o l o g y . (1) The structural h e t e r o g e n e i t y and strength variation means t h a t t h e r e is a variation in activation energy for slip (Fig.2). T h e lower b o u n d a r y m a y r e p r e s e n t h y d r o g e n b o n d s b e t w e e n h y d r a t e d mica- and c h l o r i t e - t y p e constitu e n t s while the u p p e r b o u n d a r y c o r r e s p o n d s to p r i m a r y valence b o n d s o f u n f r a c t u r e d crystal lattices. T h e s p e c t r u m will vary in the course o f creep so t h a t some energy barriers b e c o m e e n h a n c e d due to a local decrease in the deviatoric stress, while others b e c o m e operative t h r o u g h local stress rise. (2) Each r o c k e l e m e n t contains a certain n u m b e r o f slip units in a given interval o f the activation energy range at each particular time after the onset o f the creep. (3) In the course o f the creep, new slip units are g e n e r a t e d at the lower end of t h e energy barrier s p e c t r u m while the high u-end is an " a b s o r b i n g barrier". (4) J u m p s bring the given slip unit u p or d o w n against a barrier by an a m o u n t 6u higher or l o w e r t h a n the previous one. T h e m o d e l is based on the a s s u m p t i o n that slip has the c h a r a c t e r of activated a t o m i c or m o l e c u l a r j u m p s and t h a t it can be visualized as resulting f r o m shiftings o f individual a t o m s as well as o f large p a t c h e s o f such units, the shiftings over energy barriers being driven by e x t e r n a l stress. T h e n u m b e r o f p o t e n t i a l slip units per unit v o l u m e held up at barriers of height u, is n (u, t) 5 u per energy unit, w h e r e t is time after the onset o f creep, and 5 u one o f the energy intervals into which the e n e r g y s p e c t r u m is subdivided. As in the case o f creep in metals and clastic materials, a basic a s s u m p t i o n is t h a t the a t t e m p t f r e q u e n c y o f slip , ( u ) is given b y the Arrhenius rate equation: u(u)=uDexp(--u/kT),
i
I
uI
u2
ul <~ u<~ u2
Fig.2. Activation energy spectrum at a given time t after the onset of creep.
(1)
305 where u is the barrier height. Here,/~D is an atomic vibrational frequency of the order o f 1012 per second, k is Boltzmann's constant and T the absolute temperature. If slip has been activated at a certain poi nt in the crystal matrix or in a fissure or joint, i.e. a barrier has been overcome, a cont ri but i on to the overall shear is made by the associated extension of the local slip-patch. The next barrier to be en co unt e r ed by the same spreading slip zone will be either higher or lower by an average a m o u n t 5 u. The magnitude of 5 u is determined by the amplitude of the internal stress field, and by the physical nature of the barriers. Assuming an equal probability of slip, meaning t hat an activated jump of a patch brings it to a barrier which is either lower or higher than the previous one, Pusch and Feltham (1980) arrived at the relationship: ~ n / ~ t = D ( a 2 [n exp ( - - u / k T ) ] /a u 2} 1 2 D =~'D(SU )
(2)
where t = time after onset of creep and where n is short for n(u, t). If the a t t e m p t f r equency of all such points is vD e -u/~T, then the contribution by the u-interval to the flow rate in uniaxial tension or compression is: 5~ ~ uDn(u, t) exp ( - - u / k T ) S u
(3)
Now, if it is assumed that each activated j um p makes the same, average c o n t r i b u t i o n to the bulk strain, the rate of creep will be: U2
~
.[
e -u/kT . n(u, t ) d u
(4)
which yields the relationship: ~ 1 / ( t + to)
(5)
where to is an integration constant. The appropriate constant of pr opor t i ona l i ty of eq. 5 should depend on the deviator stress, t e m per a t ur e, and structural details. At low and intermediate stresses, the frequency of activated slip units should be at least a p p ro x imately proportional to the applied stress and a roughly linear relationship between stress and creep rate is expected. Similarly, for temperatures n o t exceeding r o o m t e m per a t ur e by more than a few tens of centigrades, the creep-activating effect should be very moderate. Under these conditions an a pp r o x imately linear relationship between absolute t em perat ure and creep rate is probable from a phenomenological poi nt of view. Applying these relationships eq. 5 yields: = ~ T o / ( t + to) with o = const. where f~ and to depend on the structure and involved mineral com ponent s.
(6)
306 The meaning of eq. 6 is that the creep rate is successively retarded as a consequence of the "blue-shift" of the activation energy spectrum that results when the strengthening processes outweigh the damaging ones. OBSERVATIONS The theoretically derived creep law of eq. 6 should be applicable to the conditions specified in the introduction, i.e. when the num ber of interacting units is practically constant, which is in turn determined n o t only by the stress level but also by the geometry of the system. A num ber of literaturereported creep tests referring to largely different rock volumes will be cited in this section to check the validity o f this hypothesis.
Laboratory tests In the transient creep regime, logarithmic creep, approximately linear in stress and temp er a t ur e as predicted by eq. 6, is widely observed in solids and soil materials and it also seems to be representative of m any rock types as concluded from a large n u m b e r of r e por t ed data (Langer, 1979). Thus, under uniaxial compression, small-diameter cored rock samples tend to obey this creep law as long as the applied stress is in the interval of approximately 1/3 to 2/3 of the u nconf i ne d compression strength determined in conventional rapid tests (Fig.3). It should be added, however, that many transient creep data also fit power laws {Carter and Kirby, 1978). The validity of the log-time creep law is supported by measurements of the tunnel closure rate in model tests with coal blocks in which 10--15 cm diameter tunnels had been drilled (Kaiser and Morgenstern, 1981). In addition, these tests clearly d e m o n s t r a t e d the progressive failure that is associated with large strain of a rock matrix which can expand into a cavity in the course o f the creep. Naturally, such strain is a combination of elastic deformations, microcracking and microplasticity, i.e. semibrittle behaviour.
I ,,-, .;o-' ,=+,,,,,.l \ FOR t o 0 ~ 1 ~
10"~° 10+
IN
\
10s
10e
t,s
Fig.3. Creep rate versus time at uniaxial compression test of gneiss. The axial stress corresponds to 40% of the compression strength.
307
Field load tests Plate loading tests on biotite-rich, slightly weathered gneiss with a marked subhorizontal layer-type structure, have dem o nst rat ed the applicability o f eq. 6 also for high stresses (Pusch et al., 1974). The test arrangement, which is shown in Fig.4, made it possible to record the settlement under constant load conditions of cylindrical concrete foundations with a radius of 0.35 to 1.0 m and a m a x i m u m average c ont a c t pressure of as much as 103 MPa. The load was applied in 1 MPa steps, the application being made when the settlem e n t rate under the previous load had dr oppe d to 3 • 10 -4 mm/s. At slightly less than half the m a x i m u m pressure, i.e. 42 MPa for the smallest foundation, the settlement was recorded for about 3 h. All these tests showed a log-time creep behaviour (cf. Fig. 5), the involved rock volumes being a few cubic meters. 4 m HYDRAULIC JACKS
I
I
I
i
,
t
~ :
II
:
I
TIE RODS STEEL LID
CONCRETE FOUNDATION
¢ '°° me, O. .OLES
Fig.4. Plate load arrangement for the determination of time-dependent
settlement.
Convergence o f large tunnels Several convergence measurements in tunnels, which were virtually stable and which involved several tens of cubic meters have supported the applicability o f the log time creep law (Skaanes, 1979; Ito and Hisatake, 1979; T h o m s o n and E1-Nahhas, 1980). Skaanes referred to a 3.5 m diameter tunnel in marl at a b o u t 1000 m depth, which gave the convergence rate shown in Fig.6. Similarly, Ito and Hisatake, and T h o m s o n and E1-Nahhas found that the inward m o v e m e n t o f the crown as well as the sidewalls cont i nued as a time-dependent creep that could be approximated by a linear line when
308
ld"
_ ]~. Eq(5) TFOR to=O
o~'s,;,v~o'%... mm/s ld 6
10"~02
10 3
10 4
10 5
t,s Fig.5. Creep rate r e c o r d e d o n loading of circular f o o t i n g w i t h 0.35 m d i a m e t e r o n gneiss (Pusch et al., 1974). Average g r o u n d pressure 42 MPa.
10-9
\
fEq(5)FOR
~1 10-1°
to=O
\ OBSERVED
-11
10
10
5
10
6
10 t,
7
10
8
s
Fig.6. Radial c o n v e r g e n c e rate of 3.5 m d i a m e t e r t u n n e l at 1000 m d e p t h in marl. Calculated f r o m S k a a n e s ' data (Skaanes, 1979).
plotted versus the logarithm of time for tunnels at shallow depth in andesite and a sequence of sandstone, siltstone, claystone and shale, respectively. DISCUSSION
The literature exhibits a vast number of practical creep records involving largely varying rock volumes and showing linear relationship between accumulated strain and log time in the pre-failure stage. This suggests that the basic stochastic features of the theoretically derived creep model referred to here are of fundamental importance and general applicability. It is clear, however, that in most cases the creep recording time has not been sufficient to give evidence of a continued log time creep behaviour of non-expanding systems over very long periods of time. A more rapidly retarding creep rate is actually expected at low stresses for which the healing effects outweigh the damaging ones. The derivation of a generalized creep model for this case
309 (cf. Pusch, 1979), with the same stochastic build-up but with the omission of any creep-enhancing redistribution of microstresses, yields the expression ~ (1 -~ t/to), which implies strong retardation of the strain rate after a rapid onset. It goes w i t h o u t saying that the validity of this law over very long creep periods is equally difficult to verify in practice, and the m ore conservative log time law should therefore be a safer choice in the prediction of accumulated strain. It should be m e n t i o n e d that the creep rate deceleration to a steady-state that is observed in certain rocks like salt has not been verified for hard igneous rock at low and m ode r at e shear stresses. Still, it is not excluded that this effect appears also for non-weathered granite, for example. The creep rate and accumulated strain also over long periods of time would be of negligible practical importance, however. Many o f the recorded creep curves, such as those of Figs.5 and 6, show a successive approach to the straight, theoretical curve of eqs. 5 and 6 with to = 0 in double logarithmic plottings. This manifests that to is oft en substantially higher than 0, for which there are several explanations, the c o m m o n one being that it results from a rather " f l a t " shape of the initial barrier spectrum, i.e. a rather even u{u, 0) distribution (Pusch and Feltham, 1980). COMMENT ON POSSIBLE PRACTICAL USE One of the problems in applied rock mechanics is to predict creep strain since the scale-dependence o f the involved parameters and the short time which is usually available for running creep tests make predictions very uncertain. Th e creep t h e o r y outlined here on the basis of stochastic mechanics suggests that the creep rate is approximately inversely proportional to the time after onset of the creep, provided that failure will n o t occur. This offers a possibility o f predicting the strain developed after long periods of creep of various types o f constructions and excavations in rock. Thus, if a few observed displacements at know n time intervals are at hand, the prediction of longterm creep strain should be possible, with reasonable accuracy, by fitting eq. 6 with the observed data. It is required, however, that all the assumed conditions for the validity of the t h e o r y are actually fulfilled. Thus, when t h e creep is associated with expansion and therefore with a reduction in the n u m b e r o f interacting elements, there will be a critical strain b e y o n d which the strain rate accelerates and failure is ultimately developed. REFERENCES Carter, N.L. and Kirby, S.H., 1978. Transient creep and semibrittle behavior of crystalline rocks. Pageoph, 116: 807--839. Ito, T. and Hisatake, M., 1979. Surface displacements caused by tunnel driving in anisotropic viscoelastic ground. Proc. Int. Congr. Rock Mechanics, 4th, Montreux, Vol. 1, pp.677--684. Kaiser, P.K. and Morgenstern, N.R., 1981. Time-dependent deformation of small tunnels, II. Typical test data. Int. J. Rock Mechanics, Mining Sciences and Geomechanics, 18: 141--152.
310
Langer, M., 1979. Rheologieal behavior of rock masses. Proc. Int. C o n g . Rock Mechanics, 4th, Montreux, Vol. 3, pp.29--62. Pusch, R., 1979. A physical clay creep model and its mathematical analogy. Int. Conf. Numerical Methods in Geomechanics, 3rd, Aachen, pp.485--492. Pusch, R. and Feltham, P., 1980. A stochastic model of the creep of soils. Geotechnique, 30(4): 497--506. Pusch, R., Hansbo, S., Berg, G. and Henricsson, E., 1974. B~righet och s~ittningar vid grundl~/ggning p~i berg. Rapport No 11, Svenska ByggnadsentreprenSrfSreningen, Stockholm. Skaanes, S., 1979. Mechanisches Tunnelausbruch am Beispiel Baulos Huttegg des Seelisbergtunnels. Proc. Int. Congr. Rock Mechanics, 4th, Montreux, Vol. 1, pp.551--556. Thomson, S. and El-Nahhas, F., 1980. Field measurements in two tunnels in Edmonton, Alberta. Can. Geotech. J., 17(1): 20--33.
301
C R E E P I N R O C K AS A S T O C H A S T I C P R O C E S S
ROLAND P u s c H Department of Engineering Geology, Lund University of Technology and Natural Sciences; Swedish Geological, Lund (Sweden)
(Received March 16, 1984; accepted after revision July 30, 1984)
ABSTRACT Pusch, R., 1984. Creep in rock as a stochastic process. Eng. Geol., 20: 301--310. The heterogeneous character of rock suggests the use of statistical mechanics for the formulation of models of plasticity and creep. A stochastic creep model which allows for large variations in strength and size of the involved rock mass is outlined. The model, which is analogous to a previously developed concept for clastic materials, is shown to account rather well for typical creep behaviour of rock at moderate stresses and temperatures. Thus, it yields the often observed linear relation between the creep strain and the logarithm of time. INTRODUCTION
M a n y s e d i m e n t a r y and m e t a m o r p h i c r o c k s e x h i b i t significant c r e e p t h a t a f f e c t s the stability o f r o c k s t r u c t u r e s . H a r d igneous r o c k w i t h o u t clay s e a m s a n d g r a p h i t e inclusions usually shows insignificant s h o r t - t e r m creep o n loading or u n l o a d i n g . Still, t i m e - d e p e n d e n t strain a c c u m u l a t e s in the r o o f and walls o f tunnels and caverns even in such r o c k and m a y u l t i m a t e l y yield r o c k fall a n d collapse. T h e m a t t e r is o f p a r t i c u l a r i m p o r t a n c e f o r r o c k caverns used for oil a n d gas storage a n d f o r d e p o s i t i o n o f h a z a r d o u s w a s t e p r o d u c t s . F o r m u l a t i o n o f design criteria a n d c h o i c e o f suitable stabilization t e c h n i q u e s f o r such s t r u c t u r e s require u n d e r s t a n d i n g o f t h e involved m e c h a n i s m s f o r w h i c h physical m o d e l s i m p l y i n g s t r u c t u r a l h e t e r o g e n e i t y should f o r m the basis. Such a m o d e l is o u t l i n e d here for the pre-failure stage, yielding the o f t e n o b s e r v e d log-time e x p r e s s i o n o f the c r e e p f o r m o d e r a t e stress and t e m p e r a t u r e . T h e involved m e c h a n i s m s are, in principle, similar f o r small crystal m a t r i x e l e m e n t s and large r o c k masses, w h i c h explains t h e size-invariant c r e e p b e h a v i o u r as long as t h e n u m b e r o f i n t e r a c t i n g s t r u c t u r a l c o m p o n e n t s o f each c o n s i d e r e d s y s t e m is c o n s t a n t . This c o n d i t i o n is r e a s o n a b l y : w e l l fulfilled in uniaxial ( c o m p r e s s i o n ) c r e e p tests a n d in field p l a t e l o a d tests as well as in the early p h a s e o f t u n n e l c o n v e r g e n c e w h e n strain is still small.
0013-7952/84/$03.00
© 1984 Elsevier Science Publishers B.V.
302 PHYSICAL MECHANISMS As f o r m o s t crystalline materials, t r a n s i e n t creep in r o c k can be e x p l a i n e u in t e r m s o f m i g r a t i o n and i n t e r a c t i o n o f dislocations. This is the case for individual crystals o f various silicates, for instance, and for small crystal aggregates, b u t f o r e l e m e n t s c o m p o s e d o f d i f f e r e n t minerals w i t h varying stress/strain and t h e r m a l p r o p e r t i e s and w i t h a m o r e or less h e t e r o g e n e o u s s t r u c t u r e , o t h e r m e c h a n i s m s b e c o m e decisive in t h e creep b e h a v i o u r . T h u s , a m a i n s o u r c e o f c r e e p o f f r a c t u r e - p o o r r o c k is shear d i s p l a c e m e n t along crystal lattice planes and a l o n g i n t e r f a c e s b e t w e e n c o n t a c t i n g , b o n d e d as well as n o n - b o n d e d crystals. T h e latter are r e p r e s e n t e d b y d i f f e r e n t n e i g h b o u r i n g minerals w h i c h o f t e n have i n c o m p l e t e c o m m o n b o u n d a r i e s ( F i g . l ) , a f e a t u r e w h i c h is largely responsible f o r the p o r o s i t y o f crystalline igneous and m e t a m o r p h i c r o c k and f o r t h e w e l l - k n o w n influence o f the cell pressure on the s t r e n g t h o f r o c k s a m p l e s t e s t e d u n d e r triaxial stress c o n d i t i o n s . T h e s e f e a t u r e s i m p l y a c o n s i d e r a b l e variation in b o n d strength, a p r o p e r t y which is especially o b v i o u s for larger v o l u m e s w h e r e the n u m b e r o f crystals with
Imm
a-a B
X
~/
",
~
r-.
I t
oz b
/
y
A
"E
Fig. 1. Graphical interpretation of a typical granite section. (a~a) represents a cross-section through the crystal matrix. A = polished surface of the 30 um thick section for microscopy B = actual irregular surface exposed in a joint; C = incomplete intercrystalline contacts; D = fissure through crystal; E = tightly contacting crystals.
303 different properties, orientation and spacing yields a strongly heterogeneous matrix. In even larger rock masses, fissures and joints represent the major weaknesses with a corresponding strength span and, in the case of metamorphic rock, also a marked anisotropy. Each discontinuity of this sort is characterized by asperities and a varying a m o u n t of contact "islands". The variation in strength over cross-sections or volumes of any size suggests the use of statistical mechanics for the derivation of a plausible model for creep in rock. Considering the various scale-dependent creep-producing features, such a model should account for the following. (1) The application of a sufficiently high shear stress initiates slip, i.e. local plasticity, in the form of translatory shear displacements at overstressed crystal contacts and along certain atom planes in individual crystals, as well as slip at the interface between contacting asperities in fissures and joints. (2) Redistribution of microstresses is facilitated by slip, by which higher local stresses are produced where previously they were relatively low.. This causes local stress increase and additional microstructural plasticity by which the creep rate is enhanced. (3) Deformation will induce local displacements such that stronger units will make contact and help to strengthen the structure. This involves microdilatancy and mechanical interlocking, which are the main "healing" processes by which the creep rate is retarded. In principle, creep is produced by the continuation of the slip process initiated by the activation and formation of slip units. The net effect of the various mechanisms is governed by the magnitude of the deviat)r stress and the total strain. Sufficiently high stresses produce accumulated structural breakdown by which the damaging processes outweigh the healing ones and the strain rate accelerates, leading to failure. At lower stresses, short-term creep is retarded but the geometrical conditions become decisive in the macroscopic behaviour of the system. If expansion is prevented or very moderate, such as when a heavily loaded foundation rests on a free surface, the maximum number of interacting units is constant and the creep strain rate decelerates; but if expansion can take place, disintegration eventually occurs and the number of such units diminishes. The latter case, which is represented by tunnel, shaft and slope excavation activities, ultimately leads to progressive failure. We will consider here the condition of intermediate shear stresses with a non-decreasing number of activated slip units, i.e. the early, pre-failure creep stage. Creep of rock, as well as of soils, is temperature-sensitive, which suggests that it is basically a thermally activated process. It is therefore logical to apply the concept of molecular jump activation and a rate theory according to which displacements within the crystal matrix are treated as thermally assisted passages over energy barriers aided by the external application of stress. This approach has been made by other investigators as well but without applying statistical mechanics, which is called for by the heterogeneous structure and the corresponding large spectrum of activation energies (cf. Langer, 1979).
304 MATHEMATICAL FORM
T h e following general features o f the creep m o d e l are suggested as a conseq u e n c e o f the described p h e n o m e n o l o g y . (1) The structural h e t e r o g e n e i t y and strength variation means t h a t t h e r e is a variation in activation energy for slip (Fig.2). T h e lower b o u n d a r y m a y r e p r e s e n t h y d r o g e n b o n d s b e t w e e n h y d r a t e d mica- and c h l o r i t e - t y p e constitu e n t s while the u p p e r b o u n d a r y c o r r e s p o n d s to p r i m a r y valence b o n d s o f u n f r a c t u r e d crystal lattices. T h e s p e c t r u m will vary in the course o f creep so t h a t some energy barriers b e c o m e e n h a n c e d due to a local decrease in the deviatoric stress, while others b e c o m e operative t h r o u g h local stress rise. (2) Each r o c k e l e m e n t contains a certain n u m b e r o f slip units in a given interval o f the activation energy range at each particular time after the onset o f the creep. (3) In the course o f the creep, new slip units are g e n e r a t e d at the lower end of t h e energy barrier s p e c t r u m while the high u-end is an " a b s o r b i n g barrier". (4) J u m p s bring the given slip unit u p or d o w n against a barrier by an a m o u n t 6u higher or l o w e r t h a n the previous one. T h e m o d e l is based on the a s s u m p t i o n that slip has the c h a r a c t e r of activated a t o m i c or m o l e c u l a r j u m p s and t h a t it can be visualized as resulting f r o m shiftings o f individual a t o m s as well as o f large p a t c h e s o f such units, the shiftings over energy barriers being driven by e x t e r n a l stress. T h e n u m b e r o f p o t e n t i a l slip units per unit v o l u m e held up at barriers of height u, is n (u, t) 5 u per energy unit, w h e r e t is time after the onset o f creep, and 5 u one o f the energy intervals into which the e n e r g y s p e c t r u m is subdivided. As in the case o f creep in metals and clastic materials, a basic a s s u m p t i o n is t h a t the a t t e m p t f r e q u e n c y o f slip , ( u ) is given b y the Arrhenius rate equation: u(u)=uDexp(--u/kT),
i
I
uI
u2
ul <~ u<~ u2
Fig.2. Activation energy spectrum at a given time t after the onset of creep.
(1)
305 where u is the barrier height. Here,/~D is an atomic vibrational frequency of the order o f 1012 per second, k is Boltzmann's constant and T the absolute temperature. If slip has been activated at a certain poi nt in the crystal matrix or in a fissure or joint, i.e. a barrier has been overcome, a cont ri but i on to the overall shear is made by the associated extension of the local slip-patch. The next barrier to be en co unt e r ed by the same spreading slip zone will be either higher or lower by an average a m o u n t 5 u. The magnitude of 5 u is determined by the amplitude of the internal stress field, and by the physical nature of the barriers. Assuming an equal probability of slip, meaning t hat an activated jump of a patch brings it to a barrier which is either lower or higher than the previous one, Pusch and Feltham (1980) arrived at the relationship: ~ n / ~ t = D ( a 2 [n exp ( - - u / k T ) ] /a u 2} 1 2 D =~'D(SU )
(2)
where t = time after onset of creep and where n is short for n(u, t). If the a t t e m p t f r equency of all such points is vD e -u/~T, then the contribution by the u-interval to the flow rate in uniaxial tension or compression is: 5~ ~ uDn(u, t) exp ( - - u / k T ) S u
(3)
Now, if it is assumed that each activated j um p makes the same, average c o n t r i b u t i o n to the bulk strain, the rate of creep will be: U2
~
.[
e -u/kT . n(u, t ) d u
(4)
which yields the relationship: ~ 1 / ( t + to)
(5)
where to is an integration constant. The appropriate constant of pr opor t i ona l i ty of eq. 5 should depend on the deviator stress, t e m per a t ur e, and structural details. At low and intermediate stresses, the frequency of activated slip units should be at least a p p ro x imately proportional to the applied stress and a roughly linear relationship between stress and creep rate is expected. Similarly, for temperatures n o t exceeding r o o m t e m per a t ur e by more than a few tens of centigrades, the creep-activating effect should be very moderate. Under these conditions an a pp r o x imately linear relationship between absolute t em perat ure and creep rate is probable from a phenomenological poi nt of view. Applying these relationships eq. 5 yields: = ~ T o / ( t + to) with o = const. where f~ and to depend on the structure and involved mineral com ponent s.
(6)
306 The meaning of eq. 6 is that the creep rate is successively retarded as a consequence of the "blue-shift" of the activation energy spectrum that results when the strengthening processes outweigh the damaging ones. OBSERVATIONS The theoretically derived creep law of eq. 6 should be applicable to the conditions specified in the introduction, i.e. when the num ber of interacting units is practically constant, which is in turn determined n o t only by the stress level but also by the geometry of the system. A num ber of literaturereported creep tests referring to largely different rock volumes will be cited in this section to check the validity o f this hypothesis.
Laboratory tests In the transient creep regime, logarithmic creep, approximately linear in stress and temp er a t ur e as predicted by eq. 6, is widely observed in solids and soil materials and it also seems to be representative of m any rock types as concluded from a large n u m b e r of r e por t ed data (Langer, 1979). Thus, under uniaxial compression, small-diameter cored rock samples tend to obey this creep law as long as the applied stress is in the interval of approximately 1/3 to 2/3 of the u nconf i ne d compression strength determined in conventional rapid tests (Fig.3). It should be added, however, that many transient creep data also fit power laws {Carter and Kirby, 1978). The validity of the log-time creep law is supported by measurements of the tunnel closure rate in model tests with coal blocks in which 10--15 cm diameter tunnels had been drilled (Kaiser and Morgenstern, 1981). In addition, these tests clearly d e m o n s t r a t e d the progressive failure that is associated with large strain of a rock matrix which can expand into a cavity in the course o f the creep. Naturally, such strain is a combination of elastic deformations, microcracking and microplasticity, i.e. semibrittle behaviour.
I ,,-, .;o-' ,=+,,,,,.l \ FOR t o 0 ~ 1 ~
10"~° 10+
IN
\
10s
10e
t,s
Fig.3. Creep rate versus time at uniaxial compression test of gneiss. The axial stress corresponds to 40% of the compression strength.
307
Field load tests Plate loading tests on biotite-rich, slightly weathered gneiss with a marked subhorizontal layer-type structure, have dem o nst rat ed the applicability o f eq. 6 also for high stresses (Pusch et al., 1974). The test arrangement, which is shown in Fig.4, made it possible to record the settlement under constant load conditions of cylindrical concrete foundations with a radius of 0.35 to 1.0 m and a m a x i m u m average c ont a c t pressure of as much as 103 MPa. The load was applied in 1 MPa steps, the application being made when the settlem e n t rate under the previous load had dr oppe d to 3 • 10 -4 mm/s. At slightly less than half the m a x i m u m pressure, i.e. 42 MPa for the smallest foundation, the settlement was recorded for about 3 h. All these tests showed a log-time creep behaviour (cf. Fig. 5), the involved rock volumes being a few cubic meters. 4 m HYDRAULIC JACKS
I
I
I
i
,
t
~ :
II
:
I
TIE RODS STEEL LID
CONCRETE FOUNDATION
¢ '°° me, O. .OLES
Fig.4. Plate load arrangement for the determination of time-dependent
settlement.
Convergence o f large tunnels Several convergence measurements in tunnels, which were virtually stable and which involved several tens of cubic meters have supported the applicability o f the log time creep law (Skaanes, 1979; Ito and Hisatake, 1979; T h o m s o n and E1-Nahhas, 1980). Skaanes referred to a 3.5 m diameter tunnel in marl at a b o u t 1000 m depth, which gave the convergence rate shown in Fig.6. Similarly, Ito and Hisatake, and T h o m s o n and E1-Nahhas found that the inward m o v e m e n t o f the crown as well as the sidewalls cont i nued as a time-dependent creep that could be approximated by a linear line when
308
ld"
_ ]~. Eq(5) TFOR to=O
o~'s,;,v~o'%... mm/s ld 6
10"~02
10 3
10 4
10 5
t,s Fig.5. Creep rate r e c o r d e d o n loading of circular f o o t i n g w i t h 0.35 m d i a m e t e r o n gneiss (Pusch et al., 1974). Average g r o u n d pressure 42 MPa.
10-9
\
fEq(5)FOR
~1 10-1°
to=O
\ OBSERVED
-11
10
10
5
10
6
10 t,
7
10
8
s
Fig.6. Radial c o n v e r g e n c e rate of 3.5 m d i a m e t e r t u n n e l at 1000 m d e p t h in marl. Calculated f r o m S k a a n e s ' data (Skaanes, 1979).
plotted versus the logarithm of time for tunnels at shallow depth in andesite and a sequence of sandstone, siltstone, claystone and shale, respectively. DISCUSSION
The literature exhibits a vast number of practical creep records involving largely varying rock volumes and showing linear relationship between accumulated strain and log time in the pre-failure stage. This suggests that the basic stochastic features of the theoretically derived creep model referred to here are of fundamental importance and general applicability. It is clear, however, that in most cases the creep recording time has not been sufficient to give evidence of a continued log time creep behaviour of non-expanding systems over very long periods of time. A more rapidly retarding creep rate is actually expected at low stresses for which the healing effects outweigh the damaging ones. The derivation of a generalized creep model for this case
309 (cf. Pusch, 1979), with the same stochastic build-up but with the omission of any creep-enhancing redistribution of microstresses, yields the expression ~ (1 -~ t/to), which implies strong retardation of the strain rate after a rapid onset. It goes w i t h o u t saying that the validity of this law over very long creep periods is equally difficult to verify in practice, and the m ore conservative log time law should therefore be a safer choice in the prediction of accumulated strain. It should be m e n t i o n e d that the creep rate deceleration to a steady-state that is observed in certain rocks like salt has not been verified for hard igneous rock at low and m ode r at e shear stresses. Still, it is not excluded that this effect appears also for non-weathered granite, for example. The creep rate and accumulated strain also over long periods of time would be of negligible practical importance, however. Many o f the recorded creep curves, such as those of Figs.5 and 6, show a successive approach to the straight, theoretical curve of eqs. 5 and 6 with to = 0 in double logarithmic plottings. This manifests that to is oft en substantially higher than 0, for which there are several explanations, the c o m m o n one being that it results from a rather " f l a t " shape of the initial barrier spectrum, i.e. a rather even u{u, 0) distribution (Pusch and Feltham, 1980). COMMENT ON POSSIBLE PRACTICAL USE One of the problems in applied rock mechanics is to predict creep strain since the scale-dependence o f the involved parameters and the short time which is usually available for running creep tests make predictions very uncertain. Th e creep t h e o r y outlined here on the basis of stochastic mechanics suggests that the creep rate is approximately inversely proportional to the time after onset of the creep, provided that failure will n o t occur. This offers a possibility o f predicting the strain developed after long periods of creep of various types o f constructions and excavations in rock. Thus, if a few observed displacements at know n time intervals are at hand, the prediction of longterm creep strain should be possible, with reasonable accuracy, by fitting eq. 6 with the observed data. It is required, however, that all the assumed conditions for the validity of the t h e o r y are actually fulfilled. Thus, when t h e creep is associated with expansion and therefore with a reduction in the n u m b e r o f interacting elements, there will be a critical strain b e y o n d which the strain rate accelerates and failure is ultimately developed. REFERENCES Carter, N.L. and Kirby, S.H., 1978. Transient creep and semibrittle behavior of crystalline rocks. Pageoph, 116: 807--839. Ito, T. and Hisatake, M., 1979. Surface displacements caused by tunnel driving in anisotropic viscoelastic ground. Proc. Int. Congr. Rock Mechanics, 4th, Montreux, Vol. 1, pp.677--684. Kaiser, P.K. and Morgenstern, N.R., 1981. Time-dependent deformation of small tunnels, II. Typical test data. Int. J. Rock Mechanics, Mining Sciences and Geomechanics, 18: 141--152.
310
Langer, M., 1979. Rheologieal behavior of rock masses. Proc. Int. C o n g . Rock Mechanics, 4th, Montreux, Vol. 3, pp.29--62. Pusch, R., 1979. A physical clay creep model and its mathematical analogy. Int. Conf. Numerical Methods in Geomechanics, 3rd, Aachen, pp.485--492. Pusch, R. and Feltham, P., 1980. A stochastic model of the creep of soils. Geotechnique, 30(4): 497--506. Pusch, R., Hansbo, S., Berg, G. and Henricsson, E., 1974. B~righet och s~ittningar vid grundl~/ggning p~i berg. Rapport No 11, Svenska ByggnadsentreprenSrfSreningen, Stockholm. Skaanes, S., 1979. Mechanisches Tunnelausbruch am Beispiel Baulos Huttegg des Seelisbergtunnels. Proc. Int. Congr. Rock Mechanics, 4th, Montreux, Vol. 1, pp.551--556. Thomson, S. and El-Nahhas, F., 1980. Field measurements in two tunnels in Edmonton, Alberta. Can. Geotech. J., 17(1): 20--33.