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Copyright © 1997 Elsevier Science Ltd Int. J. Rock Mech. & Min. Sci. Vol. 34, No. 34, 1997 ISSN 01489062 To cite this paper: Int. J. RockMech. &Min. Sci. 34:34, Paper No. 310
T I M E  D E P E N D E N T C O M P R E S S I V E FAILURE A R O U N D AN OPENING T.M. Tharp
Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, IN 47907, USA ABSTRACT
Compressive failure of rock at a free face is a timedependent process that proceeds outward from the free face. The model for general compressive failure is the sliding crack model of Kemeny, Cook 1987, including crack interaction and subcritical propagation of wing cracks. ]:or a rock mass with an array of propagating cracks near a free surface, moduli are different in all three orthogonal directions. Elastic constants are derived for the resulting orthorhombic symmetry. This model of identical, equally spaced interacting cracks maximizes the reduction in stiffness with crack propagation. The resulting stress reduction around an opening prevents failure even for high levels of stress. It fails to predict the commonly observed slabbing or exfoliation failures. For ~3/61 small, failure is typically more localized than assumed in the above model. This situation is modeled as propagation of a single long crack with periodic outofplane jogs with frictional sliding. Such a crack can propagate in the 61 direction at nearly constant stress. Proximity of a free face increases K I, however, for long cracks the effect of the free face is small. A small compressive 63 greatly reduces K I, and with the 63 gradient found near an opening, subcritical propagation velocity will drop precipitously away from the opening. This is consistent with failure by progressive exfoliation of thin sheets. Copyright © 1997 E l s e v i e r S c i e n c e Ltd
KEYWORDS A n i s o t r o p y • Brittle F a i l u r e • C o m p r e s s i v e Strength • C o n s t i t u t i v e R e l a t i o n s • F r a c t u r e M e c h a n i c s • Mechanical Properties • Numerical Analysis • Time Dependency INTRODUCTION
Compressive failure at a free face in a mine, civil work or borehole is a timedependent process that proceeds outward from the free face. If the rock around the opening is initially relatively homogeneous and isotropic, it typically fails by slabbing of discrete sheets. Because the process is nonlinear, it is path dependent, and time dependence governs the stressstrain path at any point in the rock. In order to model the process and understand the macroscopic failure mechanism, it is useful to model the timedependent progressive nature of the failure. The model for compressive failure used in this investigation assumes the presence of cracks oriented so as to slide in compression. Wing cracks initiate at the tips of these initial cracks and propagate in a direction that asymptotically approaches the direction of 61 , the maximum compressive principal stress. These cracks may ultimately interact and coalesce. I consider first a dense array of such short interacting
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cracks, and then an individual long crack that incorporates sliding segments. For a rock mass with a dense array of sliding cracks with wing cracks, the effective elastic moduli of the rock mass were derived by Kemeny, Cook 1991 for the transversely isotropic case. For transverse isotropy moduli are the same in two orthogonal directions and in any direction lying in that plane, with different moduli pertaining to the third orthogonal direction. For the general case of loading near a free surface, moduli will be different in all three orthogonal directions. I derive elastic constants for the orthorhombic symmetry that results. The second approach to compressive failure near an opening is to consider propagation of a long crack that approximates the roughness observed in splitting cracks in laboratory experiments. Such a crack will have frictional sliding segments along its length and will incorporate additional sliding segments by coalescence with other sliding cracks or by outofplane propagation of the crack. The time dependence of compressive failure is assumed to result from subcritical crack propagation. For small K I / K I c , velocity of crack propagation v is given by: v = Ce ( K I / K I c )
n
(I)
where KI is mode I stress intensity, KlC, is critical stress intensity for instantaneous propagation, c e is equal to v o exp (  H / R T ) , where H i s activation enthalpy, v o and n are empirical constants, Tis absolute temperature, and R is the gas constant. Values of c e and n are calculated for linear portions of the log (velocity) vs log (/(i) curves interpreted by Atkinson 1984 to correspond to region 1, where propagation velocity is controlled by reaction rate at the crack tip. Because this region corresponds to the lowest stress intensities, it may be extrapolated to lower stress intensities than have been investigated experimentally.
ORTHORHOMBIC
ELASTIC MATERIAL
In compression, wing cracks propagate from the ends of an initial flaw. Kemeny, Cook 1991 demonstrate that a wide variety of different flaws yield similar relationships between imposed stresses and KI. A straight crack is assumed to be the initial flaw. Kemeny, Cook 1987 and Kemeny 1991 give an approximate equation for KI for wing cracks propagating from initial straight sliding cracks that accounts for interaction with identical collinear cracks.
(2)
KI = 2lo 're cos 13 / (b sin (~ I / b)) w  62 (2b tan (• l / 2b)) v2 where z e is:
1
(3)
= = [(01  o 2 ) sin 2[3  Ix ( ( o l + o 2 + ( o i  o 2 ) c o s 2131 ;Z
and l o is half the length of the initial cracks, I is half the length, measured in the direction of ~1 of the initial crack plus the wing cracks, b is half the centertocenter spacing between initial cracks, 13is the angle between the initial crack and ~2, and bt is the friction coefficient (Fig. 1). The friction coefficient is assumed to be 0.6 for all analyses in this paper. Compressive failure occurs when propagation causes collinear cracks to coalesce. This model is calibrated for a particular rock by finding initial crack spacing b to allow failure at the appropriate time and load for known shortterm strength (Tharp, Holdrege 1995). An initial crack length, which is poorly known, must be assumed. However, Tharp, Holdrege
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1995 and Tharp 1995 show that, when calibrated with shortterm strength data, time to failure for a given stress varies only slightly for initial flaw size varying over 4 orders of magnitude. Indeed, Tharp, Holdrege 1995 found that when calibrated by shortterm test results time dependence is hardly affected by initial flaw size or friction on the crack. Elastic Constants f o r I n t e r a c t i n g Cracks
As the wing cracks propagate, the stiffness of the rock decreases and the decrease becomes particularly pronounced near coalescence. The procedure for deriving the elastic constants for the rock with any propagated length of wing cracks is similar to that used by Kemeny, Cook 1987, 1991. Displacements of an elastic cracked body may be found using Castigliano's theorem (Sokolnikoff 1956). From Castigliano's theorem the displacement x i resulting from load F i when the body is subject to loads F 1, F 2, ... F n is:
xi = xi
(uncracked body) +
(4)
3oUe/OFi
where Uc is the elastic strain energy of the cracked body. For a crack in plane strain (Kemeny, Cook 1991):
2(1_ 2) (5) where E is Young's modulus and v is Poisson's ratio. Integration of Eqn. 5 with K I given by Eqn. 2 gives:
U~ = (8(1  "o2)/rc E) [1o2 Xe2 cos 2 ~ In [tan (l rc / 2 b ) / t a n (Io rc / 2b)l  2 lo x, cr2 b cos [3 In [tan (rt(1 + l / b)/4) / tan (rr(1 + l o / b ) / 4 ) ] + cJz2 b 2 In [sec (I n / 2b) / sec (Iorc / 2b)]
(6)
With substitution of z e from Eqn. 3, Eqn. 6 becomes strain energy associated with a single plane strain crack as a function of the applied stresses (Yl and or2, with (Yl parallel to the direction of the wing cracks. Kemeny, Cook 1991 multiply the strain energy by N, where N is the number of cracks in the volume V. Nwould be 1 for the unit volume V = (2b) (2w) associated with a single crack. This will generally overestimate the strain energy associated with a single real crack, because the plane strain assumption requires that these cracks, short in the xy plane, have infinite extent in the z direction. The (Yl and cy2 directions refer to coordinate axes x and y respectively, here and in later equations. The plane strain assumption means that although cracks may be widely spaced in the x and y directions, the spacing between them is zero in the perpendicular direction z. N should properly be less than 1 to reflect the actual presence of intact rock between cracks in that direction. For an approximate solution to this problem I assume that a crack's length in the z direction equals its length in the xy plane, and that this equality is maintained as it propagates. I also assume that centertocenter spacing is 2b in the z direction, and in the y direction as well (i.e. 2w = 2b). With these assumptions N = l/b. To calculate elastic constants for the cracked rocks, strains 1~1 and ~2 must be found as a function of (Yl and cy2. To derive these relationships in terms of Castigliano's theorem it is convenient to state the forces acting on the unit volume in the (Yl and cy2 directions respectively a s
F 1 =
2w (Yl and F 2 = 2b cy2.
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Resulting displacements are 81 and ~2" Normal strains are e 1 = 81/2b and E2 =
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82/2wand by Castigliano's
theorem ~1  0 Ue/OF 1  (0 Ue/O(ya) (0(ya/0Fa) and ~2  (0 Ue/O(Y2) (0(Y2/OF2). After multiplying by N:
~1 = [Nl(2b 2w)] OUe/Oo~
(7)
e2 = [N/(2b 2w)l 3UJ3o2
(8)
For unit dimension in the z direction, (2b)(2w) is the volume of the unit volume and it is convenient to define initial crack density A as A = N lo:/V. The partial derivatives 0 Ue/Ocy1 and 0 UjOcy 2 may each be written as constant terms multiplying cy1 and or2, such that:
et = cl o l + c2 o2
(9)
e 2 = C 3 (~1 + (74 (~2
(10)
where c 3 = c 2. With s = sin 13 and c = cos 13:
cl = (1  ~02)/E + [ 16( 1  "02)/rt E] Ac z (sc  g c2) z in [tan (l ~ / 2b) / tan (lorc / 2b)]
(11)
c2 =  v (1 + u ) / E  [16(1  ~2)i~ E] Ac (sc  g c 2) [ ( s c + Ix s 2) cln [tan (I~ / 2b) t tan (lo ~ / 2b)] + (b / lo) In [tan (•(t + l/b)/4) / tan (~ (1 + lo / b)/4]] c4 = ( 1  "02) / E + [ 16 (t  ~02)/ ~ E] A [c 2 (sc + g s2)2 In [tan (I g / 2b) / tan (lo n / 2b)] + (2cb / lo) (sc + g s 2) In [tan (~(1 + l / b)/4) I tan (n(l + lo / b)/4)] + (b 2 //o 2) In [sec (lrt / 2b) / sec (to it / 2b)]
(12)
(I 3)
The first term in each of the equations for Cl, c 2 and c 4 represents the contribution of the uncracked rock with the assumption of plane strain. To represent the relationship between "c12and 712 (i.e. "Cxyand Yxy), the crack array is simplified, as by K e m e n y , Cook 1991, to an array of open, straight collinear cracks subject to a shear stress "c12in the plane of the cracks. The mode II stress intensity Kiiis given by (Rooke , Cartwright 1976):
KH = 'h2 (2b tan (re 11 2 b ) ) 1/2
(14)
The equation for strain energy results from replacement of K I with KII in Eqn. 5, which upon substitution of Eqn. 14 and integration yields:
Ue = [8 (1  "0z) x212 b2 / rt E] In [sec (l g / 2b) / sec (lolt / 2b)]
(15)
Following the procedures used above
7t2 = c5 x12
(16)
where, if G is the shear modulus for the uncracked rock:
c5 = I l G + [ 16(1  ~a2) b 2 t l,,2 ,t E] A In [sec (l = / 2b) I sec (/o = / 2b)]
(17)
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This simplified analysis neglects cross compliances between normal and shear components. For plane strain finite element analysis, the system of Eqns. 9, 10 and 16 is inverted to give the [D] matrix, defined in the standard way by:
ay
= [D]
Zxy
(t8)
ey Y.y
where:
C4](C1C4C 2)
 C2/(C1C4C 2)
[l)]=
0o
0
0
(19)
1/c 5
Behavior of the Interacting Crack Model
To evaluate the behavior of the model for an underground opening a circular hole with radius of 3 m is loaded with remote stresses of 60 MPa in the horizontal direction and 20 MPa in the vertical direction. The rock is assumed to have the subcritical crack propagation parameters c and n evaluated for falerans micrite (Henry et al. 1977). Effective critical stress intensity KQ as evaluated for short cracks in limestone by Tharp, Holdrege 1994 replaces Kic in Eqn. 1. For assumed initial crack length of 2 x 102 mm, b is found to equal 7.8 x 102 mm for short term strength (with 100 second ramp load to failure) of 200 MPa by the method of Tharp, Holdrege 1995. The finite element model gives a compressive stress of 173 MPa at the Gauss point nearest the top of the circular opening. This exceeds the infinite plate stress because the finite element mesh is finite in size. For a constant uniaxial compression of this magnitude time to failure would be 5 minutes (Tharp, Holdrege 1995). However, as cracks propagate, the elastic moduli decrease and stresses are relieved near the opening. Because near the opening (Yl is maximum and (Y3 is nearly zero, cracks initially propagate most rapidly there. Fig. 2 shows tangential stresses at Gauss points near the opening after 30 days under load. By this time tangential stress near the opening has dropped 27%. The rate of decrease in stress decays with time. At this location a reduction of 19% occurred in the first hour of loading, but the reduction is only 43% after 106 years. The stress redistribution occasioned by stiffness reduction causes a large zone of substantial crack propagation, as seen after 106 years under load in Fig. 3a. Another prominent feature of this model is large dilation as the cracks open and propagate under load. Dilation in the large region of crack propagation at the top of the opening reduces vertical compressive stress to the right of this region causing modest crack propagation in that area as well. After this long loading duration crack propagation has nearly stopped and wing cracks have propagated at most only 41% of the distance necessary for crack coalescence and failure. Fig. 3b shows cracking for the same loading under the assumption that elastic moduli are not affected by crack propagation. In this case stresses do not change. In Fig. 3b maximum crack length is the same as in Fig. 3a, but this was achieved with a loading duration of only 27
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hours. This is substantially longer than the 5 minutes required for failure under unconfined compression because ~3 at the Gauss point with highest stress is compression of 2.2 MPa. When stress relief is not allowed, crack propagation is concentrated in a narrow zone adjacent to the opening and propagation goes to coalescence and therefore failure.
LONG CRACK PROPAGATION Finite Element Model for Long Cracks Closely spaced, identical, interacting sliding cracks allow stress relaxation without failure, as shown above. However, this is an extreme model for failure in nearly uniaxial compression. If such failures more closely approach a weakest link type of failure, slabbing might be modeled as the propagation of a single crack. Stresses would be relaxed in the immediate vicinity of such a crack, but elastic moduli in the surrounding rock would be unaffected. Wing cracks propagating from a single sliding crack experience decreasing K I as they become long and thus eventually stop. However, a crack propagating in the cyI direction might incorporate additional sliding segments by coalescence with other sliding cracks. Localized propagation along cleavage planes or other planes of weakness might also produce outofplane crack segments that would slide after initiation of a wing crack. Peng, Johnson 1972 found large departures from planarity in splitting cracks produced experimentally in uniaxial compression. Many parts of these cracks were oriented such that closure and frictional sliding would be expected. For purposes of analysis I approximate such a crack as segments parallel to cy1 linked by jogs in the crack at an angle to cy1. These jogs could experience sliding. The crack is modeled by finite elements using a special array of 8 crack tip elements around each crack tip (Fig. 4). The elements are all standard 8node isoparametric elements, but for crack tip elements one side is collapsed to a point at the crack tip and midside nodes on adjacent sides are placed at the quarterpoint position (Barsoum 1977). K I is evaluated from nodal displacements on the crack surface (Shih et al. 1976). The potential sliding segments (jogs) are modeled with trusses of zero length (but finite stiffness) spanning the crack between its two surfaces. The trusses are oriented perpendicular to the crack faces so as to apply only normal load, and they are used to monitor normal closure and shear displacement across the crack in the sliding segments. Axial loads applied to the nodes of the trusses are adjusted iteratively to enforce a zero closure condition across the sliding segments. The load carried by the trusses and the iterative axial loads are used to calculate frictional resistance on the crack, which is applied as iteratively adjusted loads to the nodes at either end of each truss. If a truss experiences extension the crack is opening at that location and the stiffness of the truss is set to zero. If closure occurs at a later iterative step the truss is replaced. Models are run with even numbers of jogs from 2 to 20. The orientation of the jogs departs 40 ° from the general crack direction. The jogs have a length of 1.3 ram, with spacing between jogs of 3 ram. These values are suggested by the lengths observed by Peng, Johnson 1972. Distance from the last jog to the crack tip at each end equals the spacing between jogs. For regularly occurring jogs this is the greatest distance a crack would have to propagate before encountering another jog, thus it should approximate the lowest K I. Although KII may be substantial, and important in producing jogs, I consider only variation of
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K I, which is a good measure of the crack's ability to continue to propagate in its plane.
The finite element model is 50 cm long, with a constant displacement, parallel in direction to the crack and free face, applied at one end. This is calculated to apply an initial stress (Yl of 1000 MPa. The free face is a stress boundary condition, with normal stress (Y3 equal to zero o r 0 . 0 1 ( y 1. The 50 cm dimension of the model implies a spacing between cracks of 50 cm. Because the end displacement is the same in all models, the total initial strain energy available is the same, regardless of crack length. K t F o r L o n g Cracks
Fig. 5 shows K I for cracks with 2 to 20 jogs and corresponding variation in length. The value o f K I is smaller in all cases for a single jog. K I is increased by proximity to the free face as expected, but this influence is reduced as the crack increases in length. For the crack 1 cm from the free face, frictional sliding is almost entirely restricted to the jogs immediately adjacent to the crack tip. Sliding on these cracks wedges open the central region of the crack, preventing friction there. With the deeper cracks, additional jogs near the crack tips slide because the greater thickness of rock between the crack and the free face decreases the opening in the central region of the crack. As a crack grows longer, K I will increase because the magnitude of the loads at the jogs near the end changes little. But as crack length increases, some stress relaxation occurs, and this tends to decrease K I. The two influences seem to be in approximate balance for long cracks, maintaining a nearly constant K I. Because K I is nearly independent of distance from the free face for long cracks it is not surprising that multiple splitting cracks typically occur through the thickness of an unconfined compression specimen. Fig. 5 however indicates that application of a stress cy3 normal to the crack with 1/100 the magnitude of initial or1, the stress parallel to the crack, reduces K I to 60% of the uniaxial compression value for a crack 3 cm from the face. The reduction is similar for the other crack configurations. The ratio of subcritical crack propagation velocities v "and v ' f o r stress intensities Ki'and K i ' q s given by: v'/v"=
(20)
(Kt'/ Kt") n
If n has a typical value of 20, the change in K I caused by addition of cy3 =
(Yl/100
would decrease crack
propagation velocity by a factor of 4 x 10 5. A further modest increase in (Y3 would close the crack completely. This is consistent with the observation that splitting failure is replaced by shear failure at low levels of confining stress. On the surfaces of underground openings radial compression rises steadily outward, which should restrict slabbing failures to a narrow zone at the perimeter of the hole.
DISCUSSION AND CONCLUSIONS The model of equally spaced interacting cracks of equal length produces an abrupt reduction in stiffness as failure is approached (Kemeny, Cook 1991). Because all unit cells subject to a given stress behave in the same way, this computational approach maximizes the reduction in stiffness at any stage of crack propagation. The resulting stress reduction around an opening is sufficient to prevent failure even for rather high levels of stress. In particular, it fails to predict the slabbing or exfoliation failures commonly observed in isotropic rock. Better performance for this type of model might be achieved by computing a composite stiffness for a body with unit cells having initial cracks of different length.
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The lack of failure at the free face suggests that the continuum approach in the first part of this paper is too simple to yield reasonable results, at least when (y3/(Yl is very small. For this case failure is typically more localized then when (Y3 is larger. In this situation, a single long crack with periodic outofplane jogs with frictional sliding can propagate in the (Yl direction at nearly constant stress. As predicted by Odintsev 1993, the outofplane jogs near the crack tips will experience friction, while those near the center of the crack will pull apart. Proximity to a free face increases K I, particularly for short cracks. As a result long cracks will initiate more easily near a free face even in uniaxial compression. However, once the crack is relatively long, the effect of the free face on K I is small. In a typical rock with a wide distribution of initial crack lengths the resulting distribution of splitting cracks might show only a modest concentration near the free face if loaded in homogeneous uniaxial compression. Even a small compressive (Y3 greatly reduces K I for long as well as short cracks. With the (Y3 gradient generally found near an opening, subcritical propagation velocity will drop precipitously away from the opening. This is consistent with failure by progressive exfoliation of thin sheets. Given the effect of (Y3, it is probable that a particular exfoliation crack will not propagate significantly until the adjacent outlying sheet has pulled away.
FIGURES
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Paper 310, Figure 1. 0" 1
1 0"2
213
..................
,
!1 i i
i i
b
2w
Figure 1. Definition of parameters for an infinite array of sliding cracks with wing cracks (Kemeny, Cook 1987). The unit volume for computation of elastic constants is enclosed in dashed lines.
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Paper 310, Figure 2. O3
t~
,ooi\ 140.0
u3 ~3 L
120,0

1 00,o

Or3 t~
x
t80,0
\

C r~ 80,0
I
0
I
1.~ z.o Radial
[
I
I
I
3.0 4.0 s,o
s,o 7,0
Distance
(m)
Figure 2. Tangential stresses at Gauss points at the top of a circular opening of 3 m radius after 30 days under load.
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Paper 310, Figure 3a.
A
~."....
".. • '~" :"...'." \ \ I
L
Figure 3a. Relative wing crack lengths and orientations at selected Gauss points for times at which maximum wing crack length is 41% of that necessary for failure (coalescence). Maximum plotted length corresponds to actual wing crack length of 2.8 x 10 2 mm. A:
106 years under load with cracklength dependent stiffness (Eqn. 19) and resulting stress relaxation.
B:
27 hours under load giving the same crack propagation length, assuming that stiffnesses and stresses are not relaxed.
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Paper 310, Figure 3b.
B
~..,,~ .,.&
,
% \
\ \
\
Figure 3b. Relative wing crack lengths and orientations at selected Gauss points for times at which maximum wing crack length is 41% of that necessary for failure (coalescence). Maximum plotted length corresponds to actual wing crack length of 2.8 x 10 2 mm. A:
106 years under load with cracklength dependent stiffness (Eqn. 19) and resulting stress relaxation.
B:
27 hours under load giving the same crack propagation length, assuming that stiffnesses and stresses are not relaxed.
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Paper 310, Figure 4.
Figure 4. Central part of finite element mesh for long crack with 6 jogs and free face 1 cm above. The element boundaries along the jogs are 40 ° from horizontal. The continuation of the mesh to left, right and below is not shown.
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Paper 310, Figure 5. 4.0 /'"~__ l
. i~:~
I
crn om
Od
3.0
/ 1,0
3 c'm 03=0"]#1
/ I 2,0
,0
~3=0 
¢3=0
#
a_
0"30
I ~.0
1 6.0
I B,C,
I 10.0
Crack Length (cm] Figure 5. K / for long cracks with 2 jogs (shortest) to 20 jogs for initial (31 parallel to the free face of 1000 MPa. For each curve the distance to the free face and
(33
imposed at the free face are given.
References References Atkinson B.K. 1984. Subcritical crack growth in geological materials. J. Geophys. Res., 89: 40774144. Barsoum R.S. 1977. Triangular quarterpoint elements as elastic and perfectlyplastic crack tip elements. Int. J. Num. Meth. Eng,, 11: 8598. Henry J.P., Paquet J., Tancrez J.P. 1977. Experimental study of crack propagation in calcite rocks. Int. J. Rock Mech. Min. Sci. & Geomech. Abs~, 14: 8591. Kemeny J.M. 1991. A model for nonlinear rock deformation under compression due to subcritical crack growth. Int. J. Rock Mech. Min. Sci. & Geomech. Abstn, 28: 459467. Kemeny J.M., Cook N.G.W. 1987. Crack models for the failure of rock in compression. Proc. 2nd Int. Conf. Constitutive Laws for Engineering Mat., 2: 879887, Elsevier, New York. Kemeny J.M., Cook N.G.W. 1991. Micromechanics of deformation in rocks. Toughening Mechanisms in QuasiBrittle Materials', 155188, Kluwer Academic, The Netherlands. Odintsev V.N. 1993. Scale effects in rock splitting. Scale Effects in Rock Masses 93, 225231, Balkema, Rotterdam. Peng S., Johnson A.M. 1972. Crack growth and faulting in cylindrical specimens of Chelmsford Granite. Inst. J. Rock Mech. Min. Sci., 9: 3786.
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Rooke D.R, Cartwright D.J. 1976. Compendium efStress Intensity Factors, H.M. Stationery Office, London, U.K. Shih C.F., deLorenzi H.G., German M.D. 1976. Crack extension modeling with singular quadratic isoparametric elements. Int. J. Fracture, 12:647651. Sokolnikoff I.S. 1956. Mathematical Theory efEIasticity, McGrawHill, New York. Tharp T.M. 1995. Design against collapse ofkarst caverns. Prec. 5th Multidisciplinary Conf. on Sinkholes and the Environmental Impacts' e f Karst, 397406, Balkema, Rotterdam. Tharp T.M., Holdrege T.J. 1994. Fracture mechanics analysis of limestone cantilevers subjected to very longterm tensile stress in natural caves. Prec. 1st North American Rock Mech. Syrup., 817824, Balkema, Rotterdam. Tharp T.M., Holdrege T.J. 1995. Very longterm loading of roof beams in limestone caves. Prec. 35th U.S. Syrup. Rock Mech., 789794, Balkema, Rotterdam.
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