Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol.30, No.7, pp. 1351-1357, 1993 0148-9062/93 $6.00 + 0.00 Printed in Great Britain Pergamon Press Ltd

Micromechanical Modeling of Tuffaceous Rock for Application in Nuclear Waste Storage R. WANG1 J.M. KEMENYt

This paper describes the development of micromechanical models for tuffaceous rock. In particular, laboratory tests have been conducted on Topopah Spring tuff from Yucca Mountain, Nevada and Apache Leap tuff from Superior, Arizona. Topopah Spring tuff is the host rock for the proposed underground nuclear waste repository at Yucca Mountain, and Apache Leap tuff is an analog for the host rock. Based on SEM microscopy of the damaged rock specimens, the specific micro-mechanisms for deformation in tufts have been determined. Micromechanical models based on fracture mechanics theory are then developed for these specific mechanisms. The micromechanical models are able to predict the nonlinear stress-strain behavior of tuff, including strain-hardening, strain-softening, triaxial strength, and dilatation.

INTRODUCTION

Yucca Mountain, Nevada, is currently being considered as a potential site for the underground storage of high-level civilian radioactive waste [1]. The host rock surrounding the underground repository must isolate radionuclide migration to an acceptable level for tens of thousands of years. Groundwater and gases are important potential carriers for transporting radionuclides in the host rock, and preexisting fractures and pores in the rock mass surrounding the repository may provide potential pathways for contaminated water and air. Also, due to the in-situ stresses at Yucca Mountain and the high temperatures of the emplaced waste, a damage zone with an increased crack density may be created in the rock surrounding the waste canisters and the drifts [2]. Understanding the response of preexisting fractures and the potential for new crack growth due to thermal and mechanical loads are very important issues for the design of the underground repository at Yucca Mountain.

Topopah Spring tuff is the host rock for the proposed repository at Yucca Mountain. Apache Leap tuff from Superior, Arizona is another commonly tested tuff that is used as an analog material for Topopah Spring tuff. Previous experimental studies have shown that pores and inclusions are the most important microstructrues in Topopah Spring and Apache Leap tuffs [3,4]. In horizons in which large inclusions and pores exist, these features control rock strength [8]. In horizons in which these large-scale features are absent, micropores are found to control rock strength. The models presented in this paper are based on the testing of samples of Topopah Spring tuff in which no large-scale pores or inclusions were present.

Under compressive loading, microcracking has been found to initiate from sharp comers of the micropores and propagate subparallel to the maximum stress direction

~- Department of Mining and Geological Engineering University of Arizona, Tucson, AZ 85721, U.S.A.

[3,4]. Figure 1 shows microcracking in a sample of Topopah Spring tuff subjected to uniaxial loading. Also, pore collapse has been observed in post-failure specimens of tuff along localized shear zones. In such cases the zone around the crushed pore has been fragmented into a number of 'jointed microblocks', resulting in dilatancy of the rock. Also, the fracture zone around the pore has axial symmetry. Large macroscopic extensile fractures in tuff are formed under low-confinement conditions by the growth and coalescence of pore cracks. The macroscopic cracks tend to propagate sub-parallel to the maximum stress direction and along the path with the highest pore concentration [4].

In this paper, micromechanical models are presented for the microcracking processes described above. These models are based on linear elastic fracture mechanics theory. Nonlinear rock deformation is predicted from the linear theory as the cracks grow and coalesce as load is applied. The nonlinear rock deformation that can be predicted with these micromechanical models include confinement dependent rock strength, strain-hardening, strain softening, and dilatation. The next section of this paper describes micromechanical models for tuffaceous rock. The third section of the paper describes the use of these models to predict nonlinear rock deformation and failure in tuffs under compressive stresses.

MICROMECHANICAL MODELS FOR TUFFACEOUS ROCK

As discussed above, microcrack growth in tuffs subjected to compression occurs by several different mechanisms. In the past few years, mechanical models have been developed for some of these mechanisms [5-16]. In this section these models are extended to more accurately simulate extensile cracking from isolated pores, pore collapse, and the interaction and coalescence of pores by extensile crack growth.

1351

1352 ROCK MECHANICS IN THE 1990s

Figure 1. Microcracking through pores in Topopah Spring tuff. The cracking direction is subparallel to the maximum principal stress direction (vertical).

Extensile Cracking and Pore Collapse for an Isolated Pore Consider a two-dimensional cylindrical pore subjected to biaxial compressive stresses cr and ~,c as shown in Figure 2. When L <1/3, tension will occur at the boundary of the pore in the direction of maximum principal stress, resulting in extensile crack growth as the load is increased. Micromechanical models for extensile cracking from pores have been developed by many investigators. These models are based on fracture mechanics theory, and the quantity of interest is the mode I stress intensity factor, KI [17]. These models predict that K I initially increases with increasing crack length, reaches a maximum, and thereafter decreases with increasing crack length [9]. Experimental studies of acoustic emission during compressional rock deformation [18], however, indicate that many of the emission sources cannot be adequately represented by a tensile crack and suggest that shear slip is involved. Our experimental studies reveal several micromechanisms in tuff that involve shear deformation, and one of the most important mechanisms is pore collapse [2]. This phenomenon has also been observed by other researchers [19]. A model is now introduced for both extensile crack growth from pores and pore collapse.

Consider a single pore with two inclined radial cracks at an angle a , as shown in Figure 2. The modes I and II stress intensity factors are calculated for this model using the weight function method [20], which gives:

1 KU = [~(fr0+2fr2-3fr4)(1-~,)sin2a] ~ "~r-~ (2)

where l is crack length and frn are dimensionless factors that vary with the number of cracks N and the ratio of crack length to pore radius, l/R. Numerical values for frn are given in [20]. Equations (1) and (2) show how the stress intensity factors depend on the confining pressure and the crack angle a. The two specific cases of a=0 and a=45 ° are considered in detail below:

J

I 0¢

l Figure 2. Pore cracking model under biaxial compression.

Case 1: cz=0. When a=0, the crackline is parallel to the maximum stress ~, and the equations for KI and KII reduce to:

KI = [ -fro~-f2-~ 1 +~)+~fr4(t -~ , ) ] a ~ - I (3)

KII = 0 (4)

Thus for 0=0, KI is maximized and KII is identically zero. This result is consistent with the traditional "cylindrical pore model" in [6,9].

0 . s L , , , , f . . . . I . . . . t . . . . I . . . . i i i i :

0.4 .... i ................ i ................ i ............... i .............. -

. . . . . . . . . . . . . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . l . . . . i . . . . ! . . . . !

0 0.5 1 1.5 2 2.5 I/R

Figure 3. KI/O vs. crack length under different confining pressures, for the case of ct=0. The points are given by equation (3) and curves are by equations (7) and (8).

Curves of KI/~ with crack length l/R at different confining stresses for the more general case given in equation (3) are presented in Figure 3. K I / a init ially increases with increasing l, reaches a maximum, and thereafter decreases with increasing I. The maximum in each curve represents the transition from stable to unstable crack growth. Figure 3 also shows that the transition occurs at a crack length less than the pore radius, and the

ROCK MECHANICS IN THE 1990s 1353

transition decreases with increasing k. Figure 3 shows that KI decreases with crack length after the maximum to the point where KI = 0. Crack growth beyond this point is not possible regardless of how much compressive load is applied. The curves presented in Figure 3 agree with the results of [6].

1.6 . . . . . . . . . . . . I , , , t , , , L = O ! N = 2 i

1 . 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ ' ¥ ' w ' " ' " i . . . . . . . . . . . .

1 . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -, ---'-',;'" .'? N = 4 " '" '? . . . . . . . . . . . . • . " i i

l .......................... ~---;--:- ................... .* ............. i ............ • . • L = 0 . 4 i N = 2

K i l l ° 0 . 8 . . . . . . . . . . . . . . . . . . . . . . ~ ' - . . . . . . . . . . . . . . . . . . . . . . . . " i "~ "~ ' ' ' : ' ' ' - ' -~ . . . . . . . . . . . . • ! i . o ' ' . • i N = 4

( ' , / m ) 0 . 6 . . . . • : : . . . . . . . . ~ I__ i -"-----~---':-- - : . . . . . . . . . ~ . . . . . . . . . .

0 4 ........... ! , " i ............. i i ............ • l : : ' • • 6

e : : ' I I ' : 0 . 2 . . . . . i . . . . . . ! " ~ ' " ; " i / " i " ~ - * ~ ~ ' " t " l t " ' " ' ? N - ' 4 " " " t . . . . . . . . . . . .

• . : : : : " o e 9 . . . . ? 1 1 o ° , , I , , , [ , , , I , , , I , , , I , , ,

0 J ; I t t

0 0 . 4 0 . 8 1 . 2 1 . 6 2 2 . 4

1/R Figure 4. KII/C vs. l/R, for the case of ot=45 °. N=either 2 or 4.

Case 2:~=45 °. When ~=45 °, K I becomes negative and KII is maximized. A negative KI indicates that the cracks are closed. Equations (1) and (2) reduce to:

KI < 0 (5 )

KII---~,I fr0 + 2fr2 -3fr4)(1-k)cr~]'-~ (6) z . ,

Curves of Kii/cr with crack length l/R at different confining stresses determined from equation (6) are presented in Figure 4. In contrast to the results for a=0 in Figure 3, for this case KII continues to increase with crack length. Thus even for high values of ;~, KII does not become zero, which allows mode II crack growth at high confining pressures.

I

I . J ; , u r ~ : 5 . P ~ J r e c o l l a p s e m o d e l f o r f o u r r a d i a l c r a c k s a t e q u a l

"lhi~ m¢~dcl can be used to simulate the process of

pore collapse. A pore collapse model with four shear cracks (which has axial symmetry) is shown in Figure 5. If a shear crack growth criteria is used (KIIC or Gc, see [21]), the extension of the shear cracks is an approximation to the actual mechanics of pore collapse. Curves of Kii/¢r with crack length l/R at different confining stresses are presented in Figure 4. These curves are very similar to the two-crack model in the same figure.

The importance of the pore cracking model can be appreciated by comparing Figures 3 and 4. For small crack lengths, KI for or=0 is much larger than KII for ct=45 °, favoring initial extensile crack growth. At larger crack lengths, however, K I for ~.=0 becomes small and KII for a=45 ° continues to increase, favoring a transition to shear crack growth as deformation increases. Also, for high values of k, extensile crack growth is effectively shut down, again favoring the growth of shear cracking. This is in agreement with experimental results [3,4].

(5

- - R 1

Figure 6. Generic model of pore cracking under confining pressure. L -- Distance between two pore centers, C 1 -- Distance of interaction to first pore center, C 2 -- Distance of interaction to second pore center.

Interaction and linking of two pores The formation of macroscopic fractures in tuff occurs by crack interaction and coalescence. Micromechanical models that take into account crack interaction and coalescence are able to predict rock strength, strain- hardening and strain-softening, and creep rupture [7-9,22]. Many of the crack interaction models are based on the sliding crack model, which is not appropriate for tuffaceous rock. The results in the previous section indicate that for an isolated pore, extensile crack growth is initially unstable and then stabilizes when the crack length grows to about 20% of the pore size (this ignores the effect of inertia). The effect of a neighboring pore is to

1354 ROCK M E C H A N I C S IN THE 1990s

cause the tensile crack to become unstable again and to coalesce with the neighboring pore. This process is repeated, resulting in the formation of a macroscopic fracture penetrating a series of pores. A model for this process is now considered.

Consider an infinite body containing two cylindrical pores and four pore cracks as shown in Figure 6. The maximum stress direction is parallel to the centerline of the holes. The two holes will be linked as a result of the interaction and coalescence of the inner cracks. When the two pores are of equal size, a closed form solution for the stress intensity factors for the inner cracks under uniaxial compression has been given by [23]. Figure 7 shows the relationship of the stress intensity factor with crack length for this case. For the parmneters used in this example, the growth of the inner cracks is unstable.

0 . .... i ...... ...... ...... ...... + + i + . . i

. , , o 0 6 ................ i ................ ! ................ i . . . . .

0 , , , i , , . i . , . + , , , i , , , t , , , , 0 0.2 0.4 0.6 0.8 t

I/R

Figure 7. Predicted Ki lo curves for two pore model under uniaxial compressive stress. C/R=2, the points are from [23].

A model has been developed for the stress intensity factors for two unequal sized pores with radii R1 and R2 and subjected to both an axial stress o and a lateral stress ko'. This model was developed utilizing equations (1) and (2) and also the solutions given in [23]. The stress intensity factor solutions are given by:

K~I)/(o ~N~]'~II) e'4)~(e'll/Rl-e-2Oll/Rl) - 2

I- Xll -I tan t2RI(C1/RI'I)] 3~./1/R 1 (7)

8C1/R1

K~2)/(o ~.~22) e'4k(e't2/R2"e-2012/R2) - 2

I" ~12 "1 + tant2a2(C2/R2-1)] -3M2/R2 (8)

8C2/R2 where ~, is the coefficient of confining pressure, R1 and R2 are radii of two pores, ll and 12 are the crack lengths associated with pores 1 and 2, L is center distance of the two pores, and C1 and C2 are the relative distances CI=R1L/(RI+R2), C2=R2L/(RI+R2). Some results from these equations are given below:

(1). When the center distance between the pores is infinite (L/R= **), the tangent function in equations (7) and (8) becomes zero and pore interaction is effectively turned off. The KIlO curves at different values of confining pressure for this case are shown in Figure 3. The points in the figure are calculated from the weight function method

of equation (3). It is seen that the solution from the weight function method and equations (7) ~ (8) agree -,,cry well.

(2). For the case of two equal size pores under uniaxial compression, the Kilo curve for C=2 is shown in Figure 7. The plotted points in the figure are fromthe closed form solution from [23]. It is seen that equation (7)agrees very well with the results of the closed form solution.

(3). For the case of two unequal size pores with applied biaxial compressive stresses, the Ki/cr curves for different ~. are plotted in Figure 8. Figure 8 shows that the point at which the two inner cracks start to interact is not halfway between the two pores, but closer to the smaller pore. Under the same stresses, cracks from the bigger pore are longer than that from the smaller one.

1.6 . . . . . . . I . . . . . . . . I . . . . I , , ,

i [~+ i ia~: 1 i ! . . . . . . . . . . . ! . . . . . . . . . . . . [ ~ ~ . . . . . . . . . . ! . . . . . . . . . . . . . . ~ a ~ ~ z ! . . . . . . . . . . . -

........... + ............ i l ......... i . . . . . . . . . . . ............ 0.8

KIlO

('/m) 0 . 6 ? i i

0 0.5 1 1.5 2 2.5 l/ R1

Figure 8. Predicted KIlO curves at different confining pressures for the generic pore cracking model, when the two pores are different sizes.

MICROMECHANICAL MODELING OF ROCK DEFORMATION AND FAILURE

Here the nonlinear rock deformation and failure of tuffaceous rock is modeled utilizing the micromechanical models from the previous section. The first step is to calculate the strain due to an assemblage of cracks and pores. The strain due to a body containing cracks and pores includes the elastic strains due to the body without the pores and cracks, the additional strains due to the pores, and the additional strains due to the growing cracks. The crack strains are calculated using the stress intensity factor solutions along with an energy theorem such as Castigliano's theorem. Nonlinear rock deformation occurs due to the fact that cracks are growing and giving additional crack strain as the load is applied. The stress at which crack growth occurs is calculated using the stress intensity factor solutions along with the fracture toughness for the material. Nonlinear stress-strain curves are finally derived by using the equations for strain along with the conditions for crack growth.

Consider a body of width 2w, height 2h, and unit thickness, containing a single pore, and subjected to the principal stresses o and ~.o. The elastic strains in the axial (ee) and lateral (e'e) direction under the assumption of plane strain are as follows:

ee= ~,[1 - 3.1-Y~v ] (9)

ROCK MECHANICS IN THE 1990s 1355

8 e = X - (10)

where E'=E/(1-v2), E is Young's modulus and v is Poisson's ratio. It is assumed that this strain includes the strain due to the pore. The additional displacement due to the two cracks from the pore can be calculated utilizing Castigliano's theorem [9] along with the appropriate stress intensity factor solutions:

~ ' ) (KI2+KII2"~ 1 C= O p t 7 ~-7 tu j (11)

where 8 c is the additional displacement due to the cracks at the point and direction where an applied force P is applied, and the integral is over the length of the crack. Crack strains are calculated by dividing the crack displacements by the appropriate dimension (2h for axial strains and 2w for lateral strains).

For the crack strain in the axial (vertical) direction, consider a load P where (~=P/2w. Substituting equations (I) and (2) into (1 I) gives:

8c 0 ~ 2 (~/o2rA2(l+~.) 2 4 =~L~0, ) . k 4 + B2(1-~')2c°s22°t

- AB(l'~'2)c°s2~2 + C2(1-;£)~sin22°tl }d/ (12)

where A=fro + fr2; B = frO + 3fr4; C=fr0+2fr2-3fr4, frO, fr2 and fr4 are the functions of crack length 1. The derivative with respect to P can be performed first, and thus the axial strain due to crack growth is given by:

r t d j " ~J-A2(I+~,) 2 4 4 e - c ~ l J '[_ + B2(1-~') 2cos22~

k u

- AB(I'~'2)c°s2c~2 + c2(a')~'~sin22al }d/ (13)

The total axial strain is the crack strain plus the elastic strain. Also, the crack strain due to 2N cracks (from N pores) will be N times the crack strain for the two cracks from a single pore (crack interaction is not included here). The total axial strain for a body containing N pores is then given by:

o [ l ~. v ]+Nrc r~-A2(l+~.) 2 B2(1-~.)2cos22~ 8 L - l--~vJ w-f fJ 'L 4 + 4

0

- AB(l'~'2)c°s2°t2 + C2(1-~')]sin22~]dl } (14)

The crack strains from equation (14) are calculated with numerical quadrature. To calculate the lateral strain

~', consider a lateral load Q where ~.c~ = 2~h. Following the

procedures outlined above for the axial strain, the lateral strain for the body containing N pores can be expressed as:

l

8=-~{ [ ' o ~"" 1---~]v +-~-h-- J tLNrt~" (.rA2(l+k) 2 4 + B2(l'k)2c°s22~4

0

- AB ( l'~'~)c°s2cz+C2(1 -~')42sin22czld I } (15)

The next step is to calculate the stress necessary for crack growth for any given crack length. For extensile crack growth (or=0), the criterion K I = KIC is appropriate, where the KI solution in Equation (3) is used and where KIC is the fracture toughness for tuff. This gives:

KIC= [- f r0~,- f-~-~(1 +~, + ~fr4(1-~,,] o ~ (16,

A KIC value of 1.6 MPa~m has been measured for Apache Leap tuff. Equation (16) is then solved for o, which gives the stress necessary for extensile crack growth for a given set of conditions (l, KIC, R, E', etc.). For shear crack growth (cz=45°), the criterion G=Gc is appropriate, where G = KII2/E ' (when KI=0), KII from equation (6) is used, and Gc is the shear fracture energy for tuff. This gives:

G,g c E , 1 = ~fro + 2fr2 -3fr4)(1-k)a~/ (17)

Based on the laboratory results in [4] and the method given by [121, a Gc value of approximately 5x104 J/m 2 was determined for Topopah Spring tuff. Again the resulting equation is solved for c and gives the stress necessary for shear crack growth for a given set of conditions.

200 ' ' " " " ' , ' , ' , ' , ' , I ' " ' I ' , ' , ' , ' , ' , : ' , : ' ,

i 0.2 X 160 ............ ! .............. i .............. :: .............. i ..... .--0-.-1 .........

s . 0 (MPa)

80 ............ ] ............. : ......... T ............. ~ ~ o [ ............

40

0 I i i I I

0.1 0.2 0.3 0.4 0.5 0.6 Strain (%)

Figure 9. Predicted axial stress-strain curves for isolated pore model under different confining pressures.

Nonlinear stress-strain curves are derived by combining the equations for strain (equations 14 and 15) with the conditions for crack growth (equation 16 for extensile crack growth and equation 17 for shear crack growth). Details of this procedure are given in [9]. For extensile crack growth and using the material parameters given in Table 1, nonlinear stress-strain curves are presented in Figure 9. Figure 9 shows that extensile crack growth from pores without crack interaction results in initial linear stress-strain behavior followed by strain- hardening as the extensile cracks grow. A rock strength cannot be predicted from these curves since the stress continues to increase with strain. For shear crack growth

1 3 5 ~ R O C K M E C H A N I C S I N T H E 1 9 9 0 s

and using the material parameters given in Table 1, nonlinear stress-swain curves are presented in Figure 10. Figure I0 shows that shear crack growth from pores results in initial linear stress-swain behavior followed by swain-softening as the shear cracks grow. The peak stress for a given value of 7~ is the stress at which shear cracks begin to grow. This is consistent with laboratory results which indicates strain-softening after a shear fault has been formed.

Table 1. Material Properties Used in the Models Material Propertiesl For Fig. 8 For Fig. 9 I For Fig. 10 Young's Modulus 4.0xlO l° Pa 4.0xlO l° Pa 4.0xlO 1° Pa Poison's Ratio Pore Radius Dist between Pore., Initial Crack Len Crack Angle Sample Width Sample Height Number of Pores Gc Klc

0.22 1.0xl0 "5 m

0.2x10 "5 m 0 0.05 m 0.1 m 1000 1.0xl03 Jim ~

0.22 1.0x10 "5 m

0.2x10 "5 m 45 0.05 m 0.1 m 1000 1.0xl03 Jim 2

0.22 1.0x10 "5 m 4.0x10 "4 m 0.2x10 5 m 0 0.05 m 0.1 m 2000

1 6 M P s ~ m

7 0 0 ' : : I : : ] : :

6 0 0 . . . . . . . . . . ~ . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . ., . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . .

5 0 0 -

4 0 0 - S t r e s s

(Mr'a) 300-

2 0 0 -

1 0 0 -

0 4 6 8 1 0

S t r a i n ( % )

Figure 10. Predicted stress-strain curves for the pure shear cracking model at different confining pressures.

Crack interaction and coalescence will affect the stress- strain response due to crack growth. Consider a body of width 2w, height 2h and unit thickness containing two pores and subjected to principal stresses o and ~,o. Following the procedure outlined above for deriving the strain for extensile cracking from isolated pores (equations 9 to 14), the axial and lateral strains for the two pore interaction model are calculated. As before, N pairs of pores can be considered, which will give N times the crack strain of a single pair (the crack interaction between the different pairs is not taken into account). This gives:

£ = 1-

N : KT (1) . , KI (2) +w J ll(~nt~ )2cul+ J:lzt-'=~)2d/2o'q r~l 2

6 0

(18)

Nn~- : KI(I) 2 ':'. KI(2) ,.. II +-~ )/i(---~ i~--~/i d / 1 7 2 t ~ - ~ 2 j ~ (19,

where KI (I) and KI (2) are given in equations (8) and (9), respectively. The stress at which the cracks will grow is determined by setting KI = KIC. For instance, for the cracks from pore I, the condition is:

[e-4~.(e-ll/R1-e-2011/Rl) KIC=~ "

tan L2RI(CI /RI . I ) j +- 8CI/RI -3MI/R I (20)

6 0 0 i . . . . . . . . . . . . . . i . . . .

I , , , , , , , , , , , , , , , i , i , ,

L = 0 . 0 6

5 0 0 . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . : , : . . . : . . . . .

i i ? ..... / 4 0 0 . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . : . . . . . . . . . . . . ,~ - : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

s .s t ./'L-0.03 ..... "i

1 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 . . . . I ' ' ' ' ' '

0 . 5 1 . 5 2 2 . 5

S t r a i n ( % )

Figure 11. Predicted complete stress-strmn curves at different ~. (0, 0.03, 0.06) by using the pore linking model.

A similar equation holds for crack growth from pore 2. Taken together, equations (18) to (20) can be used to calculate the nonlinear stress-strain behavior due to the linking of two pores via an extensile crack. Nonlinear stress-strain curves calculated for different values of X (0, 0.03, 0.06) are presented in Figure 11. The material properties used are listed in Table 1. Figure 11 indicates that crack linking of pores is a process of initial swain hardening followed by strain softening as the two inner cracks interact and coalesce. The overall stress-swain behavior for this model agrees with experimental results on the development of a splitting crack in Topopah Spring tuff subjected to low confinement (0 to 10 MPa). The model predicts the large increase in strength with very small increases in confining stress. Also, Figure 11 shows a predicted uniaxial compressive strength of 179.6 MPa which agrees with the experimental result of 183.1 MPa given in [3]. The model does not take into account shear cracking and therefore does not predict the transition to shear faulting that occurs at higher values of confining s t r e s s .

C O N C L U S I O N S

Experimental results indicate that extensile cracking from pores and pore collapse are the dominant

ROCK MECHANICS IN THE 1990s 1357

micromechanisms for rock deformation and failure in tuffaceous rock. In particular, under low confining pressures (0 to 10 MPa), extensile cracks originating at pore boundaries connect to form macroscopic splitting cracks. At higher confining pressures, extensile cracking stabilizes prior to the formation of splitting fractures, and rock failure occurs by a process of shear localization.

Micromechanical models have been developed for the micromechanisms of extensile cracking from pores, pore collapse, and the interaction of extensile cracks to form a macroscopic splitting fracture. In particular, stress intensity factor solutions are derived using the weight function method. Crack strains are determined using these stress intensity factor solutions along with Castigliano's theorem, and the conditions for extensile and shear crack growth are developed with the fracture toughness and shear fracture energy for tuffaceous rock. Finally, the models are used to predict the nonlinear stress-strain behavior of Topopah Spring tuff and are found to agree with experimental results.

In the future, the micromechanical models developed in this paper will be used to more accurately model the deformation and failure of tuffaceous rock. In particular, a statistical distribution of pores and initial cracks will be considered based on microscopic studies in tuff. Crack interaction effects will also be improved. Finally, the micromechanical models will be implemented into large numerical codes to model the complex, site specific boundary conditions at Yucca Mountain.

Acknowledgments -- This work was supported by the National Science Foundation Solid and Geomechanics Program, under Grant No. MSS9022381.

REFERENCES

1. DOE. Site characterization plan - overview of Yucca Mountain site, Nevada research and development area, Nevada, DOE/RW-0198 (1988).

2. Kemeny J.M. and Cook N.G.W. Time-dependent borehole stability under mechanical and thermal stresses: Application to underground nuclear waste storage. Proceedings of the 33rd U.S. Rock Mechanics Symposium, Norman, OK, 977-986 (1991).

3. Wang R. and Kemeny J.M. Fracturing mechanisms of Apache Leap tuff under compressive su'ess. Fractured and Jointed Rock Masses Conference, Lake Tahoe, California (1992).

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