micromechanics-based prediction of creep failure of hard rock for long-term safety of high-level...

Micromechanics-based prediction of creep failure of hardrock for long-term safety of high-level radioactive

waste disposal system

Kazuhiko Miura a, Yoshiaki Okui b,*, Hideyuki Horii c

a Radioactive Waste Management Funding and Research Center, 2-8-10 Toranomon, Minato-ku, Tokyo 105-0011, Japanb Department of Civil and Environmental Engineering, Saitama University, 255 Shimo-Ookubo, Saitama 338-8570, Japan

c Department of Civil Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyou-ku, Tokyo 113-8656, Japan

Received 15 June 2001; received in revised form 22 January 2002

Abstract

A high-level radioactive waste disposal project is under way in Japan. Isolation of the radioactive waste in a rock

formation at great depth is considered as one of the promising disposal methods. Before proceeding with the disposal

project, however, assessment of the long-term stability of rock mass surrounding waste packages is necessary. This

paper presents a micromechanics-based model for predicting the creep failure of hard rock under compression. The

subcritical crack growth due to stress corrosion cracking and interaction effects between cracks are considered as the

main mechanisms of creep failure. An evolution problem of two interacting cracks is formulated and creep failure

following tertiary creep is represented as unstable extension of the interacting cracks in the present model. The time to

failure is calculated for different values of axial stress, confining pressure, and environmental conditions such as

temperature and presence of water. The effects of water, temperature, and stress states on the long-term behavior are

also included in the proposed model.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Creep; Subcritical crack growth; Granite; Crack interaction

1. Introduction

For disposal of high-level radioactive waste,isolating it in rock formations at great depths has

been considered one of the most effective methods.

Research and development on durable disposal

methods has been carried out worldwide, and the

validity of the methods has been presented through

environmental impact assessments in such coun-tries as the United States, Canada and Finland.

In Japan, there are 52 commercial nuclear

power plants, which produce about one third of

the total electricity production. The Japanese

government supports nuclear power as one of the

very important sources from the viewpoints of

stable energy supply and low environmental im-

pact. For disposal of radioactive waste, the Japan

*Corresponding author. Tel.: +81-48-858-3552; fax: +81-48-

858-7374.

E-mail address: [email protected] (Y. Okui).

0167-6636/03/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0167-6636 (02 )00286-7

Mechanics of Materials 35 (2003) 587–601

www.elsevier.com/locate/mechmat

mail to: [email protected]

Nuclear Cycle Development Institute (1999) re-

ported that host rock that is fully stable over a

long period of time exists in Japan, and that it is

possible to build geological disposal systems that

can guarantee long-term safety. The Nuclear

Waste Management Organization of Japan startedthe process of selecting a repository site in 2000.

Under the current concept of radioactive waste

disposal, wastes are buried in a vault excavated in

rock formations at depths of 500–1000 m below

ground level. It is important to grasp the long-term

behavior of rock masses surrounding the vault

from the following viewpoints:

• The safety of disposal facilities should be as-

sessed for a period of about 10,000 years.

Changes in the near-field environment during

the period need to be predicted. We must there-

fore examine the time-dependent behavior of

near-field rock masses and its effect on excava-

tion damage development. The near-field rock

masses are expected to act as a natural barrieragainst diffusion of radioactive substances.

Hence expansion of the damage zone can be re-

garded as deterioration of the performance of

the natural barrier.

• No timing is fixed for backfilling the vault after

the storage of radioactive waste. There is an al-

ternative of not backfilling for about 300 years

while continuing maintenance (US Departmentof Energy, 1998). Hence, it should be verified

whether or not the safety of the vault may be

jeopardized due to the creep of the rock sub-

jected to high stresses during the period.

• High-level radioactive waste produces heat,

which combines with groundwater and acceler-

ates the creep of rock. It is therefore necessary

to study the effects of heat and groundwateron the creep of rock.

In Canada, Atomic Energy of Canada Limited

(AECL) constructed the underground research

laboratory (URL) in the Lac du Bonnet batholith

to conduct in situ experiments on the long-term

behavior of heated rock masses (Read et al., 1997).

Furthermore, as a part of the investigation, Lajtaiet al. carried out a series of creep and crack ve-

locity tests of intact Lac du Bonnet granite to in-

vestigate the effects of water and temperature on

the long-term behavior (Lajtai et al., 1987).

Over the past decades, a considerable number

of constitutive models for the time-dependent be-

havior of rocks have been proposed (see Cristescu

and Hunsche, 1998 for a review) The constitutivemodels are based either on a phenomenologi-

cal approach or on a micromechanics-based ap-

proach. In most phenomenological models, since

model structures are contrived to fit the experi-

mental data, the models have an empirical char-

acter. This causes, however, great uncertainties in

the extrapolation to the distant future and to

conditions not covered in existing data. Therefore,the physical deformation mechanisms for creep

must be included in reliable models for long-term

predictions.

For crystalline rocks, nucleation of microcracks

and their subcritical crack growth due to stress

corrosion have been considered to be domi-

nant mechanisms of deformation in primary and

steady-state creep (Atkinson, 1984; Atkinson andMeredith, 1981, 1987; Kranz, 1979, 1980; Swan-

son, 1984; Wilkins et al., 1984; Meredith and At-

kinson, 1985). Tertiary creep and eventual creep

failure have been ascribed to effects of interaction

between microcracks and coalescence of micro-

cracks.

Okui et al. (1993) proposed a micromechanics-

based continuum theory for short- and long-termbehavior of hard rocks under compression (Okui

et al., 1993; Okui and Horii, 1997). The continuum

theory, called interaction field theory (IFT), is

formulated by homogenization of an elastic solid

with many cracks. Two mechanisms of crack

growth are considered, namely stress-induced

crack growth based on linear elastic fracture me-

chanics and stress corrosion cracking. In the crackgrowth criterion, the effects of interaction among

cracks are taken into account, which leads to

formulation of an integral equation that deter-

mines the interaction effects. IFT can reproduce

the behavior from primary to tertiary creep, and

accordingly can evaluate time to failure under

long-term loading. In order to obtain a solution of

the theory, however, one has to simultaneouslysolve a set of differential equations such as an

equilibrium equation as well as the integral equa-

588 K. Miura et al. / Mechanics of Materials 35 (2003) 587–601

tion by using the finite element method. Thus, it

may not be suitable to use IFT for parametric

studies to investigate long-term strength.

In this study, we propose a simple mechanism-

based model for predicting the creep failure of

rocks under compression. In this model, essentiallythe same mechanisms of crack growth as those in

IFT are considered, and by assuming a uniform

stress state, the continuum theory is reduced to an

evolution problem of two interacting cracks.

After introduction of a micromechanical model

of flaws inside hard rocks under compression, a

simple analytical model is presented for predicting

short-term strength and time to failure under long-term constant load. Then, the validity of the pre-

sent model is examined through a parametric

study and comparison with reported experimental

data.

2. Sliding crack model and its simplification

The sliding crack model shown in Fig. 1(a) has

been employed by many researchers as a reason-

able model of flaws in rocks under compression

(Brace and Bombolakis, 1963; Horii and Nemat-

Nasser, 1985a,b). The sliding crack model consists

of the initial defect of length 2c0 and two tensilecracks. The maximum compressive stress r22makes the surfaces of the initial defect come intocontact and slide. The frictional sliding of the

initial defect produces the tensile cracks from the

tips of the initial defects.

An evolution problem of the sliding crack

model can be solved analytically with the method

of distribution of dislocations. However, an ex-

plicit solution is not available, and the procedure

of the solution is complicated. Hence, further

simplified models have been proposed (Horii andNemat-Nasser, 1986; Nemat-Nasser and Obata,

1988). In this study, we employ a simplified model

shown in Fig. 1(b) as a microstructural model of a

crack. The tensile crack is assumed to be parallel

to the maximum compressive stress r22 on thebasis of microscopic observation of rocks (Kranz,

1979). The effect of the frictional sliding on the

tensile crack is modeled with a concentrated forceF . The magnitude of the concentrated force isgiven by

F ¼ 2c0h�s� þ lr�i sin h ¼ c0hk2r11 � k1r22i; ð1Þwhere h i denotes the positive part of the argu-ment,

k1 ¼ sin hfsin 2h � lð1� cos 2hÞg;k2 ¼ sin hfsin 2h þ lð1þ cos 2hÞg;

ð2Þ

and h is the inclined angle to the direction of themaximum compressive stress, r� and s� are thenormal and shear stresses on the initial defect re-spectively, and l is the friction coefficient.Note that the mode I stress intensity factor

(SIF) of the model is given by

KI ¼ffiffiffiffiffipl

pr11 þ

c0hk2r11 � k1r22iffiffiffiffiffipl

p : ð3Þ

This expression will be used in formulation of the

evolution problem in Section 5.

3. Interacting crack model

In order to develop a theoretical model for

creep of hard rocks, an evolution problem of manysimplified crack models embedded in an elastic

medium is probably the most suitable. Okui et al.

(1993) investigated this problem by transforming

the problem into a continuum theory. In this

paper, however, the evolution problem of two in-

teracting cracks is discussed for simplicity. Fig.

2(a) shows the considered problem, in which the

simplified crack models shown in Fig. 1(b) are

2C0

θ

-σ11

-σ22

-σ11

-σ22

-σ11

-σ22

-σ11

-σ22

F F 2l2C

θ

-σ11

-σ22

-σ11

-σ22

-σ11

-σ22

-σ11

-σ22

F F 2l

(a) (b)

Fig. 1. Microcrack model under compression: (a) sliding crack

model; (b) simplified model.

K. Miura et al. / Mechanics of Materials 35 (2003) 587–601 589

embedded in the elastic body, and the principal

stresses r11 and r22 are applied to the body at in-finity.

Major reasons for this simplification are as

follows:

• In reality, many cracks grow while interactingwith one another. Interaction between two ad-

jacent cracks is, however, actually most pre-

dominant, and distant cracks have little effect.

Solution of the evolution problem of two cracks

would therefore capture the essence of the prob-

lem.

• In fact, under IFT, interaction of many cracks

is represented by an integral equation for cracksthat are distributed continuously with a certain

density. When solving the equation by discretiz-

ing it by a finite element method, the structure

of the discretized equations is basically identical

to that of the governing equations in the evolu-

tion problem of two cracks.

For evaluating crack interaction, the pseudo-traction method was used (Horii and Nemat-

Nasser, 1985a,b). The original problem involving

two cracks was decomposed into a homogeneous

problem and two subproblems as shown in Fig. 2.

In the homogeneous problem there is no crack and

the same stresses as the original problem are ap-

plied at infinity, while in each subproblem a single

crack is contained without far-field stresses.In the subproblem a, �ðr22 þ rPa

22 Þ must be ap-plied to the surface of the crack a in order to

satisfy the stress free condition on crack surfaces in

the original problem, where r22 is the stress in thehomogeneous problem, and rPa

22 denotes the stress

at the position of the crack a in the subproblem b.

Since rPa22 is the stress caused by the other crack in

the other problem, the following consistency con-

dition must be satisfied:

rPa ¼ cabðr þ rPbÞ; ð4Þ

where r ¼ fr11; r22;r12gT; rPa ¼ frPa11 ; r

Pa22 ; r

Pa12g

T,

and cab is a 3 by 3 matrix whose components are

the stresses at the position of the crack a in thesubproblem b caused by application of a unit

stress to the crack b.Assuming that the pseudotractions are constant

on the crack surface and neglecting higher order

terms of ðlb=dÞ, the analytical expression of cab is

obtained by Okui et al. (1993),

cab ¼ 12

lbd

� �2 b1a1 �b2a1 a4b1a2 �b2a2 a5b1a3 �b2a3 a6

24

35 ð5Þ

with

a1 ¼ 2 cos 2/ab � cos 4/ab; b1 ¼ 1þ 2k2c0=ðplbÞ;a2 ¼ a6 ¼ cos 4/ab; b2 ¼ 2k1c0=ðplbÞ;a3 ¼ a4 ¼ sin 2/ab � sin 4/ab;

a5 ¼ sin 2/ab þ sin 4/ab;

where the definitions of d, /ab, and lb are shown inFig. 3.

σ11

σ22

=

σ11

σ22

+ +

σ11

σ22

=

σ11

σ22

(a) Original Problem (b) Homogeneous Problem (c) Subproblem β (d) Subproblem α

crack α

crack β

- -

- -

Fig. 2. Decomposition of (a) original evolution problem into (b) homogeneous problem and (c,d) subproblems.


From Eq. (4), we have a system of six linear

equations for the pseudotractions rPa, rPb:

I �cab

�cba I

� rPa

rPb

�¼ cabr

cbar

�; ð6Þ

where I stands for the 3 by 3 identity matrix. Thepseudotractions can be determined by solving Eq.

(6). Once the pseudotractions are known, the

mode I SIF of the crack a is obtained by

KIa ¼ffiffiffiffiffiffiffipla

pðr11 þ rPa

11 Þ þc0ffiffiffiffiffiffiffipla

p

hk2ðr11 þ rPa11 Þ � k1ðr22 þ rPa

22 Þi: ð7Þ

Comparing Eq. (7) with Eq. (3) for a single crack

problem, the interaction effect on the SIF is takeninto account through the pseudotractions.

The numerical result obtained from Eqs. (6)

and (7) for the stress state r11=r0 ¼ 0 and r22=r0 ¼ �1:06 is shown in Fig. 4, where r0 is the

reference stress defined by r0 ¼ KIc=ffiffiffiffiffiffiffipc0

p. In case

of lb ¼ 0, which corresponds to a single crackwithout interaction effect, the SIF of crack a de-creases with increasing crack length la under aconstant stress state. On the other hand, in case of

la ¼ lb, the SIF first decreases and then increasesas both crack lengths increase. This subsequent

increase in the SIF is due to interaction effect.

4. Crack growth criteria

In this section, crack growth criteria in terms of

SIF are introduced. Under short-term loadingwhere loads are applied instantaneously, crack

extension is assumed, based on linear fracture

mechanics, to occur when the SIF KI at the cracktip reaches the fracture toughness KIc:

KI � KIc ¼ 0; dlP 0;KI � KIc < 0; dl ¼ 0;

ð8Þ

where dl is increment of the crack length.On the other hand, for long-term loading, even

if KI < KIc, subcritical crack growth is known tooccur for crystalline rocks including granite. The

relationship between the crack growth velocity

dl=dt and the SIF can be described by an empiricalequation of the form

dldt

¼ V0KIK0

� �n

; ð9Þ

where t is time, and V0 is the crack velocity atKI ¼ K0. Theoretically, V0 and n are constants,which are affected by the rock type and environ-ment such as temperature and moisture. For

granitic rocks, reported values of n (called thesubcritical crack growth index) and the coefficient

V0 range from 30 to 70 and 10�2 to 10�9 m/s, re-

spectively (Atkinson, 1984). It is known that re-

ported values of V0 scatter considerably even in thesame rock type and environment (Atkinson and

Meredith, 1987; Swanson, 1984).

5. Evolution problem and example solution

Now we will apply these crack growth criteria

to the evolution problem of two cracks shown in

0.5

1

1.5

l c0

0.5

1

1.5

lβ/

/

c0

00.20.40.60.8

1KIα

KIc

0.5

1α c0

Fig. 4. Nondimensional SIF KIa=KIc as a function of non-dimensional crack lengths la=c0 and lb=c0 for d=c0 ¼ 1:5,/ab ¼ 150�, l ¼ 0:3, and h ¼ 45�. Interaction effect on SIF.

crack - α

crack - β

φαβ

2lβ

d

crack - α

crack - β

φαβ

2lβ

d

Fig. 3. Definition of d, /ab, and lb.


Fig. 2(a). As with loading history of a creep test,

we assume that first stress is applied until the de-

sired stress state is reached, and then the stress

state is held constant. In the numerical analysis,

first the short-term crack growth criterion (8) is

applied for the duration of increasing stress, andthen the long-term crack growth criterion is used.

Consequently, the present evolution problem is

formulated as an initial-value problem of the form:

dladt

¼ V0KIaK0

� �n

;

dlbdt

¼ V0KIbK0

� �n ð10Þ

with KIa ¼ KIb ¼ KIc at t ¼ 0. By solving this evo-lution problem, we can evaluate time to failure

under long-term loading and strength under short-

term loading as well.

Fig. 5 shows an example of solutions of theevolution problem defined by Eq. (10) under a

stress condition r11=r0 ¼ 0, r22=r0 ¼ �1:06, andV0 ¼ 0:5 10�3 m/s at K0 ¼ KIc. The other para-meters used in the calculation are listed in Table 1.

In this figure, two cases with different /ab namely

/ab ¼ 90� and 150� are compared. Since in each

case two cracks grow while the two crack lengthsremain equal: la ¼ lb, only la is plotted in Fig. 5.The difference in the crack lengths of two cases is

small at t ¼ 0, but it increases with increasing time.At the initial stage of crack growth, the interaction

effect is small because of the short length of the

cracks. As the crack lengths increase, the interac-

tion effect becomes large; accordingly, the differ-

ence in the crack lengths due to the locationbecomes large.

Fig. 5. A example solution of the proposed model. Comparison of time history of crack length la for different crack location: /ab ¼ 90�and 150� (V0 ¼ 0:5 10�3 m/s at K0 ¼ KIc. See Table 1 for other parameters).

Table 1

Parameter sets for estimation of creep failure time for Barre

granite

Fracture toughness, KIc (MPam1=2) 1.2

Initial defect half length, c0 (lm) 80

Nondimensional distance between

two cracks, d=c01.5

Coefficient of friction, l 0.3

Initial defect angle, h (�) 45

Angle between crack a and crack b,/ab (�)

150

Subcritical crack growth index, n 30 and 40

Crack velocity at KI ¼ KIc, V0 (m/s) 0:5 10�3 (pc ¼ 0MPa) and 0:5 10�6(pc ¼ 50 MPa)


In the case of /ab ¼ 90�, almost no crack

growth is observed subsequently to rapid increase

in the crack length just after t ¼ 0. For the presentcrack model shown in Fig. 1(b), the SIF at the

crack tip decreases with increasing crack length

under a constant stress state as mentioned before.

This results in transition from primary to second-

ary creep. On the other hand, in the case of/ab ¼ 150�, the cracks grow at a constant velocity,and finally the velocity reaches infinity. The even-

tual rapid increase in the crack length can be in-

terpreted as creep failure subsequently to tertiary

creep.

Fig. 6 shows the relationship between the non-

dimensional SIF KIa=KIc and the nondimensionalcrack length la=c0 of crack a under the same con-dition as case /ab ¼ 150� of Fig. 5 along with twoincreased stress states: jr22=r0j ¼ 3:32 and 2.00. Inthis figure, short-term crack growth takes place

along KIa=KIc ¼ 1, while subcritical crack growthoccurs in the region KIa=KIc < 1. The subcriticalcrack growth starts from point A, moves through

point B, and ends at point C. If the applied com-

pressive stress in a creep test increases to jr22=r0j ¼2:00, then short-term crack growth from point A

to point D takes place, and the time to failure

becomes shorter than that in the original loading

case. Furthermore, the applied stress attains

jr22=r0j ¼ 3:32, the starting point coincides with

the end point at point E. This leads to tf ¼ 0, andhence the applied stress proves to be strength in

short-term loading.

In order to illustrate anisotropy in the interac-

tion effect, the logarithms of times to failure tf inseconds are plotted in Fig. 7 for different /ab

keeping the same distance between two cracks:

d=c0 ¼ 1:5. In the analysis, the time to failureis defined as the time when the SIF attains the

0.2 0.4 0.6 0.8 1 1.2 1.4

0.2

0.4

0.6

0.8

1

Nondim. crack length lα/c0

Non

dim

. SIF

K

Iα/K

Ic

|σ22/σ0|= 3.32 2.00 1.06

Subcritical crack growth

Short-term crack growtht=tft=0

A

B

CD E

Fig. 6. Relationship between nondimensional SIF and nondimensional crack length for three stress states.

11 12 13 14 15φαβ=90˚

120˚

150˚

180˚

Log(tf) [sec]

Fig. 7. Logarithm of failure time tf as a function of /ab with a

constant crack distance d=c0 ¼ 1:5.


fracture toughness after tertiary creep. The time to

failure is minimized when the cracks are aligned in

the direction of about /ab ¼ 145–150�.In actual rock, numerous microcracks exist, and

grow under long-term loading. Owing to the in-

teraction effect, growth of microcracks in someregions is accelerated; coalescence of those micro-

cracks results in a macroscopic failure plane. This

localization of microcracking governs macroscopic

failure. Kranz (1980) reported the fracture angle,

which is the angle between a macroscopic failure

plane and the direction of the maximum com-

pressive stress, in creep tests as well as conven-

tional short-term triaxial tests of Barre granite. Itwas shown that the fracture angle decreases in

terms of /ab with increasing confining pressure.

The values of the fracture angle observed in the

experiment ranged from 145� to 165�. These valuesroughly agree with /ab ¼ 150� that gives the mini-mum failure time in the theoretical model. Hence,

in the following analyses, we use /ab ¼ 150� forthe arrangement of two cracks.

6. Parametric study on short-term strength and

creep failure time

In this section, after reviewing model para-

meters, we investigate the effects of the model

parameters on short-term strengths and times tofailure in creep tests. On the basis of this para-

metric study, we identify the model parameters to

fit short-term strengths and times to failure of

Barre granite reported by Kranz (1980). Finally,

we shall discuss the predictive capability of the

present model.

The parameters used in the model are:

• c0: the half length of the initial defect. Kranz(1980) reported the average crack length of

Barre granite in a uniaxial creep test. The re-

ported values of the average crack length ran-

ged from 54 lm for an unstressed sample to

164 lm for a sample at tertiary creep.

• d: the distance between two cracks. It representsthe density of microcracks in actual rock.

• l: the coefficient of friction on an initial defect.It has a value between 0 and 1.

• /ab: the angle between two cracks. According to

the discussion on Fig. 5, we use /ab ¼ 150�,which gives the minimum time to failure.

• h: the orientation of initial defects. Okui andHorii (1997) conducted a parametric study toexamine the effect of variation of h on the

short-term strength by using IFT. It was shown

that h ¼ 45� gives the minimum short-term

strength. Hence, h ¼ 45� is used.• KIc: the fracture toughness. The reported valuesof KIc of Barre granite lie between 0.74 and 1.5MPam1=2 (Atkinson, 1984). We use KIc ¼ 1:2MPam1=2.

• n: the subcritical crack growth index. No dataare available for Barre granite whereas values

of n for Westerly granite in air at a temperatureof 20 �C range from 31.2 to 39 (Meredith and

Atkinson, 1985). We use n ¼ 30 for Barre gran-ite at room temperature.

• V0: the crack velocity in subcritical crack growthat KI ¼ K0. In many cases, V0 values are evalu-ated from relaxation tests with a double torsion

specimen. Reported values of V0 considerablyscatter depending on the initial value of KI ineach relaxation test, for example V0 ¼ 4:5 10�9 to 1:0 10�8 m/s at KI ¼ 1:0 MPam1=2

(Atkinson, 1984), and V0 ¼ 1:0 10�7 to 1:8 10�4 m/s at KI ¼ 1:26 MPam1=2 (Swanson,

1984) for Westerly granite.

In the above model parameters, the subcritical

crack growth index n and V0 affect only long-termbehavior, while the other undetermined parame-

ters such as c0, l, and d influence both short- andlong-term behavior. First, we investigate the effect

of parameters c0, l, and d on short-term strength,and then examine the effect of the subcritical crackgrowth index n and V0 on the long-term behavior.Fig. 8 show the effect of c0, l, and d on nu-

merical short-term strength respectively. The ex-

perimental data of Barre granite reported by

Kranz are also plotted in these figures. These ex-

periments were conducted at room temperature

under dry condition. Table 1 shows the values of

the model parameters that give the best fit result.For the parametric study displayed in Fig. 8, we

used the values listed in Table 1, unless otherwise

indicated explicitly.


Among theoretical short-term strengths shown

in Fig. 8, the strength with c0 ¼ 40 lm, l ¼ 0:3,and d=c0 ¼ 1:5 is in best agreement with the ex-perimental data for confining pressures of 20–100

MPa. However, despite employing these best fitparameters, theoretical short-term strengths devi-

ate from the experimental ones with the increase

of confining pressure. This is ascribable to brittle-

ductile transition of macroscopic failure mode.

While the confining pressure is small, mode I crack

growth is dominant; almost all cracks are parallel

to the direction of the maximum compression. As

the confining pressure increases, mode II crackgrowth with frictional sliding becomes important.

In the range of confining pressure less than about

50 MPa, however, it seems possible to use the

present model to prediction of short-term strength

as shown in Fig. 9.

Next, we proceed to examine the effects of thesubcritical crack growth index n and V0 on time tofailure in creep tests. Fig. 10 shows a comparison

of time to failure with experimental data reported

by Kranz (1980) for uniaxial stress state pc ¼ 0 anda confining pressure of pc ¼ 53 MPa. The hori-zontal axes denote the normalized stress difference,

in which r1 is the axial compression stress appliedin a creep test, and the denominator ðr1 � pcÞmax isthe maximum stress difference obtained from a

Fig. 8. Comparison of short-term strength with experimental data of Barre granite by Kranz (1980); (a) effect of initial defect length,

(b) effect of friction coefficient, and (c) effect of crack distance on short-term strength.


short-term fracture test. In numerical analyses,

two values of the subcritical crack growth index nare used, namely 30 and 40, which approximately

correspond to the minimum and maximum values

observed in experiments of Westerly granite

(Meredith and Atkinson, 1985).

Since t0 ¼ c0=V0 is a reference time for the pre-sent formulation, if the value of V0 is doubled withkeeping c0 constant then the time to failure ishalved. The values of V0 used in Fig. 10 are de-

termined by fitting the numerical results to experi-

mental ones. Thus, the value of V0 required to fitexperimental data depends on confining pressure.

Variation of V0 with pressure has been observedin experiments as well. Meredith and Atkinson

(1985) conducted a series of subcritical crackgrowth tests of Westerly granite and Black gabbro

under water vapor pressures. They reported that

increasing water vapor pressures from 0.6 to 15

kPa results in approximately 30 times higher crack

growth rate for Westerly granite at room tem-

perature. However, the present numerical results

show an opposite tendency for the dependence of

V0 on confining pressure: the best-fit values of V0decrease with increasing confining pressure. One

possible explanation for this discrepancy is that,

for triaxial compression tests, increasing confining

pressure decreases crack opening inside specimens

and hence delays stress corrosion crack propaga-

tion by hindering the access of reactive species to

crack tips. On the other hand, for subcritical crack

growth tests with a double torsion specimen, watervapor pressure applies directly to the crack tip in

the specimen without any obstacles, and hence

reactive species are injected into the newly created

crack tip as the stress corrosion crack propagates.

However, since the range of water vapor pressures

used in tests by Meredith and Atkinson (1985) is

considerably smaller than the confining pressure in

0

100

200

300

400

500

0 10 20 30 40 50

Exp. [Kranz, 1980]

Present modelStr

ess

diffe

renc

e (M

Pa)

Confining Pressure (MPa)

Short-term fracture testBarre granite

Fig. 9. Close-up view of comparison of short-term strength

with best-fit parameters.

Fig. 10. Creep failure time vs. normalized stress difference: (a) uniaxial compression case pc ¼ 0; and (b) confining pressure pc ¼ 53MPa. (r1 � pc) is the stress difference at creep test; ðr1 � pcÞmax is the maximum stress difference at short-term test. Experimental datafrom Kranz (1980).


creep tests by Kranz (1980), subcritical crack

growth data under higher confining pressures are

necessary to clarify the influence of pressure on

subcritical crack growth.

7. Effects of temperature and water

Lajtai et al. (1987) conducted a series of me-

chanical tests on Lac du Bonnet granite, including

conventional short-term uniaxial compression

tests, fracture toughness tests, uniaxial creep tests,

and subcritical crack growth tests. These tests were

carried out under three environmental conditions:room temperature and humidity (designated here

as ‘‘dry 25 �C’’), room temperature with the

specimen saturated with water (wet 25 �C), and at90 �C with the specimen in water (wet 90 �C). Theyverified that creep accelerates, subcritical crack

growth index n decreases and V0 increases, whenthe specimen is under the wet 90 �C condition ascompared with the dry 25 �C condition. In this

section, we simulate these creep tests to investigate

the ability of the present model to predict envi-

ronmental effects.

First, we use the reported average values of

uniaxial strength and fracture toughness under the

dry 25 �C condition for analysis of creep tests

under the three environments. We neglect the ef-fects of environments on these parameters, because

Fig. 11. Creep failure time vs. uniaxial stress under different conditions; (a) dry 25 �C, (b) wet 25 �C, and (c) wet 90 �C.


the reduction in these parameters due to temper-

ature increase and introduction of water is less

than 5% (Lajtai et al., 1987). Second, initial defect

length c0, friction coefficient l, and distance be-tween cracks d are determined to fit the averageuniaxial strength of 225 MPa reported by Lajtaiet al. (1987). Experimental data on compressive

strength under different confining pressures are not

available. The values of these three parameters are

somewhat arbitrary, since we may have other

combinations of these parameters to fit the uni-

axial strength.

Times to failure in creep tests under the three

environments are calculated using the values of nand V0 obtained by subcritical crack growth tests,and compared with creep test results as shown in

Fig. 11. The parameters used in the analysis are

listed in Table 2. In Fig. 11, the horizontal axis

represents the stress applied in a creep test r1 afternormalization by uniaxial compression strength

rmax. The solid line in each figure indicates creepfailure time when the value of V0 obtained bysubcritical crack growth tests is used. Test results

are well reflected when the value of V0 is changedby several orders of magnitude. Such uncertainty

of V0 has been reported by Lajtai et al. (1987) andby Atkinson (1984).

In a subcritical crack growth test, a load is ap-

plied to a specimen so that the SIF at the crack tip

may be held below the fracture toughness. Lajtai

et al. (1987) pointed out that the value of V0 variesdepending on the initial SIF in a subcritical crack

growth test. Hence, they reported values of V0 lis-ted in Table 2 as approximated values with the

same initial SIF of about 2.27 MPam1=2. On the

other hand, Atkinson (1984) explained that a fac-

Table 2

Parameter sets for estimation of creep failure time for Lac du

Bonnet granite

Fracture toughness,� KIc(MPam1=2)

2.46

Initial defect half length, c0 (lm) 300

Nondimensional distance between

two cracks, d=c01.4

Coefficient of friction, l 0.2

Initial defect angle, h (�) 45

Angle between crack aand crack b, /ab (�)

150

Subcritical crack

growth index,� n96 (dry 25 �C); 54 (wet25 �C); 23 (wet 90 �C)

Crack velocity at KI ¼ 1:0MPam1=2,� V0 (m/s)

1:82 10�35 (dry 25�C); 5:90 10�25 (wet25 �C); 6:97 10�12(wet 90 �C)

*Data from Lajtai et al. (1987).

0 0.025 0.05 0.075 0.1 0.125 0.15Confining pressure Uniaxial strength

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Axi

alst

ress

/ Uni

axia

lstr

engt

h

LDB granite, Dry 25o C

Uniaxial strength = 225 MPa

106104

100

108 yr

1

/

(a) Dry 25 ºC

0 0.025 0.05 0.075 0.1 0.125 0.15Confining pressure/Uniaxial strength

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Axi

alst

ress

/Uni

axia

lstr

engt

h

LDB granite, Wet 90o C

Uniaxial strength = 225 MPa

108 yr106

104

100

1 yr

(b) Wet 90ºC

Fig. 12. Creep failure time under different stress conditions for

Lac du Bonnet granite; (a) dry 25 �C, (b) wet 90 �C.


tor significantly affecting V0 is not fully controlledin tests although the factor remains unknown. For

candidate factors, the specimen size, the shape of

groove in the specimen, and the effect of loading

method such as load relaxation are reviewed. In

conclusion, none of the candidates can immedi-ately explain uncertainty. Some basic problems

may be involved.

Finally, creep failure time is calculated for each

stress state using the identified parameters as

shown in Fig. 12. It is very important to assess the

creep failure times under different stress states and

environmental conditions, as we examine long-

term stability of facilities for radioactive waste.The assumption confirmed in the previous section

that the value of V0 would decrease with increas-ing confining pressure was ignored as being con-

servative because no relevant test results were

available. The value of V0 is determined so thatagreement can be achieved with the creep test re-

sults shown in Fig. 11. It is evident that creep

failure time accelerated owing to the effects ofwater and temperature, and thus contours shifted

rightward. Temperature and water have substan-

tial influence on creep failure time.

8. Some considerations for heated failure test

conducted by AECL

AECL conducted heated failure tests (HFT) in

the URL at a depth of 420 m below ground level

from 1993 to 1996 (Read et al., 1997). The effects

of thermal loading on progressive failure and on

excavation damage development were investi-

gated. In this study, the observed breakout phe-

nomena and reported results of stress analysis for

the observation borehole HFT1 in stage 1 are in-vestigated. The borehole HFT1 was drilled prior

to heating the rock mass. The temperature was

raised up to 85 �C. Fig. 13 shows breakouts ob-served in the borehole HFT1. Fig. 14 shows the

depths of the borehole bottom and the ends (top

and bottom) of the observed breakouts. When the

borehole was drilled up to 1.95 m, the SW break-

out was initiated at a depth of 1.90 m. On the otherhand, the NE breakout started at a depth of 1.45

m when the borehole was drilled completely up to

4.15 m. The NE breakout gradually expanded

before heating, and accelerated when the temper-

ature was raised. After the temperature reached

50 �C, the SW borehole started to expand. When

the target temperature of 85 �C was reached, bothbreakouts expanded from about 1 to 3 m.Here we consider the state of a depth of 3.0 m,

where the breakout was not observed before the

temperature was raised up to 68 �C. AECL carriedout the thermal–mechanical analysis with finite

difference and boundary element methods. The

stress state near the borehole surface where the

breakout occurs is under uniaxial compression.

The values of the maximum compressive stress atNE and SW breakouts at the depth of 3.0 m were

read from the report figures, and are shown in

Table 3. Time to failure is computed by the

method presented in previous sections for the

condition of the HFT. Table 4 shows the com-

puted time to failure under uniaxial compression

Fig. 13. Breakouts in Borehole HFT; (a) NE breakout, (b) SW

breakout. Courtesy of Ontario Power Generation Inc.


with axial stresses of 160, 170, 180, and 190 MPafor wet 25 �C and wet 90 �C. In this calculation, weuse the parameters for Lac du Bonnet granite lis-

ted in Table 2 except for V0. For values of V0, weuse V0 ¼ 5:90 10�24 and 6:97 10�12 m/s for wet25 �C and wet 90 �C, respectively on the basis offitting times to failure to the experimental results

as shown in Fig. 11.

The effect of heating is understood to increasethe axial stress due to thermal expansion and to

accelerate the creep phenomena. When the axial

stress is high and close to the compressive strength,

the creep failure time is short and the effect of

temperature is small. On the other hand, the

temperature effect is remarkable when the axial

stress is relatively low.

The axial stress in the heating failure test is highand close to the compressive strength. Hence the

comparison between the observed phenomena and

results of the current creep analysis does not pro-

vide characteristic information. However, if the

test was conducted with smaller axial stress, it is

possible to have a situation in which the breakout

cannot be explained unless the creep law at the

increased temperature is used. For example, withan axial stress of 150 MPa, the creep failure time

is 221 days at 25 �C, while it is only 3.2 hours at90 �C. If tests similar to HFT are planned, suchconditions are recommended by the present study

to validate the approach presented in this study.

9. Concluding remarks

A simple mechanism-based model has been

proposed to estimate creep failure of hard rocks

under compression. In the proposed model, both

Fig. 14. Breakout advance and temperature for Borehole HFT1 with time in AECL�s HFT. Depth of hole bottom, terminuses ofbreakouts, and temperature vs. time. Data from Read et al. (1997).

Table 3

Effect of thermal loading on the maximum compressive stress

on NE and SW sides of borehole HFT1 at a depth of 3.0 m in

Stage 1

Temperature (�C) Stress (MPa)

NE side SW side

15 171 169

32.5 176 173

50 181 181

67.5 186 184

85 191 188

Table 4

Failure time under different stress conditions

Stress (MPa) Creep failure time

Wet 25 �C Wet 90 �C

150 221 days 3.2 h

160 6.7 days 44 min

170 6.1 h 11 min

180 17 min 2.9 min

190 53 s 50 s


instantaneous and time-dependent crack growth

are taken into account. Subcritical crack growth

due to stress corrosion and crack interaction are

considered to be the main mechanism of time-

dependent behavior in steady state creep and in

tertiary creep, respectively. It is found that creepfailure time under different stress states can be

calculated by the present model. Furthermore, the

effects of water and elevated temperature on creep

acceleration are also considered in the model.

For safety evaluation of high-level waste re-

positories, it is necessary to predict creep beha-

vior over a period on the order of 10,000 years.

Although the model presented here capturesseveral essential features of creep failure of hard

rocks, the predicted results largely depend on

parameters used in the calculation. In particular,

parameters for subcritical crack growth (n andV0) are crucial for prediction of failure times. Inorder to improve the reliability of the model results,

experiments are necessary to investigate subcriti-

cal crack growth under high confining pressure,and to clarify the cause of uncertainty in sub-

critical crack tests.

References

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