Effect of the triaxial stress system on fracture and flow of rocks

1972, Phys. Earth Planet. Interiors 5, 318-324. North-Holland Publishing Company, Amsterdam

EFFECT OF THE TRIAXIAL STRESS SYSTEM ON FRACTURE AND FLOW OF ROCKS

K1YOO MOGI

Earthquake Research Institute, University of Tokyo, Tokyo, Japan

By a new triaxial compression method, in which all three principal stresses are independently controllable, the effects of stress states on fracture and flow properties of rocks were experimentally studied. Strength, ductility, strain-hardening, stress drop and fracture angle are obtained as functions of the intermediate principal stress o2 and the minimum principal stress oz. These deformational properties are affected by on, but also by a2, although these two effects are additional in some properties, but opposite in other ones.

The original Mohr theory cannot predict the observed effect of the general stress state on fracture angles, because the theory assumes that the intermediate principal stress a2 has no effect. However, a modified application of this theory may make it possible to predict the fracture angle from the strength data under general stress states.

I. Introduction

ductility, strain-hardening, stress drop and fracture angle (fig. 1).

In the Earth's interior, fracture and flow o f rocks occur under complex combined stresses. Failure behavior of rocks is greatly affected by stress systems. In order to study the effect of stress systems, a n u m b e r of laboratory experiments have been made, but most previous experiments have been made under some limited conditions of triaxial stresses, in which two of the principal stresses are equal (e.g. GR1GGS and HANDIN, 1960), and very few accurate experiments have been made under more general stress states, because of experimental difficulties (e.g. JAEGER and COOK, 1969). Very recently, new satisfactory methods for general triaxial compression tests have been developed by HOJEM and COOK (1968) and MOGI (1969). By our new triaxial compression technique, the effect of stress states on strength and ductility has been studied and a new fracture criterion has been proposed (MOGI, 1971a, b). In this paper which is a continuation of the preceding papers, a further study on failure behavior of rocks under general triaxial stresses is presented. In general, a state of stress is described by the three principal stresses, ~rl, 0"2 and 0"3. The compressive stress is taken positive, and 0"1 > 0-2 > 0"3 in this paper. The purpose of this paper is to obtain effects of tr 2 and 0"3 to important deformational properties, such as strength,

2. Experimental procedure In the conventional triaxial method, a jacketed cylindrical specimen is axially compressed by a solid piston in the fluid-confining pressure, and so the stress state is limited to the special case, in which two of the principal stresses are equal. In a new triaxial method developed by HOJEM and COOK (1968), the lateral stresses, a2 and a 3, were applied independently by two pairs of copper flat jacks, and the m a x i m u m compression al was applied by an axial piston. By this method, they measured the fracture strength under general stresses. In this case, however, the application of high lateral pressure seems to be difficult. In our method (MoGI, 1969), the m a x i m u m and the

318

Fracture Yield

(~ Fracture stress (~) Yield stress (~ Stress drop (~) Ductility (~) Elastic constant

L5

(~) Coefficient of strain hardening @

Fig. 1.

~

Axic strain

A typical stress-strain curve of rocks and definitions of some important quantities.

319

F R A C T U R E A N D F L O W OF R O C K S

intermediate principal stresses were applied independently by the axial and the lateral pistons, respectively, and the minimum principal stress, 03, was applied by the high fluid-confining pressure. In this case, there is no serious limitation of the lateral pressure. The axial and the lateral loads were measured by load cells inside the pressure vessel and the axial strain was measured by an electric resistance strain gage mounted on the specimen. The rock specimen was a rectangular prism, with steel end pieces through which a i and a 2 were applied. The frictional effect of lateral end pieces was effectively reduced by teflon sheets or thin rubbers. The specimen was jacketed by silicon rubber. The present method has made possible accurate experiments under general triaxial stresses. More detailed

Yomaguchi marble a. Effect of confining pressure

2.0kb

I

t5

description of the experimental technique is given in the preceding papers (Moc1, 1971a, b). 3. Failure criteria

Fig. 2 shows stress-strain curves of Yamaguchi marble under triaxial compression. Fig. 2a shows a series of stress-strain curves in the conventional triaxial compression test in which the intermediate principal stress 02 equals to the minimum principal stress 03, namely, the confining pressure. Strength and ductility increase with increasing confining pressure. Fig. 2b shows a series of stress-strain curves for different 02 values under the constant 03 value of 250 b. In this case, stresses at fracture and yielding appreciably increase with increasing 02, but ductility decreases with increasing a2. Thus, stresses at fracture and yielding are markedly affected not only by a 3, but also by 02. With these experimental results, previous fracture criteria were examined, but they fail to correlate the present experimental result. In the preceding papers (Moo1, 1971a, b), a new satisfactory fracture criterion, obtained by generalization of von Mises theory, was proposed. This fracture criterion is formulated by Toot = f l ( a , + 03),

\

0.25

where Zoct is the octahedral shear stress defined as • oo, =- k [ ( ¢ 1 - ¢ 2 ) 2 + ( 0 2 - ~ 3 ) 2 + ( 0 3 - ~ 1 ) 2 ]

i

2

3

E (%1

b. Effect of 0-2 (0-3:0.25kb)

t b , / ~.~,.~~.~ ' C..~

Fracture

t7 2 6

0.58 ~

I

0.25 kb

1.32 O 0

I

3

E (%)

Fig. 2. (a)Stress-strain curves o f Yamaguchi marble at different confining pressure, which are indicated by numerals in kb. (b) Stress-strain curves at various values of the intermediate principal stress 02 and the m i n i m u m principal stress o f 0.25 kb. Numerals for each curve show 02 values.

*,

(2)

and f l is a monotonic increasing function. In fig. 3, root/Co is plotted against (al+a3)/C o in logarithmic scale, where Co is a uniaxial compressive strength. Since these curves for various rocks are nearly linear, the empirical formula may be expressed approximately by the following power function: "t'oct ~

k

I

(1)

A(al+a3) n,

(3)

where A and n are constants. The value of n was 0.56 for Solenhofen limestone, 0.72 for Dunham dolomite, 0.74 for Yamaguchi marble and 0.87 for Inada granite. For the yield stress, the following formula is applicable: Zoct = f2(ax +02 + a 3 ) ,

(4)

where f2 is a monotonic increasing function (MoGI, 1971b). The yon Mises yield criterion is included in this criterion as a special case where the right side of the formula (4) is zero.

320

K I Y O O MOG1

0.4

~,e'('~e~ XO~

Fracture stress

.~ \ . f

,/-

metals. The difference in the parameter of the righthand side of the formulas (1) and (4) should be remarked. The physical interpretation of the criteria may be stated as follows: Fracture or yielding will occur when the distortional strain energy reaches a critical value. This critical energy is not constant, but monotonically increases with effective mean normal stress. In fracture, a shear faulting takes place on a plane parallel to the 0.E-direction, and so the effective normal stress may be independent on 0.2. On the other hand, yielding does not occur on such macroscopic slip planes with a definite direction, so the mean stress 1(0.1 +0"2 +0.3) is taken as the effective mean normal stress. Thus, the difference in the parameters between fracture and yielding may be reasonably interpreted.

0.8

4. Yield stress and strain-hardening

.f

(D "6

0.2

~0~

~.."" //'~\~

o~, / " .~o~w ~.~, / , -/ .". , -,/--~."

On O

o

,,"Z/

-0.2

-o.%

i

t

o'.z

o.'4

i

o'o

r

log [(G .o%)/2Co) Fig. 3. Normalized octahedral shear stress To~t/Co at fracture versus (th÷aa)/(2Co) in logarithmic scale. Co is the uniaxial compressive strength.

Thus, the criteria of fracture and yielding were obtained by generalization of yon Mises criterion, which was successfully applied to the yield stress of ductile

The stress states at yielding shows another important feature. Fig. 4 shows the differential stress a n - a 3 at yielding of Yamaguchi marble and Dunham dolomite as a function of a2. Different symbols show different 0.3. Although circles somewhat scatter, they can be approximately fitted with a single curve. This result shows that the differential stress at yielding is mainly affected

GO

O _ ~O ~ e ~ 0

Yamaguchi marble Yield stress ~

e .% 0

5

G

G

--

,e I

/

2

d

4

Dunham dolomite Yield stress

9/

® CT3,0.125 kb /

o

0.25

@

0.40

•

0.55

@

® 0-3 : 0.45 kb •

0.65

@

0.85

0.70

©

1.05

O

0.85

0

1.25

O

1.00

O

1.45

i

3 0-2

Fig. 4.

•

kb

o O-z kb

Differential stress trt --t r3 at yielding as a function of the intermediate principal stress tr2 for Yamaguchi marble and D u n h a m dolomite. Different symbols show different tra values.

FRACTURE

AND FLOW

h

321

OF R O C K S

h 0.8

0.8

Yamaguchi marble Strain-hardening

Strain-hardening

04

©

04 @ o-3=0125 kb

O@

d~<'@

-o%-

Solenhofen limestone

--;

•

0.25

@

0.70

@

@

0.85

e

1.05

O

1.00

0

1.20

2r

o

-o.%

3r

~

i

i

---..

i

r

2

t

i

4

6

O-2 kb

0-2 kb Fig. 5.

0.80

Coefficient o f s t r a i n - h a r d e n i n g h as a function o f the intermediate principal stress tTz for Y a m a g u c h i marble and Solenhofen limestone. Different s y m b o l s s h o w different t73 values.

by the intermediate principal stress 0"2, but nearly independent of the minimum principal stress 0-3. This feature was noticeable when the yield stress increases appreciably with increasing confining pressure. The coefficient of strain-hardening (h), defined as the slope of the linear part of the stress-strain curve in the post-yield region, shows a similar feature. In fig. 5, the coefficient of strain-hardening h of Yamaguchi marble and Solenhofen limestone is plotted against a2. Different symbols show different 0-3. All circles can be nearly fitted with a single curve. That is, the coefficient of strain-hardening increases monotonically with increasing 0-2, but nearly independent of a3. It is a very striking result that the shape of stress-strain curve before fracture, determined mainly by yield stress and the coefficient of strain-hardening, is markedly affected by the intermediate principal stress 0-2 and independent of the minimum principal stress 0-3, as shown in fig. 7.

the logarithm of ductility of Yamaguchi marble and Dunham dolomite is plotted against the parameter a a - ~tr2. Different symbols show different 0-3. Although circles somewhat scatter, they can be nearly fitted with a single line. Thus, the following empirical formula may be applicable:

I

Y marble Ductility

Iogen = 2 ! 0%- 025o-~ ) - 0 3

Cn O @

0

o 0-3 =O.125kb • O25 0.40

-I

0.55 0.70

~)

/,

O85

L'/~

0.'2

-0.2

0.4

(O-3-0.25O-2)

5. Ductility and stress drop at fracture

The ductility is defined as the permanent axial strain e. just before fracture, which was determined by subtraction of the elastic axial strain from the total strain (HANDIN, 1966). In the preceding papers (MoG], 1971a, b), it was shown that the ductility increases markedly with increasing 0-3, but decreases with increasing 0-2, and the ductility under general stress states is shown by ~n

~--- A ( 0 - 3

-- 0~2)

'

(5)

where f3 is a monotonic increasing function. In fig. 6,

D. dolomite Ductility

',~

Io g en = 0.8(.O-3 -0.30~ )-0.2 ~

--

G @ @@

~

•

G

.o. ® o'3 • 0.45 ~b • 0.65

•

® ®

0 @

-L

-~.5

6

o15

105 125

,.o

(O-5 - 0 3O-2 ]

Fig. 6. L o g a r i t h m o f ductility e. plotted against ~a--~t72 for Y a m a g u c h i marble a n d D u n h a m dolomite. Different s y m b o l s s h o w different cr3 values.

322

KIYOO MaGI

Effect of 0-3

Effect of 0-2

4 :5

Frecture

4,

4

F

~

~-,Z.-- ----~°'7° _.~~"~

",,

'

~)

'o55

2 I /

/

!

"

Fracture F

~ :5

--~

',"--,, 0"-3=0.25kb

E)

I

I

I

3

4

(a)

f

I

2

~

'

f

2

0-2 = 1.08 kb 0

~

'

"',,"',,,.__0.40

F

F

~,..~"s.-~'-J.....-..-"~,4, ] 4, 4, ,

,

' J - '-f- ,

K S - -

0.82

-

,

.

"

2.0 ,.67 CY2=2.31kb

O.55 ,.58 I.(31

/

0-3 = 0.55 kb

0

I

i

I

I

I

2

3

4

E (%)

~ (%)

(b)

Fig. 7. Stress-strain curves o f Y a m a g u c h i marble u n d e r triaxial compression. (a) Curves at various values o f the m i n i m u m principal stress tr3 a n d the intermediate principal stress az o f 1.08 kb. (b) Curves at various values o f a2 a n d tra o f 0.55 kb.

log e.

= gl(o- 3 -czo-2)+K2,

(6)

where K1 and K 2 a r e constants. The marked difference between the o-2 and o-3 effects on ductility may be understood from the above-mentioned experimental results on fracture stress, yield stress and the coefficient of strain-hardening. Fig. 7a ~,hows stress-strain curves of Yamaguchi marble for different 0"3 under a constant o-2-The curve shape before fracture is nearly indepcndent of the minimum principal stress o'3, as mentioned before. The fracture strength, however, markcdly increases with increasing o'3, in accordance with the fracture criterion. Consequently, the permanent strain before fracture, that is, the ductility, increases with increasing o-3. Fig. 7b shows stress-strain curves for different o-2 under a constant an. The curve shape changes markedly with the intermediate principal stress o-2, that is, the stress-strain ratio (o-1- o-3)/e increases monotonically with increasing a2. On the other hand, the fracture strength changes with increasing a 2, but the change in fracture strength is smaller in comparison with the increase of the stressstrain ratio. Therefore, the strain just before fracture, namely, the ductility, decreases with the increase of the intermediate principal stress o-2. The stress drop at fracture decreases with increasing tr 3 and increases with increasing 0"2 (MAGI, 1971a). Fig. 8 shows that the stress drop is roughly in inverse proportion to the ductility. The stress drop is the difference between the fracture strength and the resistive stress after fracture. The resistive stress after fracture

can be explained by the frictional force on the fault surface which is not influenced by the pressure in the a2-direction.

6. Fracture angle According to careful measurements of strength and fracture angles (BRACE, 1964; Mosl, 1966), the fracture angle between the o-l-direction and the fault plane can be predicted from the fracture strength data by the Mohr theory in the case of the conventional triaxial compression (a~ > 0" 2 ~ - - - 0 " 3 ) . In extension (a 1 = o-2 > o'3) , however, the Mohr theory fails to predict the fracture angle (MAGI, 1967). Fig. 9 shows the fracture angle 0 of Solenhofen lime-

t~

® 0-3 = O.06kb • 0.125 ¢ 0.25 • 0.40 o 0.55 o 0.70

Yamaguchi marble

t~

? "o

O ~ @@ @

q)

•

®

0

0

,8 0

0

C; @

0

i

i

I

2

0

Ductility (%) Fig. 8.

Stress drop at fracture versus ductility for Y a m a g u c h i marble.

FRACTURE AND FLOW OF ROCKS Solenhofen limestone

Dunham dolomite

o

o ~

5C

C~: LOkb

~

323

o

o

¢ d"O-3=L45Kb

30

0.9

2C

I0

0.8

0.6

~

0

.

4

o.e

20c

o...

IO 0.2

P

O0

l 2J

5'

0

0

i

' 2

( 0"-2- C:.~) kb

Fig. 9.

3'

4

( 0"-2- 0-3 ) kb

Fracture angle 0 as functions of the intermediate principal stress o.2 in Solenhofen limestone and Dunham dolomite. Different symbols show different values of the minimum principal stress o.3 which are indicated by numerals in kb.

stone a n d D u n h a m d o l o m i t e u n d e r general triaxial stresses as a function o f the i n t e r m e d i a t e p r i n c i p a l stress cr2. Different s y m b o l s show different values o f the m i n i m u m principal stress 0"3, which are i n d i c a t e d by n u m e r a l s in kb. The fault p l a n e is parallel to the 0-2-direction, a n d the fracture angle increases with increasing 0"3, b u t decreases with increasing 0"2, particularly u n d e r lower 0"3. T h e M o h r t h e o r y c a n n o t predict this 0"2-dependence o f the fracture angle. I f the shear strength (z) is a function o f the n o r m a l stress ( a . ) on the fault surface only a n d i n d e p e n d e n t o f the a2 value, as a s s u m e d in the M o h r t h e o r y (e.g. NADAL 1950), the fracture angle can be calculated f r o m the slope o f the z-tr, curve, namely, dr/d0", (or do'l/d0"3) (fig. 10a). T h e present experiment, however, showed t h a t M o h r ' s a s s u m p t i o n is n o t correct a n d the shear strength is affected n o t only by the n o r m a l stress, b u t also by the i n t e r m e d i a t e p r i n c i p a l stress a2. T h a t is, the z-o", relation is n o t a single curve as shown in fig. 10a,

b u t different for different tr2 values as shown in fig. 10b. However, for each T - a , curve in which tr 2 is constant, M o h r ' s analysis should be applicable, a n d so the fracture angle can be calculated f r o m the slope o f z - a , curve or (a,r/Oa,),, 2 . . . . . t,,v W h e n the strength curves as shown in fig. 11 are o b t a i n e d , the fracture angle 0 can be calculated f r o m (dal/da3),,2 . . . . . taat" According to this theory, the a b o v e - m e n t i o n e d decrease o f the fracture angle 0 with increasing a2 can be p r e d i c t e d f r o m the observed strength curve, as shown schematically in fig. 11. A s m e n t i o n e d above, the fracture angle in the

T = f (O-n,O-2)

T : f(CTn )

T

T

/

2

j

~ a ~ 4 ',

i (a) (b) Fig. 10. Shear stress (~) as functions of normal stress (o..) on the fault surface. (a) Mohr's assumption ; (b) present experimental result.

Crz

I

i Fracture -"-"---~

b

angle

'I

Fig. 11. Upper: Stress states at fracture under general triaxial compression. Typical o.1-a2 curves at fracture for various o.a values are shown schematically. A: o'2 = o.3; B: o.j = o.2. Lower: Fracture angle 0 calculated from the above stress state.

324

KIYOO

conventional triaxial compression (A in fig. 11) can be predicted by the original M o h r theory. This m a y be attributed to that d a l / d a 3 is nearly equal to (~0.1/~0.3)a2 =constant in this case. On the other hand, the original M o h r theory fails to predict the fracture angle in extension (B in fig. 11). This m a y be explained by that d a l / d a 3 is appreciably different from (OO'l/t~0.3)a2 . . . . . tant in extension. As a conclusion, the original M o h r theory is not applicable in general cases, but the above-mentioned modified application of the M o h r theory makes possible to predict the fracture angle f r o m a set o f the 0.1-63 curves under constant a2 values. 7. Summary The effect o f confining pressure on the deformational properties is typically shown in the stress-strain curves in fig. 2. The effects of the m i n i m u m and the intermediate principal stresses are shown in fig. 7a and b, respectively. It should be noticed that the 0"3 effect is markedly different from the effect of the confining pressure and the effects o f a2 and 0"3 cannot be expected from the effect of confining pressure. Previous results obtained by the conventional triaxial test, in which two of the principal stresses are equal, are undoubtedly very important, but it should be remarked that the stress state is very limited and these results are insufficient to discuss the effect of the general stress states. The effects of stress states on the deformational properties of rocks studied in this paper are summarized as follows: Case 1. The effects of a2 and 0"3 are additional. The

MOGI

fracture stress a ~ - 0 3 shows this feature, at least, except for the high tr 2 -0"3 region. Case 2. The effect of 0.2 predominates. Yield stress (0.1- ~r3) (probably at lower a3) and the coefficient of strain-hardening show this feature. Case 3. The effects of 0.2 and 0"3 are opposite. Ductility, stress drop and fracture angle show this feature. Case 4. The effect of 0"3 predominates. The differential stress after fracture show this feature. Thus, it may be stated that a general feature of rock deformation under combined stress states has been markedly clarified with the new triaxial compression test. The result shows that the main deformational properties of rocks is not only greatly affected by the m i n i m u m principal stress 03, but also markedly affected by the intermediate principal stress az.

References BRACE, W. F. (1964), Brittle fracture of rocks. In: W. R. Judd, ed., State of stress in the Earth's crust (Am. Elsevier, New York) 111-174. CRIGGS, D. and J. HANDIN, eds. (1960), Rock deformation, Geol. Soc. Am. Mem. 79, 1-382. HANDIN,J. (1966), Strength and ductility. In: S. P. Clark, Jr., ed., Handbook of physical constants, Geol. Soc. Am. Mem. 97, 239-289. HOJEM, J. P. M. and N. G. W. COOK(1968), S. African Mech. Engr. 18, 57-61. JAEGER, J. C. and N. G. W. COOK (1969), Fundamentals of rock mechanics (Methuen, London) 513 pp. MOGI, K. (1966), Rock Mech. Engr. Geol. 4, 41-55. MOGI, K. (1967), J. Geophys. Res. 72, 5117-5131. MOGI, K. (1969), On a new triaxial compression test of rocks. In: Abstr. 1969 Meeting Seismol. Soc. Japan, 3. MOGI, K. (1971a), Tectonophysics 11, 111-127. MOGI, K. (1971b), J. Geophys. Res. 76, 1255-1269. NADAI, A. (1950), Theory of flow and fracture of solids, I, 2nd ed. (McGraw-Hill, New York) 572 pp.