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Mechanism of brittle fracture of rock
Int. J. Rock Mech. Min. Sci. Vol. 4, pp. 425--430. Pergamon Press Ltd. 1967. Printed in Great Britain
MECHANISM OF BRITTLE F R A C T U R E OF ROCK PART III--FRACTURE IN TENSION AND UNDER LONG-TERM LOADING Z. T. BIENIAWSKI National Mechanical Engineering Research Institute, Pretoria, South Africa (Received 10 January 1967)
1. INTRODUCTION
IN PART I of the present paper, the theory of the fracture process was discussed and a hypothesis on the mechanism of brittle fracture of rock was propounded. In Part II, experimental verifications of this mechanism were dealt with for rock tested under compressive stress conditions. In this, Part III of the paper, the propounded mechanism is applied to rock subjected to tensile stress conditions as well as to long-term stress conditions. It will be recalled from Parts I and II of this paper that the following stages of brittle fracture are propounded to take place in rock under multiaxial compression: 1. Closing of cracks I. Crack closure 2. Linear elastic deformation II. Fracture initiation 3. Stable fracture propagation III. Critical energy release 4. Unstable fracture propagation IV. Strength failure (maximum stress) 5. Forking and coalescence of cracks V. Rupture (maximum deformation) For convenience, the above mechanism is diagrammatically represented in Fig. 1. A complete stress--strain curve for a hard rock, namely norite, in uniaxial compression is given in Fig. 2 while complete stress-strain curves for a soft rock, namely sandstone, in both uniaxial and triaxial compression are given in Fig. 3. For detailed explanations of these figures the reader is referred to Parts I and II of this paper. 2. BRITI'LE FRACTURE O F ROCK IN TENSION
The mechanism of brittle fracture of rock given in Fig. 1 is valid not only for multiaxial compression but also for multiaxial tension. In tension, however, crack closure will be absent and processes of stable and unstable fracture propagation will be of very small duration due to the fact that in tension a crack will propagate in its own plane[l]. In Fig. 4 stress-strain curves obtained on norite specimens in uniaxial tension are given. The special experimental techniques used for this study are fully described elsewhere[2]; It will be seen 425
426
Z. T. B1ENIAWSKI
50,000 40000
~0,000
10,0013
0
2000
4000
5000
8000
io,0oo
FzG. 2. Completestress-strain curve for norite in uniaxial compression.
m o
1200C /
4000 /
~
~.~
k=40
....
• Uniaxial ion )
0
4oo0
800o I2000 troo00 Axial stroin, ~in/in
2OO00
FxG.3. Completestress-strain curves for sandstonein uniaxialand triaxial compression.
]acing page 420
doJnin, crock~
~roin shatte
R.M.
Grain bou~dories in rock oct"ing os pre-existing (Griffifh) crocks
I
:~
Ttorminol vel~ci÷v v .
~5
FIo. 1. Mechanism of Brittle fracture of rock in multlaxial compression.
O"
MOh.r envelope
;rocfure propogafion m Critical energy release (long-term ;table ~ _4GcE/Tc strength) :mcfure propagation eli Fracture initiation ~=J2yE/~c • rfectly elastic or o-t= ~ 3 ( ~ / v i ' ~ ) + o'c leformaflon : ............ =T CroCkclosure ;losincj OTcraCKS
Insfable
• TV Sfrengfh failure
MOhr envelope parabola
Volumetric s'Hain
k=15
/
//
MECHANISM OF BRITTLE FJRACTURE OF ROCK-PART
III
427
from Fig. 4 that the determination of fracture initiation as well as the onset of unstable fracture propagation may be affected in the same manner as that obtained in case of compression specimens as discussed in Part II of this paper. However, comparison of fracture processes in norite in compression (see Fig. 5) and in tension (see Fig. 4) reveals that for norite fracture initiation takes place in compression at 35.0 per cent of maximum load while in tension at 94.5 per cent of maximum load. Further, unstable fracture propagation occurs in compression at 73.0 per cent of nbaximum load while in tension at 96.5 per cent of maximum load.
failure ruptwe fracture womaation n initiation’
Stroln, FIG. 4.
Stress-strain
curves
pin/in
for norite in uniaxial tension.
Consequently, for practical purposes, it may be assumed that, in tension, fracture initiation and strength failure occur simultaneously, with the process of fractu#e propagation virtually non-existent. It also means that the Griffith hypothesis applicable anly to fracture initiation, may be used as a strength failure criterion in the case of fracture under tensile stress conditions, a procedure unacceptable in the case of fracture under compressive stress conditions[3]. On the other hand, because of the close proximity of stress levels for fracture initiation and rupture in tension it is difficult to determine the complete stress-strain aurves for rock in tension and so far no comparison can be made with the curves given in Figs. 2 and 3 obtained for compression. It must be stated, however, that such curves for tension would be of academic interests only since, because of predominance of compressive sttess conditions in rock mechanics problems, Figs. 2 and 3 are of particular interest in practical applications. 3. BRITTLE FRACTUREl OF ROCK UNDER LONG-TERM
LOADING
It has been stated in Part II of this paper, Section 3.6, that the onset of u&table fracture propagation is an important stage in the fracture processes since as from dtis instant the fracture process is self-maintaining. At the onset of unstable fracture pdopagation the
428
z.T. BIENIAWSKI
energy released during fracturing, G, reaches its critical value, Go, and further the curve of the crack velocity versus crack length (see Fig. 18, Part II) changes its sign. It is believed that the stress level at the onset of unstable fracture propagation corresponds to the long-term strength of the material. Full discussion of this concept can be found elsewhere[3]. The onset of unstable fracture propagation may be determined from the plot of volumetric strain versus axial stress and is applicable to both uniaxial compression and tension. It can also be determined equally well for triaxial compression from the plot of volumetric strain versus maximum stress difference. 50
40
~) -
-J- Strength failure = 39,100 psi
30 B
d
Critical energy release G=Gc onset of unstable fracture
propagation O'B=28,500 psi =73%
20
,°// 0
-500
Fracture initiation ! o~ = 13,700 psi = 3,5 % crc
-I000 -1500 -200o Volumetric strain, /xin/in
FIG. 5. Relationship between axial stress and volumetric strain for norite in uniaxial compression.
In Fig. 5, a typical plot is given for norite in uniaxial compression. Point B in this figure represents the onset of unstable fracture propagation. It will be noted from this figure that up to point B the volumetric strain decreases with increasing compressive stress. Beyond point B, the volumetric strain increases with increasing compressive stress. Since it is also known from independent studies[4] that a volume increase characterizes the long-term strength of the material, it stands to reason that the stress level at point B in Fig. 5 may represent the long-term strength of this rock. If it is so, then the author's hypothesis will offer a convenient and quick determination of long-term strength of rock which will be of considerable value in practical applications. In order to verify the above reasoning long-term tests have been conducted on norite subjected to a uniaxial compressive test under constant temperature and humidity conditions. A rock 'creep' testing machine illustrated in Fig. 7 was used for this study. Details of the apparatus and techniques employed are given elsewhere[2]. Typical results are shown in Fig. 6. In this figure the ratio of time-dependent strength over uniaxial compressive strength of samples is plotted versus time. The testing procedure was as follows. Rock samples of norite were loaded at various constant uniaxial compressive stress levels as a percentage of the uniaxial compressive strength. The samples were left in the machine until failure occurred, the time at failure being automatically recorded.
FIG. 6. 15-ton compression testing machine for long-term testing of rock saml~les.
facing page 428 R.M.
MECHANISM OF BRITTLE FRACTURE OF ROCK--PART IIl
429
It will be seen from the comparison of the results in Figs. 7 and 5 that the long-term strength of norite (Fig. 7) approaches asymptotically the stress level of 74 per cent of its uniaxial compressive strength ae which agrees favourably with the stress level obtained at point B in Fig. 5 (73 per cent ac).
0 9~
0,90
0.85 1
IS
0.8C "
k.
o
0.75
0"74 .
.
_ .
.
.
.
.
.
.
.
.
.
.
II 0"70~
I
2
3
4
Time,
5
hr
(17d0y$~
FIG. 7. Relationship between time-dependent strength of norite and time.
It may be concluded therefore that the author's hypothesis offers a convenient means for determining the long-term strength of rock. Confirmation of this hypothesis has also been recently obtained by WILD[5] for other rock materials. 4. C O N C L U S I O N S
In this study, presented in a series of three papers, a detailed mechanism of rock fracture in compression and tension has been propounded. On the basis of this mechanism it is possible to explain all the significant processes taking place in rock from initial load application to complete separation of the specimen. The fracture mechanism described in this series has been developed for brittle fracture in general although the emphasis of the experimental investigations was on hard rocks. The propounded mechanism applies to uniaxial as well as biaxial and triaxial compression. The mechanism is also valid for multiaxial tension and long-term load applications. In tension, however, crack closure will, of course, be absent and processes of Stable and unstable fracture propagation will be of very small duration. For practical purposes, it may therefore be assumed that, in tension, fracture initiation and strength failure occur simultaneously. The propounded hypothesis on the mechanism of rock fracture also enables quick determination of the long-term strength of rock. Acknowledgements--This work forms part of an extensive rock mechanics research p r o g r a n ~ e being carried out by the South African Council for Scientific and Industrial Research on behalf of th~ Transvaal and Orange Free State Chamber of Mines. The author is indebted to these organizations f6r permission to publish the material contained in this paper. The author wishes to thank Dr. H. G. DENKHAUS, Director of the Institute for his valuable criticism and encouragement and to Messrs. U.W.O.L. VOGLER and M. N. MARAIS for their assistance in carrying out the tests.
430
z . T . BIENIAWSK1 REFERFaNCES
1. HOEK E. and BIENIAWSKIZ. T. Brittle fracture propagation in rock under compression. Int. J. Fracture Mech. 1, (3) 137-155 (1965). 2. BIENIAWSKI Z. T. Determination of Rock Properties. Report of the South African Council of Scientific and Industrial Research No. M E G 518, January (1967). 3. BIENIAWSKIZ. T. Mechanism of Rock Fracture in Compression. Report of the South African Council of Scientific and Industrial Research No. M E G 459, June (1966). 4. GLOCKLICH J. The influence of sustained loads on the strength of concrete. Rilem Bull. No. 5, pp. 14-17, December (1959). 5. W1LD B. L. The Time-Dependent Behaviour of Rock; Considerations with Regard to a Research Programme. Report of the South African Council of Scientific and Industrial Research. No. M E G 514, December (1966).
MECHANISM OF BRITTLE F R A C T U R E OF ROCK PART III--FRACTURE IN TENSION AND UNDER LONG-TERM LOADING Z. T. BIENIAWSKI National Mechanical Engineering Research Institute, Pretoria, South Africa (Received 10 January 1967)
1. INTRODUCTION
IN PART I of the present paper, the theory of the fracture process was discussed and a hypothesis on the mechanism of brittle fracture of rock was propounded. In Part II, experimental verifications of this mechanism were dealt with for rock tested under compressive stress conditions. In this, Part III of the paper, the propounded mechanism is applied to rock subjected to tensile stress conditions as well as to long-term stress conditions. It will be recalled from Parts I and II of this paper that the following stages of brittle fracture are propounded to take place in rock under multiaxial compression: 1. Closing of cracks I. Crack closure 2. Linear elastic deformation II. Fracture initiation 3. Stable fracture propagation III. Critical energy release 4. Unstable fracture propagation IV. Strength failure (maximum stress) 5. Forking and coalescence of cracks V. Rupture (maximum deformation) For convenience, the above mechanism is diagrammatically represented in Fig. 1. A complete stress--strain curve for a hard rock, namely norite, in uniaxial compression is given in Fig. 2 while complete stress-strain curves for a soft rock, namely sandstone, in both uniaxial and triaxial compression are given in Fig. 3. For detailed explanations of these figures the reader is referred to Parts I and II of this paper. 2. BRITI'LE FRACTURE O F ROCK IN TENSION
The mechanism of brittle fracture of rock given in Fig. 1 is valid not only for multiaxial compression but also for multiaxial tension. In tension, however, crack closure will be absent and processes of stable and unstable fracture propagation will be of very small duration due to the fact that in tension a crack will propagate in its own plane[l]. In Fig. 4 stress-strain curves obtained on norite specimens in uniaxial tension are given. The special experimental techniques used for this study are fully described elsewhere[2]; It will be seen 425
426
Z. T. B1ENIAWSKI
50,000 40000
~0,000
10,0013
0
2000
4000
5000
8000
io,0oo
FzG. 2. Completestress-strain curve for norite in uniaxial compression.
m o
1200C /
4000 /
~
~.~
k=40
....
• Uniaxial ion )
0
4oo0
800o I2000 troo00 Axial stroin, ~in/in
2OO00
FxG.3. Completestress-strain curves for sandstonein uniaxialand triaxial compression.
]acing page 420
doJnin, crock~
~roin shatte
R.M.
Grain bou~dories in rock oct"ing os pre-existing (Griffifh) crocks
I
:~
Ttorminol vel~ci÷v v .
~5
FIo. 1. Mechanism of Brittle fracture of rock in multlaxial compression.
O"
MOh.r envelope
;rocfure propogafion m Critical energy release (long-term ;table ~ _4GcE/Tc strength) :mcfure propagation eli Fracture initiation ~=J2yE/~c • rfectly elastic or o-t= ~ 3 ( ~ / v i ' ~ ) + o'c leformaflon : ............ =T CroCkclosure ;losincj OTcraCKS
Insfable
• TV Sfrengfh failure
MOhr envelope parabola
Volumetric s'Hain
k=15
/
//
MECHANISM OF BRITTLE FJRACTURE OF ROCK-PART
III
427
from Fig. 4 that the determination of fracture initiation as well as the onset of unstable fracture propagation may be affected in the same manner as that obtained in case of compression specimens as discussed in Part II of this paper. However, comparison of fracture processes in norite in compression (see Fig. 5) and in tension (see Fig. 4) reveals that for norite fracture initiation takes place in compression at 35.0 per cent of maximum load while in tension at 94.5 per cent of maximum load. Further, unstable fracture propagation occurs in compression at 73.0 per cent of nbaximum load while in tension at 96.5 per cent of maximum load.
failure ruptwe fracture womaation n initiation’
Stroln, FIG. 4.
Stress-strain
curves
pin/in
for norite in uniaxial tension.
Consequently, for practical purposes, it may be assumed that, in tension, fracture initiation and strength failure occur simultaneously, with the process of fractu#e propagation virtually non-existent. It also means that the Griffith hypothesis applicable anly to fracture initiation, may be used as a strength failure criterion in the case of fracture under tensile stress conditions, a procedure unacceptable in the case of fracture under compressive stress conditions[3]. On the other hand, because of the close proximity of stress levels for fracture initiation and rupture in tension it is difficult to determine the complete stress-strain aurves for rock in tension and so far no comparison can be made with the curves given in Figs. 2 and 3 obtained for compression. It must be stated, however, that such curves for tension would be of academic interests only since, because of predominance of compressive sttess conditions in rock mechanics problems, Figs. 2 and 3 are of particular interest in practical applications. 3. BRITTLE FRACTUREl OF ROCK UNDER LONG-TERM
LOADING
It has been stated in Part II of this paper, Section 3.6, that the onset of u&table fracture propagation is an important stage in the fracture processes since as from dtis instant the fracture process is self-maintaining. At the onset of unstable fracture pdopagation the
428
z.T. BIENIAWSKI
energy released during fracturing, G, reaches its critical value, Go, and further the curve of the crack velocity versus crack length (see Fig. 18, Part II) changes its sign. It is believed that the stress level at the onset of unstable fracture propagation corresponds to the long-term strength of the material. Full discussion of this concept can be found elsewhere[3]. The onset of unstable fracture propagation may be determined from the plot of volumetric strain versus axial stress and is applicable to both uniaxial compression and tension. It can also be determined equally well for triaxial compression from the plot of volumetric strain versus maximum stress difference. 50
40
~) -
-J- Strength failure = 39,100 psi
30 B
d
Critical energy release G=Gc onset of unstable fracture
propagation O'B=28,500 psi =73%
20
,°// 0
-500
Fracture initiation ! o~ = 13,700 psi = 3,5 % crc
-I000 -1500 -200o Volumetric strain, /xin/in
FIG. 5. Relationship between axial stress and volumetric strain for norite in uniaxial compression.
In Fig. 5, a typical plot is given for norite in uniaxial compression. Point B in this figure represents the onset of unstable fracture propagation. It will be noted from this figure that up to point B the volumetric strain decreases with increasing compressive stress. Beyond point B, the volumetric strain increases with increasing compressive stress. Since it is also known from independent studies[4] that a volume increase characterizes the long-term strength of the material, it stands to reason that the stress level at point B in Fig. 5 may represent the long-term strength of this rock. If it is so, then the author's hypothesis will offer a convenient and quick determination of long-term strength of rock which will be of considerable value in practical applications. In order to verify the above reasoning long-term tests have been conducted on norite subjected to a uniaxial compressive test under constant temperature and humidity conditions. A rock 'creep' testing machine illustrated in Fig. 7 was used for this study. Details of the apparatus and techniques employed are given elsewhere[2]. Typical results are shown in Fig. 6. In this figure the ratio of time-dependent strength over uniaxial compressive strength of samples is plotted versus time. The testing procedure was as follows. Rock samples of norite were loaded at various constant uniaxial compressive stress levels as a percentage of the uniaxial compressive strength. The samples were left in the machine until failure occurred, the time at failure being automatically recorded.
FIG. 6. 15-ton compression testing machine for long-term testing of rock saml~les.
facing page 428 R.M.
MECHANISM OF BRITTLE FRACTURE OF ROCK--PART IIl
429
It will be seen from the comparison of the results in Figs. 7 and 5 that the long-term strength of norite (Fig. 7) approaches asymptotically the stress level of 74 per cent of its uniaxial compressive strength ae which agrees favourably with the stress level obtained at point B in Fig. 5 (73 per cent ac).
0 9~
0,90
0.85 1
IS
0.8C "
k.
o
0.75
0"74 .
.
_ .
.
.
.
.
.
.
.
.
.
.
II 0"70~
I
2
3
4
Time,
5
hr
(17d0y$~
FIG. 7. Relationship between time-dependent strength of norite and time.
It may be concluded therefore that the author's hypothesis offers a convenient means for determining the long-term strength of rock. Confirmation of this hypothesis has also been recently obtained by WILD[5] for other rock materials. 4. C O N C L U S I O N S
In this study, presented in a series of three papers, a detailed mechanism of rock fracture in compression and tension has been propounded. On the basis of this mechanism it is possible to explain all the significant processes taking place in rock from initial load application to complete separation of the specimen. The fracture mechanism described in this series has been developed for brittle fracture in general although the emphasis of the experimental investigations was on hard rocks. The propounded mechanism applies to uniaxial as well as biaxial and triaxial compression. The mechanism is also valid for multiaxial tension and long-term load applications. In tension, however, crack closure will, of course, be absent and processes of Stable and unstable fracture propagation will be of very small duration. For practical purposes, it may therefore be assumed that, in tension, fracture initiation and strength failure occur simultaneously. The propounded hypothesis on the mechanism of rock fracture also enables quick determination of the long-term strength of rock. Acknowledgements--This work forms part of an extensive rock mechanics research p r o g r a n ~ e being carried out by the South African Council for Scientific and Industrial Research on behalf of th~ Transvaal and Orange Free State Chamber of Mines. The author is indebted to these organizations f6r permission to publish the material contained in this paper. The author wishes to thank Dr. H. G. DENKHAUS, Director of the Institute for his valuable criticism and encouragement and to Messrs. U.W.O.L. VOGLER and M. N. MARAIS for their assistance in carrying out the tests.
430
z . T . BIENIAWSK1 REFERFaNCES
1. HOEK E. and BIENIAWSKIZ. T. Brittle fracture propagation in rock under compression. Int. J. Fracture Mech. 1, (3) 137-155 (1965). 2. BIENIAWSKI Z. T. Determination of Rock Properties. Report of the South African Council of Scientific and Industrial Research No. M E G 518, January (1967). 3. BIENIAWSKIZ. T. Mechanism of Rock Fracture in Compression. Report of the South African Council of Scientific and Industrial Research No. M E G 459, June (1966). 4. GLOCKLICH J. The influence of sustained loads on the strength of concrete. Rilem Bull. No. 5, pp. 14-17, December (1959). 5. W1LD B. L. The Time-Dependent Behaviour of Rock; Considerations with Regard to a Research Programme. Report of the South African Council of Scientific and Industrial Research. No. M E G 514, December (1966).