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Mechanism of brittle fracture of rock
Int. J. Rock Mech. Min. Sci. Vol. 4, pp. 395--406. Pergamon P r e a Ltd. 1967. Printed in Great Britain
MECHANISM OF BRITTLE F R A C T U R E OF ROCK P A R T I - - T H E O R Y O F T H E F R A C T U R E PROCESS Z. T. BIENIAWSKI National Mechanical Engineering Research Institute, Pretoria, South Africa (Received 10 January 1967)
Synopsis--This paper presents results of investigations aimed at establishing a mechanism of brittle fracture of rock in compression and tension. Results of fracture studies on specimens of different shapes and subjected to different loading conditions are presented. Fraction initiation, fracture propagation, strength failure and rupture are discussed. A hypothesis on the mechanism of rock fracture is propounded, all significant failure processes taking place in rock from initial load application to complete failure being dealt with. This paper is published in three parts. Part I deals with the theory of the fracture process while Parts II and III provide experimental verification of the postulated mechanism of brittle fracture of rock. 1. INTRODUCTION AN UNDERSTANDING of the fracture mechanism of rock is an essential prerequisite for designing mining excavations and civil engineering structures, for developing rock-breaking processes such as drilling or blasting, and for devising methods to prevent such hazards as rockbursts. Increased knowledge of the fracture mechanism can make a considerable contribution towards improving the efficiency of mining operations with regard to their economy and safety. Although much research has been done on the fracture of rock[l], mainly during the last few years, very little is known at present about its actual mechanism. Studies of failure of solids stem from the attempts to predict the strength of materials and structures. These studies have resulted in a number o f failure hypotheses[2] which are applied in practice with a varying degree o f substantiation. In the case of brittle fracture it has been found that only three such hypotheses may be applicable, namely, those by MOHR[2], GgIFFn'n[3, 4] and PO~CELET[5]. The first two hypotheses have been accepted by various workers as being applicable to rock[6] while the hypothesis by Poncelet has been strongly criticized[7]. A clear distinction must be made between a phenomenological failure criterion and a genetic failure mechanism. A failure criterion simply provides a formula enabling predicting the strength values for all states of multiaxial stress from a critical quantity which may be determined in one type of test, e.g. the uniaxial tensile or compression test. A failure mechanism describes the processes taking place in the material in the course of loading and eventually leading to failure. Preferably a failure criterion should be based upon knowledge of the failure mechanism, but this is not always so. In fact, m a n y failure hypotheses have been propounded as a result of theoretical reasoning only and could not be verified by experimental evidence. The M o h r 395
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Z. T. BIENIAWSKI
hypothesis, although it has been shown to fit experimental data approximately, is not based on a failure mechanism. The Griffith hypothesis, on the other hand, is based on a genetic concept, namely the existence of small cracks or flaws, but not on a complete failure mechanism, although it offers possibilities in this respect. It should be noted, however, that the Griffith hypothesis refers to fracture initiation only which is not the same as fracture[8]. This series of three papers presents results of investigations, on the basis of which a hypothesis on the mechanism of brittle fracture in rock under compression and tension is propounded. 2. DEFINITIONS Since much confusion is caused by the use of various terms in different contexts in the literature dealing with brittle fracture*, definitions of certain terms used in the present paper are given below. FAILURE is a process by which a material changes from one state of behaviour to another one. The more important types of failure are yield, strength failure, fracture and rupture. YIELD is the failure process by which a material changes from a state of predominantly elastic behaviour to one of predominantly plastic behaviour. STRENGTH FAILURE is the failure process by which a material changes f r o m a state in which its load-bearing capacity is either constant or increases with increasing deformation to a state in which its load-bearing capacity is decreased or has even vanished. FRACTURE is the failure process by which new surfaces in form of cracks are formed in a material or existing crack surfaces are extended. Various conditions and stages of fracture can be visualized, namely: Crack Initiation is the failure process by which one or more cracks are formed in a material hitherto free from any cracks (Poncelet concept). Fracture Initiation is the failure process by which one or more cracks pre-existing in a material start to extend (Griflith concept). Fracture Propagation is the failure process by which cracks in a material are extending, thus it is a stage subsequent to fracture initiation. It may be distinguished between two types of fracture propagation, namely stable and unstable. Stable fracture propagation is the failure process of fracture propagation in which the crack extension is a function of the loading and can be controlled accordingly. Unstable fracture propagation is the failure process of fracture propagation in which the crack extension is also governed by factors other than the loading, thus becomes uncontrollable. RUPTURE is the failure process by which a structure (e.g. a specimen) disintegrates into two or more pieces.I" In addition to the above definitions the term brittle fracture and not brittle material will be used in this paper. BRIYrLE FRACTURE is defined as fracture that exhibits no or little permanent (plastic) deformation. The opposite term is ductile fracture which is preceded by appreciable plastic deformation. The definition of brittle fracture implies that the material behaves elastically (but not necessarily linearly) up to fracture. Since, for example, brittle fracture * For example, crack initiation, crack enlargement, crack growth, crack increment, slow crack propagation, fast propagation, rapid propagation, failure growth, etc. In addition, fracture and fracture initiation have the same connotation for some authors. t Sometimes called by some authors 'final fracture' or 'ultimate failure.'
MECHANISM OF BRITTLE FRACTURE OF R O C K - - P A R T I
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can occur in steel ('ductile' material) and ductile fracture in rock ('brittle' material) it is more correct to say that it is the fracture and not the material that is brittle or ductile respectively. 3. THEORY OF THE FRACTUREPROCESS It is clear from the definitions given in the preceding section, that fracture propagation and strength failure should be distinguished from fracture initiation, the ~ reason being that fracture initiation does not necessarily render a structure or a material useless but strength failure may do so while fracture propagation may cause an appreciable change in the state of the material. Moreover, strength failure usually takes place at an appreciably higher stress level than fracture initiation[8].
3.1 Fracture initiation It has been found[9] that the Griflith hypothesis furnishes a satisfactory criterion for fracture initiation. The question now arises whether this hypothesis also yields a strengthfailure criterion. Under certain conditions fracture initiation and strength failure may occur nearly simultaneously and in such cases this criterion specifies, for practical purposes, strength failure of the material. Many materials, however, amongst them rock, under certain conditions, exhibit a definite process of fracture propagation and the strength failure of the material cannot be predicted from a fracture initiation criterion. It has been recently shown by BARENnLAa'r[10]that serious errors are introduced if fracture propagation considerations are not included in a fracture criterion. The above observations are of particular importance for fracture under compressive stress conditions where, without doubt, the stress required for fracture initiation is different from that causing strength failure. The reason for this is that, in compression, a crack does not propagate in its own plane[8], as is the case in tension, and, further, stabilization of fracture propagation may take place[8] under certain stress conditions. Consequently a fracture initiation criterion cannot be expected to predict the compressive strength of rock[11 ]. Considering brittle fracture on the basis of the Griffith hypothesis, it should be noted that the GritFith criterion has been derived utilizing two methods of approach, namely (i) by considering the stress field near the tip of a pre-existing (Griffith) crack (ii) by considering the energy balance for a pre-existing (Griffith) crack. Obviously both methods of approach yield the same criterion but the one may be more convenient than the other when it is desired to use the hypothesis for a strength failure criterion. This matter will now be discussed in greater detail. 3.1.1 The stress approach. GRIF~TH[3, 4] postulated that the presence of small cracks or flaws existing in almost any material causes large tensile stress concentrations at the tips of these (Griffith) cracks when the material is stressed. He determined the relationship between the applied stress field and the tensile stress at the crack tip assuming the crack having the shape of a flat ellipse. He furthermore postulated that the crack :would start to extend (fracture initiation) when the tensile stress at or near its tip attained a certain critical value. It has been propounded by OROWAN[ 12] that this critical value represents the molecular cohesive strength of the material. Since the latter is difficult to determine by direct physical measurements, the critical value of the tensile stress at the crack tip may be expressed[9]
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z. T. BIENIAWSKI
by a corresponding critical value of the applied stress for the case of uniaxial tension utilizing the relationship between applied stress and tensile stress at the crack tip. Thus, a formula is obtained which relates the principal stress components of any applied stress field at the stage of fracture initiation (not strength failure) to the uniaxial tensile strength of the material.* This, of course, yields a fracture initiation criterion which reads as follows: (¢l -
~3)2/(~1 +
as) :
-
8at
(1)
where al and az are the major and minor principal components of the applied stress and at is the uniaxial tensile strength o f the material. Griffith's analysis, which is reviewed in detail elsewhere[9], refers to an open crack, that is, it does not make provision for the effect that closure of the crack (contact between opposite faces) might have on the tensile stress at the crack tip. The phenomenon of crack closure occurs, however, if the applied stress field is compressive. The original Griffith hypothesis therefore does not lend itself to applications for compressive applied stresses. MCCLINTOCK and WALSrI[13] propounded a modification to Griffith's hypothesis, accounting for crack closure (closed cracks) by introducing a coefficient of internal friction between crack faces.l" The modified Griffith theory therefore includes two critical quantities, namely, the critical tensile stress at the crack tip, expressed by the value of the uniaxial tensile strength of the material (as in the original Griffith hypothesis) and the coefficient of internal friction between crack faces: 4~t gl : (1 -- oa/ol)V(1 d- /z2) --/~(1 -b ~a/trl) -
-
(2)
where tz is the coefficient of internal friction of the material. In spite of the theoretical advantage of expressing the fracture initiation criterion in terms of the uniaxial tensile strength, the practical difficulties associated with the accurate determination of this strength value for rock impose serious limitations upon the usefulness of the formula. Since, in the case of rock, the uniaxial compressive strength is an easier strength value to be determined experimentally, it may be used as the critical quantity in place of the uniaxial tensile strength. Equation (2) then reads as follows[14]: at : aa[~¢/(1 -~-/z2 -+-/z)/~¢/(1 -q-/~2 __/z)] -1- (re
(3)
where ae is the uniaxial compressive strength of the material. Recent analysis[14] has shown that the modified Griffith criterion can be satisfactorily applied to predicting brittle fracture initiation in compression. While the uniaxial compressive strength can easily be determined, means for determining the internal crack friction coefficient directly are not available. The coefficient may, however, be determined in the following indirect manner. A series of tests for different multiaxialities of the state of applied stress is carried out and the values of the principal components of the applied stress fields at the onset of crack extension, i.e. at fracture initiation, are noted. The data are presented in form of 'Mohr circles at fracture initiation' and a * Strictly speaking, this is not quite correct since the uniaxial tensile strength refers to strength failure and not to fracture initiation in uniaxial tension. Since however, in tension (N.B. not in compression), both stress levelsare close to each other, the error is not great and the concept is acceptable for practical purposes. t This coefficientof internal friction is not the same as that used in soil mechanics.
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399
'Mohr envelope for fracture initiation' is fitted to the family of circles. Then a family of such Molar envelopes, based on the modified Griffith criterion with different friction coefficients, is superimposed on the Mohr diagram. The friction coefficient that yields an envelope almost coinciding with the envelope found experimentally from the test series is then taken as the internal crack friction coefficient for the particular material[8]. Ho~K and BIENIAWSKI[9] applied the modified Griffith fracture initiation criterion to triaxial test data for strength failure. By applying the same method as above for determining the friction coefficient they, of course, found--purely phenomenologically--that the criterion is applicable to strength failure, if a friction coefficient is assumed which is different from the internal crack friction coefficient valid for fracture initiation for the same material[9, 14]. The coefficient valid for strength failure was termed 'fracture surface coefficient of friction'. This coefficient can also be measured, on specimens fractured into pieces, by letting both fracture fragments slide relative to each other along their fracture faces[15]. In addition, it can be determined from the stress-strain curve[16]. The results of all three methods broadly agree with one another. Ho~K and BIENIAWSKI[9]also gave some genetic explanation for the possible validity of the modified Griffith hypothesis as a strength-failure criterion, but subsequent considerations cause the author of the present work to adopt revised views to be discussed in Part II of this paper. Although the modified Griffith hypothesis has been adopted by some research workers in rock mechanics as a phenomenological strength-failure criterion, it does not constitute a hypothesis for the mechanism of brittle fracture propagation succeeded by strength failure[8]. 3.1.2 The energy approach. The Griffith criterion can also be derived from the energy balance for a pre-existing (Griffith) crack[3]. The concept of the original Grittith hypothesis is based upon the condition that the energy W applied by loading the structure is balanced by the elastic strain energy We stored in the structure and the surface energy Ws in the free faces of the pre-existing crack, thus:
W = W e + We.
(4)
If the load is increased, the corresponding increase d W in the applied energy W may be balanced either (i) by an increase d We in the strain energy We only or (ii) by an increase d Ws in the crack surface energy We only or (iii) partly by an increase d We and partly by an increase d Ws. In the first case (dW : dWe, dWe = 0) the crack does not extend. In the two otn~:~ cases (dWs =~ 0) the crack surface energy can only increase if the crack extends, that is, if the half-length of the crack increases from c to (c q- dc). Thus, the balance for the energy increases reads:
dW--- dWe + dWe or It can be shown* that
dW dWe dWs . . . . + de dc dc dW dWe de = 2 d e "
(5)
(6)
* L o w A . F. H. A Treatise on the Mathematical Theory o f Elasticity, 2nd e.dn., Cambridge University
Press, London (1906). R.M~2C
400
Z . T . BIENIAWSKI
Substituting (6) into (5) yields dive dc
d Ws dc "
(7)
For a thin plate subjected to uniaxial tension under plane stress conditions the elastic strain energy stored was calculated by GRWFITH[3] as: We = ~rc2 o~/E
(8)
W8 = 4yc
(9)
and the crack surface energy:
where
~ is the applied uniaxial tensile stress E is the modulus of elasticity 9' is the specific surface energy, i.e. the surface energy per unit length of crack surface.
Differentiating equations (8) and (9) with respect to c and substituting the results in equation (7) yields: (10) This, in fact, is the condition for the onset of crack extension, that is, fracture initiation. The corresponding condition may be derived from multiaxial states of stress and when substituted in equation (8) will yield the same Griffith criterion for fracture initiation as that obtained from the stress approach as given by equation (1). The condition given in equation (10) implies that for ,, < [v'(2~,E/~ c) = ,,r~]
the crack does not extend. This is the first ease (d Ws = 0) mentioned above. Furthermore, the unequity
represents fracture propagation. This will be discussed in the next section of this paper. Equation (10) is, in fact, a hypothesis for a fracture initiation mechanism postulating that fracture is initiated when the applied stress attains a critical value cri~ which depends upon the length 2c of the pre-existing crack, the specific surface energy y and the modulus of elasticity of the material. Its practical application is limited, however, because of the difficulty of determining experimentally the specific surface energy ~, of the material. 3.2 Fracture propagation Once fracture has been initiated as a result of the applied stress attaining the value given by equation (10), the stage of fracture propagation is reached. Experience has shown[8] that two types exist, namely, stable and unstable fracture propagation. The fracture propagation is stable, as long as there is a definite relationship between the half-length c of the crack and the applied stress ~ and the condition o > crier is maintained.
MECHANISM OF BRITTLE FRACTURE OF R O C K - - P A R T I
401
Such a relationship would be valid for applied stress increases only, but not for decreases, in other words the process is irreversible: it would permit calculating the increase Ac in crack hail-length resulting from an increase + Aa in the applied stress but a decrease in stress (--A~) would not yield a decrease (--Ac) in crack half-length since it cannot be expected that a crack heals due to deloading. Such a relationship has been proposed by IRWIN[17] for brittle fracture of metals but has not, as yet, been adopted in rock mechanics. Irwin's relationship reads = a/(GE/~c)
(1 I)
where G is the energy released per unit crack surface area. This formula is based upon the concept that fracture propagation is due to the fact that a certain amount of energy, represented by G, is released from the stored elastic energy We and used to form additional crack surface area. The energy release from W6 takes place at the same rate as the energy absorption by crack extension. It will be noted that, in essence, Irwin's formula takes the same appearance as equation (10), 2 ~, being replaced by G, but while equation (10) is a formula specifying a criterion, equation (i 1) constitutes a functional relationship between c and u as explained above. Equation (11) is therefore a hypothesis describing stable fracture propagation. Fracture propagation is unstable when a unique relationship between e and ~ ceases to exist, that is, when other quantities, e.g. the crack growth velocity, also play a role and fracture propagation cannot be controlled any more by the applied load. While in stable propagation the crack growth can be stopped by stopping load increases, this does not hold for unstable fracture propagation; the fracture then propagates uncontrollably although the stress may be kept constant. Stable fracture propagation is usually a slow process while unstable fracture propagation is fast[18]. The question now arises whether a criterion exists which determines transitio/t from stable to unstable fracture propagation. IRWIN[17] has used equation (11) as such a criterion. He postulated that fracture propagation becomes unstable when the energy released per unit crack surface, G, attains a critical value, Ge, which is a characteristic property of the material. Thus a>~ [V(GeE/~rc) = otrvv] (12) for unstable fracture propagation. The value of Ge for a particular material may be determined by measuring the applied stress aer and the crack half-length Cer at the onset of unstable fracture propagation and making use of equation (12) as follows: 6 , = . O o r ~ cor/E.
(13)
Values of Ge determined for various materials are listed in Table 1. A more detailed description of this concept as applied to rock is given by the author elsewhere[l~8]. The Irwin concept accounts for the total energy released which is 'absorbed ia the process of fracturing' but does not specify the different forms of energy into which the energy released is converted. Useful information may be gained by considering a more detailed energy balance for the process of fracture propagation.
402
Z . T . BIENIAWSKI
TABLE1. CI~rrICAL~ROY RELEASEDG¢FORVARIOUSMATERIALS Material
Gc lb-in/sq, in.
Glass Concrete Quartzite Ship steel Rotor steel
0.08 0.1 3.5 80.0 135.0
Source
IRw~[l 6] IC~* BmNInWSm[18] IRwin[16] Wim,~ and WtmDTt
* K ~ t ~ N M. F. Crack propagation and fracture of concrete. J. Am. Concrete Inst. 58, (5) 591-610 (1961).. t Wr~r~mD. H. and W ~ D T B. M. Application of the Griffith-Irwin theory of crack propagation to the bursting behaviour of discs. Tram. Am. Soc. Mech. Engrs 80, 1643-1651 (1958).
The Griffith energy balance according to equation (7) accounts for the elastic energy stored (We) and the crack surface energy (W,) only; several other forms of energy losses, however, into which part of W6 is transformed must also be considered. The following 'losses' in addition to Ws may be listed here: (i) Kinetic energy (ii) Plastic energy ('tii) Energy dissipated on the breakdown of atomic bonds at the tips of extending cracks (iv) Energy changes due to mining such as caused by artificial rock breaking, heat removal due to ventilation, etc.[19] As far as (iii) is concerned, it is doubtful whether energy dissipated on the breakdown of atomic bonds plays any significant part[20]; energy changes due to mining (iv) are neglected for the present time. Plastic energy losses (ii) which also include visco-plastic losses exhibited very distinctly at loci of high stress concentrations, e.g. at the tips of cracks may also be neglected since brittle fracture (i.e. absence of plastic deformation in the ideal case) is dealt with here. Thus, there only remains the kinetic energy (i) associated with the movement of the faces of the extending crack to be considered. B ~ I ~ , [ 2 1 ] has shown that the amount of elastic strain energy (We) transformed into such kinetic energy and eventually converted into heat is appreciable and emphasized that a basic limitation of the Griffith hypothesis is due to the fact that kinetic energy is not accounted for in Griffith's energy balance. Equation (7) should therefore be rewritten as follows: dWedWs
dc
dWk
dc
+ --
dc
04)
where We is the kinetic energy loss. The kinetic energy has been evaluated by MOTT[22] in unlaxial tension under plane stress conditions: Wk = kpc ~ v ~ o~/2E ~
(15)
MECHANISM OF BRITrLE FRACTURE OF ROCK--PART I
where
403
k is a constant proportionality factor the density of the material c the crack half-length v the crack velocity the applied stress E the modulus of elasticity.
p
The constant k was evaluated by ROBOTS and WELLS[23]. The crack velocity can be expressed as follows[24]: v
= 0-38 Eh,
0
-
(16)
co~c)
where co is the initial crack half-length, i.e. the half-length of the pre-existing (Griffith) crack. It is obvious from equation 06) that the velocity will, with increasing crack length, 2c, approach the asymptotic value 0.38 v ' E / p = vT
(17)
which is called the terminal velocity. It will be noted from equation (17) that the terminal velocity is a characteristic property of the material; it is a fraction of the velocity of the longitudinal stress wave in a rod made of this material. Equation (16) is graphically presented as the dotted curve in Fig. 1. In the same figure are plotted experimental results obtained by BIS~WSKI[18] for rock. Similar results have also been obtained by SCHARDrN[25]for glass. It must be concluded from Fig. 1 that refinements are required to the hypothesis on which equation (16) is based in order to achieve agreement between theoretical and experimental results.* 200C ~•.... 150(: E /
>~oc
I I I I !
/
"-~'-Theo;eticol
/
~
mental
I
5
I0
15 Ro+io
20 25 50 Crock holf.len~tth c Originol crock holf-lenqth Co
35
FIG. 1. Crack velocityrelated to crack length ratio; theoretically according to equation (16) and experimentally for norite rock according to BmNXAWSKI[18]. * This will be the subject of a subsequent publication by the author.
404
Z. T. BIENIAWSKI
It will be noted from the results given in Fig. 1 that fracture propagation starts with low crack velocity. Further, according to both SCHARDIN[25]and BIENIAWSlO[26]approximately up to the turning point of the curve, the elastic energy released by crack extension is not sufficient to maintain fracture. At a later stage, when elastic energy released is able to maintain fracture, the crack velocity increases rapidly to a limit where it attains a constant value. Consequently, the turning point of the curve, that is where c/co ---- c o t / c o , ( d 2 v / d c 2 = 0), marks the transition from stable to unstable crack propagation. It may also be shown mathematically[18] that when c/co -----cer/co then the energy release G = Ge. It may also be concluded[18] that while the influence of crack velocity may be neglected during stable fracture propagation, it will be the governing factor in the process of unstable fracture propagation. This statement is also supported by the results of an analysis by CRAOOS[27] who based his considerations on dynamic stresses created by a propagating crack without reference to the extended Griffith energy balance. Craggs has shown that, as crack velocity increases, the force required to maintain crack propagation decreases. Using Craggs analysis it may be shown[18] that on the onset of unstable fracture propagation the fracture process will become self-maintaining. On the basis of considerations of velocity aspects in fracture propagation, a detailed review of which is given by the author elsewhere[28], it may be noted that once the crack approaches its terminal velocity the kinetic energy associated with crack extension will also approach a constant value. The released energy increases with crack length, however, and in order to dissipate the additional energy, the crack tends to increase its surface area and hence its surface energy by forking (bifurcation or branching) to form additional cracks at an angle to the original crack. This conclusion can also be substantiated by Cragg's analysis who has shown analytically that crack forking will occur once the terminal velocity is reached. The onset of forking represents a transition within the process of unstable crack propagation. The condition which determines this transition is v --- v~,.
(18)
It has been recently shown[8] that this transition coincides with strength failure of the material. Once this transition has taken place successive forking will lead to coalescence of propagating fracture culminating in rupture of the material. 4. P O S T U L A T E D M E C H A N I S M OF BRITTLE ROCK FRACTURE OF ROCK
On the basis of the above considerations and experimental results to be presented in Parts II and III of this paper, the following stages of brittle fracture of rock in multiaxial compression are postulated: 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
JCrack lengi'h C
~
14 ]R.M.
... f ~
Linear strain
loop
/~"
Grain boundariesin rack acting as pre-existing (Griffith) cracks
~ / " ,~o-~
I
_
~,~
~ E of solid IF ~ ' ~ ~
_~I - - f - - - - - - - - - - ~
~
I~._1 I~ ) ~ J /~
i
=E E._c ,~
Iik ~
I
J J ~
Nm
/
/
=
~-
~
T
~=~2yE/.c
~
I
)+o'c
o"
Mohr envelope
1
Crackcl°sure I
°-,=o ' 3 ( ~ / ~
or
.e11 Fractureinitiation
~ Closing "°I of cracks
t
~ ,~ Perfectlyelastic (~)1 def°rmation
-- -f-~
~ 1 ,ructure~o~ga,ion
Unstable ~ I fracture propagation - ~ ]]I Critical energyrelease(long-term "
",~ ' ".i
.-oI~ Strength failure
I
| |
FIG. 2. Mechanism of brittle fracture of rock in multiaxial compression.
Volumetricstrain
/
........... ~ ' 7
I Maximumstress 100% / J~_S . - ' ~ P / i. ~i J / ~ / ~ ~ [/ ~ I " ~ % m'~ ~Ru'pfure . . . . . . . . . ,.~ _~. . . . . . . . . . . . .
/ ~ / ......... /F~king andcrockcoalescence / ~ / ~/energy balancedepends I- . / /upon sfructure / IProcTuredirect.i^/ /
-J=~----"---'~'JCcr
Mdr~r envelope
k=oo
-.
./
-
/
/ //
Volumetric strain
~
k=l.3
/
/ .~I" ~o~P ~0"15 _i.-'"~'~ ~ _ - -~' ,f l ~ ¢ - ~J F.~,~..,~-F
i/
o~,///
//~=6
MECHANISM OF BRITIT,E FRACTURE OF ROCK--PART I
405
IV. Strength failure ( m a x i m u m s t r e s s ) - o n s e t o f f o r k i n g S. F o r k i n g a n d coalescence o f cracks V. R u p t u r e ( m a x i m u m d e f o r m a t i o n ) . The a b o v e m e c h a n i s m is d i a g r a m m a t i c a l l y represented in Fig. 2. T h e stages o f brittle fracture o f r o c k s h o w n in this figure also generally a p p l y for tension. I n tension, however, c r a c k closure will, o f course, be a b s e n t a n d processes o f stable a n d u n s t a b l e fracture p r o p a g a t i o n will be o f very small d u r a t i o n d u e to the fact that, in tension, a c r a c k will p r o p a g a t e in its o w n plane. E x p e r i m e n t a l verification o f this m e c h a n i s m will be dealt with in P a r t s I I a n d I I I o f this p a p e r where certain details will also be e l a b o r a t e d on.
REFERENCES 1. BXE~AWAS~E. M. A Bibliography on Fracture of Rock. Report of the South African Council of Scientific and Industrial Research No. MEG 355, May (1965). 2. NADAIA. L. Theory of Fracture and Flow of Solids, Vol. 1, McGraw-Hill (1950). 3. GR~FrrH A. A. The phenomena of rupture and flow in solids. Phil. Trans. A221, 163-198 (1921). 4. GRn~vllt~ A. A. Theory of Rupture. Proceedings of the First International Congressfor Applied Mechanics, (Bienzeno and Burgess, Eds.) pp. 53-64. J. Waltman Jr. Press, Delft (1925). 5. PONCEL~rE. F. Fracture and comminution of brittle solids. Metals Technol. 11, Teeh. Pub. 1684, April (1944). 6. BRACEW. L. Brittle fracture of rocks. State of Stress in the Earth's Crust (W. Judd, Ed.) pp. 111-180, Elsevier, New York (1964). 7. BRACEW. L. Discussion to paper by E. F. Poncelet. Theoretical aspects of rock behaviour under stress. Proceedings of the Fourth Symposium on Rock Mechanics, Bull. Miner. Industr. Expt. Stn. Penn. St. Univ. No. 76, 65-71 (1961). 8. BmmAWSKIZ. T. Mechanism of Rock Fracture in Compression. Report of the South African Council of Scientific and Industrial Research No. MEG 459, June (1966). 9. HOEK E. and BmNmws~ Z. T. Brittle fracture propagation in rock under compression. Int. J. Fracture Mechanics 1, (3) 139--155 (1965). 10. BAI~NSLATr(3. I. Brittle Fracture, Royal Institute of Technology Publications, StockhoLm, No. 149 (1966). 11. BRACEW. L., PAULDINGB. W. and SCHOLZC. Dilatancy in the fracture of crystalline rocks. J. geophys. Res. 71, (16) 3939-3953 (1966). 12. OROWANE. Fracture and strength of solids. Rep. Prog. Phys. 12, 185-232 (1949). 13. MCCLINTOCKF. A. and WAL~ J. B. Friction on Griffith Cracks in Rocks under Pressure. Proceedings of the Fourth U.S. Congress on Applied Mechanics, pp. 1015-1021. American Society of Mechanical Engineers, New York (1963). 14. HOEKE. and Bm~O_AWSKIZ. T. Fracture Propagation Mechanism in Hard Rock. Proceedings of the First Congress of the International Society of Rock Mechanics, Lisbon, September, 1966, Vol. I, pp. 243-249. 15. JAEOL~J. C. Brittle Fracture of Rock.Proceedings of the Eighth Symposium on Rock Mechanics, University of Minnesota, Minneapolis, September, 1966. In press. 16. WALSHJ. B. The effect of cracks in the uniaxial elastic compression of rocks. J. geophys. Res. 70, (2) 399-411 (1965). 17. I a w ~ G. R. Fracture mechanics. StructuralMechanics, (Goodier and Hoff, Eds.) pp. 557-592, Pergamon Press (1960). 18. BIEmAWS~Z. T. Stable and Unstable Fracture Propagation in Rock. Report of the South African Council of Scientific and Industrial Research No. MEG 493, October (1966). 19. BLACKR. A. L. and S T ~ A. M. A Dynamic or Energy Approach to Strata Control Theory and practice. Proceedings of the Fourth International Conference on Strata Control and Rock Mechanics, pp. 450-462, Columbia University Press, New York (1965). 20. ZWICKVF. Die Reissfestigkeit von Steinsals. Phys. Z. 24, 131-137 (1923). 21. BAR~I,a~VG. M. The Mechanism of Energy Dissipation at Brittle Fracture. Proceedings of the International Conference on Fracture, Sendal, Japan, September, 1965, Vol. A, No. 25, pp. 445-453. 22. Mo'rr N. F. Fracture of metals: theoretical considerations. Engineering 165, 15-18 (1948). 23. ROn~.RTSD. K. and WELLSA. A. The velocity of brittle fracture. Engineering 178, 820-821 (1954). 24. DtmAt,~" E. N. and BI~CE W. F. Velocity behaviour of a growing crack. J. appl. Phys. 31, (12) 2233-2236 (1960).
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Z . T . BIENIAWSKI
25. SCHARDINH. Velocity effects in fracture. Fracture (B. L. Averbach et al., Eds.) pp. 297-329, Wiley, N.Y. (1959). 26. BmNIAWSKIZ. T. Fracture Velocity of Rock. Report of the South African Council of Scientific and Industrial Research No. MEG 517, December (1966). 27. CRAGGS J. W. On the propagation of a crack in an elastic-brittle material. J. Mech. Phys. Solids 8, 66-75 (1960). 28. BIm,nAWSKIZ. T. A Literature Survey on Fracture of Rock under Dynamic Stress Conditions. Report of the South African Council of Scientific and Industrial Research No. MEG 340, March (1965).
MECHANISM OF BRITTLE F R A C T U R E OF ROCK P A R T I - - T H E O R Y O F T H E F R A C T U R E PROCESS Z. T. BIENIAWSKI National Mechanical Engineering Research Institute, Pretoria, South Africa (Received 10 January 1967)
Synopsis--This paper presents results of investigations aimed at establishing a mechanism of brittle fracture of rock in compression and tension. Results of fracture studies on specimens of different shapes and subjected to different loading conditions are presented. Fraction initiation, fracture propagation, strength failure and rupture are discussed. A hypothesis on the mechanism of rock fracture is propounded, all significant failure processes taking place in rock from initial load application to complete failure being dealt with. This paper is published in three parts. Part I deals with the theory of the fracture process while Parts II and III provide experimental verification of the postulated mechanism of brittle fracture of rock. 1. INTRODUCTION AN UNDERSTANDING of the fracture mechanism of rock is an essential prerequisite for designing mining excavations and civil engineering structures, for developing rock-breaking processes such as drilling or blasting, and for devising methods to prevent such hazards as rockbursts. Increased knowledge of the fracture mechanism can make a considerable contribution towards improving the efficiency of mining operations with regard to their economy and safety. Although much research has been done on the fracture of rock[l], mainly during the last few years, very little is known at present about its actual mechanism. Studies of failure of solids stem from the attempts to predict the strength of materials and structures. These studies have resulted in a number o f failure hypotheses[2] which are applied in practice with a varying degree o f substantiation. In the case of brittle fracture it has been found that only three such hypotheses may be applicable, namely, those by MOHR[2], GgIFFn'n[3, 4] and PO~CELET[5]. The first two hypotheses have been accepted by various workers as being applicable to rock[6] while the hypothesis by Poncelet has been strongly criticized[7]. A clear distinction must be made between a phenomenological failure criterion and a genetic failure mechanism. A failure criterion simply provides a formula enabling predicting the strength values for all states of multiaxial stress from a critical quantity which may be determined in one type of test, e.g. the uniaxial tensile or compression test. A failure mechanism describes the processes taking place in the material in the course of loading and eventually leading to failure. Preferably a failure criterion should be based upon knowledge of the failure mechanism, but this is not always so. In fact, m a n y failure hypotheses have been propounded as a result of theoretical reasoning only and could not be verified by experimental evidence. The M o h r 395
396
Z. T. BIENIAWSKI
hypothesis, although it has been shown to fit experimental data approximately, is not based on a failure mechanism. The Griffith hypothesis, on the other hand, is based on a genetic concept, namely the existence of small cracks or flaws, but not on a complete failure mechanism, although it offers possibilities in this respect. It should be noted, however, that the Griffith hypothesis refers to fracture initiation only which is not the same as fracture[8]. This series of three papers presents results of investigations, on the basis of which a hypothesis on the mechanism of brittle fracture in rock under compression and tension is propounded. 2. DEFINITIONS Since much confusion is caused by the use of various terms in different contexts in the literature dealing with brittle fracture*, definitions of certain terms used in the present paper are given below. FAILURE is a process by which a material changes from one state of behaviour to another one. The more important types of failure are yield, strength failure, fracture and rupture. YIELD is the failure process by which a material changes from a state of predominantly elastic behaviour to one of predominantly plastic behaviour. STRENGTH FAILURE is the failure process by which a material changes f r o m a state in which its load-bearing capacity is either constant or increases with increasing deformation to a state in which its load-bearing capacity is decreased or has even vanished. FRACTURE is the failure process by which new surfaces in form of cracks are formed in a material or existing crack surfaces are extended. Various conditions and stages of fracture can be visualized, namely: Crack Initiation is the failure process by which one or more cracks are formed in a material hitherto free from any cracks (Poncelet concept). Fracture Initiation is the failure process by which one or more cracks pre-existing in a material start to extend (Griflith concept). Fracture Propagation is the failure process by which cracks in a material are extending, thus it is a stage subsequent to fracture initiation. It may be distinguished between two types of fracture propagation, namely stable and unstable. Stable fracture propagation is the failure process of fracture propagation in which the crack extension is a function of the loading and can be controlled accordingly. Unstable fracture propagation is the failure process of fracture propagation in which the crack extension is also governed by factors other than the loading, thus becomes uncontrollable. RUPTURE is the failure process by which a structure (e.g. a specimen) disintegrates into two or more pieces.I" In addition to the above definitions the term brittle fracture and not brittle material will be used in this paper. BRIYrLE FRACTURE is defined as fracture that exhibits no or little permanent (plastic) deformation. The opposite term is ductile fracture which is preceded by appreciable plastic deformation. The definition of brittle fracture implies that the material behaves elastically (but not necessarily linearly) up to fracture. Since, for example, brittle fracture * For example, crack initiation, crack enlargement, crack growth, crack increment, slow crack propagation, fast propagation, rapid propagation, failure growth, etc. In addition, fracture and fracture initiation have the same connotation for some authors. t Sometimes called by some authors 'final fracture' or 'ultimate failure.'
MECHANISM OF BRITTLE FRACTURE OF R O C K - - P A R T I
397
can occur in steel ('ductile' material) and ductile fracture in rock ('brittle' material) it is more correct to say that it is the fracture and not the material that is brittle or ductile respectively. 3. THEORY OF THE FRACTUREPROCESS It is clear from the definitions given in the preceding section, that fracture propagation and strength failure should be distinguished from fracture initiation, the ~ reason being that fracture initiation does not necessarily render a structure or a material useless but strength failure may do so while fracture propagation may cause an appreciable change in the state of the material. Moreover, strength failure usually takes place at an appreciably higher stress level than fracture initiation[8].
3.1 Fracture initiation It has been found[9] that the Griflith hypothesis furnishes a satisfactory criterion for fracture initiation. The question now arises whether this hypothesis also yields a strengthfailure criterion. Under certain conditions fracture initiation and strength failure may occur nearly simultaneously and in such cases this criterion specifies, for practical purposes, strength failure of the material. Many materials, however, amongst them rock, under certain conditions, exhibit a definite process of fracture propagation and the strength failure of the material cannot be predicted from a fracture initiation criterion. It has been recently shown by BARENnLAa'r[10]that serious errors are introduced if fracture propagation considerations are not included in a fracture criterion. The above observations are of particular importance for fracture under compressive stress conditions where, without doubt, the stress required for fracture initiation is different from that causing strength failure. The reason for this is that, in compression, a crack does not propagate in its own plane[8], as is the case in tension, and, further, stabilization of fracture propagation may take place[8] under certain stress conditions. Consequently a fracture initiation criterion cannot be expected to predict the compressive strength of rock[11 ]. Considering brittle fracture on the basis of the Griffith hypothesis, it should be noted that the GritFith criterion has been derived utilizing two methods of approach, namely (i) by considering the stress field near the tip of a pre-existing (Griffith) crack (ii) by considering the energy balance for a pre-existing (Griffith) crack. Obviously both methods of approach yield the same criterion but the one may be more convenient than the other when it is desired to use the hypothesis for a strength failure criterion. This matter will now be discussed in greater detail. 3.1.1 The stress approach. GRIF~TH[3, 4] postulated that the presence of small cracks or flaws existing in almost any material causes large tensile stress concentrations at the tips of these (Griffith) cracks when the material is stressed. He determined the relationship between the applied stress field and the tensile stress at the crack tip assuming the crack having the shape of a flat ellipse. He furthermore postulated that the crack :would start to extend (fracture initiation) when the tensile stress at or near its tip attained a certain critical value. It has been propounded by OROWAN[ 12] that this critical value represents the molecular cohesive strength of the material. Since the latter is difficult to determine by direct physical measurements, the critical value of the tensile stress at the crack tip may be expressed[9]
398
z. T. BIENIAWSKI
by a corresponding critical value of the applied stress for the case of uniaxial tension utilizing the relationship between applied stress and tensile stress at the crack tip. Thus, a formula is obtained which relates the principal stress components of any applied stress field at the stage of fracture initiation (not strength failure) to the uniaxial tensile strength of the material.* This, of course, yields a fracture initiation criterion which reads as follows: (¢l -
~3)2/(~1 +
as) :
-
8at
(1)
where al and az are the major and minor principal components of the applied stress and at is the uniaxial tensile strength o f the material. Griffith's analysis, which is reviewed in detail elsewhere[9], refers to an open crack, that is, it does not make provision for the effect that closure of the crack (contact between opposite faces) might have on the tensile stress at the crack tip. The phenomenon of crack closure occurs, however, if the applied stress field is compressive. The original Griffith hypothesis therefore does not lend itself to applications for compressive applied stresses. MCCLINTOCK and WALSrI[13] propounded a modification to Griffith's hypothesis, accounting for crack closure (closed cracks) by introducing a coefficient of internal friction between crack faces.l" The modified Griffith theory therefore includes two critical quantities, namely, the critical tensile stress at the crack tip, expressed by the value of the uniaxial tensile strength of the material (as in the original Griffith hypothesis) and the coefficient of internal friction between crack faces: 4~t gl : (1 -- oa/ol)V(1 d- /z2) --/~(1 -b ~a/trl) -
-
(2)
where tz is the coefficient of internal friction of the material. In spite of the theoretical advantage of expressing the fracture initiation criterion in terms of the uniaxial tensile strength, the practical difficulties associated with the accurate determination of this strength value for rock impose serious limitations upon the usefulness of the formula. Since, in the case of rock, the uniaxial compressive strength is an easier strength value to be determined experimentally, it may be used as the critical quantity in place of the uniaxial tensile strength. Equation (2) then reads as follows[14]: at : aa[~¢/(1 -~-/z2 -+-/z)/~¢/(1 -q-/~2 __/z)] -1- (re
(3)
where ae is the uniaxial compressive strength of the material. Recent analysis[14] has shown that the modified Griffith criterion can be satisfactorily applied to predicting brittle fracture initiation in compression. While the uniaxial compressive strength can easily be determined, means for determining the internal crack friction coefficient directly are not available. The coefficient may, however, be determined in the following indirect manner. A series of tests for different multiaxialities of the state of applied stress is carried out and the values of the principal components of the applied stress fields at the onset of crack extension, i.e. at fracture initiation, are noted. The data are presented in form of 'Mohr circles at fracture initiation' and a * Strictly speaking, this is not quite correct since the uniaxial tensile strength refers to strength failure and not to fracture initiation in uniaxial tension. Since however, in tension (N.B. not in compression), both stress levelsare close to each other, the error is not great and the concept is acceptable for practical purposes. t This coefficientof internal friction is not the same as that used in soil mechanics.
MECHANISM OF BRITTLE FRACTURE OF ROCK--PART I
399
'Mohr envelope for fracture initiation' is fitted to the family of circles. Then a family of such Molar envelopes, based on the modified Griffith criterion with different friction coefficients, is superimposed on the Mohr diagram. The friction coefficient that yields an envelope almost coinciding with the envelope found experimentally from the test series is then taken as the internal crack friction coefficient for the particular material[8]. Ho~K and BIENIAWSKI[9] applied the modified Griffith fracture initiation criterion to triaxial test data for strength failure. By applying the same method as above for determining the friction coefficient they, of course, found--purely phenomenologically--that the criterion is applicable to strength failure, if a friction coefficient is assumed which is different from the internal crack friction coefficient valid for fracture initiation for the same material[9, 14]. The coefficient valid for strength failure was termed 'fracture surface coefficient of friction'. This coefficient can also be measured, on specimens fractured into pieces, by letting both fracture fragments slide relative to each other along their fracture faces[15]. In addition, it can be determined from the stress-strain curve[16]. The results of all three methods broadly agree with one another. Ho~K and BIENIAWSKI[9]also gave some genetic explanation for the possible validity of the modified Griffith hypothesis as a strength-failure criterion, but subsequent considerations cause the author of the present work to adopt revised views to be discussed in Part II of this paper. Although the modified Griffith hypothesis has been adopted by some research workers in rock mechanics as a phenomenological strength-failure criterion, it does not constitute a hypothesis for the mechanism of brittle fracture propagation succeeded by strength failure[8]. 3.1.2 The energy approach. The Griffith criterion can also be derived from the energy balance for a pre-existing (Griffith) crack[3]. The concept of the original Grittith hypothesis is based upon the condition that the energy W applied by loading the structure is balanced by the elastic strain energy We stored in the structure and the surface energy Ws in the free faces of the pre-existing crack, thus:
W = W e + We.
(4)
If the load is increased, the corresponding increase d W in the applied energy W may be balanced either (i) by an increase d We in the strain energy We only or (ii) by an increase d Ws in the crack surface energy We only or (iii) partly by an increase d We and partly by an increase d Ws. In the first case (dW : dWe, dWe = 0) the crack does not extend. In the two otn~:~ cases (dWs =~ 0) the crack surface energy can only increase if the crack extends, that is, if the half-length of the crack increases from c to (c q- dc). Thus, the balance for the energy increases reads:
dW--- dWe + dWe or It can be shown* that
dW dWe dWs . . . . + de dc dc dW dWe de = 2 d e "
(5)
(6)
* L o w A . F. H. A Treatise on the Mathematical Theory o f Elasticity, 2nd e.dn., Cambridge University
Press, London (1906). R.M~2C
400
Z . T . BIENIAWSKI
Substituting (6) into (5) yields dive dc
d Ws dc "
(7)
For a thin plate subjected to uniaxial tension under plane stress conditions the elastic strain energy stored was calculated by GRWFITH[3] as: We = ~rc2 o~/E
(8)
W8 = 4yc
(9)
and the crack surface energy:
where
~ is the applied uniaxial tensile stress E is the modulus of elasticity 9' is the specific surface energy, i.e. the surface energy per unit length of crack surface.
Differentiating equations (8) and (9) with respect to c and substituting the results in equation (7) yields: (10) This, in fact, is the condition for the onset of crack extension, that is, fracture initiation. The corresponding condition may be derived from multiaxial states of stress and when substituted in equation (8) will yield the same Griffith criterion for fracture initiation as that obtained from the stress approach as given by equation (1). The condition given in equation (10) implies that for ,, < [v'(2~,E/~ c) = ,,r~]
the crack does not extend. This is the first ease (d Ws = 0) mentioned above. Furthermore, the unequity
represents fracture propagation. This will be discussed in the next section of this paper. Equation (10) is, in fact, a hypothesis for a fracture initiation mechanism postulating that fracture is initiated when the applied stress attains a critical value cri~ which depends upon the length 2c of the pre-existing crack, the specific surface energy y and the modulus of elasticity of the material. Its practical application is limited, however, because of the difficulty of determining experimentally the specific surface energy ~, of the material. 3.2 Fracture propagation Once fracture has been initiated as a result of the applied stress attaining the value given by equation (10), the stage of fracture propagation is reached. Experience has shown[8] that two types exist, namely, stable and unstable fracture propagation. The fracture propagation is stable, as long as there is a definite relationship between the half-length c of the crack and the applied stress ~ and the condition o > crier is maintained.
MECHANISM OF BRITTLE FRACTURE OF R O C K - - P A R T I
401
Such a relationship would be valid for applied stress increases only, but not for decreases, in other words the process is irreversible: it would permit calculating the increase Ac in crack hail-length resulting from an increase + Aa in the applied stress but a decrease in stress (--A~) would not yield a decrease (--Ac) in crack half-length since it cannot be expected that a crack heals due to deloading. Such a relationship has been proposed by IRWIN[17] for brittle fracture of metals but has not, as yet, been adopted in rock mechanics. Irwin's relationship reads = a/(GE/~c)
(1 I)
where G is the energy released per unit crack surface area. This formula is based upon the concept that fracture propagation is due to the fact that a certain amount of energy, represented by G, is released from the stored elastic energy We and used to form additional crack surface area. The energy release from W6 takes place at the same rate as the energy absorption by crack extension. It will be noted that, in essence, Irwin's formula takes the same appearance as equation (10), 2 ~, being replaced by G, but while equation (10) is a formula specifying a criterion, equation (i 1) constitutes a functional relationship between c and u as explained above. Equation (11) is therefore a hypothesis describing stable fracture propagation. Fracture propagation is unstable when a unique relationship between e and ~ ceases to exist, that is, when other quantities, e.g. the crack growth velocity, also play a role and fracture propagation cannot be controlled any more by the applied load. While in stable propagation the crack growth can be stopped by stopping load increases, this does not hold for unstable fracture propagation; the fracture then propagates uncontrollably although the stress may be kept constant. Stable fracture propagation is usually a slow process while unstable fracture propagation is fast[18]. The question now arises whether a criterion exists which determines transitio/t from stable to unstable fracture propagation. IRWIN[17] has used equation (11) as such a criterion. He postulated that fracture propagation becomes unstable when the energy released per unit crack surface, G, attains a critical value, Ge, which is a characteristic property of the material. Thus a>~ [V(GeE/~rc) = otrvv] (12) for unstable fracture propagation. The value of Ge for a particular material may be determined by measuring the applied stress aer and the crack half-length Cer at the onset of unstable fracture propagation and making use of equation (12) as follows: 6 , = . O o r ~ cor/E.
(13)
Values of Ge determined for various materials are listed in Table 1. A more detailed description of this concept as applied to rock is given by the author elsewhere[l~8]. The Irwin concept accounts for the total energy released which is 'absorbed ia the process of fracturing' but does not specify the different forms of energy into which the energy released is converted. Useful information may be gained by considering a more detailed energy balance for the process of fracture propagation.
402
Z . T . BIENIAWSKI
TABLE1. CI~rrICAL~ROY RELEASEDG¢FORVARIOUSMATERIALS Material
Gc lb-in/sq, in.
Glass Concrete Quartzite Ship steel Rotor steel
0.08 0.1 3.5 80.0 135.0
Source
IRw~[l 6] IC~* BmNInWSm[18] IRwin[16] Wim,~ and WtmDTt
* K ~ t ~ N M. F. Crack propagation and fracture of concrete. J. Am. Concrete Inst. 58, (5) 591-610 (1961).. t Wr~r~mD. H. and W ~ D T B. M. Application of the Griffith-Irwin theory of crack propagation to the bursting behaviour of discs. Tram. Am. Soc. Mech. Engrs 80, 1643-1651 (1958).
The Griffith energy balance according to equation (7) accounts for the elastic energy stored (We) and the crack surface energy (W,) only; several other forms of energy losses, however, into which part of W6 is transformed must also be considered. The following 'losses' in addition to Ws may be listed here: (i) Kinetic energy (ii) Plastic energy ('tii) Energy dissipated on the breakdown of atomic bonds at the tips of extending cracks (iv) Energy changes due to mining such as caused by artificial rock breaking, heat removal due to ventilation, etc.[19] As far as (iii) is concerned, it is doubtful whether energy dissipated on the breakdown of atomic bonds plays any significant part[20]; energy changes due to mining (iv) are neglected for the present time. Plastic energy losses (ii) which also include visco-plastic losses exhibited very distinctly at loci of high stress concentrations, e.g. at the tips of cracks may also be neglected since brittle fracture (i.e. absence of plastic deformation in the ideal case) is dealt with here. Thus, there only remains the kinetic energy (i) associated with the movement of the faces of the extending crack to be considered. B ~ I ~ , [ 2 1 ] has shown that the amount of elastic strain energy (We) transformed into such kinetic energy and eventually converted into heat is appreciable and emphasized that a basic limitation of the Griffith hypothesis is due to the fact that kinetic energy is not accounted for in Griffith's energy balance. Equation (7) should therefore be rewritten as follows: dWedWs
dc
dWk
dc
+ --
dc
04)
where We is the kinetic energy loss. The kinetic energy has been evaluated by MOTT[22] in unlaxial tension under plane stress conditions: Wk = kpc ~ v ~ o~/2E ~
(15)
MECHANISM OF BRITrLE FRACTURE OF ROCK--PART I
where
403
k is a constant proportionality factor the density of the material c the crack half-length v the crack velocity the applied stress E the modulus of elasticity.
p
The constant k was evaluated by ROBOTS and WELLS[23]. The crack velocity can be expressed as follows[24]: v
= 0-38 Eh,
0
-
(16)
co~c)
where co is the initial crack half-length, i.e. the half-length of the pre-existing (Griffith) crack. It is obvious from equation 06) that the velocity will, with increasing crack length, 2c, approach the asymptotic value 0.38 v ' E / p = vT
(17)
which is called the terminal velocity. It will be noted from equation (17) that the terminal velocity is a characteristic property of the material; it is a fraction of the velocity of the longitudinal stress wave in a rod made of this material. Equation (16) is graphically presented as the dotted curve in Fig. 1. In the same figure are plotted experimental results obtained by BIS~WSKI[18] for rock. Similar results have also been obtained by SCHARDrN[25]for glass. It must be concluded from Fig. 1 that refinements are required to the hypothesis on which equation (16) is based in order to achieve agreement between theoretical and experimental results.* 200C ~•.... 150(: E /
>~oc
I I I I !
/
"-~'-Theo;eticol
/
~
mental
I
5
I0
15 Ro+io
20 25 50 Crock holf.len~tth c Originol crock holf-lenqth Co
35
FIG. 1. Crack velocityrelated to crack length ratio; theoretically according to equation (16) and experimentally for norite rock according to BmNXAWSKI[18]. * This will be the subject of a subsequent publication by the author.
404
Z. T. BIENIAWSKI
It will be noted from the results given in Fig. 1 that fracture propagation starts with low crack velocity. Further, according to both SCHARDIN[25]and BIENIAWSlO[26]approximately up to the turning point of the curve, the elastic energy released by crack extension is not sufficient to maintain fracture. At a later stage, when elastic energy released is able to maintain fracture, the crack velocity increases rapidly to a limit where it attains a constant value. Consequently, the turning point of the curve, that is where c/co ---- c o t / c o , ( d 2 v / d c 2 = 0), marks the transition from stable to unstable crack propagation. It may also be shown mathematically[18] that when c/co -----cer/co then the energy release G = Ge. It may also be concluded[18] that while the influence of crack velocity may be neglected during stable fracture propagation, it will be the governing factor in the process of unstable fracture propagation. This statement is also supported by the results of an analysis by CRAOOS[27] who based his considerations on dynamic stresses created by a propagating crack without reference to the extended Griffith energy balance. Craggs has shown that, as crack velocity increases, the force required to maintain crack propagation decreases. Using Craggs analysis it may be shown[18] that on the onset of unstable fracture propagation the fracture process will become self-maintaining. On the basis of considerations of velocity aspects in fracture propagation, a detailed review of which is given by the author elsewhere[28], it may be noted that once the crack approaches its terminal velocity the kinetic energy associated with crack extension will also approach a constant value. The released energy increases with crack length, however, and in order to dissipate the additional energy, the crack tends to increase its surface area and hence its surface energy by forking (bifurcation or branching) to form additional cracks at an angle to the original crack. This conclusion can also be substantiated by Cragg's analysis who has shown analytically that crack forking will occur once the terminal velocity is reached. The onset of forking represents a transition within the process of unstable crack propagation. The condition which determines this transition is v --- v~,.
(18)
It has been recently shown[8] that this transition coincides with strength failure of the material. Once this transition has taken place successive forking will lead to coalescence of propagating fracture culminating in rupture of the material. 4. P O S T U L A T E D M E C H A N I S M OF BRITTLE ROCK FRACTURE OF ROCK
On the basis of the above considerations and experimental results to be presented in Parts II and III of this paper, the following stages of brittle fracture of rock in multiaxial compression are postulated: 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
JCrack lengi'h C
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MECHANISM OF BRITIT,E FRACTURE OF ROCK--PART I
405
IV. Strength failure ( m a x i m u m s t r e s s ) - o n s e t o f f o r k i n g S. F o r k i n g a n d coalescence o f cracks V. R u p t u r e ( m a x i m u m d e f o r m a t i o n ) . The a b o v e m e c h a n i s m is d i a g r a m m a t i c a l l y represented in Fig. 2. T h e stages o f brittle fracture o f r o c k s h o w n in this figure also generally a p p l y for tension. I n tension, however, c r a c k closure will, o f course, be a b s e n t a n d processes o f stable a n d u n s t a b l e fracture p r o p a g a t i o n will be o f very small d u r a t i o n d u e to the fact that, in tension, a c r a c k will p r o p a g a t e in its o w n plane. E x p e r i m e n t a l verification o f this m e c h a n i s m will be dealt with in P a r t s I I a n d I I I o f this p a p e r where certain details will also be e l a b o r a t e d on.
REFERENCES 1. BXE~AWAS~E. M. A Bibliography on Fracture of Rock. Report of the South African Council of Scientific and Industrial Research No. MEG 355, May (1965). 2. NADAIA. L. Theory of Fracture and Flow of Solids, Vol. 1, McGraw-Hill (1950). 3. GR~FrrH A. A. The phenomena of rupture and flow in solids. Phil. Trans. A221, 163-198 (1921). 4. GRn~vllt~ A. A. Theory of Rupture. Proceedings of the First International Congressfor Applied Mechanics, (Bienzeno and Burgess, Eds.) pp. 53-64. J. Waltman Jr. Press, Delft (1925). 5. PONCEL~rE. F. Fracture and comminution of brittle solids. Metals Technol. 11, Teeh. Pub. 1684, April (1944). 6. BRACEW. L. Brittle fracture of rocks. State of Stress in the Earth's Crust (W. Judd, Ed.) pp. 111-180, Elsevier, New York (1964). 7. BRACEW. L. Discussion to paper by E. F. Poncelet. Theoretical aspects of rock behaviour under stress. Proceedings of the Fourth Symposium on Rock Mechanics, Bull. Miner. Industr. Expt. Stn. Penn. St. Univ. No. 76, 65-71 (1961). 8. BmmAWSKIZ. T. Mechanism of Rock Fracture in Compression. Report of the South African Council of Scientific and Industrial Research No. MEG 459, June (1966). 9. HOEK E. and BmNmws~ Z. T. Brittle fracture propagation in rock under compression. Int. J. Fracture Mechanics 1, (3) 139--155 (1965). 10. BAI~NSLATr(3. I. Brittle Fracture, Royal Institute of Technology Publications, StockhoLm, No. 149 (1966). 11. BRACEW. L., PAULDINGB. W. and SCHOLZC. Dilatancy in the fracture of crystalline rocks. J. geophys. Res. 71, (16) 3939-3953 (1966). 12. OROWANE. Fracture and strength of solids. Rep. Prog. Phys. 12, 185-232 (1949). 13. MCCLINTOCKF. A. and WAL~ J. B. Friction on Griffith Cracks in Rocks under Pressure. Proceedings of the Fourth U.S. Congress on Applied Mechanics, pp. 1015-1021. American Society of Mechanical Engineers, New York (1963). 14. HOEKE. and Bm~O_AWSKIZ. T. Fracture Propagation Mechanism in Hard Rock. Proceedings of the First Congress of the International Society of Rock Mechanics, Lisbon, September, 1966, Vol. I, pp. 243-249. 15. JAEOL~J. C. Brittle Fracture of Rock.Proceedings of the Eighth Symposium on Rock Mechanics, University of Minnesota, Minneapolis, September, 1966. In press. 16. WALSHJ. B. The effect of cracks in the uniaxial elastic compression of rocks. J. geophys. Res. 70, (2) 399-411 (1965). 17. I a w ~ G. R. Fracture mechanics. StructuralMechanics, (Goodier and Hoff, Eds.) pp. 557-592, Pergamon Press (1960). 18. BIEmAWS~Z. T. Stable and Unstable Fracture Propagation in Rock. Report of the South African Council of Scientific and Industrial Research No. MEG 493, October (1966). 19. BLACKR. A. L. and S T ~ A. M. A Dynamic or Energy Approach to Strata Control Theory and practice. Proceedings of the Fourth International Conference on Strata Control and Rock Mechanics, pp. 450-462, Columbia University Press, New York (1965). 20. ZWICKVF. Die Reissfestigkeit von Steinsals. Phys. Z. 24, 131-137 (1923). 21. BAR~I,a~VG. M. The Mechanism of Energy Dissipation at Brittle Fracture. Proceedings of the International Conference on Fracture, Sendal, Japan, September, 1965, Vol. A, No. 25, pp. 445-453. 22. Mo'rr N. F. Fracture of metals: theoretical considerations. Engineering 165, 15-18 (1948). 23. ROn~.RTSD. K. and WELLSA. A. The velocity of brittle fracture. Engineering 178, 820-821 (1954). 24. DtmAt,~" E. N. and BI~CE W. F. Velocity behaviour of a growing crack. J. appl. Phys. 31, (12) 2233-2236 (1960).
406
Z . T . BIENIAWSKI
25. SCHARDINH. Velocity effects in fracture. Fracture (B. L. Averbach et al., Eds.) pp. 297-329, Wiley, N.Y. (1959). 26. BmNIAWSKIZ. T. Fracture Velocity of Rock. Report of the South African Council of Scientific and Industrial Research No. MEG 517, December (1966). 27. CRAGGS J. W. On the propagation of a crack in an elastic-brittle material. J. Mech. Phys. Solids 8, 66-75 (1960). 28. BIm,nAWSKIZ. T. A Literature Survey on Fracture of Rock under Dynamic Stress Conditions. Report of the South African Council of Scientific and Industrial Research No. MEG 340, March (1965).