- Email: [email protected]
The effect of stress wave duration on brittle fracture
Ltt. J. Rock Mech. Min. Sci. Vol. 3, pp. 191-203. Pergamon Press Ltd 1966. Printed in Great Britain
THE EFFECT OF STRESS WAVE D U R A T I O N ON BRITTLE F R A C T U R E M. H. MILLER Ingersoll-Rand Research Center, Bedminster, New Jersey (Received 31 January 1966)
Abstract-A qualitative theoretical examination of the fracture of heterogeneous brittle materials by a stress wave indicates that the duration of the wave significantly influences the amount of energy needed to fracture a unit volume of material. It is concluded that the specific fracture energy will be much lower for a high-amplitude, short-duration wave than it is for the loads imparted by mechanical methods presently used for drilling rock. The results of experiments employing an underwater discharge to produce high-intensity 5 tzsecpressure pulses seem to confirm the theoretical conclusions. 1. INTRODUCTION ALTHOUGH a number of investigators have given considerable attention to the effects of load magnitude on the failure of brittle materials, very little attention has been devoted to the effects of load duration. There has been some work which has shown that the specific fracture energy (energy required to fracture a unit volume of material) is essentially the same for dynamic [drop tower] and static loads. One might draw the conclusion from these results that there is, therefore, no load-duration effect. This conclusion would be based on incomplete information. Drop-tower tests and static loading machines both produce loads which have durations far greater than the time required for brittle fracture mechanisms to operate. It is conceivable, therefore, that significant amounts of energy are being wasted by both static and present dynamic loading devices. This possibility is investigated in Section 2. An underwater electrical discharge system which is capable of generating very shortduration (5 ~sec) loads is described in Section 3. Some preliminary results are presented in Section 4 and the author's conclusions may be found in Section 5. 2. SHORT-DURATION LOADING 2.1 Power vs. energy as failure criterion
One may ask if there is an optimum power (energy transmission rate) for the transmission of a given amount of energy into a brittle material so as to cause fracture. This is equivalent to asking if there is any difference in the fracturing produced by a high-intensity, short-duration pulse as opposed to a lower-intensity, long-duration pulse, where the product of intensity and duration is constant. One may also ask if the amount of material breakage is proportional to the magnitude of the stress rather than the total energy imparted to the material. That is, for a load of a given amplitude, does the amount of fracturing continue to increase with the increased duration of the load ? In order to keep the relationships between these various quantities clear, it is recalled 191
192
M . H . MILLER
that the strain energy in the interior o f l o a d e d h o m o g e n e o u s isotropic elastic material is defined by e
½a~jeijdV
.....
+
V
2E
aija~j
dV
(1)
where 0 ~ cq~ - - ~ll + e22 + ea3. We define an effective stress ~ by the e q u a t i o n :
o, [
2E=
+
2E
1
crij cT~j
(2)
a n d assume that each stress c o m p o n e n t is uniform t h r o u g h o u t the volume o f interest. Therefore, if2
e := 2E A V
(3)
where c~ij eij v e c~ E /x V
.... = == ---
stress c o m p o n e n t s strain c o m p o n e n t s Poisson's ratio strain energy 'effective' stress elastic m o d u l u s volume o f material being stressed.
Henceforth, the discussion will be limited to a one-dimensional stress field caused by o n e - d i m e n s i o n a l p r o p a g a t i o n o f a longitudinal stress wave. This restriction will not limit the generality o f the conclusions. The effective stress reduces to actual stress for a uniaxial stress field. The mean p o w e r is defined by P e/r (4) where T - d u r a t i o n o f loading. The volume being stressed by a pulse is a function o f the d u r a t i o n o f the load. This may be expressed by ± v --- K T
(5)
where K is a function o f the geometry o f the p r o b l e m but is not a function o f the duration. Therefore, K e - - 2E ~2T (6) and K P = ~2E ~2.
(7)
F o r a one-dimensional stress field and wave p r o p a g a t i o n , if the l o a d is a p p l i e d over an area, A, on the surface o f the material, K
Ac
where c - : velocity o f wave p r o p a g a t i o n in the material. Therefore, e -~ P T
THE EFFECT OF STRESS WAVE DURATION ON BRITTLE FRACTURE
and
Ac
2
P - 2E ~ "
193 (8)
2.2 Failure in an incremental volume Consider a volume element of material just large enough to contain the incipient cracks needed for Griffith's failure criterion, or large enough to allow a continuum point of view as required for application of the C o l o u m b - M o h r fracture criteria. Within this volume element, failure will occur when a component of stress reaches a critical value. For Griffith's criterion, the magnitude of the stress must be sufficiently large so that the decrease in the strain energy caused by an infinitesimal extension of an incipient crack is greater in absolute value than the increase in the new surface energy required for this infinitesimal extension. F o r the modified C o u l o m b - M o h r criteria [1], failure will occur when a combination of stresses known as the virtual shear stress reaches a critical value or when a tensile stress reaches the tensile strength of the material. Similarly, all other failure theories can be described by defining suitable critical stresses. For a non-homogeneous material such as rock, this critical stre~s will vary from point to point, which has led various investigators to discuss a statistical (usually Weibull) distribution of critical stresses. Nevertheless, failure will commence when the local critical stress is exceeded. That is, as each elemental volume is loaded it may or may not fail depending upon whether its critical stress is exceeded, and its behaviour will not affect the possibility of failure of some other incremental volume provided that the magnitudes of the transmitted stresses are unchanged. If a sufficient number of incremental volume elemenls fail, then we consider the material to have failed. Insofar as the individual volume elements are concerned, the amplitude of an approaching stress pulse determines if failure will occur and the duration of the pulse is of no consequence. As the loading of an entire body can be considered to be a sequential loading of a number of elemental volumes, the failure of the entire body is also dependent upon the magnitude of the stress wave and not its duration, provided that the duration is sufficiently long to allow crack propagation. It will be shown in Section 2.3 that even durations down to 5 ~sec will allow significant crack propagation. Thus, referring to equations (6) and (7), it is the stress, and hence the power, and not the energy of a stress wave which governs fracture. In a real material, the magnitude of a stress wave will be a~tenuated by dispersion, frictional dissipation, and by entrapment as some of the wave energy is caught in a volume which has broken away from the main body. In this situation fracture will continue to occur as long as the magnitude of the stress wave is larger than the local critical stress. Again, the duration of the wave is of no consequence. Thus, it would seem to be preferable to start a wave with as large an amplitude as possible so as to obtain a maximum amount of breakage. This leads to the conclusion that for a given amount of energy, maximum breakage is obtained by imparting this energy to the rock in a wave which has the largest amplitude and, hence, shortest duration possible. There is, however, a limit to the stress amplitude which is desirable. If this amplitude far exceeds the largest critical stress value found in the entire body and the duration is very short, then the first volume element which is encountered will definitely fail and the entire wave may be trapped in this volume element. This first volume element will be pulverized, but no more damage will be done to the remainder of the rock. It should also be noted that short-duration waves probably suffer viscous-type dissipation to a greater extent than longduration waves, as such losses tend to increase with frequency.
194
M. t l . M I L L I : R
2.3 Failure clue to c o m p r e s s i r e w a r e Since, for simplicity, we are limiting our discussion to one dimension, we consider a plane wave which has an initial compressive a m p l i t u d e or0 a n d a d u r a t i o n T which is travelling along a r o d made up o f a n u m b e r o f volume elements as shown in Fig. 1. The width o f 4
cT - -
-~i
Sc ~
Sc z
Sc 3
hl
Fl~. 1. Stress wave entering a rod. the rod, h, is d e t e r m i n e d by the distance that a crack can p r o p a g a t e during the time in which its immediate n e i g h b o u r h o o d is stressed by a travelling stress wave. Thus h -- CeT
where Ce - - velocity o f crack p r o p a g a t i o n . If we assume that the lower b o u n d on physically o b t a i n a b l e stress wave d u r a t i o n s is 5 l~sec and we use the crack p r o p a g a t i o n velocity f o u n d in the literature [2] for glass, then the m i n i m u m width will be h = 2.19 km/sec 5 ~( 10 6 sec
10.95 mm.
Each volume element has a characteristic length (.,~i) and a critical stress, in this case the compressive strength Sei. W h e n the wave enters volume / / 1 , if ~0 --< S,:l failure will not occur a n d the wave will pass on to volume # 2 . In the process o f travelling t h r o u g h volume / / I , the a m p l i t u d e will have been a t t e n u a t e d to C~h a n d so failure in volume # 2 will occur if~x ~ Sc2. This process will continue until the a m p l i t u d e o f the a t t e n u a t e d wave exceeds the compressive strength o f some volume element and so failure will occur.
Vol. ~I
T
Vol. #2
Scq . . . .
FI~. 2. Stress wave fracture. Let us now assume t h a t this failure occurs in volume ~ 1 , that is, that ~r0 ~:- Sel. T h a t part o f the stress wave which precedes the p o i n t c~ - - S~1 (shown as point 1 in Fig. 2) will pass into volume # 2 before failure occurs in volume ~ 1 . This leading part o f the stress wave will continue to travel d o w n the b o d y a n d it m a y do a d d i t i o n a l d a m a g e if it encounters a weaker volume element.
T H E EFFECT OF STRESS W A V E D U R A T I O N ON BRITTLE FRA(_TURE
195
The remainder of the stress wave will cause fracturing to occur in volume # 1 . Again, for the sake of simplicity, we will assume that all of the fracturing is concentrated on the boundary between volume elements 1 and 2. If, as a result of this fracturing, a void develops between volume ~ 1 and volume ~ 3 , then volume ~ 1 will be bounded by a free surface and the remainder of the initial compressive stress wave will be reflected from this free surface as a tensile wave which will unload volume ~/1. If the duration of this initial wave is sufficiently short, the wave wilt go into net tension, and the rock in volume ~ 1 , which is much weaker in tension than in compression, will be pulverized. This is shown in Fig. 3. Thus a very large-amplitude, very short-duJation pulse could be entirely trapped in volume ~1.
Vol. '#1
¢//"if FIG. 3. R e f l e c t e d w a v e ,
Now let us consider the possibility that the initial fracture will occur along the interface between volumes 1 and 2, but that this volume will remain in contact. In this case the remainder of the stress wave which follows point ~ 1 will pass essentially unchanged into volume # 2 . As some energy is required to cause the initial fracture, we may assume that the wave will be slightly attenuated as it enters volume ~ 2 , but this fracture energy is probably very small [5] compared to the initial energy in the wave, and so the wave will not be greatly attenuated. We have now considered the case in which essentially all of the wave passes to the next volume element and the case where none of the wave passes to the next element. As is always the case in real life, the truth lies somewhere between these two extremes. We expect that when a fracture occurs, the wave following point ~ 1 will be partially reflected and partially refracted, secondary shear waves will form, and energy will be lost in dissipation, dispersion and in creating new surfaces. In the midst of these many alternative phenomena, let us concentrate our attention on the initial compressive wave. As it travels down the rod generating new waves and creating new surfaces, it is constantly being attenuated. Thus to maximize total rock breakage, we again conclude that we should maximize the initial amplitude of the wave. And what of the wave's duration? At best, the duration is of no consequence, as was previously suggested. We can consider here the wave which has some duration (any duration) which is continually attenuated as it travels along the rod until it can no longer cause even the weakest volume element to fail.
196
M . H . MILLER
At worst, however, a long duration can cause useful energy to be lost. After the initial fracture occurs, the shear and tension waves which probably follow will cause additional fractures to occur in the same neighbourhood or else cause the initial crack to widen. Sooner or later an essentially free boundary will be presented to the oncoming remainder of the initial compressive wave and this remainder will then be trapped. Only if the duration of the initial wave is relatively short, can most of the wave possibly pass to the next incremental volume before the free surface can sufficiently develop. Thus, we conclude that a high-amplitude, short-duration wave would probably maximize failure due to a compressive load. 2.4 Failure due to reflected ware Let us now consider a compressive wave which has travelled down a rod passing through a number of volume elements which are too strong for it to break, until it reaches a sufficiently weak element. A fracture then occurs and some part of the wave is then reflected. We will assume that the fracture has generated an essentially free surface and so the reflected wave will be tensile. In this section we will concentrate our attention on this reflected tensile wave. The magnitude of the reflected wave is proportional to the magnitude of the initial compressive wave. As the previous comments about alnplitude decay due to dispersion, dissipation and entrapment of compressive waves also applies to the reflected tensile wave, it is again concluded that maximum rock breakage generally is obtained by maximizing the amplitude of the initial compressive wave. With regard to the duration of the reflected wave, the shorter the duration, the sooner the stressed volume will go into net tension. This is important because, even though the tensile wave is travelling through volume elements which the compressive wave just failed to fracture, brittle materials such as rock are much weaker in tension than in compression and so the material may very well begin to fail as soon as the wave achieves net tension. Even after the wave is entirely reflected and hence is entirely tensile, a long duration will just increase the possibility of entrapment of the end of the wave. When fracture is caused by the leading edge of the tensile wave, a reflccting surface will begin to grow. The trailing end of the tensile wave will then be again reflected, probably as an essentially compressive wave which will accomplish absolutely no additional cracking until it is again reflected as a tensile wave, by which time it may be entirely dissipated. Therefore, we may conclude that the best type of reflected tensile wave also has a high amplitude and short duration. The duration of the reflected wave is not determined only by the duration of the incident wave. One must also take into account the generation of the reflecting surface as was discussed in Section 2.3. In Fig. 2 that part of the compressive incident wave in front of point 7//I will not be reflected and hence the duration of the reflected wave would be shortened.
3. E X P E R I M E N T A L
SET-UP
Typical mechanical devices such as percussive drills will load the rock for a period of 300-1000 ~sec. This load duration is governed by the length of the bit which strikes the rock, and the boundary conditions of the bit, particularly at the bit-rock interface. As was discussed in Section 2, it is possible that the duration of loading may influence the specific fracture energy (energy required per volume of rock removed). The discussion in
197
THE EFFECT OF STRESS WAVE D U R A T I O N ON BRITTLE FRACTURE
Section 2 indicates that it would be desirable to load the rock for a much shorter period of time than is done by a percussive drill. MARTIN [3] has shown that an underwater electrical discharge (hydrospark) can produce a shockwave which has a duration of only 5 ~sec. This period can be extended by increasing the inductance in the discharge circuit which would permit the growth of a spark bubble and subsequently permit a high pressure to endure for up to a millisecond. Thus, the hydrospark could be considered as a research tool, a sort of unique high-speed compression machine which is capable of investigating the possibly large saving in energy which is indicated by the arguments presented in Section 2. It was, therefore, decided that we would build a 1 ~F, 100 kV underwater discharge system. This would yield 5000 J per discharge. It was further decided that we would begin by using a relatively low power (550 W input to capacitors) system. Although this decision eliminated the possibility of studying the effect of high repetition rate, it was felt that such a system would be adequate to investigate the damage caused by a single discharge. The general circuit is shown in Fig. 4, which is both a schematic drawing and an indication of the actual physical positions of the various components. The discharge circuit designed for this programme is shown in Fig. 5. The capacitor
°l°
s,
PWR. SUPPLY
-i-c,
T
_.L - GI
FIG. 4. General circuit.
I
oloSi
-I-c,
ii__t
E
t
o-
....
--220
El
-
GI
FIG. 5. O u t d o o r d i s c h a r g e circuit.
bank, Cx, consisted of two capacitors, each rated at 2 ~tF and 50 kV. These capacitors were connected in series yielding 1 ~F at 100 kV. The leads from the capacitor through the air gap switch, $1, and to the discharge electrodes, Ea, were 1-ft wide copper sheet, which had a thickness of 0.075 in. This sheet conductor was used to minimize inductance and resistance. All connexions in the discharge circuit leads were silver soldered.
198
M.H. MILLER
The discharge switch, $1, is essentially an insulating lucite plate held between two copper cylinders. Each cylinder had a 3-in. outer diameter which prevented corona at 100,000 V. The lucite plate was k-in. thick, which insured that it would not break down under a 100 kV potential and its other dimensions were 18 in. × 24 in. in order to prevent a leak from the high-voltage side to ground before the insulating sheet was removed. The underwater discharge electrodes were made of 1-in. diameter copper bar stock which was machined to a point at the ends. They were attached to the copper sheet leads by gradually reducing the effective width of the copper sheet leads, thereby minimizing inductance, and joined by silver soldering. The gap between the electrodes was pre-set before each test. This distance was usually between ~ in. and 1 in. The electrodes were immersed in water in a tank and, depending upon the experiment, a rock sample might also be placed in the tank. An alternate discharge circuit is also shown in Fig. 5. A coaxial cable type RG19AU, 4~-in. diameter inner conductor and ~-in. insulation, was attached to a lug on the highvoltage electrode lead. The ground shield of this cable was connected directly to the system ground. A Teflon insulating sheet was placed between the indoor electrodes, El, in order to prevent a discharge at this point. With the system in this configuration, when switch $1 was closed, the inner conductor of the coaxial cable assumed a potential of 100 kV above the shield conductor of that cable. The end of this cable, where the discharge occurred, was immersed in a hole previously drilled in a rock specimen and filled with water. This system could be converted back to its original 'short-lead' configuration by simply removing the Teflon insulator, 11, and breaking the connexion at the lug on the highvoltage electrode lead. 3.1 Instrumentation The instrumentation system used to monitor the discharge current is shown in Fig. 6. When dielectric breakdown occurs in the air gap and water gap, these gaps begin to conduct and so they can be considered to be resistors, Re and R3, which have time-dependent resistances. As the caphcitor discharges, the current in the circuit will oscillate and will result in a damped sine wave. This current can be monitored by measuring the voltage drop across a resistor in the circuit. This 'sampling resistor' must, of necessity, be very small, as the current amplitude can reach 100,000 A, which would provide a very large voltage drop across even a very small resistor. It was, therefore, decided to monitor the voltage drop across a very small piece of the copper sheet lead. It was not possible to determine the actual resistance of this 'sample resistor' even under d.c. conditions. The voltage drop across this resistor, R4, was monitored by using a Tektronix 555 oscilloscope with a high-frequency, fast rise-time L plug-in. In order to insure an impedance match in the instrumentation lead from resistor R4 to the oscilloscope, a 50 ~ impedance coaxial cable was used and this was terminated at both ends by 50 ~--)resistors, R5 and R6. The oscilloscope was then connected to the terminal J1. The results of these tests are discussed in Section 4. After a number of attempts, efforts to directly measure, by means of a piezoelectric transducer, the pressure in the liquid surrounding the discharge were terminated because of the interference in the probe leads due to the very large electromagnetic field generated by the discharge. Previous investigators have used the plastic flow of a thin metal plate [4] and velocity measurements [3] as an indication of the shock pressure, but these methods were not considered to be applicable to the present experimental set-up.
THE EFFECTOF STRESSWAVE DURATIONON BRITTLEFRACTURE
L
199
o'vv~ R2 R3
-CI m
R5 R4
CABLE
~x/x,
I
R6
FIG. 6. Instrumentation circuit. 4. EXPERIMENTAL RESULTS 4.1 Current measurement The circuits shown in Fig. 6 produce very good pictures of the current wave. One example of the 'short-lead' discharge is shown in Fig. 7. It can be seen in Fig. 7 that the ringing fi'equency is
f~150
kc.
Since 1 f : 2~r~/LC
and C:
10-6F
then L=
1.1 × 10 - 6 H .
The envelope of the current wave is given by = exp(
Rt/2L)
200
M.H. MILLER
where R = Rz q- R3 + R4. If we assume that R is a constant 11 I2
R exp [-- 2L (tl
t,~)].
From Fig. 7 9
4.5 = exp [
(-20
× 10-6)].
Hence R =
1.1 ~ 10 6 10 -'~, (0-694) = 0.0764 ~.
The cable discharge rig produced current wave forms such as that shown in Fig. 8. In this case the ringing frequency has been reduced to f - , : 83.3 kc. Thus, the inductance has been increased to L
3.62 x 10 -6 H.
The ratio of amplitude is
5 1 ::-exp[
R( 2L
48 "J 10-6)]. ....
Hence R = 1.61
7.24 x 10 -6 = 0.234 ~2. 48 × 10 -6
It is seen that both the inductance and resistance of the cable discharge circuit are approximately three times larger than in the short-lead circuit. The primary difference between these two circuits is the 20 ft o f coaxial cable. 4.2 Outdoor drilling tests In September of 1964, initial tests were begun with the cable discharge circuit shown in Fig. 5. A hole approximately l~-in, diameter and 3½-in. deep was excavated in a piece of Reserve Taconite using an oxyacetylene torch. The actual profile o f this hole is shown as the solid line curve in Fig. 9. A series of three tests was then conducted in order to determine the geometry of failure using the underwater discharge. The first test consisted of 10 discharges, each at 25 kV. The results of this first test are shown in Fig. 9 also, and it is seen in this figure that only a small a m o u n t of rock was removed in this first attempt. The second test consisted o f 60 discharges, again at 25 kV per discharge. It was noted that the cable was moved a little bit by each discharge and it took between 5 and 8 discharges to knock it completely out of the hole. Since the electrode was being moved away from the hole b o t t o m during these tests, it must be assumed that a great deal of energy was harmlessly dissipated in the water and never reached the rock walls. The results of this test are also shown in Fig. 9. It should also be noted that small cracks seemed to be propagating out from the hole on the surface o f the rock.
I
0
~-~ •J '
r
o
LEGEND ORIGINALHOLE TEST ONE TESTTWO TESTTHREE
°~
/
f
i
l
SCALE
e
s
.
.
i-"'" - ' ~ f " ~.~LL
FIG. 9. Successive hole p
- "
.
,
°
t,J
>
© Z
z
<
m
,-.1 0 ~n
rn r~
,.-]
202
M . H . MILLER
The third and final test consisted of 100 discharges again at 25 kV. In this last test a weight was used to prevent the cable from bouncing out over the hole. The results are also seen in Fig. 9. A test was then undertaken to directly measure the specific fracture energy of taconite. One-hundred discharges at 25 kV took place with a total change in rock volume of 50 cm a. This produced a specific fracture energy of 625 J/cm 3 or 90,7000 psi. Tests have indicated that the compressive strength of this rock is in the order of 50,000 psi.* By way of comparison, it should be noted that the Russian investigator, TITKOV [6] reported a specific fracture energy of 415 J/cm 3 (60,000 in lb/in a) for dolomite, which has a compressive strength of 14,200 psi. It would, therefore, seem that the present results are at least as good as, if not better than, the Russian experiments based upon the ratio of specific fracture energy (energy required to break a unit volume of rock) to compressive strength. Of course the Russian experimenters were able to advance their experimental programme to the point of designing and constructing a prototype drill head and were thus able to obtain actual drilling rates. It is particularly important to note that investigators such as MARTIN [3] and FARMER and ATTEWELL [4] have indicated that at most 5 per cent of the energy stored on the capacitors will actually be transferred into the shock wave. This implies that the actual specific fracture energy of taconite, using the hydrospark, is 31 J/cm a, or 4500 psi. As the specific fracture energy using standard mechanical Ioadings is considered to be in the order of the compressive strength of a rock, it is seen that the hydrospark seems capable of breaking rock with far less expenditure of mechanical energy than mechanical drills. 5. CONCLUSIONS A qualitative examination of the failure caused in a brittle material by the passage of a stress wave has led to the conclusion that specific fracture energies can be greatly reduced by reducing the duration of the load. It has been experimentally determined that the specific fracture energy (based upon input power) using the hydrospark rig is of the same order of magnitude as the results obtained by the Russians and is roughly equivalent to the energy required to fracture rock using present-day mechanical devices. If one notes that there is probably a very large energy loss in the air gap switch and that perhaps only 5 per cent of the remaining energy will end up in the shock wave, then the mechanical energy supplied to the rock per unit volume broken becomes much smaller with the hydrospark method than with present-day methods. Although this indicates that there is indeed an advantage to a short-duration pressure pulse, the technical problems of reducing the loss in the air gap switch and in increasing the amount of power which goes into the shock wave are indeed formidable. Section 2 reviews the arguments for using a short-duration high-intensity pressure pulse. It was noted that a short-duration pulse is preferable for compressive failure of rock. This should be very important in either saving energy or increasing penetration rate in mechanical drills. It was also noted that a short-duration pulse was preferable when fracture is to be obtained by tensile failure after the reflection of a compressive wave. This is very important in blasting, excavation and many continuous mining concepts. The experimental rig designed and constructed as part of this project can well be considered as a basic tool for investigating the effects of short-duration loads on materials. * Private correspondence from James Paone, Minneapolis Mining Research Center, Bureau of Mines.
THE EFFECT OF STRESS WAVE DURATION ON BRITTLE FRACTURE
203
T o be m o s t effective, s o m e a d d i t i o n a l effort w o u l d be r e q u i r e d in o r d e r to a c c u r a t e l y m e a s u r e t h e d u r a t i o n a n d i n t e n s i t y o f the p r e s s u r e pulse.
Acknowledgement--The author wishes to acknowledge the suggestions of Dr. BURTONPAUL, of the IngersollRand Company who kindly reviewed the text. REFERENCES 1. PAUL B. A modification of the Coulomb-Mohr theory of fracture, J. appl. Mech. 28, 259-268 (1961). 2. IRWIN G. R. Fracture, Encyclopedia of Physics, Vol. VI (1958). 3. MARTIN E. A. Experimental investigation of a high-energy density, high-pressure arc plasma, J. appl. Phys. 31,255-267 (1960). 4. FARMERI. and ATTEWELLP. Experiments with water as a dynamic pressure medium, Mine Quarry Engng 29, 524-530 (1963). 5. SIMON R. Energy balance in rock drilling, Proceedings of the First Conference on Drilling and Rock Mechanics, University of Texas, January (1963). 6. TITKOV N. I. et al. Drilling by means of electric discharge in a liquid medium, Neft. Khoz. 35 (10) 5-10 (1957).
THE EFFECT OF STRESS WAVE D U R A T I O N ON BRITTLE F R A C T U R E M. H. MILLER Ingersoll-Rand Research Center, Bedminster, New Jersey (Received 31 January 1966)
Abstract-A qualitative theoretical examination of the fracture of heterogeneous brittle materials by a stress wave indicates that the duration of the wave significantly influences the amount of energy needed to fracture a unit volume of material. It is concluded that the specific fracture energy will be much lower for a high-amplitude, short-duration wave than it is for the loads imparted by mechanical methods presently used for drilling rock. The results of experiments employing an underwater discharge to produce high-intensity 5 tzsecpressure pulses seem to confirm the theoretical conclusions. 1. INTRODUCTION ALTHOUGH a number of investigators have given considerable attention to the effects of load magnitude on the failure of brittle materials, very little attention has been devoted to the effects of load duration. There has been some work which has shown that the specific fracture energy (energy required to fracture a unit volume of material) is essentially the same for dynamic [drop tower] and static loads. One might draw the conclusion from these results that there is, therefore, no load-duration effect. This conclusion would be based on incomplete information. Drop-tower tests and static loading machines both produce loads which have durations far greater than the time required for brittle fracture mechanisms to operate. It is conceivable, therefore, that significant amounts of energy are being wasted by both static and present dynamic loading devices. This possibility is investigated in Section 2. An underwater electrical discharge system which is capable of generating very shortduration (5 ~sec) loads is described in Section 3. Some preliminary results are presented in Section 4 and the author's conclusions may be found in Section 5. 2. SHORT-DURATION LOADING 2.1 Power vs. energy as failure criterion
One may ask if there is an optimum power (energy transmission rate) for the transmission of a given amount of energy into a brittle material so as to cause fracture. This is equivalent to asking if there is any difference in the fracturing produced by a high-intensity, short-duration pulse as opposed to a lower-intensity, long-duration pulse, where the product of intensity and duration is constant. One may also ask if the amount of material breakage is proportional to the magnitude of the stress rather than the total energy imparted to the material. That is, for a load of a given amplitude, does the amount of fracturing continue to increase with the increased duration of the load ? In order to keep the relationships between these various quantities clear, it is recalled 191
192
M . H . MILLER
that the strain energy in the interior o f l o a d e d h o m o g e n e o u s isotropic elastic material is defined by e
½a~jeijdV
.....
+
V
2E
aija~j
dV
(1)
where 0 ~ cq~ - - ~ll + e22 + ea3. We define an effective stress ~ by the e q u a t i o n :
o, [
2E=
+
2E
1
crij cT~j
(2)
a n d assume that each stress c o m p o n e n t is uniform t h r o u g h o u t the volume o f interest. Therefore, if2
e := 2E A V
(3)
where c~ij eij v e c~ E /x V
.... = == ---
stress c o m p o n e n t s strain c o m p o n e n t s Poisson's ratio strain energy 'effective' stress elastic m o d u l u s volume o f material being stressed.
Henceforth, the discussion will be limited to a one-dimensional stress field caused by o n e - d i m e n s i o n a l p r o p a g a t i o n o f a longitudinal stress wave. This restriction will not limit the generality o f the conclusions. The effective stress reduces to actual stress for a uniaxial stress field. The mean p o w e r is defined by P e/r (4) where T - d u r a t i o n o f loading. The volume being stressed by a pulse is a function o f the d u r a t i o n o f the load. This may be expressed by ± v --- K T
(5)
where K is a function o f the geometry o f the p r o b l e m but is not a function o f the duration. Therefore, K e - - 2E ~2T (6) and K P = ~2E ~2.
(7)
F o r a one-dimensional stress field and wave p r o p a g a t i o n , if the l o a d is a p p l i e d over an area, A, on the surface o f the material, K
Ac
where c - : velocity o f wave p r o p a g a t i o n in the material. Therefore, e -~ P T
THE EFFECT OF STRESS WAVE DURATION ON BRITTLE FRACTURE
and
Ac
2
P - 2E ~ "
193 (8)
2.2 Failure in an incremental volume Consider a volume element of material just large enough to contain the incipient cracks needed for Griffith's failure criterion, or large enough to allow a continuum point of view as required for application of the C o l o u m b - M o h r fracture criteria. Within this volume element, failure will occur when a component of stress reaches a critical value. For Griffith's criterion, the magnitude of the stress must be sufficiently large so that the decrease in the strain energy caused by an infinitesimal extension of an incipient crack is greater in absolute value than the increase in the new surface energy required for this infinitesimal extension. F o r the modified C o u l o m b - M o h r criteria [1], failure will occur when a combination of stresses known as the virtual shear stress reaches a critical value or when a tensile stress reaches the tensile strength of the material. Similarly, all other failure theories can be described by defining suitable critical stresses. For a non-homogeneous material such as rock, this critical stre~s will vary from point to point, which has led various investigators to discuss a statistical (usually Weibull) distribution of critical stresses. Nevertheless, failure will commence when the local critical stress is exceeded. That is, as each elemental volume is loaded it may or may not fail depending upon whether its critical stress is exceeded, and its behaviour will not affect the possibility of failure of some other incremental volume provided that the magnitudes of the transmitted stresses are unchanged. If a sufficient number of incremental volume elemenls fail, then we consider the material to have failed. Insofar as the individual volume elements are concerned, the amplitude of an approaching stress pulse determines if failure will occur and the duration of the pulse is of no consequence. As the loading of an entire body can be considered to be a sequential loading of a number of elemental volumes, the failure of the entire body is also dependent upon the magnitude of the stress wave and not its duration, provided that the duration is sufficiently long to allow crack propagation. It will be shown in Section 2.3 that even durations down to 5 ~sec will allow significant crack propagation. Thus, referring to equations (6) and (7), it is the stress, and hence the power, and not the energy of a stress wave which governs fracture. In a real material, the magnitude of a stress wave will be a~tenuated by dispersion, frictional dissipation, and by entrapment as some of the wave energy is caught in a volume which has broken away from the main body. In this situation fracture will continue to occur as long as the magnitude of the stress wave is larger than the local critical stress. Again, the duration of the wave is of no consequence. Thus, it would seem to be preferable to start a wave with as large an amplitude as possible so as to obtain a maximum amount of breakage. This leads to the conclusion that for a given amount of energy, maximum breakage is obtained by imparting this energy to the rock in a wave which has the largest amplitude and, hence, shortest duration possible. There is, however, a limit to the stress amplitude which is desirable. If this amplitude far exceeds the largest critical stress value found in the entire body and the duration is very short, then the first volume element which is encountered will definitely fail and the entire wave may be trapped in this volume element. This first volume element will be pulverized, but no more damage will be done to the remainder of the rock. It should also be noted that short-duration waves probably suffer viscous-type dissipation to a greater extent than longduration waves, as such losses tend to increase with frequency.
194
M. t l . M I L L I : R
2.3 Failure clue to c o m p r e s s i r e w a r e Since, for simplicity, we are limiting our discussion to one dimension, we consider a plane wave which has an initial compressive a m p l i t u d e or0 a n d a d u r a t i o n T which is travelling along a r o d made up o f a n u m b e r o f volume elements as shown in Fig. 1. The width o f 4
cT - -
-~i
Sc ~
Sc z
Sc 3
hl
Fl~. 1. Stress wave entering a rod. the rod, h, is d e t e r m i n e d by the distance that a crack can p r o p a g a t e during the time in which its immediate n e i g h b o u r h o o d is stressed by a travelling stress wave. Thus h -- CeT
where Ce - - velocity o f crack p r o p a g a t i o n . If we assume that the lower b o u n d on physically o b t a i n a b l e stress wave d u r a t i o n s is 5 l~sec and we use the crack p r o p a g a t i o n velocity f o u n d in the literature [2] for glass, then the m i n i m u m width will be h = 2.19 km/sec 5 ~( 10 6 sec
10.95 mm.
Each volume element has a characteristic length (.,~i) and a critical stress, in this case the compressive strength Sei. W h e n the wave enters volume / / 1 , if ~0 --< S,:l failure will not occur a n d the wave will pass on to volume # 2 . In the process o f travelling t h r o u g h volume / / I , the a m p l i t u d e will have been a t t e n u a t e d to C~h a n d so failure in volume # 2 will occur if~x ~ Sc2. This process will continue until the a m p l i t u d e o f the a t t e n u a t e d wave exceeds the compressive strength o f some volume element and so failure will occur.
Vol. ~I
T
Vol. #2
Scq . . . .
FI~. 2. Stress wave fracture. Let us now assume t h a t this failure occurs in volume ~ 1 , that is, that ~r0 ~:- Sel. T h a t part o f the stress wave which precedes the p o i n t c~ - - S~1 (shown as point 1 in Fig. 2) will pass into volume # 2 before failure occurs in volume ~ 1 . This leading part o f the stress wave will continue to travel d o w n the b o d y a n d it m a y do a d d i t i o n a l d a m a g e if it encounters a weaker volume element.
T H E EFFECT OF STRESS W A V E D U R A T I O N ON BRITTLE FRA(_TURE
195
The remainder of the stress wave will cause fracturing to occur in volume # 1 . Again, for the sake of simplicity, we will assume that all of the fracturing is concentrated on the boundary between volume elements 1 and 2. If, as a result of this fracturing, a void develops between volume ~ 1 and volume ~ 3 , then volume ~ 1 will be bounded by a free surface and the remainder of the initial compressive stress wave will be reflected from this free surface as a tensile wave which will unload volume ~/1. If the duration of this initial wave is sufficiently short, the wave wilt go into net tension, and the rock in volume ~ 1 , which is much weaker in tension than in compression, will be pulverized. This is shown in Fig. 3. Thus a very large-amplitude, very short-duJation pulse could be entirely trapped in volume ~1.
Vol. '#1
¢//"if FIG. 3. R e f l e c t e d w a v e ,
Now let us consider the possibility that the initial fracture will occur along the interface between volumes 1 and 2, but that this volume will remain in contact. In this case the remainder of the stress wave which follows point ~ 1 will pass essentially unchanged into volume # 2 . As some energy is required to cause the initial fracture, we may assume that the wave will be slightly attenuated as it enters volume ~ 2 , but this fracture energy is probably very small [5] compared to the initial energy in the wave, and so the wave will not be greatly attenuated. We have now considered the case in which essentially all of the wave passes to the next volume element and the case where none of the wave passes to the next element. As is always the case in real life, the truth lies somewhere between these two extremes. We expect that when a fracture occurs, the wave following point ~ 1 will be partially reflected and partially refracted, secondary shear waves will form, and energy will be lost in dissipation, dispersion and in creating new surfaces. In the midst of these many alternative phenomena, let us concentrate our attention on the initial compressive wave. As it travels down the rod generating new waves and creating new surfaces, it is constantly being attenuated. Thus to maximize total rock breakage, we again conclude that we should maximize the initial amplitude of the wave. And what of the wave's duration? At best, the duration is of no consequence, as was previously suggested. We can consider here the wave which has some duration (any duration) which is continually attenuated as it travels along the rod until it can no longer cause even the weakest volume element to fail.
196
M . H . MILLER
At worst, however, a long duration can cause useful energy to be lost. After the initial fracture occurs, the shear and tension waves which probably follow will cause additional fractures to occur in the same neighbourhood or else cause the initial crack to widen. Sooner or later an essentially free boundary will be presented to the oncoming remainder of the initial compressive wave and this remainder will then be trapped. Only if the duration of the initial wave is relatively short, can most of the wave possibly pass to the next incremental volume before the free surface can sufficiently develop. Thus, we conclude that a high-amplitude, short-duration wave would probably maximize failure due to a compressive load. 2.4 Failure due to reflected ware Let us now consider a compressive wave which has travelled down a rod passing through a number of volume elements which are too strong for it to break, until it reaches a sufficiently weak element. A fracture then occurs and some part of the wave is then reflected. We will assume that the fracture has generated an essentially free surface and so the reflected wave will be tensile. In this section we will concentrate our attention on this reflected tensile wave. The magnitude of the reflected wave is proportional to the magnitude of the initial compressive wave. As the previous comments about alnplitude decay due to dispersion, dissipation and entrapment of compressive waves also applies to the reflected tensile wave, it is again concluded that maximum rock breakage generally is obtained by maximizing the amplitude of the initial compressive wave. With regard to the duration of the reflected wave, the shorter the duration, the sooner the stressed volume will go into net tension. This is important because, even though the tensile wave is travelling through volume elements which the compressive wave just failed to fracture, brittle materials such as rock are much weaker in tension than in compression and so the material may very well begin to fail as soon as the wave achieves net tension. Even after the wave is entirely reflected and hence is entirely tensile, a long duration will just increase the possibility of entrapment of the end of the wave. When fracture is caused by the leading edge of the tensile wave, a reflccting surface will begin to grow. The trailing end of the tensile wave will then be again reflected, probably as an essentially compressive wave which will accomplish absolutely no additional cracking until it is again reflected as a tensile wave, by which time it may be entirely dissipated. Therefore, we may conclude that the best type of reflected tensile wave also has a high amplitude and short duration. The duration of the reflected wave is not determined only by the duration of the incident wave. One must also take into account the generation of the reflecting surface as was discussed in Section 2.3. In Fig. 2 that part of the compressive incident wave in front of point 7//I will not be reflected and hence the duration of the reflected wave would be shortened.
3. E X P E R I M E N T A L
SET-UP
Typical mechanical devices such as percussive drills will load the rock for a period of 300-1000 ~sec. This load duration is governed by the length of the bit which strikes the rock, and the boundary conditions of the bit, particularly at the bit-rock interface. As was discussed in Section 2, it is possible that the duration of loading may influence the specific fracture energy (energy required per volume of rock removed). The discussion in
197
THE EFFECT OF STRESS WAVE D U R A T I O N ON BRITTLE FRACTURE
Section 2 indicates that it would be desirable to load the rock for a much shorter period of time than is done by a percussive drill. MARTIN [3] has shown that an underwater electrical discharge (hydrospark) can produce a shockwave which has a duration of only 5 ~sec. This period can be extended by increasing the inductance in the discharge circuit which would permit the growth of a spark bubble and subsequently permit a high pressure to endure for up to a millisecond. Thus, the hydrospark could be considered as a research tool, a sort of unique high-speed compression machine which is capable of investigating the possibly large saving in energy which is indicated by the arguments presented in Section 2. It was, therefore, decided that we would build a 1 ~F, 100 kV underwater discharge system. This would yield 5000 J per discharge. It was further decided that we would begin by using a relatively low power (550 W input to capacitors) system. Although this decision eliminated the possibility of studying the effect of high repetition rate, it was felt that such a system would be adequate to investigate the damage caused by a single discharge. The general circuit is shown in Fig. 4, which is both a schematic drawing and an indication of the actual physical positions of the various components. The discharge circuit designed for this programme is shown in Fig. 5. The capacitor
°l°
s,
PWR. SUPPLY
-i-c,
T
_.L - GI
FIG. 4. General circuit.
I
oloSi
-I-c,
ii__t
E
t
o-
....
--220
El
-
GI
FIG. 5. O u t d o o r d i s c h a r g e circuit.
bank, Cx, consisted of two capacitors, each rated at 2 ~tF and 50 kV. These capacitors were connected in series yielding 1 ~F at 100 kV. The leads from the capacitor through the air gap switch, $1, and to the discharge electrodes, Ea, were 1-ft wide copper sheet, which had a thickness of 0.075 in. This sheet conductor was used to minimize inductance and resistance. All connexions in the discharge circuit leads were silver soldered.
198
M.H. MILLER
The discharge switch, $1, is essentially an insulating lucite plate held between two copper cylinders. Each cylinder had a 3-in. outer diameter which prevented corona at 100,000 V. The lucite plate was k-in. thick, which insured that it would not break down under a 100 kV potential and its other dimensions were 18 in. × 24 in. in order to prevent a leak from the high-voltage side to ground before the insulating sheet was removed. The underwater discharge electrodes were made of 1-in. diameter copper bar stock which was machined to a point at the ends. They were attached to the copper sheet leads by gradually reducing the effective width of the copper sheet leads, thereby minimizing inductance, and joined by silver soldering. The gap between the electrodes was pre-set before each test. This distance was usually between ~ in. and 1 in. The electrodes were immersed in water in a tank and, depending upon the experiment, a rock sample might also be placed in the tank. An alternate discharge circuit is also shown in Fig. 5. A coaxial cable type RG19AU, 4~-in. diameter inner conductor and ~-in. insulation, was attached to a lug on the highvoltage electrode lead. The ground shield of this cable was connected directly to the system ground. A Teflon insulating sheet was placed between the indoor electrodes, El, in order to prevent a discharge at this point. With the system in this configuration, when switch $1 was closed, the inner conductor of the coaxial cable assumed a potential of 100 kV above the shield conductor of that cable. The end of this cable, where the discharge occurred, was immersed in a hole previously drilled in a rock specimen and filled with water. This system could be converted back to its original 'short-lead' configuration by simply removing the Teflon insulator, 11, and breaking the connexion at the lug on the highvoltage electrode lead. 3.1 Instrumentation The instrumentation system used to monitor the discharge current is shown in Fig. 6. When dielectric breakdown occurs in the air gap and water gap, these gaps begin to conduct and so they can be considered to be resistors, Re and R3, which have time-dependent resistances. As the caphcitor discharges, the current in the circuit will oscillate and will result in a damped sine wave. This current can be monitored by measuring the voltage drop across a resistor in the circuit. This 'sampling resistor' must, of necessity, be very small, as the current amplitude can reach 100,000 A, which would provide a very large voltage drop across even a very small resistor. It was, therefore, decided to monitor the voltage drop across a very small piece of the copper sheet lead. It was not possible to determine the actual resistance of this 'sample resistor' even under d.c. conditions. The voltage drop across this resistor, R4, was monitored by using a Tektronix 555 oscilloscope with a high-frequency, fast rise-time L plug-in. In order to insure an impedance match in the instrumentation lead from resistor R4 to the oscilloscope, a 50 ~ impedance coaxial cable was used and this was terminated at both ends by 50 ~--)resistors, R5 and R6. The oscilloscope was then connected to the terminal J1. The results of these tests are discussed in Section 4. After a number of attempts, efforts to directly measure, by means of a piezoelectric transducer, the pressure in the liquid surrounding the discharge were terminated because of the interference in the probe leads due to the very large electromagnetic field generated by the discharge. Previous investigators have used the plastic flow of a thin metal plate [4] and velocity measurements [3] as an indication of the shock pressure, but these methods were not considered to be applicable to the present experimental set-up.
THE EFFECTOF STRESSWAVE DURATIONON BRITTLEFRACTURE
L
199
o'vv~ R2 R3
-CI m
R5 R4
CABLE
~x/x,
I
R6
FIG. 6. Instrumentation circuit. 4. EXPERIMENTAL RESULTS 4.1 Current measurement The circuits shown in Fig. 6 produce very good pictures of the current wave. One example of the 'short-lead' discharge is shown in Fig. 7. It can be seen in Fig. 7 that the ringing fi'equency is
f~150
kc.
Since 1 f : 2~r~/LC
and C:
10-6F
then L=
1.1 × 10 - 6 H .
The envelope of the current wave is given by = exp(
Rt/2L)
200
M.H. MILLER
where R = Rz q- R3 + R4. If we assume that R is a constant 11 I2
R exp [-- 2L (tl
t,~)].
From Fig. 7 9
4.5 = exp [
(-20
× 10-6)].
Hence R =
1.1 ~ 10 6 10 -'~, (0-694) = 0.0764 ~.
The cable discharge rig produced current wave forms such as that shown in Fig. 8. In this case the ringing frequency has been reduced to f - , : 83.3 kc. Thus, the inductance has been increased to L
3.62 x 10 -6 H.
The ratio of amplitude is
5 1 ::-exp[
R( 2L
48 "J 10-6)]. ....
Hence R = 1.61
7.24 x 10 -6 = 0.234 ~2. 48 × 10 -6
It is seen that both the inductance and resistance of the cable discharge circuit are approximately three times larger than in the short-lead circuit. The primary difference between these two circuits is the 20 ft o f coaxial cable. 4.2 Outdoor drilling tests In September of 1964, initial tests were begun with the cable discharge circuit shown in Fig. 5. A hole approximately l~-in, diameter and 3½-in. deep was excavated in a piece of Reserve Taconite using an oxyacetylene torch. The actual profile o f this hole is shown as the solid line curve in Fig. 9. A series of three tests was then conducted in order to determine the geometry of failure using the underwater discharge. The first test consisted of 10 discharges, each at 25 kV. The results of this first test are shown in Fig. 9 also, and it is seen in this figure that only a small a m o u n t of rock was removed in this first attempt. The second test consisted o f 60 discharges, again at 25 kV per discharge. It was noted that the cable was moved a little bit by each discharge and it took between 5 and 8 discharges to knock it completely out of the hole. Since the electrode was being moved away from the hole b o t t o m during these tests, it must be assumed that a great deal of energy was harmlessly dissipated in the water and never reached the rock walls. The results of this test are also shown in Fig. 9. It should also be noted that small cracks seemed to be propagating out from the hole on the surface o f the rock.
I
0
~-~ •J '
r
o
LEGEND ORIGINALHOLE TEST ONE TESTTWO TESTTHREE
°~
/
f
i
l
SCALE
e
s
.
.
i-"'" - ' ~ f " ~.~LL
FIG. 9. Successive hole p
- "
.
,
°
t,J
>
© Z
z
<
m
,-.1 0 ~n
rn r~
,.-]
202
M . H . MILLER
The third and final test consisted of 100 discharges again at 25 kV. In this last test a weight was used to prevent the cable from bouncing out over the hole. The results are also seen in Fig. 9. A test was then undertaken to directly measure the specific fracture energy of taconite. One-hundred discharges at 25 kV took place with a total change in rock volume of 50 cm a. This produced a specific fracture energy of 625 J/cm 3 or 90,7000 psi. Tests have indicated that the compressive strength of this rock is in the order of 50,000 psi.* By way of comparison, it should be noted that the Russian investigator, TITKOV [6] reported a specific fracture energy of 415 J/cm 3 (60,000 in lb/in a) for dolomite, which has a compressive strength of 14,200 psi. It would, therefore, seem that the present results are at least as good as, if not better than, the Russian experiments based upon the ratio of specific fracture energy (energy required to break a unit volume of rock) to compressive strength. Of course the Russian experimenters were able to advance their experimental programme to the point of designing and constructing a prototype drill head and were thus able to obtain actual drilling rates. It is particularly important to note that investigators such as MARTIN [3] and FARMER and ATTEWELL [4] have indicated that at most 5 per cent of the energy stored on the capacitors will actually be transferred into the shock wave. This implies that the actual specific fracture energy of taconite, using the hydrospark, is 31 J/cm a, or 4500 psi. As the specific fracture energy using standard mechanical Ioadings is considered to be in the order of the compressive strength of a rock, it is seen that the hydrospark seems capable of breaking rock with far less expenditure of mechanical energy than mechanical drills. 5. CONCLUSIONS A qualitative examination of the failure caused in a brittle material by the passage of a stress wave has led to the conclusion that specific fracture energies can be greatly reduced by reducing the duration of the load. It has been experimentally determined that the specific fracture energy (based upon input power) using the hydrospark rig is of the same order of magnitude as the results obtained by the Russians and is roughly equivalent to the energy required to fracture rock using present-day mechanical devices. If one notes that there is probably a very large energy loss in the air gap switch and that perhaps only 5 per cent of the remaining energy will end up in the shock wave, then the mechanical energy supplied to the rock per unit volume broken becomes much smaller with the hydrospark method than with present-day methods. Although this indicates that there is indeed an advantage to a short-duration pressure pulse, the technical problems of reducing the loss in the air gap switch and in increasing the amount of power which goes into the shock wave are indeed formidable. Section 2 reviews the arguments for using a short-duration high-intensity pressure pulse. It was noted that a short-duration pulse is preferable for compressive failure of rock. This should be very important in either saving energy or increasing penetration rate in mechanical drills. It was also noted that a short-duration pulse was preferable when fracture is to be obtained by tensile failure after the reflection of a compressive wave. This is very important in blasting, excavation and many continuous mining concepts. The experimental rig designed and constructed as part of this project can well be considered as a basic tool for investigating the effects of short-duration loads on materials. * Private correspondence from James Paone, Minneapolis Mining Research Center, Bureau of Mines.
THE EFFECT OF STRESS WAVE DURATION ON BRITTLE FRACTURE
203
T o be m o s t effective, s o m e a d d i t i o n a l effort w o u l d be r e q u i r e d in o r d e r to a c c u r a t e l y m e a s u r e t h e d u r a t i o n a n d i n t e n s i t y o f the p r e s s u r e pulse.
Acknowledgement--The author wishes to acknowledge the suggestions of Dr. BURTONPAUL, of the IngersollRand Company who kindly reviewed the text. REFERENCES 1. PAUL B. A modification of the Coulomb-Mohr theory of fracture, J. appl. Mech. 28, 259-268 (1961). 2. IRWIN G. R. Fracture, Encyclopedia of Physics, Vol. VI (1958). 3. MARTIN E. A. Experimental investigation of a high-energy density, high-pressure arc plasma, J. appl. Phys. 31,255-267 (1960). 4. FARMERI. and ATTEWELLP. Experiments with water as a dynamic pressure medium, Mine Quarry Engng 29, 524-530 (1963). 5. SIMON R. Energy balance in rock drilling, Proceedings of the First Conference on Drilling and Rock Mechanics, University of Texas, January (1963). 6. TITKOV N. I. et al. Drilling by means of electric discharge in a liquid medium, Neft. Khoz. 35 (10) 5-10 (1957).