Single-particle fracture under impact loading

Int. J. Miner. Process. 54 Ž1998. 1–28

Single-particle fracture under impact loading L.M. Tavares, R.P. King

)

UniÕersity of Utah, Comminution Center, 306 Browning Building, Salt Lake City, UT 84112, USA

Abstract Particle fracture is the elementary process that governs comminution. Understanding this elementary fracture process is indispensable to the development of improved size reduction techniques. The Ultrafast Load Cell ŽUFLC., has been used to investigate the deformation and fracture of single particles subject to impact. The theoretical background of the measurements is presented in detail. The Ultrafast Load Cell allows the measurement of three fundamental deformation and fracture characteristics of particulate materials: the particle fracture energy, the particle strength and the particle stiffness. These measurements can be made under impact loading similar to the conditions in typical industrial ball mills. The effect of material type, particle size and particle shape on the three fundamental fracture characteristics of brittle materials is investigated. The fracture characteristics were found to be independent of the rate of strain at rates expected to be found in industrial comminution equipment. Two practical applications of the measurements are presented and the use of the Ultrafast Load Cell to study thermally assisted comminution of quartz is discussed in detail. q 1998 Elsevier Science B.V. All rights reserved. Keywords: single-particle fracture; impact; particle fracture energy; fast load cell

1. Introduction Comminution of brittle particles is a complex process that has not yielded particularly well to quantitative theoretical treatment in spite of the research effort that has been devoted to this topic for more than a century. There are many difficulties that have hindered the development of a good understanding of the basic fracture mechanisms that underlie these important comminution processes. Because of the importance of the comminution industry, most researchers have understandably focussed attention on the process engineering of comminution, and the successes enjoyed by the population )

Corresponding author. Tel.: q1 Ž801. 585-3113; Fax: q1 Ž801. 581-8119; E-mail: [email protected]

0301-7516r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 1 - 7 5 1 6 Ž 9 8 . 0 0 0 0 5 - 2

2

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

balance method in providing a macroscopic description of the evolution of particle population statistics in comminution operations bear witness to this research effort. Important as it is to be able to describe the population statistics quantitatively, it is clear that a more detailed and fundamental understanding of the basic particulate fracture processes is required to ensure that the technology will develop in a way that will lead to greater operating efficiencies. This awareness has led to a resurgence of interest in fracture physics as it can be applied to particle breakage mechanisms. Fracture mechanics of brittle materials is a well-developed science and its application to particle failure has been fruitful in recent years. Comminution concerns the breakage of brittle particles under conditions of applied compressive stress. The nature of the failure mechanisms is governed by the material properties of the particulate material and by the nature of the stress field around and within individual particles. The response of the particulate material to the stress field is largely elastic but significant non-elastic behavior occurs, particularly at the tips of growing cracks where large quantities of energy are dissipated when the criteria for fracture are met. The dissipation at the crack tip of the stored elastic energy in the particle turns out to be of critical interest in industrial comminution machines where energy efficiency is of major consequence because of its economic importance. Industrial comminution processes are typically inefficient in their use of energy in the sense that considerably more energy is consumed by the operating equipment than is actually required to break the particles. In spite of the importance of this observation, it has not been possible to calculate precisely how much energy is actually required. Early attempts were invariably based on estimates of the increased particle surface area that is produced during comminution and the theoretical specific surface energy of the material. These estimates are inevitably low because of the impossibility of measuring particle surface area in any absolute sense and because considerable additional energy must be expended by inelastic deformation in very small volumes at the growing tips. Even Ž1988. has calculated that energy efficiencies are allowing for this latter effect Schonert ¨ in the range 4–8% when assessed on the basis of the surface area that is produced. Any useful assessment of energy efficiency should clearly be based on the minimum energy required to break the particles and not on the surface area produced. The specific particle fracture energy has been measured directly under conditions of slow compression, indirectly from breakage probability measurements under single point ballistic impact loading, and indirectly from breakage probability measurements in drop-weight tests ŽHildinger, 1969; Baumgardt et al., 1975; Krogh, 1980.. However, prior to the development of the UFLC it was not possible to measure particle fracture energy directly under two-point impact loading conditions. The accurate measurement of this important quantity is the main focus of this paper. It has been known for many years that the energy utilization, measured as surface area increase per unit of energy expended, is significantly greater under slow compressive loading than that measured under single point ballistic impact loading and, furthermore, the energy utilization increases significantly as the applied energy decreases, at least under slow compression ŽSchonert, 1972.. This has major potential ¨ significance for industrial comminution since it indicates that the best conditions for energy efficiency occur when the energy is applied slowly under low intensity Ž1 Jrg or

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

3

less. — certainly under conditions far removed from the type of impact experienced in a large ball mill where a 10-cm diameter steel ball falling 3 m would impact the load with a specific impact energy of approximately 15–50 Jrg Ž4–15 kWhrton.. By contrast under conditions of extremely rapid loading by ballistic impact, the optimum energy application is between 5 and 10 Jrg ŽSchonert, 1972. which is closer to energy loads ¨ found in ball mills. Since the average rate of loading typically experienced by particles in a ball mill must lie between the two extremes of slow compression and ballistic impact, it is important to establish the relationship between energy utilization and the intensity of energy application under conditions of rapid two-point impact loading. The meager data that are available indicate that the best energy efficiencies are found in the region between 0.1 and 2 Jrg ŽRumpf, 1973.. Furthermore these data suggest that energy utilization can vary by a factor of 2 or 3 as the energy intensity varies. This provides a strong motivation for measuring the particle fracture energy under precisely controlled two-point impact loading conditions and for investigating the mechanism of particle fracture under these conditions. Specific fracture energy of the particles is not the only fundamental property that is important: the particle strength also plays a significant role in determining the overall comminution properties of the material. A particle will be broken only if it is stressed beyond its strength which is determined by the intrinsic properties of the material, the presence of microflaws which act as stress raisers when the particle is under load and the state of stress that is experienced by the particle.

2. Single-particle fracture Brittle particles are fractured by compressive loading and the fundamental properties of the fracture process can be studied most effectively by well-controlled experiments on single particles. Single-particle fracture studies have provided the basis for particle comminution research since the 1960s, particularly in the United States ŽGilvarry and Bergstrom, 1961; Bergstrom, 1962. and in Germany ŽRumpf, 1973.. Three different types of compressive loading have been used in these experimental studies: slow controlled compressive loading in conventional mechanical testing machines, drop-weight loading and ballistic impact loading. Each of these methods allows investigation over a restricted range of strain rate and, because the size distribution of the progeny is dependent on the strain rate, the experimental method must be carefully chosen. The mechanism of the particle fracture process is largely understood and there is general consensus in the literature on the basic model description. From the instant of initial impact the particle is stressed and energy is stored as elastic strain energy in the particle. The small amount of plastic deformation at and around the loading points is generally considered to be insignificant. The deformation of the particle is generally considered to be described by the Hertzian model during this initial period. When the criteria for the failure of a brittle flaw within the body of or on the surface of the particle are met, the flaw grows unstably and increasingly rapidly as an advancing crack which may or may not branch. When the crack or cracks emerge from the particle, several

4

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

progeny particles are formed. The number and size of the progeny particles depends on the size and location of the initiating flaw or flaws and on the extent of crack branching that occurs. The initiation of unstable crack growth is governed by the Griffith criterion but conventional fracture mechanics techniques cannot be used in any quantitative way because of the great difficulty in calculating the stress field inside an irregularly shaped particle. Fracture mechanics dictates that if the crack grows unstably the energy release rate G must be greater than the crack resistance, R, which is twice the surface-specific fracture energy of the material ŽAnderson, 1995, p. 47; Lawn, 1995, p. 41; Cottrell and Mai, 1996, p. 31.. The essential feature of this model of particle fracture is that the energy required to drive the crack comes entirely from the stored elastic energy which is available at crack initiation, in other words, the particle fracture energy. This idea is based on the observation that no other energy source is available to deliver energy at the required rate during crack growth which is several orders of magnitude larger than the rate at which the falling weight is delivering energy to the particle. A necessary consequence of this model is that the stored particle fracture energy must be at least as large as the total energy that is dissipated at the growing crack tip. In general the particle fracture energy will exceed the dissipated energy by significant amounts and the unused energy is distributed after fracture is complete, mostly as kinetic energy of the progeny fragments which in turn can result in further breakage depending on the physical configuration of particle and stressing tools. Any calculation of comminution efficiency that is based on the product of surface area produced and the area-specific fracture energy can give only a lower bound and it is the particle fracture energy that should be used as the reference point for the efficiency calculations. The single-particle fracture process does not terminate after first failure at a flaw because the falling weight usually will still possess considerable kinetic energy which must be dissipated during the second stage of the process after the initial fracture of the particle. This is accomplished by secondary fracture of the initial progeny and possibly several further stages of sequential fracture as well. Thus the fracture event has two distinct stages. The first stage extends from the instant of impact until the instant of first fracture and the second stage extends from the instant of first fracture until all breakage of progeny particles has ended. If the initial energy of the falling weight exceeds the total energy required in stage 1 and stage 2, the excess energy is dissipated as wasteful steel-on-steel impact which is the primary cause of steel consumption in industrial comminution machines. The final size distribution that results from the impact is governed largely by what happens during stage 2. The UFLC can be used to study both stages independently but this paper is concerned exclusively with stage 1. The UFLC also has the flexibility to allow the study of comminution of particle beds. The UFLC is unique in allowing the single-particle fracture energy of individual particles to be measured under two-point impact loading. The fundamental importance of the particle fracture energy was first described by Baumgardt et al. Ž1975. and we demonstrate some of its practical uses in this paper. The measurement of the specific particle fracture energy in samples of industrially important materials under impact loading using the UFLC represents the first actual measurement of the amount of energy required for comminution under conditions approximating those

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

5

in industrial ball mills and at last provides an absolute measurement of this important quantity. It offers the possibility that empirical measures such as the Bond Work Index can be replaced by direct measurements of the fundamental material properties that govern industrial comminution processes. In this paper, the use of the UFLC to characterize the mechanical behavior of brittle materials up to fracture under impact loading is investigated. Applications of the measurements to comminution are presented and the use of the UFLC to study thermally assisted comminution of quartz is discussed as an example of the utility of these measurements. In the present work the particle fracture energy Žand the breakage probability function. is studied as a function of material composition, microstructure type, particle size and shape. Also, the particle stiffness, a measure related to the deformation behavior and the internal integrity of the particle is introduced and is shown to be particularly useful to assess the effect of pretreatment on comminution. The particle stiffness can also be used to quantify deviations of the particle behavior from the Griffiths model during stage one of the particle fracture process. Fragmentation and the relationship between progeny size distribution and the input or net absorbed energy will be subject of a future publication. The theoretical background of UFLC experiments as well as the sensitivity of the measurements to several factors such as particle shape, impact velocity and the configuration used in the experiment are studied in detail.

3. The Ultrafast Load Cell The Ultrafast Load Cell was developed at the Utah Comminution Center by Weichert ŽWeichert and Herbst, 1986. as a result of the hybridization of the simple drop-weight apparatus and the Hopkinson pressure bar. It consists of a long steel rod equipped with strain gauges on which a single particle or a bed of particles is placed and impacted by a falling steel ball ŽFig. 1.. The compressive wave resulting from the impact travels down the rod and is sensed by the solid-state strain gauges. This results in a voltage change in the Wheatstone bridge, which is then recorded as a function of time using a digital oscilloscope. Given the mechanical and physical properties of the rod as well as the bridge and gauge factors, the individually measured outputs are then transformed into force–time histories ŽKing and Bourgeois, 1993a.. Compression experienced by a particle sitting on top of the anvil is not determined directly. It is calculated from the momentum balance of the falling steel ball and the deformation of the steel rod as follows. The motion of the striking ball during the impact can be determined from its momentum balance: mb

d2 ub dt2

s yFb q m b g

Ž 1.

where u b is the position of the center of gravity of the ball, m b its mass, Fb is the force

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

6

Fig. 1. Schematic outline of the Ultrafast Load Cell. The laser beam is set 1 mm above the top of the particle to measure the velocity of the striker immediately before impact. The UFLC rod is 5 m long to ensure that the stress wave which is reflected from the bottom end of the rod and travels back up the rod does not reach the strain gauges before the original wave has passed completely.

exerted by the particle on the ball and g is the acceleration of gravity. Integrating Eq. Ž1. subject to the initial conditions at the instant of contact Ž t s 0.: d ub dt

s Õo and Fb s 0

Ž 1a .

gives: d ub dt

s Õo q gt y

1 mb

t

H0 F Ž t . d t b

Ž 2.

where Õo is the velocity of the striker at the instant of contact. Typically free-fall conditions prevail so that the velocity of the striker is calculated as Õo s Ž2 gh.1r2 , where h is the initial distance between the bottom of the ball and the top of the particle. This assumption was confirmed by measuring the time taken for the ball to travel the last 1 mm before impacting the particle. Assuming one-dimensional wave propagation in the rod, the forces and deformations of the top of the rod are related by ŽGoldsmith, 1960.: d ur dt

1 s

rr A r Cr

Fr Ž t .

Ž 3.

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

7

where rr , A r and Cr are the density, the cross-sectional area and the wave velocity of the rod, respectively. By neglecting the inertia of the particle during impact, force continuity exists at the surfaces in contact Ž Fr s Fb s F ., and subtraction of Eqs. Ž2. and Ž3. and integration gives:

a Ž t . s Õo t q

gt 2

1 y

2

mb

t

1

t

t

H0 H0 F Ž tˆ . dtˆ dt y r A C H0 F Ž t . dt r

r

Ž 4.

r

where a s u b y u r , as illustrated in Fig. 2. Eq. Ž4. gives an expression for the net approach between the center of gravity of the falling ball and a point in the rod distant from the point of impact. a therefore corresponds to the overall deformation in the vicinity of the contact, resulting from the compression of the particle and the local indentations of the ball and the anvil. It is calculated using the initial impact velocity, the ball mass and the experimentally determined force–time profile. Eq. Ž4. is valid until the arrival of the reflected strain waves at the strain gauge location. Typical force–time profiles are presented in Fig. 3. The initiation of the first fracture is identified by a rapid reduction of the force exerted by the particle on the top of the rod.

Fig. 2. Illustration of the principle used to calculate the deformation experienced by a particle of brittle material during impact on the UFLC. u b is calculated from a momentum balance of the falling ball and u r is calculated from the measured compression of the rod.

8

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

Fig. 3. Six typical force–time profiles recorded for 1.00–1.18 mm quartz particles impacted using a 0.0283 kg steel ball at a velocity of 1.16 mrs. Arrows show the instants of fracture. The variation of the response among individual particles is clearly evident. For clarity, only the traces during stage 1 are recorded for all but one of the tests. Stage 2 traces are quantitatively similar to the one shown here.

The UFLC device used in this work consists of a 5 m long, 19 mm diameter steel rod coupled to an oscilloscope with 12-bit, 10 MHz digitizers. This high-resolution setup allows a time resolution of 100 ns and a force resolution of 0.2 N. Detailed calibration work has been carried out and results are presented elsewhere ŽKing and Bourgeois, 1993a.. A strength of the UFLC is that Eq. Ž4. contains no empirical constants and consequently absolute calibration is possible. Animated highspeed video images of the impact process are available at the Utah Comminution Center web site http:rrwww.mines.utah.edur;wmwucc. The information presented there confirms the validity of the momentum balance equation that is used to describe the process.

4. Materials used Narrow-size samples of a variety of minerals and ores ŽTable 1 . have been prepared by stage-crushing — except for ore samples which are used as received — and screening using round-mesh precision microsieves. Densities Ž r . were determined using a Helium pycnometer and shape factors Ž b . were calculated according to the expression m p s br d p3 where d p is the geometric mean size. Assuming that the shape factor is statistically independent of particle size, b was determined by plotting the ratio between the average particle mass for each size class and the material density versus the representative particle size. The intercept gives the shape factor. Ore samples consisted of an iron ore from Iron Ore Company of Canada, a taconite ore from Eveleth Mines,

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

9

Table 1 Summary of sample characteristics Sample

Density, r Žgrcm3 .

Shape factor, b

Alumina Glass Apatite Barite Chromite Corundum Feldspar Fluorite Galena Gilsonite Halite Hematite Magnetite Pyrite Quartz Resinite Sphalerite Basalt Coal Copper ore Gold ore Iron ore Limestone Taconite

3.60 2.25 3.21 4.38 4.42 4.00 2.61 3.10 7.40 1.07 2.32 5.24 5.12 4.90 2.65 1.09 4.10 2.59 1.40 2.83 2.85 3.80 2.72 3.65

0.524 0.524 0.450 0.450 0.446 0.444 0.489 0.392 0.505 0.367 0.510 0.408 0.520 0.411 0.476 0.404 0.406 0.367 0.448 0.413 0.407 0.395 0.386 0.363

Minnesota, a gold ore from near Las Vegas, Nevada and a copper ore from Bingham Canyon Mine, Utah. Alumina and glass samples consisted of spheres provided by the manufacturers.

5. Measures of fracture characteristics Data measured using the UFLC were used to calculate the following fundamental characteristics associated with the deformation and fracture of particulate materials. 5.1. Particle fracture energy The particle fracture energy EX ŽBaumgardt et al., 1975. corresponds to the energy stored by the particle until the instant of fracture and corresponds to the area below the force–deformation curve: EX s

Dc

H0

F dD

Ž 5.

where D is the particle deformation and Dc is the deformation at fracture. The UFLC gives the overall deformation in the vicinity of the contact and substituting a from Eq.

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

10

Ž4. for D in Eq. Ž5. we have an expression for the energy required to break the particle using the UFLC: E s Õo

tc

tc

1

ž

tc

H0 F Ž t . dt q gH0 F Ž t . t dt y 2 m H0 F Ž t . dt b

1

y

tc

HF r AC 0 r

r

2

2

/

Ž t . dt

Ž 6.

r

Setting a s D in Eq. Ž5. requires that the local indentations in the ball and the UFLC rod are negligible. This assumption is discussed later. The mass-specific particle fracture energy is given by Em s Erm p and the volume-specific particle fracture energy is calculated by Ev s Em r . 5.2. Particle strength The strength of a particle cannot be unequivocally defined as its internal state of stresses is not known a priori ŽSchonert et al., 1962.. Using photoelastic methods, ¨ Ž . Hiramatsu and Oka 1966 showed that the stress states of a sphere, a prism and a cube subject to a pair of concentrated loads are similar. They analyzed the stresses of an elastic sphere subject to point-load compression and, after simplifications, obtained an expression for the tensile strength, given by: 2.8 Fc

sp s

p d p2

Ž 7.

where Fc is the fracture load and d p the distance between loading points. The validity of this expression was then verified by comparison of strengths calculated using Eq. Ž7. from compression of irregularly shaped specimens and tensile strengths estimated using the Brazilian test, and good correspondence was observed ŽHiramatsu and Oka, 1966.. Eq. Ž7. is used in this work to estimate the strength of a particle subject to impact on the UFLC. For convenience, d p in Eq. Ž7. is estimated as the geometric mean size of the passing and retained sieves of the monosize. 5.3. Particle stiffness The particle stiffness ŽTavares and King, 1995. is determined on the basis of the Hertzian contact theory ŽGoldsmith, 1960.. The relationship between force and deformation for a spherical or nearly spherical particle compressed between a falling ball and the rod is given by: K e d 1r2 p

a 3r2 3 K e is the local deformation coefficient of the Hertzian contact, given by: Fs

Ke s

k p k b ,r k p q k b ,r

Ž 8.

Ž 9.

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

11

where k b,r is the stiffness of the ball or rod and k p is the particle stiffness given by: kp s

Yp

and

1 y m2p

k b ,r s

Y b ,r 1 y m2b ,r

Ž 10 .

where Y is Young’s modulus and m Poisson’s ratio. If the modulus of elasticity and Poisson’s ratio of the particle are known, the particle stiffness can be directly calculated using Eq. Ž10.. Otherwise a simple procedure described as follows can be used. Substituting Eq. Ž8. in Eq. Ž5. and integrating, the particle fracture energy can be related to the deformation at fracture and the local deformation coefficient: 5r2 E s 152 d 1r2 p K e ac

Ž 11 .

Alternatively, K e can be related to the critical load and the particle fracture energy by substituting Eq. Ž8. in Eq. Ž11. and rearranging:

ž

K e s 0.576

Fc5 dp E3

1r2

/

Ž 12 .

then the stiffness of the particle can be calculated by solving Eq. Ž9. giving: kp s

K e k b ,r

Ž 13 .

K e y k b ,r

Eqs. Ž12. and Ž13. shows that the local deformation coefficient of the Hertzian contact can be estimated simply using the critical load and the particle fracture energy. Both the UFLC rod and the drop weights are made of stainless steel, with Young’s modulus of 210 GPa and Poisson’s ratio of 0.3, so that k b,r s 230 GPa. Finally, by substituting Eq. Ž7. in Eq. Ž12. and given that Em s ErŽ br d p3 . we obtain:

sp s K e2r5 Ž Em br .

3r5

Ž 14 .

Eq. Ž14. shows that the three measures discussed in this work, namely the particle fracture energy, the particle strength and the particle stiffness are uniquely related, given the stiffness of the tools. This is illustrated in Fig. 4 where fracture data of two sizes of glass beads are plotted in log–log axes. The data are concentrated along a line of slope 3r5 as predicted by Eq. Ž14.. From the intercept, the particle stiffness is estimated which gives 77.5 GPa. Undamaged glass typically has a Young modulus of 70 GPa and a Poisson ratio of 0.16 so that its stiffness, calculated using Eq. Ž10., is 73 GPa, which is close to the experimental value. As a consequence of Eq. Ž14. it suffices to use only two of the three measures discussed in the present work, given their unique inter-relationship. The procedure described above applies strictly to elastic spherical particles. For irregularly shaped particles, a precise constitutive equation is not defined a priori. Testing of a large number of non-spherical but relatively uniform aspect particles on the UFLC showed that discrepancies with the theory were mainly in the initial portion of the profiles, particularly due to surface crushing as shown in Fig. 5. This suggests that a

12

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

Fig. 4. Relationship between specific particle fracture energy and particle strength for glass spheres of different sizes.

Fig. 5. Force–deformation profile during stage one of the particle breakage process for a 4.4 mm limestone particle subject to impact loading as measured on the UFLC. The broken line shows the theoretical Hertzian contact relationship.

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

13

relatively isometric particle can be modeled as a hard brittle core surrounded by a thin crumbly skin. The degree of surface roughness has been shown to have a significant effect when comparing the stiffness of samples of smooth and rough cubic particles of the same size. Provided that the particles are approximately isometric, Eq. Ž12. can be used to estimate the stiffness of irregularly shaped particles. For some materials, however, inelastic deformation immediately before fracture gives rise to a significant discrepancy. This is clearly evident in Fig. 5 where the experimental force profile deviates from the ideal Hertzian profile during the last 15 mm of deformation before fracture. In that case Eq. Ž8. must be modified to account for the accumulation of damage during deformation of the particle. Details are describe elsewhere ŽTavares and King, 1998.. Fig. 2 also shows that although the bulk of the deformation during an UFLC experiment is induced on the particle, the particle itself also indents the tools in response. The extent of that deformation will depend on the mechanical properties of the bodies in contact. Assuming perfect elasticity and lubrication of the bodies in contact the ratio between the compression of the particle D and the overall deformation in the vicinity of the contact a can be estimated using Hertzian theory which gives:

D

k b ,r

s

a

Ž 15 .

k b ,r q k p

Eq. Ž15. is relevant to UFLC measurements as it can be used to estimate the fraction of the measured energy calculated using Eq. Ž6. that is actually absorbed by the particle. Substituting Eq. Ž15. in Eq. Ž11. and rearranging it gives: EX s E

ž

1 k prk b ,r q 1

/

Ž 16 .

6. Statistical analysis of fracture 6.1. Distribution of fracture characteristics Whenever the fracture characteristics of individual particles in a sample of particles are measured, a large scatter of the data results and must be described statistically. A method commonly used to deal with such variable data is based on order statistics. This method consists of ranking the test results in ascending order and then assigning i s 1,2, . . . , N to the ranked observations, where N is the total number of tests performed. The cumulative probability distribution for the measured variable Že.g. particle fracture energy. is approximated by: P Ž Em ,i . s

i y 0.5 N

Ž 17 .

The resulting pairs w Em ,i , P Ž Em ,i .x can then be fitted to the appropriate statistical distribution using standard least-squares analysis. Several statistical distributions have

14

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

been proposed. The Weibull distribution ŽWeibull, 1951. is often favored in materials testing particularly for strength measurements and has been used by Weichert Ž1990. to describe particle fracture energy data of glass beads. The log-normal distribution, on the other hand, has been successfully used to describe particle fracture energy Žor fracture probability. data of a variety of irregularly shaped brittle materials ŽBaumgardt et al., 1975; Dan and Schubert, 1990; King and Bourgeois, 1993a. and is given by: P Ž Em . s 12 1 q erf

ž

ln Em y ln Em 50

(2 s

E

/

Ž 18 .

where Em ,50 and s E2 are respectively the median and the geometric variance of the distribution. Distributions of particle fracture energy and particle stiffness for selected minerals are presented in Figs. 6 and 7, respectively. It is evident that the UFLC can discriminate between materials and that the data can be effectively described by the log-normal distribution. Particle strength data have also been well described by the log-normal distribution but results are omitted for brevity. The parameters Em ,50 and s E are estimated from the least-squares best straight line as shown in Figs. 6 and 7. A summary of the fracture characteristics of 2.0–2.8 mm size fraction of various materials is presented in Fig. 8. The lines of constant stiffness are calculated using Eq. Ž14. and have a slope 3r5. This plot gives an indication of the inherent resistance of

Fig. 6. Particle fracture energy distributions of 2.0–2.8 mm particles of various minerals. Lines represent the least-squares fit to the log-normal distribution.

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

15

Fig. 7. Particle stiffness distributions of 2.0–2.8 mm particles of various minerals. Lines represent the least-squares fit to the log-normal distribution.

geological materials to mechanical size reduction by impact. For the same particle strength, rocks and ores have usually higher fracture energies than pure minerals due to their lower stiffness. This shows that materials commonly encountered in industrial comminution are comparatively tough. The fraction of the measured energy consumed by the particle relative to the tools calculated using Eq. Ž16. is also shown in Fig. 8 and the ratio EXrE is constant along the lines of constant stiffness. Except for tests on corundum, glass and alumina less than 18% of the total strain energy during the impact is consumed by the tools confirming that the assumption of negligible indentation in the tools used to derive Eq. Ž6. is valid under most circumstances. 6.2. Effect of particle size on fracture characteristics The data in Fig. 8 can be used only within a limited size range. This is due to the fact that particle strength and particle fracture energy are strongly affected by particle size as shown in Fig. 9 for quartz particles. A decrease in the particle size resulted in a shift of the distributions of particle fracture energy of quartz to higher values. This increase in strength with the decrease in size is commonly observed in brittle materials and is explained on the basis of the Griffith model of brittle fracture. The size of the largest microcrack present in a particle must decrease as the size of the particle decreases and, if the fracture toughness of the material remains constant, this leads to an increase in particle strength.

16

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

Fig. 8. Relationship between mean particle fracture energy and mean particle strength for 2.0–2.8 mm particles of various materials. Lines of constant particle stiffness are calculated from Eq. Ž14. and have slope 3r5.

Particle fracture energy distributions of a variety of materials over a range of sizes have been measured and typical results are summarized in Fig. 10. It is evident that data for ores at finer sizes and for minerals can be generally well described by a power law. Power law relationships between the mass-specific particle fracture energy and the particle size were derived independently by Yashima et al. Ž1987. and by Weichert Ž1990. based on Hertzian contact theory and Weibull’s weakest link criterion ŽWeibull, 1951.. Careful inspection of Fig. 10, however, shows that as particle size increases, the measured energies tend towards a constant value which is specific to each material. A model that describes the data well is: Em 50 s Em ,` 1 q Ž d p ,ord p .

f

Ž 19 .

Em ,` , d p,o and f are model constants that are estimated from the data in Fig. 10 by least-squares estimation. Em ,` represents the residual particle fracture energy of the material at larger size and d p,o is a characteristic size of the material microstructure. Eq. Ž19. has been fitted to particle fracture energy data on a variety of materials over a range of sizes and results are summarized in Table 2 and Fig. 10. Values of f vary between 1 and 2.7. The residual particle fracture energy Em ,` of minerals is consistently

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

17

Fig. 9. Particle fracture energy distributions for quartz particles of various sizes. Smaller sizes have larger specific fracture energies than larger particles which is reflected in greater specific energy consumption for the production of finer powders by comminution.

lower than that of ores and rocks, which indicates the higher toughness of the latter. Also, the critical size d p,o is significantly larger for minerals than for ores, reflecting the fine microstructures of the latter.

Fig. 10. Variation of mean volume-specific particle fracture energy with particle size for various materials. The lines represent Eq. Ž19. for each material.

18

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

Table 2 Parameters of the particle fracture energy versus particle size model ŽEq. Ž19.. Material

Em ,` ŽJrkg.

d p,o Žmm.

f

Size range Žmm.

Apatite Galena Gilsonite Quartz Sphalerite Magnetite Copper ore Iron ore Limestone Marble Taconite ŽBM. Taconite ŽCC.

1.50 3.19 5.50 43.4 7.00 9.56 96.1 47.3 114.2 45.9 235.9 163.3

19.3 7.31 7.03 3.48 8.24 3.93 1.17 1.08 0.490 0.882 0.803 0.856

1.62 1.03 1.60 1.61 1.16 1.96 1.26 2.30 2.05 2.66 1.42 1.76

0.25– 8.00 0.70– 7.60 1.18–10.0 0.25– 4.75 0.35–10.0 0.25– 7.20 0.25–15.8 0.25–15.0 0.35– 5.60 0.50–15.0 0.35– 6.00 0.35–10.0

Fig. 11 shows the effect of the size on the measured distribution of particle stiffness for quartz. Unlike particle strength and particle fracture energy, particle stiffness data for all sizes fall within a narrow band, suggesting that stiffness is a material property and is essentially independent of particle size. This independence of size on particle stiffness has also been observed for other materials. Mean measured particle stiffness is shown as a function of particle size for a number of materials in Fig. 12. In general, the data show a slight decrease in stiffness as particle size increases although this trend appears to be

Fig. 11. Particle stiffness distributions for quartz particles of various sizes. The variation with particle size is considerably less than the variation of particle fracture energy or particle strength. Compare with Fig. 9.

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

19

Fig. 12. Variation of mean particle stiffness with particle size for various materials measured on the Ultrafast Load Cell.

reversed for the copper ore that was studied. At a microscale level, the modulus of elasticity Žand stiffness. depends on the atomic and molecular structure and is an intrinsic property of the material ŽDieter, 1986.. At the macroscopic level, however, the stiffness depends upon the cumulative compliance of structure elements and therefore would be affected by microstructural features such as pores, cracks, grain boundaries, parting planes, etc. Summarizing, particle fracture energy and particle strength are structure-sensitive properties as they are strongly affected by the presence of critical flaws and cracks in zones of high stress in the material. As particle size decreases cracks progressively disappear which results in increases in both strength and particle fracture energy. Particle stiffness, on the other hand, is not a structure-sensitive property, as it depends only on the cumulative effect of the deformations in individual portions of the particle. As a result it can also increase or remain constant with a reduction in particle size. These different pieces of information provided by the measures of fracture Žparticle fracture energy and particle strength. and deformation Žparticle stiffness. provide important and independent information on the mechanical response of materials subject to impact in the UFLC. 6.3. SensitiÕity of the measurements In UFLC experiments, particles of approximately uniform aspect are usually tested. In order to investigate the sensitivity of the measurements in respect to particle shape, fracture characteristics of river-bed quartz particles, which naturally occur in nearly

20

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

Table 3 Effect of particle shape on fracture characteristics of 1.00–1.18 mm quartz Particle shape

Nearly round Irregular Flat

Shape factor, b

0.476 0.373 0.305

Particle fracture energy Particle strength

Particle stiffness

Em 50 ŽJrkg.

s E2

sp 50 ŽMPa.

ss2

k p 50 ŽGPa.

s k2

381.1 328.1 364.8

0.364 0.345 0.433

63.5 39.0 27.4

0.164 0.268 0.196

57.7 34.2 12.1

0.262 0.540 0.563

spherical shapes, have been compared to those measured for irregular quartz fragments produced by crushing ŽTable 3.. The measured fracture energies were shown to be drawn from identical statistical populations with 95% confidence ŽMilin and King, 1994.. On the other hand, particle strength and particle stiffness clearly decrease as particles become more irregular. The reduction in the measured particle strength can be partially explained by the fact that as particles become more flat, their vertical dimension when positioned in a stable manner on the UFLC anvil becomes significantly smaller than the particle size estimated by the geometric mean sieve size. On the other hand, particle shape influences particle stiffness due to the undetermined radius of curvature of flatter particles and its poor correlation with the geometric mean particle size. It has been observed from testing of regularly shaped specimens such as cylinders, cubes and spheres that when the radius of

Fig. 13. Effect of impact velocity on particle fracture energy distributions of different materials. Vertical lines represent the standard deviations of the distributions.

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

21

curvature of the particle is known, calculated stiffness values are statistically independent of specimen shape. Further, the chipping of corners and edges, and the relocation of the particle on the anvil that occur particularly during loading of irregularly shaped particles also contribute for additional dissipation of energy which results in underestimation of particle stiffness. In practice, this sensitivity of the measured particle stiffness to particle shape means that it can be compared only for samples with statistically similar shapes. The sensitivity of the UFLC measurement to the configuration used in the experiments has also been investigated. Fig. 13 shows particle fracture energy measurements of various materials as a function of impact velocity. It shows that although the magnitude of the impact velocity is essential to calculating the deformation of particles during impact ŽEq. Ž4.. and consequently the calculation of the particle fracture energy, it does not affect the particle breakage behavior. This is in agreement with results from Ž1991. who noted that within the range of Krogh Ž1980. and observations from Schonert ¨ stressing velocities of interest to comminution — a few centimeters to a few meters per second — the stressing rate can be assumed to have no effect on the fracture characteristics of brittle materials. Further, the loading geometry used in the experiment, i.e. ball size and configurations such as ball–ball, ball–flat and flat–flat were found to have no effect on the fracture characteristics.

7. Applications Measurement of fundamental fracture properties of particles on the UFLC can be used to throw light on a number of problems that are of importance in industry. A partial list of interesting investigations includes the following points. Ž1. Establish a baseline for comparison of industrial comminution processes: singleparticle comminution is the most efficient method of size reduction, as no losses due to particle–particle interactions exist. A realistic definition of the comminution energy efficiency can be established. Ž2. Predict optimum comminution routes or optimum energy utilization for an ore. Ž3. Characterize the grindability of different feed materials to a grinding circuit: UFLC measurements can be used as a fast and effective alternative to traditional Bond batch milling experiments to assess the relative resistance of different materials to mechanical size reduction by impact ŽKing et al., 1997.. Ž4. Model and simulate comminution machines: the use of UFLC data along with the knowledge of the impact energy spectrum in tumbling mills ŽMishra and Rajamani, 1992. can be used to compute selection and breakage functions for modeling and simulation of comminution machines ŽKing and Bourgeois, 1993b.. Ž5. Assess the effect of predamage on comminution: the UFLC can be used to study the effect of damage resulting from blasting and explosives ŽKing and Zhu, 1994; McCarter, 1996., thermal treatment using conventional and microwave heating ŽTavares and King, 1995; Tavares, 1997. and crushing and grinding equipment that leave residual damage ŽTavares, 1997..

22

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

Fig. 14. Variation of mean mass-specific particle fracture energy with particle size of taconite samples from the fresh feed and the discharge of an industrial ball mill. Particles that survive the milling operation are on average tougher than particles of the same size in the feed.

Two specific examples are given below. 7.1. Comminution circuit analysis As part of a study to investigate the efficiency of a taconite grinding circuit, the distribution of particle fracture energy was measured in several streams in the circuit. The effect of particle size on the mean particle fracture energy of the fresh feed Žmagnetic concentrate. and the ball mill discharge of a circuit is shown in Fig. 14 . The data show that particles being discharged from the closed-circuit ball mill are on the average tougher than those entering the circuit. Considering that compositions were not dramatically different ŽKing and Schneider, 1994., this would likely be the result of the lower breakage rates of tougher particles in the mill. The tough particles survive more readily in the mill and therefore build up in the circulating load. Indeed, circulating loads tougher than ball mill feed are commonly observed in milling circuits. Notice how this effect is evident only for particle sizes larger than 400 mm. 7.2. Study of thermal-shock damage quartz The application of heat followed by thermal shock can weaken brittle particulate materials. This weakening results primarily from the cracking induced during rapid contraction. As a common gangue mineral, quartz has been extensively investigated regarding its response to thermal pretreatment ŽChakravarti and Jowett, 1966; Geller and Tervo, 1975; Kanellopoulos and Ball, 1975; Hariharan and Venkatachalam, 1977; Fitzgibbon and Veasy, 1990; Pocock et al., 1998.. Although the effect of pretreatment

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

23

on the comminution behavior of quartz has been determined, experimental techniques used in the majority of these studies were seriously limited by the lack of sensitivity to changes in the amount of energy required for comminution particularly at different particle sizes. Single-particle fracture studies in the UFLC are conducted under controlled, reproducible experimental conditions and are ideally suited to assess the effect of thermal pretreatment. They require minimum sample preparation and can be performed with small sample volumes so that several variables influencing thermal pretreatment such as material type, preheat temperature, quenching medium and particle size can be readily investigated. The UFLC allows the measurement of the reduction in the specific particle fracture energy and it can also be used to assess the effect of pretreatment on the internal material integrity. It is known that the presence of cracks increases the compressibility of a solid. By measuring the stiffness of a material before and after pretreatment, the fractional reduction of the cross-sectional area that has been stress-relieved by cracks and that is unable to withstand load can be estimated by ŽTavares and King, 1995.: kd Ds1y Ž 20 . ku where k u and k d are the stiffness of the material before and after pretreatment, respectively. Quartz particles were subjected to heating in a laboratory batch furnace and quenching in water and the fracture characteristics were measured. The effect of pretreatment temperature on the particle fracture energy and on the damage parameter D is shown in Fig. 15. Thermal shock induced by water quenching following pretreatment at lower temperatures was able to reduce the particle fracture energy but not to affect the material integrity due to the limited and superficial nature of the cracks induced. Thermal shock

Fig. 15. Effect of pretreatment temperature on fracture characteristics of 4.00–4.75 mm quartz.

24

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

following heating above 6008C, on the other hand, produced both a significant reduction in particle fracture energy and an increase in volume cracking which resulted in an increase in damage. This has been confirmed by inspection of polished sections of the damaged material under the optical microscope ŽTavares and King, 1995.. The simultaneous measurement of particle fracture energy and particle stiffness makes it possible to make qualitative predictions of the potential benefits of pretreatment to industrial milling operations. The presence of a few effective flaws near the particle surface can reduce the particle fracture energy and therefore increase the breakage rate of the parent particle. However, as soon as fresh surfaces are created, the effect of pretreatment rapidly vanishes. On the other hand, massive volume cracking such as that induced in quartz above 6008C increases the breakage rate of both the parent and the progeny as exposed surfaces contain cracks themselves. This suggests that reduction in particle fracture energy alone would have only a limited impact on the energy consumption in comminution, while reduction in particle fracture energy with a significant increase in damage is likely to translate to noticeable energy savings in subsequent size reduction operations. A comprehensive evaluation of thermal pretreatment will be published elsewhere.

8. Conclusions The UFLC is an accurate, reproducible, fast and inexpensive device that can be used to characterize the mechanical behavior of brittle particulate material subject to impact fracture. It can discriminate between materials and therefore classify materials easily and quickly with respect to their resistance to comminution. The UFLC enables a clear distinction to be made between the primary and secondary stages of particle comminution. It is the behavior of the particle during the first stage that governs its probability of fracture and is the subject of this paper. Two of the parameters measured using the UFLC are of special significance: the specific particle fracture energy determines the minimum energy required to fracture a brittle particle, and the particle stiffness represents the rate at which strain energy is stored in the material with deformation prior to fracture. The specific particle fracture energy is the main variable that determines the energy requirement by the material for comminution and it establishes the base value from which the energy efficiency of a comminution machine can be established. The particle fracture energy has a particular significance in comminution as it represents the minimum energy required to fracture a brittle particle. Its distribution has been demonstrated to be equivalent to the probability of fracture ŽKing and Bourgeois, 1993a. and it can be used to estimate the selection functions in a ball mill ŽKing and Bourgeois, 1993b.. Methods are now being developed to estimate the conventional work index form measurements made on the UFLC ŽKing et al., 1997.. Data obtained from a variety of mineralogical materials confirm that particle fracture is described most convincingly by the Griffith model of fracture. Particle failure is catastrophic when the fracture criterion is met. However, there is evidence that some damage accumulation, as postulated by the continuum damage mechanics model, does

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

25

occur immediately prior to fracture in some polycrystalline materials ŽFig. 5.. Furthermore, damage can accumulate in a particle due to a variety of events including repeated loading, thermal shock, and stressing under multiple-point compressive loading in a compressed bed of particles. The crystalline microstructure exerts a significant effect on the fracture properties of a single particle. It is not sufficient to consider only the relative hardness or softness of the material as has been done in the past. The UFLC can be used to make quantitative measurements of damage induced by pretreatment such as that resulting from blasting, thermal pretreatment and crushing and grinding equipment that leave residual damage. Potentially the most rewarding application of UFLC measurements is the prediction of the selection and breakage functions in ball mills without the need for the laborious and uncertain parameter estimation techniques that are in common use today ŽKing and Bourgeois, 1993b.. 9. Nomenclature Cross-sectional area of the rod Žm2 . Wave velocity in the rod Žmrs. Pretreatment damage constant Žy. Particle size Žmm. Energy required to break the particle on the UFLC; for most materials E s EX X E Particle fracture energy ŽJ. Em , Em 50 Mass-specific particle fracture energy and median of the log-normal distribution, respectively ŽJrkg. Ev Volume-specific particle fracture energy ŽJrcm3 . Em ,` , d p,o , f Constants of the particle fracture energy versus size model F, Fb , Fr Forces on the particle, on the ball and on the rod, respectively ŽN. Fc Critical load on the particle at the instant of fracture g Gravitational constant Žmrs 2 . h Distance between the bottom of the ball and the top of the particle Žm. Ke Local deformation coefficient of the Hertzian contact ŽGPa. kd , k u Stiffness of the damaged and of the intact material respectively ŽGPa. k p , k p 50 Particle stiffness and mean of the log-normal distribution, respectively ŽGPa. k b,r Stiffness of the UFLC ball and rod ŽGPa. mb , mp Mass of the ball and the particle, respectively Žkg. N Number of particles tested in a UFLC experiment Žy. P Cumulative distribution Ž%. P Ž Em . Cumulative distribution of specific particle fracture energy in a sample of particles s probability that a particle from the sample will fracture when subject to an impact of energy Em t Elapsed time from contact tc Time at fracture Žs.

Ar Cr D dp E

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

26

ub , ur Õo Yp , Y b,r

Position of the center of gravity of the ball and bulk deformation of the rod Žm. Velocity of ball at instant of impact Žmrs. Modulus of elasticity of the particle and of the ball and the rod, respectively ŽGPa.

9.1. Greek letters

a , ac b D, Dc m , m b,r r , rr sp s E2 , ss2 , s k2 t

Local deformation and local deformation at fracture, respectively Žmm. Particle shape factor Žy. Particle deformation and particle deformation at fracture, respectively Žmm. Poisson’s ratio of the particle and of the ball and the rod, respectively Žy. Density of the particle and of the rod, respectively Žkgrm3 . Particle strength and mean of the log-normal distribution ŽMPa. Variances of the mass-specific particle fracture energy, particle strength and particle stiffness distributions, respectively Integration variable Žy.

Acknowledgements This research has been supported by the Department of the Interior’s Mineral Institute program administered by the U.S. Bureau of Mines through the Generic Mineral Center for Comminution under grant numbers G1125249, G1135249, G1145249 and G1155249.

Appendix A. Definition of terms The term particle fracture energy is given a precise definition in this paper ŽSection 5.1. but there may be some confusion with similar terms used in fracture mechanics literature. The term particle fracture energy is used here to mean the strain energy that is stored in the particle from the instant of impact until the instant of failure. It is the minimum amount of energy required to fracture the particle under the particular orientation and loading conditions of the test. Since it depends on orientation and loading rate, and on the location, orientation and size of internal microflaws, any one particle does not have a unique fracture energy. This quantity is called the wmassx specific fracture energy by Yashima et al. Ž1987., the comminution energy by BaumŽ1995.. The term particle gardt et al. Ž1975. and the breakage energy by Schonert ¨ fracture energy is preferred for the following reasons. The term comminution generally refers to the entire size reduction process including sequential breaking of parents and successive generations of progeny. Thus the term comminution energy is reserved for the total energy, conventionally expressed as kWhrton, that is required to reduce a given material from a feed size distribution to the final ground product. The term

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

27

specific breakage energy is reserved for the energy consumed during the entire singleparticle breakage event including both first and second stages. These separate stages can be easily identified from the experimental traces in Fig. 3. Mass-specific fracture energy or specific fracture energy is too close to the term surface-specific fracture energy used in the fracture mechanics literature. There is some slight chance that the term specific particle fracture energy can be confused with the term surface-specific fracture energy used in the conventional fracture mechanics literature Žoften shortened to fracture energy.. This latter usage is reserved for a material-specific constant which is the sum of the surface energy and the plastic work per unit of surface created ŽAnderson, 1995; Cottrell and Mai, 1996, p. 3.. This quantity Ž1979.. However, if the designation ‘particle’ has been called crack energy by Schonert ¨ is always made or implied the chance of confusion is slight.

References Anderson, T.L., 1995. Fracture Mechanics: Fundamentals and Applications Ž2nd. ed... CRC Press, Boca Raton, FL. Baumgardt, S., Buss, B., May, P., Schubert, H., 1975. On the comparison of results in single grain crushing under different kinds of load. Proc. 11th Int. Miner. Process. Congr., Cagliari, pp. 3–32. Bergstrom, B.H., 1962. Energy and size distribution aspects of single particle crushing. Proc. 5th Symp. Rock Mechanics, pp. 155–172. Chakravarti, A., Jowett, A., 1966. Aspects of comminution by heating. Proc. 2nd Eur. Symp. Comminution, Amsterdam, pp. 583–604. Cottrell, B., Mai, Y.W., 1996. Fracture Mechanics of Cementitious Materials. Blackie, Glasgow. Dan, C.C., Schubert, H., 1990. Breakage probability, progeny size distribution and energy utilization of comminution by impact. Aufbereit.-Tech. 31, 241–247. Dieter, G.E., 1986. Mechanical Metallurgy Ž3rd ed... McGraw-Hill, New York. Fitzgibbon, K.E., Veasy, T.J., 1990. Thermally assisted liberation — a review. Miner. Eng. 3, 181–185. Geller, L.B., Tervo, R.O., 1975. Grinding of preheated rocks. Trans. Inst. Min. Metall. 84, C25–C33. Gilvarry, J.J., Bergstrom, B.H., 1961. Fracture of brittle solids, II. Distribution function for fragment size in single fracture Žexperimental.. J. Appl. Phys. 32, 400–410. Goldsmith, W., 1960. Impact. Edward Arnold, London. Hariharan, K., Venkatachalam, S., 1977. Influence of thermal treatment upon grindability of quartz. Min. Mag. 136, 105–108. Hildinger, P., 1969. Festigkeitsuntersuchungen an Glas-Kugeln sowie Quarz-Zementklinker- und KalksteinKornen durch Fallkorperbeanspruchung. Chem.-Ing.- Tech. 41, 278–281. ¨ ¨ Hiramatsu, Y., Oka, Y., 1966. Determination of the tensile strength of rock by compression test of an irregular test piece. Int. J. Rock Mech. Min. Sci. 3, 89–99. Kanellopoulos, A., Ball, A., 1975. The fracture and thermal weakening of quartzite in relation to comminution. J. S. Afr. Inst. Min. Metall. 76, 45–52. King, R.P., Bourgeois, F., 1993a. Measurement of fracture energy during single-particle breakage. Miner. Eng. 6, 353–367. King, R.P., Bourgeois, F., 1993b. A new conceptual model for ball milling. Proc. XVIII Int. Miner. Process. Congr., Vol. 1, Sydney, pp. 81–86. King, R.P., Schneider, C.L., 1994. Computer simulation of taconite grinding and concentration circuits. Comminution Center, University of Utah, Salt Lake City. King, R.P., Zhu, Y., 1994. Measurement of fracture properties of explosively treated particles using the ultrafast load cell. Comminution Center, University of Utah, Salt Lake City. King, R.P., Tavares, L.M., Middlemiss, S., 1997. Establishing the energy efficiency of a ball mill. In:

28

L.M. TaÕares, R.P. King r Int. J. Miner. Process. 54 (1998) 1–28

Kawatra, S.K. ŽEd., Comminution Practices. Soc. Mining Metall. Exploration Inc, Littleton, CO, pp. 311–316. Krogh, S.R., 1980. Crushing characteristics. Powder Technol. 27, 171–181. Lawn, B., 1995. Fracture of Brittle Solids Ž2nd ed... Cambridge University Press. McCarter, M.K., 1996. Effect of blast preconditioning on comminution for selected rock types. Proc. 22nd Annu. Conf. Explosives and Blasting Techniques, Orlando, FL, pp. 119–129. Milin, L., King, R.P., 1994. Testing the equality of two distributions of data measured on the ultrafast load cell. Comminution Center, University of Utah, Salt Lake City. Mishra, B.K., Rajamani, R.K., 1992. The discrete element method for the simulation of ball mills. Appl. Math. Modeling 16, 598–604. Pocock, J., Veasy, T.J., Tavares, L.M., King, R.P., 1998. The effect of heating and quenching on grinding characteristics of quartzite. Powder Technol. 95, 137–142. Rumpf, H., 1973. physical aspects of comminution and new formulation of a law of comminution. Powder Technol. 7, 145–159. Schonert, K., 1972. Role of fracture physics in understanding comminution phenomena. Trans. Soc. Min. Eng. ¨ AIME 252, 21–26. Schonert, K., 1979. Aspects of the physics of breakage relevant to comminution, 4th Tewkesbury Symposium, ¨ Melbourne, pp. 3.1–3.30. Schonert, K., 1988. Fundamentals of particle breakage. Course notes, University of the Witwatersrand, ¨ Division of Continuing Engineering Education, Johannesburg, Section F6. Schonert, K., 1991. Advances in comminution fundamentals and impacts on technology. Aufbereit.-Tech. 32, ¨ 487–494. Schonert, K., 1995. Comminution from theory to practice. Proc. 19th Int. Mineral Processing Congr., San ¨ Francisco, Vol. 1, pp. 7–14. Schonert, K., Umhauer, H., Rumpf, H., 1962. Die Festigkeit kleiner Glaskugeln. Glastech. Ber. 35, 272–278. ¨ Tavares, L.M., 1997. Microscale Investigation of Particle Breakage applied to the Study of Thermal and Mechanical Predamage. Ph.D. dissertation, Univ. of Utah, Salt Lake City. Tavares, L.M., King, R.P., 1995. Microscale investigation of thermally assisted comminution. Proc. 19th Int. Mineral Processing Congr., Vol. 1, San Francisco, pp. 203–208. Tavares, L.M., King, R.P., 1998. Continuum damage modeling of particle breakage. In prep. Weibull, W., 1951. A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293–297. Weichert, R., 1990. Fracture physics in comminution. Proc. 7th Eur. Symp. Comminution, Ljubljana, Vol. 1, pp. 3–20. Weichert, R., Herbst, J.A., 1986. An ultrafast load cell device for measuring particle breakage. Prepr. 1st World Congr. Particle Technology, Nurnberg II, 3. ¨ Yashima, S., Kanda, Y., Sano, S., 1987. Relationship between particle size and fracture as estimated from single particle crushing. Powder Technol. 51, 277–282.