crack propagation in britle materials · conventional comminution processes. the stress-induced...

IJRRAS 13 (2) ● November 2012 www.arpapress.com/Volumes/Vol13Issue2/IJRRAS_13_2_05.pdf

406

CRACK PROPAGATION IN BRITLE MATERIALS:

RELEVANCE TO MINERALS COMMINUTION

Desmond Tromans

Department of Materials Engineering,

University of British Columbia, Vancouver BC, Canada V6T 1Z4

E-mail: [email protected]

ABSTRACT

A simple bond breaking model is developed to estimate the limiting Mode I crack speed, climit in polycrystalline

materials of different chemistry, including oxides, sulphides and halides. It utilizes previously developed ionic and

covalent model equations to determine the non-linear extension of a stretched bond pair at the crack tip and

calculates the shortest time taken to break the bond, after which the crack tip advances to the next atom pair. It is

shown that climit is always less than both the transverse acoustic wave speed cT and the Rayleigh surface wave

velocity cR. Increasing the strain rate at the tip of a limiting crack requires additional means to release the locally

increasing strain energy. This may be achieved by crack branching (bifurcation), leading to fragmentation... The

effect of loading rate upon the branching and fragmentation process is examined and shown to be relevant to the

energy efficiency of mineral comminution.

Keywords: Crack branching, elastic modulus, fragmentation, limiting crack velocity, strain rate, surface energy.

1. INTRODUCTION

Mineral comminution is a primary process in the mining industry whereby crushing and grinding operations are

used for the breakage of large mineral rock particles into smaller particles for further processing. These subsequent

processes include the extraction of valuable metal-containing components via thermal (pyrometallurgy) and aqueous

(hydrometallurgy) methods, or the extraction of valuable rock inclusions by physical separation procedures (e.g.

diamonds), and crushing of lime for cement manufacture. Comminution operations consume large amounts of

energy, much of which is released as heat, and utilise a significant percentage of the overall total national energy

consumption in countries where mining is practiced. Estimated percentages of the overall energy consumption in

mining countries with freely available information are ~1.86% for Canada, 1.48% for Australia, 1.8% for South

Africa and 0.39% for the USA [Tromans, 2008]. The origin of the large energy consumption is the use of

compressive loading methods to propagate pre-existing cracks (flaws) in mineral particles, the energy efficiency

being of the order of <1% to ~2% when based on the ratio of the surface energy required to generate new fracture

surfaces relative to the work input [Furstenau and Abouzeid, 2002; Tromans and Meech, 2002, 2004]. The

maximum ideal limiting efficiency has been estimate to be ~5% to ~9% based on compressive loading of a single

particle containing a central crack [Tromans, 2008], although such efficiencies have not been realised in

conventional comminution processes.

The stress-induced propagation behaviour of pre-existing cracks (flaws) plays a major role in the comminution

process. For example, in a situation where a particle contains several flaws, only one favourably sized and oriented

flaw may undergo initial propagation. As the propagating crack approaches the vicinity of other non-propagating

flaws it may interact (join) and activate them, thereby generating an increase in new fracture surface area without

further increase in load. Interaction (coalescence) with nearby flaws is more likely if branching (bifurcation) of the

propagating crack occurs before it has traversed the particle. Interactions (coalescence) between flaws is greatly

enhanced if different-sized flaws are induced to propagate simultaneously and undergo crack branching. Such

considerations are relevant to particle fragmentation and the fraction of strain energy used to generate new surface

area (i.e., energy efficiency). Consequently, the critical factors in the comminution of brittle minerals are those

relating to crack speed, crack branching and loading rate effects on crack propagation.

IJRRAS 13 (2) ● November 2012 Tromans ● Crack Propagation in Britle Materials

407

2. CRACK PROPAGATION

Comminution of minerals invariably occurs under compressive loading conditions such as those found in crushers,

ball mills, rod mills and grinding rolls. Natural minerals contain flaws and planar defects some of which, under

suitable conditions of load and orientation relative to the load, will experience tensile stress components sufficient

to propagate as cracks. Obviously, the generation of tensile stresses under compressive loading conditions is not a

very energy efficient method of inducing crack propagation and is a situation discussed previously [Tromans 2008].

A schematic of the loading situation is shown in Fig. 1.

P P

Figure 1. Schematic of mineral rock particle with crack-like planar defects subjected to compressive load P.

The required critical tensile stress c, acting normal to the crack plane, necessary for initiating propagation of a

stationary penny shaped planar crack under opening mode conditions (Mode I) is given by the well known fracture

mechanics condition in Eq. (1) [Broek 1982]:

2/12/12 )()1( ICGEKIC (1)

where E is the tensile elastic modulus (Young’s modulus), is the Poisson ratio, and GIC is the static critical energy

release rate per unit area of crack plane at the onset of cracking due to formation of new upper and lower crack

surfaces. For an internal circular crack of radius a, or a semicircular edge crack of radius a, the corresponding

critical static stress intensity KIC is obtained from Eq. (2):

2/1)/(2 aK cIC (2)

If a stationary crack is induced to propagate as a crack it is desirable that it reaches its limiting crack speed climit

because this represents the maximum speed at which the strain energy at the tip of a propagating crack may be

released by the generation of new crack surface area. Under these conditions, further increases in strain energy at the

crack tip have to be released by other means, in particular by a more rapid increase in surface area associated with

crack branching (bifurcation). This is precisely the desirable situation for comminution as it leads to fragmentation.

2. 1 Limiting crack speed climit

Freund [1972a, 1972b, 1973, and 1990] has analysed a single propagating Mode I (opening mode) crack in

considerable and rigorous detail for a linearly elastic isotropic continuum. He determined the relationship between

the instantaneous dynamic energy release rate per unit area of crack plane G(d) (J m-2

) and the instantaneous dynamic

stress intensity KI(d) (Pa m1/2

) of the moving crack, as shown in Eq. (3) [Freund [1990]:

2

)()(

22

)(

2

)( )()1(

)()1(

sIcdId KgE

KE

G (3)

where KI(s) (Pa m1/2

) is the equivalent static stress intensity factor KIC of a stationary crack having the same

instantaneous length as the dynamic crack (c.f. Eq. 1). The function g(c) is the complex universal function of crack

speed which may be approximated by the simple relation in Eq. (4) [Freund 1972a, 1990; Rose 1976]:

R

crR

R

crc

c

cc

c

cg

1)( (4)

where ccr is the crack tip speed and cR is the Rayleigh surface wave velocity.


408

The value of cR for homogeneous isotropic solids in a medium having little or no inertia, such as air, may be

calculated via Eq. [5] (e.g. Vinh and Ogden [2004], where cT is the transverse (shear) acoustic body wave velocity,

is the density, G is the shear (rigidity) modulus and E is the tensile (Young’s) modulus:

21

2

221

2

22

2

2

1142

/

T

R

/

T

R

T

R

c

bc

c

c

c

c

(5)

where 2/1/ GcT with 1)/()(0 22 TR cc : )G/(Gb 2 with :b 10 )3/()2( EGGEG

N.B. A useful approximation is )1/()14.1862.0( TR cc ], Bergmann [1954] and Achenbach [1973].

Equation (4) is precise at the limits g(c) = 1 as ccr → 0, and g(c) = 0 as ccr → cR, indicating the theoretical limiting

speed of a propagating crack is cR, which is approached asymptotically when (if) G(d) remains unchanged. However,

there are two restrictions on Eqs. (3) and (4). In the first restriction G(d) does not remain constant. It increases with

increasing crack velocity and is possibly related to the observations that crack surfaces of brittle materials exhibit

increasing roughness with increasing crack velocity [Fineberg et al. 1991; Parisi and Ball 2005; Buehler and Gao

2006], and roughness increases continuously along the crack path [Ravi-Chandar 1998]. Empirically, it seems

reasonable to treat the influence of increasing surface roughness area on G(d) as the product of GIC and a factor that

increases with crack speed:

crR

RICd

cc

cGG )( (6)

where G(d = GIC when ccr = 0 and G(d > GIC for ccr > 0.

Fracture surfaces with a rough faceted appearance are readily obtained after crushing minerals, as shown in the

scanning electron microscope(SEM) images of crushed tennantite (a copper ore belonging to the tetrahedrite group)

and ilmenite (a titanium ore and major source of TiO2 pigment) in Fig. 2. Secondary micro-cracks penetrating the

primary fracture surface, indicative of crack branching, are arrowed.

Tennantite (Cu,Fe,Zn)12As4S13) Ilmenite (FeTiO3)

Figure. 2. SEM micrographs of crushed mineral particle surfaces.

The second restriction on Eq. (3) and Eq. (4) is that even though limiting crack velocities occur, as first observed in

glasses by Schardin and Struth [1938], the limiting terminal velocity climit of a Mode I crack is always lower than cR

[Rose 1976, Parisi and Ball 2005, Ravi-Chandar 1998, Sherman 2005]. The failure to achieve cR is generally

attributed to limitations of the continuum mechanics approach used to derive Eq. (1), which ignores the discrete

atomic microstructure at the crack tip as noted by Sieradzki et al. [1998] and Gilman [1988]. Other workers have

recognized the importance of atomistic phenomena and used generalized models of crack propagation based on

2m

m

2m

m


409

idealised lattices and non-linear elastic effects [Marder and Gross, 1995, Holian and Ravelo, 1995, Gao 1996,

Holian et al. 1997, Gumbsch and Cannon 2000, Buehler and Gao 2006]. While these models have improved the

general conceptual understanding of fast cracking, it should be noted that care is required with cracking models that

associate limiting crack velocities with Rayleigh wave behaviour or transverse acoustic wave velocities in solids.

Rayleigh surface waves and transverse acoustic waves both involve small linear elastic inter-atom displacements

normal to the plane of travel [Russell 2006], whereas inter-atom crack tip displacements normal to the fracture plane

are large and non-linear. Thus, inertia controlled bond breaking processes may impose greater limitations on crack

propagation than the release of crack tip strain energy via the generation of Rayleigh surface waves during passage

(propagation) of the crack tip.

2.2. Polycrystalline minerals

Most fundamental research on crack propagation behaviour has been directed towards cleavage fracture in single

crystals or crack propagation in brittle amorphous materials. Less attention has been applied towards the behaviour

of polycrystals. Mineral rock sizes encountered in comminution operations are usually of sufficient size to be treated

as polycrystal aggregates composed of individual monocrystals (grains). In the current study, crack propagation is

treated on the assumption that the individual grains (crystals) are small relative to the polycrystal and exhibit a

completely random orientation such that the overall linear elastic behaviour of the polycrystal aggregate is isotropic.

Under these circumstances the resulting behaviour may be determined from knowledge of the monocrystal

compliance constants Smn and stiffness constants Cmn:

S (7)

C (8)

Equation (7) corresponds to a uniaxial stress () situation with a three dimensional strain () where the constant of

proportionality is the compliance S (N.B. for tensile stresses S is the reciprocal of Young’s modulus E). Equation (8)

represents a three dimensional stress state with uniaxial strain and stiffness constant C (Nye 1985, Tromans 2011).

The first averaging method for obtaining polycrystal compliance and stiffness constants from monocrystal values

was developed by Voigt [1889] with a later summary on pages 962-963 of his Lehrbuch [Voigt 1928]. He based his

analysis on the assumption of homogeneous strain throughout the stressed polycrystal and, using the stiffness

constants Cmn he derived two general relation ships: (i) one for the bulk modulus K involving volume change

without shape change and (ii) one for the shear (rigidity) modulus G based on shape change without volume change,

as shown in Eq. (9), where the subscript V refers to the Voigt analysis:

15/)333(

9/)222(

665544133312332211

132312332211

CCCCCCCCCG

CCCCCCK

V

V

(9)

Later Reuss [1929] derived equations for K and G based on the compliances Smn and the assumption of uniform

stress throughout the polycrystal to produce the relationships in Eq. (10), where the subscript R refers to the Reuss

analysis:

15/)333444444(/1

)222(/1

665544133312332211

132312332211

SSSSSSSSSG

SSSSSSK

R

R

(10)

Equations (9) and (10) are then used to obtain the corresponding Voigt and Reuss values of Poisson’s ratio and

Young’s modulus E from standard relationships for isotropic materials in Eq. (11) [e.g. Hibbeler 1997],

KGEEG

EGEK

G

E

9

1

3

11;

)39()21(3;

12

; (11)

The required elastic constants Smn and Cmn for application of Eqs. (9) and (10) to some selected minerals are listed in

Table 1. The sources of the data, and the crystal structures, are indicated in the table footnote. Diamond is included

in Table 1 for comparison with other minerals although it usually occurs naturally as a monocrystal.


410

Table 1. Elastic compliance constants (Smn) and stiffness constants (Cmn) for selected minerals.

Mineral TPa-1

TPa-1

TPa-1

TPa-1

TPa-1

TPa-1

TPa-1

TPa-1

TPa-1

S11 S12 S13 S22 S23 S33 S44 S55 S66

(1) Periclase MgO 4.01

-0.960 -0.960 4.01

-0.960 4.01 6.47 6.47 6.47

(1) Lime CaO 5.05 -1.08 -1.08 5.05 -1.08 5.05 12.4 12.4 12.4

(2) Sphalerite ZnS 19.5 -7.60 -7.60 19.5 -7.60 19.5 22.5 22.5 22.5

(2) Galena PbS 8.46 -1.38 -1.38 8.46 -1.38 8.46 45.0 45.0 45.0

(2) Fluorite CaF2 6.93 -1.52 -1.52 6.93 -1.52 6.93 29.5 29.5 29.5

(1) Halite NaCl 22.9 -4.80 -4.80 22.9 -4.80 22.9 78.3 78.3 78.3

(1) Anorthite Al2Si2O8 11.2 -3.00 -2.35 6.00 -0.730 7.60 42.6 27.3 24.1

(3) Forsterite Mg2SiO4 3.35 -0.800 -0.730 5.86 -1.60 4.97 15.2 12.3 12.4

(1) Spinel MgAl2O4 5.80 -2.05 -2.05 5.80 -2.05 5.80 6.49 6.49 6.49

(1) Corundum Al2O3 2.38 -0.700 -0.700 2.38 -0.700 2.38 7.03 7.03 7.03

(3) Pyrite FeS2 2.80 -0.300 -0.300 2.80 -0.300 2.80 9.70 9.70 9.70

(4) Chalcopyrite CuFeS2 26.5 -8.60 -12.7 26.5 -12.7 28.7 39.2 39.2 35.9

(1) α-Quartz SiO2 12.8 -1.74 -1.32 12.8 -1.32 9.75 20.0 20.0 29.1

(1)Moissanite-6H,α-SiC 2.08 -0.37 -0.171 2.08 -0.171 1.80 5.92 5.92 4.90

(1) Diamond C 1.01 -0.140 -0.140 1.01 -0.140 1.01 1.83 1.83 1.83

(6) Borazone BN 1.336 -0.251 -0.251 1.336 -0.251 1.336 2.0833 2.0833 2.0833

(2) Ice-Ih (270 K) H2O 105 -44.0 -23.0 105 -23.0 86.0 338.0 338.0 298.0

Minerals 109 Pa 10

9 Pa 10

9 Pa 10

9 Pa 10

9 Pa 10

9 Pa 10

9 Pa 10

9 Pa 10

9 Pa

C11 C12 C13 C22 C23 C33 C44 C55 C66

(1) Periclase [C] 294 93 93 294 93 294 155 155 155

(1) Lime [C] 224 60 60 224 60 224 80.6 80.6 80.6

(2) Sphalerite [C] 102 64.6 64.6 102 64.6 102 44.6 44.6 44.6

(2) Galena [C] 127 24.4 24.4 127 24.4 127 23 23 23

(2) Fluorite [C] 165 47 47 165 47 165 33.9 33.9 33.9

(1) Halite [C] 49.1 12.8 12.8 49.1 12.8 49.1 12.8 12.8 12.8

(1) Anorthite [M] 124 66 50 205 42 156 23.5 40.4 41.5

(3) Forsterite [O] 328.4 63.9 68.8 199.8 73.8 235. 65.9 81.2 80.9

(1) Spinel [C] 282 154 154 282 154 282 154 154 154

(1) Corundum [Trg] 495.00 160.00 115.00 495.00 115.00 497.00 146.00 146.00 167.50

(3) Pyrite [C] 366 49 49 366 49 366 103 103 103

(4) Chalcopyrite [Tet] 89.8 61.3 66.9 89.80 66.90 94.1 25.5 25.50 89.8

(1) α-Quartz [Trg] 86.6 6.7 12.60 86.60 12.60 106.10 57.8 57.80 39.95

(1) Moissanite-6H [H] 502 95 96 502.00 96.00 565 169 169.00 203.50

(1) Diamond [C] 1040 170 170 1040 170 1040 550 550 550

(5) Borazone [C] 820 190 190 820 190 820 480 480 480

(2) Ice-Ih (270 K) [H] 13.70 7.00 5.60 13.70 5.60 14.70 2.96 2.96 3.35 (1) Hearmon [1979]: (2) Hearmon [1984]: (3) Gerbrande [1982]: (4) Sirota and Zhagasbekova [1991] with reservations by

Łażewski et al. [2004]: [5] Grimsditch et al. [1994.

[C]-cubic: [H]-hexagonal:[M]-monoclinic: [O]-orthorhombic: [Tet]-tetragonal: [Trg]=trigonal.

Hill [1952] critically examined the analyses of Voigt [1889, 1928] and Reuss [1929] and determined that Voigt’s

assumption of constant strain disallowed equilibrium of the forces between grains, whereas the Reuss assumption of

homogeneous stress would not allow distorted grains to fit together. Based on a consideration of energy densities

Hill [1952] concluded that the Voigt moduli EV and GV should exceed the Reuss moduli ER and GR and both were

approximations defining upper and lower bounds of E and G with a mean value between. Results of the Voigt and

Reuss averaging methods for the minerals in Table 1 are listed in Table 2, together with the mean GV-R, EV-R and V-R

averages.


411

Table 2. Isotropic polycrystal moduli based on Voigt (V) averaging, Reuss (R) averaging and mean (V-R) values

Mineral

GV

(GPa)

EV

(GPa)

GR

(GPa)

ER

(GPa)

GV-R

(GPa)

EV-R

(GPa) V-R

Periclase MgO 133.2 312.80 127.26 301.57 130.23 307.18 0.1794

Lime CaO 81.16 197.0 81.01 196.93 81.09 196.96 0.2145

Sphalerite ZnS 34.24 89.47 28.43 75.99 31.33 82.73 0.3202

Galena PbS 34.32 86.14 28.68 73.94 31.50 80.04 0.2706

Fluorite CaF2 43.94 112.70 40.88 105.82 42.41 109.26 0.2881

Halite NaCl 14.94 37.35 14.46 36.39 14.70 36.87 0.2539

Anorthite Al2Si2O8 42.88 110.84 37.0 96.05 39.94 103.44 0.2951

Forsterite Mg2SiO4 82.73 204.96 79.44 197.0 81.09 200.98 0.2393

Spinel MgAl2O4 118 295 98.29 252.65 108.14 273.83 0.2660

Corundum Al2O3 165.03 406.35 159.07 393.18 162.05 399.77 0.2335

Pyrite FeS2 125.2 295.79 120.48 285.71 122.84 290.75 0.1834

Chalcopyrite CuFeS2 21.0 57.54 18.62 51.48 19.81 54.51 0.3759

α-Quartz SiO2 47.60 100.84 40.97 90.16 44.28 95.50 0.0782

Moissanite-6H, α-SiC 193.77 457.26 195.06 451.81 194.41 454.54 0.1690

Diamond 504 1107.52 495.54 1091.70 499.77 1099.61 0.1001

Borazone BN 414 923.42 396.85 894.67 405.43 909.05 0.1211

Ice-Ih (270 K) H2O 3.447 9.138 3.359 8.918 3.403 9.028 0.3265

2.3 A bond-breaking model for climit.

The development of a simple bond by bond breaking model for brittle crack propagation in polycrystalline minerals

is shown Figs .3. and 4. Models of this type, where the crack advances step by step, are often referred to as lattice

trapping models [Gumbsch and Cannon 2000, Zhu et al. 2006]. For purposes of analysis, the average equilibrium

spacing R0 between atoms in the polycrystalline mineral is taken to be equal to N-1/3

, where N is the total number of

atoms per unit volume (m-3

). The atoms are then considered to be distributed over a cubic lattice of average

equilibrium atom spacing R0, as shown in Figure 3. A tensile stress σ is applied in the Z direction normal to the X-Y

plane.

R o

R o

R oR o

x

z

y

Figure 3. N atoms/m3 distributed over a cubic lattice with equilibrium spacing Ro (Ro = N-1/3)

A propagating planar Mode I crack is introduced in the lattice, lying in the X-Y plane, with the remote tensile stress

σ acting in the Z-direction. The situation at the crack tip is shown schematically in Figs. 4(a) and 4(b). In Fig. 4(a)

bonds across atom pairs A-A and B-B have broken during passage of the crack. These atoms now lie on a newly

formed crack surface and are no longer subjected to large forces (f) acting across the crack plane in the Z-direction.

The atom pair C-C at the crack tip is on the verge of separation under the maximum bond breaking force, fmax, with a

corresponding atom separation distance of Rmax. The distances R1, R2 and R3 between atom pairs D-D, E-E and F-F

ahead of the crack tip are considered to remain sufficiently close to R0 so that the separating forces f1, f2, and f3

acting across them are negligibly small relative to fmax. The local force acting on each of these atoms atom is simply

σ/N2/3

in units of Newtons. Similarly, where max is the maximum theoretical tensile stress (cohesive stress) for bond

breaking, 3/2

maxmax / Nf .


412

x

z

y

(a) f max

A

A B

B

D FC

CD E

E

F

Rmax

Ro

Crack Propagation

Ro

R3

f max

R1 R2

f1 f3

f1

f2

f3f2

Ro

x

z

y

(b) f max

f max

C

C

ED

DE F

F

Rmax

Ro

R1 R2

A

A B

B

Crack Propagation

f2f1

f1 f2

Ro

Figure 4.(a) and (b): Progress of crack tip from situation in (a) to (b) in presence of a remote

tensile stress showing local forces f on atoms.

Referring to Fig. 4(a), the essence of the model is to recognize that as soon as the bond across the crack tip pair C-C

breaks, as in Fig 4(b), the maximum load fmax is transferred to the next atom pair D-D which, while initially under

the force f1 << fmax), is suddenly subjected to a force fmax. Thus, an atom in the D-D pair experiences an initial

acceleration (accI) of ~ fmax/mc relative to the other atom in the pair, where mc is the average mass of an atom in the

crystal obtained from the density (kg m-3

) and N (m-3

):

2

maxI sm ,/NN/Nm/facc /

max

/

maxc

3132 (12)

As the D-D bond stretches, the accelerating force decreases from fmax at R = R0 to zero at R=Rmax where the bond

breaks. Thus, from Eq. (12), the acceleration accR at any stretch position between R0 and Rmax is given by Eq. (13):

2sm ,/N)(acc /

RmaxR

31 (13)

The function σR describes the relationship between tensile stress and atom separation R at the crack tip. Atom pairs

A-A, B-B, C-C, in the wake of the crack tip at fmax continue to experience a decreasing attraction force that reaches

a negligible value after a distance of a few multiples of R0 from the crack tip, as shown by Tromans and Meech

[2002, 200]. These forces play no role in the crack tip acceleration accR.

The σR function for a macroscopically isotropic brittle material (i.e. a randomly oriented polycrystalline aggregate)

at 298 K has been calculated previously for different ionic crystals with a Born-type model [Tromans and Meech

2002], and for covalent crystals with a Morse-type model [Tromans and Meech 2004]). The ionic model in SI units

is given by Eq. (14):

Pa)(

)(

)(

1

4)(

3

211

1

0

2

23/2

n

n

RR

R

R

eMN

0E (14)

where M is a modified Madelung constant; e is the elementary charge (1.602177 x 10-19

C); E0 is the permittivity in

vacuum (8.854188 x 10-12

CV-1

m-1

); and n is a number > 1 related to the bulk modulus. Values of N (atoms per m3),

R0, Mn, and the enthalpy of formation Hfgas ions to solid) for polycrystals treated as ionic are listed in Table 3,

together with calculated EV-R and V-R values from Table 2. The Hf was used in the calculation of M and n.


413

Table 3. Values of polycrystalline parameters for Equation 14, ionic model [Tromans and Meech, 2002].

Mineral N

1028

m-3

R0

10-10

m

n

Hf

kJ mol-1

EV-R

GPa V-R

Periclase MgO 10.7159 2.1054 3.9154 4.1259 -3915.74 307.18 0.1794

Lime CaO 7.1872 2.4052 3.8322 4.9177 -3527.9 196.96 0.2145

Sphalerite ZnS 5.0649 2.70278 4.519 4.5320 -3621.36 82.73 0.3202

Galena PbS 3.8249 2.9680 4.049 5.3476 -3082.62 80.04 0.2706

Fluorite CaF2 7.3614 2.3861 1.7565 7.1862 -2641.76 109.26 0.288

Halite NaCl 4.4585 2.8201 0.9164 7.7245 -786.51 36.87 0.254

Anorthite CaAl2Si2O8 7.7705 2.3434 3.537 3.7892 -20069.79 103.28 0.295

Forsterite Mg2SiO4 9.6555 2.180 2.233 6.1216 -8337.33 201.39 0.242

Spinel MgAl2O4 10.6062 2.1126 5.872 3.581 -19485.91 273.83 0.266

Corundum Al2O3 11.7819 2.0398 6.256 3.6989 -15547.20 399.76 0.2335

Pyrite FeS2 7.5502 2.3660 1.931 10.0622 -3063.90 296.05 0.1553

Chalcopyrite CuFeS2 5.4877 2.6315 5.4317 3.5212 -8214.51 54.51 0.3759

The covalent model in SI units is given by Eq. (15):

Pa)(exp)(2exp3

)21(2σ

3/1

e RRRRN

Uococ

cR

(15)

The Ue is the equilibrium binding energy (J m-3

) for N atoms at R = R0 and αc is a covalent bonding constant (m-1

).

Mineral values for N, R0, Ue , c and enthalpy of formation Hf (gas atoms to solid), for polycrystals treated as

covalent, are listed in Table 4 together with calculated EV-R and V-R values from Table 2.

Table 4. Values of polycrystalline parameters for Equation 15, covalent model [Tromans and Meech 2004].

Mineral N

1028

m-3

Ro

10-10

m Hf

kJ mol-1

Ue

108 Jm

-3

c

1010

m-1

EV-R

GPa V-R

α-Quartz SiO2 7.96684 2.32402 -1859.05 -0.81968 0.61936 95.5 0.078

Moissanite-6H, α-SiC 9.72992 2.17419 -1238.585 -1.0004 1.4757 454.54 0.169

Diamond C 17.6377 1.78315 -714.786 -2.0932 1.7603 1099.6 0.100

Borazone BN 16.91876 1.80805 -1288.178 -1.8093 1.7442 909.05 0.121 †Ice-Ih (270 K) H2O 9.1959 2.2155 -977.215 -0.49734 0.3998 9.028 0.326

(GPa)

2E-10 2.4E-10 2.8E-10 3.2E-100

10

20

30

40

Periclase MgO

R (m)

max

Rmax

E

R 0

(GPa)

2.4E-10 2.8E-10 3.2E-10 3.6E-100

5

10

15

20

R (m)

Alpha-quartz SiO2

max

Rmax

R 0

E

Figure 5. Atom separation (R) versus tensile stress () of MgO and SiO2.

Examples of R curves based on the ionic model for periclase (MgO) and the covalent model for -quartz (SiO2) are

shown in Fig. 5. The curves are non linear between R0 and Rmax, departing significantly from the linear elastic

behaviour depicted by the tensile modulus line labelled E, and reach a maximum tensile stress max at Rmax. The

limiting crack velocity is obtained by inserting Eqs. (14) or (15) into. Eq. (13) and calculating by iteration the total


414

time ttotal for the crack tip bond (e.g. D-D in Fig. 4) to stretch from R0 to breakage at Rmax. At this point, the load is

transferred to the bond E-E and the crack tip moves forward a distance of R0. Thus, the maximum limiting crack

velocity is obtained:

totallimit tRc /0 (16)

The iteration procedure for ttotal involves calculating the acceleration and velocity of the separating atom for small

ΔR increments of 1 x 10-13

m. First, the initial acceleration accI is calculated from Eq. (12) for R = R0. The accI is

assumed to be constant over the first increment (ΔR)1. This allows calculation of the time increment (Δt)1 to move

from R0 → R0 + (ΔR)1 and calculation of the velocity V1 of the separating atom at the end of (ΔR)1. A second

increment of separation (ΔR)2 is now added with an initial velocity V1. The acceleration acc2 at the beginning of the

increment is calculated from Eq. (13) for R = R0 + (ΔR)1. Again acc2 is assumed to be constant over the second

increment (ΔR)2. The time increment (Δt)2 to stretch from R0 + (ΔR)1 → R0 + (ΔR)1+ (ΔR)2 is calculated from the

conjoint effects of acc2 and V1, and the velocity V2 at the end of the second increment (ΔR)2 is calculated. The

iteration procedure is repeated for n increments until

nn

n

n

1

0max )( RRRR and

nn

n

n

1

)( tttotal . Throughout this

period of time the acceleration of the separating atom decreases to zero at R=Rmax and its velocity increases from

zero to a maximum value at R=Rmax The final climit values obtained from Eq. (16) are shown in Table 5 for all

minerals in Tables 3 and 4, together with the final separation velocity Vmax of crack tip atoms normal to the crack

plane.

Table 5. Calculated limiting crack speeds climit and atom separation velocity Vmax in polycrystalline minerals.

Mineral Ro

(10-10

m)

Rmax

(10-10

m)

ρ

(103 kg m

-3)

max

(GPa)

climit

(m s-1

)

Vmax

(ms-1

)

*Periclase MgO 2.1054 2.8450 3.585 32.817 2751 1278

*Lime CaO 2.4052 3.1725 3.346 19.130 2282 962.9

*Sphalerite ZnS 2.70278 3.6243 4.097 8.990 1360 610.0

*Galena PbS 2.9680 3.8711 7.597 7.4122 965 388.3

*Fluorite CaF2 2.3861 2.9965 3.181 8.4620 1737 587.2

*Halite NaCl 2.8201 3.5108 2.163 2.727 1221 395.1

*Anorthite CaAl2Si2O8 2.3434 3.2049 2.761 11.530 1816 882.8

*Forsterite Mg2SiO4 2.180 2.7932 3.222 17.222 2349 874.0

*Spinel MgAl2O4 2.1126 2.9125 3.579 31.448 2595 1299

*Corundum Al2O3 2.0398 2.7993 3.989 45.175 2972 1463

*Pyrite FeS2 2.3660 2.8575 5.013 18.347 2255 617.9

*Chalcopyrite 2.6315 3.6366 4.180 6.313 1071 540.9 †α-Quartz SiO2 2.3240 3.4432 2.649 16.587 1999 1297

†Moissanite-6H α-SiC 2.1742 2.6439 3.239 35.417 3947 1150

†Diamond C 1.7832 2.1769 3.517 87.579 5894 1755

†Borazone BN 1.8081 2.2055 3.486 72.063 5383 1595

†Ice-Ih (270 K) H2O 2.2136 3.9514 0.9195 2.601 1052 1113

*Ionic model Eq. (14): †Covalent model Eq. (15).

2. 4. Velocity of transverse and longitudinal acoustic waves in polycrystals

The velocity cT of transverse acoustic waves in isotropic materials is dependent upon the density and the elastic

stiffness shear constant C44 of the material:

2/1

44 )/( CcT (17)

Similarly the velocity cL of longitudinal acoustic waves in isotropic materials is dependent upon the density and

the elastic stiffness tensile constant C11 of the material:

2/1

11 )/( CcL (18)


415

Matrices of the elastic stiffness (C) and compliance (S) constants for isotropic materials are well established and

contain only two independent stiffness constants and two independent compliance constants, as shown in Eq. (19)

[e.g. Nye 1985, Tromans 2011]:

44

44

44

111212

121112

121211

C

C

C

CCC

CCC

CCC

44

44

44

111212

121112

121211

S

S

S

SSS

SSS

SSS

(19)

2/)( 121144 CCC )(2 121144 SSS

Clearly, 1/G =S44=2(S11-S12), and since S11=1/E, then S12= (1/E)-(1/2G) and the stiffness constants of an

isotropic polycrystal may be obtained from the compliances via matrix inversion. This has been done, based on the

EV-R ( 1/S11) and GV-R ( 1/S44) moduli in Table 2, and the results listed in Table 6 which show that C44 is equal to

the corresponding GV-R, consistent with Eq. (5), and C11 > EV-R. The corresponding cT and cL, based on Eqs. (17 and

(18) and density data in Table 5, are included in Table 6, together with the Rayleigh surface wave speed cR based on

Eq. (5). The calculated limiting crack velocities climit from Table 5 are included Table 6.

Table 6. Independent elastic stiffness constants and acoustic wave speeds in poly crystalline minerals

Mineral

C44

( G )

GPa

C11

GPa cT

m s-1

cL

m s-1

cR

m s-1

climit

m s-1

climit/cT

climit/cL

climit/cR

*Periclase MgO 130.23 333.32 6027.1 9642.4 5472.79 2751 0.4564 0.2853 0.5027

*Lime CaO 81.09 223.08 4922.9 8165.2 4500.2 2282 0.4635 0.2795 0.5071

*Sphalerite ZnS 31.33 118.5 2765.3 5378.1 2573.61 1360 0.4918 0.2529 0.5284

*Galena PbS 31.50 100.12 2036.3 3630.3 1879.97 965 0.4739 0.2658 0.5133

*Fluorite CaF2 42.41 142.5 3651.3 6693.1 3380.78 1737 0.4757 0.2595 0.5138

Halite NaCl 14.70 44.59 2606.9 4540.4 2400 1221 0.4684 0.2689 0.5088

*Anorthite CaAl2Si2O8 39.94 137.33 3803.4 7052.6 3525.61 1816 0.4775 0.2575 0.5151

*Forsterite Mg2SiO4 81.09 236.58 5016.7 8568.9 4606.73 2349 0.4682 0.2741 0.5099

*Spinel MgAl2O4 108.14 339.3 5496.8 9736.7 5070.98 2595 0.4721 0.2665 0.5117

*Corundum Al2O3 162.05 466.06 6373.7 10809.1 5846.76 2972 0.4663 0.2750 0.5083

*Pyrite FeS2 122.84 316.87 4950.2 7950.4 4498.45 2255 0.4555 0.2836 0.5013

*Chalcopyrite CuFeS2 19.81 99.57 2176.7 4880.0 2042.66 1071 0.4920 0.2195 0.5243

†α-Quartz SiO2 44.28 96.79 4088.5 6044.7 3631.12 1999 0.4889 0.3307 0.5505 †Moissanite-6H, SiC 194.41 488.10 7747.4 12275.8 7020.2 3947 0.5095 0.3215 0.5622 †Diamond 499.77 1124.66 11920.6 17882.3 10642.1 5894 0.4944 0.330 0.5538 †Borazone BN 405.43 940.43 10784.4 16424.8 9673.62 5383 0.4991 0.3277 0.5565 †Ice-Ih (270 K) H2O 3.403 13.21 1923.8 3790.3 1792.0 1052 0.5468 0.2776 0.5871

*Ionic av. Ratios climit/cR = 0.512 0.008: climit/cT = 0.472 0.012: climit/cL = 0.266 0.018 †Covalent av. Ratios climit/cR = 0.562 0.015: climit/cT = 0.508 0.023: climit/cL = 0.278 0.023

Average climit/cR, climit/cT and climit/cL ratios based on the ionic model and covalent model are included at the base of

Table 6, together with their standard deviations (STD). It is evident that covalent ratios tend to exceed ionic ratios.

This may be due to approximations in the different models or the possibility that covalent bonding tends to produce

higher climit values. The lowest deviations are exhibited by the climit/cR ratios.

2.4 Model testing with glassy (amorphous) materials.

No data are available on limiting crack velocities in mineral polycrystals with which to check the usefulness of Eqs.

(14) and (15). However, there are limited data on some glassy materials to which the covalent model based on Eq.

(15) may be applied. Compositions and properties of these are listed in Table 7 and the resulting parameters required

for the covalent model are listed in Table 8.


416

Table 7. Composition and properties of some glassy materials

Material

Composition

mol %

ρ

(kg

m-3

)

G

GPa

E

GPa

Silica glass SiO2, Spinner [1954] 100 SiO2 2202 31.39 72.94 0.162

Vitreous Carbon, Dobbs [1974] 100 C 1514 12.953 30.25 0.1676

Soda Lime Float Glass.

Congleton and Petch [1967]

Küppers [1967], Dobbs [1974]

Typical compositions: 71.4 SiO2,

13.19 Na2O, 8.77 CaO, 5.625

MgO, 0.64 Al2O3, 0.375 K2O

2530 29.3 72 0.23

Glass #17 Schardin [1959] 81.19 SiO2, 15.5 PbO, 3.31 Na2O 4790 22.32 55.358 0.24

Table 8. Values of glassy materials data for covalent model based on Equation (15).

Mineral

N

1028

m-3

Ro

10-10

m Hf

kJ mol-1

Ue

108 Jm

-3

c

1010

m-1

Rmax

10-10

m max

GPa

Silica glass SiO2 6.62155 2.4718 -1849.53 -0.677874 0.630862 3.57054 11.6875

Vitreous Carbon C 7.59156 2.3617 -713.09 -0.898927 0.368959 4.24035 8.6789

Soda lime glass 7.37676 2.3844 ------------ -0.747192 0.686154 3.39459 11.00209

Glass #17 9.60601 2.1835 ------------ -9.16878 0.604403 3.33033 10.4868

For silica glass, the enthalpy of formation f was based on the on the difference between the sum of the enthalpies

of the individual atoms (Si and O) in the gas phase and the crystal phase, using the data compiled by Cox et

al.[1989], and corrected for the enthalpy difference of 9 kJ mol-1

between crystalline quartz and the glass phase

obtained from Holm et al. [1967]. The enthalpy of formation of vitreous carbon was based on the difference

between the enthalpies of the C atoms in the gas phase and the graphite phase, using the data compiled by Cox et al

[1989], and corrected for an enthalpy difference of 3.585 kJ mol-1

between graphite and vitreous carbon obtained

from Takahashi and Westrum [1970)]

The, overall Ue of the soda lime glass was estimated by dividing it into the corresponding mole proportions of SiO2,

Na2SiO3, K2SiO3, CaAl2Si2O8, CaMgSi2O6, CaSiO3. Enthalpy differences between gas atoms and the three

components CaAl2Si2O8, CaMgSi2O6, CaSiO3 were obtained from data compiled by Barin [1995], and enthalpy

differences between gas atoms and the Na2SiO3 and K2SiO3 components were obtained from Roine [1999]. The

Schardin #17 glass was treated as if it was composed of SiO2, Na2SiO3 and PbSiO3, using the data of Roine [1999]

to obtain enthalpy differences between gas atoms and the Na2SiO3 and PbSiO3 components. Predicted estimates of

limiting crack velocities in glassy materials based on Eq. (15) and the data in Table 8 are listed in Table 9, together

with reported experimental values.

Table 9. Predicted limiting crack behaviour of glasses.

Glassy Material climit

(m s-1

)

Vmax

(m s-1

)

Experimental climit

(m s-1

)

Silica glass SiO2 1915.5 1147.6 2200 Schardin and Struth [1938]; 2155 Schardin [1959]

Vitreous Carbon C 1487.6 1594.9 1400 Dobbs [1974]

Soda lime glass 1776.1 988.8 1550 Dobbs [1974; 1525 Küppers [1967]

~1750 Congleton and Petch [1967]

Glass #17 1131.7 801.1 750 Schardin [1959]

There is a sufficiently fair agreement between predicted and experimental limiting crack velocities in Table 9 to give

encouraging support for the model equations employed for estimation of limiting crack behaviour in polycrystal

mineral aggregates in Table 5. For example, the limiting crack velocity of ice in Table 5 is 1052 m s-1

, which

compares favourably with a measured value of 1050 ± 130 m s-1

by Arakawa et al. [1995].

2.5. Energy release rate at climit. With reference to the model in Fig. 3, where N atoms per m3 are distributed over a

cubic lattice, it is evident from Fig. 4 that the number of broken bonds per unit length of crack front is 1/R0=N1/3

as

atoms D-D are stretched from the initial separation R0 through R1 to final separation at Rmax. At climit the total energy

released per unit area of crack propagation is the sum of the kinetic energy (KE) of N2/3

separating atoms of mass

moving at Vmax (see Table 5), and the work done W in separating N2/3

atoms from R0 to Rmax in increments of

R under the accelerating force f. The KE is obtained from Eq. (20) and W is obtained from Eq. (21).Both values


417

are listed in Table 10. [N.B. Eq. (21) applies to a planar atom surface area; it will be larger for crack plane areas with

rough surfaces, i.e. larger effective atom surface area per unit area of crack plane.]

)N/()V(KE /

max

312 2 (20) max

/R

R

)R(fNW

0

32 (21)

Table 10. Calculated values of kinetic energy (KE) and Work (W) done at climit

Mineral

KE

J m-2

W

J m-2

Mineral

J m-2

KE

J m-2

W

J m-2

Mineral

J m-2

KE

J m-2

W

J m-2

MgO 0.6140 0.6135 CaAl2Si2O8 0.2510 0.2510 -SiO2 0.5181 0.5164

CaO 0.3714 0.3711 Mg2SiO4 0.2670 0.2665 α-SiC 0.4653 0.4618

ZnS 0.2051 0.2051 MgAl2O4 0.6355 0.6351 Diamond 0.9653 0.9566

PbS 0.1693 0.1693 Al2O3 0.8670 0.8664 BN 0.8016 0.7944

CaF2 0.1303 0.1300 FeS2 0.2254 0.2246 Ice-Ih 0.1261 0.1258

NaCl 0.0474 0.0473 CuFeS2 0.1602 0.1603 *Glass 0.2945 0.2934

Soda lime glass from Table 9 is included in Table 10 for comparison with the minerals. Within the limits of the

iteration procedures, the KE and W values for each mineral in Table 10 are equal, as expected. The total energy

release rate per unit area of crack plane per second at climit is obtained from Eq. (22) and listed in Table 11.

12sJ.m limitc)WKE(T (22)

Table 11. Calculated values of total energy release rate T per (metre)2 per second at clmit.

Mineral

T

J.m-2

s-1

Mineral

J m-2

s-1

T

J.m-2

s-1

Mineral

J m-2

s-1

T

J.m-2

s-1

Periclase MgO 3377 Anorthite CaAl2Si2O8 911.6 -Quartz SiO2 2068

Lime CaO 1694 Forsterite Mg2SiO4 1253 Moissanite-6H α-SiC 3659

Sphalerite ZnS 557.9 Spinel MgAl2O4 3297 Diamond C 11328

Galena PbS 326.7 Corundum Al2O3 5152 Borazone BN 8591

Fluorite CaF2 452.1 Pyrite FeS2 1015 Ice-Ih (270 K) H2O 265

Halite NaCl 115.6 Chalcopyrite CuFeS2 343.5 Soda Lime Plate glass 1044

It is important to recognize that the sum of the KE and W in Table 10 and Eq. (22) is not the actual (conventional)

free surface energy of the mineral even though it has the same units (J m-2

). It is the critical energy released during

the bond breaking process at the crack tip under climit conditions. Crack surfaces immediately behind the crack tip are

still opening under decreasing stress conditions and the energy released during this process contributes to the overall

energy required to produce new stress-free surfaces, as evident in the analyses of Tromans and Meech [2002, 2004].

Consequently, the total energy rate T in Eq. (22) and Table 11 is an energy release rate per second associated

specifically with the crack tip bond breaking process. In effect, it is a critical rate process at the crack tip allowing

climit cracking to occur, being a mechanical process somewhat analogous to activation energy in a chemical reaction.

3. CRACK GROWTH BEHAVIOUR

3.1 Growth at constant stress c. When a mineral particle is subjected to compressive loading during comminution some defects (cracks) experience

tensile stress components that are dependent upon their orientation with respect to the compression axis [Tromans

2008]. Crack propagation occurs if the tensile component reaches a critical value c that is sufficient to generate the

critical static stress intensity factor KIC at the crack tip as indicated in Eqs. (1) and (2). The initial propagation rate is

likely to be < climit and increase towards climit as the crack extends and the load increases. However, if the load (i.e.

stress c) is maintained constant once the cracking is initiated, the crack will continue to extend with increasing

velocity towards climit due to the increasing stress intensity associated with the increasing crack length ai. It is

possible to estimate the instantaneous growth behaviour of a penny shaped crack of initial radius a as the crack

velocity increases from zero to climit , after crack growth starts at an initial stress intensity KI(s) and the stress is

maintained constant at c (see Eq. (2)). This is achieved by combining Eqs. (3), (4) and (6) and replacing KI(s)


418

by 2/1)/(2 ic a , where ia is the instantaneous crack length (radius) due to crack growth and ccr is the corresponding

crack tip speed, so obtaining Eq. (23):

ic

R

crR

crR

RICd

a

c

cc

Ecc

c 22

)(

)(4)1(GG …for ia > a and ccr ≤ climit (23)

After rearrangement, Eq. (23) becomes:

ic

R

crRIC

a

c

cc

E

21

2 )(4)1(G …for ia > a and ccr ≤ climit (24)

Noting that ccr = 0 when aai , it is evident from Eq. (24) that

1)/()/(

crRRi cccaa for ccr ≤ climit (25)

It seems reasonable to assume that by the time ccr = climit the effective surface area has increased by a factor > 1

(i.e. ICd GG /)( > 1), and )(dG has reached a limiting value ( )(dG )limit , so that Eq. (6) becomes

)d(IC)d( (/ GGG )limit / ICG = [cR /(cR - climit)]

If cR, climit and β are known, crack growth behaviour at constant stress is determined via Eq. (25), as shown in Fig. 5.

C

(m/s

)cr

a /ai

1 2 3 4 5 6 7 8 9 100

250

500

750

1,000

1,250

1,500

1,750

climit

= 1.25

Kuppers data..

= 1.5

= 1

Figure. 5. Comparison between Küppers [1967] experimental crack growth data and Eq.( 25) with different .

At this point, β is uncertain because is uncertain. However, based on the roughened (faceted) fracture surfaces of

tennanite and ilmenite in Fig. 2, it is reasonable to assume is of the order of ~2, allowing β to be calculated from

Eq. (26). A β-value that is > 1 < 2 tends to be supported by Küppers [1967] work on glass plates, where ai/a

behaviour at constant stress (σc) was measured for initial crack lengths between 2 mm and 8 mm, giving a climit of

~1525 m s-1

. Assuming it was a typical soda lime glass with the properties listed in Table 7, a calculated value of

3117 m s-1

for cR was obtained from Eq. (5). The corresponding values of β calculated from Eq. (26) are 1 when =

1.96, 1.25 when = 2.316, and 1.5 when = 2.74. The resulting ai/a crack growths predicted by Eq. (25), using

these β values, are plotted in Fig. 5.

Küppers data are included in Fig. 5, where the shaded region encompasses his reported variance of 6%. An average

value of 1.25 for β appears to provide the most satisfactory overall fit to the data. Assuming this is a reasonable

average value for many minerals it is now possible to use Eq. (25) to estimate the ai/a behaviour of other minerals

listed in Table 6. Examples are shown in Fig. 6(a) for constant stress (c) crack growth in galena, sphalerite and -

quartz using the climit values listed in Table 6. It is evident that large initial cracks (large a) have to propagate larger


419

distances (larger ai) to achieve the same crack speed. An example of the roughened (faceted) fracture surface

topography of sphalerite (ZnS, an important zinc ore) is shown in Fig. 6(b) where micro cracks penetrating the main

fracture surface are indicated by arrows.

C (

m/s

)cr

a /ai

-Quartz

Sphalerite

Galena

c = ccr limit

branching

1 2 3 4 5 6 7 8 9 10 11 12 13 140

500

1,000

1,500

2,000

2,500

Figure. 6: (a) Crack growth behaviour: Equation (25), = 1.25 (b) Sphalerite (ZnS) fracture topography.

It must be recognized that that although = 1.25 is a useful approximation for modelling crack propagation, it is

possible there is a continuous increase from = 1 at the onset of cracking to > 1 throughout subsequent crack

propagation, leading to even further increases in the surface roughness factor given by Eq. (6) and a more significant

increase in surface roughness along the crack path as climit is approached. This suggestion is consistent with

observations of an initial mirror-like fracture surface on glassy fractures that changed to a “mist” appearance

followed by a hackle region as the crack propagated [Mecholsky et al., 1974: Dobbs, 1974]. Originally, Shand

[1959] described the mist regions as being stippled. Most importantly, Mecholsky [2009] has shown that on the

micro scale the appearance of the mist region is similar to the hackle region, the only difference being one of scale,

and noted that the mirror region also exhibited small surface perturbations.

3.2. Dynamic crack branching (bifurcation) at climt. In the treatment of crack branching the term bifurcation is

used to describe the situation where a crack travelling at climit branches into two cracks each travelling at climit .It is

evident that the dynamic energy release rate (G(d))limit, as defined by Eq. (26), increases no further despite an

increasing instantaneous stress intensity associated with ongoing crack propagation (increasing ai). It is not possible

for a single crack tip to release the strain energy any faster. Further increases in local strain energy must be

accommodated (released) in some other manner, possibly by generation of heat (phonons) or branching attempts at

the crack front to provide increased surface area. However, it is not possible for the crack to branch (bifurcate) into

two main secondary cracks, each travelling at climit, until the strain energy associated with increasing ai reaches a

value of 2(G(d))limit. For example, in the case of -quartz, when the strain energy to be released is 1.5(G(d))limit and the

primary crack continues at climit the secondary branch crack speed is only ~0.39 climit (see Fig 6(a) and Eq.(26) for =

1.25). Therefore, the likely scenario is the continuous formation and dying of microcracks where the primary crack

travels at climit and the microcracks move very small distances at speeds << climit before dying as they fall behind the

strain field of the more rapidly advancing crack tip which eventually bifurcates at 2(G(d))limit. Subsequently, in turn,

each main secondary branch crack may reach a length where the increasing strain at its crack tip allows another

stable bifurcation propagation of two new secondary branches moving at climit.

The concept of crack branching to maintain energy balance is not new [Beauchamp, 1974], but it is most important

to recognize that the dynamic stable bifurcation condition occurs at the climit speed and the branches also propagate at

climit, as is clearly evident from the glass fracture studies of Schardin [1959]. Any attempt at stable bifurcation below

climit will result in two cracks with decreased equal crack speeds, for which there appears to be little evidence.

Therefore, applying a (G(d))limit to Eq. (23), where alimit is the crack length when the condition ccr = climt first occurs,

and applying the 2(G(d))limit condition to Eq. (23) where abranch is the crack length at bifurcation where the condition

ccr = climt still holds, a simple relationship is obtained:

10m


420

2itlim

branch

itlim

)d(

branch

)d(

a

a

aa

GG2 (27)

Hence the ratio ai /a at bifurcation branching is twice the ai /a ratio at climit. Based on Eq. (27) and the calculated

limiting crack velocities in Table 6, the ai /a ratios for climit and bifurcation conditions in -quartz, sphalerite and

galena are indicated on Fig. 6. Furthermore, the estimated ai /a ratio for bifurcation in Küppers [1967] glass (see Fig.

5) is between ~8.8 and 9.8. Since his reported data did not exceed ai /a > 9 and branching was unreported, it is

reasonable to assume no bifurcation branching was observed below ~9, consistent with Eq. (27).`

3.2.1.Biurcation angles. Consider an isotropic body with a crack lying in the x-y plane, as shown in Fig. 7(a),

where the orthogonal axes are the x, y. z directions .The crack front AB is parallel to the y-axis, the tensile stress

normal to the x-y plane is z and the tensile stress parallel to the x-direction is x. The direction r lies in the x-z plane

with its origin on the crack front The angle between r and the x-axis is θ degrees. The total angle subtended by a

crack bifurcation is 2θ, as shown in Fig. 2(b).

(a)

(b)

x

y

z

A

B

z

x

r

crack

bifurcation

(c)

degreesN

orm

alis

ed

N

0 2 4 6 8 10 12 141

1.01

1.02

1.03

1.04

1.05

Figure 7. (a) Stresses in cracktip region; (b) symmetrical bifurcation: (c) Normalized tensile stress in z-x plane.

The tensile stress z at any position (r, θ) along r for small r is given by Eq. (28), and the tensile stress x at (r, θ) is

given by Eq. (29), according to Paris and Sih [1964], where KI is the opening mode I stress intensity factor. [N. B.

Paris and Sih identified the z-axis as the y-axis and the y-axis as the z-axis, whereas in the present study the usual

convention of crystallography was followed where by the vertical axis is the z-axis and the x-axis is horizontal].

2

3

21

22 21sinsincos

)r(

K/

Iz (28)

2

3

21

22 21sinsincos

)r(

K/

Ix (29)

At any angle θ, the component of z normal to r in the x-z plane is zcos(θ) and the component of x normal to r in

the x-z plane is xsin(θ). Consequently, the normalized value of the total tensile stress acting normal to r in the x-

z plane, relative to the value when θ is zero, is readily obtained from Eqs. (28) and (29) to give Eq. (30):

0 )/()sincos( zxzN (30)

where the resulting behaviour of N for θ values from zero fifteen degrees is shown in Fig. 7(c).

It is implicit in Eqs. (28) to (30) that the stress field equations represent the situation for a stationary crack, i.e. the

condition where KI < KIC. However for a crack tip travelling at velocity ccr under a stress intensity >KIC it seems

reasonable to assume that Eq. (30) provides a reasonable approximation to the moving crack situation provided ccr ≤

climit ≤ 0.55cT, as indicated in Table 6. Hence, Fig. 7(c) shows that for θ angles ≤ 150 the value of N increases by no


421

more than ~4%, implying that a single crack tip travelling at climit can branch into two cracks both travelling at climit

provided the condition for Eq. (27) is satisfied and the subtended angle is 2θ ≤ 300. It is intuitive that the momentum

of a moving crack tip prior to bifurcation will attempt to keep each new crack tip moving in the same direction,

whereas the interaction of strain fields associated with formation each new crack will counteract travel on parallel

crack planes, leading to an optimum angle of bifurcation. The photographs of Schardin [1959] on bifurcation in

glass plates indicate angles of the order of 2θ ≈ 300.

It is possible to make an estimation of bifurcation angles based on the cracking model in Fig. 4, using the schematic

diagram of the instant of bond breaking in Fig. 8, where the bond separation distance at fracture is Rmax and the

neighbouring unbroken bond distance R1 is approximated to Ro. It is evident that the resulting local strain field is

likely to be related to the angle , which is obtained from Eq. (31):

00

1 2R/RRtan max (31)

Rmax Ro

Ro

Figure 8. Schematic of angle with respect to Rmax and Ro in Figure 4.

After inserting calculated values of Rmax and R0 for soda lime glass listed in Table 5, Eq. (31) produces a value

of 120 for and a corresponding value of 24

0 for 2. This is sufficiently close to the subtended bifurcation angles 2θ

≈ 300 in Schardin’s photographs in glass plates to suggest that Eq. (31) provides reasonable estimates of bifurcation

angles. Equation (31) was applied to the R0 and Rmax data in Tables 5 and 8, and results are listed in Table 12.

Table 12. Estimated subtended angles 2θ (degrees) during bifurcation in isotropic (polycrystalline) minerals and materials.

Mineral 2θ = 2 Mineral 2θ = 2 Mineral 2θ = 2

*Periclase MgO ~20 *Forsterite Mg2SiO4 ~16 †Diamond C ~12.5

*Lime CaO ~18 *Spinel MgAl2O4 ~21 †Borazone BN ~13

*Sphalerite ZnS ~19 *Pyrite FeS2 ~12 †Ice-Ih (270 K) ~43

*Galena PbS ~17.5 *Chalcopyrite ~21.5 Silica glass SiO2 ~25

*Fluorite CaF2 ~14.5 *Corundum Al2O3 ~21 Vitreous Carbon C ~43

*Halite NaCl ~14 †α-Quartz SiO2 ~27 Soda Lime Glass ~24

*Anorthite CaAl2Si2O8 ~21 †Moissanite-SiC ~12.5 Glass #17 ~29.5

*Ionic polycrystals: †Covalent polycrystals

3.4. Step loading to c. If the stress is rapidly (instantly) stepped from zero to a value c, where and c is

the critical tensile stress at fracture as defined in Eq. (2), then Eq. (23) becomes:

ic

R

crR

crR

RIC)d(

a)(

c

cc

E

)(

cc

c 22 41GG …for ia > a and ccr ≤ climit (32)

The step is equivalent to multiplying the right hand side of Eq. (23) by χ2 so that each ai value (each ai/a) behaves as

if it is χ2 times larger, causing (G(d))limit (defined by the condition ccr = climit) to be reached more quickly at lower ai/a.

The resulting effect on Eq. (23) after replacing ai/a by leads to Eq. (33):

1

)/()/(

crRRi cccaa for ccr ≤ climit (33)

The consequences of Eq. (33), using =1.25, are commencement of cracking at a finite velocity (which may be as

high as climit), together with crack branching at lower ai/a values (even at onset of cracking). These behaviours are

demonstrated for α-quartz and sphalerite in Fig. 9 for χ values of 1.25, 1.75, 2.25, and 3.5.


422

1 2 3 4 5 6 7 80

500

1,000

1,500

2,000

2,500C (

m/s

)cr

-Quartz

= 1.25

= 1.75

= 2.25

= 3.5

branch

=3.5branch

=2.25branch

=1.75

branch

=1.25

c = 1999 m/slimit

a /ai

1 2 3 4 5 6 70

500

1,000

1,500

2,000

C (

m/s

)cr

a /ai

Sphalerite

= 1.25

= 1.75

= 2.25

= 3.5

branch

=3.5branch

=2.25branch

=1.75

branch

=1.25

c = 1360 m/slimit

Figure 9. Effect of step loading χc on cracking behaviour of -Quartz and Sphalerite.

Figure 9 shows that for -Quartz, as χ increases from zero, the initial crack velocity at ai/a =1 is raised from zero to

650 m s-1

at χ = 1.25, from zero to 1420 m s-1


at χ = 2.25, and from zero to climit

at χ = 3.5 Also, crack branching occurs earlier in the crack growth process (lower ai/a) so that when χ = 3.5 crack

branching occurs immediately upon initial loading. Similarly, for sphalerite, the initial crack velocity is raised from

zero to 460 m s-1



at χ = 2.25, and from zero

to climit at χ = 3.5 Also, in both minerals, crack branching occurs earlier in the crack growth process (lower ai/a) so

that when χ = 3.5 crack branching occurs immediately upon initial loading. When χ = 2.46 (not shown in Fig. 9)

cracking commences immediately at climit but branching does not occur until ai/a = 2, based on Eq. (27).]. These

observations are generally consistent with bifurcations (branching) obtained by Schardin [1959] after striking

constant stressed glass plates with a knife edge.

The earlier onset of cracking and branching (bifurcation) at lower ai/a as the step loading increases has implications

for improved efficiency of comminution because it indicates that more cracking and crack branching

(fragmentation) may be activated almost simultaneously in rock particles containing a family of different sized

defects (cracks). For example, the consequences of a large step loading (e.g. χ = 3.25) is to activate propagation of

more cracks, with more branching events, so that increased interaction and coalescence of growing cracks occurs

and more effective fragmentation is obtained. The ratio of the strain energies per unit volume at stresses of 3.25(c)1

and (c)1 is 10.563. If the surface area of fragmented particles at 3.25(c)1 is >10.563 times that at (c)1 then the

energy of comminution has improved.

3.5 Strain rate effects. It is virtually impossible to instantaneously step from a zero load to a finite load as it implies

infinitely fast strain (loading) such as explosive and impulse loading, together with stress wave analyses, areas

where Grady and Kipp [1979, 1980] and Grady [1982] have contributed much to the fundamental of the processes

involved. For the present study, it is useful to extend the procedures used for constructing Fig. 9 to the effect of high

strain rates where time is included in the analysis. Commencing with the simple linear elastic relationship between

tensile stress ( and strain (), the differential with respect to time (t) is represented by Eq. (34):

)t(/)(E)t(/)( , more conveniently stated as E (34)

where units of are Pa s-1

and units of are s-1

.

Semicircular edge cracks or circular internal (penny shaped cracks) of radius (a) were considered in the analyses of

strain rate behaviour. The first step required calculation of the time tstart for the rising stress to reach the critical

stress c for the onset of cracking. The c was obtained from Eq. (2) and then Eq. (34) was used to obtain the final

equation for tstart:

21

2

1/

ICcstart

a

K

EEt

(35)


423

where values of E (i.e. EV-R) are listed in Table 2 and estimated values of KIC have been reported previously by

Tromans and Meech [2002, 2004].

Calculation of the increase in crack velocity ccr up to the limiting speed climit was obtained by replacing in Eq.

(33) by 2

cc / , where is an incremental increase in associated with the strain rate in Eq. (34), to

yield Eq. (36), and using a value of 1.25 for (see Eq. (33)):

1212 )cc/(ca/a/)cc/(ca/a crRRicccrRRi (36)

where ai contains the associated incremental increase in crack length a, (i.e. aaai .

An iteration procedure was used to obtain increments as the time dependent strain (stress) increases in

increments of t. Thus,

Et and crc/at (37).

Also, as ccr increases incrementally with in Eq. (36) it also increases incrementally with t , as shown in Eq.

(37), until it reaches climit

The final result at climit is obtained:

12 )cc/(ca/aa/ itlimRRcc (38)

where the sum of the increments required to reach climit and a is the sum of the associated crack tip

growth increments at climit.

The iteration procedures were conducted by allowing increments of a = 1x 10-6

m to occur from which

increments were obtained from Eq. (37). The ccr was obtained from Eq. (36) using calculated values of cR in

Table 6. The t was calculated from a and the calculated ccr. In this manner crack tip growth curves expressed as

summated growth increments minus a (ai -a) could be obtained as function of time up to climit. Thereafter, crack

tip growth is constant at climit and crack branching (bifurcation) eventually occurs (see Figs 6 and 9). Based on Eq.

(27), the first bifurcation occurs at twice the ai/a ratio at climit. Consequently, when the two branch tips (travelling at

climit) are unable to dissipate the increasing crack tip strain energy associated with increasing crack length, they each

must undergo another bifurcation at twice the ai/a ratio at which the first bifurcation occurs (i.e. four times the ai/a

ratio at climit). A subsequent (third) bifurcation occurs at the twice the ai/a ratio at which the second bifurcation

occurred (i.e. eight times the ai/a ratio of the first bifurcation). A subsequent fourth bifurcation occurs at sixteen

times the ai/a ratio of the first bifurcation). Branching behaviour is summarized in Eq. (39).

(ai/a)branch 4 =2(ai/a)branch 3 = 2(ai/a)branch 2 = 2(ai/a)branch 1 = 2(ai/a)Climit (39)

Crack growth analyses were conducted for sphalerite and -quartz at strain rates = 1 s-1

and 102 s

-1, using penny

shaped cracks of initial radii, a, of 1 mm and 10 mm, as shown in Fig. 10. The crack start time was obtained from

Eq. 35 using estimated KIC values of 0.588 MPa m1/2

for sphalerite [Tromans and Meech, 2002] and 0.715 MPa m1/2

for -quartz [Tromans and Meech, 2004], together with corresponding EV-R values in Table 2. Thus, c was

obtained as: (i) 16.478 MPa and 5.211 MPa for 1 mm radius and 10 mm radius cracks, respectively, in sphalerite

and (ii) 20.038 MPa and 6.337 MPa for 1 mm radius and 10 mm radius cracks, respectively, in-quartz. Crack

starts, E/t cstart , are indicated on the diagrams in Fig. 10. Each climit is identified by an open square symbol □

and occurrence of crack branching (bifurcation) is indicated by closed square symbols ■.


424

(a) (b)

0 50 100 150 200 250 3000

20

40

60

80

100

Time (s)

1 mm crack

10 mm crack

= 1 s-1-Quartz

c 1999 m/slimit

branching

10 mm start 1 mm start

Cra

ck g

row

th (

mm

)

2nd.

3rd.1st.

1st.

2nd.4th.

climit

climit

0 1 2 3 4 50

1

2

3

4

5

Time (s)

1 mm crack

10 mm crack

= 10 s2 -1-Quartz

c 1999 m/slimit

branching

10 mm start 1 mm startCra

ck g

row

th (

mm

)

1st.1st.

2nd. 2nd.

climit

climit

(c)

(d)

0 50 100 150 200 250 3000

20

40

60

80

100

Time (s)

10 mm crack

1 mm crack

= 1 s-1Sphalerite

c 1360 m/slimit

10 mm start 1 mm start

Cra

ck g

row

th (

mm

)

2nd.

3rd.1st.

1st.

2nd.

4th.

3rd.

climit

climit

branching

0 2 4 6 80

2

4

6

8

Time (s)

1 mm crack

10 mm crack

= 10 s2 -1Sphalerite

c 1360 m/slimit

branching

10 mm start 1 mm startCra

ck g

row

th (

mm

)

1st.

1st.

2nd.

2nd.

3rd.

4th.

climit

climit

Figure 10.Time-dependent crack growth behaviour of 1mm and 10 mm cracks:(a) and (b) -quartz;(c) and (d) sphalerite.

At strain rates of 1 s-1

it is clearly evident from Figs. 10(a) and 10(c) that crack growth of the 10 mm cracks in both

a-quartz and sphalerite have reached the 3rd

branching condition before the 1mm cracks have commenced

propagation., There is a ~160 s separation time between crack initiation of 10m and 1mm cracks indicating that the

larger cracks may propagate completely through a mineral particle before small cracks have been activated. In

contrast, at strain rates of 102 s

-1 it is equally evident from Figs. 10(b) and 10(d) that the 10 mm and 1 mm cracks

commence propagation almost simultaneously, within ~1.5 s of each other, and crack branching events are

occurring in both cracks within the first 6 mm of crack growth.

Based on Fig. 10(b) and 10(d), it is evident that very high strain rate loading has the same effect as high step loading

in its ability to promote simultaneous cracking and crack branching in mineral particles with different crack sizes.

Furthermore, even if enhanced crack branching does not lead to full fragmentation in a single comminution process,

an increased number of cracks have been generated for propagation during subsequent comminution processing.

This is similar to one of the advantages of controlled blasting of rocks to produce extra fractures (cracks) within

resulting particles to reduce the energy required for subsequent downstream crushing and grinding processes

[Workman and Eloranta, 2003].


425

Sphalerite 1 mm Crack

= 10 s2 -1

Sphalerite 10 mm Crack

crack startclimit

2nd..

1st.

climit 1st.crack start

2nd..

3rd.

4th.

Figure 11.Scaled comparison of cracking and branching (bifurcation) appearance in Figure 10 (d).

The branching pattern of co-propagating cracks at high strain rates is dependent upon the initial crack size and the

subtended angle 2 degrees between bifurcation branches (see Table 12). This is illustrated in Fig. 11 which shows a

scaled comparison of the crack propagation patterns of the 10mm and 1mm cracks in sphalerite, based on Fig. 10(d).

The propagation distance of the crack tip from the crack start time to the first branch is greater for the larger initial

crack (10 mm) than the smaller initial crack (1 mm) Also, distances between successive branches (1st, 2

nd 3

rd etc) are

larger for the 10 mm initial crack. However, the subtended bifurcation angles are similar for both cracks, being ~19o

for sphalerite based on Table 12. The result is a more compact crack branching network for the initial 1 mm crack.

Such crack patterns are similar to those observed in brittle fracture of glass by Schardin [1959]. Consequently, for a

family of initial cracks with size range between 1 mm and 10 mm in sphalerite, a strain rate of = 102 s

-1 will

produce essentially simultaneous propagation of all cracks in the range with crack networks whose appearances

range between those in Fig. 11. The conclusion of such analyses is that the application of high strain rate

comminution processing should lead to more effective particle fracture and fragmentation and, as a consequence,

improve the energy efficiency of particle size reduction by decreasing the number and frequency of unsuccessful

impacts with mineral particles which consume strain energy without producing fracture [Tromans, 2008].

The analyses on dynamic crack propagation in Figs. 9 to11 used sphalerite and -quartz as examples only. There are

sufficient data in the current study to conduct similar analyses on all materials listed in Tables 6 and 7, using

values of 1 s-1

and 102 s

-1, and KIC values from Tromans and Meech [2002, 20024], in order to arrive at the same

conclusion that high strain rate comminution should have a beneficial influence on the energy efficiency of particle

size reduction. However these higher will require significant innovative changes in the design of conventional

crushing and grinding processes, which typically involve lower strain rates of 10-5

to 10-1

s-1

[Sadrai et al., 2011],

whereas normal rock blasting operations utilize rates of ~1s-1

to 103 s

-1 [Grady and Kipp, 1980].

4. CONCLUSIONS

The mineral comminution process is inherently inefficient because only a small fraction of the strain energy

introduced during loading is utilised in generation of fracture surface area. Dynamic crack propagation modelling of

important parameters relating to fracture of polycrystalline minerals, including average elastic modulus, surface

energy, limiting crack velocity, Rayleigh surface wave velocity and crack bifurcation have been used in conjunction

with dynamic fracture mechanics equations to model dynamic crack propagation under high strain rate (high

loading) conditions. The modelling shows that high strain rates (e.g. = 102 s

-1) offer a possible way to improve the

energy efficiency of mineral comminution by increasing the fracture surface area via the propagation of a larger

range of existing cracks sizes (defects) in the initial crack population. Improved efficiency is related to the

promotion of more crack branching together with more opportunities for crack interactions (crack coalescence),

leading to increased formation of fragmented particles


426

5. ACKNOWLEDGEMENTS

The author wishes to thank the Natural Sciences and Engineering Research Council (NSERC) of Canada for

providing funding and Prof. J.A. Meech for his ongoing interest in the study.

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