IJRRAS 13 (2) ● November 2012 www.arpapress.com/Volumes/Vol13Issue2/IJRRAS_13_2_05.pdf
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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.
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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.
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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
IJRRAS 13 (2) ● November 2012 Tromans ● Crack Propagation in Britle Materials
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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.
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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.
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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 .
IJRRAS 13 (2) ● November 2012 Tromans ● Crack Propagation in Britle Materials
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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.
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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
IJRRAS 13 (2) ● November 2012 Tromans ● Crack Propagation in Britle Materials
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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)
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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.
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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
IJRRAS 13 (2) ● November 2012 Tromans ● Crack Propagation in Britle Materials
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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)
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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
IJRRAS 13 (2) ● November 2012 Tromans ● Crack Propagation in Britle Materials
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
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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
IJRRAS 13 (2) ● November 2012 Tromans ● Crack Propagation in Britle Materials
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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.
IJRRAS 13 (2) ● November 2012 Tromans ● Crack Propagation in Britle Materials
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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 = 1.75, from zero to 1860 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 = 1.25, from zero to 1010 m s-1
at χ = 1.75, from zero to 1330 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)
IJRRAS 13 (2) ● November 2012 Tromans ● Crack Propagation in Britle Materials
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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 ■.
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(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].
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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
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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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