Fundamental Studies on Fracture Mechanics of Minerals:

Application to Energy Reduction in Comminutionby

Desmond Tromans

CERM 3

UBC

Presented at 1st. Annual UBC-CERM 3 Workshop SymposiumVancouver, B.C., September 23, 2002

OUTLINE

*Particle fracture - important factors(particle flaws, fracture toughness, surface energy)

*Limiting fine particle size(surface steps and dissolution)

*Prediction of fracture toughness - procedures(bonding models in crystals, ideal brittle fracture)

*Toughness values of minerals(computed from first principles)

*Estimated comminution efficiency(ratio energy used/energy to create new surface area)

Induced stresses-compressive load P

P

P

P

a2

a1

2a3

2a4

a51

2 3

45

KI=Yi(ai)1/2

At fracture:

KIC=Yic(ai)1/2

where

KIC=(EGIC)1/2

GIC= Fracture ToughnessKI=Stress intensity (at fracture KI=KIC, i=ic)i=Tensile stress, ai=crack length Y= Geometrical factor E=Tensile modulus, GIC= critical energy release rate/m2

P

P

D

(a)

P

kP kPP

kP kP

2a

(b)

2a

P

P

D

P

kP kPP

kP kP

Schematic of particle containing a crack (flaw) of radius Schematic of particle containing a crack (flaw) of radius aa subjected to compressive force subjected to compressive force PP. .

i=P(kCos-Sin) KI=Y P(kCos-Sin)a1/2

**At fracture KI=KIC and limiting average fine particle size is

D ~ (KIC/kP)2 (= 0)

Fracture Surface Topography- Dissolution Terrace-Step-Kink (TSK) Structure of a Mineral Surface

Terrace Site Step Site Kink Site

Reactant Species

Ratio ofReaction Rates on Step vs. Terrace

Sites is

1800 to 4

Overall reaction rate is increased

significantly whenparticle size drops

below 1 microneven when the

fraction of step sites is less than 1%

5 m

Microtopography of Sphalerite ZnS

Fraction of dissolution sites on step edges ~ 8.5 x 10-4

KIC=(EGIC)1/2 - Calculation of GIC for MineralsFirst principles calculation

INPUT *Crystal bonding model (ionic, covalent).

*Crystal bonding energy UR (J/m3).

*Average atom spacing R in crystal.

*Change in UR with R.

*Calculation of tensile stress to extend perfect crystal to fracture.

Units

KIC - Critical stress intensity for crack propagation (Pa m1/2)

GIC - Critical crack energy release rate (J m-2)

- Surface energy (J m-2) = (GIC /2)

R (m)

+

-

0

U e

R O

0

U ( Jm )R-3

Crystal energy per unit volume vs. average atomic spacing Crystal energy per unit volume vs. average atomic spacing

Ue Equilibrium crystal binding energy at Ro (J m-3)

Ro Average distance between atoms in unstrained crystal (m)

(GPa)x

0 5E-10 1E-9 1.5E-9 2E-9 2.5E-90

10

20

30

40

50

xR (m)

1

2

3

4

5

6

7

8

1. Cuprite2. Galena3. Zincite4. Pyrite5. Hercynite6. Periclase7. Rutile8. Corundum

Computed uniaxial tensile stress behaviour Computed uniaxial tensile stress behaviour of defect-free minerals of defect-free minerals

(GPa)x

0 5E-10 1E-9 1.5E-9 2E-9 2.5E-90

5

10

15

20

25

30

1

2

3

4

5

6 1. Halite2. Anhydrite3. Fluorite4. Anorthite5. Forsterite6. Pyrope7. Andalusite

xR (m)

7

oxides and sulphidesoxides and sulphides halides, sulphates and silicateshalides, sulphates and silicates

itx

ox

RR

RRxxIC R

lim2mJ2G

Mineral FormulaIntragranular crack

(J m-2)

Grain boundary crack Gb

(J m-2)G IC

(J m-2)

KIC

(MPa m1/2)

(G IC)Gb

(J m-2)

(KIC)Gb

(MPa m1/2)Cuprite Cu2O 0.886 0.163 0.4428 0.769 0.152 0.117

Periclase MgO 13.704 2.052 6.852 12.293 1.943 1.411Lime CaO 8.335 1.281 4.1680 7.382 1.206 0.953

Barium oxide BaO 4.274 0.623 2.137 3.768 0.590 0.506Wustite FeO 4.885 0.789 2.443 4.333 0.743 0.552

Cobalt oxide CoO 7.449 1.188 3.725 6.624 1.120 0.825Nickel Oxide NiO 9.964 1.522 4.982 8.921 1.440 1.043

Bromellite BeO 17.238 2.613 8.619 15.545 2.481 1.693Zincite ZnO 5.599 0.842 2.799 4.990 0.795 0.609Rutile TiO2 18.445 2.293 9.223 16.885 2.194 1.560

Cassiterite SnO2 14.845 1.974 7.423 13.438 1.878 1.407Corundum Al2O3 19.250 2.774 9.625 17.387 2.636 1.863Hematite Fe2O3 20.750 2.099 10.375 19.438 2.032 1.312Eskolaite Cr2O3 14.617 2.144 7.309 13.135 2.033 1.482

Titanium oxide Ti2O3 12.269 1.732 6.135 11.059 1.644 1.210Spinel MgAl2O4 14.014 1.959 7.007 12.675 1.863 1.339

Hercynite FeAl2O4 10.687 1.541 5.344 9.624 1.462 1.063Chromite FeCr2O4 12.209 1.811 6.104 10.934 1.714 1.275

Nickel Chromite NiCr2O4 3.080 0.573 1.540 2.684 0.535 0.396Zinc ferrite ZnFe2O4 11.420 1.660 5.710 10.249 1.572 1.171Magnetite Fe3O4 12.897 1.724 6.449 11.700 1.642 1.197

Chrysoberyl BeAl2O4 18.624 2.694 9.312 16.828 2.561 1.796Galena PbS 3.736 0.547 1.868 3.278 0.512 0.458

Sphalerite ZnS 4.180 0.588 2.090 3.712 0.554 0.468Metacinnabar -HgS 2.316 0.335 1.158 2.037 0.314 0.279Greenockite CdS 2.289 0.327 1.145 2.017 0.307 0.272

Wurtzite ZnS 4.536 0.628 2.268 4.037 0.592 0.499Pyrite FeS2 6.143 1.349 3.072 5.233 1.245 0.910

Computed toughness values of oxide and sulphide minerals at 298 KComputed toughness values of oxide and sulphide minerals at 298 K

Mineral FormulaIntragranular crack

(J m-2)

Grain boundary crack Gb

(J m-2)G IC

(J m-2)

KIC

(MPa m1/2)

(G IC)Gb

(J m-2)

(KIC)Gb

(MPa m1/2)

Computed toughness values of halide, sulphate and silicate minerals at 298 KComputed toughness values of halide, sulphate and silicate minerals at 298 K

Halite NaCl 1.155 0.206 0.577 0.993 0.191 0.162

Sylvite KCl 0.758 0.135 0.379 0.647 0.125 0.111

Cesium chloride CsCl 0.676 0.131 0.338 0.570 0.120 0.106

Fluorite CaF2 3.179 0.589 1.589 2.754 0.548 0.425

Barite BaSO4 1.203 0.269 0.602 1.021 0.248 0.182

Anhydrite CaSO4 1.805 0.366 0.902 1.550 0.340 0.255

Nepheline NaAlSiO4 6.412 0.698 3.206 5.933 0.672 0.479

Cobalt olivine Co2SiO4 4.570 0.864 2.285 3.973 0.806 0.597

Liebenbergite Ni2SiO4 5.450 1.058 2.725 4.729 0.985 0.721

Fayalite Fe2SiO4 3.924 0.730 1.962 3.414 0.681 0.510

Monticellite CaMgSiO4 4.665 0.813 2.332 4.081 0.760 0.584

Forsterite Mg2SiO4 6.329 1.129 3.164 5.542 1.056 0.787

Wadsleyite - Mg2SiO4 6.792 1.374 3.396 5.872 1.278 0.920

Ringwoodite - Mg2SiO4 6.685 1.397 3.343 5.756 1.297 0.929

Andalusite Al2SiO5 10.130 1.580 5.065 9.011 1.491 1.119

Anorthite CaAl2Si2O8 5.478 0.752 2.739 4.930 0.714 0.548

Grossularite Ca3Al2Si3O12 7.356 1.392 3.678 6.396 1.298 0.960

Pyrope Mg3Al2Si3O12 7.348 1.306 3.674 6.452 1.224 0.896

Almandine Fe3Al2Si3O12 7.102 1.310 3.551 6.208 1.225 0.894

Andradite Ca3Fe2Si3O12 7.154 1.252 3.577 6.271 1.172 0.883

Mineral

(J m-2

(103 kg m-3)/

(10-4 J m kg-1)Normalised

/Normalised

FractureEnergy†

Galena (PbS) 1.868 7.597 2.459 0.102 ~0.1

Sphalerite (ZnS) 2.090 4.097 5.1 0.21 ~0.2

Corundum(Al2O3)

9.625 3.989 24.13 1.0 1

King et al. (1997)

Normalized for galena, sphalerite and corundum-comparison with measured fracture energy for single

particle fracture (King et al)

Estimated crushing and grinding efficiencyEstimated crushing and grinding efficiency

*Bond’s data (1961): †Anorthite chosen as representative: ‡Grossularite chosen as representative

Mineral

(J m )-2

(103 kg m-3)Wi

(kWh/ton)*(Wi)SI(kJ kg )-1

Efficiency(%)

Feldspar† 2.739 2.761 11.67 46.31 0.97Galena 1.868 7.597 10.19 40.43 0.27Garnet‡ 3.678 3.597 12.37 49.1 0.94Hematite 10.375 5.270 12.68 50.31 1.76Magnetite 6.449 5.197 10.21 40.51 1.38Pyrite 3.072 5.013 8.9 35.32 0.78Rutile 9.223 4.250 12.12 48.1 2.03Fluorite 1.589 3.181 9.76 38.73 0.58Pyrite 3.072 5.013 8.9 35.32 0.78Quartz (alpha) 2.678 2.649 12.77 50.67 0.9Silicon Carbide 2.34 3.216 26.17 103.8 0.32Dolomite 1.34 2.863 11.31 44.88 0.47

CONCLUSIONS*Toughness predicted for over 50 homogeneous minerals.

*Impact efficiency is related to pre-existing crack sizes and their orientation

(elastic deformation with no fracture).

*Limiting average fine particle size is related to fracture toughness via (KIC)2.

*The energy efficiency of crushing and grinding processes is very low ~1%.

*A small (few%) improvement in efficiency will produce relatively large

cost ($) savings (comminution consumes 65-80% of the energy in mine-mill

operations.)

Continuing Comminution ResearchContinuing Comminution Research

*Toughness estimates for increased numbers of minerals

*Application to heterogeneous minerals (rocks)

*Crack branching (production of fines)

*Impact efficiency during comminution -( crack distribution, crack orientation and length)

*Improved understanding of current processes(reduction of inherent inefficiencies)

*New methods (more efficient production of small particles).

*Collaboration between CERM 3 and CSIRO on energy use in comminution systems (on-line monitoring).