Minerals Engineering 46–47 (2013) 68–75

Contents lists available at SciVerse ScienceDirect

Minerals Engineering

journal homepage: www.elsevier .com/ locate/mineng

Image based modeling of rock fragmentation

Nenad Djordjevic ⇑JKMRC, SMI, University of Queensland, 40 Isles Road, Indooroopilly, Qld 4068, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 November 2012Accepted 7 March 2013Available online 2 May 2013

Keywords:ImageRock fragmentationModelingOreLiberationFracture

0892-6875/$ - see front matter � 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.mineng.2013.03.002

⇑ Tel.: +61 7 3365 5888; fax: +61 7 3365 5999.E-mail address: n.djordjevic@uq.edu.au

In the mineral extraction industry, comminution modeling is not only interested in maximum rockstrength, but also, or much more, in the energy required to induce rock fracture and, most significantly,into the effect of energy application on the produced rock fragments size distribution. An additionalaspect of rock breakage, specific to the mineral extraction industry, is the modeling of liberation of par-ticular mineral grains from the host rock matrix. These aspects of rock behavior make comminution mod-eling a unique field of rock mechanics.

From a traditional engineering point of view (mining and civil), rock samples are considered to behomogenous. Although the mechanical properties of individual minerals can vary significantly, the prop-erties of the minerals and of the mineral boundaries interact randomly enough to assume that in the sizeof rock samples mechanical properties can be considered homogenous. However, from a comminutionpoint of view, heterogeneity caused by a difference in the properties of minerals are crucial and thereforerock material, even in the scale of a few centimeters, should be considered as heterogeneous. The com-minution response of such rock will be influenced by the textural parameters of the rock as well asmechanical properties of constitutive mineral grains.

Image based numerical modeling is a useful tool for investigation of the pattern and dynamics of therock breakage process. Its usefulness rests on the fact that a difficult step of building a faithful modelof rock texture and composition, as a pre-requisite for modeling of rock breakage, is removed. Numericalmodeling based on the use of classified digital image of the rock surface, could be particularly effective inthe mineral extraction industry, where one of the key objectives is liberation of specific minerals, by pro-viding inside view of mechanisms that are responsible for liberation of valuable minerals embedded intospecific ore matrix.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

One way to investigate the effect of different textural classes isto perform numerical modeling of the effect of certain texturaltypes on rock fragmentation and mineral liberation. Textural clas-ses can be the product of mathematical modeling or can be a digitalimage of a typical rock texture of a certain rock type. Regardless ofhow a particular ore texture is generated it is of interest to inves-tigate the spatial and temporal pattern of the rock fragmentation.

Recent developments in computing power have created theopportunity for more rock specific modeling of rock fragmentation.This is based on the application of object-oriented finite elementmodeling of the rock deformation and breakage. In this context,the object represents a specific type of mineral or structural fea-tures. This modeling has been performed using the OOF (ObjectOriented Finite) element code, (Langer et al., 2001). The proposedapproach is particularly attractive, due to the nature of its input,which is a digital image of the rock surface.

ll rights reserved.

The objective is to establish a cause-effect relationship betweenthe presence of certain mineral grains and their pattern of distribu-tion on the propensity for liberation of another mineral type, i.e., amineral which is of economic interest. One of the most significantobjectives is to establish the maximum size of the rock fragmentsrequired for the full or partial liberation of the mineral of interest.Obviously, critical fragment size is dependent on the textural fea-tures of the rock, mechanical properties of individual minerals,and the nature of the stress field to which a rock sample is exposed.

2. Modeling methodology

Before we continue, we should remind ourselves what rock tex-ture is. According to Encyclopedia Britannica, the texture of a rockis the size, shape, and arrangement of the grains (for sedimentaryrocks) or crystals (for igneous and metamorphic rocks). Therefore,the definition of texture is essentially geometric and pictorial, innature. The classical definitions of texture do not include informa-tion about the mechanical/physical properties of grains or crystals.However, from the point of view of rock comminution in the

Fig. 1. Nano-indentation testing of sphalerite (Rosebury).

N. Djordjevic / Minerals Engineering 46–47 (2013) 68–75 69

context of the recovery of particular minerals, the mechanicalproperties of minerals are of great importance. Another parameterwhich is not explicitly considered in traditional definitions of tex-ture is the presence and nature of voids (pores and microcracks).

The modeling methodology of rock mass behavior is stronglyinfluenced by the nature of the rock mass. In the case of rocks asnatural geologic materials, physical and mechanical propertiesneed to be determined rather than selected through a manufactur-ing process. In the case of man-made materials, the common pur-pose of routine mechanical testing is not to gain new knowledge,but to provide quality control and verification.

Rock mass in situ is characterized by unknown structural prop-erties, the unknown state of stress and unknown details of themechanical properties. From a practical point of view, all theseproperties are not just unknown in detail but they are frequentlypractically unknowable. To a lesser extent, the same is applicablefor the modeling rock matrix (in the scale of rock samples). Thislack of precise information indicates that the approach to rockmodeling should be different from one used for the modelingbehavior of known materials (man-made), whose properties aregenerally known.

Fig. 2. Nano-indentation testing of sphalerite (Broken Hill).

Fig. 3. Normalised (non-dimensional), value of the elastic modulus as a function ofthe assumed Poisson’s ratio, using for normalization value of elastic modules thatcorrespond to the Poisson ratio of 0.25.

3. Determinations of the mechanical properties of the minerals

Input parameters for OOF modeling are mechanical parametersof the minerals identified in the image. Among them are Young’smodule of elasticity, Poisson’s ratio, and compressive and tensilestrength. These parameters are determined either from the avail-able published information or they are measured through nano-(micro) indentation testing. In terms of the published information,they tend to be restricted to the modulus of elasticity and Poisson’sratio. The strength properties of minerals are rarely available.

Strength properties as well as elastic properties are determinedthrough instrumented, computer controlled, nano-indentationtesting. Indentation or hardness testing has been used for a longtime for material characterization. Traditional hardness testingconsists of the application of a single static force for a specifiedtime. Depending on the shape and tip material of the indenter,the dimensions of the impression created will be in order of milli-meters. The output of the traditional hardness tester is typically asingle indentation hardness value that is a measure of the relativepenetration depth.

In contrast to traditional hardness testing, instrumented inden-tation testing allows the application of a specified force or dis-placement history. Force and displacement are measuredcontinuously over a complete loading cycle. For the purpose ofinstrumented nano-indentation we used the UMIS nano-indenta-tion instrument from the CSIRO. UMIS measures elastic, plastic,strain hardness, creep, fracture and other mechanical propertiesof a material surface. The UMIS offers in situ observation of theindentation process, and specimen positioning to within 0.1 lm.

Testing has been performed using a Berkowich diamond tip in-denter, with constant force (5–10 tests per mineral sample). In thecase of homogenous minerals, testing produced an indent of highlyreproducible size and shape, Figs. 1 and 2. From the unloading partof the load-deformation curve, the instrument calculates the elas-tic modulus of the indented surface.

Elastic modulus is calculated with the assumption that Poissonratio is equal to the mean value published in the literature for aparticular mineral. This assumption introduces error. However,the magnitude of the error in most cases is not high. Due to thehigh elastic modulus of the diamond tip used for indentation,and the nature of the testing method, the calculated values of theelastic modulus of the minerals are within the range of ±3% ofthe value if the exact value of the Poisson ratio of the mineral is

known. For instance, an error of 20% in the assumed value of thePoisson ratio of the mineral results in an error of 3% in the calcu-lated value of the elastic modulus, Fig. 3.

Elastic properties of the same mineral vary from mine to mine,and probably within the same mine, depending on the specific his-tory of mineralization. This is illustrated in the case of sphalerite, inwhich Young’s modulus varies substantially between Roseburyand Broken Hill mines (Australia). Therefore, using the propertiesof Rosebury sphalerite to model behavior of Broken Hill ore mayproduce incorrect results, (see Fig. 4) Fig. 9.

However, within the same mineralization the mechanical prop-erties of a particular mineral, tend to vary in a relatively narrow

Fig. 4. Elastic modulus of sphalerite from Rosebury and Broken Hill mine.

70 N. Djordjevic / Minerals Engineering 46–47 (2013) 68–75

range. This is illustrated in the case of chalcopyrite from Mt. IsaMine, Fig. 5.

4. Modeling the influence of texture on rock strength

One of the key properties of rock as a natural material is its het-erogeneity. The onset of crack initiation and crack propagation isheavily influenced by the heterogeneity of the rock material.

There is a significant body of experimental results which showsthat the texture of rock will have an effect on rock strength. For in-stance, Everitt and Lajtai (2004) investigated the contribution ofrock fabric on excavation damage development in the granite ofthe Lac du Bonnet Batholith. Data collected in their study demon-strated the highly heterogeneous internal structure characteristicof apparently homogenous batholiths, and the influence of thisheterogeneity on rock mass response. Grain size had a clear impacton the strength. With an increase in the grain size of main mineralstypes (quartz, feldspars), rock strength decreased.

Different techniques were used to introduce rock heterogeneityinto the numerical models. Garboczi and Day (1995) used meshwith a random geometry, but equal properties for the elements.The authors developed an algorithm based on FEM applied to dig-ital images for computing the linear elastic properties of heteroge-neous materials. Schlangen and Garboczi (1997) applied anapproach where the generated microstructure is projected to reg-ular elements to which different properties are assigned accordingto their position.

Blair and Cook (1998a,b) developed a non-linear rule-basedmodel for the fracture in compression of heterogeneous brittlematerials such as rock, and used it to study crack nucleation andpropagation at the grain scale. The authors used the model to sim-ulate uniaxial compression tests of rock samples. The resultsunderscored the importance of crack interaction in tensile crackingof rock in compression even at low crack densities. The model pro-duces non-linear stress–strain behavior similar to that observed inlaboratory tests. They performed a parameter-sensitivity analysis

Fig. 5. Elastic modulus of chalcopyrite from Mt. Isa Mine.

to evaluate the relative importance of different types of grain-scaleheterogeneity on fracture processes and the compressive strengthin simulated compression tests. The results presented indicate thatheterogeneity in a local stress field (due to grain shape and load-ing) has a first-order effect on macroscopic properties and is muchmore important than heterogeneity initial strength (i.e. strengthsof mineral grains).

Tang et al. (2000) developed RFPA2D a finite element code (RockFailure Process Analysis). The code was developed by consideringthe deformation of an elastic material containing an initial randomdistribution of micro-features. By introducing the heterogeneity ofrock properties into the model, the RFPA code can simulate non-linear deformation of a quasi-brittle behavior with an ideal brittleconstitutive law for the local material elements.

Langer et al. (2001) developed Object Oriented Finite elementcode (OOF) for modeling macroscopic properties from images ofreal or simulated microstructures. OOF takes a non-reductionist,brute force approach in a user-friendly way. The user starts witha digitized image of the rock texture and builds a data structureon top of it. Tools are provided to allow the user to graphically se-lect features in the microstructural image of the rock and specifytheir properties. For OOF, the microstructure is a data structurecomposed of image and property data.

The mechanical properties of mineral grains or mineral phaseswhich are required for modeling, depend on the intended type ofmodeling. In cases of modeling the elastic properties of rock andthe stress distribution within a rock, resulting from the applicationof an external force, the only parameters required are Young’smodulus of elasticity and Poisson’s ratio. If anisotropy coefficientsof individual minerals are known, the coefficient of anisotropy canalso be used.

In cases when modeling is concerned with some form of rockfracturing or rock damage, then parameters which describe min-eral strength (compressive and tensile) need to be introduced. Be-sides strength limits, it is also necessary to introduce parameterswhich control post-peak strength behavior of the particular min-eral. Fragment size distribution which comes as a result of OOFmodeling is determined using the Image Tools code. Consideringthat OOF modeling is in 2D domain, the calculated fragment distri-bution is relative to the initial area of the sample (i.e., due to con-fining effect of 3rd dimension, real fragmentation will be to someextent coarser).

5. Influence of the mechanical properties of minerals on thefragmentation model

It is of interest to know how different a mechanical propertyneeds to be to cause a visible difference in the output of the frag-mentation model. In the case of samples from Ernest Henry Mine(Queensland, Australia), we noticed that the presence of magnetiteis of critical significance for fragmentation. The influence of mag-netite appears to be particularly strong in comparison with pyrite.This would seem surprising considering that pyrite has a higher va-lue of elastic modulus than magnetite and will, therefore, absorbhigher stresses. The high stress focusing abilities of pyrite, shouldresult in efficient debonding of pyrite from the host matrix, dueto yielding of relatively soft Feldspar, which dominates the matrix.

In terms of mechanical properties, the main difference betweenmagnetite and pyrite is in the value of the Poisson ratio. Magnetitehas a much higher value of the Poisson ratio than pyrite (0.25 vs.0.17), which indicates that the ability of magnetite to deform andtransfer stress into the neighboring mineral grains will be higher.We investigated the effect of Poisson ratio on the patterns of frag-mentation of the sample, while all other properties of minerals arekept constant.

Fig. 6. Case of uniformly distributed magnetite (black) and chalcopyrite (red)within K-spar matrix (sample 1). (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

N. Djordjevic / Minerals Engineering 46–47 (2013) 68–75 71

Of particular relevance to texture, is the effect of the shape ofthe grains and their distribution, on the fracture initiation andpropagation. Previous research (Blair and Cook, 1998a,b) has indi-cated that stress amplification associated with the shape of theminerals is a more significant factor than the mechanical proper-ties of minerals (particularly of presence of relatively corners/tips,which acts as stress amplification points).

In the first phase, we modeled the mechanical response of therock texture, which was composed from the grains extracted fromthe images of real rock texture. The intention was to developclearly different textural types and to investigate sensitivity ofthe methodology. For this purpose we used texture from ErnestHenry mine, which was then simplified to the level that texturalmodel was composed from 1 to 3 mineral types. Models were com-posed from three common minerals, K-spar, chalcopyrite and mag-netite. Elastic properties of minerals are accepted based on thepublished values. Strength properties are determined based onthe published value for elastic modulus of elasticity and approxi-mate relationship that compressive strength of materials tends tobe approximately 1/400th of the elastic modulus, and that tensilestrength is 10% of the compressive strength, Table 1.

From the point of view of the influence of texture, two cases areof particular interest: first when chalcopyrite and magnetite areuniformly distributed within the host matrix composed from K-spar, and a second case where magnetite and chalcopyrite are clus-tered into separate clusters, Figs. 6 and 7. Properties of gangueminerals are kept same.

The application of a vertical load, modeling resulted in fragmen-tation patterns which are clearly different, as shown in Figs. 8 and9.

From the modeled fragmentation pattern, the fragment size dis-tribution is calculated. Fragment size distribution shows that theclustered magnetite will produce finer fragmentation, Figs. 10and 11.

However, from the point of view of liberation of chalcopyrite itis clear that liberation of chalcopyrite will be stronger, in the mod-eled case of texture where chalcopyrite and magnetite were mixedtogether. This is indicated based on the improved ability for gener-ation of relatively small fragments, in the second case. Improvedability for generation of small fragments comes as result of interac-tion between relatively hard magnetite grains and relatively softchalcopyrite grains. In the modeled case, difference in pattern ofliberation of chalcopyrite is also affected by the clustering’s of chal-copyrite and position of the cluster. However, in any case resultsshows that, depending on the textural features of rock, good rockfragmentation and good liberation of valuable minerals may notoccur simultaneously.

6. Influence of grain boundaries

At present all of the digital images which were used as input fornumerical modeling using OOF code, came as classified images ofmineral phases present in the polished section of the ore. In manycases, some minerals such as K-spar are not represented as distinc-tive singular mineral grains, but as a continuous area, composed

Table 1Mechanical properties of minerals used in modeling.

Mineral Elasticmodulus(GPa)

Poisson’sratio

Compressivestrength (MPa)

Tensilestrength(MPa)

K-spar 39.7 0.33 99 9.9Chalcopyrite 84.6 0.28 210 21Magnetite 230.6 0.25 570 57Pyrite 292 0.17 720 72

from many mineral grains of K-spar. On the other hand, mineralssuch as magnetite tend to present themselves in the polished sec-tion as the singular grains. Another, potentially significant aspectof modeling, is that interfaces between different minerals are con-sidered to be continuous, and there is no special weakening of thematerials as the result of discontinuity in the type of materials

Fig. 7. Case where magnetite and chalcopyrite and separated into distinctiveclusters (sample 2).

Fig. 8. Fragmentation pattern for the case of homogenous distribution of magnetiteand chalcopyrite (sample 1) with black regions indicates fractures.

Fig. 9. Fragmentation pattern for the case of clustered distribution of magnetiteand chalcopyrite (sample 2) with black regions indicates fractures.

Fig. 10. Fragment size distributions of modeled compression tests (based offragment area).

Fig. 11. Fragment size distributions of modeled compression tests, based onfragment volume.

Fig. 12. Relationship between grain boundary fracture toughness and transgranularfracture toughness for a range of minerals (modified after Tromans and Meech,2002).

72 N. Djordjevic / Minerals Engineering 46–47 (2013) 68–75

(from one mineral grain to another). Or in other words, even for thecase of mono-minerals, multi-grain material, mineral boundariesare not considered to be weaker than the main body of the mineral.

However, it is commonly believed that the interface betweentwo mineral grains is characterized with lower strength than themineral grain itself. This is confirmed with modeling. Results pre-sented by Tromans and Meech (2002) show that, on average, grain

boundary fracture toughness is lower than trans-granular fracturetoughness. By fitting numerical values presented by these authors,we concluded that grain boundary fracture toughness is lower onaverage by about 9% than trans-granular fracture toughness,Fig. 12.

The implications of ignoring systematic reduction in strength,associated with grain boundaries, are investigated by modelingmechanical response and fragmentation for two cases of synthetictexture. In one case a model is built with minerals without reduc-tion in mechanical proprieties along the grain boundaries, while inthe other case strength of the material along a relatively thick grainboundary region is reduced by 10%, relative to the strength of min-eral (K-spar) which comprised the dominant mineral phase in themodel, Fig. 13.

N. Djordjevic / Minerals Engineering 46–47 (2013) 68–75 73

The same modeling is repeated, without specific grain bound-aries. The results obtained demonstrate that in the case wherethe second phase is a mineral which is very different (harder) thanthe host minerals, the effect of grain boundaries appears to beinsignificant, Figs. 14 and 15.

In the case where textures are composed from the minerals inwhich the mechanical properties are similar, assuming that shapeof grains is the same or similar, it is reasonable to expect thatreduction in strength associated with grain boundaries will be ofgreater consequence. Considering, the random nature of the orien-tation of the load and the orientation of the grain boundaries, it isnot surprising that in such cases liberation of any particulate min-erals will appear as a random process.

This is common for waste rock where minerals tend to be ofsimilar mechanical properties. In the case of ore, where frequentlyvaluable minerals are of strikingly different mechanical propertiesfrom associated gangue minerals, randomness associated with ori-entation of grains in relation to the direction of pressure applica-tion, is reduced. In such cases we can argue that preferentialliberation of minerals will occur more frequently.

Fig. 13. Synthetic texture used to investigate effect of grain boundaries (black) formodel composed from K-spar (gray) and magnetite (blue). (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

Fig. 14. Fragmentation pattern for the case with grain boundaries, with blackregions indicates fractures.

7. Influence of texture vs. mechanical properties of minerals onthe rock fragmentation pattern

To further investigate the sensitivity of fragmentation modelingon the variation in mechanical properties of minerals, we modeleda case where rock is composed from matrix and one type of min-eral in which properties were varied. In both cases matrix is repre-sented with K-spar, while the second phase was either magnetite,pyrite or chalcopyrite. Mechanical properties of matrix were keptconstant, while properties of 2nd mineral phase were selected asper Table 1. Samples were loaded along vertical axis in compres-sion on a fixed base. In the case of pyrite vs. magnetite, the finalpattern of fracturing is almost identical, Figs. 16 and 17.

The results show that differences in elastic properties betweenpyrite and magnetite are not sufficient to produce a noticeabledifference in fragmentation pattern where the host matrix is K-spar.

In the case where the second phase is chalcopyrite, the patternof fragmentation at the same level of strain, differs significantlyfrom the one observed in the case of pyrite and magnetite, Fig. 18.

Fig. 15. Fragmentation pattern for the case without grain boundaries, black regionsindicate fractured rock.

Fig. 16. Fragmentation for the case of pyrite as second phase, black indicatesfractures.

Fig. 17. Fragmentation for the case of magnetite as second phase, black indicatesfractures.

Fig. 18. Fragmentation for the case of chalcopyrite as second phase, black indicatesfractures.

Fig. 19. Pattern of fracturing for the case of pyrite as second phase minerals, blackindicates fractures.

Fig. 20. Pattern of fracturing for the case of chalcopyrite as the second phasemineral, under same loading conditions, black indicates fractures.

Fig. 21. Case of large grains, largely of regular shape (EH2), black indicatesfractures.

74 N. Djordjevic / Minerals Engineering 46–47 (2013) 68–75

The difference in properties between pyrite and chalcopyritebecome significant, in terms of level of damage that can be inducedin the rock, for a given amount of introduced energy. For the sameamount of strain, pyrite as the second phase will cause substantialweakening of K-spar matrix while, in the case of chalcopyrite, onlyminor debonding will occur, Figs. 19 and 20.

It is quite informative to compare the final pattern of fragmen-tation, in the case of two radically different textures, but with thesame second phase mineral (magnetite), Figs. 17 and 21.

Although fraction of the second phase minerals was not identi-cal, differences in pattern of fragmentation for the same amount ofstrain are large. This is clearly visible in the fragment size distribu-tions, Fig. 22.

These results indicate that the difference in texture may bemore significant for the final pattern of fragmentation, than smallor even moderate differences in the mechanical properties of min-erals. The case where second phase mineral is widely distributedwithin a relatively soft matrix (of identical properties in bothcases), is conducive for fine fragmentation, while the opposite isthe case where second phase, where relatively hard mineral, ispresent in the form of a few relatively large grains. The implica-tions are that any third mineral phase homogenously distributed

Fig. 22. Fragment size distribution curve for two radically different textures (withsame properties of the second phase mineral under identical loading conditions).

N. Djordjevic / Minerals Engineering 46–47 (2013) 68–75 75

within matrix, which may be of economical interest, is much morelikely to be liberated when hard mineral phase is also relativelywidely distributed within matrix. This indicates that liberation ofthe 3rd phase (chalcopyrite) as result of proximity to harder min-erals (pyrite, magnetite) is strong function of distance betweenthose grains. More even distribution of hard grains will be moreconducive to liberation of chalcopyrite grains dispersed in theirvicinity.

8. Conclusions

Image based numerical modeling is a useful tool for investiga-tion of the pattern and dynamics of the rock breakage process.Its usefulness rests on the fact that a difficult step of building afaithful model of rock texture and composition, as a pre-requisitefor modeling of rock breakage, is removed. Images based numericalmodeling could be particularly effective in the mineral extractionindustry, where the key objective of crushing and grinding is thefull or partial liberation of specific minerals.

One of the stumbling blocks in the implementation of the methodis the determination of the necessary input parameters, such asYoung’s modulus of elasticity and the strengths of minerals. In caseswhere reliable information cannot be found in the literature, thisobstacle can be removed by micro-indentation testing of the polishedsamples of rock. Micro-indentation testing is a fast and sufficientlyaccurate method for the determination of the key mechanical param-eters of minerals required for numerical modeling.

In cases where matrix is of significantly lower elasticity andstrength than the included minerals, modeling results indicate thateven moderate variation in the mechanical properties of valuableminerals will not change the final pattern of fragmentation. Tex-tural parameters of the rock, such as size, shape and distributionof minerals are more important than their mechanical properties

within the range investigated. This may not be the case when min-erals are of similar mechanical properties to the properties ofmatrix.

The modeling approach used so far, mimics slow compressionloading. In real crushing and grinding environments, the rate ofloading is faster; therefore there is a possibility that some selectiv-ity, in terms of breakage and liberation, seen in this modeling, maynot be present in cases of more dynamic modeling. To investigatecases of the dynamic breakage of rock, it is necessary to use differ-ent types of FE codes. This may be possible by transferring rock im-age based, structured FE mesh from the OOF code into compatibledynamic FE codes.

Acknowledgments

This research was part of collaborative geometallurgical projectbeing undertaken at CODES (University of Tasmania) and JKMRC(SMI, University of Queensland). The author would like to acknowl-edge help of Dr. Steven Walters who created images of syntheticrock texture and Dr. Luke Keeney who performed micro-indenta-tion testing. Thanks are due to Dr. Rob Morrison for insightful com-ments and Mrs. K. Holtham for editing help. The authoracknowledges financial support and permission to publish fromindustry sponsors of the AMIRA International GEM Project.

References

Blair, S.C., Cook, N.G.W., 1998a. Analysis of compressive fracture in rock usingstatistical techniques: Part I. A non-linear rule-based model. InternationalJournal of Rock Mechanics and Mining Sciences 35 (7), 837–848.

Blair, S.C., Cook, N.G.W., 1998b. Analysis of compressive fracture in rock usingstatistical techniques: Part II. Effect of microscale heterogeneity on macroscopicdeformation. International Journal of Rock Mechanics and Mining Sciences 35(7), 849–861.

Everitt, R.A., Lajtai, E.Z., 2004. The influence of rock fabric on excavation damage inthe Lac du Bonnett granite. International Journal of Rock Mechanics and MiningSciences 41(8). Rock Mechanics Results from the Underground ResearchLaboratory, Canada, December 2004, pp. 1277–1303.

Garboczi, E.J., Day, A.R., 1995. An algorithm for computing the effective linear elasticproperties of heterogeneous materials: three-dimensional results forcomposites with equal phase Poisson’s ratios. Journal of the Mechanics andPhysics of Solids 43 (9), 1349–1362.

Langer, S.A., Fuller Jr., E.R., Carter, W.C., 2001. OOF: an image-based finite-elementanalysis of material microstructures. Computing in Science & Engineering 3 (3),15–23.

Schlangen, E., Garboczi, E.J., 1997. Fracture simulations of concrete using latticemodels: computational aspects. Engineering Fracture Mechanics 57 (2–3), 319–332.

Tang, C.A., Liu, H., Lee, P.K.K., Tsui, Y., Tham, L.G., 2000. Numerical studies of theinfluence of microstructure on rock failure in uniaxial compression – Part I:Effect of heterogeneity. International Journal of Rock Mechanics and MiningSciences 37 (4), 555–569.

Tromans, D., Meech, J.A., 2002. Fracture toughness and surface energies of minerals:theoretical estimates for oxides, sulphides, silicates and halides. MineralsEngineering 15 (12), 1027–1041.