Eurock '96, Barla (ed.) © 1996 Balkema, Rotterdam. ISBN 90 5410 843 6

Analysis of rock fragmentation with the use of the theory of fuzzy sets

L'analyse de la destruction de la rüche a la base de la theorie des ensembles flous

Analyse der Gesteinzerstörung mit Verwendung der Theorie der Fuzzy Mengen

L. L. Mishnaevsky Jr & S. Schmauder - MPA, University of Stuttgart, Germany

ABSTRACT: The damage evolution in loaded rock is modeled on the basis of the methods of the theoryof fuzzy sets. The heterogeneity of rock as well as damage are characterized by the membership functionsof the rock into fuzzy sets of materials with given properties and failed materials, respectively. On thebasis of developed concept of the fuzzy damage parameter, the damage evolution in heterogeneous rockis simulated. The influence of initial rock heterogeneity on its strength is studied numerically, and it isconcluded that the more heterogeneous is the rock, the less its strength.

RESUME: L'evolution des endommagements dans la roche heterogene est etudie. Un modele mathe­matique de la evolution des endommagements dans la roche , que s' etablit a la base de la theorie desensembles flous , est elabore. L'influence de la heterogenite de la roche a la resistance limite est analyse.On montre, que plus heterogene est une roche, moins la resistance limite de la roche.

ZUSAMENFASSUNG: Die Schädigungsevolution in Gestein wird mit Methoden der Theorie von FuzzyMengen analysiert. Das Schädigungsniveau im Gestein wird mit Fuzzy Mitgliedschaftsfunktion characte­risiert. Auf Grund des entwickelten Modells des Fuzzy Schädigungsparameter, wird dieSchädigungsevolution in heterogenen Gestein simuliert. Der Einfluß der anfänglichen Heterogenitätdes Gesteins auf seine Festigkeit wird numerisch untersucht. Es wird gezeigt, daß mit steigender He­terogenität des Gesteins die Festigkeit abnimmt.

1 INTRODUCTION

The efficiency of rock drilling is determined by theprocesses of rock fragmentation to a large extent.The fragmentation of rock depends on the conditi­ons of loading as weHas on the rock properties (inparticular, strength). Any data about the strengthof rock are averaged and uncertain; it is a typicalcase, when the difference between the measuredand predicted properties of rock is more than40% , or when the strength of two specimensfrom the same material differ sufficiently. It iscaused by the fact that rocks are usually highlyheterogeneous materials, consist on different mi­nerals and their formation during many thousandsof years was determined by random processes.

In order to improve the efficiency of rock frag-

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mentation, mathematical models of this process'which describe all peculiarities of the rock destruc­tion, are needed. Yet, only the models of singlecrack propagation (fracture mechanics) or the evo­lution of distributed microcracks (the continuumdamage mechanics, see Lemaitre, 1992) can be ap­plied efficiently to describe the rock destruction.Yet, such processes as rock crushing or cracks sy­stem evolution are of great importance in drilling(see review by Mishnaevsky Jr 1995b). For exam­pIe, the energy consumption in drilling is determi­ned by the rock crushing to a large extentj about80 % of the loading energy is spent just for it .One of the ways to improve the efficiency of dril­ling is to use the interaction between large cracksunder cuts (Mishnaevsky Jr 1994). So, the mo­dels of rock destruction which take into account

not only single objects (like the fracture mecha­nies), and are applicable not only in the regionup to crack formation (like the continuum damagemechanics) are needed to describe the rock frag­mentation in drilling. Here, it is suggested to usethe concept of the theory of fuzzy sets to modelthe destruction of heterogeneous rocks, and to in­vestigate interrelations between rock strength andheterogeneity.

Traditionally, the methods of the theory of fuzzysets are applied to operate quantitatively withsome linguistic, inexact or uncertain values (seeYao, 1979, Brown, 1980, etc). For example, Brown(1979) has obtained theoretically the safety mea­sure (probability of failure), which includes bothsubjective and objective information about con­sidered system, and the subjective information isdetermined with the use of the fuzzy sets theoryon the basis of the linguistic data.

Yao (1980) has suggested to use the descriptivewords of specialists to 0btain the membership func­tion of a structure in a corresponding fuzzy set .Blockley and Baldwin (1987) have considered anapplicability of the fuzzy logic models in computerknowledge bases.

Lu Ping and Hudson (1993) have used the fuzzysets in order to evaluate the inexact data aboutthe stability of underground excavation. Shiraishiand Furuta (1983) developed a methodology forthe evaluation of subjective uncertain factors (likemistakes,etc) on the basis of the fuzzy sets theory.So, the main direction of the application of thefuzzy sets theory in the engineering science is touse the descriptive, subjective or inexact data inorder to obtain the quantitative assessments of sa­fety, reliability or other characteristics of objects.

Yet, the theory of fuzzy sets can be used not onlyto relate some qualitative and quantitative values,but also to describe uncertain relations or charac­teristics which are available in nature. One of theexamples of an application of the fuzzy sets con­cept to describe a natural uncertainty. is the modelof cyclic plasticity developed by Klicinski (1988).This paper seeks to develop a model of rock da­mage, which allows for the natural uncertainty oflocal rock properties as well as the microfractureprocesses, and to study the influence of the rockheterogeneity on the efficiency of drilling.

2 HETEROGENEITY AND UNCERTAINTYOF ROCK PROPERTlES

Rocks are highly heterogeneous materials, andtheir heterogeneity has a pronounced effect on theefficiency of drilling. To study the destruction ofsuch material, one needs to develop a method ofdescription of the heterogeneity. Consider a ho­mogeneous material with strength u. It can betaken as an element of a set of such materials withthe membership degree 1. Yet, if one considers arock, this set presents a fuzzy set: the strength ofroc.k depends on size, conditions of loading, com­ponents, history of rock formation, origin, etc. So,the membership degree of a given rock specimeninto the set of the rocks with given properties isless than 1. One can define the membership func­tion X(u), which characterizes the closeness of be­haviour (strength) of the rock to a homogeneousmaterial with given properties.

Usually, rocks consist on a number of com­ponents (minerals): for example, granite consistson grains of quartz, feldspar, etc. If one designa­tes the strengths of each mineral as U1, U2, etc,the properties of rock depend on all these values,and are similar to a homogeneous materials withthe strength U1 or U2, etc to a some (most often,very small) degree. So, one can define a member­ship nmction X(Ui) , which characterizes the roleof each component in the rock behaviour.

The heterogeneity of a material can be charac­terized as well by the statistical entropy of localproperties (strength) of the material. Based onthe results by Kauffmann (1977), one can writethe relation between the function X( u) and thisstatistical entropy:

H = -(l/N) LPi (Ui) In Pi (ai) (1)i

where H - the statistical entropy of local proper­ties (strength) of the material, Ui - the strength ofeach mineral, Pi(Ui) = X(Ui)/L:i(X(Ui). It is evi­dent, that the more heterogeneous is the rock, theless is the value X(u), and the greater the statisti­cal entropy of the material properties.

Thus, the function X(Ui) and the statisticalentropy of local strength of the rock characteri­zes the heterogeneity of the rock, and the role ofeach constituent of the rock in its behaviour underloading.

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3 DAMAGE PARAMETER AS A MEMBER­SHIP FUNCTION OF MATERIAL INTOFUZZY SET OF FAILED STATE

The damage mechanies, as differentiated fromother, more traditional methods of modelling ofdestruction (the fracture mechanics or statisticaltheory of strength, for example) allows to describethe dependence of rock destruction on time, andthermodynamical aspects of destruction. Yet, thedamage evolution is described in the framework ofthe damage mechanies only up to the formationof a macrocrack (Lemaitre 1992). As said above,the efficiency of rock fragmentation in drilling isinfluenced mainly by the rock crushing or crackssystem evolution, what is outside the region of theapplicability of the damage mechanics. Here, itis suggested to formulate the concept of damage,which is based on the fuzzy sets approach, and isapplicable not only to the evolution of distributedmicrocracks.

Consider the region of behaviour of rock, whichis limited by the condition of local fracture. It isclear, that the condition of local fracture of hete­rogeneous rock is uncertain. So, this region pres­ents a fuzzy set. The complement of this set (Le.a region of failed state of rock) is a fuzzy set asweIl. One can define the membership function ofthe rock into the fuzzy set of failed state " whichdepends on the applied stress, and characterizesthe degree of c!oseness of the body to failure (Mis­hnaevsky Jr 1995a). In the continuum damage me­chanics, the damage parameter is considered oftenas a measure of closeness of loaded body to fai­lure. This meaning of the damage parameter corre­sponds to this membership function, which we justdefined, and that is why we prefer to use the shortterm "fuzzy damage parameter" instead of "mem­bership function of the material into the fuzzy setof failed state" .

The function , can be considered as some kindof generalization of the probability of failure, butit characterizes not the possible, but present stateor kind of behaviour of rock, and this state of rockcan lie not only between elastic and failed one, butbetween cracked and crushed rock as weIl.

In general, the function , can be related withany continuously growing from monolithic to failedstate value (Mishnaevsky Jr 1995a), or determinedwith the use of people experience and knowledge(Yao 1980, Brown 1979, etc). Yet, it follows fromthe physical nature of fracture, that the change of

the nmction , proceeds only in the direction from,=0 to 1. So, this value characterizes the degree ofirreversibility of the evolution of the material, andcan be related with the accumulated entropy perunit volume of material (Mishnaevsky Jr 1995a).

4 DAMAGE EVOLUTION AND HETEROGE­NIZATION OF LOADED VOLUME

Suppose that the fuzzy damage parameter of a rockvolume is equal to ,I. Then, if this volume ofrock is loaded, and this load can produce in non­damaged rock the fuzzy damage ,2, the summarydamage in the rock is calculated by the followingformula (Kauffmann 1977):

(2)

where MAX is the maximal value from 'I and ,2'As differentiated from the probabilities offailure orreliabilities of elements, membership functions arenot added. It means that if the rock specimen con­tains some number of microcracks and some num­ber of microcracks is added, the state of rock doesnot changed. This conclusion (which is evidentlynot correct) is drawn, since eq.(2) does not allowfor the change of rock properties due to the da­mage accumulation and the fact that the function, depends on the local strength of material.

Taking into account the dependence of , onlocal strength, one can present this membershipnmction as a conditional one, depending on thestrength of rock. So, the fuzzy damage parameterin heterogeneous rock is determined by the for­mula of conditional membership function (Kauff­mann 1977):

,= 1- MIN[l-,(la),x(a)] (3)

where ,(la) is a conditional fuzzy damage para­meter for given strength of rock, x(a) is the mem­bership function, which characterizes the hetero­geneity of the strength of rock.

Eq.(2) gives the relation between the heteroge­neity of material properties (x(a)) and the dama­

ged state in the m,aterial.

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1.0

~ 0.8cu

~ 0.6"0>- 0.4NNLt 0.2

0.00.0 2.0 4.0 6.0 8.0 10.0

Time (timesteps)

Figure 1. Fuzzy damage versus time

Consider the damage evolution in rock underloading. Let the initial heterogeneity of the rockbe characterized by the function Xo (o"i). Due tothe damage evolution, the local strength of rock ischanged as weil (Lemaitre 1992).

0.149:c >-g- 0.144•...

-CQ)äi

0.139- (j)

0.134-0.51.5 3.5 5.5 7.5 9.5

Time (timesteps)

Figure 2. Stat. entropy of strength versus time

So, the heterogeneity of damaged rock is deter­mined both by the initial heterogeneity and by theheterogeneity caused by the damage distribution.

Ir the rock is homogeneous initially, and only theheterogeneity of damage distribution determinesthe heterogeneity of loaded rock, the fuzzy set ofgiven properties of rock presents a complement ofthe fuzzy set of the damaged state of rock, andfunction X can be determined as folIows: x(ai) =1- ,(lai).

Ir the rock heterogeneity depends both on the

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initial (Xo) and damage-induced (1 - ,) heteroge­neity, the function x(ai) is determined as an alge­braic sum of these two functions. On rearrange­ments, one can obtain:

Eq.(4) relates the heterogeneity of rock and thedamage distribution.

Substituting eq.(4) into eq.(3), one can obtainthe damage evolution law in recursive form. In sodoing, one should note that the values , in rightand left sides of eq.(4) correspond to different in­stants of time. So, the damage evolution law shoudbe rewritten as folIows:

where t and t + At mean two successive timesteps.Fig.l shows the fuzzy damage parameter versus

time (in timesteps). The value, was calculated bythe formula (5); the rock was supposed to consistson 15 different component. The values X and , forone of these components are equal to 0.5 and 0.002,respectively (it means, that this constituent is rat­her strength and determines the rock properties toa large extent). For all other minerals, , and X

are equal to 0.04 and 0.06 (Le., these minerals arerelatively soft).

For each timestep, the statistical entropy of localstrength of rock was calculated by the formula (1).

Fig.2 shows the statistical entropy versus timedependence. One can see that the heterogeneity ofrock increases with increasing damage parameter.

5 INFLUENCE OF ROCK HETEROGENEITYON STRENGTH

The developed model allows to investigate nu­merically the influence of rock heterogeneity onstrength.

We considered about 250 types of rock, with dif­ferent properties and distributions of constituentsin them.

Each rock consists on 15 components; each com­ponent is characterized by the values X (Le. thedegree of variation of the strength of the compo­nent from the rock strength) and , (i.e. the fuzzydamage parameter for this mineral at given load).The values , for each constituent were taken asrandom numbers; in order to ensure varying theentropy of rock strength, the values X were taken

to be proportional to im, where i - the numberof considered constituent of the rock, i =1,2, ...15,and m - the number of considered type of rock,m = 1,2,...250. The statistical entropy of strengthdistribution and the fuzzy damage parameter werecalculated by formulas (1) and (3).

Fig.3 shows the fuzzy damage parameter of rock(at given load) versus the statistical entropy H oflocal rock strengths.

1.0

~ 0.8CU

~ 0.6"0>- 0.4NNLt 0.2

0.00.00 0.05 0.10 0.15 0.20

Rock strength entropy

Figure 3. Fuzzy damage versus rock strengthentropy

One can see that the more heterogeneous is therock, the eloser it to failure, at the same loads.

It is of interest to consider the scale efffect onrock fracture in the context of the fuzzy concept ofdamage. One can see that this effect can not be ex­plained directly, like it can be done in the statisti­cal theory of strength: the membership function ofa system from independently loaded elements withthe same membership functions does not dependon the number of elements (Kauffmann 1977) (incontrast to the reliability or probability of failureof this system). Yet, taking into account that thefuzzy damage parameter increases with increasingthe heterogeneity of rock, one can suppose thatthe increase of size of a specimen leads most oftento the increase in the rock heterogeneity : rockshave heterogeneous structure at severallevels (forexample, anisotropy, granularity, texture, layers,etc), and the greater the size of used specimen, themore levels of structure influence on the strengthof material and form its heterogeneity. Thus, thegreater is the specimen, the greater is its hetero­geneity and, consequently, the less its strength.

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6 HETEROGENEITY OF ROCK AND EFFI­CIENCY OF ROCK FRAGMENTATION

It follows from Fig.I-2, that the damage growthrate decreases when the fuzzy damage parameter israther large and exceeds approximately 0.85. Theregion in which the damage growth rate is maxi­mal, corresponds to the maximal heterogenizationof loaded volume of rock due to the damage accu­mulation.

It can be compared with the description of thestages of rock fragmentation, given in the reviewby Mishnaevsky Jr (1995b): the heterogenizationof rock in damage accumulation, which is provedhere theoretically, was observed in experimental in­vestigations as weIl. The loaded volume of rock atlast stages of rock deformation contains the zoneof crushed rock, the system of large cracks and dis­tributed microcracks. About 70-80 of the energyof loading goes into the zone of crushed rock . Itmeans that the heterogenization of rock volumedue to the damage localization (Le. the dividingof loaded volume into the zones with different de­

gree of destruction) leads to the increase of theenergy consumption in drilling, and, consequently,decreases of the efficiency of drilling.

One can see from Fig.l, that the damage growthrate decreases at large damage parameter. This ef­feet can not be predicted on the basis of Lemaitre'sor Kachanov's damage evolution laws (Lemaitre1992), but the analogous result was obtained byRossmanith et al (1995) on the basis of the com­puter simulation of the impact destruction of rockby the method of discrete element. The same de­pendence between the degree of rock destructionand the rate of damage growth was obtained expe­rimentally (see review, Mishnaevsky Jr 1995b): atlast stages of rock destruction, the zone of crushedrock is formed in the vicinity of the rock/tool con­tact surface and almost all energy of loading goesinto this zone; the specific energy needed for rockcrushing is much more than that for crack growth,and that is why the damage growth slows down.

It is seen from Fig.3 that a heterogeneous rockis destructed more intensively than homogeneousor less heterogeneous one, at the same load.

Since the heterogeneity of rock can be achie­ved through the heterogeneous damage distribu­tion (or it can be heterogeneous initially, what cannot be controlled), one can conelude that in orderto achieve the maximal degree of rock fragmen­tation, one should form maximum heterogeneous

stress field in loaded rock.This conclusion can be compared with the re­

sults of Mishnaevsky Jr (1996), who has shownthat the high efficiency of drilling can be achievedby high heterogeneity of conditions of loading.

Practically, in order to obtain the heterogeneousdistribution of damage in rock volume, which is tobe destructed and removed, one can load this va­lume before drilling (pre-Ioading) and form cracksin it (Mishnaevsky Jr 1994). So, the improvementof efficiency of rock fragmentation can be achievedby pre-cracking or pre-Ioading of the rock volume.

7 CONCLUSIONS

It is shown that the methods and concepts ofthe theory of fuzzy sets can be efficiently appliedto study the damage evolution in heterogeneousrocks.

The damage accumulation in loaded rock leadsto the heterogenization of the rock, i.e. to the for­mation of zones with sufficiently different level ofdamage (zone of crushed rock, cracked zone, etc,see Mishnaevsky Jr 1995b). The heterogenizationof rock, especially, the formation of the zone ofcrushed rock, gives rise to the energy consumptionin drilling, and decreases the efficiencyof rock frag­mentation.

It is shown that the more homogeneous is rock,the more its strength. Practically, it means that inorder to increase the efficiency of rock fragmenta­tion, the rock should be made more heterogeneousby formation of damage or cracks in the rock va­lume, which should be removed. Such pre-crackingallows to improve the efficiency of rock fragmenta­tion.

ACKNOWLEDGEMENTS

The author (L.M.) is grateful to the Alexandervon Humboldt Foundation for the possibility tocarry out the research project in the University ofStuttgart, MPA (Germany).

The author (L.M. also) was first introdu­ced to the theory of fuzzy sets by Doz. Dr.H.P.Rossmanith, Institute of Mechanies, TechnicalUniversity of Vienna (Austria). Interesting discus­sions with Dr.Rossmanith during my work in thePhoto and Fracture Mechanics Laboratory, TU Vi­enna, and his valuable advices are gratefully ack­nowledged.

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