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 theory
of fuzzy sets. The heterogeneity of rock as well as damage are characterized by the membership functions
of the rock into fuzzy sets of materials with given properties and failed materials, respectively. On the
basis of developed concept of the fuzzy damage parameter, the damage evolution in heterogeneous rock
is simulated. The influence of initial rock heterogeneity on its strength is studied numerically, and it is
concluded 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 des
ensembles 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 Fuzzy
Mengen analysiert. Das Schädigungsniveau im Gestein wird mit Fuzzy Mitgliedschaftsfunktion characterisiert. Auf Grund des entwickelten Modells des Fuzzy Schädigungsparameter, wird die
Schädigungsevolution in heterogenen Gestein simuliert. Der Einfluß der anfänglichen Heterogenität
des 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 the
processes of rock fragmentation to a large extent.
The fragmentation of rock depends on the conditions of loading as weHas on the rock properties (in
particular, strength). Any data about the strength
of rock are averaged and uncertain; it is a typical
case, when the difference between the measured
and predicted properties of rock is more than
40%
, or when the strength of two specimens
from the same material differ sufficiently. It is
caused by the fact that rocks are usually highly
heterogeneous materials, consist on different minerals and their formation during many thousands
of years was determined by random processes.
In order to improve the efficiency of rock frag-
735
mentation, mathematical models of this process'
which describe all peculiarities of the rock destruction, are needed. Yet, only the models of single
crack propagation (fracture mechanics) or the evolution of distributed microcracks (the continuum
damage 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 about
80 % 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 cracks
under 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 region
up to crack formation (like the continuum damage
mechanics) are needed to describe the rock fragmentation in drilling. Here, it is suggested to use
the concept of the theory of fuzzy sets to model
the destruction of heterogeneous rocks, and to investigate interrelations between rock strength and
heterogeneity.
Traditionally, the methods of the theory of fuzzy
sets are applied to operate quantitatively with
some linguistic, inexact or uncertain values (see
Yao, 1979, Brown, 1980, etc). For example, Brown
(1979) has obtained theoretically the safety measure (probability of failure), which includes both
subjective and objective information about considered system, and the subjective information is
determined with the use of the fuzzy sets theory
on the basis of the linguistic data.
Yao (1980) has suggested to use the descriptive
words of specialists to 0 btain the membership function of a structure in a corresponding fuzzy set .
Blockley and Baldwin (1987) have considered an
applicability of the fuzzy logic models in computer
knowledge bases.
Lu Ping and Hudson (1993) have used the fuzzy
sets in order to evaluate the inexact data about
the stability of underground excavation. Shiraishi
and Furuta (1983) developed a methodology for
the evaluation of subjective uncertain factors (like
mistakes,etc) on the basis of the fuzzy sets theory.
So, the main direction of the application of the
fuzzy sets theory in the engineering science is to
use the descriptive, subjective or inexact data in
order to obtain the quantitative assessments of safety, reliability or other characteristics of objects.
Yet, the theory of fuzzy sets can be used not only
to relate some qualitative and quantitative values,
but also to describe uncertain relations or characteristics which are available in nature. One of the
examples of an application of the fuzzy sets concept to describe a natural uncertainty. is the model
of cyclic plasticity developed by Klicinski (1988).
This paper seeks to develop a model of rock damage, which allows for the natural uncertainty of
local rock properties as well as the microfracture
processes, and to study the influence of the rock
heterogeneity on the efficiency of drilling.
2 HETEROGENEITY AND UNCERTAINTY
OF ROCK PROPERTlES
Rocks are highly heterogeneous materials, and
their heterogeneity has a pronounced effect on the
efficiency of drilling. To study the destruction of
such material, one needs to develop a method of
description of the heterogeneity. Consider a homogeneous material with strength u. It can be
taken as an element of a set of such materials with
the membership degree 1. Yet, if one considers a
rock, this set presents a fuzzy set: the strength of
roc.k depends on size, conditions of loading, components, history of rock formation, origin, etc. So,
the membership degree of a given rock specimen
into the set of the rocks with given properties is
less than 1. One can define the membership function X(u), which characterizes the closeness of behaviour (strength) of the rock to a homogeneous
material with given properties.
Usually, rocks consist on a number of components (minerals): for example, granite consists
on 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 with
the 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 role
of each component in the rock behaviour.
The heterogeneity of a material can be characterized as well by the statistical entropy of local
properties (strength) of the material. Based on
the results by Kauffmann (1977), one can write
the relation between the function X( u) and this
statistical entropy:
H
= -(l/N)
Li
Pi (Ui) In Pi (ai)
(1)
where H - the statistical entropy of local properties (strength) of the material, Ui - the strength of
each mineral, Pi(Ui) = X(Ui)/L:i(X(Ui).
It is evident, that the more heterogeneous is the rock, the
less is the value X(u), and the greater the statistical entropy of the material properties.
Thus, the function X(Ui) and the statistical
entropy of local strength of the rock characterizes the heterogeneity of the rock, and the role of
each constituent of the rock in its behaviour under
loading.
736
3 DAMAGE PARAMETER AS A MEMBERSHIP FUNCTION OF MATERIAL INTO
FUZZY SET OF FAILED STATE
The damage mechanies, as differentiated from
other, more traditional methods of modelling of
destruction (the fracture mechanics or statistical
theory of strength, for example) allows to describe
the dependence of rock destruction on time, and
thermodynamical aspects of destruction. Yet, the
damage evolution is described in the framework of
the damage mechanies only up to the formation
of a macrocrack (Lemaitre 1992). As said above,
the efficiency of rock fragmentation in drilling is
influenced mainly by the rock crushing or cracks
system evolution, what is outside the region of the
applicability of the damage mechanics. Here, it
is suggested to formulate the concept of damage,
which is based on the fuzzy sets approach, and is
applicable not only to the evolution of distributed
microcracks.
Consider the region of behaviour of rock, which
is limited by the condition of local fracture. It is
clear, 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 as
weIl. One can define the membership function of
the rock into the fuzzy set of failed state " which
depends on the applied stress, and characterizes
the degree of c!oseness of the body to failure (Mishnaevsky Jr 1995a). In the continuum damage mechanics, the damage parameter is considered often
as a measure of closeness of loaded body to failure. This meaning of the damage parameter corresponds to this membership function, which we just
defined, and that is why we prefer to use the short
term "fuzzy damage parameter" instead of "membership function of the material into the fuzzy set
of failed state" .
The function , can be considered as some kind
of generalization of the probability of failure, but
it characterizes not the possible, but present state
or kind of behaviour of rock, and this state of rock
can lie not only between elastic and failed one, but
between cracked and crushed rock as weIl.
In general, the function , can be related with
any continuously growing from monolithic to failed
state value (Mishnaevsky Jr 1995a), or determined
with the use of people experience and knowledge
(Yao 1980, Brown 1979, etc). Yet, it follows from
the physical nature of fracture, that the change of
737
the nmction , proceeds only in the direction from
,=0 to 1. So, this value characterizes the degree of
irreversibility of the evolution of the material, and
can be related with the accumulated entropy per
unit volume of material (Mishnaevsky Jr 1995a).
4 DAMAGE EVOLUTION AND HETEROGENIZATION OF LOADED VOLUME
Suppose that the fuzzy damage parameter of a rock
volume is equal to ,I. Then, if this volume of
rock is loaded, and this load can produce in nondamaged rock the fuzzy damage ,2, the summary
damage in the rock is calculated by the following
formula (Kauffmann 1977):
(2)
'I
where MAX is the maximal value from
and ,2'
As differentiated from the probabilities offailure or
reliabilities of elements, membership functions are
not added. It means that if the rock specimen contains some number of microcracks and some number of microcracks is added, the state of rock does
not changed. This conclusion (which is evidently
not correct) is drawn, since eq.(2) does not allow
for 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 , on
local strength, one can present this membership
nmction as a conditional one, depending on the
strength of rock. So, the fuzzy damage parameter
in 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 damaged state in the m,aterial.
1.0
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:
~ 0.8
cu
~
0.6
Lt
0.2
"0
>N 0.4
N
Eq.(4) relates the heterogeneity of rock and the
damage distribution.
Substituting eq.(4) into eq.(3), one can obtain
the damage evolution law in recursive form. In so
doing, one should note that the values , in right
and left sides of eq.(4) correspond to different instants of time. So, the damage evolution law shoud
be rewritten as folIows:
0.0
0.0 2.0 4.0 6.0 8.0 10.0
Time (timesteps)
Figure 1. Fuzzy damage versus time
Consider the damage evolution in rock under
loading. Let the initial heterogeneity of the rock
be characterized by the function Xo (o"i). Due to
the damage evolution, the local strength of rock is
changed as weil (Lemaitre 1992).
--
C
(j)
Q)
:c äi
•...
g>-
0.139
0.149
0.144
0.134
-0.51.5
3.5 5.5 7.5 9.5
Time (timesteps)
where t and t + At mean two successive timesteps.
Fig.l shows the fuzzy damage parameter versus
time (in timesteps). The value, was calculated by
the formula (5); the rock was supposed to consists
on 15 different component. The values X and , for
one 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 to
a large extent). For all other minerals, , and X
are equal to 0.04 and 0.06 (Le., these minerals are
relatively soft).
For each timestep, the statistical entropy of local
strength of rock was calculated by the formula (1).
Fig.2 shows the statistical entropy versus time
dependence. One can see that the heterogeneity of
rock increases with increasing damage parameter.
5 INFLUENCE OF ROCK HETEROGENEITY
ON STRENGTH
Figure 2. Stat. entropy of strength versus time
So, the heterogeneity of damaged rock is determined both by the initial heterogeneity and by the
heterogeneity caused by the damage distribution.
Ir the rock is homogeneous initially, and only the
heterogeneity of damage distribution determines
the heterogeneity of loaded rock, the fuzzy set of
given properties of rock presents a complement of
the fuzzy set of the damaged state of rock, and
function X can be determined as folIows: x(ai) =
1- ,(lai).
Ir the rock heterogeneity depends both on the
738
The developed model allows to investigate numerically the influence of rock heterogeneity on
strength.
We considered about 250 types of rock, with different properties and distributions of constituents
in them.
Each rock consists on 15 components; each component is characterized by the values X (Le. the
degree of variation of the strength of the component from the rock strength) and , (i.e. the fuzzy
damage parameter for this mineral at given load).
The values , for each constituent were taken as
random numbers; in order to ensure varying the
entropy of rock strength, the values X were taken
to be proportional to im, where i - the number
of 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 strength
distribution and the fuzzy damage parameter were
calculated by formulas (1) and (3).
Fig.3 shows the fuzzy damage parameter of rock
(at given load) versus the statistical entropy H of
local rock strengths.
1.0
~ 0.8
CU
~
0.6
Lt
0.2
"0
>- 0.4
N
N
0.0
0.00 0.05 0.10 0.15 0.20
Rock strength entropy
Figure 3. Fuzzy damage versus rock strength
entropy
One can see that the more heterogeneous is the
rock, the eloser it to failure, at the same loads.
It is of interest to consider the scale efffect on
rock fracture in the context of the fuzzy concept of
damage. 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 of
a system from independently loaded elements with
the same membership functions does not depend
on the number of elements (Kauffmann 1977) (in
contrast to the reliability or probability of failure
of this system). Yet, taking into account that the
fuzzy damage parameter increases with increasing
the heterogeneity of rock, one can suppose that
the increase of size of a specimen leads most often
to the increase in the rock heterogeneity : rocks
have heterogeneous structure at severallevels (for
example, anisotropy, granularity, texture, layers,
etc), and the greater the size of used specimen, the
more levels of structure influence on the strength
of material and form its heterogeneity. Thus, the
greater is the specimen, the greater is its heterogeneity and, consequently, the less its strength.
739
6 HETEROGENEITY OF ROCK AND EFFICIENCY OF ROCK FRAGMENTATION
It follows from Fig.I-2, that the damage growth
rate decreases when the fuzzy damage parameter is
rather large and exceeds approximately 0.85. The
region in which the damage growth rate is maximal, corresponds to the maximal heterogenization
of loaded volume of rock due to the damage accumulation.
It can be compared with the description of the
stages of rock fragmentation, given in the review
by Mishnaevsky Jr (1995b): the heterogenization
of rock in damage accumulation, which is proved
here theoretically, was observed in experimental investigations as weIl. The loaded volume of rock at
last stages of rock deformation contains the zone
of crushed rock, the system of large cracks and distributed microcracks. About 70-80 of the energy
of loading goes into the zone of crushed rock . It
means that the heterogenization of rock volume
due to the damage localization (Le. the dividing
of loaded volume into the zones with different degree of destruction) leads to the increase of the
energy consumption in drilling, and, consequently,
decreases of the efficiency of drilling.
One can see from Fig.l, that the damage growth
rate decreases at large damage parameter. This effeet can not be predicted on the basis of Lemaitre's
or Kachanov's damage evolution laws (Lemaitre
1992), but the analogous result was obtained by
Rossmanith et al (1995) on the basis of the computer simulation of the impact destruction of rock
by the method of discrete element. The same dependence between the degree of rock destruction
and the rate of damage growth was obtained experimentally (see review, Mishnaevsky Jr 1995b): at
last stages of rock destruction, the zone of crushed
rock is formed in the vicinity of the rock/tool contact surface and almost all energy of loading goes
into this zone; the specific energy needed for rock
crushing 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 rock
is destructed more intensively than homogeneous
or 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 can
not be controlled), one can conelude that in order
to achieve the maximal degree of rock fragmentation, one should form maximum heterogeneous
stress field in loaded rock.
This conclusion can be compared with the results of Mishnaevsky Jr (1996), who has shown
that the high efficiency of drilling can be achieved
by high heterogeneity of conditions of loading.
Practically, in order to obtain the heterogeneous
distribution of damage in rock volume, which is to
be destructed and removed, one can load this valume before drilling (pre-Ioading) and form cracks
in it (Mishnaevsky Jr 1994). So, the improvement
of efficiency of rock fragmentation can be achieved
by pre-cracking or pre-Ioading of the rock volume.
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7 CONCLUSIONS
It is shown that the methods and concepts of
the theory of fuzzy sets can be efficiently applied
to study the damage evolution in heterogeneous
rocks.
The damage accumulation in loaded rock leads
to the heterogenization of the rock, i.e. to the formation of zones with sufficiently different level of
damage (zone of crushed rock, cracked zone, etc,
see Mishnaevsky Jr 1995b). The heterogenization
of rock, especially, the formation of the zone of
crushed rock, gives rise to the energy consumption
in drilling, and decreases the efficiency of rock fragmentation.
It is shown that the more homogeneous is rock,
the more its strength. Practically, it means that in
order to increase the efficiency of rock fragmentation, the rock should be made more heterogeneous
by formation of damage or cracks in the rock valume, which should be removed. Such pre-cracking
allows to improve the efficiency of rock fragmentation.
ACKNOWLEDGEMENTS
The author (L.M.) is grateful to the Alexander
von Humboldt Foundation for the possibility to
carry out the research project in the University of
Stuttgart, 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, Technical
University of Vienna (Austria). Interesting discussions with Dr.Rossmanith during my work in the
Photo and Fracture Mechanics Laboratory, TU Vienna, and his valuable advices are gratefully acknowledged.
740
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