Multiscale models of failure and percolation

Physics of the Earth and Planetary Interiors, 61(1990) 36—43 Elsevier Science Publishers B.V., Amsterdam — Printed in The Netherlands

36

Multiscale models of failure and percolation S.A. Molchanov Department of Mathematics, Moscow State University, Leninskye Gory, Moscow 119899 (U.S.S.R.)

V.F. Pisarenko and A.Ya. Reznikova International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Academy of Sciences of the U.S.S.R., Warshavskoye sh. 79, k2, Moscow 113556 (U.S.S.R.)

Molchanov, S.A., Pisarenko, V.F. and Reznikova, A.Ya., 1990. Multiscale models of failure and percolation. Phys. Earth Planet. Inter., 61: 36—43. Multiscale hierarchical percolation and failure models are introduced, and a wide class of discrete hierarchical models and their asymptotical properties are studied. This asymptotical analysis is carried out by the help of the ideas of scaling. The necessary and sufficient conditions for failure and percolation are discovered. For continuous spherical hierarchical percolation models the classical percolation theory is developed. It is shown that by increasing the number of scales and the scale step the critical volume concentration can be made to approach unity.

1. Introduction The modern concentration theory of failure of a mechanical continuum is based on the idea of accumulation of microfractures. The theoretical and experimental background of this theory has been developed in papers of Academician Zhurkov and his colleagues (see Zhurkov et al., 1977; Zhurkov, 1983). The percolation theory is an adequate technique in this field. The papers of Chelidze and his colleagues (e.g. Chelidze and Kolesmkov, 1983; Chelidze, 1987) suggested different models of failure of geophysical media, according to which the formation of the main fracture is to be understood as the last stage of percolation phase transition. In all these studies the hypothesis of a single-scale ensemble of microcracks is implied. In accordance with this hypothesis, many experimental works on failure under load (see Zhurkov et al., 1977; Zhurkov, 1983) dealt with comparatively small samples of homogeneous solids (concrete, glass and so on). How0031-9201/90/$03.50 ~ 1990

—

Elsevier Science Publishers B.V.

ever, after Sadovsky’s studies (Sadovsky, 1979; Sadovsky et al., 1987), geophysical media have often been regarded as a hierarchy of different scales. These ideas lead to a generalization of the percolation theory to deal with the case of many scales. This generalization gives us a more cornplex mathematical theory and another qualitative description of the formation of macrofractures. The multiscale model of failure was considered perhaps for the first time by Allegre et al. (1982). Our paper has two purposes. First, the wide class of models generalized by Allegre et al. (1982), and their asymptotic properties, will be studied (using ideas on scaling from Shldovsky and Efros (1979)). Second, the classical percolation theory (its basic models) will be developed for multiscale models. The principal qualitative conclusion is as follows: the percolation process is mainly controlled by the largest scales. Therefore, the popular belief in the universality of the value of the so-called coordinational number at the percolation threshold in the multiscale case is not true. In our

37

MULTISCALE MODELS OF FAILURE AND PERCOLATION

work irreversible (non-stationary) processes are studied. We have not considered the problems of consolidation (healing) in geophysical media.

type as for earthquakes (see Gutenberg and Richter, 1954). Let us introduce a concept of the failure criterion K, using the ideas of Allegre et al. (1982). The white block (cube) of the (N 1)th rank contains elementary blocks (cubes) of the Nth rank. The smallest cubes were painted black or white independently with probabilities p and g respectively. The set of all configurations (its number is 2”) is divided into two classes. Class K is called the class of failure configurations; the other class consists of non-failure configurations. In this section, we shall use the following failure criterion: a configuration is a failure if there exists a pair of opposite sides connected by a ‘surface’ formed by black cubes. In other words, a configuration is a non-failure if for all pairs of opposite sides there exists a path of white cubes joining these sides. —

2. Multiscale failure models A general d-dimensional model of this type depends on three parameters: N is the number of scales or ranks, p is intensity of microfractures of each rank, and v is the step of scaling. The defective set is constructed by induction. At the beginning (step zero), the d-dimensional space is divided into unit cubes (blocks) of basic rank 10 = 1. For each cube, we choose its colour (black or white) independently, with probabilities p and g = 1 p, respectively. Black cubes are interpreted as belonging to the defective set. The set of black (defective) cubes is denoted by D0 its —

volume density in the standard ergodic sense is P• In the next step, all the cubes of zero rank are divided into v’~equal cubes of the first (l~ 1/v) rank. Each cube of the first rank is painted black or white with the same probabilities p and g. All the cubes of the first rank form the set D1, and its volume density is pg. Repeating this process, we construct defective sets D2,..., DN formed by black cubes of ranks 12 = 1/v2,..., ‘N = 1/EN Thus, the volume con=

N

centration of the whole defective set D = U D~is 2+ +pgN = 1 gN+~ i=0 p =p +pg +pg The volume concentration of the ‘normal’ (white) part of the space is gN+ 1. Let us compare the characteristics of this model with the frequency—magnitude relation (see Gutenberg and Richter, 1954). As the volume density of defective cubes of the k th rank is pgk, the number of such cubes Nk in some large volume V will be approximately equal to VpgI~~vI~i and when V—~’oc, ln Nk C 0 + C1k. The energy Ek of transition of a cube from the normal state to the defective state is taken to be proportional to the volume: Ek c2v’~”; hence ln Ek C3 k. Therefore in Nk depends on in Ek linearly. Thus, the frequency—magnitude relation is of the same ...

—

In section 3 we shall introduce another defimtion of class K. The choice of class K depends on the specific conditions and enters the model definition. Thus, every failure model is characterized by the set d, N, and K. The polynomial ~‘

1(P)

p (a)

=

0

K

will be called a generating polynomial of class K; here p(a) —p’~°~(l _p)Pd_k(o), and k(a) is the number of black cubes in the configuration a. Failure in model is (reco4ing). defined by aInsuccessive procedure of this ‘repainting’ the first step, we keep without alteration all the black cubes of the (N 1)th scale, and the white cubes of the (N 1)th scale containing the configuration a E K of black cubes of the Nth scale (rank) are recoded black. As a result of this procedure, defects of the (N 1)th scale are excluded from consideration. The intensity of black blocks of the (N 1)th rank is now F(p) =p + (1 —p)f(p) —

—

—

—

—

.

We apply the same process of repainting (recoding) of cubes of the (N 1)th rank to new defects of tj~p(N 2)th ra~. 4~~is stage, th~ defects of the (N 1)th rank are excluded, and the defects of the (N 2)th rank become the —

—

—

—

38

S.A. MOLCHANOV ET AL.

smallest elements of our system. The intensity of ‘new’ defects of the (N 2)th rank is

In these terms, the criterion (1) has the form

p2=F2(p)=F(p1)=F[F(p)]

where D is a defective set.

Repeating this procedure, we reach the defects of zero (main) rank, whose intensity is

These results were obtained implicitly in the paper of Allegre et al. (1982). Let us consider the simplest general properties of functions 1(p) and F( p). We note that the degree of the polynomial f( p) is not greater than ~ because f(p)=~0~~pIkIgPd_IkI• It is evident that k I ~ v”_~ for a E K. Therefore, f(p) = 0( p” ‘)for p 0. A more precise analysis shows that f(p) Cp””’ where the constant C is of a combinatorial nature. We shall analyze the asymptotic behaviour of the function f(p) for three situations: (1) ~ is fired, N no; (2) N is fixed, v no; (3) p no, N no in some co-ordinated way. (1) From the previous discussion it follows that

—

FN (P)

=

PN

=

F( PN- 1) = F( F[... F ( P)] N times

}

If the final set of black cubes of zero rank percolates at infinity, we consider our (N + 1~ scale model to describe failure. If we make the centres of zero-rank cubes black or white (depending on the cube colour), then the percolation problem for a system of cubes of zero rank becomes equivalent to a d-dimensional site-percolation problem with the parameter PN~ It is known (see Shldovsky and Efros, 1979) that the percolation condition in a d-dimensional site-problem has the form

> p~

J

—*

—

—‘ —‘

—,

—

F(p) =p

PN >pcr(d)

where 2) Pcr(d) 0.59, Pcr(3) is the critical 0.31; for defect higher concentration, dimensions Pcr(

p(D)

+

gf(p) =p

+

cp~d~I + o(p~”’+1)

From the inverse function theorem it follows that F~’(p) —p Cp”~+ —

1

(

Per d)

2d

=

—

1

Taking into account the definition of the failure criterion and the formula for PN’ we find that the necessary and sufficient failure condition in the (N + 1)-scale model has the form FN(p)

>pcr(d)

(1)

We note that pcr(d) depends on the space dimension only, the function F( p) depends on the dimension, the scale step v and the failure criterion K. The root of the equation FN (p) = Per ls denoted by PCr~Then p~ = F 1~(Pcr)~ Formula (1) may be written in the form

We shall use the following well-known the asymptotic properties of iterations tion (see De Brein, 1961). If g(x) 1 —‘

g~(x)=g(...g(x))-

[n(k

—

=

1

—

[i

_p,~’)J N+1

—

—

N times

Hence it follows that

c, (N)

Per

—

Nth/(~~”’1)1

where the constant C1 may be expressed through the combinatorial constant C in the asymptotic formula for f( p) as p no. It is easy to evaluate the critical concentration: (N) 1 [1 ~( N)] N —,

P~7~ Therefore, p~7)is a failure threshold in the initial model. It should be noted that the corresponding critical volume concentration is >

fact about of a funcx E (0, 1), = x Cx”~ x>0

—

Pcr

{

—

—

—

Cr

[

=

1

—

exp

=

1

—

exp(

—

—

.1

1

Np~7 [1 +

0(1)1

}

C1N1_1~~’(P”’_1)

39

MULTISCALE MODELS OF FAILURE AND PERCOLATION

As in papers of Molchanov et al. (1986) and Golosov et al. (1987), the critical volume concentration of the defective set tends to unity at the percolation threshold. (2) Here we cannot calculate the functions 1(p) = f,,( p), F(p) exactly (from exact formulae a solution of the percolation problem would follow). In this case, general physical considerations (scaling hypothesis) may be used, and the results are obtained by computer. According to Shklovsky and Efros (1979), constants p, y and C exist, depending on the dimension and failure criterion, such that 1(p) exp C,,( Po p)~where —

—

x

=

+

< 00

—

3. Multiscale percolation models The percolation theory deals with the topological properties of random sets (first of all, the connectivity) in the theory of multiphase systems. The constructions of section 1 give us a typical random two-phase system: a black set (the failing cubes of different scales) andcritical a complementary white set. The question of the characteristics of the black set near the percolation threshold

~ > ‘

X~ X

This means that, for d = 3, p~ ~ for large v and N = 0[exp(C,,)J. This result means that three-dimensional failure is connected with a two-dimensional failure surface and this is much less probable than one-dimensional percolation.

It is easy to see that p~= 1

r0, where r0 is a percolation threshold in the d-dimensional siteproblem; for d = 2, Po 0.41, and for d = 3, Po 0.69. The function F(p) has almost a piecewiselinear graph. The problem of iteration of F( p) is not difficult; the answer depends on the relationship between Po and ~ For d = 2, ~ Po 0.41 when v no and N is fixed; for d = 3, 0.69> 0.31 Per’ ~ Pcr 0.31, when no and N is fixed. Thus, for large v and fixed N, multiscaling is not essential, (3) This case (when p, N no in a coordinated way) can be investigated with the help of the previous formulae. Let us consider the case d = 3. It is easy to see that, for P
is interesting from the mathematical point of view. In geophysical applications the natural characteristics are the volume concentration of the black set and the co-ordinational number. How do these characteristics depend on d, v and N? In the simplest percolation models (N = 1) the contour method is used to answer this question. This method originated from statistical mechanics. We use a generalized version of the contour method (marked contours). However, it is possible to apply a direct method: an iteration approach as in section 2.

within exponential accuracy. Hence

obtained there. Let Po’ Pi’.” PN be the probabilities of defects at the corresponding scale in the model described above (in the preceding discussion we had Po = Pi = = PN = p). The defective set D is divided as usual into non-intersecting clusters D. We shall use the weakest form of connectivity: two defective cubes are called con-

—

—p

~‘

—~

—*

‘~ —

—

—

—

—

~

(2)

r

•y

1

C,, ( Po Pcr) I = Per ( exp C1v) If k << exp C1i’ then the exponent is practically unchanged, i.e. Per — exp

Per

(k)

Per

—

(k—i) <

Per

—

exp

—

c ~‘

We have the following result: if N << exp C1v

=

exp [C,,(po

7]

—

3.1. The contour method

In its simplest form, this method was put forward by Molchanov et al. (1986); however, the obvious sufficient percolation conditions were not

nected if their sides have at least one common Molchanov al., 1986) gives the estimate point. The etsingle-scale contour method Per> (see

~pcr)

then

l/( 3d 2). Indeed, the number of self-avoiding contours of length n with a fixed initial cube is (3d 1)(3d 2)”’, and the probability of such —

~(N)

~Pcr

—

(N

—

1) exp(— C1v)

~

—

—

40

S.A. MOLCHANOV ET AL.

~( p3”— 2y

a path is p~.From the convergence of series with the help of the Borel—Cantelli

estimation improves. According to the Borel— Kantelli lemma it is sufficient for non-percolation

lemma, it follows that the defect cluster containing the fixed cube has finite volume with probability of unity. In the multiscale situation the application of the contour method has one complication: contours must be marked. A direct count of the path numbers (see Molchanov et al,, 1986) gives only very rough qualitative estimates. We shall demonstrate one version of the contour method for the two-scale problem. For simplicity, we assume d=2, N=2 and Po, Pi
that the series converges: ~P(B) = ~ n

n,k ~

k l~

p~Cpt~l +

Po) ~ (1 i =

1

k

pkpli+

~ 11

~ ~1=p

~ [P (4v + 4)] k k

F Li

[

~ i

(7p1~(1 —

k

—

Po)

7p1(1 —po)

1/p 1/v2

]

.1

[Po (4v + 4)(7p1 ) “(1 Po )i/v 7Pj (i Po) = k 1 1/v2 Thus the sufficient condition for non-percolation has the form —

L

—

—

p0(4v+4)(7p1)”(i 1”’po) <1 1 7p~(1 Po) or 7 [~o~ (4v + 4)] (1 Po )1/V2 —

~

<

—

~

~

—

1/v

In a less exact but simpler form, this inequality can be rewritten as follows:

+tk7/1712

7p

tk

2)(4v + 4)” 0)~~~V Here the summation is performed over all ~ Ik v. The factor (4v + 4)k appears because each zero-rank square has 4v + 4 neighboring first-rank squares, and the condition I, v, i = 1,..., k, means that between two different nonneighboring zero-rank squaresthese theretwo are zero-rank at least first-rank squares connecting squares. The reason for the appearance of factor (1 p 2 is explained above. If among zerorank 0)E1i’v squares there exist any neighbors, then our 7tk(1

=

11\ 7p1(1—p0)--~~i

k

~.[p

1/v

n

8.7’~p~(1 ~ p The second inequality follows from the fact that if there exists a path of n defects of the first rank, then it crosses more than n/v2 non-defective squares of zero-rank. The probability P(B~~) is estimated as follows. Let ~ l,,~ be the lengths of subpaths of firstrank squares between zero-rank squares. Then P(Bflk)

7/k

2)(4v + ~

=

—

through the Borel—Kantelli lemma. Let B~denote the case where there exists a path of defective squares of different ranks. We subdivide B~into subevents B~ corresponds to k zero-rank 0,B~1,...,Ba,, squares in where this path. Bflk We have 1p~ P(B P(B~~) 8.7” 0~)

. . .

1/v

0(4v+4)] 1/(32

7/1712

k

(1

2), that is, non-percolation conditions are fulfilled for each scale separately. Sufficient conditions for non-percolation will be derived =

+ ‘k

l~ v

_p

“

1”’
(8p0v)”’” < ~ 2,..., N/v” are very As thewhen exponents i/v, 2/v small p is large, the main condition for non-percolation is reduced to

—

7PN

<

1.

41

MULTISCALE MODELS OF FAILURE AND PERCOLATION

is to be noted that the volume concentration of the defective set is

as either this square is defective as a zero-rank defective square or is covered by first-rank defec-

1

tive squares. We reduce the defective cluster if we leave only defective squares of zero-rank. If the new defective cluster 2)percolates to infinity (and > Per = 0.41), then surely this happens when ~( the original cluster will percolate too. For an (N + 1)-scale model similar arguments can be used. We define functions

It

—

(1 —p0)(1

~Pi)

(~ p~~)

...

If Po = = PN = p~the sufficient condition for non-percolation takes the form 7p(8pv)v/[(v 1)2] <1 —

regardless of N. The volume concentration of the defective set in this case is equal to 1 (1 p)N~ To sum up, we obtain a stronger result than that obtained by Molchanov et al. (1986); if v is fixed and p is sufficiently small (so that 7p(8pv)v/~~1)2 <1) the volume concentration of the defective set can be made as close to one as desired only by increasing the number of scales N—p no. —

—

~(2) =PN-I +

(1 ~PN-1)P

~(3) =PN—2 +

(1

+

~(N)~

(1

N

PN_2)(P~1)

_p 1)(p(~’~’—l)y

Example Let us take

~(N+I)

2 < 1; p3 X 7 X 162<1, 2; 7p(16p) that is, =

1 1/3 <(7.256) Then

+

(1

—po)(p~”~)”

jf ~~(N+1) > 0.41, percolation (N =+ 1)-scale model. In the case whereexists Po in Pi the = = p the problem is reduced to iteration of the function

PN

F(p) —p + (1 —p)p”2 In fact, it is possible to substitute polynomial

0.082

—

1 (1 _~)N = ( So far, a value of two092)N for dimension d has been used. For arbitrary d, the sufficient condition for non-percolation takes the form

Per>

Po

—

d— I/v

2,defective where H(p) corresponds configurations of subsquares of thetoprevious H(p) for p” rank that would guarantee defectiveness of the main square. For example, it is sufficient that H( p) is the generating function of the configuration set whose connected defective part inside

(

1)pNlP] 2

3d2)p[(3d [(

2(d—1)/v

—

1)pN...2v]

[(3d

—

i)p0v] N(d—

3.2.

The recoding method (cf Section 2)

3d

...

l)/9N

<~

We shall now derive sufficient conditions for percolation for a general discrete model using some ideas from Allegre et al. (1982). We now explain our approach to the simplest situation. We consider again the two-scale model and suppose that Po = Pi = p. Then the probability ~(i) that some zero-rank square is wholly defective is given by 2 = p~ + (1 Po )P~ —

each square-side in more than half of its elementary cells of the previous rank. morethe general sufficient condition for percolationA has following form. We construct the sequence of values: = PN_ + (1 PN —~) H( PN) —

~(I)

=PN....2 +

~N+1

p

+

1 —pN_l)H(p)

(1

po)H(p~)

The sufficient condition for percolation is ~( N+ 1)> 041 It is for not non-percolation difficult to formulate sufficient tion in thethe same terms, condiwhich

42

S.A. MOLCHANOV ET AL.

are applicable in a more general situation than under the conditions formulated above, Let i? be a failure criterion or recoding rule~ which declares as non-defective all configurations which contain no connected defective paths from one side to the opposite side, and which declares as defective all other configurations. Let I?( p) be the corresponding generating function. Then the sufficient condition for non-percolation has the form ~ <0.41, where p(N+I) is defined by

—

In fact, it is possible to consider our model as generated by only one random flow of microdefects with intensity A and with random r; the density function of r is N

p(x)

=

~

k=1

A~p~(x)

The density p(x) is a multimodel function with

PN =PN_1 + (1 =PN—2 +

(1

+

(1

(N+1)

let ak and 0k be the corresponding means and standard deviations such that I ~ ak+1 I ~

P Po p(N+l)
PN—1)~’(PN)

PN—2)11[p]

_~ 0)~[p(N)]

4. Continuous models The main negative feature of discrete models is the existence of a rigid lattice carcass whose subsets form defective and non-defective clusters. However, this deficiency permits one to use cornbinatorial methods and simplify the estimation. In continuous models neither percolation nor failure combinatorial techniques are applicable.

maxima near {ak}. The idea of hierarchical structure of geophysical media suggested by Sadovsky (1979) can be expressed as follows: intensities Ak and characteristic sizes ak form a geometrical progression 2A,..., AN= pNA, v>i A1 = pA, A2 = v a 2,..., a~=agN, g<1 1 =ag, a2=ag Here a and A are constants, and p and g are scale factors. For simplicity in estimation, as a rule we substitute ak for rk in the following equations. This is justified because ak ‘az 1. Let us estimate the volume occupied by all defective spheres p and separately by spheres of k th rank Pk’ k = 1,..., N. It is easy to obtain

=

=

4.1. Model description We assume d= 3. Let A

=

—

exp

[

—

~Af

—

exp

[

—

4s-A
I

P= 1 1 1

r3p (r) d r

I-

—

expi

L

—

1

Ak
~

k

j

=

1

A 1 + A2 +

...

+ +

A N be the total intensity of N independent Poisscm point processes, Ak = (x~}, k = 1,..., N. Each point is the center of a microdefect of some rank. The defect itself is a sphere of random radius r,~. This sphere can be interpreted as a ‘zone of influence’ of a particular defect. In material mechanics, a defect is usually interpreted as a plane microcrack. According to Zhurkov’s theory, two microcracks coalesce when their ‘zones of influence’ intersect, Let the random variables rk have the probability density function pk(x), k = 1,..., N. Further

= (1 p)(exp[4~Ak
Pk

—

—

= 41TAk
Bk

43

MULTISCALE MODELS OF FAILURE AND PERCOLATION

exponential statistical moments. If Bk > Bcr(d), there appears a unique infinite cluster. The critical co-ordinational numbers Bcr( d) are obtained by numerical methods: Bcr(2) 4.1; Bcr(3) 2.8. These values give rise to the following defective densities: Pcr(2) 0.64; Pcr(3) 0.3. As discrete models show (see sections 2 and 3) both coordinational number B~, and defective clusterdensity Per in the multiscale situation increase, and Pk can be made as close to unity as desired. Technically, the study of multiscale continuous models is based on a ‘halo’ method, which permits as in sections 2 and 3 to eliminate successively minor scales and to reduce the problem to the one-dimensional situation. The procedure is as follows. We take N-rank spheres. ~ (4/3)~rAN(r,~) = BN> 2.8, there exists percolation already through the N-rank spheres. If BN < 2.8, the defective set DN is divided into connected finite components. Those which do not intersect with spheres of ranks 1,2,..., N — 1 cannot enter the resulting infinite cluster. Let us consider a sphere Sk of radius rk, k N 1, and the set of N-rank spheres that either intersect it or are connected to it by a path of intersecting N-rank spheres. They constitute a ‘halo’ of Sk. The ‘thickness’ of this halo can be estimated by direct probabilistic methods, for example, by branching process techniques. Let ~k be a sphere with the same center as Sk, which includes the halo of Sk. The set of ~k, k = 1, 2,..., N 1, has a number of scales one less than the original spheres. Estimating characteristic sizes dk of ~k and the conesponding probability density of Pk’ and repeating the reasoning with N replaced by (N 1), etc., we reduce our problem to a one-dimensional situation, where the classical conditions for percolation are valid. Applying this technique to the case where ak/(ak 1) = v>> 1 and Bk < Ber, it is possible to show that for the percolation threshold and when p no —

—

—

—

—*

Ber

~

no,

Per -~ 1

Technically, this analysis is rather complicated, and it has been carried out by Golosov et al.

(1987). From the results formulated above, an important practical conclusion can be drawn; a simple application of the conditions for percolation and non-percolation (concentration conditions) from a single-scale situation to a multi-scale situation is not correct.

References Allegre, C.J., Le Mouel, J.L. and Provost, A., 1982. Scaling rules in rock fracture and possible implications for earthquake prediction. Nature, 297: 47—50. Cheidze, T.L., 1987. Percolation Methods in the Mechanics of Geomaterials. Nauka. Moscow. Chelidze, T.L. and Kolesnikov, Yu.M., 1983. The modelling and prediction of failure process in the framework of percolation theory. Izv. Akad. Nauk SSSR, Fiz. Zemi. 5: 24—34. Dc Brein, MV., 1961. The Asymptotical Methods in Analyses. IL, Moscow, 209 pp. Golosov, A.O., Molchanov, S.A. and Reznikova, A.Ya., 1987. Continuous percolation models in failure theory. In: V.!. Keilis-Borok and AL. Levshin (Editors), Numerical Modelling and Analysis of Geophysical Processes Nauka, Moscow, pp. 66—73. Gutenberg, B. and Richter, C., 1954. Seismicity of the Earth and Associated Phenomena. Princeton University Press, Princeton, NT, 324 pp. Menshikov, M.V., Molchanov, S.A. and Sidorenko, A.F., 1986. On percolation approach in failure theory. In: R.V. Gamkreidze (Editor), Itogi Nauki i Techniki, ser. Teorija verojtanostei, Mat. Statistica, Teor. Kibernetics VINITI, Moscow, pp. 53—130. Molchanov, S.A., Pisarenko, V.F. and Reznikova, A.Ya., 1986. On percolation approach in failure theory. In V.1. KeilisBorok and A.L. Levshin (Editors), Mathematical Methods in Seismology and Geodynamics Nauka, Moscow, pp. 3—8. Sadovsky, M.A., 1979. On the natural piecewise structure of rocks. Dokl. Akad. Nauk SSSR, 247: 829—840. Sadovsky, M.A., Bolhovitinov. L.G. and Pisarenko, V.F., 1987. The Deformation of Geophysical Media and Seismic Process. Nauka, Moscow, 100 pp. Shklovsky, B.N. and Efros, A.L., 1979. Electronic Properties of alloy semiconductors. Nauka, Moscow, 416 pp. Zhurkov, S.N., 1983. Kinetic conception of the strength of solids. Vestn. Akad. Nauk SSSR, 3: 46—55. Zhurkov, SN., Kuksenko, VS. and Petrov, V.A., 1977. On the prediction of failure. Izv. Akad. Nauk SSSR, Fiz. Zemi., 6: 8-20.