Failure criterion of a solid with a hierarchy of defects of thesame physical type

A.K. Stepanov 1

Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow distr., Russian Federation

Abstract

A multi-scale percolation model of solids failure under loading has been proposed. It is associated with an in®nite

cluster of microdefects of the same physical type and takes into account a random distribution and size of defects in a

space volume. A percolation criterion is given in terms of defect size distribution and defect scale correlation. Analyzed

is the validity of scaling defects in solids undergoing failure. Ó 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction

Percolation theory is applied to study the failureof solids. Loading of a solid gives rise to micro-cracks of various forms, sizes and orientationswhich appear in its volume. When two micro-cracks are close to each other, they interact toform a joint one. Closeness of microcracks meansthat there are some domains in a space volumewhich surround each microcrack such that inter-section of domains implies a merging of micro-cracks. The size of the domain is proportional tothe size of microcracks [1]. These domains are re-ferred to as microdefects. It is obvious that in-creasing of microdefect concentration leads tofailure. It has been shown in Ref. [1] that the fail-ure of a solid could be related to a critical numberof microdefect concentration. Such an observationis in agreement with the percolation theory [2]. Inother words, failure corresponds to a cluster of in-tersecting microdefects in a space volume. Experi-mental estimates of critical concentration [1] canbe compared with those in Ref. [3] for the one-size

(or scale) microdefect model of percolation theory.The one-scale model assumes that all microdefectsare spheres of the same radius with centers that areuniformly distributed in a space volume of con-stant density. Since the size of the microdefectsmay vary, a multi-scale (or-size) model would bemore realistic. Care, however, should be taken tomodelling an in®nite cluster of microdefects whichmay not prevail even when the density of the mic-rodefects may be close to 100% [3].

The objective is to develop a percolation criteri-on for the multi-scale microdefect case that appliesto the failure of solids. A hierarchy of microdefectscale is invoked by means of correlation. The mod-el also applies to microdefects generated from im-perfections di�erent from microcracks. It is validonly if failure of the solid is caused by defects ofthe same physical type such that their interactionscould be described by microdefect intersections.

2. Model description

Consider a hierarchical structure of the space.Decompose the space by cubes Q1 of the ®rst rank

Theoretical and Applied Fracture Mechanics 29 (1998) 219±222

1 E-mail: mileiko@issp.ac.ru.

0167-8442/98/$19.00 Ó 1998 Elsevier Science Ltd. All rights reserved.

PII: S 0 1 6 7 - 8 4 4 2 ( 9 8 ) 0 0 0 3 3 - 0

with the edge length s1. Unite 33 cubes of the ®rstrank having common points with Q1 to cube Q2 ofthe second rank with the edge length l2 � 3l1.Using this principle to all cubes of the ®rst rank,decomposition of the space can be made by cubesof the second rank. Prolonging a decomposition ofthe space by this procedure, the hierarchical struc-ture of the space can be developed with the kthrank cubes Qk uniting 33 cubes of �k ÿ 1�-th rankand having the edge length lk � 3kÿ1l1. Now, eachcube Qk of kth rank will be treated independentlyof the other cubes of the same rank; those withprobability qk will be painted black and those withprobability 1ÿ qk will be painted white. In termsof microdefects, it implies that in a solid volumethere are uniformly distributed microdefects ofkth scale, lk-th size and concentration 1ÿ qk. Fur-ther, each cube Qkÿ1 of �k ÿ 1�-th rank shall betreated independently of the other cubes of thesame rank; those with probability qkÿ1 and1ÿ qkÿ1 shall be painted black and white, respec-tively. Painting all cubes of all ranks to the ®rstone, there result various con®gurations of spacedecomposed by cubes of various ranks paintedblack and white. Let V be ®nite space volume. Ac-cording to a space hierarchy it may be painted indi�erent con®gurations of black and white cubesof various ranks. The total number N of all possi-ble con®gurations in V is ®nite if V is ®nite. Blackand white cubes of each con®guration in V will benumerated in some order. Hence, to each con®gu-ration in V , there results a sequence gi �gi�V �; i � 1; . . . ;N , of black and white cubes. Toeach con®guration gi, there is the correspondingprobability P �gi� being the product of probabilitiesqi and 1ÿ qi in the sequence gi. The collection ofcon®guration probabilities P�gi� over all sequencesgi gives the probability distribution PV in a ®nitespace volume V . It follows that the collection of ®-nite probabilities distribution PV over all ®nite vol-umes V gives the probability distribution in thespace according to the lawX1i�1

qi � 1: �1�

3. Percolation criterion

According to the statements of percolation the-ory [4], it can be said that failure of solids takesplace when there prevails an in®nite (white) clusterof microdefects with a probability of one.

The connectivity of a cluster means that eachtwo points of a cluster can be joined by a linewhich belongs to a cluster. Let gn be a polyhedralsurface formed by n faces of black cubes of variousranks, which is homeomorphic to a sphere of thespace with center at a ®xed point. Without lossin generality, assume that gn is a sequence of blackcubes which contains a ®xed point of the space.Each pair of neighboring cubes of gn has accordingto the connectivity rule at least one common edge.Suppose that each black cube of gn does not be-long to a black cube of greater rank than itsown. Apply the duality theorem [5] to the sequenc-es gn. If there is no in®nite sequence gn, then thereexists an in®nite sequence of white cubes that haveat least a common point. In microdefect terms, thenonexistence of the in®nite sequence gn means theexistence of solid destruction. The goal is to provethe nonexistence of the sequence gn or more pre-cisely to ®nd such conditions on probabilities qi

that result in the nonexistence of gn.

3.1. Proof

Let Pgn be the probability of the existence of thesequence gn. Then according to the de®nitions,there results

Pgn �Xfgng

qj1. . . qjn : �2�

where the summation in Eq. (2) is over all se-quences gn. Let s be some ®nite rank. Denote bygn;s such sequences gn which contain black cubeswith ranks not greater than s. Let ik �ik�gn;s�; k � 1; . . . ; s, be the number of black cubesof gn;s with the rank k. It is obvious that ik numbersadmit the inequalityXs

k�1

ik 6 n; 06 ik 6 n: �3�

Let Nik be the number of allocations in sequencegn;s of ik black cubes of kth rank and let

220 A.K. Stepanov / Theoretical and Applied Fracture Mechanics 29 (1998) 219±222

nk;1 � nk;1�gn;k�; nk;2 � nk;2�gn;k� be the numbers oftransitions in gn;k between black cubes with de-creasing and increasing of a rank accordingly.Then Eq. (2) can be rewritten as

Pgn;s �X

i1�����is�n

Ys

k�1

Nik �nk;1nk;2qk�ik : �4�

It is clear that the number Nik is not greater thanthe number of ways to choose ik black cubes fromnÿPj6 k ij; i.e.

Nik 6CiknÿP

j6 kij: �5�

Now, estimate the numbers nk;1, nk;2. According tothe de®nition, the number nk;2 is not greater thanthe number k ÿ 1 of ways to choose a di�erentrank multiplied by the number 4 of choosing aneighbor cube of a greater rank, i.e.

nk;26 4�k ÿ 1�; k P 2: �6�Further, consider the transitions in gn;k from thecube of kth rank to the cube of ith rank, i6 k.The number of the transitions is less than the num-ber �lk=li�2 multiplied by the number 6 of cube fac-es or the number �lk=lkÿ1� multiplied by thenumber 12 of cube edges. Therefore, for k P 2.

nk;16Xkÿ1

i�1

�6 � 32�kÿi� � 12 � 3kÿi�

� 6Xkÿ1

i�1

32i 1� 2

3i

� �6 10

Xkÿ1

i�1

32i

6 10 � 32�kÿ1�X1i�1

3ÿ�2iÿ2�6 5

432k: �7�

From Eqs. (4)±(7), it is found that

Pgn;s 6X

i1�����is�n

Ci1;...;is qi11

Ys

k�2

5�kÿ ÿ1� 32k qk

�ik ; �8�

where Ci1;...;is are the polynomial coe�cients. Fromthe polynomial theorem it follows that the right-hand side of Eq. (8) equals to

5n q1 �Xs

k�2

�k ÿ 1� 32k qk

!n

�9�

and thus

Pgn;s 6 5 q1 �Xs

k�2

�k ÿ 1� 32k qk

!" #n

: �10�

In the limit as s !1 in Eq. (10), the requiredprobability is obtained:

Pgn � lims!1

Pgn;s 6 5 q1 �X1k�2

�k "

ÿ1� 32k qk

!#n

: �11�

From Eq. (11), a failure criterion can be estab-lished.

3.2. Failure criterion

Consider the inequality

q1 �X1k�2

�k ÿ 1� 32k qk <1

5: �12�

A probability of one would correspond to failureof the solid. When the concentration of microde-fects 1ÿ qk satis®es the inequality as Eq. (12), thenEq. (11) leads toX1n�1

Pgn <1: �13�

The Borel±Cantelli Lemma in Eq. (13) implies thatwith a probability of one there exist only a ®nitenumber of polyhedral surfaces gn which are home-omorphic to a sphere of the space with center at a®xed point. Thus, there exists an in®nite sequenceof white cubes (in®nite defect cluster) containing a®xed point of the space. The ful®llment of Eq. (12)implies the occurrence of failure of a solid.

4. Conclusions

The following conclusions can be made:· The single-scale model may be extended to that

for multi-scale by altering a known failure crite-rion. For example, if the volume concentrationand mean number of neighboring microdefectsincrease, then a failure is caused by defects oflarger scales. But taking into account scales oftwo or three orders, a given failure criterionmay thus be corrected for the multi-scale model.

A.K. Stepanov / Theoretical and Applied Fracture Mechanics 29 (1998) 219±222 221

· It is possible to correct failure theory of solidswith impurities via probabilities 1ÿ qk speci®edin terms of real parameters such as size of a de-fect, loading, strength, concentration of impuri-ties etc. The multi-scale percolation approachpermits the use of known notions on local char-acteristics of a failure for modelling of a failurefor the whole space-volume of a solid.

Acknowledgements

The work was supported by the Russian Foun-dation for Fundamental research under grant 93-013-16742.

References

[1] S.N. Zhurkov, V.S.Kuksenko, V.A. Petrov et al., To the

question of the rock-mountains fracture prognose, in:

Proc. AS of the USSR. Earth Physics, vol. 6, 1977, pp.

118±130.

[2] B.N. Shklovskii, Electron Properties of Doped Semicon-

ductors, Nauka, Moscow, 1979, p. 418.

[3] S.A. Molchanov, V.F. Pisarenko, A.Ja. Resnikova, About

percolation in fracture theory, Mathematical methods in

seismology and geodynamics, in: Computation Seismolo-

gy, vol. 19, Nauka, Moscow, 1986, pp. 3±8.

[4] S.A. Molchanov, A.K. Stepanov, Percolation of random

®elds. I, Teor. Mat. Fiz. 55 (3) (1983) 319±340.

[5] A.K. Stepanov, Duality principle for percolation of ran-

dom ®elds on a plane, Teor. Mat. Fiz. 67 (1) (1986) 32±

39.

222 A.K. Stepanov / Theoretical and Applied Fracture Mechanics 29 (1998) 219±222