Weibull-like failure distribution induced by fluctuations in percolation

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Weibull-like failure distribution induced by fluctuations in percolation

D. Sornette

To cite this version:

D. Sornette. Weibull-like failure distribution induced by fluctuations in percolation. Journal de

Physique, 1988, 49 (6), pp.889-896. �10.1051/jphys:01988004906088900�. �jpa-00210776�

Short communication

Weibull-like failure distribution induced by fluctuations in percolation

D. Sornette

Laboratoire de Physique de la Matière Condensée, Faculté des Sciences, ^Parc Valrose, ^{06034 Nice} Cedex, ^France

(Re§u ^{le 29 octobre 1987, révisé le 19} février 1988, accepti ^le 24 ^Mars 1988)

Résumé.2014 On obtient l’expression de la distribution des seuils de rupture ^en percolation ^{discrète et}

continue dans le régime critique. L’existence de fluctuations des résistances des micro-liens en percola-

tion continue change la distribution d’une forme très abrupte (au ^moins exponentielle d’exponentielle)

en une distribution de type ^Weibull (exponentielle ^d’une puissance). ^De plus, ^{on obtient un effet de}

taille du seuil de rupture analogue ^{à ceux} existant dans les matériaux fragiles ^{observés en} mécanique

de la rupture.

Abstract.2014 The form of the failure probability distribution for discrete and continuum percolation ⁱⁿ

the critical region ^is obtained. The existence of bond-strength fluctuations in continuum percolation changes the distribution from a very steep ^form (at ^{least an} exponential ^{of an} exponential) ^{to a}

Weibull-like form (exponential ^{of a} powerlaw). Futhermore, ^{a size} effect ^{of the} typical ^failure strength analogous ^to that of brittle materials is derived.

Classification

Physics ^Abstracts

05.50 - 05.70 - 62.20M

1. Introduction.

Rupture phenomena ⁱⁿ disordered systems

are associated with the statistics of extremes [1].

In a macroscopic ^brittle system ^{submitted to a} given ^stress in mechanic ^{or to an} applied ^current density ⁱⁿ electricity, the weakest part ^{of the} sys- tem fails first and this often leads ^to the com-

plete ^{failure of the} system [2]. ^{In order to model}

these complex phenomena, ^recent study [3] ^has

used the percolation ^{model as a} paradigm ^for

disordered flawed media. Scaling ^laws govern-

ing the ensemble _average of the failure thresh- old in the mechanical and electrical context have been presented both for discrete lattice [3-5] ^and

continuum (Swiss-cheese) percolation [6]. ^How-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004906088900

890

ever, it is well known that averages do not tell the whole story especially ⁱⁿ quenched strongly

disordered systems ^{and that a} complete ^knowl- edge comes, for example, from the full ensem-

ble probability distribution of each variable of interest. Reference [7] calculates the failure dis- tribution in an electrical fuse percolation ^model

of breakdown in the regime of low concentration of defects (p ^--; 1-). The method which is used relies _on, 1) the estimation of the probability ^of finding ^the largest defect cluster in a finite _per- colation network and 2) the calculation of the

corresponding ^stress (voltage) enhancement at the tip ^{of this most} critical defect. The distri- bution is found to be of the form exponential ^of

an exponential [7b]. In the critical percolation regime (p ^- Pc) ^{,no results are presented since}

the identification of a largest ^most dangerous

defect is clearly ^not straightforward ^{and a dif-}

ferent method seems to be needed.

In this letter, ^{the form of} the failure prob- ability distribution for discrete and continuum

percolation ^{in the critical} region [8] ^{is derived}

using the node-link-blob picture of the criti- cal percolation backbone. The existence of in-

termittency in continuum percolation changes

the distribution from having ^a very steep ^form

(at ^{least an} exponential ^{of an} exponential) ^to

a smoother Weibull-like form (exponential ^{of a} powerlaw). Furthermore, ^the dependence ^{of the}

failure strength ^with respect ^to systems ^{size is} obtained. A typical feature of brittle strong ^ma- terials known as the size effect ^{where the failure} strength ^{decreases as the} system size increases is thus recovered.

An example of this size effect ^is given by

a long glass rod of unit cross-section subjected

to a tension. Experimentally, ^{the ensemble av-} eraged failure threshold X expressed ^{in unit of}

force _per unit sectional ^area is found to decrease with the total volume V of the rod according ^to

the powerlaw

where 3 p 20 typically for brittle materi- als. This stems from the Weibull distribution of strength ^obtained experimentally ^{under the}

form [9]

where xo is a scale parameter.

The well-known node-link-blob picture ^of

the percolation ^backbone [9, 11] consists in

a network of quasi-one-dimensional string seg- ments tying together ^{a set of} nodes whose

typical separation ^{is the} percolation length

£ m (p - Pc)-V . ^{Each string} consists of several sequences of singly connected bonds of total number L1~ (p - Pc)-d ^{with d=1} [11], ^{in series}

with thicker regions ^{or blobs. A} distribution of rupture thresholds may appear ^as the result of the existence of fluctuations 1) of the mesh size L of the macro-network around the typical size e

and 2) ^of the individual strength 8 ^{of the} singly

connected micro-bonds along ^a macro-link.

The mesh size or macro-node separation ^L

is distributed according ^{to a} probability p(L)

whose form is not rigorously ^{known at} present but can be inferred from the following argu- ment. From reference [12], ^{it is proven that}

there exists an essential singularity ^{for the} prob- ability ^P (n, p > pc ) m exp {-bn(d-1)/d} that ^a

given ^site belongs ^{to a} finite cluster of n sites,

where b is a constant dependent ^on p. Since at p>pc, finite clusters ^must lie inside the mesh of the node-tink-btob percolation macro-structure, this implies ^a lower bound (which ^{we assume to}

be exact) for the distribution of mesh sizes. In-

deed, the finite clusters have an isotropic shape

on the _average and a fractal structure such that their typical ^{radius L} is related to the number of their sites by ^{n~LD with} D=d - (3 Iv ^{as ob-}

tained from hyperscaling [13] (,Q ^is the critical exponent ^{of the order} parameter). This leads to

For d=2, v=413 ^and !3==5/36 ^{which yields} D(d-l)/d=91/96 ^{leading to an extended expo-}

nential for p (L, p > pc). ^For d=3, v ^{~0.98 and} (3 ^{~0.46 which} yields D(d-1)ld~1.7. ^{This law} (3) ^should be valid in the regime L> > C. ^The

constant c is an unknown function of _p.

In discrete lattice percolation, the individ- ual strength of the micro-bonds are all equal

and no fluctuations occur. In continuum _per-

colation, using ^the mapping of the Swiss-cheese

[14] ^or of the blue-cheese [15] ^model (see ^{also the}

brief discussion of paragraphe 4) ^{onto a discrete}

random network, ^{one can} identify ^the strength

of ^a bond i as controlled by the channel widths

8i (see below). ^{The 6’s are} fluctuating quantities

which ^{a rP} distributed according ^{to a continuous} probability distribution p(8) approaching ^{a fi-}

nite limit p(O) ^{for 8 -+ 0+} [14]. ^{For small 6, I}

therefore assume

Equation (4) ^{shows the extreme} importance ^of

the tail of the distribution p(8) ^{which does not}

vanish as 8 ^- 0+ [14]. These fluctuations lead to profound changes in the critical behaviour of transport and failure. In the following, ^we

consider that the only relevant fluctuations in the characteristics of the bonds are in the neck size. Fluctuations in bond length 1(6) -- 8 1/2 ^ex-

ist and are taken into account implicitely ^{via the} dependence l( 8) ^{which enters the micro-bond}

conductance. Fluctuations in singly-connected

bond separation exist also but do not contribute

to the failure fluctuations if the failure criterion involves only ^a local mechanism at the scale of each bond.

For the sake of simplicity ^and illustration,

I will consider the models of rupture ^studied

in [6b] ^{according to which} electrical (mechani- cal) ^{rupture is said to occur in any bond if} i)

the Ohmic loss (elastic energy) in this bond be-

comes larger ^{than a threshold value} Oc, ^or ii)

if the voltage drop (strain) ^{v across this bond}

becomes larger ^{than a threshold vc. These two}

criteria lead to equivalent ^{results in the case of a} delta-singular channel-width distribution. ^{For a}

general distribution p(8), ^{the correct expression}

of the failure criterion should embody, ^{at the}

scale of ^a single homogeneous bond, ^the precise physical ^{cause for the failure to occur. A gen-}

eral discussion would lead ^{us too} far in view of the large number of different physical situations.

As an illustration, ^{we can however consider the}

case when electric failure occurs in a bond of width 8, length 1 ^and conductance g(8) ^{when its}

temperature ^{T is raised above a} given ^threshold Tc ^which may be the melting temperature ^{of the} material constituting the bond. Under ^a current

I, ^{the ohmic loss} dissipated ^{in the bond is} g-1 I2

which, ^{at thermal} equilibrium, ^is equal ^{to the}

amount of _energy diffusing away from the bond

across its surface ~ 8d- 21 . Assuming ^{that the}

bond is in contact with ^a thermal bath whose

temperature ^is To, ^the equilibrium temperature of the bond is

The rupture ^{criterion T =} Tc yields

Irupture~(8d-21g)1/2_ ^{8d-3/2 with} gm 8d-3/2

and l~ 81/2 [14]. ^{This is} equivalent ^{to crite-}

rion ii) (see paragraphe 3.1). ^{Other cases will}

amount to criterion i) ^{or to} ohter criterions.

However, ^{each case can} be addressed specifically

with the tools presented ^{in this letter.}

I present the electrical failure of discrete lat- tices in paragraphe 2, of the Swiss-cheese model in paragraphe 3, briefly discuss the case of me-

chanical failure and conclude in paragraphe ^4.

2. Discrete lattice percolation.

Due to the geometric fluctuations in the mesh sizes L of the macro-structure of the _per- colation system, fluctuations of the current in-

tensity flowing in macro-links _appear. When the system is submitted to a current den-

sity j, strictly ^above pc, numerical simulations

[16, 17] ^{suggest that the} probability distribution of bonds currents for the square-lattice ^random

resistor network has ^an exponential ^tail

where

Expression (5) should be valid in the extreme

region I> 1* (p) ^{since this current scale corre-}

sponds ^{to a} size scale L> C ^{at which the} perco-

lating ^{network can} be considered homogeneous

on the _average. In a system ^{of linear size} .c> > ç

and volume V~Ld, ^I make the reasonable hy- pothesis that the tail of the current distribution

essentially samples ^the population ^of (.c / ç)d ^£1

singly connected bonds in the system. Using

the results of the appendix (Eq. (A2)) ^applied

to the random variable I and with equation (5),

the probability that Ibe less than 7m ^{in a} system of size Vm £/ , ^reads

where c is ^a numerical factor. Equation (7) ^iden-

tifies the probability distribution of the maxi-

mum value of the current found in a microscopic singly ^{connected bond.} In discrete lattice _per-

colation, both failure criteria i) ^and ii) ^{yield the}

same result and amount to say that rupture ^oc-

curs when the current in a micro-bond becomes

892

larger ^{than a threshold 7c.} Identifying ^{Im with I,}

and with equation (6), ^we obtain the following

form for the distribution of failure thresholds

Another type ^of argument may be put ^for- ward for deriving P(j) ^{directly from the geo-}

metrical distribution of mesh sizes L discussed in paragraphe ^{1. The} relationship ^{between the}

current flowing ^{into a} macro-link in a large ^mesh

and the size L of the mesh is difficult to obtain due to the subtle balance between two compet- ing effects : in the _wrong picture ^{of a constant}

current density, ^the macro-link in the largest

mesh must carry the largest ^{current. But the}

macro-link in the largest mesh is also the longest

link and this increases its resistance diminishing

the current flowing through ^{it. The} distribution of the currents in the percolation ^{structure is}

therefore ^a subtle problem involving delocalized interactions. However, ^{let us make the} guess that one can still relate the current through ^a

macro-link of size L via a relation of the form

where u=d -1 for Lm £ ^{and can} be different for

L» ç ^{due to the} scaling of the macro-link re-

sistance and the interactions between different macro-links. For a given L, ^{failure occurs at} j ⁼ jr ^{such that} IL ⁼ Ie yielding

Now, using equation (3) ^{for the} distribution of the mesh sizes L and the appendix (Eq. (A2)),

we obtain the probability distribution of the maximum value of L, ^{in a} system of linear size t i.e. out of roughly (tjç)d ^{trials, under the}

form :

Inverting equation (10) ^for j, (L ⁼ Lm) ^yields

the probability distribution of the current den-

sity ^{failure threshold}

with a ⁼ D(d - 1)/ud. Expression (12) ^has

the same exponential ^{of an} exponential depen-

dence as equation (8) ^{with however an expo-}

nent D(d - 1)/ud apparently ^{different from 1.}

This _may come from the fact that the _expo- nential form of the tail (5) ^{is not strictly ex-}

act and that the subtle essential singularity

observed in the distribution of finite cluster

size[12] ^{lets a hallmark on the current distribu-}

tion. It is indeed quite possible ^{that the numer-}

ical simulations presented ⁱⁿ [16,17] ^{have not}

been able to distinguish ^{between a} true expo- nential and an extended exponential of the form exp {-(I/I*(P))’V} ^{with v 1.}

Note that P( j) ^{has a} typical step-like shape going ^{from 1 to 0 rather} abruptly as j ^increases

above j > defined below. From equation (12),

one obtains the typical and ensemble _average failure thresholds which should scale as

I have no convincing arguments ^that 0=(d -1)v which would recover the scaling ^{of refer-}

ences [3 - 5] . The difference comes from the

large weight attributed to extreme fluctuations

leading ^to equation (12) ^{which was} neglected

in references [3 - 5] . ^In addition, j ^decreases slowly ^{as the size of the} system increases. This could be _very costly ^{but not} impossible ^to verify numerically.

Let us now summarize the main point ⁱⁿ

this discussion. The precise ^{form of the cur-}

rent failure threshold in discrete percolation ^is

very difficult to sort out but should be of the

abrupt ^{form of an} exponential ^{of an} exponen- tial of a powerlaw (this ^{result seems} robust with respect ^{to the} analysis !). ^{the exact solution}

of this problems remains however a stimulating challenge ^for future works. Note that the _expres- sions (7) ^or (12) ^are reminiscent of the result ob- tained in [7] , ^{in the p - 1-} limit. The physical ingredients ^{are however} quite different since the p - 1- limit, ^{it is} possible ^to identify ^a largest defect ^cluster of ^{size s} and, ^{from the current en-}

hancement at the tip, ^to derive the probability

of failure. In the critical region [8] , the notion of

a largest defect cluster has no meaning ^{and one}

has to rely ^{either on the} node-link-blob picture

of the percolation ^{structure or on the} knowledge

of the tail of the current distribution.

Above the percolation ^threshold pc, ^one

must also consider the influence of the bottle-

neck width fluctuations along ^a macro-link. In-

deed, ^reference [18] ^presents numerical results and a bubble model showing ^{that the} probabil- ity that the width of the narrowest bottleneck in the system ^be equal ^to unity ^{takes an} exponen- tial of ^an exponential ^{form as a function of} p.

Above _Pc, the fluctuations of the width of the bottlenecks therefore control the failure distri- bution. It is interesting ^{to note} that the fluctu- ations of the bottleneck widths along ^{a macro-}

link above Pc play ^{the role of the} fluctuations of the singly-connected micro-bonds in the critical

region p --+ Pc of the continuum percolation

models (discussed ^{in the following} section).

Note that in case when ç ^becomes larger

than the linear system ^size ,C, ⁱⁿ general only

a single macro-link connects the borders. The number of singly-connected bonds is then L1 N

.c1/v ^as _{proved by Conigl:o} [11] . ^{In a discrete}

lattice, all identical singly-connected ^micro-

bonds _carry the same total current j .cd-1. ^Fail-

ure occurs with no more fluctuations (the ^failure

distribution is a Dirac function) ^at the threshold

jr ~ L- (d- 1)

3. Swiss-cheese continuum percolation.

3.1 WEAKEST BOND FAILURE DISTRIBUTION.-

In Swiss-cheese continuum percolation [14] , spherical empty ^{holes of radius a are} randomly

distributed with concentration N in an otherwise uniform medium. Local hole density ^fluctua-

tions determine bond fluctuations which change

the failure thresholds. From [6b] , ^{the Ohmic}

loss in a single ^{bond of} width 8 reads O ~ E-112 with £ m bd-3/2 and I N jçd-l by neglecting

the fluctuations of the mesh size L [19] . ^This

leads to a failure current density ^{for a bond of}

width 8 which scales as

a ⁼ (d- 3/2)/2 ^{for criterion} i) ^{and a =} (d- 3/2)

for criterion ii) [6b] . Failure will occur on the smallest 8 in the system. Using ^{the result of}

the appendix applied ^to the random variable 6

(Eq. (A3)) ^{and with equation} (4), ^the probabil- ity, ^{that 8 be} larger ^{than 81n in a set} of R trials reads

where c" is a numerical factor. For the com-

putation ^of transport properties, ^{one is inter-}

ested in the fluctuations of 8 over the L1 singly-

connected micro-bonds of ^a macro-link, ^each

macro-link being treating separately. ^{This im-} plies the choice R ⁼ L1 (L) wich is the number of

singly-connected micro-bonds on a macro-link of

length ^L. Inserting R ⁼ L1(L) ⁱⁿ equation (15)

recovers the result already ^{used in} [14] .

For rupture phenomena, ^{one is} interested in the weakest micro-bond under stress in the total system ^{of size V =} ,Cd. This leads to the choice R m £1 (,C / ç)d-1 . Inverting (14) ^{and replac-}

ing ⁱⁿ (15) yields ^{therefore the} failure threshold

probability distribution of the weakest bond in the total system

The typical and ensemble _average failure Thresholds scales as

this leads to a scaling form of the first bond

rupture threshold : j >~ (p - pc)E.e ^with

F ⁼ (d-l)v(l-a)/a+l, ^{which is very different}

from the result of [6b] : ^{with a =} (d - 3/2) /2 ^for

criterion i), ^{one has} F ^{= 3v + 1 = 5 in 2d and} F ⁼ 2vl3 ^{+ 1 ~ 1.6 in 3d. With a =} (d - 3/2)

for criterion ii), F ^{= v + 1 = 2.3 in 2d and} F = - 2v /3 + ^{1 ~ 0.6 in 3d. It is} important ^to

realize that these scaling ^{laws are valid as} long

as the system ^size remains much larger ^than

the percolation length C. ^{In this} regime .c> > ç,

the current density ^threshold given by (17) ^is

smaller than that obtained in [6b] by ^{the factor}

(d- 1)

Note that in the ^case when C ^becomes larger

than the linear system ^size L, ⁱⁿ general only

a single macro-link connects the borders. The number of singly-connected ^{bonds is then} L1 ~

L’I’ ^as proved by Coniglio [11] . ^{In the contin-}

uous system, ^all singly-connected micro-bonds carry the same total current j£d-1 but, ^{in con-}

trast to the discrete lattice _case, they ^{are not}

identical. Failure occurs with fluctuations con-

trolled completely by ^the fluctuations of micro- bonds widths b. With R m £1 ~ L’lv this leads

to a failure distribution

and a typical failure threshold

894

This regime ç> ^L exemplifies the role of fluctu- ations present only in the continuous model.

3.2 MACROSCOPIC FAILURE DISTRIBUTION.-

Will the first micro-bond failure lead to a macro-

scopic ^{failure ? In} [16] , ^{it is argued} that the first

microscopic failure threshold could in general ^be

distinct from the macroscopic ^{failure for random}

fuse networks _away from _p ⁼ _Pc (see ^{also the}

argument ⁱⁿ [15] discussing the existence or ab-

sence of a cascade of the failure). ^In the critical

regime, ^{one can} argue that in the limit of large

mesh sizes ç, ^{links are} approximately indepen- dent, ^and macroscopic ^{failure occurs when each}

macro-link has suffered failure. This hypothesis

avoids the difficult problem ^{of the} determination of the load redistribution after successive bond failure : it can be viewed as giving an’ upper bound for the failure threshold and therefore a

lower bound for the failure exponent.

The corresponding ^failure probability ^distribu-

tion is therefore obtained from equation (15)

with R m L1 (L). ^{For L N ç, L1} (L) N çI/v ^{is the}

average number of singly ^connected micro-bonds in a macro-link of size ç [11] . ^{For L} distinct from

ç, ^{I assume the} scaling ^ansatz L1 (L)~ ^{L1l " to}

be valid. If one believe in equation (11), ^{L is} typically ^{of order}

with the exponent y depending ^{on the} precise

form of te distribution (11). Inverting equation

(14) ^{with £ replaced by} Ltyp given by equation

(18) ^{and replacing in} (15) yields ^the macroscopic

failure threshold distribution

The typical and ensemble _average macroscopic

failure thresholds scale ^as

Expression (20) ^recovers the results of [6b] ^with

the addition of ^a system ^size dependence ^char-

acteristic of brittle materials. The slow logarith-

mic dependence (Logl)-Y{(d-1)+a/v} ^{is rather}

slow and will be difficult to verify numerically

or experimentally.

The existence of bond strength fluctuations

or intermittency ^{leads to a} different critical fail-

ure exponent ^as already ^{stressed in} [6b] ^{but also}

to a much broader Weibull-like failure distribu- tion given by equations (16) ^and (19). ^{It is inter-}

esting ^{to note} the difference between equations

(8) ^and (12) for discrete lattice and equations

(16) ^and (19) ^{for continuum} percolation. ^Due

to the intermittency in continuum percolation,

the failure distribution takes ^a form _very similar

to the Weibull distribution (2) (exponential ^{of a} powerlaw) ^with p=lla. (Note ^{that the} strength

distribution is by definition one minus the fail-

ure distribution P(j)). ^{I obtain} p=4/3 ^{in 3d and}

p=4 in 2d for criterion i), ^and p=2/3 ^{in 3d and}

p=2 in 2d for criterion ii).

Note that the shape ^{of the} distribution P(j) depends ^on the value taken by ^{a. For} a> 1,

P( j) ^{has a} typical step-like shape ^{similar to but}

smoother than the discrete lattice case and _goes from 1 to 0 rather abruptly as j increases above

j >. This occurs for d ⁼ 2 with both fail-

ure criteria and for d ⁼ 3 with criterion i). ^For

a > 1, the inflexion point disappears ^and P( j)

takes the form of an extended exponential. ^This

occurs for d = 3 with the second criterion ii).

3.3 MECHANICAL FAILURE DISTRIBUTION.-

mechanical failure is complicated by ^{the tenso-}

rial nature of the stress and strain. The typ- ical form of the failure criterion is generally ^a complex function of the normal stress, ^{the shear}

stress and of the flexural and torsional moments.

However, simplifying hypotheses ^are possible ⁱⁿ

the limit p --+ p,, for which it has been _recog- nized that flexural moments dominate the elas- tic [20] ^and rupture [5] behaviour. In this limit,

the analysis for the mechanical failure follows a

similar line of reasoning ^{to that} exposed ^{for elet-}

rical failure. The results obtained for the electric

case can be readily transposed ^to the mechanical

case provided ^the following changes of variables

are made (see [6b] ) :

current density j ^# stress a,

current I iLd-1 => torque M -- aL d

Ohmic loss 0 => bending elasticity,

single ^bond conductivity E - Sd-3/2 ^z bond-bending ^force constant q m Sd+1/2,

failure criterion i) => bending ^elastic energy stored in a bond larger ^{than a} given threshold,

failure criterion ii) ^{# angle by which the}

neck is bent larger ^{than a} threshold, ^{a =} (d - 3/2)/2 for criterion i) ^{and a =} (d - 3/2) ^for

criterion ii) ^» Q!=(c!+l/2)/2 for criterion i) ^and

a ⁼ (d + 1/2) for criterion ii).

The discussion of the correct failure criterion re-

mains a difficult point ^{and I have} chosen the two

possibilities i) ^and ii) ^for illustration and simpli-

fication.

4. Conclusion.

In a discrete finite lattice, the distribution of failure thresholds is created by the intrinsic flu- cuations in the mesh sizes L around the typical

value equal ^{to the} percolation correlation length C. This leads to a very steep failure distribution

(at ^least exponential ^{of an} exponential).

On the other hand, ^a continuous distri- butions of holes in the continuous percolation

model creates another source of fluctuations : the width and therefore strength ^{of the micro-}

bonds in the percolating structure. In contrast to the discrete-lattice _case, the weakest bond fluctuations dominate over the intrinsic mesh size fluctuations and controls the failure thresh- old distribution. Therefore, ^not only ^{are the fail-}

ure threshold critical exponents changed ^{in con-}

tinuum percolation [14] ^{but the ensemble dis-}

tribution also transforms from an exponential

of an exponential ^{to an} exponential ^{of a} pow- erlaw (Weibull-like distribution). The weakest bond failure threshold is also controlled by ^scal- ing laws different from that of the macroscopic

failure threshold. Since empirical distributions also take a Weibull form, ^this suggests ^{that con-} tinuous statistical models of heterogeneous sys- tems are more relevant than discrete models.

These ideas also apply ^{to a different contin-}

uous percolation model, the blue-cheese model,

where empty needle-like holes in two dimensions

or plate-like ^{holes in} three dimensions with ^a small or vanishing ^{thickness are} randomly ^dis-

tributed in an otherwise uniform medium. The ensemble _average transport and failure _proper- ties and failure threshold distributions are pre- sented in [15] . ^{The results are very similar to}

those presented ^{in the} present ^{letter on the im-} portance ^of bond-strength fluctuations and the

corresponding appearance of ^a Weibull distribu- tion.

The result of this _paper can be generalized :

the appearance of a Weibull failure distribution

can be tracked back to the existence of a power- law distribution of micro-bonds strengths (elec-

trical conductance or mechanical elastic con-

stant). ^{In the case} of continuum percolation, ^the powerlaws ^{are created} by the flat bond-width distribution in _presence of the powerlaw depen-

dence of the stresses applied ^to micro-bonds with respect ^to the micro-bond strengths.

Aknowledgements.

I am grateful ^{to S. Roux and B.} Souillard for

stimulating discussions, the referees for perti-

nent criticism and S. Roux for a critical reading

of the manuscript.

Appendix.

Consider ^a random variable x taking ^{values in}

the interval [0, +oo] ^and governed by ^{a normal-}

ized probability distribution p(x). ^{Let us note}

PN (x xm) ^the probability that, ^{in the set of}

N trials of the random variable, ^{no values of x}

are found larger ^than xm. PN (x xm) obeys

the following ^{recursion relation :}

since 1- 1:,,:00 p(z)dz) ^{is the} probability

that x be less than _xm at the N + 1 trials. For

large N, ^the solution ^of equation (Al) ^is

This result has already ^been presented ⁱⁿ [7] ^us- ing ^{a different} scaling argument. ^The transpo- sition of this result to the determination of the

probability that, ^{in the set of N trials of the ran-}

dom variable, ^{no values of x are} found smaller

than xI" is straightforward ^{and reads}

896

References

[1] GUMPEL, E.J., Statistics of extremes (New York ; ^Columbia University Press) ^1958.

[2] ^de ARCANGELIS, L., REDNER, ^{S. and} HERMANN, H., ^J. Phys. Lett. France 46

(1985) ^L585.

[3] DUXBURY, P.M., BEALE, ^{P.D. and} LEATH, P.L., Phys. Rev. Lett. 57 (1986) ^1052.

[4] GILABERT, A., VANNESTE, C., SORNETTE,

D. and GUYON, E., ^J. Phys. ^{France 48}

(1987) ^763.

[5] GUYON, E., ROUX, ^{S. and} BERGMAN, D.J.,

J. Phys. ^{France 48} (1987) ^903.

[6a] LOBB, C.J., HUI, ^{P.M. and} STROUD, D., Phys. ^{Rev. B 36} (1987) ^1956.

[6b] SORNETTE, D., ^J. Phys. ^{France 48} (1987)

1843.

[7a] DUXBURY, P.M., LEATH, ^{P.L. and} BEALE, P.D., Phys. Rev. B 36 (1987) ^367.

[7b] DUXBURY, ^{P.M. and} LEATH, P.M., ^J. Phys.

A 20 (1987) ^L411.

[8] The critical region ^is usually defined, ⁱⁿ

critical phenomena, ^via Ginzburg ^criterion

which amounts to estimate the corrections to the susceptibility ^induced by the fluctu- ations compared ^to the corrections induced

by ^the leading ^{nonlinear term. In} percola- tion, various methods can be used [ADLER,

J., MOSHE, ^{M. and} PRIVMAN, V., ^Correc-

tions to scaling ^for percolation ^{in Percola-}

tion structures and processes, Ann. Israel

Phys. ^{Soc. 5} (1983) ^{Eds. G.} Deutscher, ^R.

Zallen and J. Adler] ^{and lead to} the follow-

ing ^universal expression for various scaling quantities

~ ~ (p - pc)-03B3 [1 + a~ (p - pc)0394 + ...]

where 0394 is positive (~1.25±0.15 ^{in two di-} mensions). ^{The critical region is then de-}

fined as the (p 2014 pc)-interval ^{such that the}

leading correction a~(p - pc)0394 ^{is smaller}

than one.

[9] ^{SKAL, A. and} SHKLOVSKI, B., ^Sov. Phys.

Semicond. 8 (1976) 1029 ;

de GENNES, P.G., ^J. Phys. Lett. France 37

(1976) ^{L-1 ; STANLEY, H.E., J. Phys. A 10} (1977) ^L211.

[10] ^{HARLOW, D.G. and} PHOENIX, S.L., ^Adv.

Appl. ^{Prob. 14} (1982) ^68-94.

[11] CONIGLIO, A., Phys. ^{Rev. Lett. 46} (1981)

250 ;

PIKE, ^{R. and} STANLEY, H.E., ^J. Phys. A

14 (1981) ^L169.

[12] KUNZ, ^{H. and} SOUILLARD, B., ^{J. Stat.}

Phys. ¹⁹ (1978) ^77.

[13] STAUFFER, D., in "on growth ^and forms",

Eds. H.E. Stanley ^{and N.} Ostrowsky (Mar-

tinus Nijhoff publishers) 1986, ^{and refer-}

ences therein.

[14] HALPERIN, B.I., FENG, ^{S. and} SEN, P.N., Phys. Rev. Lett. 54 (1985) ^2391.

[15] SORNETTE, D., Transport ^{and failure ex-} ponents ^{in continuum crack} percolation, ^to

appear.

[16] LI, ^{Y.S. and} DUXBURY, P.M., Phys. ^Rev.

36 (octobre 1987).

[17] VANNESTE, C., GILABERT, ^{A. and} SORNETTE, D., ⁱⁿ preparation.

[18] KAHNG, B., BATROUNI, ^{G.G. and} REDNER, S., ^J. Phys. A ²⁰ (1987) ^L827.

[19] The failure threshold is the extreme of a

function of two fluctuating ^{variables L and}

03B4. From the expressions (11) ^and (15), ^{it is}

easy ^{to see} that the probability distribution of extremes for 03B4 is much broader than that for L. Therefore, ^the probability ^distribu-

tion of the failure thresholds is controlled

by ^the largest fluctuation field ^03B4.

[20] ^{see for} example GUYON, ^{E. and} ROUX, S.,

The physics ^{of random} matter, ⁱⁿ "chance and matter", ^Les Houches, ^Session XLVI,

Eds. J. Souletie, ^J. Vannimenus and R.

Stora (Elsevier ^Science Publisher) 1987,

and references therein.