Difference between lattice and continuum failure threshold in percolation

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Difference between lattice and continuum failure threshold in percolation

D. Sornette

To cite this version:

D. Sornette. Difference between lattice and continuum failure threshold in percolation. Journal de

Physique, 1987, 48 (11), pp.1843-1847. �10.1051/jphys:0198700480110184300�. �jpa-00210625�

Difference between lattice and continuum failure threshold in percolation

D. Sornette

Laboratoire de Physique ^{de la Matière} Condensée, ^{CNRS UA} 190, ^{Faculté des} Sciences, ^Parc Valrose,

06034 Nice Cedex, ^France

(Regu ^{le !4} juillet 1987, accepti ^{le 1 ?’} septembre 1987)

Résumé.2014 L’étude de Halperin, Feng ^{et Sen sur les} exposants critiques de la conductivité électrique

et de la constance élastique ^est reprise pour déterminer le comportement critique des seuils de rupture électrique ^et mécanique proche du seuil de percolation pour ^une classe de systèmes ^continus désordonnés

(modèles ^de type "gruyère"). ^Au dessus de la dimension dc

3/2 ^{pour le} problème électrique et pour toute dimension dans le cas mécanique, ^les exposants ^{sont nettement} plus grands que ^ceux correspondant

aux réseaux discrets de percolation. ^{L’effet est} plus ^fort que pour les propriétés ^de transport ^{à cause de} l’extrême fragilité des liens les plus ^faibles.

Abstract.- A scaling analysis introduced by Halperin, Feng ^{and Sen to estimate critical} exponents for the electrical conductivity and elastic constant is extended to determine the critical behaviour of electrical and mechanical failure near the percolation threshold for a class of disordered continuum systems (Swiss-cheese models). ^{Above the dimension dc}

3/2 for the electrical problem ^and at any dimension for the mechanical _case, the exponents ^are significantly larger than their counterpart ^{in the} discrete lattice percolation ^{network. The effect is more} pronouced ^{than for} transport properties ^{due to}

the extreme brittleness of the weakest bonds.

Classification

Physics ^Abstracts

05.50 - 05.70J - 62.20M

1. Introduction.

In breakdown phenomena, the weakest part ^of the system fails first and the effect of disorder is markedly ^more pronounced ^{than in} transport phenomena. The failure threshold is dominated

by ^extreme fluctuations in contrast to transport and elastic coefficients which are related typically

to the second moment of the distributions.

This crucial sensitivity ^is particularly ^obvi-

ous in the difference between lattice and contin-

uum failure thresholds in percolation. ^{In this} note, ^I illustrate this point by determining ^fail-

ure thresholds near the percolation transition of

a class "Swiss-cheese" continuum models, ^where spherical empty holes of radius a are randomly

distributed in an otherwise uniform electric or

elastic medium. The exponents governing ^the

electrical and mechanical rupture properties ^of

these percolation structures are shown to be qui-

te different from the corresponding ^{ones in the}

conventional discrete-lattice percolation ^models.

This results from the existence of _very weak bonds which control the rupture ⁱⁿ constrast to the discrete case for which the weakest link is a

lattice bond. A similar statement has recently

been presented ^for transport exponents [1]. ^The

difference between continuum and lattice _expo- nents is all the more pronounced ^for rupture ^and in particular, ⁱⁿ 2D, ^even when the conductiv-

ity exponent ^{is the same for both cases} [1], ^the

failure exponent is different and greater ^{in the} continuous lattice case (see below).

The analysis adapts the discussion of refer-

ence [1] ^{to the case} of failure. It relies on the well-known node-link-blob picture ^{of the} perco- lation backbone [2] which consists of a network of

quasi-one-dimensional string segments tying ^to- gether ^{a set of} nodes whose typical separation

is the percolation length / m (p - Pc) - v . ^Each

string consists of several _sequences of singly ^con-

nected bonds of total number L1 N (p - Pc)’"

with d

1 [3], in series with thicker regions ^or

blobs. Using ^the mapping of the Swiss-cheese

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480110184300

1844

model onto a discrete random network as dis- cussed in [1], ^{one can identify the} channel width

bt which controls the strength of the bond i. The essential ingredient ^{is to} recognize ^{that the 6}

are distributed according ^{to a} continuous proba- bility distribution p(b) ^which approaches ^{a finite}

limit p(O) ^{for 8}

0+. This allows to estimate the

typical minimum value bmin ^{of 8} along ^a string ^of Li singly connected bonds using ^{a Griffith} type of argument [1] : ^the probability ^that 8min > e

is 1- fo ^p(6) dS} L1 e-p(O)orLl. e: is therefore of order L11 ^{which yields}

This argument ^{can be casted} slightly differently :

fo ^{p(8) d8 is the} probability that 8 e. The con-

dition L1 f o ^{p(6) d8 1} writes that less than one

event 6e has occurred over L1 ^{trials. This al-}

lows to identify ^{6min as} Li 1.

2. Electrical failure.

2.1 CASE OF AN APPLIED CURRENT DENSITY.- Let us first examine the electrical failure and treat first the case in which the system ^{is sub-} mitted to a given ^current density j. Using ^the

node-link-blob picture ^of percolation, one can re-

late the current I flowing ^{into a} macro-bond of

length C ^{to the current} density j imposed ^{on the} boundary by [4-6]

where d is the _space dimension.

At this stage, ^{one must take into account} the specificity ^{of the} systems ^{which are reflected} in their rupture criterion. We can distinguish

between essentially ^three rupture ^{criteria :}

i) ^{A first} natural criterion is that rupture will occur in _any bond of the system ^{if the Ohmic} losses in that bond become larger ^{than a thresh-}

old value Pc, leading ^{to the} burning ^{out of the}

bond. This criterion seems very natural and should apply ^{to a} large ^{class of} systems.

ii) ^A second criterion is that for _any bond, rupture ^{occurs if V}

Vac, ^{where V} is the volt- age drop ^{over the two} ends of the bond. This criterion involves _processes ressembling ^dielectric

breakdowns.

iii) ^{Finally, one can pose that for any bond,}

rupture ^{occurs if I}

7c. This last criterion does not seem to encapture ^the physics of weak bonds

since it is insensitive to 8 (see below). ^{In this}

case, ^we therefore recover the discrete case result.

Let us consider first the case i). ^{The Ohmic}

loss in a single ^{bond of} conductance E is typically

E is controlled by ^the "strength" of the bond. A bond corresponding ^{to a narrow} neck of width 6

can be approximated ^{as a thin} parallelogram ^of

width 8 and length t N (a8) 1/2 ^{where a is the}

hole radius. The corresponding conductance is

(1J

Inserting (2) ^and (4) ⁱⁿ (3) yields ^{the Ohmic}

losses for the weakest singly connected bond :

According ^{to criterion} i), ^the rupture ^will develop macroscopically ^when Psc

P, yielding ^{the fail-}

ure threshold for the current density

with

For d

2, f

19/12 ^{with v}

4/3. ^{For d}

3,

f -- 2.51 with v -- 0.88 [7].

Note that the discrete case is obtained by posing

d

0 recovering f

(d - 1)v [4-6]. ^{The dif-}

ference between the continuum and the discrete

case is front - fdisc

(d - 3 /2) d/2 and is relevant for d > 3/2.

Let us now consider the second criterion ii).

The voltage drop ^{V over a} singly connected bond is

Writing ^V

Vc ^{leads to} the failure threshold (6)

with

In this case ii), the correction to the discrete ^re- sult is (d - 3/2)d which is twice as large ^{as the}

correction obtained with the first criterion i). ^For

d

2, this leads to f

11/6 ^{with v}

4/3. ^For

d

3, f = ^{3.26 with v}

^0.88 [7].

With the third criterion iii), ^{one recovers the}

lattice result f

(d -1)v whatever d.

2.2 CASE OF AN APPLIED VOLTAGE GRADIENT.- Another case of interest is when the system is submitted to a given voltage gra- dent avo. ^The voltage drop ^{over a macro-link}

is V = ç ^{avo and the} current I ^{flowing in it is} I ~ ^R V4 ^{where R =} G-1 ç-(d-2) is the total resistance of a macro-link with G = (p - p,)t ^is

the conductivity ^{of the} percolating system. ^This yields I avo ç(d-l) ^G which is similar _{to equa-} tion (2) with j replaced by avoG. ^The preceding reasoning ^{can be} reproduced ^{and we obtain the}

failure threshold avo (p - pc) ^{with a criti-}

cal exponent fv

f - t ^where f is obtained from

expression (7) for the criterion i) ^{and from equa-}

tion (9) for the criterion ii). ^{In d}

^2, i) f" N ^0.3

and ii) fv m ^{0.5 where I} have used t N 1.3 [ ]. ^In

d

3, i) ^{fv m} 0.1, ii) fv m ^{0.9 where I have used}

the continuum exponent t m tl ⁺ 1 /2 m 1.9+1/2

with the correction 1/2 ^to the lattice exponent calculated in [1].

3. Mechanical failure.

3.1 CASE OF AN APPLIED STRESS.- Similarly, ⁱⁿ

the elastic problem ^{if a stress u is} applied ^{at the} boundary, the force F supported by ^{a macro-link}

is

In addition, ^the torque ^M which is transmitted in the macro-link is [6]

At this stage, similarly ^to the electric _case, one can distinguish ^between essentially ^three rupture criteria :

i) ^A first criterion is that rupture ^{will occur} in _any bond of the system ^{if the} bending ^elastic energy E in that bond becomes larger ^{than a}

threshold value Ec.

ii) ^A second criterion is that for any bond, rupture ^{will occur if the} angle ⁰ by ^{which the}

neck is bent is larger ^{that a} threshold value 8c .

iii) Finally, ^{one can pose that for any} bond,

M

Mc (for torques) ^{leads to the rupture of}

that bond.

All these criteria ^are in agreement with the fact that near threshold angular deformations indu- ced by torques ^dominate [8]. ^{Using the expres-}

sion E N (1/2) ^{-yO2 for the bending energy of the}

bond, ^where 7 is the bond-bending ^{force con-} stant, ^the relation M a E /89 yields ^{a relation}

between the moment and the bend angle

1 is controlled by ^the "strength" of the bond.

Following [1], ^{one has}

as obtained from a bent-beam problem [9].

Let us now consider the first criterion i). ^In-

serting (12), ^which yields ^{M as a function} of 8,

in the expression ^{of the} bending energy leads to

where we have used (11) ^and (13). Equating ^E

with Ec, ^{leads to} the failure threshold

with

For d

2, ^F

47/12 ^{with v}

4/3. ^{For d}

3,

F = 4.39 with v = 0.88 [7].

Note that the discrete case is obtained by posing

d

0 which recovers F = dv [6]. The difference between the continuum and the discrete case is

Feont - Fdise

(d + 1/2)4/2.

Consider now the second criterion ii). ^In-

serting (11) ^and (13) ⁱⁿ (12) ^{yields the bend an-}

gle for the weakest singly ^{connected bond :}

The rupture ^will develop macroscopically ^when 8sc = Oc yielding ^{the stress} failure threshold (15)

with

In this case ii), the correction to the discrete re-

sult is (d ⁺ 1/2)d which is twice as large ^{as the}

correction obtained with the first criterion i). ^For

d

2, ^F

31/6 ^{with v}

4/3. ^{For d}

3,

F = 6.14 with v = 0.88 [7].

Finally, with the third criterion iii) ^{one re-}

covers the lattice result F

dv.

1846

3.2 CASE OF AN APPLIED STRAIN.- Another ca- se of interest is when the system is submitted to

a given ^{strain e.} The force exerted on a mac-

robond is therefore F = Ke C, where K is the force constant of the string ^of length e [1]. ^{K is}

related to the macroscopic ^{elastic constant Y =}

(p - Pe)T by ^{Y m K} -(d-2) [1]. ^This yields

F e e(d- 1) ^Y which is similar to expression

(10) ^{with or} replaced by ^{eY. The} preceding ^rea- soning ^{can be} reproduced ^{and we} obtain the fail-

ure threshold ^{e ;ze} (p - Pe)Fe ^{with a critical expo-}

nent Fe

^{F-T where F} is obtained from _expres- sion (16) for the criterion i) ^{and from} equation

(18) for the criterion ii). ^{In d}

2, i) ^{Fe is neg-}

ative indicating ^{that a constant} strain is unable to induce rupture ^when p - Pc and ii) ^{F, ;ze 0}

where I have used T ;ze Ti+3/2 m dv+1+312 -- 3.7+3/2 ^{taking into account} the increment 3/2

of T from Tl [14] ^{induced by the} intermittency

of the bonds strengths [1]. ^{In d}

^3, i) ^{Fe is} again negative indicating ^{that a constant strain}

is unable to induce rupture ^when p - Pe and

ii) ^{Fe m 0 to} within the _accuracy ±0. 1 on the exponent Tl ^{where I have used T r.-,} Tl + 5/2 --

dv+1 + 5/2 ;ze ^{3.6 +} 5/2 taking ^{into account the}

increment 5/2 ^{of T induced} by ^the intermittency

of the bonds strengths [1]. Therefore, ^break-

down under an applied ^{strain is not} brittle in constrast to the case of an applied ^{stress. This}

is due to the decrease of the macroscopic ^elas-

tic constant Y induced by ^the intermittency ^of

the bonds strengths which leads to a force and a

moment exerted on the weakest bonds which de-

crease more rapidly ^{than its} strength ^as p --> Pc.

4. Effect of correlations.

These results rely ^on the essential ingredient ^that

6 is distributed according ^{to a} continuous proba- bility distribution p(b) ^which approaches ^{a finite}

limit p(0) ^{for 6}

^{0+. It is} interesting ^{to dis-}

cuss briefly ^{other cases} taking ^{into account the}

existence of correlations in the position ^{of the}

holes. Let us consider first the case where p(6)

goes to ^{zero as} 6x when 6

0+ with x > 0.

The same type of Griffith argument ^{as the one} leading ^to equation (1) ^{allows to} estimate the

typical minimum value 6min ^{of 6} along ^a string

of L1 singly connected bonds : the probabil- ity ^that 6min > e reads I, - L

exp {-e3:+1 L1}. ^{This yields} fo- p(6)d8l - ^sz-,

This shows that previous ^{results must be chan-}

ged by replacing ^d by dl(x ⁺ 1) ⁱⁿ equations (7), (9) ^and (16), (18). Note that the _expo- nents are modified but the lower critical dimen- sion dc = 3/2 ^for the electrical problem ^{is not} changed. ^The previous ^{case of a} finite value for

p(0) corresponds ^{to x}

^0.

Another case is p(6) m ^{e-1/6 as 6}

^{0+ corre-}

sponding ^{to x --+ +oo. This} yields bmin ⁼ Log (Ll) -1 ^{leading to the same scaling as for the}

discrete lattice with the addition of logarithmic

corrections. These examples demonstrate the ex-

treme sensitivity of the failure threshold scaling

to the _presence of even very ^rare weak bonds.

5. Concluding ^remark.

1) The results of this note and of reference

[1] ^{for the transport exponents can be contrasted}

with the exponents ^for geometrical properties

such as v which have been confirmed numerically

to be identical for both discrete and continuum systems [13]. ^This difference stems from the fact that transport and failure properties ^{are influ-}

enced by the weakest bonds of the distribution of bonds strengths. The continuum case is obtained from the discrete case when the lattice mesh b goes to ^zero. Since the radius a of the holes are

fixed and finite, the continuous case corresponds

to the limit b/a

^{0 in} constrast to the discrete

case where bla m ^{1. We can thus trace back the}

non-universality ^{of the} transport ^and failure crit- ical properties ^to the different geometrical ^limit b/a -+ ⁰ (continuum) ^and bla m ¹ (discrete).

2) The difference between the electric and elastic behaviour already emphasized ⁱⁿ [6] ^{is also}

very clear within the three rupture criteria which have been used.

3) ^{It is} interesting ^to remark that in the dis- crete lattice _case, the three rupture criteria lead to the same scaling for the failure threshold in agreement with the results of [4-6]. ^{In contrast,}

this degeneracy ^is suppresssed by the "intermet-

tency" of the weakest bonds in the continuum

case as seen from equation (7) and(9) ^{for the}

electrical case and equation (16) ^and. (18) ^{for the}

mechanical case. The values of the exponents f

and F are dependent upon the precise ^criterion

which appplies ^{to the} system ^under study ^which

reflects its microscopic ^nature.

4) The critical exponents f ^and F given by

(7), (9) ^and (16), (18) should be observed only ^for

p very close _{to Pe} (p - Pe 0.1) , ^as suggested by

recent experimental and numerical works [4,10]

(see also the discussion of [11]). ^{This implies to}

facing the difficulty problem of finite size effects.

Note that the holes must be really distributed ^at random (and ^{not located} randomly ^{as selected}

sites of a regular ^{lattice as in} [12]) ^{in order to}

create the narrow necks which are so crucial for the distinction between continuum and discrete

.systems [15]. ^See [16] ^{for a} careful discussion of the numerous experimental pitfalls ^{that must be}

avoided the critical transport.

I am grateful ^to P. Barois and C. Vanneste for useful comments.

References

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