Damage Growth in Random Fuse Networks F. Reurings and M. J. Alava Helsinki University of Technology, Laboratory of Physics, P.O.Box 1100, FIN-02015 HUT, Finland (Dated: February 2, 2008) 4 The correlations among elements that break in random fuse network fracture are studied, for 0 disorder strong enough to allow for volume damage before final failure. The growth of microfrac- 0 tures is found to be uncorrelated above a lengthscale, that increases as the the final breakdown is 2 approached. Sincethefusenetworkstrengthdecreaseswithsamplesize,asymptoticallytheprocess n resembles more and more mean-field-like (“democratic fiber bundle”) fracture. This is found from a the microscopic dynamics of avalanches or microfractures, from a study of damage localization via J entropy,andfromthefinaldamageprofile. Inparticular,thelastoneisstatisticallyconstant,except 8 exactly at thefinalcrack zone (in contrast to recent resultsby Hansenet al., Phys. Rev. Lett. 90, 2 045504 (2003)), in spite of thefact that thefracture surfaces are self-affine. ] PACSnumbers: 62.20.Mk,62.20.Fe,05.40.-a,81.40Np h c e I. INTRODUCTION democratic FBM’s [22], so that one has for the probabil- m ity distribution of the number of fuses (∆) blown in one - ’event’, for current-control, t The scaling properties of fracture processes continue a t to attract interest from the statistical mechanics com- P(∆)∼∆−5/2. (1) s munity. Key quantities are the geometric properties of . at fracture surfaces and statistics of acoustic emission, or, The correspondingAE energyexponentisaboutβ =1.7 m inanalogytoothersystems,“cracklingnoise”. Thepoint [5, 23], as expected based on the exponent relation β = is that in failure of brittle materials the elastic energy of (5/2+1)/2[23]. However,evenforstrongdisorderfinally - d a sample is released in bursts. These “avalanches” often stressenhancementscomeinto play,andthe samplefails n turn out to have scale-invariant statistics with respect catastrophically, with the elastic modulus (conductivity, o toe.g. theprobabilitydistributionofthereleasedenergy in the RFN case) having a first-order drop. c [1,2,3,4,5]. Likewise,cracksurfacesareoftenself-affine The purpose of this article is to investigate the devel- [ (with an empirical roughness exponent ζ) [6, 7, 8]. The opment of the damage, between the MF-limit valid at 1 understandingoftheoriginsofsuchcritical-likestatistics the very initial stages of fracture and the final critical v wouldperhaps be ofinterest to engineers (“how to make crack growth. Figure 1 shows an example of the transi- 2 tougher materials”) but would also mean the solution of tion. Thesubplotsdepicttheindividualfusesthatfailin 9 a very complicated many-particle system. subsequent parts of a stress-strain- (or current-voltage) 5 1 In this respect, among the simplest models that are history. Clearly, initially the damage is random (unless 0 available are mean-field like fiber bundle models (FBM) proven differently by more sophisticated analysis), and 4 [9, 10] and random fuse networks (RFN’s) [11, 12]. The in the last panel it concentrates on the vicinity and at 0 former describe democratic or global load sharing, and the final crack. / t thus do not have anything close to the stress enhance- In this respect, it is an important question how the a mentsofrealcracks(thoughonecanintroducelocalload pre-critical damage reflects the self-affine properties of m sharing to fiber bundles, and interpolate between these the final fracture surface. Recently, Hansen and co- - two limits as well). Such stress effects are to be found in workers have attempted to relate its formation to a self- d n a natural way in fuse networks, that simplify real elas- consistently developed damage profile that extends over o ticity by considering the electrostatic analogy. RFN’s all the sample [24, 25]. The scaling of the profile with c have two natural limits: weak disorder, when cracks are the system size would then explain the roughness and v: nucleated quickly and brittle failure takes place without its exponent. Clearly, this should also be visible in the i muchprecursoractivity,andstrongdisorder(withoutin- dynamics of failure also prior to the end of the process. X finitely strong elements), where damage develops before Another analogy is given by dynamics in dipolar ran- r macroscopic failure [11, 13]. dom field magnets, which can account for the symmetry a Thesamesignaturesarefoundinthelatter,RFN,case, breaking(assignaledbytheformationofthe finalcrack) that also characterizeexperimental systems: rough, self- due to shielding in the direction of the external voltage affine cracks and microcracking that corresponds to the andforstressenhancementsthatdrivecracksmostlyper- acousticemission. Theroughnessexponentisintheprox- pendicular to it [26]. imity of ζ ∼ 0.7, in 2d, tantalizingly close to the We study these aspects by concentratingon two kinds RFN minimum energy surface exponent, exactly 2/3. This re- of quantities: those that characterize the spatial distri- sult holds also for e.g. ’weak’ disorder [14, 15, 16] and is butionof damageinsamples,andthose that analyzethe closeto whatis seenin experiments [17, 18,19, 20]. The temporalcorrelationsin individual failure events (as e.g. damage develops in avalanches [5, 10, 21], in analogy to during an avalanche, or series of fuse failures due to the 2 (a) (b) about critical damage clusters in a more elaborate fiber 60 60 bundle-typemodel: thedynamicsisbasedondemocratic load sharing in spite of the presence of stress enhance- 40 40 ments [27]. It also pertains to the question of the exis- y y tence of“representativevolumeelements”[28]orcoarse- 20 20 graininginfusenetworks[29],relatedtothegeneralques- tionofhowtoaccountformicroscopicdynamicsandphe- nomenawithcoarse-grainedvariablesandequations. Be- 0 0 0 20 40 60 0 20 40 60 yond any such correlation length ξ as may exist within x x avalanchesthenetworklookshomogeneous,ifinaddition (c) (d) thedamagedensityisstatisticallyhomogeneous. Finally, 60 60 wefinishthepaperinSec. IVwithasummaryandsome open prospects. 40 40 y y 20 20 II. DISTRIBUTION OF DAMAGE 0 0 The RFN’s, as electrical analogues of (quasi-)brittle 0 20 40 60 0 20 40 60 x x media consist of fuses with a linear voltage-current re- (e) (f) lationship until a breakdown current ib. A stress-strain 60 60 test can be done by using adiabatic fracture iterations: thecurrentbalanceissolved,andateachroundthemost 40 40 strained fuse is chosen according to min(ij/jc,j), where i is the local current and j the local threshold). Cur- y y j c,j rents and voltages are solved by the conjugate-gradient 20 20 method. Inthe followingwe useforP(i )aflatdistribu- c tion P(i )=[1−R,1+R], with the disorder parameter c 0 0 0 20 40 60 0 20 40 60 R chosen as unity. The simulations are done in 2d, in x x the (10) lattice orientation,with periodic boundary con- ditions in the transverse direction (y). Square systems FIG. 1: Snapshots of subsequent damage patters in a failure upto 1002 have been studied; notice that the damage is of a RFN (each sub-plot having the same number of failed in practice volume-like,and thus thousands of iterations fuses, separately). R=1, “strong disorder”, L=60. are needed per a single system for L∼100. Studies of the break-down current I as a function b affirm the expected outcome of a logarithmic scaling increaseofacontrolparameter). SectionIIconsidersthe [12,16],resultingfromextremalstatistics(Ib ∼L/lnL). former,andusesasthemaintoolentropy,comparingthe Thisimpliesinthemean-fieldlimitthatn ∼ L2 ,where b logL damageintegratedoverwindowsoftime and/orspaceto n is the average number of broken fuses in a system. b that in, spatially, completely randomdamageformation. To analyze the spatial of distribution of damage it is We also study the damage profiles of completely failed usefulfirsttotakenoteofthefactthatinthelatterstages samples. From both kinds of analysis emerges a picture the system behaves anisotropically: just before the for- of crack development, that is mean-field -like beyond a mation of a critical crack the spatial density of the bro- finite interaction range, and until the final breakdown is kenfusesshouldbeastochasticvariable,withaconstant induced by rare event statistics [12]. This in particu- mean in the transverse direction to the external voltage. lar includes the fact that except in the “fracture process However,along the voltage direction differences may en- zone”, i.e. in the vicinity of the final crack, the damage sue. To study such trends in the damage mechanics and isstatisticallyhomogeneous. Thusinthisparticularcase the localization we consider the entropy of the damage ofRFN’sthetheoryproposedbyHansenetal. seemsun- averagedover y in each sample (this is in analogy to the likely to be the explanation for the self-affine geometry procedure used with AE experiments of Guarino et al. of cracks. [3]). The networkisdividedalongthecurrentflowdirec- In Section III, the internal dynamics of avalanches is tion into sections, and the entropy S defined as considered. We look at the probability distributions and average values of “jumps” (relative changes in the po- S =− q lnq , (2) sition of subsequent failures). It transpires that there X i i i is a smooth development, in which the these quantities exhibit a cross-over from the FBM/MF-like lack of spa- where q is the fraction of burned fuses in section i. S is i tialinhomogeneitytowardslocalizedcrackgrowthwithin normalized by S , the entropy of a random, on the aver- e a scale ξ. This resembles some observations by Curtin age homogeneous distribution of failures (of equal total 3 damage). Thus the extreme limits are zero and unity, 1 corresponding to completely localized damage and com- plete random one, respectively. The final crack extends 0.99 between y =0 and y =L, and a sensible choice is to use for the section width δx a value larger than the typical interface width w, 0.98 L) w=h(hy −h¯)2i1/2, (3) S( 0.97 where h denotes the crack location and h¯ its mean po- y sition in the x-direction. Since w ∼Lζ, with ζ <1, it is clear that using a constant number of sections will with 0.96 increasing L localize the fracture zone either entirely in- side one, or between two neighboring ones. For better 0.95 statistics it is preferable to have δx >> 1, though the 0 0.002 0.004 0.006 0.008 0.01 0.012 1/(L ln L) interpretation is perhaps more difficult than for the ex- treme value δx = 1, say. The width of the sections used FIG. 2: S vs. 1/(LlogL) in networks with strong disorder in computing the entropies was chosen to be δx=L/10. (R = 1). The straight line is a linear least squares fit that In this discrete form, the entropy reads intersects theS-axis at S =0.9943. k n n i i S =− ln , (4) X N N 1 i=1 where k = L/δx is the number of sections, n the num- i ber of burnt fuses in the i’th section, and N = k n Pi=1 i is again the total. Note that the absolute value of S is 0.9 dependent on the choice for δx. S can now be used to craotnesliyd,eorrdtihffeerfiennatlpdaarmtsagoef tphaettsetrrne.ss-strain curve, sepa- S(i) Fig. 2 shows the total entropy versus system size. The best kind of linearity with regards the data is ob- 0.8 tained with a scaling variable 1/(LlogL). This can be considered with the following Ansatz. Assume, that the fractures are distributed otherwise randomly (n ), b,i except the one containing the final crack, which has 0.7 1 2 3 4 5 n +∆n . Take ∆n ≪ hn i, which implies approxi- i (interval) b,k b,k b b mately S ≈1−lnk(∆n /n ). Noticing the logarithmic b,k b scaling of damage, it follows that S ∼ −1/(LlogL), if FIG.3: EntropyS versustimeinterval∆ni. L=100,R=1. and only if ∆n scales as ∆n ∼L/(logL)2. We have Average over20 realizations. b,k b,k notcheckedexplicitly thatthis holds; note thatthe frac- turesurface,beingself-affine,isthensupposedtocontain La fuses, with 1 < a ≪ 2, but the fracture process zone itself, contains other damage(brokenfuses) contributing in the y-direction as a function of the normalized coor- to ∆n (see Fig. 1 again, and the last panel in particu- dinate transform χ = (x−x +L)/2L, where the point b c lar). Inanycase,it is obviousthatthe entropyincreases x is chosen as the one with the maximum damage, and c with system size, indicating more and more completely is located in practice at the final fracture line, x ≃ h¯. c randomdamage. ThelimitingvalueofS isslightlybelow After this shift, the average density is computed taking unity; it is hard to say whether this difference is due to care that it is normalized correctly since the number of the choice used for computing S or a real one. samplescontributingforeachχvarieswiththefinalcrack In figure 3 an example is shown of how the damage location - x is a random variable. We also have added c actually localizes when the fracture process is divided the averagefracture line width w(L=100)as a compar- into sequential slices. It is clear that initially most of ison (from ref. [16]). It can be seen that outside of the thefusesbreakrandomly,andonlyinthelastonestrong immediate vicinity of the fractureprocess zone the dam- localization takes place. Similar sampling can be done age is constant. Notice the errorbarsof the data points, also with the external voltage or current as control pa- andthatthedatapoints locatedfarawayfromthe crack rameters,thedifferencebetweenthesetwobeingthatthe line suffer from the presence of less data points as seen fracture is more abrupt in the latter case. from the error bars. It would be interesting to analyze This lack of the localization of damage is reflected in in detail the functional shape of hρi(χ) in the proximity Fig. 4. The damage density hρi(χ) has been averaged of the crackline, χ = 0.5. The implication of the results 4 0.3 between consecutive fractures, 0.25 ∆x=|xi+1−xi| (5) ∆y =|y −y |, (6) i+1 i 0.2 where x and y are the x- and y-coordinates of the ith i i ρ〉χ()0.15 fracture. Another one choice is given by the average 〈 distances between consecutive fractures belonging to the 0.1 same avalanche(ie. induced by a single increment of the control parameter) 0.05 ∆−1 1 2w ∆x = |x −x | (7) 0 avalanche ∆−1 X i+1 i 0 0.2 0.4 0.6 0.8 1 χ i=1 ∆−1 1 FIG. 4: Averaged damage density hρi(χ). L=100, R=1. ∆yavalanche = ∆−1 X|yi+1−yi|. (8) i=1 The MF theory predicts P(∆x = k)= 2(L−k)/L2 and P(∆y =k)=2/L, if the boundary conditions used here is that the density canbe written as a sumof a constant aretakenintoaccount,and∆isagainthe avalanchesize (L-dependent)background,andatermthathastodecay measured in the number of fuses broken during it. (perhaps exponentially) within a finite lengthscale from Fig. 5 depicts, as a comparison for the mean-field re- the crack. This decay length in turn may depend on L. sults, the average distances in the x- and y-dimensions Suchanobservationisincontradictiontotheproposed between consecutive broken fuses belonging to the same “self-consistent”quadraticfunctionalform,byHansenet avalanche. h∆x i and h∆y i, are shown, al. [14]. This would imply hρi(χ) = pf −A(cid:16)L(2lχx−1)(cid:17)2, respectively, asaavafluanncchtieon of the sayvsatleamnchseize. Both are which is clearly not the case. In the light of the picture linear like in the MF theory, but with a smaller slope discussed below about the internal dynamics of microc- withL. Thismeans,thatthedamagecreatedbyatypical racks or avalanches, the interpretation is that the final avalanche(microcrackcreation,crackadvanceetc.) islo- crackisformedheresimilarlytoweakdisorderina“crit- calized compared to the MF-prediction, but nevertheless ical” manner. That is, once a damage density sufficient thelocalizationdoesnotgetstrongerwithL. Oneshould for “nucleation” is established the largest crack becomes notethat the damageassuchisalmostvolume-like. The unstable. Priortothatthecorrelationsinthedamageac- resultisthus notsurprisinginthe sensethata reduction cumulated canfor allpurposes be neglected. This would oftheslope(sublinearbehavior,say,h∆xavalanchei∼Lα, in turn to imply that the origin of the self-affine crack with α < 1) would imply concomitant faster average roughness in fuse networks is not dependent on whether crack growth, which would be in contradiction with the there is “strong” or “weak” disorder, as long as there damage scaling. are no infinitely strong fuses, or as long as the process To understand in detail the dynamics of microcracks does not resemble e.g. percolation due to the complete is a difficult task. This is since the growth dynamics domination of zero-strengthfuses. is not local: the burned fuses do not have to form con- nected clusters by any remotely easy criterion. It is easy to comprehendthat the driving force for the localization is standardstress-enhancement,but as is true for RFN’s III. AVALANCHES crack shielding and arrest (due to strong fuses, in the early stages of fracture) play a role. One may set aside Next we consider the correlations in the dynamics of for the sake of discussion the separation of the events individual fuse failures. Recall that the MF-limit states into avalanches, and just consider the distances between that consecutive ones should not be spatially correlated consecutiveburnedfuses. Figure6 demonstratesthe dif- in any fashion; the opposite limit is given by the growth ferencebetweentwoprobabilitydistributionsP(∆x),av- ofa linearcrackinwhichitis alwaysthe one adjacentto eragedoverthefirst1/8ofthetypicalfailureprocessand the crack tip to fail next. In the case that the growing the last, respectively,for a fixed L. As one could expect, crack is “rough” one expects that the subsequent failure there is a peak inthe distribution (this holds for both x- takesplaceinside afractureprocesszone,analogouslyto and y-directionsseparately),which is greatly suppressed normalfracturemechanics,oneofthefollow-upquestions in the first part closest to the MF-limit. being how the size and the shape of this zone vary with Thus one may conclude, that there is a continuous system size and as the crack grows [15]. The simplest cross-overfrompurelyMF-likebehaviortoacomplicated quantities to compute, to examine localization and spa- non-local growth dynamics. This is also exhibited by tial correlations between fractures, are the 1d distances such distributions P(∆x), P(∆y). The analysis of the 5 100 τ = 1/2 30 τ = 2/3 ∆y 10 L) ∆x y(20 MF, y ∆ MF, x L), ξ x ( ∆ 1 10 0 0 0.0 0.2 0.4 0.6 0.8 0 20 40 60 80 100 120 damage L FIG.7: Thescalingofξy(seeEq.(9))withincreasingdamage FIG.5: Averageone-dimensionaldistancesh∆x iand avalanche for L = 100. The total number of broken bonds has been h∆y i between consecutive broken fuses belonging to avalanche divided into ten consecutive windows, and in each of these the same avalanche vs. linear dimension of system L. R=1. hdyi has been computed, and ξy using Eq. (9). For τ two The solid lines are linear least squares fits with slopes 0.018 guesses (1/2, 2/3) are used, note that 0<τ <1. for h∆x i and 0.055 for h∆y i. The dashed avalanche avalanche lines, linear with slopes 0.33 for h∆x i and 0.25 for avalanche h∆y i,correspondtothedistancespredictedbymean- avalanche fieldtheory. Statisticalerrorsaresmallerthanorequaltothe correlation length), ξ or ξ . Second, one should recall size of thedata points. x y the scaling of the strength with L: catastrophic crack growthtakesplace earlierandearlierwith respectto the 0.25 intensive variable, current. This means that the RFN’s first eighth of fractures last eighth of fractures resemble, in the thermodynamic limit, more and more MF 0.2 the mean-field-case in their fracture properties in spite of the stress- (or more exactly current-) enhancements ∆P(x)0.15 that the model contains. bility Again, our data does not allow us to conclude firmly a ob 0.1 howsuchcorrelationlengthsbehavefor∆xor∆y small- pr how the associateddistributions P would scale for small 0.05 arguments that is. One may however simply use an Ansatz that P ∼ ∆y−τ upto ξ , say and MF-like for y 00 20 40 60 80 100 larger ∆y [29]. This defines the correlationlength ξy for distance between consecutive fractures in x−direction ∆x a given damage density ρ. Using now the distribution P allowstocomputeh∆yiandrelateittoξ ,validforsuch y FIG. 6: Distribution of average distances in the x-dimension deviations from the uncorrelated fracture process, that st between consecutive broken fuses computed over the 1 and ξ ≪ hL/4i. The result is in analogy to Delaplace et al. th th y 8 8 of the fractures of 20 realizations. L = 100, R = 1. [29] that The solid lines correspond to mean-field predictions. 1−τ (2−τ)L2+8τξ2 y detailed shape of the small-argument part of the proba- hdyi= . (9) 4(2−τ) (1−τ)L+2τξ y bilitydistributionswouldbe aninterestingchallenge. To first order, the result is a convolution of a microcrack size distribution and the corresponding stress enhance- In the opposite limit, ξ ∼ 1 the correlations are badly y mentfactor,suchthatthedistributionP evolvesaccord- defined since the model is discrete. Figure 7 shows an ing to the growth indicated by the involved probability example of the ensuing scaling with different guesses for distributions. Given the simple forms of say P(∆x) for τ, for L = 100. The main observation is an exponential smallargumentstheremightbesomehopefordeveloping (perhaps) increase of ξ with damage. Again, note that y analytical arguments. with still larger system sizes the total damage is dimin- When considered as a function of L it becomes imme- ished, which in turn implies that the maximal correla- diately apparent why the avalanche statistics resembles tionsinthedamageaccumulationbecomeweaker. Please theMF-casesomuch. Thisisduetotwoseparatefactors: observethatwehavenotstudiedindetailtheotherpossi- first, the growth is clearly in the sample case of Fig. 6 bility,ξ sincethemaininterestlieswiththecorrelations x (orFig. 1again)localoveracertainlengthscale(damage in the crack-growthdirection. 6 IV. SUMMARY and one often has just the propagation, and formation of the final crack. Nevertheless of this complication, the In this article we have studied the distributions and measuredroughnessexponentsfromsuchsimulationsare development of damage in random fuse networks, with close to that in the case here at hand, but note also that “strong”disorder. Our aim has been to understand pos- such exponents are notoriously hard to measure numeri- sible deviations from mean-field theory, and the associ- cally in finite sized samples: more extensive work in this atedcorrelations. Thisisofrelevancebothasregardsthe respect would certainly be desirable. statisticalmechanicsoffractureingeneral,andinpartic- The internal correlations of the avalanches become ular also the growth and formation of self-affine cracks. more and more important as damage grows, but in line In other words, we have concentrated on the “approach with the fact that the statistics is close to the mean- tothecriticalpoint”ifthefailuretransitionisconsidered field case the growth is never very far from the MF: for as an analogue of ordinary phase transitions. all phases studied there are remote,brokenfuses instead An analysis of the localization of damage both during of ones localized close to the last growth event. There the fracture process and a posteriori reveals that in the is an associated lengthscale that can be roughly defined case studied the correlations are very weak, are formed based on the x- and y-dependent results, but of course mostly in the last catastrophic phase of network failure, one could go further and look at the radial probability after the maximum current Imax, and do not have any distribution P(~r), with ~r = (xi+1,yi+1) − (xi,yi) (for global correlations. The localization is centered in and which one would presumably need still much larger sys- around the final crack surface, or what may be called tems to getdecent averaging). One centrallessonis that as the total volume encompassed by a “fracture process localizationwilldiminishwithsystemsizeduetothenor- zone”. We would like to note that this is in contrast to malvolume effect ofstrength, decreasingwith L. In this the recent theory of self-organized damage percolation, respect, fuse networks are not unique, and other simu- of Hansen et al. ([24]) devised to explain the formation lation models of brittle fracture should exhibit the same of self-affine cracks in fuse networks, and in related ex- behavior. Toconclude,eveninourcasewithquitestrong periments. In particular it should be stressed that there disorder the failure process consists of weakly correlated is no evidence of a global, non-trivial damage profile as damage growth and a final catastrophic crack propaga- contained in that proposition. Recent numerical results tion phase, that induces a first-order drop in the elastic of Nukala et al. [30] with much better statistics than modulus. what is the case here or with other, earlier authors also seem to imply the same. 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