Damage in Fiber Bundle Models

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Damage in fiber bundle models

Ferenc Kun1,2, Stefano Zapperi3,4, and Hans. J. Herrmann1,3

1 ICA 1, University of Stuttgart, Pfaffenwaldring 27, 70569 Stuttgart, Germany2 Department of Theoretical Physics, University of Debrecen, P.O.Box: 5, H-4010 Debrecen, Hungary,3 PMMH-ESPCI, 10 Rue Vauquelin, 75231 Paris CEDEX 05, France,4 INFM sezione di Roma 1, Universita ”La Sapienza”, P.le A. Moro 2 00185 Roma, Italy.

February 7, 2008

Abstract. We introduce a continuous damage fiber bundle model and compare its behavior with that ofdry fiber bundles. Several interesting constitutive behaviors, such as plasticity, are found in this modeldepending on the value of the damage parameter and on the form of the disorder distribution. We comparethe constitutive behavior of global load transfer models, obtained analytically, with local load transfermodels numerical simulations. The evolution of the damage is studied analyzing the cluster statistics fordry and continous damage fiber bundles. Finally, it is shown that quenched random thresholds enhancedamage localization.

PACS. 4 6.50.+a, 62.20.Fe, 62.20.Mk

1 introduction

The rupture of disordered media has recently attractedmuch technological and industrial interest and has beenwidely studied in statistical physics. It has been suggestedby several authors that the failure of a disordered materialsubjected to an increasing external load shares many fea-tures with thermodynamic phase transitions. In particu-lar, a stressed solid can be considered to be in a metastablestate [1] and the point of global failure can be seen as a nu-cleation process in a first order transition near a spinodal[2,3,4]. Thus, the power law behavior observed experimen-tally in the acoustic emission before failure [5,6,7] has beencompared with the mean-field scaling expected close to aspinodal point [8]. In analogy with spinodal nucleation[9], scaling behavior can only be seen when long-range in-teractions are present, as it is the case for elasticity, butshould not be observable when the stress transfer functionis short ranged. This observation is confirmed in fracturemodels with short-range elastic forces, which usually donot show scaling [10,11].

Most of the theoretical investigations in this field relyon large scale computer simulation of lattice models wherethe elastic medium is represented by a spring (beam) net-work, and disorder is captured either by random dilutionor by assigning random failure thresholds to the bonds[12]. The failure rule usually applied in lattice models isdiscontinuous and irreversible: when the local load exceedsthe failure threshold of a bond, the bond is removed fromthe calculations (i.e. its elastic modulus is set to 0). Fur-thermore, failed bonds are never restored (no healing).Very recently, a novel continuous damage law has been

introduced in lattice models [13]. In the framework of thismodel when the failure threshold of a bond is exceededthe elastic modulus of the bond is reduced by a factor a(0 < a < 1), furthermore, multiple failures of bonds areallowed. This description of damage in terms of a contin-uous parameter corresponds to consider the system at alength scale larger than the typical crack size. Computersimulations have revealed some remarkable features of themodel: after some transients the system tends to a steadystate which is macroscopically plastic, and is characterizedby a power law distributed avalanches of breaking events.

A very important class of models of material failure arethe fiber bundle models (FBM) [14,15,16,17,18,19,20,21,22,23,24,25,26,27which have been extensively studied during the past years.These models consists of a set of parallel fibers having sta-tistically distributed strength. The sample is loaded paral-lel to the fibers direction, and the fibers fail if the load onthem exceeds their threshold value. In stress controlledexperiments, after each fiber failure the load carried bythe broken fiber is redistributed among the intact ones.Among the several theoretical approaches, one simplifica-tion that makes the problem analytically tractable is theassumption of global load transfer, which means that aftereach fiber breaking the stress is equally distributed on theintact fibers neglecting stress enhancement in the vicinityof failed regions [14,15,16,17,18,19,20,21,22,23,29,30,31]. Therelevance of FBM is manifold: in spite of their simplicitythese models capture the most important aspects of mate-rial damage and due to the analytic solutions they providea deeper understanding of the fracture process. Further-more, they serve as a basis for more realistic damage mod-els having also practical importance. The very successful

http://arXiv.org/abs/cond-mat/9908226v2

2 Ferenc Kun et al.: Damage in fiber bundle models

micromechanical models of fiber reinforced composites areimproved variants of FBM taking into account stress lo-calization (local load transfer) [19,20,21,24], the effect ofmatrix material between fibers [20,21,22,23,29,30,31], andpossible non-linear behavior of fibers [16]. Previous stud-ies of FBM addressed the macroscopic constitutive behav-ior, the reliability and size scaling of the global materialstrength, and the avalanches of fiber breaks preceding ul-timate failure [10,11,32,33,34,35,36].

In this paper we generalize the FBM applying a contin-uous damage law for the elements in the spirit of Ref. [13].Emphasis is put on the microstructure of damage andits evolution with increasing load. For the case of globalload transfer in the continuous damage model, we derivean exact analytic expression for the constitutive behaviorand show that the system reaches asymptotically a steadystate, which is macroscopically plastic. It is demonstratedthat the continuous damage model provides a broad spec-trum of description of materials by varying its parametersand for special choices of the parameter values the modelrecovers the dry FBM and other micromechanical modelsof composites known in the literature. Next, we present atheoretical investigation of damaging in FBM by study-ing the ’dry’ FBM varying the range of load transfer. Inthe case of global load transfer the model approaches thefailure point by scaling laws, analogous to those observedclose to a spinodal [8]. However, scaling is not observedfor local load transfer FBM [10,11] as it is expected for aspinodal instability, which can only be observed in mean-field theory. Increasing the range of interactions, one canobserve that the spinodal point, defined in the global loadtransfer FBM, is approached. The evolution of damage iscompared in the local load transfer dry and continuousdamage FBM. Finally, we analyze the effect of quenchedrandom threshold on damage localization.

The paper is organized as follows: in section 2 we de-scribe the FBM and derive their constitutive behavior insection 3. In section 4 we discuss the local load transferFMB focusing on the constitutive behaviour and on thecluster analysis, and in section 5 we explore the role of thetype of disorder in the evolution of damage. Section 6 isdevoted to discussion and conclusions.

2 Models

The system under consideration is composed of N fibersassembled in parallel on a two dimensional square latticeof side length L, i.e. N = L2. The geometrical structureof the model is illustrated in Fig. 1. The square latticecorresponds to a cross section of a unidirectional fiberensemble. In FBM, the fibers are considered to be lin-early elastic until breaking (brittle failure) with identicalYoung-modulus Ef but with random failure thresholds di,i = 1, . . . , N . The failure strength di of individual fibersis supposed to be independent identically distributed ran-dom variables with a cumulative probability distributionP (d). The fiber bundle is supposed to be loaded uniax-ially, and load F applied parallel to the fibers gives riseto a strain f of the bundle. When a fiber experiences a

local load larger than its failure threshold the fiber fails.In dry FBM there is no matrix material present, whichimplies that broken fibers do not support load any more,and their load is redistributed to the surviving fibers.

In the global load transfer FBM, after failure the loadis transfered equally to all the remaining intact fibers, sothat the load on fiber i is simply given by Fi = F/ns(F ),where ns(F ) is the total number of surviving fibers for aload F . This also implies that the range of interaction be-tween fibers is infinite, and hence, the global load transfercorresponds to the mean field treatment of FBM. In thelocal load transfer FBM the load is transfered equally onlyto the surviving nearest neighbor fibers, giving rise to highstress concentration around failed regions (see also Fig. 1).We also study intermediate situations in which the load istransfered to a local neighborhood surrounding the failedfiber (i.e. a square of radius R centered on the failed fiber,see Fig. 1). This model interpolates between the nearestneighbor local FBM and the global FBM as the range ofinteraction is increased.

Next we generalize the model replacing the brittle fail-ure of fibers by a continuous damage parameter [13]. Whenthe load on a fiber reaches the threshold value di the stiff-ness Ef of the fiber is reduced by a factor 0 < a < 1. Thecharacterization of damage by a continuous parameter cor-responds to describe the system on length scales largerthan the typical crack size. This can be interpreted suchthat the smallest elements of the model are fibers and thecontinuous damage is due to cracking inside fibers. How-ever, the model can also be considered as the discretiza-tion of the system on length scales larger than the size ofsingle fibers, so that one element of the model consists ofa collection of fibers with matrix material in between. Inthis case the microscopic damage mechanism resulting inmultiple failure of the elements is the gradual cracking ofmatrix and the breaking of fibers.

In the following the elements of the continuous damageFBM will be called fibers, but we have the above two pos-sible interpretations in mind. Once the fiber i has failedits load is reduced to afi and the rest of the load (1−a)fi

is distributed equally among all the other fibers (globalstress transfer) or among the neighboring fibers (localstress transfer). In principle, a fiber can now fail morethan once and we define kmax as the maximum numberof failures allowed per fiber. We will first study the modelfor finite kmax and eventually take the limit kmax → ∞.It is important to note that once a fiber has failed, we caneither keep the same failure threshold (quenched disorder)or chose a different one of the same distribution (annealeddisorder), which can model microscopic rearrangementsin the material. The failure rules of the model in the twocases are illustrated in Fig. 2. In the following sections wewill analyze both cases, showing that there are differencesin the microstructure of damage between quenched andannealed disorder in this problem.

Ferenc Kun et al.: Damage in fiber bundle models 3

3 Constitutive laws

Here we derive the constitutive law for continuous dam-age FBM and show how the FBMs used in the literaturecan be recovered in particular limits. We first considerthe case in which fibers are allowed to fail only once: theconstitutive equation reads as

F

N= f(1 − P (f)) + afP (f), (1)

where P (f) and 1 − P (f) are the fraction of failed andintact fibers, respectively, and the Young-modulus Ef ofintact fibers is taken to be unity. In Eq. (1) the first termprovides the load carried by intact fibers while the secondterm is the contribution of the failed ones. Note that thisparticular case together with the parameter choice a =0 (i.e. broken fibers carry no load) corresponds to thedry FBM [14,15,17,18], while setting a = 0.5 in Eq. (1)we recover the so-called micromechanical model of fiberreinforced ceramic matrix composites (CMC’s), which hasbeen extensively studied in the literature [25,26,27,28]. InCMC’s the physical origin of the load bearing capacity offailed fibers is that in the vicinity of the broken face ofthe fiber the fiber-matrix interface debonds and the stressbuilds up again in the failed fiber through the sliding fiber-matrix interface.

When the fibers are allowed to fail more than oncewe have to distinguish between quenched and annealeddisorder.

(i) Quenched disorder: When the fibers are allowed tofail twice the constitutive equation can be written as

F

N= f(1 − P (f)) + af [P (f) − P (af)] + a2fP (af), (2)

where [P (f) − P (af)] is the fraction of those fibers whichfailed only once, and P (af) provides the fraction of fiberswhich failed already twice. In the general case, when fibersare allowed to fail kmax times, where kmax can also go toinfinity, the constitutive equation can be cast into the form

F

N= f(1 − P (f)) +

kmax−1∑

i=1

aif[

P (ai−1f) − P (aif)]

+akmaxfP (akmax−1f) (3)

(ii) Annealed disorder: As in the previous case we con-sider first the case in which fibers are allowed to fail twice,obtaining

F

N= f(1 − P (f)) + afP (f)(1 − P (af)) + (4)

+a2fP (f)P (af),

where P (f)(1−P (af)) is the fraction of fibers which failedonly once, and P (f)P (af) is the fraction of fibers whichfailed already twice. Finally, when fibers are allowed tofail kmax times, where kmax can also go to infinity, the

constitutive equation is given by

F

N=

kmax−1∑

i=0

aif[

1 − P (aif)]

i−1∏

j=0

P (ajf) + (5)

+akmaxf

kmax−1∏

i=0

P (aif).

In Fig. 3 we show the explicit form of the constitutivelaw for quenched disorder for different values of kmax inthe case of the Weibull distribution

P (d) = 1 − exp(−(d/dc)m), (6)

where m is the Weibull modulus and dc denotes the char-acteristic strength of fibers. It is important to remark thatthe constitutive laws derived above are exact only in theinfinite size limit (N → ∞), while fluctuations in the valueof the failure stress Fc have been observed and studied forfinite size bundles. For this reason, we compare the theo-retical results with numerical simulations of bundles of sizeN = 1282. The agreement between simulations and theoryturns out to be satisfactory both for quenched (Fig. 3) andannealed disorder and reflects the fact that for global loadsharing finite size fluctuations, for instance for Fc, shouldscale as 1/

√

(N). This behavior has to be contrasted withlocal-load sharing fiber bundles where finite size effects arevery strong, as we will discuss in the following.

In Fig. 3a the fibers are supposed to have akmax resid-ual stiffness after having failed kmax times, which givesrise to hardening of the material, i.e. the F/N curvesasymptotically tend to straight lines with slope akmax . In-creasing kmax the hardening part of the constitutive be-havior is preceded by a longer and longer plastic plateau,and in the limiting case of kmax → ∞ the materials be-havior becomes completely plastic (see Fig. 3). A similarplateau and asymptotic linear behavior has been observedin brittle matrix composites, where the multiple crackingof matrix turned to be responsible for the relatively broadplateau of the constitutive behavior, and the asymptoticlinear part is due to the linear elastic behavior of fibersremained intact after matrix cracking [40].

In order to describe macroscopic cracking and globalfailure instead of hardening, the residual stiffness of thefibers has to be set to zero after a maximum number k∗

of allowed failures [13]. In this case the constitutive lawcan be obtained from the general form Eqs. (3) and (5)by skipping the last term corresponding to the residualstiffness of fibers, and by setting kmax = k∗ in the re-maining part. A comparison of the constitutive laws ofthe dry and continuous damage FBM is presented in Fig.3b for the case of quenched disorder. Annealed disorderyields similar results. One can observe that the dry FBMconstitutive law has a relatively sharp maximum, how-ever, the continuous damage FBM curves exhibit a plateauwhose length increases with increasing k∗. Note that themaximum value of F/N corresponds to the macroscopicstrength of the material and in stress controlled experi-ments the plateau and the decreasing part of the curves


cannot be reached. However, by controlling the strain f ,the plateau and the decreasing regime can also be realized.The value of the driving stress σ ≡ F/N corresponding tothe plastic plateau is determined by the damage parame-ter a, while the length of the plateau is controlled by kmax

and k∗.In Fig. 4, we directly compare the constitutive law for

quenched and annealed disorder and confirm that the dif-ferences between the cases are very small. In particular,all the basic constitutive behavior are reproduced in thetwo cases.

It is important to remark that the behavior of the dryFBM model (a = 0) under unloading and reloading to theoriginal stress level is completely linear, since no new dam-age can occur during unloading-reloading sequences andthe effect of the matrix material is completely neglected.This also implies that in each damage state the model iscompletely characterized by the Young modulus defined asthe slope of the unloading curve. If the value of the dam-age parameter is larger than 0 (a > 0) the behavior of thesystem under unloading and reloading is rather compli-cated. Due to the sliding of broken fibers with respect tothe matrix, hysteresis loops and remaining inelastic strainoccur (for examples see Ref. [38] and references therein).

4 Local load transfer FBM

4.1 Constitutive behaviour

To study the effect of stress enhancement around failedfibers on the damage evolution and on the macroscopicconstitutive behavior we employ local load transfer forthe stress redistribution after fiber failure [20,25,26,27].Since this case cannot be treated analytically, we performnumerical simulations in the dry FBM model: after fiberfailure the load is redistributed on the intact nearest neigh-bors of the failed fiber using periodic boundary conditionon the square lattice (see also Fig. 1). For simplicity, in thiscase the strength of fibers di has a uniform distributionbetween 0 and 1. The algorithm to simulate the loadingprocess is as follows: (i) we impose on all the fibers thesame load, equal to the smallest failure threshold, whichresults in breaking of the weakest element. (ii) The loadcarried by the failed fiber is redistributed on the intactnearest neighbors, and the load of the broken fiber is setto zero. (iii) After the stress redistribution, those fiberswhose load exceeds their failure threshold di are identi-fied and removed from the calculation, and the simulationis continued with point (ii). If the configuration obtainedafter the stress redistribution is stable, the global load isincreased to cause the failure of one more fiber and thesimulation is continued with point (ii). This proceduregoes on until all fibers are broken. The applied stress justbefore global failure is considered to be the failure strengthof the model solid. Simulations were performed with sys-tem sizes L = 16, 32, 64, 128.

The constitutive behavior of the local and global loadtransfer dry FBM is compared in Fig. 5. For clarity, the

total force F (instead of stress) is presented as a func-tion of strain f , for several different system sizes L. Notethat N = L2 is chosen the same for global and local loadsharing simulations. In the case of global load transfer theN → ∞ constitutive law can be obtained exactly by sub-stituting the cumulative probability distribution P (f) = fof the uniform distribution into the general form Eq. (1)and setting the damage parameter a = 0:

F = Nf(1 − P (f)) = Nf(1 − f), f ∈ [0, 0.5], (7)

the strain corresponding to macroscopic failure is fc = 0.5.It can be seen in Fig. 5 that the macroscopic constitu-tive behavior for local load transfer always coincides withthe global FBM solution, however, the macroscopic failurestrength is substantially reduced in the local case givingrise to more brittle constitutive behavior. It is interest-ing to note that increasing the system size L the failurestrength of the local FBM decreases, showing the logarith-mic size effect also found in the one dimensional local FBM[11] and in two dimensional fuse networks [37], while theglobal load transfer case does not have size dependence.

To get a deeper understanding of the behavior of thesystem as a function of the range R of load redistribution,we perform simulations by redistributing the load afterfiber failure on the intact fibers in a square of side length2R + 1 centered on the failed fiber. The range of load re-distribution R is varied between 1 and (L − 1)/2. Notethat R = 1 corresponds to local load transfer on near-est and next-nearest neighbors, while the limiting case ofR = (L − 1)/2 recovers the ’infinite’ range global loadtransfer. The comparison of the constitutive behavior inthe local and global load transfer case is presented inFig. 6. Simulations reveal that the constitutive laws ob-tained at different R values always fall onto the curve ofthe global load transfer case and the macroscopic failurestrength increases with increasing range of interaction Rapproaching the strength of global FBM. For clarity, inFig. 6 we indicated by vertical dashed lines the positionof global failure at different values of R.

To characterize the elastic response of the dry FBMmodel in a given damage state, we compute the Youngmodulus Y , defined as the slope of unloading curves as afunction of the driving stress σ ≡ F/N :

Y (σ) =σ

f(σ)= 1 − P (f(σ)) = 1 − f(σ). (8)

Using the constitutive law Eq. (7) for the global loadtransfer case, Y can be written into a closed form as afunction of stress

Y (σ) =1

2

[

1 +√

1 − 4σ]

. (9)

The results on Y for global and local load transfer areshown in Fig. 7, where the vertical dashed lines indicatethe position of macroscopic failure at different values ofthe redistribution range R. It can be seen that at the fail-ure point Y (σ) has a discrete jump, the size of which de-creases with increasing R, but it remains finite in the limit


of global load transfer. Increasing R gives rise to increas-ing slope of Y (σ) at the failure point, and in the limitof infinite range interaction the tangent of Y (σ) becomesvertical at the point of failure.

4.2 Cluster analysis

One of the most interesting aspects of the damage mech-anism of disordered solids is that the breakdown is pre-ceded by an intensive precursor activity in the form ofavalanches of microscopic breaking events. Under a givenexternal load F a certain fraction of fibers fails immedi-ately. Due to the load transfer from broken to intact fibersthis primary fiber breaking may initiate secondary break-ing that may also trigger a whole avalanche of breaking. IfF is large enough the avalanche does not stop and the ma-terial fails catastrophically. It has been shown by analyticmeans that in the case of global load transfer the size dis-tribution of avalanches follows asymptotically a universalpower law with an exponent −5/2 [11,32], however, in thecase of local load transfer no universal behavior exists,and the avalanche characteristic size is bounded [10,11].This precursory activity can also be observed experimen-tally by means of the acoustic emission analysis. Acousticemission measurements have revealed that for a broad va-riety of disordered materials the response to an increasingexternal load takes place in bursts having power law sizedistribution over a wide range [5,6,7].

In this section we analyze the evolution of damage inlocal load transfer FBM, comparing dry and continuousdamage models. Instead of avalanches of fiber failures, wefocus on the properties of clusters of broken fibers whichare much less explored. In the following simulations weemploy a uniform distribution for the thresholds di. Theload after fiber failure is redistributed on the survivingnearest neighbors. As the load is increased, fiber breaksand clusters of broken fibers are formed due to the spatialcorrelation introduced by the local load transfer. Theseclusters of broken fibers can be identified as microcracksformed in the plane perpendicular to the load direction.We monitor the damage evolution by taking snapshots ofthe clusters at different loads. In Fig. 8 the damage evolu-tion is shown in the dry FBM. One can observe the nucle-ation and gradual growth of clusters with increasing loadF . We find that the clusters are small compared to thesystem size even before global failure (see Fig. 8(d)), inaccordance with the first-order transition scenario. In thecontinuous damage FBM with local load transfer the clus-ters of failed fibers are defined as connected sets of fibershaving the same number of failure, taking into accountonly nearest neighbor connections. In these calculationswe set kmax = ∞, and the simulations are stopped whenthe plastic regime is reached.

To obtain quantitative informations on the damageevolution, we measure the cluster probability distributionn(s, F ), defined as the number of clusters formed by sneighboring broken fibers when the applied load is F [8].The moments (Mk(F ) ≡

∫

skn(s, F )ds is the k-th mo-ment) of n(s, F ) contain most of the information on the

evolution of the damage. We determine n(s, F ) for differ-ent system sizes L by averaging over the disorder. Thetotal number of clusters nc ≡ M0 as a function of theload is presented in Fig. 9. The increasing part of nc asa function of F is due to the nucleation of new microc-racks, and the short plateau or decreasing regime in thevicinity of global failure is caused by the coalescence ofgrowing cracks. The inset of Fig. 9 demonstrates that indry FBM nc obeys a simple scaling law nc = L2g(F/L2)implying that the clusters are homogeneously scatteredthrough the lattice. A similar scaling is observed for thecontinuous damage annealed FBM in Fig. 10. The insetshows that the scaling function is independent of the dam-age parameter a.

Next, we measure the average cluster size defined asS ≡ M2/M1 and show that it approaches a value whichdecreases with system size (Fig. 11(a)). It can be seenthat for a given system size the S(F ) curves have tworegimes: a slowly increasing initial regime due to the nu-cleation and growth of clusters, and a rapidly increasingpart close to global failure which is caused by the coa-lescence of growing cracks. In dry FBM, we can simplyrescale the data according to the law S(F, L) = s(F/L2)and obtain a good collapse (Fig. 11(b)). Similar results areobtained for the continuous damage case, but we see thatthe rescaled curves depend on a (Fig. 12). The larger a is,the smaller the clusters are, since the stress concentrationdecreases with increasing a, and the disorder gets moredominating. These results demonstrate that global failureis initiated once the crack size reaches a critical size sc af-ter which a crack becomes unstable. However, the averagecluster size S does not provide a reliable estimate of sc,which can be obtained instead monitoring the size of thelargest cluster Smax as a function of the load. It can beseen in Fig. 13 that Smax reaches a value that increaseswith L, but for large L this value seems to saturate. Therapid increase of Smax close to the failure point is due tothe coalescence of clusters and is thus produced by a verysmall amount of fiber failures.

5 damage localization: effect of the quenched

disorder

In the previous section, we analyzed the damage structurein the local load transfer models. In the dry FBM and inthe case of continuous damage FBM with annealed disor-der we do not expect to find any non trivial damage local-ization for global load transfer rules, since these modelsbehave effectively like in mean-field theory. On the otherhand, quenched disorder can lead to localized structurefor continuous damage FBM even in global load transferconditions. Weak fibers are expected to fail more timesgenerating an inhomogeneous damage pattern.

In order to analyze this effect, we measure k(i), thenumber of failures at fiber i, when there is exactly onefiber, which has reached kmax. In the bottom part of Fig. 14we plot the value of k(i) and the corresponding value ofthe threshold d(i). The ’damage’ k(i) shows a very irreg-ular pattern, which should be compared with a roughly


uniform structure expected for annealed disorder. In theupper part of the figure we display the decay of k as afunction of d, showing how weak fibers break more oftenthan strong ones. The decay is more pronounced when ais close to one, and becomes less important for smallera. It is straightforward to obtain an analytic expressionfor k(d) which is compared with the numerical results inFig. 15.

Finally, we expect that quenched disorder should havean effect also on the cluster structure of local load transfermodels. In order to confirm this point, we compare thenumber of clusters nc(F ) and the average cluster size Sfor quenched and annealed disorder. The results, shownin Figs. 16 and 17, indicate that damage is more localizedwhen the disorder is quenched.

6 conclusions

We have proposed a continuous damage version of theFBM, that can be used to model a wide variety of con-stitutive behaviors. We have analyzed the development ofdamage in FBM under different conditions and comparedthe local load transfer model with different interactionranges with the global load transfer model that can besolved exactly in the limit N → ∞.

From the theoretical point of view, the cluster analysisshows the analogies between the failure in the FBM andnucleation in first-order phase transition. The failure pointin global load transfer FBM plays the role of a spinodalpoint. The strain f carried by the fibers, proportional tothe fraction of intact fibers [8], close to the failure load Fc

has a diverging derivative df/dF ∼ (Fc−F )−1/2. A similarbehavior is observed close to a spinodal instability in first-order phase transitions. One should note that the spinodalpoint and its associated scaling is a mean-field property,obtained in the limit N → ∞, and in general is not ob-served for finite dimensional short-ranged models wherenucleation occurs much before reaching the spinodal. Sim-ilarly, in local-load transfer (i.e. short range) FBM failureoccurs much before the corresponding global load sharinginstability (i.e. the spinodal) and the avalanche character-istic size is bounded [32,10,11].

We have shown that increasing the range of interactionthe failure point is shifted towards the spinodal. A similarbehavior is observed for instance in Ising systems whenthe range of interactions is increased [39]. In models withlong range stress transfer, as for instance in elastic or elec-tric networks, it is possible to observe the spinodal scalingeven in finite dimensional systems [8,11]. The presence of aspinodal instability could explain the observation of scal-ing properties in acoustic emission experiments [5,6,7].

Our continuous damage FBM can reproduce a widevariety of elasto-plastic constitutive behaviors. A remark-able feature of the model is that multiple failure of theelements results in ductile macroscopic behavior in spiteof the brittleness of the constituents. Similar ductile be-havior has been observed experimentally in fiber rein-forced composites made of brittle constituents [42,43,44].

Experiments revealed that the mechanism of this ductil-ity, which is called pseudo-strain hardening, is the mul-tiple failure of the material [42,43,44]. Our continuousdamage model, recovering as special cases the dry bundlemodel and micromechanical models known in the litera-ture, could provide a general framework for the statistical-micromechanical modeling of the behavior of fiber rein-forced composites. The fitting of experimental results inthe framework of our model will be presented in a forth-coming publication.

Acknowledgment

This work was supported by the project SFB381. F. K.acknowledges financial support from the Alexander vonHumboldt Foundation (Roman Herzog Fellowship). F. K.is grateful to I. Sajtos for the valuable discussions. S. Z.acknowledges financial support from EC TMR ResearchNetwork under contract ERBFMRXCT960062.

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fiber reinforced cement composites, (E & FN Spon, Lon-don, 1995).

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Fig. 1. The geometry of the FBM. The uniaxial fiber bundle(left) is modeled on a square lattice (right) corresponding toa cross section of the specimen. The black plaquette indicatesa broken fiber, its nearest neighborhood is shadowed, and thebold line shows the neighborhood for the range of interactionR = 2.

f

F

d

F

f

d

d

d i

i1

i3

2

a)

b)

i

Fig. 2. The constitutive behavior of a single fiber of the con-tinuous damage model when multiple failure is allowed. (a)Quenched disorder: the horizontal line indicates the damagethreshold di, which is constant in time for each fiber. (b) An-nealed disorder: a new threshold is extracted at random aftereach failure.


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

f

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0F

/N

a)

F/N=akmax f

kmax =

kmax = 8

kmax = 4

kmax = 2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

f

0.0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

F/N

b)

CDFBM, k*=

CDFBM, k*=8

CDFBM, k*=4

dry FBM, a=0

Fig. 3. (a) The constitutive law for the global stress transfercontinuous damage FBM (CDFBM) for different values of kmax

for m = 2.0, a = 0.8 and quenched disorder. Plastic behavioris obtained in the limit of kmax → ∞. (b) Comparison betweendry and continuous damage FBM. If we allow for brittle failureafter k∗ damage events, we obtain a plastic plateau followedby brittle failure. Symbols refer to simulations of bundles ofsize N = 1282 and lines to the analytic calculations.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

f

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

F/N

quenched

annealed

a = 0.8

kmax = k*

= 8

Fig. 4. Comparison of the constitutive laws of the quenchedand annealed case for a = 0.8, kmax = 8 and k∗ = 8.

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

f

0

500

1000

1500

2000

2500

3000

3500

4000

4500

F

exact

L=128

L=64

L=32

Fig. 5. The constitutive law for the global stress transfer dryFBM (solid line) is compared with the local stress transfermodel. Increasing the system size the failure stress decreases.


0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

f

0.0

0.05

0.1

0.15

0.2

0.25

0.3

F/N

Fig. 6. The constitutive law for the global stress transferdry FBM (solid line) is compared with the local stress trans-fer FBM for different interaction ranges R. The values ofR corresponding to the consecutive vertical dashed lines are1, 3, 5, 11, 15 from left to right, and the system size L = 128was chosen. Increasing the interaction range the failure stressincreases approaching the value predicted by the global stresstransfer model.

0.0 0.05 0.1 0.15 0.2 0.25 0.30.0

0.2

0.4

0.6

0.8

1.0

Y

Fig. 7. The Young modulus for the global stress transfer dryFBM is compared with the local stress transfer model for dif-ferent interaction ranges R. The values of R corresponding tothe vertical dashed lines are the same as in Fig. 6.

Fig. 8. Snapshots of the damage in the dry FBM model ona square lattice of size L = 128 for different values of theload: a) F/Fc = 0.153 b) F/Fc = 0.468, c) F/Fc = 0.796 d)F/Fc = 0.997.

0 500 1000 1500 2000 2500 3000 3500 4000

F

0

300

600

900

1200

1500

1800

nc

L=128

L=64

L=32

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

F/L2

0.0

0.02

0.04

0.06

0.08

0.1

0.12

nc/

L2

Fig. 9. The number of clusters as a function of the load F inthe local stress transfer dry FBM for different system sizes. Inthe inset the rescaled plot is presented, where also the systemL = 16 is shown (square).


0 1000 2000 3000 4000 5000 6000 7000 8000

F

0

1000

2000

3000

4000

5000

6000

nc

L=128

L=64

L=32

L=16

a=0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

F/L2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

nc/

L2 a=0.001

a=0.5

a=0.95

Fig. 10. The number of clusters as a function of the load Fin the local stress transfer continuous damage annealed FBMfor different system sizes at a = 0.5. In the inset we show therescaled plot for different values of a and L.

0 500 1000 1500 2000 2500 3000 3500 4000

F

1

3

5

7

9

11

13

S

a)

L=128

L=64

L=32

L=16

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

F/L2

1

3

5

7

9

11

13

S

b)

L=128

L=64

L=32

L=16

Fig. 11. (a) The average cluster size as a function of the loadF in the local stress transfer dry FBM for different system sizesL and the corresponding rescaled plot (b).

0 1000 2000 3000 4000 5000 6000 7000 8000

F

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

S

a)

L=128

L=64

L=32

L=16

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

F/L2

1

2

3

4

5

6

7

8

S

b)a=0.001

a=0.5

a=0.95

L=128

L=64

L=32

L=16

Fig. 12. (a)The average cluster size as a function of the load Fin the local stress transfer continuous damage annealed FBMwith a = 0.5 for different system sizes L, and (b) the corre-sponding rescaled plot for different values of L and a.

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

F/L2

10

20

30

40

50

60

70

Sm

ax

L=128

L=64

L=32

L=16

Fig. 13. The size of the largest cluster Smax as a function ofF/L2 in the local stress transfer dry FBM for different systemsizes L.


0.0 0.2 0.4 0.6 0.8 1.0

d

0

10

20

30

40

50

60

70

80

90

100

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0 50 100 150 200 250 300 350 400 450 500

i

0

10

20

30

4050

60

70

80

90

100

k(i

)

0.1

0.2

0.3

0.40.5

0.6

0.7

0.8

0.9

1.0

d(i

)

Fig. 14. The number of failures k of the fibers as a functionof their threshold d (distributed uniformly) in the quenchedglobal load transfer continuous FBM (top). The parametersused are kmax = 100, N = 500 and a = 0.4, 0.8, 0.9, 0.95. Thethreshold d(i) as a function of i is compared with the ’damage’k(i) for a = 0.9 (bottom).

Fig. 15. The number of failures k of the fibers as a functionof their threshold d (distributed according to the Weibull dis-tribution with m = 1.5) in the quenched global load transfercontinuous FBM: comparison between simulations and analyticresults for a = 0.95, 0.9, 0.4.

0.0 0.1 0.2 0.3 0.4 0.5 0.6

F/L2

0.0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4n

c/L

2

annealed

quenched

Fig. 16. The number of clusters as a function of the rescaledload F/L2 in the local stress transfer continuous damagequenched and annealed FBM for different system sizes ata = 0.5. Note that the number of clusters increses faster forannealed disorder indicating a smaller degree localization.


0.0 0.1 0.2 0.3 0.4 0.5 0.6

F/L2

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5S

annealed

quenched

Fig. 17. The average cluster size as a function of the rescaledload F/L2 in the local stress transfer continuous damagequenched and annealed FBM for different system sizes ata = 0.5. Note that the cluster size increses faster for quencheddisorder indicating a larger degree localization.

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Damage in Fiber Bundle Models

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