Damage in fiber bundle models

Transcription

1 Eur Phys J B 17, (2) THE EUROPEAN PHYSICAL JOURNAL B c EDP Sciences Società Italiana i isica Springer-Verlag 2 Damage in iber bunle moels Kun 1,2,a, S Zapperi 3,4, an HJ Herrmann 1,3 1 ICA 1, University o Stuttgart, Paenwalring 27, 7569 Stuttgart, Germany 2 Department o Theoretical Physics, University o Debrecen, PO Box 5, 41 Debrecen, Hungary 3 PMMH-ESPCI, 1 rue Vauquelin, Paris Ceex 5, rance 4 INM sezione i Roma 1, Università La Sapienza, Ple A Moro Roma, Italy Receive 27 March 2 Abstract We introuce a continuous amage iber bunle moel an compare its behavior with that o ry iber bunles Several interesting constitutive behaviors, such as plasticity, are oun in this moel epening on the value o the amage parameter an on the orm o the isorer istribution We compare the constitutive behavior o global loa transer moels, obtaine analytically, with local loa transer moels numerical simulations The evolution o the amage is stuie analyzing the cluster statistics or ry an continous amage iber bunles inally, it is shown that quenche ranom threshols enhance amage localization PACS 465+a racture mechanics, atigue an cracks 622e Deormation an plasticity (incluing yiel, uctility, an superplasticity) 622Mk atigue, brittleness, racture, an cracks 1 Introuction The rupture o isorere meia has recently attracte much technological an inustrial interest an has been wiely stuie in statistical physics It has been suggeste by several authors that the ailure o a isorere material subjecte to an increasing external loa shares many eatures with thermoynamic phase transitions In particular, a stresse soli can be consiere to be in a metastable state [1] an the point o global ailure can be seen as a nucleation process in a irst orer transition near a spinoal [2 4] Thus, the power law behavior observe experimentally in the acoustic emission beore ailure [5 7] has been compare with the mean-iel scaling expecte close to a spinoal point [8] In analogy with spinoal nucleation [9], scaling behavior can only be seen when longrange interactions are present, as it is the case or elasticity, but shoul not be observable when the stress transer unction is short range This observation is conirme in racture moels with short-range elastic orces, which usually o not show scaling [1,11] Most o the theoretical investigations in this iel rely on large scale computer simulation o lattice moels where the elastic meium is represente by a spring (beam) network, an isorer is capture either by ranom ilution or by assigning ranom ailure threshols to the bons [12] The ailure rule usually applie in lattice moels is iscontinuous an irreversible: when the local loa excees the ailure threshol o a bon, the bon is remove rom the calculations (ie its elastic moulus is set a eri@tpatomkihu to ) urthermore, aile bons are never restore (no healing) Very recently, a novel continuous amage law has been introuce in lattice moels [13] In the ramework o this moel when the ailure threshol o a bon is exceee the elastic moulus o the bon is reuce by aactora ( <a<1), urthermore, multiple ailures o bons are allowe This escription o amage in terms o a continuous parameter correspons to consier the system at a length scale larger than the typical crack size Computer simulations have reveale some remarkable eatures o the moel: ater some transients the system tens to a steay state which is macroscopically plastic, an is characterize by a power law istribute avalanches o breaking events A very important class o moels o material ailure are the iber bunle moels (BM) [14 31], which have been extensively stuie uring the past years These moels consists o a set o parallel ibers having statistically istribute strength The sample is loae parallel to the ibers irection, an the ibers ail i the loa on them excees their threshol value In stress controlle experiments, ater each iber ailure the loa carrie by the broken iber is reistribute among the intact ones Among the several theoretical approaches, one simpliication that makes the problem analytically tractable is the assumption o global loa transer, which means that ater each iber breaking the stress is equally istribute on the intact ibers neglecting stress enhancement in the vicinity o aile regions [14 23,29 31] The relevance o BM is maniol: in spite o their simplicity these moels capture the most important aspects o material amage an ue

2 27 The European Physical Journal B to the analytic solutions they provie a eeper unerstaning o the racture process urthermore, they serve as a basis or more realistic amage moels having also practical importance The very successul micromechanical moels o iber reinorce composites are improve variants o BM taking into account stress localization (local loa transer) [19 21,24], the eect o matrix material between ibers [2 23,29 31], an possible nonlinear behavior o ibers [16] Previous stuies o BM aresse the macroscopic constitutive behavior, the reliability an size scaling o the global material strength, an the avalanches o iber breaks preceing ultimate ailure [1,11,32 36] In this paper we generalize the BM applying a continuous amage law or the elements in the spirit o reerence [13] Emphasis is put on the microstructure o amage an its evolution with increasing loa or the case o global loa transer in the continuous amage moel, we erive an exact analytic expression or the constitutive behavior an show that the system reaches asymptotically a steay state, which is macroscopically plastic It is emonstrate that the continuous amage moel provies a broa spectrum o escription o materials by varying its parameters an or special choices o the parameter values the moel recovers the ry BM an other micromechanical moels o composites known in the literature Next, we present a theoretical investigation o amaging in BM by stuying the ry BM varying the range o loa transer In the case o global loa transer the moel approaches the ailure point by scaling laws, analogous to those observe close to a spinoal [8] However, scaling is not observe or local loa transer BM [1,11] as it is expecte or a spinoal instability, which can only be observe in mean-iel theory Increasing the range o interactions, one can observe that the spinoal point, eine in the global loa transer BM, is approache The evolution o amage is compare in the local loa transer ry an continuous amage BM inally, we analyze the eect o quenche ranom threshol on amage localization The paper is organize as ollows: in Section 2 we escribe the BM an erive their constitutive behavior in Section 3 In Section 4 we iscuss the local loa transer MB ocusing on the constitutive behaviour an on the cluster analysis, an in Section 5 we explore the role o the type o isorer in the evolution o amage Section 6 is evote to iscussion an conclusions 2 Moels The system uner consieration is compose o N ibers assemble in parallel on a two imensional square lattice o sie length L, ie N = L 2 The geometrical structure o the moel is illustrate in igure 1 The square lattice correspons to a cross section o a uniirectional iber ensemble In BM, the ibers are consiere to be linearly elastic until breaking (brittle ailure) with ientical Young-moulus E but with ranom ailure threshols i, i =1,,N The ailure strength i o iniviual ibers ig 1 The geometry o the BM The uniaxial iber bunle (let) is moele on a square lattice (right) corresponing to a cross section o the specimen The black plaquette inicates a broken iber, its nearest neighborhoo is shaowe, an the bol line shows the neighborhoo or the range o interaction R =2 is suppose to be inepenent ientically istribute ranom variables with a cumulative probability istribution P () The iber bunle is suppose to be loae uniaxially, an loa applie parallel to the ibers gives rise toastrain o the bunle When a iber experiences a local loa larger than its ailure threshol the iber ails In ry BM there is no matrix material present, which implies that broken ibers o not support loa any more, an their loa is reistribute to the surviving ibers In the global loa transer BM, ater ailure the loa is transere equally to all the remaining intact ibers, so that the loa on iber i is simply given by i = /n s ( ), where n s ( ) is the total number o surviving ibers or a loa This also implies that the range o interaction between ibers is ininite, an hence, the global loa transer correspons to the mean iel treatment o BM In the local loa transer BM the loa is transere equally only to the surviving nearest neighbor ibers, giving rise to high stress concentration aroun aile regions (see also ig 1) We also stuy intermeiate situations in which the loa is transere to a local neighborhoo surrouning the aile iber (ie a square o raius R centereontheaileiber, see ig 1) This moel interpolates between the nearest neighbor local BM an the global BM as the range o interaction is increase Next we generalize the moel replacing the brittle ailure o ibers by a continuous amage parameter [13] When the loa on a iber reaches the threshol value i the stiness E o the iber is reuce by a actor <a<1 The characterization o amage by a continuous parameter correspons to escribe the system on length scales larger than the typical crack size This can be interprete such that the smallest elements o the moel are ibers an the continuous amage is ue to cracking insie ibers However, the moel can also be consiere as the iscretization o the system on length scales larger than the size o single ibers, so that one element o the moel consists o a collection o ibers with matrix material in between In this case the microscopic amage mechanism resulting in multiple ailure o the elements is the graual cracking o matrix an the breaking o ibers In the ollowing the elements o the continuous amage BM will be calle ibers, but we have the above two possible interpretations in min Once the iber i has aile

3 Kun et al: Damage in iber bunle moels 271 a) b) i 3 i 1 i 2 i intact ibers is taken to be unity In equation (1) the irst term provies the loa carrie by intact ibers while the secon term is the contribution o the aile ones Note that this particular case together with the parameter choice a =(ie broken ibers carry no loa) correspons to the ry BM [14,15,17,18], while setting a = 5 in equation (1) we recover the so-calle micromechanical moel o iber reinorce ceramic matrix composites (CMC s), which has been extensively stuie in the literature [25 28] In CMC s the physical origin o the loa bearing capacity o aile ibers is that in the vicinity o the broken ace o the iber the iber-matrix interace ebons an the stress buils up again in the aile iber through the sliing iber-matrix interace When the ibers are allowe to ail more than once we have to istinguish between quenche an anneale isorer (i) Quenche isorer: When the ibers are allowe to ail twice the constitutive equation can be written as ig 2 The constitutive behavior o a single iber o the continuous amage moel when multiple ailure is allowe (a) Quenche isorer: the horizontal line inicates the amage threshol i, which is constant in time or each iber (b) Anneale isorer: a new threshol is extracte at ranom ater each ailure its loa is reuce to a i an the rest o the loa (1 a) i is istribute equally among all the other ibers (global stress transer) or among the neighboring ibers (local stress transer) In principle, a iber can now ail more than once an we eine k max as the maximum number o ailures allowe per iber We will irst stuy the moel or inite k max an eventually take the limit k max It is important to note that once a iber has aile, we can either keep the same ailure threshol (quenche isorer) or chose a ierent one o the same istribution (anneale isorer), which can moel microscopic rearrangements in the material The ailure rules o the moel in the two cases are illustrate in igure 2 In the ollowing sections we will analyze both cases, showing that there are ierences in the microstructure o amage between quenche an anneale isorer in this problem 3 Constitutive laws Here we erive the constitutive law or continuous amage BM an show how the BMs use in the literature can be recovere in particular limits We irst consier the case in which ibers are allowe to ail only once: the constitutive equation reas as = (1 P ()) + ap (), (1) N where P () an1 P () are the raction o aile an intact ibers, respectively, an the Young-moulus E o N = (1 P ()) + a [P () P (a)] + a2 P(a), (2) where [P () P (a)] is the raction o those ibers which aile only once, an P (a) provies the raction o ibers which aile alreay twice In the general case, when ibers are allowe to ail k max times, where k max can also go to ininity, the constitutive equation can be cast into the orm kmax 1 N = (1 P ()) + a i [ P (a i 1 ) P (a i ) ] i=1 +a kmax P(a kmax 1 ) (3) (ii) Anneale isorer: As in the previous case we consier irst the case in which ibers are allowe to ail twice, obtaining = (1 P ()) + ap ()(1 P (a)) (4) N +a 2 P()P (a), where P ()(1 P (a)) is the raction o ibers which aile only once, an P ()P (a) is the raction o ibers which aile alreay twice inally, when ibers are allowe to ail k max times, where k max can also go to ininity, the constitutive equation is given by kmax 1 N = a i [ 1 P (a i ) ] i 1 P (a j ) (5) i= +a kmax k max 1 i= P (a i ) j= In igure 3 we show the explicit orm o the constitutive law or quenche isorer or ierent values o k max in the case o the Weibull istribution P () =1 exp( (/ c ) m ), (6)

4 272 The European Physical Journal B a) k max =k * =8 a = 8 anneale quenche 6 5 /N k max =2 k max =4 k max =8 k max = /N=a k max /N b) ig 4 Comparison o the constitutive laws o the quenche an anneale case or a =8, k max =8ank =8 4 /N ry BM, a= CDBM, k * =4 CDBM, k * =8 CDBM, k * = ig 3 (a) The constitutive law or the global stress transer continuous amage BM (CDBM) or ierent values o k max or m =2, a =8 an quenche isorer Plastic behavior is obtaine in the limit o k max (b) Comparison between ry an continuous amage BM I we allow or brittle ailure ater k amage events, we obtain a plastic plateau ollowe by brittle ailure Symbols reer to simulations o bunles o size N = an lines to the analytic calculations where m is the Weibull moulus an c enotes the characteristic strength o ibers It is important to remark that the constitutive laws erive above are exact only in the ininite size limit (N ), while luctuations in the value o the ailure stress c have been observe an stuie or inite size bunles or this reason, we compare the theoretical results with numerical simulations o bunles o size N = The agreement between simulations an theory turns out to be satisactory both or quenche (ig 3) an anneale isorer an relects the act that or global loa sharing inite size luctuations, or instance or c, shoul scale as 1/ (N) This behavior has to be contraste with local-loa sharing iber bunles where inite size eects are very strong, as we will iscuss in the ollowing In igure 3a the ibers are suppose to have a kmax resiual stiness ater having aile k max times, which gives rise to harening o the material, ie the /N curves asymptotically ten to straight lines with slope a kmax Increasing k max the harening part o the constitutive behavior is precee by a longer an longer plastic plateau, an in the limiting case o k max the materials behavior becomes completely plastic (see ig 3) A similar plateau an asymptotic linear behavior has been observe in brittle matrix composites, where the multiple cracking o matrix turne to be responsible or the relatively broa plateau o the constitutive behavior, an the asymptotic linear part is ue to the linear elastic behavior o ibers remaine intact ater matrix cracking [4] In orer to escribe macroscopic cracking an global ailure instea o harening, the resiual stiness o the ibers has to be set to zero ater a maximum number k o allowe ailures [13] In this case the constitutive law can be obtaine rom the general orm equations (3, 5) by skipping the last term corresponing to the resiual stiness o ibers, an by setting k max = k in the remaining part A comparison o the constitutive laws o the ry an continuous amage BM is presente in igure 3b or the case o quenche isorer Anneale isorer yiels similar results One can observe that the ry BM constitutive law has a relatively sharp maximum, however, the continuous amage BM curves exhibit a plateau whose length increases with increasing k Note that the maximum value o /N correspons to the macroscopic strength o the material an in stress controlle experiments the plateau an the ecreasing part o the curves cannot be reache However, by controlling the strain, the plateau an the ecreasing regime can also be realize The value o the riving stress σ /N corresponing to the plastic plateau is etermine by the amage parameter a, while the length o the plateau is controlle by k max an k In igure 4, we irectly compare the constitutive law or quenche an anneale isorer an conirm that the ierences between the cases are very small In particular,

5 Kun et al: Damage in iber bunle moels exact ig 5 The constitutive law or the global stress transer ry BM (soli line) is compare with the local stress transer moel Increasing the system size the ailure stress ecreases all the basic constitutive behavior are reprouce in the two cases It is important to remark that the behavior o the ry BM moel (a = ) uner unloaing an reloaing to the original stress level is completely linear, since no new amage can occur uring unloaing-reloaing sequences an the eect o the matrix material is completely neglecte This also implies that in each amage state the moel is completely characterize by the Young moulus eine as the slope o the unloaing curve I the value o the amage parameter is larger than (a >) the behavior o the system uner unloaing an reloaing is rather complicate Due to the sliing o broken ibers with respect to the matrix, hysteresis loops an remaining inelastic strain occur (or examples see Re [38] an reerences therein) 4 Local loa transer BM 41 Constitutive behaviour To stuy the eect o stress enhancement aroun aile ibers on the amage evolution an on the macroscopic constitutive behavior we employ local loa transer or the stress reistribution ater iber ailure [2,25 27] Since this case cannot be treate analytically, we perorm numerical simulations in the ry BM moel: ater iber ailure the loa is reistribute on the intact nearest neighbors o the aile iber using perioic bounary conition on the square lattice (see also ig 1) or simplicity, in this case the strength o ibers i has a uniorm istribution between an 1 The algorithm to simulate the loaing process is as ollows: (i) we impose on all the ibers the same loa, equal to the smallest ailure threshol, which results in breaking o the weakest element (ii) The loa carrie by the aile iber is reistribute on the intact nearest neighbors, an the loa o the broken iber is set to zero (iii) Ater the stress reistribution, those ibers whose loa excees their ailure threshol i are ientiie an remove rom the calculation, an the simulation is continue with point (ii) I the coniguration obtaine ater the stress reistribution is stable, the global loa is increase to cause the ailure o one more iber an the simulation is continue with point (ii) This proceure goes on until all ibers are broken The applie stress just beore global ailure is consiere to be the ailure strength o the moel soli Simulations were perorme with system sizes L =16, 32, 64, 128 The constitutive behavior o the local an global loa transer ry BM is compare in igure 5 or clarity, the total orce (instea o stress) is presente as a unction o strain, or several ierent system sizes L Note that N = L 2 is chosen the same or global an local loa sharing simulations In the case o global loa transer the N constitutive law can be obtaine exactly by substituting the cumulative probability istribution P () = o the uniorm istribution into the general orm equation (1) an setting the amage parameter a =: = N(1 P ()) = N(1 ), [, 5], (7) the strain corresponing to macroscopic ailure is c =5 It can be seen in igure 5 that the macroscopic constitutive behavior or local loa transer always coincies with the global BM solution, however, the macroscopic ailure strength is substantially reuce in the local case giving rise to more brittle constitutive behavior It is interesting to note that increasing the system size L the ailure strength o the local BM ecreases, showing the logarithmic size eect also oun in the one-imensional local BM [11] an in two-imensional use networks [37], while the global loa transer case oes not have size epenence To get a eeper unerstaning o the behavior o the system as a unction o the range R o loa reistribution, we perorm simulations by reistributing the loa ater iber ailure on the intact ibers in a square o sie length 2R + 1 centere on the aile iber The range o loa reistribution R is varie between 1 an (L 1)/2 Note that R = 1 correspons to local loa transer on nearest an next-nearest neighbors, while the limiting case o R =(L 1)/2 recovers the ininite range global loa transer The comparison o the constitutive behavior in the local an global loa transer case is presente in igure 6 Simulations reveal that the constitutive laws obtaine at ierent R values always all onto the curve o the global loa transer case an the macroscopic ailure strength increases with increasing range o interaction R approaching the strength o global BM or clarity, in igure 6 we inicate by vertical ashe lines the position o global ailure at ierent values o R To characterize the elastic response o the ry BM moel in a given amage state, we compute the Young moulus Y, eine as the slope o unloaing curves as a unction o the riving stress σ /N: Y (σ) = σ =1 P ((σ)) = 1 (σ) (8) (σ)

6 274 The European Physical Journal B /N Y ig 6 The constitutive law or the global stress transer ry BM (soli line) is compare with the local stress transer BM or ierent interaction ranges R The values o R corresponing to the consecutive vertical ashe lines are 1, 3, 5, 11, 15 rom let to right, an the system size L = 128 was chosen Increasing the interaction range the ailure stress increases approaching the value preicte by the global stress transer moel ig 7 The Young moulus or the global stress transer ry BM is compare with the local stress transer moel or ierent interaction ranges R The values o R corresponing to the vertical ashe lines are the same as in igure 6 a b Using the constitutive law (7) or the global loa transer case, Y can be written into a close orm as a unction o stress Y (σ) = 1 [ ] σ (9) 2 The results on Y or global an local loa transer are shown in igure 7, where the vertical ashe lines inicate the position o macroscopic ailure at ierent values o the reistribution range R It can be seen that at the ailure point Y (σ) has a iscrete jump, the size o which ecreases with increasing R, but it remains inite in the limit o global loa transer Increasing R gives rise to increasing slope o Y (σ) at the ailure point, an in the limit o ininite range interaction the tangent o Y (σ) becomes vertical at the point o ailure 42 Cluster analysis One o the most interesting aspects o the amage mechanism o isorere solis is that the breakown is precee by an intensive precursor activity in the orm o avalanches o microscopic breaking events Uner a given external loa a certain raction o ibers ails immeiately Due to the loa transer rom broken to intact ibers this primary iber breaking may initiate seconary breaking that may also trigger a whole avalanche o breaking I is large enough the avalanche oes not stop an the material ails catastrophically It has been shown by analytic means that in the case o global loa transer the size istribution o avalanches ollows asymptotically a universal power law with an exponent 5/2 [11,32], however, in c ig 8 Snapshots o the amage in the ry BM moel on a square lattice o size L = 128 or ierent values o the loa: a) / c = 153 b) / c = 468, c) / c =796, ) / c =997 the case o local loa transer no universal behavior exists, an the avalanche characteristic size is boune [1, 11] This precursory activity can also be observe experimentally by means o the acoustic emission analysis Acoustic emission measurements have reveale that or a broa variety o isorere materials the response to an increasing external loa takes place in bursts having power law size istribution over a wie range [5 7] In this section we analyze the evolution o amage in local loa transer BM, comparing ry an continuous amage moels Instea o avalanches o iber ailures, we

7 Kun et al: Damage in iber bunle moels n c S n c /L /L ig 9 The number o clusters as a unction o the loa in the local stress transer ry BM or ierent system sizes In the inset the rescale plot is presente, where also the system L = 16 is shown (square) n c n c /L a=95 a=5 a= /L 2 a= ig 1 The number o clusters as a unction o the loa in the local stress transer continuous amage anneale BM or ierent system sizes at a =5 In the inset we show the rescale plot or ierent values o a an L ocus on the properties o clusters o broken ibers which are much less explore In the ollowing simulations we employ a uniorm istribution or the threshols i The loa ater iber ailure is reistribute on the surviving nearest neighbors As the loa is increase, iber breaks an clusters o broken ibers are orme ue to the spatial correlation introuce by the local loa transer These clusters o broken ibers can be ientiie as microcracks orme in the plane perpenicular to the loa irection We monitor the amage evolution by taking snapshots o the clusters at ierent loas In igure 8 the amage evolution is shown in the ry BM One can observe the nucleation an graual growth o clusters with increasing loa We in that the clusters are small compare to the system size even beore global ailure (see ig 8), in S /L 2 ig 11 (a) The average cluster size as a unction o the loa in the local stress transer ry BM or ierent system sizes L an the corresponing rescale plot (b) accorance with the irst-orer transition scenario In the continuous amage BM with local loa transer the clusters o aile ibers are eine as connecte sets o ibers having the same number o ailure, taking into account only nearest neighbor connections In these calculations we set k max =, an the simulations are stoppe when the plastic regime is reache To obtain quantitative inormations on the amage evolution, we measure the cluster probability istribution n(s, ), eine as the number o clusters orme by s neighboring broken ibers when the applie loa is [8] The moments (M k ( ) s k n(s, )s is the kth moment) o n(s, ) contain most o the inormation on the evolution o the amage We etermine n(s, )or ierent system sizes L by averaging over the isorer The total number o clusters n c M as a unction o the loa is presente in igure 9 The increasing part o n c as a unction o is ue to the nucleation o new microcracks, an the short plateau or ecreasing regime a) b)

8 276 The European Physical Journal B a) 6 5 S 3 25 S max S /L 2 a=1 a=5 b) a=95 ig 12 (a) The average cluster size as a unction o the loa in the local stress transer continuous amage anneale BM with a =5 or ierent system sizes L, an (b) the corresponing rescale plot or ierent values o L an a /L 2 ig 13 The size o the largest cluster S max as a unction o /L 2 in the local stress transer ry BM or ierent system sizes L goo collapse (ig 11b) Similar results are obtaine or the continuous amage case, but we see that the rescale curves epen on a (ig 12) The larger a is, the smaller the clusters are, since the stress concentration ecreases with increasing a, an the isorer gets more ominating These results emonstrate that global ailure is initiate once the crack size reaches a critical size s c ater which a crack becomes unstable However, the average cluster size S oes not provie a reliable estimate o s c,which can be obtaine instea monitoring the size o the largest cluster S max as a unction o the loa It can be seen in igure 13 that S max reaches a value that increases with L, but or large L this value seems to saturate The rapi increase o S max close to the ailure point is ue to the coalescence o clusters an is thus prouce by a very small amount o iber ailures in the vicinity o global ailure is cause by the coalescence o growing cracks The inset o igure 9 emonstrates that in ry BM n c obeys a simple scaling law n c = L 2 g(/l 2 ) implying that the clusters are homogeneously scattere through the lattice A similar scaling is observe or the continuous amage anneale BM in igure 1 The inset shows that the scaling unction is inepenent o the amage parameter a Next, we measure the average cluster size eine as S M 2 /M 1 an show that it approaches a value which ecreases with system size (ig 11a) It can be seen that or a given system size the S( ) curves have two regimes: a slowly increasing initial regime ue to the nucleation an growth o clusters, an a rapily increasing part close to global ailure which is cause by the coalescence o growing cracks In ry BM, we can simply rescale the ata accoring to the law S(, L) = s(/l 2 )anobtaina 5 Damage localization: eect o the quenche isorer In the previous section, we analyze the amage structure in the local loa transer moels In the ry BM an in the case o continuous amage BM with anneale isorer we o not expect to in any non trivial amage localization or global loa transer rules, since these moels behave eectively like in mean-iel theory On the other han, quenche isorer can lea to localize structure or continuous amage BM even in global loa transer conitions Weak ibers are expecte to ail more times generating an inhomogeneous amage pattern In orer to analyze this eect, we measure k(i), the number o ailures at iber i, when there is exactly one iber, which has reache k max In the bottom part o igure14weplotthevalueok(i) an the corresponing

9 k() (i) k(i) i ig 14 The number o ailures k o the ibers as a unction o their threshol (istribute uniormly) in the quenche global loa transer continuous BM (top) The parameters use are k max = 1, N = 5 an a =4, 8, 9, 95 The threshol (i) as a unction o i is compare with the amage k(i) or a =9 (bottom) value o the threshol (i) The amage k(i) shows a very irregular pattern, which shoul be compare with a roughly uniorm structure expecte or anneale isorer In the upper part o the igure we isplay the ecay o k as a unction o, showing how weak ibers break more oten than strong ones The ecay is more pronounce when a is close to one, an becomes less important or smaller a It is straightorwar to obtain an analytic expression or k() which is compare with the numerical results in igure 15 inally, we expect that quenche isorer shoul have an eect also on the cluster structure o local loa transer moels In orer to conirm this point, we compare the number o clusters n c ( ) an the average cluster size S or quenche an anneale isorer The results, shown in Kun et al: Damage in iber bunle moels 277 k() a=95 a=9 a= ig 15 The number o ailures k o the ibers as a unction o their threshol (istribute accoring to the Weibull istribution with m =15) in the quenche global loa transer continuous BM: comparison between simulations an analytic results or a =95, 9, 4 n c /L anneale /L 2 quenche ig 16 The number o clusters as a unction o the rescale loa /L 2 in the local stress transer continuous amage quenche an anneale BM or ierent system sizes at a =5 Note that the number o clusters increases aster or anneale isorer inicating a smaller egree localization igures 16 an 17, inicate that amage is more localize when the isorer is quenche 6 Conclusions We have propose a continuous amage version o the BM, that can be use to moel a wie variety o constitutive behaviors We have analyze the evelopment o amage in BM uner ierent conitions an compare the local loa transer moel with ierent interaction

10 278 The European Physical Journal B S anneale ranges with the global loa transer moel that can be solve exactly in the limit N rom the theoretical point o view, the cluster analysis shows the analogies between the ailure in the BM an nucleation in irst-orer phase transition The ailure point in global loa transer BM plays the role o a spinoal point The strain carrie by the ibers, proportional to the raction o intact ibers [8], close to the ailure loa c has a iverging erivative / ( c ) 1/2 A similar behavior is observe close to a spinoal instability in irst-orer phase transitions One shoul note that the spinoal point an its associate scaling is a mean-iel property, obtaine in the limit N, an in general is not observe or inite imensional shortrange moels where nucleation occurs much beore reaching the spinoal Similarly, in local-loa transer (ie short range) BM ailure occurs much beore the corresponing global loa sharing instability (ie the spinoal) an the avalanche characteristic size is boune [1, 11, 32] We have shown that increasing the range o interaction the ailure point is shite towars the spinoal A similar behavior is observe or instance in Ising systems when the range o interactions is increase [39] In moels with long range stress transer, as or instance in elastic or electric networks, it is possible to observe the spinoal scaling even in inite imensional systems [8, 11] The presence o a spinoal instability coul explain the observation o scaling properties in acoustic emission experiments [5 7] Our continuous amage BM can reprouce a wie variety o elasto-plastic constitutive behaviors A remarkable eature o the moel is that multiple ailure o the elements results in uctile macroscopic behavior in spite o the brittleness o the constituents Similar uctile behavior has been observe experimentally in iber reinorce composites mae o brittle constituents [42 44] Experiments reveale that the mechanism o this uctility, which is calle pseuo-strain harening, is the multiple ailure o the material [42 44] Our continuous amage moel, recovering as special cases the ry bunle moel an micromechanical moels known in the literature, coul provie a general ramework or the statistical-micromechanical moeling o the behavior o iber reinorce composites The itting o experimental results in the ramework o our moel will be presente in a orthcoming publication /L 2 quenche ig 17 The average cluster size as a unction o the rescale loa /L 2 in the local stress transer continuous amage quenche an anneale BM or ierent system sizes at a = 5 Note that the cluster size increases aster or quenche isorer inicating a larger egree localization This work was supporte by the project SB381 K acknowleges inancial support rom the Alexaner von Humbolt ounation (Roman Herzog ellowship) K is grateul to I Sajtos or the valuable iscussions SZ acknowleges inancial support rom EC TMR Research Network uner contract ERBMRXCT9662 Reerences 1 A Buchel, JP Sethna, Phys Rev Lett 77, 152 (1996); Phys Rev E 55, 7669 (1997) 2 JB Runle, W Klein Phys Rev Lett 63, 171 (1989) 3 RLB Selinger, Z-G Wang, WM Gelbart, A Ben-Saul, Phys Rev A 43, 4396 (1991); Z-G Wang, U Lanman, RLB Selinger, WM Gelbart, Phys Rev B (1991) 4 RLB Selinger, Z-G Wang, WM Gelbart, J Chem Phys 95, 9128 (1991) 5 A Garcimartín, A Guarino, L Bellon, S Ciliberto, Phys Rev Lett 79, 322 (1997); A Guarino, A Garcimartín, S Ciliberto, Eur Phys J B 6, 13 (1998) 6 C Maes, A van Moaert, H reerix, H Strauven, Phys Rev B 57, 4987 (1998) 7 A Petri, G Paparo, A Vespignani, A Alippi, M Costantini, Phys Rev Lett 73, 3423 (1994) 8 S Zapperi, P Ray, HE Stanley, A Vespignani, Phys Rev Lett 78, 148 (1997); Phys Rev E 59, 549 (1999) 9 C Unger, W Klein, Phys Rev B 29, 2698 (1984); ibiem 31, 6127 (1985) or a review on spinoal nucleation see L Monette, Int J Mo Phys B 8, 1417 (1994) 1 MKloster,AHansen,PCHemmer,PhysRevE56, 2615 (1997) 11 A Hansen, PC Hemmer, Phys Lett A 184, 394 (1994) 12 Statistical moels or the racture o isorere meia, eite by HJ Herrmann, S Roux (North-Hollan, Amsteram, 199) 13 S Zapperi, A Vespignani, HE Stanley, Nature (Lonon) 388, 658 (1997) 14 HE Daniels, Proc R Soc Lonon A 183, 45 (1945) 15 BD Coleman, J Appl Phys 29, 968 (1958) 16 D Krajcinovic, MAG Silva, Int J Solis Struct 18, 551 (1982) 17 D Sornette, J Phys A 22, L243 (1989) 18 D Sornette, J Phys rance 5, 745 (1989) 19 C Moukarzel, PM Duxbury, J Appl Phys 76, 1 (1994) 2 DG Harlow, SL Phoenix, J Composite Mater 12, 195 (1978)

11 Kun et al: Damage in iber bunle moels RL Smith, SL Phoenix, J Appl Mech 48, 75 (1981); DG Harlow, SL Phoenix, J Mech Phys Solis 39, 173 (1991) 22 SL Phoenix, M Ibnabeljalil, C-Y Hui, Int J Solis Struct 34, 545 (1997) 23 SL Phoenix, Raj, Acta Metall Mater 4, 2813 (1992) 24 IJ Beyerlein, SL Phoenix, J Mech Phys Solis 44, 1997 (1996) 25 WA Curtin, J Am Ceram Soc 74, 2837 (1991) 26 WA Curtin, J Mech Phys Solis 41, 217 (1993) 27 SJ Zhou, WA Curtin, Acta Metal Mater 43, 393 (1995) 28 Hil, A Burr, A Leckie, Eur J Mech A 13, 731 (1994) 29 WA Curtin, N Takea, J Comp Matls 32, 242 (1998) 3 WA Curtin, J Am Ceram Soc 74, 2837 (1991) 31 WA Curtin, Phys Rev Lett 8, 1445 (1998) 32 PC Hemmer, A Hansen, J Appl Mech 59, 99 (1992) 33 WI Newman, AM Gabrielov, TA Duran, SL Phoenix, DL Turcotte, Physica D 77, 2 (1994) 34 WI Newman, DL Turcotte, AM Gabrielov, Phys Rev E 52, 4827 (1995) 35 Y Moreno, JB Gomez, A Pacheco, Physica A 274, 4 (1999) 36 R a Silveira, Phys Rev Lett 8, 3157 (1998) 37 P Duxbury, PD Beale, PL Leath, Phys Rev Lett 57, 152 (1986) 38 Kun, HJ Herrmann, appearing in J Mater Sci 39 D Heerman, W Klein, D Stauer, Phys Rev Lett (1982); T Ray, W Klein, J Stat Phys 61, 891 (199) 4 AG Evans, W Zok, J Mater Sci 29, 3857 (1994) 41 SM Spearing, W Zok, J Eng Mater Technol 115, 314 (1993) 42 AE Naaman, HW Reinhart, High perormance iber reinorce cement composites (E & N Spon, Lonon, 1995) 43 AG Evans, JM Domergue, E Vagaggini, J Am Ceramic Soc 77, 1425 (1994) 44 T Kanaa, VC Li, J Eng Mech 125, 29 (1999)