Record statistics of bursts signals the onset of acceleration towards failure
aa r X i v : . [ c ond - m a t . d i s - nn ] O c t Record statistics of bursts signals the onset ofacceleration towards failure
Vikt ´oria K ´ad ´ar , Gerg ˝o P ´al , and Ferenc Kun Department of Theoretical Physics, Doctoral School of Physics, Faculty of Science and Technology, University ofDebrecen, H-4010 Debrecen, P.O.Box: 5, Hungary Institute of Nuclear Research (Atomki), H-4026 Debrecen, Poroszlay ´ut 6/c, Hungary * [email protected] ABSTRACT
Forecasting the imminent catastrophic failure has a high importance for a large variety of systems from the collapse of engi-neering constructions, through the emergence of landslides and earthquakes, to volcanic eruptions. Failure forecast methodspredict the lifetime of the system based on the time-to-failure power law of observables describing the final acceleration to-wards failure. We show that the statistics of records of the event series of breaking bursts, accompanying the failure process,provides a powerful tool to detect the onset of acceleration, as an early warning of the impending catastrophe. We focuson the fracture of heterogeneous materials using a fiber bundle model, which exhibits transitions between perfectly brittle,quasi-brittle, and ductile behaviors as the amount of disorder is increased. Analyzing the lifetime of record size bursts, wedemonstrate that the acceleration starts at a characteristic record rank, below which record breaking slows down due to thedominance of disorder in fracturing, while above it stress redistribution gives rise to an enhanced triggering of bursts andacceleration of the dynamics. The emergence of this signal depends on the degree of disorder making both highly brittlefracture of low disorder materials, and ductile fracture of strongly disordered ones, unpredictable.
Introduction
Forecasting failure is a long standing problem which has an utmost importance to mitigate the consequences of the col-lapse of engineering constructions and of natural catastrophes like landslides, earthquakes, volcanic eruptions, rock and snowavalanches . Fracture processes of heterogeneous materials occurring under constant or slowly varying external loads playa decisive role for the emergence of catastrophic failures. The micro and meso-scale heterogeneity of materials has the con-sequence that their fracture process is accompanied by crackling noise, i.e. fracture proceeds in intermittent bursts of localbreakings which generate acoustic emissions . Cracking bursts can be considered as precursors of the ultimate failure ofthe system, so that they can be exploited to forecast the impending catastrophic event . After a longer period of steadyevolution, failure is approached through an acceleration of the dynamics which is indicated by the increasing rate of defor-mation, and acoustic or seismic signals . Failure forecast methods (FFM) rely on the analogy of failureand critical phenomena which implies time-to-failure power laws of observables in the acceleration regime making possibleto predict the lifetime of the system .Disorder is an inherent property of natural and most of the artificially made materials. Depending on the relevant lengthscale, it appears in the form of dislocations, microcracks, flaws, grain boundaries, which affect the nucleation and propaga-tion cracks. Experimental and theoretical studies have revealed that the intensity of the precursory activity, and hence, thepredictability of failure, depends on the degree of materials’ disorder. In the limiting case of zero disorder, the ultimate failureoccurs in an abrupt way with hardly any precursors . However, higher disorder makes possible to arrest propagating cracksgiving rise to a gradual accumulation of damage with a growing rate of breaking bursts as failure is approached . Thiseffect has recently been precisely quantified by experiments performed on the compressive failure of porous glass sampleswhere the degree of heterogeneity could be well controlled during the sample preparation . It has been demonstrated thatthe accuracy of the lifetime prediction of FFM rapidly improves with increasing mesoscale heterogeneity of the material underlining the key importance of heterogeneity for forecasting failure.Instead of the failure time, here we focus on the onset of acceleration which marks the start of the critical regime of theevolution of the fracture process, and hence, can serve as an early warning of the imminent failure, similar to the entropychange of seismicity under time reversal suggested earlier for earthquakes . To generate fracture processes of hetero-geneous materials we use a fiber bundle model , which has the advantage that varying the amount of microscale disorder, itexhibits transitions between distinct phases of perfectly brittle, quasi-brittle, and ductile fracture. To quantify how the degreeof disorder determines the predictability of failure, we investigate the internal structure of the sequence of breaking bursts m a x / m i n BrittleQuasi-brittle max c Figure 1.
Phase diagram of the model on the µ − ε max plane (for derivation see SI1). Increasing the amount of disorder thesystem undergoes brittle, quasi-brittle, or ductile fracture. The curve of ε cmax ( µ ) gives the phase boundary between perfectbrittleness and quasi-brittle fracture, while ductility is obtained in the limit ε max → + ∞ . (GLE 4.2.5,URL:http://glx.sourceforge.net/)analyzing the record size events. Records are bursts of size greater than any previous crackling event of the fracture processso that they can easily be identified by experimental techniques both in laboratory and in field measurements above the noisybackground. The record statistics (RS) of stochastic time series has attracted a great attention due to its relevance for weather,climate and earthquake research . The RS analysis has proven very successful to reveal trends, correlations, and spatio-temporal clustering of events in complex evolving systems . For fracture processes, here we demonstrate that the waitingtime between consecutive record breakings, i.e. the lifetime of records, is very sensitive to the details of the fracture processproviding a clear signal of the acceleration of the dynamics towards ultimate failure. In particular, we show the existence of acharacteristic record rank k ∗ which marks the onset of acceleration of record breaking: below k ∗ record breaking slows downdue to the dominance of disorder in the fracture process, while above it the stress redistribution gives rise to an enhancedtriggering of bursts after breaking events. Detecting k ∗ can be exploited as an early signal of the imminent ultimate failureof the system, however, the significance of the accelerating regime strongly depends on the degree of disorder. Most notably,we show that the highly brittle fracture of low disorder materials and the ductile failure of strongly disordered ones are bothunpredictable due to the absence of accelerated record breaking. Our results imply the existence of a lower and upper boundof the amount of materials’ disorder beyond which no meaningful failure prediction is possible. Results
Disorder driven transition between brittle, quasi-brittle, and ductile fracture
To investigate the evolution of fracture processes leading to ultimate failure, we use a generic fiber bundle model (FBM) whichhas proven successful in reproducing both the constitutive response and the intermittent bursting dynamics of heterogeneousmaterials on the macro- and microscales, respectively . The model is composed of N parallel fibers, similar to a rope,which have a linearly elastic behavior. Materials’ disorder is represented by the random strength ε ith ( i = , . . . , N ) of fiberswith a fat-tailed probability distribution, i.e. a power law distribution is considered p ( ε th ) = D ε − ( + µ ) th , (1)over the range ε min ≤ ε th ≤ ε max . In the calculations the lower cutoff is fixed to ε min = ε max and the exponent µ of the distribution Eq. (1), which control the range of variabilityof fibers’s strength and the decay of weaker against stronger fibers, respectively. The cutoff strength ε max takes values inthe range ε min < ε max ≤ + ∞ , while the disorder exponent µ is selected in the interval 0 ≤ µ <
1. At these µ values, in thelimiting case of an infinite upper cutoff ε max → + ∞ the disorder is so high in the system that no finite average fiber strengthexists. Hence, varying the control parameters ε max and µ of the model, the degree of disorder can be tuned between theextremes of zero and infinity. Computer simulations were performed by slowly increasing the external load, which generatesan intermittent dynamics where fibers break in bursts analogous to acoustic outbreaks in real experiments (see Methods for igure 2. Event sequence of bursts, i.e. the size of bursts ∆ is presented as a function of their order number n for two valuesof the cutoff strength ( a ) ε max = ( b ) ε max = + ∞ at the disorder exponent µ = . N = fibers. The yellow line indicates the moving average of burst sizes h ∆ i calculated over 50 consecutive events, while the redbars highlight record size events of the series. The definition of the record size ∆ kr and the waiting time m k between records isalso illustrated. (GLE 4.2.5, URL:http://glx.sourceforge.net/)further details of the model construction). The size ∆ of bursts is determined as the number of fibers breaking in the correlatedtrail of avalanches.It is a crucial feature of our FBM that varying the amount of disorder transitions occur between distinct phases of perfectlybrittle, quasi-brittle, and ductile behaviors with qualitative differences in the macroscopic response and in the precursorybursting activity. This enables us to quantify how the degree of disorder affects the details of burst sequences and the fore-castability of failure in different types of fracture. The phase diagram in Fig. 1 delineates the overall behavior of the systemon the µ − ε max plane based on analytical calculations (see Supplementary information (SI1) for details). The figure demon-strates that at any value of the disorder exponent in the range 0 ≤ µ <
1, for a sufficiently low cutoff strength ε max of fibers ε max < ε cmax ( µ ) , the bundle exhibits a perfectly brittle response, i.e. a linearly elastic behavior is obtained up to the breaking ofthe weakest fiber which then triggers a catastrophic burst leading to an immediate abrupt failure. Above the phase boundary ε cmax ( µ ) a quasi-brittle phase emerges, where global failure is approached through intermittent breaking avalanches. A repre-sentative example of the sequence of bursts up to failure can be seen in Fig. 2 ( a ) , where the size of bursts ∆ is presented asa function of their order number n , similarly to the natural time analysis of time series . The size of bursts ∆ fluctuatesdue to the disorder of fibers’ strength, however, the evolution of the overall structure of the sequence can be inferred: at thebeginning of the fracture process the moving average of the burst size h ∆ i remains nearly constant which shows the highdegree of stationarity of the crackling activity over a broad range. However, close to failure the rapidly increasing averageburst size h ∆ i and the growing fluctuations of ∆ indicate the acceleration of the fracture process. This generic trend of thecracking sequence is also confirmed by the behavior of the average number a of fiber breakings triggered immediately afterthe breaking of a single fiber at the strain ε (for derivation see SI2) a ( ε ) = µ − (cid:16) εε max (cid:17) µ . (2)The expression of a ( ε ) has to be evaluated over the range ε min ≤ ε ≤ ε c , where ε c denotes the critical strain ε c = ε max / ( − µ ) / µ , where failure occurs. As the system approaches the critical point ε c , the value of a monotonically increases to 1indicating the acceleration of the fracture process and the instability emerging at the critical point. Ultimate failure occursin the form of a catastrophic avalanche a ≥
1, where all the remaining intact fibers break in one event. Varying the control igure 3.
Average size (cid:10) ∆ kr (cid:11) ( a ) and lifetime h m k i ( b ) of records as a function of their rank k for several values of thedisorder exponent µ at an infinite cutoff strength ε = + ∞ . ( c ) The average number of records h N n i that occurred until n bursts are generated during the fracture process. The dashed line represents a logarithmic function. (GLE 4.2.5,URL:http://glx.sourceforge.net/)parameters ε max and µ inside the quasi-brittle phase, the qualitative structure of the event series remains the same, however,the extension of the accelerating regime and the magnitude of acceleration, which are essential features for forecasting, arechanging. The overall behaviour of the sequence of crackling events of our model has a nice qualitative agreement withthe accelerating acoustic and seismic activity accompanying the creep rupture and compressive failure ofvarious types of disordered materials, and the failure phenomena of geosystems such as volcanic eruptions , cliff collapse ,breakoff of hanging glaciers , and landslides .Our system has the remarkable feature that in the limit of very high disorder ε max → + ∞ , the acceleration of the dynamicsdisappears a = µ , and the entire series of crackling events remains stationary until the last avalanche. Figure 2 ( b ) illustratesthat the stability of the failure process is retained until the end, and no catastrophic avalanche emerges, hence, this type offracture process is considered to be ductile in the model. The underlying mechanism is that due to the slow decay of thefat-tailed distribution of fibers’ strength Eq. (1), there are sufficiently strong fibers in the bundle which can always stabilizethe fracture process. Note that for µ > ε max (seeFig. 1).This powerful model allows us to unveil how the changing degree of disorder affects the predictability of the ultimatefailure of the system. In the limiting case of perfectly brittle fracture, the system collapses when its load reaches the strengthof the weakest fiber. Since it is not known a priori, the failure point has a great uncertainty. In the ductile phase of highdisorder, the stationary bursting activity does not provide any hint of the imminent failure. Tuning the amount of disorderby varying µ and ε max , we can drive the system between these two limits giving a precise quantitative characterization of thestrength of acceleration of the precursory activity, and hence, the forecastability of failure. Instead of the time of ultimatefailure, we focus on the onset of acceleration, analyzing the statistics of record size events of the crackling sequence. Approaching failure through record breaking bursts
A record of crackling noise is a burst which has a size ∆ r greater than any previous event, similarly to sports like athletics,where records are the best results of a discipline . Assuming that the first burst is a record, record breaking (RB) events forma monotonically increasing sub-sequence during the fracture process, as it is illustrated in Fig. 2. RB events are identified bytheir rank k = , , . . . , which occurred as the n k th event of the complete sequence with size ∆ kr . As fracture proceeds, recordsget broken after a certain number of bursts giving rise to new records. The number of events, one has to wait to break the k threcord, defines the waiting time m k m k = n k + − n k , (3)which can also be considered as the lifetime of the record. The definition of the record characteristics ∆ kr and m k is illustratedin Fig. 2 ( a ) . It can be observed that in the accelerating regime of the quasi-brittle fracture process in Fig. 2 ( a ) , records rapidlyfollow each other reaching the total number N totn =
22, while in the stationary burst sequence of ductile failure in Fig. 2 ( b ) the value of N totn remains significantly lower ( N totn = m k in Fig. 2 ( a ) . igure 4. Probability distribution of the lifetime p ( m ) ( a ) and size p ( ∆ r ) ( c ) of records for several values of the disorderexponent µ at ε = + ∞ . Rescaling the distributions with proper powers α and β of the distance from the critical point 1 − µ ,distributions obtained at different µ values can be collapsed on a master curve. In ( b ) and ( d ) the straight lines representpower laws of exponent z = τ r =
1, respectively. The legend is presented in ( b ) for all the figures. (GLE 4.2.5,URL:http://glx.sourceforge.net/)However, in the stationary regime of the beginning of the fracture process, record breaking slows down with increasing valuesof m k , similarly to ductile fracture in Fig. 2 ( b ) . To quantify these important trends and correlations in burst sequences, weanalyze the statistics of the size and lifetime of records, and their evolution with the record rank as the system approachesfailure at different degrees of disorder. Record statistics in the limit of high disorder
Figure 3 demonstrates that in ductile fracture ε max = + ∞ the statistics of records is entirely consistent with the behavior ofsequences of independent identically distributed (IID) random variables : The average record size (cid:10) ∆ kr (cid:11) is of course amonotonically increasing function of the record rank k = , , . . . for all values of the disorder exponent µ (Fig. 3 ( a ) ). Theaverage lifetime of records h m k i also increases with k , since it gets more and more difficult to break the growing records of astationary sequence (Fig. 3 ( b ) ). It follows that as fracture proceeds, the number of records N n slowly increases with the totalnumber of bursts n . For IID sequences a universal logarithmic dependence has been derived for the average number of records h N n i that occurred until the n th event of the sequence . Figure 3 ( c ) shows that the IID result perfectly describes the recordbreaking process of ductile fracture h N n i ≈ A + B ln n , (4)with the additional feature that the multiplication factor B of the logarithmic term depends on µ . Reducing the amount ofdisorder in the ductile phase by increasing the exponent µ towards the critical point of perfectly brittle failure µ → µ c ( ε cmax =+ ∞ ) =
1, the qualitative behavior of the curves in Fig. 3 remains the same, however, record breaking accelerates: the asymp-totic value of the record size in Fig. 3 ( a ) rapidly increases, while the asymptotic record lifetime in Fig. 3 ( b ) tends to zeroas µ approaches 1. We quantified this behavior by calculating the average value of the largest record size h ∆ maxr i and largestwaiting time h m max i that occurred up to failure as function of µ . Both quantities proved to have a power law dependence onthe distance from the critical point h ∆ maxr i ∼ ( µ c − µ ) − α , h m max i ∼ ( µ c − µ ) β , (5)with the critical exponents α = . ± .
05 and β = . ± .
05 (for figure see SI3). The acceleration of record breaking hasalso the consequence that the number of records h N n i grows faster with the event number n for higher µ , i.e. the multiplication igure 5. ( a ) Average lifetime of records h m k i as a function of the record rank k for several values of the cutoff strength λ . ( b ) Average number of records h N n i formed during the fracture process as a function of the event number n varying λ in abroad range. The value of the disorder exponent is fixed to µ = .
7. (GLE 4.2.5, URL:http://glx.sourceforge.net/)factor B in Eq. (4) increases to 1 as µ approaches µ c = m of records. The probability distribution p ( m ) of the record lifetime m has a power law functional form p ( m ) ∼ m − z , (6)with an exponent z , which has a universal value z = (see Fig. 4 ( a ) ). For the size distributionof records p ( ∆ r ) a non-universal behavior is expected in the sense that p ( ∆ r ) depends on the underlying distribution of burstsizes . In our system a power law distribution is obtained p ( ∆ r ) ∼ ∆ − τ r r , (7)with an exponent τ r =
1, which does not depend on the value of µ (see Fig. 4 ( c ) ). It is important to emphasize that approachingthe critical point of brittle failure the cutoff of the size distributions p ( ∆ r ) diverges, while the one of the lifetime distribution p ( m ) tends to zero, in agreement with the behavior of the high rank limit of the average size and lifetime of records in Figs.3 ( a , b ) . Figures 4 ( b , d ) demonstrate that rescaling the distributions p ( ∆ r ) and p ( m ) according to the scaling laws Eq. (5), thecurves obtained at different µ exponents can be collapsed on the top of each other. The good quality data collapse confirmsthe consistency of the results. The excellent agreement of the record statistics of the burst sequence of ductile fracture withthe behavior of IID sequences implies that, in spite of the increasing external load on the bundle, the entire fracture process iscontrolled by the microscale disorder of the system. Approaching failure through accelerated record breaking
Inside the quasi-brittle phase, when the amount of disorder is reduced by the finite cutoff strength ε max of fibers, the evolutionof burst sequences substantially changes since the initial stationary regime is followed by acceleration in the vicinity of failure(see Fig. 2 ( a ) ). To facilitate the comparison of results obtained varying the cutoff strength ε max at different exponents µ , weintroduce the parameter λ = ( ε max − ε cmax ) / ε cmax , which characterizes the relative distance λ > ε cmax ( µ ) at a given value of µ . In order to determine how the precise amount of disorder controls the onset andsignificance of acceleration, and hence, the forecastability of ultimate failure, we performed computer simulations varying thevalue of λ and the disorder exponent µ in broad ranges 0 . ≤ λ ≤ + ∞ and 0 . ≤ µ ≤
1, respectively. At each parameterset averages were calculated over 6000 samples (for details of averaging see Methods).Of course, the average size of records (cid:10) ∆ kr (cid:11) has the same monotonically increasing trend with the record rank k as inthe ductile phase for all values of λ and µ (see SI4 for figure). However, the average lifetime of records h m k i exhibits anastonishing behavior: As a representative example, Fig. 5 ( a ) demonstrates for µ = . λ > . h m k i curves develop a maximum at a characteristic record rank k ∗ . For low rank records k < k ∗ the RB processslows down due to the stationarity of the beginning of the fracture process, while beyond the maximum k > k ∗ , the decreasinglifetime indicates that the approach to failure is accompanied by an accelerated record breaking. Note that as the amount ofdisorder grows with increasing λ , the position of the maximum k ∗ slightly shifts to higher values, while the maximum getsgradually less pronounced and eventually disappears when approaching ductile fracture. Surprisingly, in the opposite limit igure 6. ( a ) Size distribution of record events p ( ∆ r ) for several values of λ . The straight line represents a power law ofexponent τ r = ( b ) Probability distribution of the lifetime of records p ( m ) for cutoff strengths of fibers λ where the averagewaiting time in Fig. 5 ( b ) solely exhibits slowdown. The straight line represents a power law of exponent z = ( c ) Lifetimedistributions p ( m ) for intermediate λ values where the RB process accelerates prior to failure. The two straight linesrepresent power laws of exponent z a = . z a = .
15. The disorder exponent µ has the same value µ = . λ → h m k i tends to a monotonically increasing formshowing the slowdown of record breaking, in spite of the rapid collapse of the highly brittle system.The acceleration of the RB process results in a rapid increase of the number N n of records close to failure. Figure 5 ( b ) shows that at early stages of the fracture process the average record number h N n i increases logarithmically with the numberof bursts n in agreement with Eq. (4), however, at a characteristic event index n k ∗ , corresponding approximately to the rank k ∗ of maximum lifetime, the functional form of h N n i changes to a rapid increase. It is important to emphasize that varying theamount of disorder by λ , as the accelerating regime diminishes in the limits of highly brittle ( λ →
0) and ductile ( λ → + ∞ )fracture, the deviations from the generic logarithmic trend of IIDs gradually disappear apart from some fluctuations.Also the overall statistics of record sizes and lifetimes is sensitive to the degree of disorder. Figure 6 ( a ) shows that closeto the phase boundary of perfect brittleness λ → p ( ∆ r ) is identical with the correspondingdistribution of ductile fracture (see Fig. 4 ( c ) for comparison), i.e. a power law distribution Eq. (7) is obtained with a universalexponent τ r =
1. Further from the phase boundary, the change in the dynamics of fracture from a steady evolution to accelera-tion gives rise to a crossover of p ( ∆ r ) between two regimes of different functional forms: for small records, generated duringearly stages of the fracture process, the distribution remains similar to its ductile counterpart, however, beyond a characteristicrecord size a steeper power law is formed. Simulations revealed that the crossover point practically coincides with the averagerecord size (cid:10) ∆ kr (cid:11) at the rank k ∗ of the maximum lifetime in Fig. 5 ( a ) . The underlying mechanism of the emergence of thecrossover of the distribution of record sizes is the crossover of the entire burst size distribution , similar to the so-calledb-value anomaly observed for event magnitude distributions in geosystems .The lifetime distribution p ( m ) shows the same qualitative behavior as λ is varied. For clarity, the distributions p ( m ) arepresented separately for the parameter ranges where the acceleration of record breaking is absent and present, in Fig. 6 ( b ) and ( c ) , respectively. At low and high disorder, where no acceleration occurs, the waiting time distributions are consistent with theIID behavior Eq. (6) (Fig. 6 ( b ) ). For intermediate λ , when accelerated record breaking emerges, p ( m ) exhibits a crossoverbetween two power laws of different exponents (Fig. 6 ( c ) ). For short lifetimes, typical for the vicinity of failure, the exponent z a is smaller z a = .
7, while, for large waiting times characteristic for the initial slowdown of the process, the exponent ishigher z a = .
15 than the IID result z =
1. It can be observed in Figs. 6 ( b , c ) that the particular value of λ inside the tworegimes only affects the crossover point and the cutoff of the distributions.Simulations revealed that varying the disorder exponent µ the qualitative behaviour of the average waiting time h m k i and event number h N n i , and of the distributions of record sizes p ( ∆ r ) and lifetimes p ( m ) , remains the same, they undergoonly quantitative changes what we analyze in the next section. The results demonstrate that the statistics of records is verysensitive to the details of the crackling sequence providing a powerful tool to identify the onset of acceleration towards failure.Additionally, comparing the results to IID event sequences, our analysis proves that before acceleration the precursory burstingactivity is dominated by the microscale disorder giving rise to stationarity. Acceleration starts when the stress enhancementsgenerated by the redistribution of load after breaking bursts, can enhance the triggering of further avalanches. < k > < m k * > , < m k m a x > -3 -1
10 10 a)b) Figure 7. ( a ) Average value of the difference h δ k i of the highest record rank k max and the position of the maximum k ∗ ofthe record lifetime as a function of λ for several values of the disorder exponent µ . ( b ) The average of the maximum lifetime h m k ∗ i (open symbols) and the lifetime of the last record h m k max i (filled symbols) as a function of λ . For clarity, pair of curvesare presented only for four values of the disorder exponent µ . The vertical dashed lines indicate the λ window ofacceleration for µ = .
9. (GLE 4.2.5, URL:http://glx.sourceforge.net/)
The effect of disorder on the onset of acceleration
The presence of a sufficiently broad accelerating regime in the series of crackling events is of ultimate importance to obtainan early warning of the imminent failure and to forecast the final collapse of the evolving system with a good precision. Theposition of the maximum k ∗ of the average record lifetime h m k i and the corresponding event index n k ∗ provide an excellentsignal of the start of acceleration towards failure after a longer period of stationary evolution. However, it can be observedin Fig. 5 ( a ) that both the extension of the accelerating regime and the magnitude of acceleration depend on the degree ofdisorder in the system. To assess the predictability of the ultimate failure, the significance of the acceleration regime has to becharacterized.To quantify the extension of the accelerating regime we determined the difference δ k of the highest record rank k max obtained up to failure and the position of the maximum k ∗ of the record lifetimes in single samples. The average of thisquantity h δ k i = h k max − k ∗ i (8)over a large number of simulations is presented in Fig. 7 ( a ) as a function of λ for several values of the disorder exponent µ .For each value of µ a well defined range of the cutoff strength λ can be identified where a significant difference h δ k i > k ∗ and k max . Both for a high degree of brittleness λ → λ → + ∞ , the value of h δ k i is practically zero, which shows that no acceleration can be detected. Increasing the amount of disorder by decreasingthe exponent µ , the acceleration regime h δ k i > λ .The magnitude of acceleration of the RB process can be characterized by comparing the last record lifetime m k max andthe maximum lifetime of records m k ∗ . Figure 7 ( b ) presents the average values h m k max i and h m k ∗ i , using the same scale of λ on the horizontal axis as in Fig. 7 ( a ) to facilitate the comparison with the behavior of h δ k i . In the vicinity of µ = h m k max i and h m k ∗ i practically coincide for all values of the cutoff strength λ . However, lowering the exponent µ , abroader and broader range of λ emerges where a significant difference h m k ∗ i ≫ h m k max i is obtained, indicating the presenceof a dominating maximum of the record lifetime. The λ window of h δ k i > h m k ∗ i ≫ h m k max i verifying m k k k -3 -1
10 10 a) b) Figure 8. ( a ) The lifetime of consecutive records m k , obtained during the evolution of a single system, as a function of theirrank k for several values of λ at the same disorder exponent µ = . ( a ) for the sample averaged curves. ( b ) Thedifference δ k of the highest record rank k max and the position of the maximum k ∗ of the record lifetime for a single sample asa function of λ for several values of the disorder exponent µ . (GLE 4.2.5, URL:http://glx.sourceforge.net/)the existence of a well defined range of disorder where failure can be foreseen. Outside this window the degree of disorder iseither too low or too high so that no signal of the imminent failure can be identified. This λ window of significant accelerationis highlighted by the vertical dashed lines in Figs. 7 ( a , b ) for µ = . ( a ) presents the lifetime of consecutive records m k as a functionof their rank k obtained by the analysis of a single system for several values of λ at the same disorder exponent µ = . ( a ) . It is important to emphasize that apart from fluctuations records of the single sample exhibit the same overallbehaviour as the sample averaged curves in Fig. 5 ( a ) , i.e. m k has a well defined maximum in the quasi brittle regime of thefracture process so that the value of the record rank k ∗ and the corresponding event index n k ∗ can be obtained in a reliableway. This result is further supported by the behaviour of δ k in Fig. 8 ( b ) which shows that the window of forecastability ofa single system, where a significant accelerating regime can be detected by the method of record statistics, agrees very wellwith the sample averaged result of Fig. 7 ( a ) . Simulations showed that the relative value of sample-to-sample fluctuations ofrecord quantities vary between 0.05 and 0.2, typically increasing with the record rank k , making the method efficient for singlesamples undergoing quasi-brittle fracture. Discussion
Catastrophic failure of engineering constructions and of geosystems is often caused by the fracture of disordered materials.Under a constant or slowly varying external load, the inherent disorder of materials gives rise to a jerky evolution of thefracture process accompanied by a sequence of acoustic outbreaks. Methods of failure forecasting predict the lifetime of theevolving system by exploiting the power law acceleration of the precursory crackling activity prior to ultimate failure. Herewe focused on the acceleration preceding the final collapse, and proposed a method to identify the onset of this critical regimeof the dynamics which can be used as an early warning of the imminent failure. Based on a fiber bundle model of fracturephenomena, we studied the statistics of record size events to reveal how the sequence of breaking bursts evolves as the systemapproaches failure. We demonstrated that the small subset of record bursts grasps essential features of the dynamics leadingto ultimate failure, with the additional advantage that they are relatively easy to identify even experimentally over the noisybackground.As an important outcome of the work, we showed that during quasi-brittle fracture, where failure proceeds through anintense precursory activity, the acceleration of the crackling sequence towards failure is accompanied by an accelerated recordbreaking. The onset of acceleration can be identified by the record which has the longest lifetime so that its rank k ∗ , andthe corresponding event index n k ∗ provide a reliable signal in the burst sequence where the critical regime of the dynamicsstarts. Before this characteristic record, the process of record breaking slows down and the statistics of records proved to beequivalent with the behavior of event sequences of IIDs. These results imply that the beginning of fracture is dominated bythe disorder of the material in spite of the increasing external load and decreasing load bearing capacity of the system. Beyond k ∗ , acceleration is caused by the enhanced triggering of bursts following the stress redistribution after breaking events.Early warning and forecastability of the imminent failure requires a sufficiently broad critical regime with a considerablemagnitude of acceleration. To asses the effect of disorder on the forecastability of failure, we made a quantitative charac-terization of the significance of acceleration in terms of record statistics. Most notably, we showed that the highly brittle igure 9. Three-dimensional representation of the h δ k ( µ , λ ) i function, which gives an overview of the forecastability offailure in terms of disorder. The bold red line highlights the ridge of the surface, where the broadest acceleration regime isobtained. (MATLAB 2018a, URL:https://uk.mathworks.com/)fracture of low disorder materials, and the ductile failure of the strongly disordered ones, are both unpredictable. In spiteof the considerable number of bursts generated, the absence of acceleration limits forecastability to a well defined range ofdisorder on the phase diagram of the system. This is illustrated in Fig. 9 which presents the value of h δ k i over the µ − λ plane.It can be observed that for the significance of the accelerating regime, disorder has an optimum amount, i.e. the combinationof the disorder exponent µ and of the cutoff strength λ of fibers determining the ridge of the h δ k i surface provides the bestpredictability. Note that lowering the exponent µ results in a broadening of the λ range where a significant acceleration occursso that the range of predictability tends to infinity in terms of λ for µ →
0. Our results imply that former conclusions in theliterature that increasing disorder improves forecastability is generally not valid for fat-tailed disorder. Increasing the cutoffstrength λ beyond the ridge of h δ k i ( µ , λ ) at fixed exponents µ in Fig. 9, disorder becomes disadvantageous. The absoluteupper bound of predictability is defined by the line on the µ − λ plane, where h δ k i ≈ λ regime (see Fig.9). This bound emerges due to the slow decay of the fat-tailed threshold distribution: since the tail of the strength distributionEq. (1) is efficiently sampled even at small system sizes, at sufficiently high upper cutoffs λ there will be so strong fibers inthe system which can stabilize the fracture process till the end. As a consequence, ductile behaviour can already be reached atfinite λ values. Predictability has also a lower bound which extends from λ ≈ .
01 to λ ≈ . µ increases from 0 to 1 (seeFig. 9).To analyze the evolution of the failure process, we only considered the magnitude of crackling events. The RB analysis issimilar to the natural time analysis in the sense that the physical time of events is ignored and only the magnitude of events isconsidered as a function of their order parameter. For practical applications, the adaptation of our method is straightforwardin those cases, where the system approaches failure through an increasing activity of acoustic or seismic events. Laboratoryexperiments have revealed an accelerating rate of acoustic events accompanied by an increasing average event magnitudeduring the tertiary regime of the creep failure of heterogeneous materials , and for the approach to failure of porousrocks under a slowly increasing compressive load . A similar behaviour of the rate of acoustic or seismic signals hasbeen observed for the failure of geosystems such as volcanic eruptions , cliff collapses , the breakoff of hanging glaciers ,and in some cases also for landslides . In these systems, records either of the fluctuating daily (hourly) rate of seismic(acoustic) events, or of the magnitude of individual signals can serve as the starting point of the analysis of record statistics.After setting the first record of the time window of the analysis, the rest of the record events can be unambigously identifiedsince the largest events of the sequence have to be found. Our method suggests that the event index n k ∗ of the longest livingrecord, and its corresponding physical time, provide the onset of acceleration of the record breaking process, and hence, ofthe start of the critical regime of the approach to failure. Failure forecast methods (FFM) predict the time of failure based onthe power law acceleration of a characteristic quantity of the time evolution of the system. Our method can also be used tocomplement FFMs by conditioning a series of discrete events to identify the time window where the assumption of a powerlaw acceleration is applicable . ur study is based on a fiber bundle model with equal load sharing which is essentially a mean field approach to fracture.The model has the advantage that solely one source of disorder is present in the system, i.e. the random strength of fibers. Inmore realistic situations stress fluctuations occur around cracks. Recently, we have shown in Ref. that when the strengthdisorder is very high in the system with fat-tailed local strength distributions, stress concentration around cracks have a minoreffect on the breakdown process. Hence, we conjecture that our statements have a broader validity, they are not limited by theassumption of the homogeneous stress field. Methods
To study the fracture of heterogeneous materials we use a fiber bundle model which provides a straightforward way to revealthe role of microscale disorder in fracturing. The model is composed of N parallel fibers with a perfectly brittle response, i.e.the fibers exhibit a linearly elastic behavior up to breaking at a critical deformation ε th . Materials’ disorder is introduced bythe random strength of fibers ε ith ( i = , . . . , N ) for which we considered a power law distribution over a finite range. Computersimulations were performed by slowly increasing the external load on the bundle to provoke the breaking of a single fiber. Theload of broken fibers is overtaken by the remaining intact ones. We assume equal load sharing after breaking events whichensures that all fibers keep the same load during the entire breaking process. Since no stress fluctuations can arise, the randomstrength of fibers is the only source of disorder in the system.After load redistribution, the excess load may trigger additional breakings, which is again followed by load redistribution.Eventually, such repeated cycles of breaking and load redistribution steps give rise to bursts of breakings in the model whichare analogous to the acoustic outbreaks of real experiments. The size of bursts ∆ is defined as the number of fibers breakingin the avalanche. In all the simulations presented in the manuscript the number of fibers was fixed N = × and averagingwas performed over 6000 samples, which provided a sufficient precision for the data analysis.Characteristic quantities of records such as the average size (cid:10) ∆ kr (cid:11) and lifetime h m k i were obtained by averaging over thesamples at fixed record ranks k . The same type of sample average was calculated for the average number of records h N n i and h δ k i at fixed event numbers n and upper cutoffs λ , respectively. Probability densities p ( ∆ r ) and p ( m ) were obtained for theentire ensemble of samples at given values of the control parameters µ and λ . References Voight, B. A method for prediction of volcanic eruptions.
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Acknowledgments
The work is supported by the EFOP-3.6.1-16-2016-00022 project. The project is co-financed by the European Union andthe European Social Fund. This research was supported by the National Research, Development and Innovation Fund ofHungary, financed under the K-16 funding scheme Project no. K 119967. The research was financed by the Higher EducationInstitutional Excellence Program of the Ministry of Human Capacities in Hungary, within the framework of the Energeticsthematic program of the University of Debrecen.
Author contributions statement
VK carried out computer simulations. VK, GP, and FK performed the data analysis. FK conceived of and designed the study,and drafted the manuscript. All authors read and approved the manuscript.
Additional information
Competing interests