Record statistics of bursts signals the onset of acceleration towards failure

Forecasting the imminent catastrophic failure has a high importance for a large variety of systems from the collapse of engineering constructions, through the emergence of landslides and earthquakes, to volcanic eruptions. Failure forecast methods predict the lifetime of the system based on the time-to-failure power law of observables describing the final acceleration towards 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 focus on 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, we demonstrate that the acceleration starts at a characteristic record rank, below which record breaking slows down due to the dominance of disorder in fracturing, while above it stress redistribution gives rise to an enhanced triggering of bursts and acceleration of the dynamics. The emergence of this signal depends on the degree of disorder making both highly brittle fracture of low disorder materials, and ductile fracture of strongly disordered ones, unpredictable.


SI1: Derivation of the brittle -quasi-brittle phase boundary
The probability density p(ε th ) of the breaking thresholds of fibers has a power law functional form over the range ε min ≤ ε th ≤ ε max . The cumulative distribution of thresholds reads as Since equal load sharing is assumed after fiber breakings, the constitutive equation σ(ε) of the bundle can be obtained from the general form σ(ε) = Eε[1 − P (ε)] by substituting the distribution function P (x) from Eq. (S2) [P1,P2] For finite cutoff strength ε max < +∞ the constitutive curve σ(ε) has a maximum whose position ε c and value σ c define the fracture strength of the bundle. The critical strain and stress depend on the disorder parameters of the model and The bundle exhibits a perfectly brittle behaviour if the breaking of the weakest fiber triggers immediate abrupt failure. This occurs when the critical strain ε c coincides with the lowest breaking threshold ε min . It follows that at any µ exponent there exists a critical upper bound below which the bundle fails abruptly. The above equation defines the phase boundary between the brittle and quasi-brittle phases of the model in Fig. 1 of the manuscript.
SI2: Derivation of the average number of breaking fibers triggered by the breaking of a single fiber at a strain ε The load σ = Eε dropped by the broken fiber is equally shared by the intact ones of number N [1 − P (σ)], giving rise to the stress increment ∆σ = σ/N [1 − P (σ)]. Multiplying ∆σ with the probability density p(Eε) of failure thresholds and with the total number of fibers N the average number of triggered breakings a can be cast into the form which is Eq.
(2) of the manuscript. Note that the Young modulus is E = 1.
SI3: Supplementary figure to illustrate the behaviour of the average value of the largest record size and lifetime as function of the distance from the critical point of brittle failure In the ductile phase of the system ε max = +∞, the record breaking process accelerates as the disorder exponent approaches the critical point of brittle failure µ → µ c (ε max = +∞) = 1. To quantify this behaviour we determined the average value of the largest record size ∆ max r and of largest waiting time m max that occurred up to failure as function of µ. Figure S1 presents the two quantities as function of the distance from the critical point µ c − µ with µ c = 1. Both curves can be very well described by straight lines, which implies the power law functional forms with the critical exponents α = 1.8(5) and β = 1.0(5).
SI4: Supplementary figure of the average size of records ∆ k r at finite cutoff strength Figure S2 illustrates that in the quasi-brittle phase of the system, when the cutoff strength of fibers is finite, the average size of records monotonically increases with the record rank similarly to the ductile phase.