Intermittent flow under constant forcing: Acoustic emission from creep avalanches Open Access

Crackling noise occurs under many scenarios, in magnetization processes,¹ martensitic transitions,² plastic deformation in solids,³ gravitational wave detections,⁴ avalanches during stellar evolution,⁵ and material failure.^6–9 Quantitative analysis is commonly undertaken with constant rates of external forcing, with the highest experimental resolution for stress collapse of porous materials.^10–16 All previous experiments focused on the uniaxial compression with a constant loading rate and identified the energy exponents to range between 1.33 and 1.97. The waiting time distributions are power law forms with an exponent τ near 2 at long waiting times and 1 for small waiting times.¹⁷ This includes the de-twinning of martensites,^10,18 the emission during relaxor freezing,^19,20 temperature driven ferroelectric transitions,²¹ and the Burridge-Knopoff model of earthquakes and Barkhausen instabilities.²² We know no experimental work for constant loading (creep scenario) where materials transform after a finite time, even if the applied field is below their coercive field.²³ Sub-critical fracture²⁴ of this kind is widely observed in physical,²⁵ geological,²⁶ and biological systems.²⁷ Microcrack nucleation, nano-structural relaxation, viscoelasticity, and healing of microcracks are typical in sub-critical processes.^28–31 Theoretically, sub-critical crackling noise was studied in an extended fiber bound model^24,32 where time-dependent damage of fibers has been introduced for immediate breaks to capture the cumulative effect of the loading history. We argue in this paper that this scenario is born out in the collapse of weakly bound quartz grains³³ under creep conditions.

Another motivation for this work stems from the requirement to measure noise exponents under extremely slow forcing to avoid overlap effects of individual avalanches (“jerks”). Only in such cases could the signal profile, say in a memory device, be predicted.³⁴ Not only mean field solutions⁶ but also a wide scale of non-mean field exponents^35,36 were observed with the largest dataset for SiO₂ based vycor¹⁷ (ε = 1.39) and natural SiO₂ based materials, like sandstone, with ε = 1.5¹¹ measured at very slow strain rates. The recent studies from laboratory³⁷ and simulation^38,39 show that variations in ε may indicate combinations of two break mechanisms.³⁷ Superposition leads to characteristic mixing of power laws and was identified using Maximum Likelihood methods.³⁵ We now ask what happens if the rate becomes zero (creep)? Is the response zero, stationary, or non-stationary? and what are the exponents if there are signals at all?

Finally, creep conditions were used in recent simulations where supersonic kinks were generated under constant loading.⁴⁰ This effect cannot be observed electronically because available detection devices cannot measure on such short timescales. We reduce the timescale by using collapse materials, which fulfil similar criteria for creep dynamics.

We use sandstone from the Sichuan province in China. The density and porosity of this sample are 2.2 g/cm³ and 18% determined by the wax seal method and mercury intrusion analysis, respectively. The shape of the sandstone sample is cylindrical with a diameter of 50 mm and a length of 100 mm. To avoid background acoustic emission (AE) noise induced by fraction and concentrating stress between sample surfaces and loading equipment, the sides of the specimen were smooth, and the ends of the specimens were flat within ±0.02 mm (according to ISRM testing guidelines, Ref. 41).

The creep experiment was performed using the loading equipment described in Ref. 37. The constant load is provided by weights in a container. The sandstone samples were placed between the lower tilting beam and a static support. AE signals were measured during creep by two piezoelectric sensors (NANO-30 Physical Acoustics Company) fixed on the sample's round surface by rubber bands. The sensors were acoustically coupled to the sample by a thin layer of grease. The acoustic signal was pre-amplified (40 dB) and transferred to the AE analysis system (DISP from American Physical Acoustics Company). The threshold for detection was chosen as the threshold of an empty experiment (45 dB). The Peak Definition Time (PDT) is 35 μs, the Hit Definition Time (HDT) is 150 μs, and the Hit Lockout Time (HLT) is 300 μs (definitions in Fig. 1).

The energies (1 aJ = 10⁻¹⁸J) of AE signals during creep are shown as functions of time in Fig. 2(a). AE energy spectra show cycle modes with a series of active periods. The shape of AE energy spectra in active periods is triangular [Fig. 2(b)]. This kind of cyclic mode has been captured by fiber bound model simulation under sub-critical fracture.²⁴ Four active periods were found before the final collapse. This behavior resembles two rupture mechanisms in a fiber breaking model, namely, fiber breaks instantaneously at time t when its local load exceeds the strength of the fiber and those fibers that remained intact undergo a damage accumulation process due to the additional load. Equivalent models can be imagined for SiO₂ spheres that are weakly bonded to form sandstone. In all cases, slowly proceeding damage sequences can trigger bursts of breaking (active period). These active periods are identified by high average AE activities. When the sample survives the external constant loading, the system transfers to the next slowly proceeding damage sequences until the final collapse (the final active period with significantly increased average AE energy). We mark the four observed active periods as n = 1, 2, 3, and 4 [Fig. 2(a)], where n = 4 is the final collapse sequence [Fig. 2(c)], which extends over a very short time interval.

The probability distribution functions, PDFs, of AE energies derived from active periods n = 1 and n = 4 are derived by logarithmic binning of the event energies and shown in Fig. 3(a). A power law dependence, P(E) ∼ E^-ε, is observed for n = 1 and n = 4 for more than 6 decades. ε at the critical stage (n = 4) is consistent with the result of Vycor.¹⁷ Vycor showed stationary behavior over more than 6 decades.¹⁷ The exponent ε is larger in other natural SiO₂ materials (ε closer to 1.6). The double-logarithmic plots show power law distributions in good approximation. We apply the maximum likelihood method⁴² to study the distributions in more detail. The results for all active periods during creep are shown in Fig. 3(b), where ML shows plateaus that define the optimum exponent ε for each dataset. The plateau for n = 4 is well developed, while others are slightly inclined, which may be due to damping effects.³⁵ Similarly, inclined plateaus were observed in Refs. 10 and 43. In the uniaxial compressive test of sandstone in Ref. 11, no flat plateau was found. In these cases, the relevant exponent was taken as the onset of the plateau, namely, the first kink in the ML curve. ML of n = 4 confirms a plateau with ε = 1.39. The other active periods (n = 1, 2, 3) have ε near 1.65, i.e., they do not follow mean field behavior.^36,37 Previous measurements at a constant stress rate^36,37 showed mixtures of exponents,³⁵ while mixing was found inside intervals.

Figure 4 shows the superposition of datasets of early stages (with large exponent, L data points) and final stage (with small exponent, S data points). Because the final stage is short, the amount S of data is much smaller than that in early stages. This allows us to use different amounts of data in the early stage to mix with the data in the final stage. In Fig. 4, we plot the mixing results with the data number ratio of early and final stages, namely, L/S are 1:1, 2:1, 3:1, …, 14:1, and 15:1. The plateau of mixing increases with the increasing ratio. These results are similar to the results from different sandstone in earlier “driven” experiments.¹¹ This observation may explain the variance of exponents in SiO₂ based materials: the nano-porous material of Vycor is in the critical state through the whole compression. The natural sandstones contain non-stationary, non-critical parts so that the experimental exponents stem from a superposition of critical and sub-critical intervals. Similar sequences were also encountered in collapse experiments in porous alumina,¹³ berlinite,¹² and goethite.¹⁴ This may indicate that AE experiments at very low stress rates may comprise segments of creep and field driven data. From our experimental findings, it appears that the creep exponents outside the final collapse area are larger than mean field exponents and that mixing may explain some earlier field driven results.³⁵ 

Sub-critical rupture has been investigated by computer simulation^24,32 so that we can compare our PDF of AE amplitudes (rather than energies) in Fig. 5(a). At the critical stage (n = 4), the exponent ν is near the FBM (fibre bundle model) and MF (mean field) result (ν = 1.5).^24,44–46 The exponent ν of AE events far from the final collapse point is ν = 2.0, while the exponent of FBM simulations is slightly higher (ν = 5/2).

The distribution of waiting time is shown in Fig. 5(b). We observe a change of power law exponent from 2.2 to 1.32. This agrees well with the local load share fiber bound model, with exponents of 2.0 and 1.4.³² The change of exponent is determined by the damage accumulation (λ) that controls the sensitivity of the system to stress inhomogeneity. At the critical stage, the damage accumulation process becomes so fast that the inhomogeneous stress distribution dominates the failure process.

Båth's law^47,48 states that the average ratio of the energy magnitudes of one MS (mainshock) and its AS* (largest aftershock) is 1.2. Figure 6 shows  ⟨ΔM⟩ = log(E_MS/E_AS*) as a function of the mainshock energy which agrees with Båth's law for seven decades.

In summary, creep leads to non-stationary AE in our samples. The final collapse shows mean field behavior, while greater energy and amplitude exponents are characterized in earlier stages. The AE distribution of the final stage agrees well with previous results in nano-crystalline Vycor compression at low rates. The distribution of AE events and waiting times agree well with the crackling noise from the local load share (LLS) fibre bound model with sub-critical rupture, and the relative magnitude agrees with Båth's law.

E.K.H.S is grateful to EPSRC (EP/K009702/1) and the Leverhulme Trust (EM-2016-004) for support. H.L. acknowledges financial support from the National Natural Science Foundation of China (Grant No. 51304256). D.J. is grateful to the National Science and Technology Major Project (No. 2016ZX05045001-005).