The plots derived from our series of experiments are typically like the one shown in Fig. 2a. A very large peak occurs at low energies, indicating a large number of such events, and a much lower number of signals at higher energies. An important break of the functional distribution curves occurs in Fig. 2a at 3 aJ. At higher energies relating to ‘louder’ signals, we find a power-law dependence.

a Typical crackling noise distributions with power-law dependencies at high energies and a peak at low energies. The crossover energy is indicated by the vertical line. This was recorded from measurements on the weddellite stone (A-series). b Uric acid stone (C-series) shows mild signals below 210 aJ where the low-energy signals show a log-normal or Gamma function [26] while the high-energy signals follow a power law. Similar behaviour was found in the B (cystine) series but with extremely sparse signals in the power-law regime

Below the crossover at 3 aJ, the greater amount of signals peak near 0.6 aJ. The signals stem from local strain release under stress. Their two different fingerprints relate to random, independent events in the peak-regime at low energies and have correlated collapse by networks of cracks in the power-law regime [19, 34,35,36,37]. The boundary between the two regimes is referred to as the crossover point and separates the conditions where strain release occurs in a correlated manner (wild) and the low-energy regime where events are local and uncorrelated (mild) [38].

Figure 2b shows a similar curve of energy against probability for one of the C-series stones, composed of uric acid. It shows mild avalanche signals below 210 aJ where low-energy signals conform to a log-normal or Gamma function, while the high-energy signals show a power law. Similar behaviour was found in the B-series (cystine) but with extremely sparse signals in the power-law regime. The exponents can be extracted from the power-law regime and are shown in Table 1 for three representative stones (samples A2, B7, and C2) from each of the three stone classes examined in this study. Values for the crossover point (\(E_\textrm\)) from mild (uncorrelated), to wild (avalanche-like) for the all stones are given in Table 2.

For the sake of simplicity, data sets for samples A2, B7, and C2 are displayed in Figs. 3, 4, 5, 6, 7 and 8 as being representative of each of the kidney stone types. Figures 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 and 22 in the appendix include additional energy probability distribution functions (PDFs) and maximum likelihood (ML) analysis which show the results for all samples summarised in Table 2. Crackling noise spectra, i.e. the experimental jerk spectra, for samples A2, B7 and C2 are shown in Fig. 3.

Energy jerk spectra for samples A2, weddellite (a), B7, cystine (b), and C2, uric acid (c) showing energy on the y-axis (aJ) vs time (s) on the x-axis

The jerk distribution is unusual for typical crack propagation in minerals where the density of jerks per time interval is fairly constant. The ‘standard’ behaviour is classified as ‘stationary’ because the number of events per time interval is roughly constant throughout the experiment [39, 40]. In Fig. 3 the behaviour of our samples is shown to be non-stationary with some bursts separated by a multitude of very weak signals and quiet periods. This sets the scene for the discussion of avalanches: the big peaks are the wild avalanches (highly non-stationary) and the weak, stationary background signals indicate mild events. Furthermore, Fig. 3 shows that bursts of the kind shown in Fig. 3a are slightly more numerous with shorter intervals in between (i.e. slightly more stationary) than in Fig. 3b and Fig. 3c. The bursts in Fig. 3b are extremely sparse and non-stationary, while the low-energy signals are abundant and stationary. We find an extreme predominance of mild events in this case.

The jerk spectra show typical characteristics which are quantified in Fig. 4. The wild, power-law distributed region is marked by large jerks and are dominant in the A-series, (weddellite). The power-law regime is much shorter in the B (cystine)- and C (uric acid)-series where the mild events dominate. The crossover points between mild and wild events are indicated by vertical black lines in Fig. 4. A low crossover energy means that the collapse occurs predominantly by collective avalanches, which are typical for hard brittle materials, while a high crossover energy means that the dominant collapse mechanism is by mild events, similar to the behaviour of soft, non-brittle materials.

Energy and amplitude probability distribution functions highlighting the power-law regime, i.e. for \(E \ge E_\textrm\) and \(A \ge A_\textrm\) for the representative stones from the A-series weddellite stones (a, b), B-series cystine stones (c, d), and C-series uric acid stones (e, f). The energy and amplitude crossovers are 3 aJ and 35 \(\upmu\)V for sample A2, 700 aJ and 150 \(\mu\)V for B7, and 250 aJ and 250 \(\mu\)V for C2. The energy exponents for samples A2, B7 and C2 are \(\epsilon = 1.55\). \(\epsilon = 1.3\), and \(\epsilon = 1.5\), respectively. The amplitude exponents are \(\tau = 2.1\), \(\tau = 1.6\), and \(\tau = 2.0\), respectively

The probability distributions in Fig. 4a (A-series) range from 1 to 10,000 aJ. The B- and C-series have shorter power-law regimes and the determination of the exponents is difficult, comparatively. The amplitudes and the energies of wild avalanches are highly correlated. Depending on the waveform of the emitted signal, the most common correlation is \(E \propto SA^x\), where the exponent x takes values around 2. The correlation between energies and amplitudes of all series follows the predicted scaling \(E \propto A^2\) rather well (Fig. 5) [41] with ca. 6 orders of magnitude of the power laws in the A-series.

Energy and amplitude scaling for each of the representative kidney stones showing the approximate scaling \(E\propto A^2\). Sample A2 is shown in red, B7 in blue and C2 in black

The scaling prefactor S varies between the data sets when very different mechanisms are superimposed. This effect is called multi-branching and has been commonly observed, for example, in Refs. [21, 42, 43].

In Fig. 6 we show the correlation curve which indicates that the scaling is approximately uniform and, therefore, multiple collapse mechanisms do not contribute significantly to the overall signal. The variation which is observed is the same for collapse mechanisms under compression in vastly different materials. An advanced method to display the power-law behaviour is the maximum likelihood (ML) estimate [44, 45]. If the AE parameter is power-law distributed, the ML curve shows a plateau, the value of which is equal to the power-law exponent [46].

Maximum likelihood analysis of samples A2 (a), B7 (b), and C2 (c) showing energy exponents of \(\epsilon =1.55\), \(\epsilon =1.3\), and \(\epsilon =1.5\), respectively

Figure 6a shows the largest plateau over seven decades, observed in weddellite stones. The exponent is \(\epsilon = 1.55\). The B-series has a slightly lower exponent of \(\epsilon = 1.3\), which is remarkably close to the ‘critical mean field’ value of 1.33 [26]. Despite the sparse statistics, the B-series samples B7 and B5 provide sufficiently good data to perform statistical analysis. The length of the plateau is surprisingly large so that the energy exponents are fully confirmed. The results for the wild avalanches are summarised in Table 1. The estimated errors are \(\pm \). The self-consistency of the exponents is confirmed by the scaling relationship [26] \(2(\epsilon -1) = (\tau -1)\) in mean field theory. Our experimental results fulfil this relationship reasonably well.

Wild avalanches take some time to develop. The duration of an avalanche is defined as the total time during which an avalanche persists. It is measured in an AE experiment between the first threshold crossing of an acoustic signal, and the last threshold crossing of the emitted acoustic wave. The points shown in Fig. 3 are examples of classic singular acoustic emissions. The distribution of avalanche durations is also a power law with an exponent \(\alpha\). However, while the duration statistics are commonly well defined, kidney stones of the B- and C-series prove problematic. The origin of this difficulty is explained in the present section.

Figure 7a–c shows classical acoustic emissions. However, avalanches can generate aftershocks, which means multiple acoustic waves will not cross the minimum threshold resulting in an inflated duration as shown in Fig. 7d, e. Furthermore, avalanches overlap and individual avalanches are hard to identify. Figure 7d–f, shows typical spectra with multiple avalanches, where individual points on the energy spectra should be treated as an event with a short duration. However, because there is no minimum threshold crossing, multiple avalanches are bundled together resulting in very high nominal values of their duration. Samples A2 and A3 have fewer bundled avalanches with an exponent of \(\alpha \approx 2\). The B- and C-series have many avalanche bundles. Samples C3 and C4 with duration exponents of \(\alpha =2.3\), and \(\alpha =2.8\), respectively, are typical for wild avalanches. Lower values near 1.5 represent bundled avalanches of the type shown in Fig. 7d, e.

A more obvious way to present this phenomenon is to plot the cross correlation between durations and amplitudes (or energies) of wild avalanches as shown in Fig. 8. The scaling relationship shows correlation exponents of amplitudes versus durations near 1.5, which is the theoretical value for wild avalanches [21]. However, the data are very noisy because the mild events strongly interfere with the jerk distributions. The distribution is somewhat constrained in Fig. 8a and almost random in Fig. 8b, c. On closer inspection, we find that all data with higher durations than those indicated by the lines are bundles and, hence, artificially long. Conversely, Fig. 7a–c shows single events following points along the line \(X = 1.3\) in Fig. 8a.

Individual waveforms from sample A2 along the line \(X = 1.3\) (a–c), and \(X = 1.7\) (d, e). The peak amplitude and durations are: a \(A = 474.9\) \(\upmu\)V, \(D = 588.0\) \(\upmu\)s; b \(A = 1942.4\) \(\upmu\)V, \(D = 1502.5\) \(\upmu\)s; c \(A = 17,734.0\) \(\upmu\)V, \(D = 7786.7\) \(\upmu\)s; d \(A = 150.7\) \(\upmu\)V, \(D = 6162.1\) \(\upmu\)s; e \(A = 692.6\) \(\upmu\)V, \(D = 19,056.9\) \(\mu\)s; and f \(A = 4644.4\) \(\upmu\)V, \(D = 46,684.1\) \(\upmu\)s

Amplitude–duration scaling revealing exponents \(X=1.3\), and \(X=1.7\) for A2 (a), \(X = 1.5\) for B7 (b), and \(X = 1.5\) for C2 (c)

Data for sample A1 in Fig. 9 reveal an energy exponent of \(\epsilon = 1.6\) over four decades with a small hump and slight reduction in \(\epsilon\) to 1.5 at \(10^5\) aJ. Sample A2 has an energy exponent of \(\epsilon = 1.55\) over four decades and reduces to \(\sim 1.4\) again at \(\sim 10^5\) aJ (Fig. 5b). Sample A3 has some oscillations but in the low-energy regime is stable at \(\epsilon =1.7\) between 500 and 50 aJ. Upon increasing energy, there is an increase of \(\epsilon\) to 1.8; however, at energies greater than \(10^4\) aJ the error increases to ± 0.25, as shown in Fig. 10, making it difficult to determine if the higher energy exponent is meaningful. Sample A4 has an energy exponent of \(\epsilon =1.57\) between 100 and 100,000 aJ. Between \(10^4\) and \(10^5\) aJ, there is a continuous decrease in \(\epsilon\) from 1.6 to 1.4 shown in Fig. 11. Sample A5 has an energy exponent of \(\epsilon =1.55\) between 100 and 100,000 aJ (Fig. 12). Between \(10^3\) and \(10^7\) aJ, there is a gradual decrease in \(\epsilon\) from 1.55 to 1.35. Figure 13 shows energy PDF and ML for sample A6 revealing an energy exponent of \(\epsilon =1.53\) stable over four decades. There is, however, a subtle decrease in \(\epsilon\) from 1.53 to 1.5 at \(10^4\) aJ. Sample A7 has an energy exponent of \(\epsilon =1.5\) stable over three decades. There is a sharp decrease in \(\epsilon\) from 1.5 to 1.2 between \(10^5\) and \(10^6\) aJ (Fig. 14).

The only experiments in the B-series that generated enough data to perform meaningful statistics were from samples B5 and B7. This is a result of the much finer time scales over which the cystine stones generated AEs. They occurred at some duration less than the limit of detection, i.e. \(< 25\) ns, owing to the sparse data sets in comparison to the weddellite, and uric acid stones. Sample B5 gave an exponent of \(\epsilon =1.2\) (Fig. 15) and was stable over a narrow region at low energies. The ML mimics the behaviour seen in exponentially damped power-law systems [44, 47,48,49] and is marked by a gradual increase in \(\epsilon\) as the energy increases. The same dynamics are seen in sample B7, except the exponent is 1.3 and it is stable over a slightly larger energy range between \(10^3\)–\(10^4\) aJ before continually increasing. This indicates that there was only one mechanism, and it occurred over short time scales (\(D < 50\) ns; most likely crack propagation).

Figure 16 shows energy PDF and ML for sample C1 revealing an energy exponent of \(\epsilon = 1.45\) stable between 250 and 1000 aJ. For energy values greater than 1000 aJ the energy exponent gradually increases to 1.6 at \(E = 5500\) aJ but reduces again to 1.4 at 12,000 aJ. Sample C2 yielded an exponent of \(\epsilon =1.5\) which is stable over the entire energy spectrum with no indication of multiple failure mechanisms at play. Sample C3 gave an energy exponent of \(\epsilon =1.6\). There is a small hump in \(\epsilon\) between 400 and 1,200 aJ and a decrease in \(\epsilon\) from 1.6 to 1.4 at \(10^5\) aJ. C4 gave perhaps the most obvious change in exponent, as shown in Fig. 19. \(\epsilon =1.6\) for energies larger than 600 aJ, and there is a sharp increase in energy exponent reaching a maximum of \(\epsilon =1.9\) for energies less than 600 and greater than 30 aJ. The results from kidney stone sample C4 were discussed in detail in Ref. [20]. C5 gave a relatively constant exponent of \(\epsilon =1.6\) until \(10^4\) aJ and there are no signs of multiple processes. Similarly, C6 gave only one exponent of \(\epsilon =1.6\) and no AEs with \(E>10^5\) aJ. This means that for this experiment, the low-energy failure mechanism was all that was needed to instigate failure. C7 gave two energy exponents of \(\epsilon =1.9\) for \(E<10^3\), and \(\epsilon =1.5\) for \(E>10^3\) (Fig. 22. A summary of these results is shown in Table 2.

The total number of uncorrelated events in our study was, in general, much greater than the correlated events where only the correlated events represent the breaking of the hard and brittle parts of the kidney stones. It is, therefore, important to distinguish between the mild events and wild avalanches very carefully. Note that this behaviour has been observed before in a few other materials [50] but it is certainly not typical for crack-induced collapse of materials. The jerk energies and amplitudes in the wild regime are power law distributed with energy and amplitude exponents \(\epsilon\) and \(\tau\): \(P(E) \approx E^\) and \(P(A) \approx A^\).