# Acoustic Emission Spectroscopy: Applications in Geomaterials and Related Materials
## 4 Avalanche Mixing  
A crucial outcome of previous research is that failure does generally not follow a single mechanism. This surprising result can be exemplified by the observation of a collapse processes in metals [2] where local porous collapse and the movement of dislocations combine to achieve the final, global collapse. In other cases, the porous collapse entails the avalanche movements of twin boundaries [58], or a phase transition triggers catastrophic changes of microstructures related to structural transformations [65]. During avalanche evolution we find that often one instability triggers another instability. This means that several avalanche mechanisms interact and mix dynamically [66]. In this situation, probability densities are no longer simple power laws but show the typical up-wards curvature of mixed power laws. Figure 5: Energy exponent as a function of the lower energy cut-off determined by maximum likelihood (ML) estimate. (**a**) Energy exponents determined by the maximum likelihood method for k-intervals 13-14 and 20-21 with error bars and estimate values; (**b**) energy exponents for all intervals between k = 13 and k = 21. Sparse data sets lead to a less defined plateaus. We used data near 103 aJ to determine the energy exponents. (**c**) Evolution of the waiting time renormalization factor \(\lambda\) and the energy exponent \(\varepsilon\) with increasing rank k of superjexts. Two plateaus can be distinguished for large and small \(\lambda\) and exponents near \(\varepsilon\) = 1.5 and \(\varepsilon\) = 1.32. Reproduced with permission from [59]. American Physical Society, 2017.  
laws [67]. [PERSON] et al. [67] extended the analysis of mixing to the use of the'maximum likelihood estimation method', ML [68; 69; 70]. They derived the characteristic ML profiles for single, underdamped avalanches (a plateau), a damped avalanche (a monotonically increasing profile), and mixed avalanches (a peak before a plateau). The upwards bent probability density distribution translates into a steep increase in the ML estimates for small avalanche energies which peaks at a maximum value, which is bounded by the energy exponent of the avalanche with the higher exponent. It then decays towards the exponent of the second avalanche. This rather complex ML profile was identified in several experimental investigations, which shows that avalanche mixing is indeed a much more common phenomenon than previously assumed [6]. In order to confirm the mixing mechanism for different avalanches during a collapse process, [PERSON] and his coauthors designed a simple and instructive experimental arrangement. They used two materials with different strengths and superimposed them in the same stress device (Figure 6) [71]. The stress was identical for both materials, but the avalanche dynamics were different. This experiment combined one porous material at the early stage of uncorrelated, random damage mode while the second material was more highly damaged within the correlated failure state. The common stress created two avalanche processes simultaneously and ensured mixing between them. In the compression experiment, they used coal and sandstone as test materials. As predicted by theory, the probability distribution of the AE energy of the composite rock was a nonpower law with an upwards bent of the probability distribution function (PDF) in double logarithmic display. The maximum likelihood estimation curve showed the predicted increase, peak and decrease towards the lower exponent (Figure 7). Several other compression experiments show that the mixing effect of different avalanche mechanisms is a common phenomenon. For example, compression experiments on granite show that the crumbling and fracture mechanism of granite mineral particles can form a state of avalanche mixing [72]. High temperature annealing (near the \(\alpha\)-\(\beta\) phase transition temperature of quartz) reduces the internal constraints of granite and weakens the effect of fracture, so that the mixing between crumbles and fracture is observed (Figure 8).