PSI - Issue 47

Oleg Naimark et al. / Procedia Structural Integrity 47 (2023) 782–788 / Structural Integrity Procedia 00 (2019) 000–000

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Organized Criticality (SOC). It is characteristic for an open nonlinear systems, which can self-organize to a critical state (associated with, the so-called, “strange attractor”). The power-law nature of the energy distribution function in a wide range of values (~4 orders of magnitude) is a characteristic of SOC. The distribution of time intervals between subsequent events is also close to a power law for a certain range of values. The analysis of sensitivity of AE waveforms to damage accumulation was proposed by Tanvir, Sattar, Mba and Edwards (2020) and is based on the calculation of the cumulative Renyi entropy as a function of time for each channel for a sequence of events. Figure 3b shows the cumulative Rényi entropy, which does not reflect the transitions between different regimes of the material's nonlinear behaviour due to damage stages. This indicates the need to use more sensitive signal analysis methods capable of characterizing various modes of observed signals and transitions between these modes. It has been established, that a number of time intervals between successive events are not sufficiently sensitive to the features of the nonlinear behaviour of the system; therefore, the energies of AE events were analysed further. The possibility of correspondence of the process of damage accumulation in composites to the class of dynamic systems with the property of "deterministic chaos" was studied. To filter the noise, a low-pass filter with an angular frequency of 200 kHz was applied to each waveform, the energy of each event was calculated, and a non-uniform time series of fiber signal energies was obtained. By calculating the average in a sliding window (1 second long, without overlap), a uniform time series of fiber signal energies is obtained. The main task of signal analysis was to determine the critical points separating the characteristic stages of damage development using Recurrent Quantitative Analysis (RQA) developed by Marwan (2008), Marwan and Webber (2015). The Recurrent Matrix (RM) shows the times at which the reconstructed phase trajectory returns to a small neighborhood of some point, and is defined as:   ˆ ˆ , , 1,2,..., ij i j R x x i j M       . (1) Here ε is the radius of a small neighbourhood, indices i and j count discrete time, ˆ ˆ i j x x  is the distance between phase trajectories at times i and j . For the quantitative analysis of RM in a sliding window, we used two characteristics: divergence, the reciprocal of the maximum length of the diagonal, and determinism, the proportion of points that form only diagonal lines

N 

( )

D l H l 

1 (1 R )(1 R ) R l N           1, 1 j 1, 1 j i i

l d 

, where

(2)

( )

DET

D H l



min

, i k j k  

N

, 1   i j

R

, 1 

i j

0

k



ij

Fig. 4 shows the RQA results for two samples. As initial data for RQA on the piezoelectric channel, we used a median frequency series characterizing the distribution of the released energy over frequencies. The RQA results reflect a successive abrupt change in dynamic regimes, much more detailed than the analysis of flow velocity curves gives. On the piezoelectric channel, there is a sharp dip in the determinism parameter and a simultaneous jump in divergence around 250 and 200 seconds for the first and second samples, respectively, which corresponds to an increase in the degree of stochasticity and unpredictability of the dynamics. Further, a gradual increase in determinism and a decrease in divergence are observed, which is a sign of sequential ordering, subordination of the system to collective relaxation modes up to the development of a main crack and failure. A qualitatively similar picture is observed also on the optical channel, with the difference that the jump in divergence is smoother and more gradual. This may be due to the higher sensitivity of the optical scheme for recording the dynamics of damage accumulation in this experiment. The laminarity parameter introduced on the basis of the RQA method and calculated from the data for different channels showed the presence of two critical points which