PSI - Issue 25

Andronikos Loukidis et al. / Procedia Structural Integrity 25 (2020) 195–200

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2 A. Loukidis et al. / Structural Integrity Procedia 00 (2019) 000–000 Sachse et al., 1991). Being able to determine the level of damage accumulated in infrastructure is imperative especially when human activity is involved such as bridges, monuments etc. In this context, several AE parameters have been used as indices for the impending failure e.g., see Shiotani (1994); Kourkoulis et al. (2018); Triantis and Kourkoulis (2018). Nowadays, natural time analysis (NTA) has been proven to allow the optimal extraction of signal information especially in systems approaching criticality (Varotsos et al., 2011b). Therefore, analysing the AE energy in natural time, another parameter emerges, namely the variance κ 1 , which can be considered as prefailure indicator. The presented AE data were recorded from two marble specimens cut from Greek Dionysos marble (Perdikatsis et al., 2006; Kourkoulis et al., 1999, 2010), which is exclusively used in the restoration project of the Acropolis monuments. The marble specimens were subjected to uniaxial compression and three-point bending. In addition, Ordinary Portland Cement (OPC) specimens (Young, 2001) were subjected to three point bending. The purpose of this work is to investigate whether the scaled distributions of the variance κ 1 of natural time (Varotsos et al., 2002a, 2011b), obtained by the AE time-series of the above-mentioned experiments, exhibit the characteristic exponential tail, found in the universal curve of seismicity (Varotsos et al., 2005; Sarlis et al., 2011a; Sarlis and Christopoulos, 2012).

2. Theoretical background 2.1. Natural time

In a time series comprising N acoustic events, the natural time of the k -th event of AE energy A k (Rao, 1990) is defined in accordance with Varotsos et al. (2001, 2002a,b, 2011b) by χ k = k / N . In natural time analysis (NTA), the pair ( χ k , p k ) , where p k = A k / ∑ N n = 1 A n is the normalized energy, is studied. The variance κ 1 of natural time χ weighted for p k is given by

2 k −   N ∑ k = 1

p k χ k   2

N ∑ k = 1

(1)

p k χ

κ 1 =

and has been proposed (Varotsos et al., 2005) as an order parameter for seismicity. The quantity κ 1 may identify the approach of the dynamical system to a critical point (Varotsos et al., 2011a), when the condition κ 1 = 0 . 070 (2) holds. One of the reasons, that the variance κ 1 of natural time has been proposed as an order parameter of seismicity was that the distribution of the scaled fluctuations of κ 1 exhibit a behaviour similar to that of the order parameter of several equilibrium and non-equilibrium systems (Varotsos et al., 2005). In particular, the scaled distibution of κ 1 is defined as the distribution of the z-score z = [ κ 1 − µ ( κ 1 ) ] /σ , i.e., its fluctuations relative to the standard deviation of its distribution (Varotsos et al., 2005). The calculation of the order parameter κ 1 was performed using the sliding window calculation algorithm (e.g., see section IV of Sarlis and Christopoulos, 2012): Starting from the first AE event, the κ 1 values were calculated using a window of 6 consecutive events (including the first one). Next, the window increases by one AE event, and a new value of κ 1 is calculated by taking into consideration also the 7th event. We keep increasing the window until the window length becomes 40. Then, we repeat the calculation starting from the second AE event, etc. After sliding, event by event, through the whole time-series, the calculated κ 1 values enable the construction of the probability density function P ( κ 1 ) (Varotsos et al., 2005; Sarlis et al., 2011a). The study of the scaled distribution σ ( κ 1 ) P ( y ) vs. y = [ κ 1 − µ ( κ 1 ) ] /σ ( κ 1 ) for the long-term seismicity has revealed (Varotsos et al., 2005) an exponential tail similar to that obtained upon studying the order parameter fluctuations for several equilibrium systems like the 2D Ising model (Zheng and Trimper, 2001; Zheng, 2003; Clusel et al., 2004) and non-equilibrium systems such 3D ricepiles, turbulent flow, magnetic vortices penetration into type II superconductors and other self-organised critical systems (Sarlis et al., 2011a).