PSI - Issue 25

Andronikos Loukidis et al. / Procedia Structural Integrity 25 (2020) 195–200 A. Loukidis et al. / Structural Integrity Procedia 00 (2019) 000–000

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4

1

0.1

0.01

σ ( κ 1 )P(y)

0.001

MARBLE CEM BK CEM OFC CSR ß=0.4707 L=256

0.0001

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-3

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-1

0

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y=( μ ( κ 1 )- κ 1 )/ σ ( κ 1 )

Fig. 4. The scaled distribution σ ( κ 1 ) P ( y ) versus y = ( µ ( κ 1 ) − κ 1 ) /σ ( κ 1 ) for the results obtained from the natural time analysis of the AE events of the marble and cement mortar specimens. The red circles correspond to the scaled distribution of the order parameter κ 1 of the notched beam-shaped marble specimen, labelled MARBLE. The blue open squares correspond to the scaled distribution of the order parameter κ 1 of the cement mortar specimen, labelled CEM, when the sample mimics the behaviour found when analysing the Burridge-Knopoff train model in natural time (Sarlis et al., 2006; Varotsos et al., 2010; Loukidis et al., 2019), whereas the cyan squares correspond to the scaled distribution of the order parameter κ 1 of the same cement mortar specimen CEM BK, when the system mimics the results found for the Olami-Feder-Christensen model (Olami et al., 1992) in natural time (Sarlis et al., 2011b; Loukidis et al., 2019), see also pp.350-358 of Varotsos et al. (2011b). The green triangles correspond to the scaled distribution of the order parameter κ 1 of the semi ring- shaped marble specimen, labelled CSR. The solid brown line corresponds to the scaled distribution of the order parameter for the 2D Ising model of linear dimension L =256 at inverse temperature parameter β =0.4707 (Zheng, 2003; Varotsos et al., 2005).

acoustic signals produced. On the other hand, one acoustic R15 α sensor was mounted on the cement mortar specimen.

4. Results and Discussion Figure 4 depicts the scaled distributions obtained from the analysis in natural time of the AE data of the aforementioned marble and mortar cement specimens, together with that obtained from a finite 2D Ising system, see Varotsos et al. (2005), see also Zheng and Trimper (2001); Zheng (2003); Clusel et al. (2004). We notice that the scaled distributions of the variance κ 1 of natural time share for -at least- three orders of magnitude an exponential tail already observed for the scaled distribution of the order parameter of the 2D Ising model. If we also take into account the fact that κ 1 abruptly turns to zero upon failure, the results of Fig. 4 reveal that κ 1 can serve as an order parameter for the AE that precedes rupture. Let us now comment on the generality of the left exponential tail shown in Fig.4 whose practical meaning is that an extreme fluctuation is orders of magnitude more probable than it would be if Gaussian statistics were valid. As concerns seismicity such a tail has been first observed by Tanaka et al. (2004), see their Fig. 8, and later studies (Varotsos et al., 2005) revealed its similarity with the fluctuations of the order parameter in other equilibrium and non-equilibrium critical systems (Bramwell et al., 1998, 2000, 2001; Zheng and Trimper, 2001; Zheng, 2003; Clusel et al., 2004). Interestingly, the analysis in natural time of self-organized critical systems (for their relation to seismicity, e.g., see Varotsos et al., 2010) has also shown (Sarlis et al., 2011a) this characteristic left exponential tail which was also observed when studying the fluctuations of the order parameter of seismicity before strong earthquakes in California(see Fig.1(d) of Varotsos et al., 2012).