PSI - Issue 13

I.Yu. Smolin et al. / Procedia Structural Integrity 13 (2018) 1059–1064 I.Yu. Smolin et al. / Structural Integrity Procedia 00 (2018) 000 – 000

1062

4

Fig. 3. Spectrum of the time series: (a) marble; (b) ceramics.

The spectral representation of a time series or a signal is usually provided by the fast Fourier transform (FFT). Figure 3 shows the result of the FFT: the spectrums of our entire signals. One can see that the main energy contribution to the signals is produced by low frequencies. In the case of marble, the spectrum is characterized by the existence of frequency ranges rather than peaks corresponding to some leading frequencies. A more interesting analysis may be performed using the two components of the complex FFT vector. In Fig. 4a one can see a 2D hodograph of the complex FFT vector for the signal shown in Fig. 1a. It is presented in the complex plane. Values of the real (cosine) and imaginary (sine) parts of the FFT are plotted along the axes of the graph, the signal frequency is a parameter. The hodograph reflects the multiscale character of deformation with fracture as evidenced by the fractal property of the enlarged fragment of the plot shown in Fig. 4b. Spectral analysis of the time series characterizes the overall spectral composition of the series in its entirety and is therefore insensitive to local (short-term) inhomogeneities of the series. To detect local inhomogeneities, it is advisable to use wavelet analysis. In this case, the sensitivity of the wavelet analysis to local inhomogeneities depends on the type of the wavelet used. Let us see how the wavelet analysis works for the time series shown in Fig. 1a using Symlet sym2 (a symmetrical Daubechies wavelet) as an example. Figure 5 shows the investigated fragment of the signal before the start of its sharp increase and the brightness diagram of the coefficients of the wavelet decomposition depending on the width of the wavelet (level) and its location b on the time axis. It is seen that the wavelet decomposition of the signal reveals short signal spikes in the first and last quarters of the signal graph, which correspond to a slight local fracturing of the specimen. In the wavelet decomposition in Fig 5, they are reflected as bright stripes whose width corresponds to the scale of the inhomogeneity of the signal.

Fig. 4. (a) 2D hodograph of the FFT complex vector for marble specimen; (b) its zoomed part.