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

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1. Introduction

The mathematical theory of evolution of solids proposed by Makarov (2008) treats a sample or a medium under loading as a nonlinear dynamical system. Investigations by Makarov (2007, 2008, 2010) have shown that solids under loading manifest the main features of nonlinear dynamical system evolution such as instabilities, self organization, processes localization in space and time. In terms of the states of stress and strain, it means that at the stage of stable quasi-static loading the average trend of the stress and strain changes smoothly in time while during the failure the deformation in the sample or media increases abruptly. In other words (in terms of the nonlinear dynamics theory), the blow-up regime follows the stationary stage of the evolution of the dynamical system. Papers by Stefanov (2008), Smolin et al. (2016), Smolin et al. (2017), Kocharyan et al. (2016), Eremin et al. (2016) describe both experimental evidence and theoretical study of these phenomena for rock specimens. The most interesting aspect of this nonlinear behavior is the change of evolution stages and the question of whether there is a precursor of the upcoming catastrophic failure at the relatively steady quasi-stationary deformation stage. To answer this question additional investigations are needed. Some attempts to do this are made in this paper. The idea of the study is to apply a method analogous to the analysis of the Earth surface deformation by seismographs for a small solid specimen. In our case, we investigated the deformation evolution of loaded marble and ceramic specimens by measuring the specimen surface velocity using the laser interferometry method. The registered data represent the so-called time series. Thus, to find some features of the deformation evolution we can employ the time series analysis methods available in some software codes like MATLAB, Octave or R. As a matter of fact, these methods are statistical by nature and allow for revealing some regularity in the quasi-random behavior of complex objects. The specimens for uniaxial compression tests were shaped to cubes measuring 15×15×15 mm 3 in the case of marble and 12.5×12.5×12.5 mm 3 in the case of alumina-based ceramic. The specimens were subjected to uniaxial compression at a constant load up to their macroscopic failure using a DVT GP D NN tensile-and-compression testing machine (Devotrans, Inc.). A simultaneous registration of the surface velocity was carried out, including the ultrafast catastrophic stage of failure, using a Polytec laser Doppler vibrometer. The method of laser Doppler vibrometry allows measuring the velocity parallel to the direction of lasing within the laser beam spot. The laser in the experiments was adjusted perpendicular to the lateral surface of the specimen, so the experimentally obtained velocity values corresponded to the normal component of the lateral surface velocity. The frequency recorded during the experiments ranged from 48 to 250 kHz, the precision of measurements allows for a registration of the velocity to 0.1 µm/s, and the laser spot diameter in the gage site was about 50 µm. For statistical computing here we use the codes by R Core Team (2018) and MATLAB (2016). 3. Experimental results and their analysis The obtained sequence of velocity measurements is a time series x ( t j ) whose sequence of values was recorded at discrete instants of time t j = t 0 + j ⋅ Δ t , j = 1, 2, ..., n, where Δ t is the time step determined by the frequency measurement. In Fig. 1, one can see two parts of the time series to analyze, which correspond to marble and ceramic specimens and pass to the catastrophic stage at the end. One can see that these time series are essentially non stationary. In the case of marble at the final time interval there is a sharp rise in velocity amplitude of the same sign (Fig. 1a), which reflects the transition of the deformation to the catastrophic stage of failure. In the matter of ceramic one can see more rapid oscillation amplitude increase with the alternating velocity sign (Fig. 1b). These are examples of two different evolution scenarios at the final stage of deformation just before failure. Let us consider the evolution of the specimen surface velocity during loading using the methods of mathematical statistics. We start with the autocorrelation functions for this time series that are shown in Fig. 2. The shapes of these curves indicate that the process in both cases has a high degree of autocorrelation, though including the inhomogeneous last parts of the signals. Autocorrelation decreases with increasing time shifts (lags) but still exceeds the significance bounds. 2. Materials and methods