Revista Mexicana de F´ısica S 58 (1) 116–120

JUNIO 2012

Correlations and scaling in a simple sliding spring-block model C.A. Vargas Departamento de Ciencias B´asicas, Universidad Aut´onoma Metropolitana-Azcapotzalco, Av. San Pablo No. 180, Col. Reynosa, M´exico D.F. 02200, M´exico. e-mail: [email protected] E. Basurto Departamento de Ciencias B´asicas, Universidad Aut´onoma Metropolitana-Azcapotzalco, Av. San Pablo No. 180, Col. Reynosa, M´exico D.F. 02200, M´exico L. Guzm´an-Vargas Unidad Profesional Interdisciplinaria en Ingenier´ıa y Tecnolog´ıas Avanzadas, Instituto Polit´ecnico Nacional, Av. IPN No. 2580, L. Ticom´an, M´exico D.F. 07340, M´exico, e-mail: [email protected] F. Angulo-Brown Departamento de F´ısica, Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional, Edif. No. 9 U.P. Zacatenco, M´exico D.F., 07738, M´exico. Recibido el 23 de Marzo de 2010; aceptado el 27 de Abril de 2011 In this work we analyze some statistical properties of both sliding size sequences and interevent time series of an experimental spring–block array. This experiment is used to mimic certain dynamical characteristics of actual seismic faults. In a previous work we showed that this spring-block system reproduces Gutenberg-Richter type laws for the sliding size distributions and that also mimics other important features of the real seismicity. In the present work we show some further characteristics by means of two methods stemming from nonlinear dynamics: The detrended fluctuation analysis and the Higuchi’s fractal dimension. Keywords: Spring-block; seismicity; scaling properties. En este trabajo analizamos algunas propiedades estad´ısticas tanto de tama˜no de deslizamiento como de series de tiempo interevento de un arreglo experimental resorte–bloque. Este experimento se us´o para mimetizar ciertas caracter´ısticas de fallas s´ısmicas reales. En un trabajo previo mostramos, que el sistema resorte–bloque reproduce leyes de tipo Gutenberg–Richter para las distribuciones de taman˜ o de deslizamiento y que tambi´en mimetiza m´as caracter´ısticas importantes de la sismicidad real. En este trabajo mostramos otras caracter´ısticas obtenidas por medio de dos m´etodos provenientes de la din´amica no lineal: DFA y la dimensi´on fractal de Higuchi. Descriptores: Bloque–resorte; sismicidad; propiedades de escalamiento. PACS: 68.35.Ja; 68.35.Ct; 91.30.Ga; 91.32.Jk

1. Introduction Recently, we published an article [1] where we used a simple spring-block system as a qualitative analogous of a seismic fault. This experimental set up was constructed to mimic the general dynamical behavior of a seismic fault based on the pioneering work by Burridge and Knopoff [2], and similar to the experimental arrangement proposed by Feder and Feder [3]. The system studied by Burridge and Knopoff was a linear spring-block array described by means of a set of differential equations. Later, many authors have extended this idea towards two an three dimension arrays treated by means of cellular automata [6-16]. Most of the articles were based on the concept of self-organized criticality (SOC). The idea of SOC was introduced by Bak et al. [4], as a general organizing principle governing the behavior of open spatially extended dynamical systems with both temporal and spatial degrees of freedom. According to this principle, composite open systems having many interacting elements organize themselves into a stationary critical state with no length of

time scales others than those imposed by the finite size of the systems. The critical state is characterized by spatial and temporal power laws. In such a state, a smaller event often begins a chain reaction that can lead to a catastrophe. This behavior is reminiscent of the dynamical mechanism of seisms. Nowadays some seismologists [5], assert that SOC is a feasible dynamical explanation of the seismic process. Thus, we can say that SOC is perharps the first solid basis underlying the empirical power-laws of seismology. As asserted by Feder and Feder [3], the simple stick–slip process of dragging a block over a rough surface can qualitatively mimic the general dynamic mechanism of earthquakes. Seismic slip is associated to earthquakes and refers to motion due to a frictional instability between the two sides of a fault. After undergoing seismic slip, the formely sliding rock experiences an interval of little or no motion during which the stress recharges. The elastic strain monotonically increases on a fault, resulting in an increasing of stress. Once the stress accumulates to the breaking strength, this region becomes unstable and rapidly rebounds or slips to a lower, more stable

CORRELATIONS AND SCALING IN A SIMPLE SLIDING SPRING-BLOCK MODEL

state. All this process occurs in a spring–block system. In our previous work [1] we found that an experimental springblock array has some general properties of actual seismicity. In the present work we analyze the statistics of the sliding size distribution and also the interevent times for the case of homogeneous interfaces between the sliding block and the surface where the sliding occurs by means of two methods: The detrented fluctuation analysis (DFA) and the so called Higuchi’s fractal dimension. The paper is organized as follows: In Sec. 2, we describe the experimental setup of the spring-block array; in Sec. 3 we present the methods used to study the statistics of sliding data; in Sec. 4 we discuss the results and finally we present the concluding remarks.

2.

3.1.

F IGURE 1. Schematic top view of the experimental setup. a) Sliding block with sand-paper in the bottom surface, b) load cell, c) rough surface mounted in an aluminum base, d) suspension glass plates, e) steel spheres, f) DC motor, with gear mechanism, g) elastic rope.

Experimental Setup

The experiment series reported in this work has been performed in a simple spring-block system that we have devised in order to model a seismic fault. We have included a complete description of this apparatus in Ref. 1. Such apparatus is presented schematically in Fig. 1. The interfaces of interest that act as the seismic fault surfaces have been constructed with one aluminum block which is a solid square (0.1 m (sides) and 0.025 m (high)) with a mass of 0.5 kg, supported by an aluminum plate. Both pieces have been recovered with sandpaper of the same grade on their respective contact surfaces. The aluminum plate is sustained over a very low friction table which was constructed with a pair of glass plates with steel balls between them. The previous system is held fixed on a heavy metal frame to provide it mechanical stability. The aluminum block is pulled at constant speed of 2.22 × 10−4 ms−1 with an elastic rope (fishing line) attached to two series reducing system 60:1 variable speed DC motor (Baldor CD5319). We fixed a charge cell (Omega LCL) to the metal frame in order to measure the force exerted by the aluminum plate over it. That is, the charge cell acts like a bumper between the aluminum plate and the metalic frame. During the experimental run the system has succesive stick– slip events like those shown in Fig. 2.

3.

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F IGURE 2. Time series (force as a time function) recorded with a charge cell.

after doing this, the resulting series is divided into boxes of size n. For each box, a straight line is fitted to the points, yn (k). Next, the line points are subtracted from the integrated series, y(k), in each box. The root mean square fluctuation of the integrated and detrended series is calculated by means of v u N u1 X 2 F (n) = t [y(k) − yn (k)] , (1) N k=1

Methods Detrended Fluctuation Analysis

In Ref. 17 Peng et al. introduced detrended fluctuation analysis (DFA) to detect correlations in irregular and nonstationary signals. The DFA method is based on the classical random walk variations. To illustrate DFA, we depart from an initial time series B(i) (of length N ), with Bave =

N 1 X B(i). N j=1

First, this series is integrated, y(k) =

k X i=1

[B(i) − Bave ] ,

this process is taken over several scales (box sizes) to obtain a power law behavior F (n) ∝ nγ , with γ an exponent which reflects self-similar properties of the signal. It is known that γ = 0.5 corresponds to white noise (non correlated signal), γ = 1 corresponds 1/f noise and γ = 1.5 represents a Brownian motion [18]. The DFA-exponent γ is related with the spectral exponent β by means of β = 2γ − 1 [18], thus also with the fractal dimension D (see next Subsection) by γ = 3 − D. 3.2.

Higuchi Method

One method which is particularly appropiated to estimate the fractal dimension of short nonstationary time series is the Rev. Mex. Fis. S 58 (1) (2012) 116–120

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Higuchi method [19,20]. In Ref. 21 Higuchi proposed a method to calculate fractal dimension of self-affine curves in terms of the slope of the straight line that fits the length of the curve versus the time interval (the lag) in a double log plot. The method consists in considering a finite set of data taken at ν1 , ν2 , ...νN . From this series we construct new time k series, νm , defined as ¸ ¶ µ · N −k ·k ; ν(m), ν(m + k), ν(m + 2k), ..., ν m + k with

m = 1, 2, 3, ..., k. (2)

where [ ] denotes Gauss’ notation, that is, the bigger integer and m and k are integers that indicate the initial time and the k interval time, respectively. The length of the curve νm , is defined as: " [ N −m ] k 1 ³ X Lm (k) = |ν(m + ik) k i=1 ´ N −1 − ν(m + (i − 1)k)| £ N −m ¤ k

F IGURE 4. Log F (n) vs. log n from sliding size sequences for several runs.

# k

(3)

and the term (N − 1)/[(N − m)/k]k represents a normalization factor. Then, the length of the curve for the time interval k is given by hL(k)i : the average value over k sets Lm (k). Finally, if hL(k)i ∝ k −D , then the curve is fractal with dimension D [21]. The fractal dimension is related to the spectral exponent β by means of β = 5 − 2D [21]. Note that this relationship is valid for 1 < D < 2 and 1 < β < 3. F IGURE 5. Log F (n) vs. log n from inter-event time sequences for several runs.

4. Data Analysis In Fig. 2 we show a segment of a typical time series of force values registred at the charge cell. In this figure we can see how seccesively the force between the rugged surfaces increases (with a saw profile) when the blocks sticks to the rug-

F IGURE 3. Example of a typical graph of the sliding size sequence as a function of time.

ged track and abruptly decreases when the block slips. We construct the cummulative distribution of sliding events N (s) with amplitude larger than s, and in all experimental runs we find Gutenberg–Richter (G-R) type laws of the form N (S > s) ∼ S −b (see Figs. 3, 4, 5 and 6 of Ref. 1). In Fig. 2, the interfaces have both sandpapers with the same grade (homogeneous case). Figure 3, shows a time chart of sliding sizes against time. In what follows we analyze some statistical propierties of five succesive runs without changing the sandpapers at the block-track interfaces; that is, we study a kind of aging effect similar to that reported in our Ref. 1, but by means of two new methods: The DFA and Higuchi’s fractal dimension. Here we focus our attention on scaling properties of sliding and interevent time sequences for homogeneous interfaces. In Fig. 4, we show a log-log plot of F(n) (Eq. (1)) against the box sizes n, for the sliding sizes. As we can see, when the sandpapers (Fandelli’s 80 grade) are new (run 1) the DFA-exponent is γ ≈ 0.6 indicating the presence of peristent long-range correlations, meaning that a large (compared to the average) value is more likely to be followed by a large value or viceversa [17]. In Fig. 4 we also

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see that for runs from 2 to 5, that is, with the aging effect of the sandpapers the DFA-exponent tends to γ ≈ 0.5, that is, a white noise with loss of correlations. The larger the wear of the asperities the larger the loss of memory in succesive slidings. This result is very consistent with real seismicity, where tipically after a great earthquake a sequence of aftershocks evidently correlated with the mainshock occurs. The big earthquakes are linked with fault interfaces dominated by big asperities. When the asperities are of small size the upper limit for the magnitude of a mainshock disminishes. For the case of the interevent times (Fig. 5) the values of the DFAexponents go from γ ≈ 0.5 up to γ ≈ 0.55; that is, from uncorrelated sequences to slightly persistent fluctuations, indicating the presence of positive correlations. It is convenient to remark that in both cases (Fig. 4 and Fig. 5) the dynamical behavior is relatively near to uncorrelated signals (white noise). For the case of the Higuchi’s method, we first integrate both the sliding size and the interevent time series. As we can see in Fig. 6, for the fifth run the fractal dimension is D = 1.5 (Brownian noise) corresponding to the original white noise signal which once integrated gives Brownian noise. That is, a result completely consistent with γ = 0.5 for the most wore interface. For the first run D = 1.46 corresponding to a persistent sequence of sliding sizes. If one uses the simple relationship α + 1 = 3 − D [22], we get α ≈ 0.54, which is in reasonable agreement with the value obtained by means of DFA (α = 0.6). In Fig. 7 we depict the log-log plot whose slope gives the Higuchi’s fractal dimension for the sequence of interevent times. In this case, the first run leads to D ≈ 1.5 which corresponds to integrated uncorrelated fluctuations whereas for the fifth run, the value is D = 1.4 which is also consistent with the results from DFA analysis.

F IGURE 6. Log-log plot of hL(k)i vs k for sliding size sequences. We observe that the fractal dimension decreases as the run number increases. The original series was integrated to improve the resolution.

F IGURE 7. Log-log plot of hL(k)i vs k for interevent time sequences. The original series was integrated to improve the resolution.

5.

Concluding remarks

In our previous work [1], we reported that the experimental array consisting in a simple a spring–block system presents an aging effect in the sandpapers, that is, as the number of experimental runs increases, the largest characteristic event diminishes. We also found that the largest characteristic event depends on the mass of the sliding block. The greater the mass is, the greater the maximum event is. In all the experimental runs a G–R type law was found. In the present article we have analyzed only the case of homogeneus interfaces (the same sandpaper covering the sliding block and the fixed track). This analysis was made by means of two additional methods: The DFA and the Higuchi’s fractal dimension. The sandpaper used was of Fandelli’s 80 grade, that is a sandpaper with moderate size of grain. This fact is reflected in Fig. 4 where for the first run (new sandpaper) the sliding size sequence shows only a moderate correlation (α ≈ 0.6) corresponding to the presence of moderate mains events. However, when the sandpapers suffer an aging effect, that is, run 5, the DFA–exponent is α ≈ 0.5, that is, a sliding sequence without correlations (withe noise) due to the deterioration of the asperities, resulting in a sliding sequence without main events and therefore without aftershock clusters correlated with main slidings. This same behavior is reflected through the Higuchi’s dimension D, which goes from D=1.46 (correlated case) up to D=1.5, a Brownian motion corresponding to the integration of the white noise present in the original sliding size sequence. We also analyze the interevent sequences. In this case we found that the DFA-exponents only exhibit a behavior very close to white noise, that is the relative small size of the grains (Fandelli’s 80 grade) does not produce big events enough to clearly identify time correlations as those appearing in interfaces with big

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asperities. That is the resolution of interevent times is not of good quality. In summary, by means of DFA and Higuchi’s methods we observe some of main properties of seismicity only corresponding to sliding size sequences, but for interval times our results are not clear, and asperities of bigger sizes will be necessary.

Acknowledgments LGV and FAB thank COFA-EDI-IPN for partial financial support

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