Functional-Friction Networks: New Insights on the Laboratory Earthquakes

We report some new applications of functional complex networks on acoustic emission waveforms from frictional interfaces. Our results show that laboratory faults undergo a sequence of generic phases as well as strengthening, weakening or fast-slip and slow-slip leading to healing. Also, using functional networks, we extend the dissipated energy due to acoustic emission signals in terms of short-term and long-term features of events. We show that the transition from regular to slow ruptures can have an additional production from the critical rupture class similar to the direct observations of this phenomenon in the transparent samples. Furthermore, we demonstrate detailed sub-micron evolution of the interface due to the short-term evolution of rupture tip, which is represented by phenomenological description of the modularity rates. In addition, we found nucleation phase of each single event for most amplified events follows a nearly constant time scale, corresponding to initial strengthening of interfaces.


1) Introduction -
The 2011 Tohoku-Oki earthquake (M w =9.0) showed how a slow slip phenomenon can 1 lead to destructive ruptures with emerging successive rupture transitions from slow to sub-Rayleigh and su-2 per-shear ruptures [1][2][3]. Such a dramatic transition from creep fronts or slow slip events (SSE) to fast and 3 sometimes critical ruptures has been reported in laboratory earthquakes: slow phase to fast, slow to critical 4 phase and regular (fast) ruptures to super shears have been examined in transparent and rock materials [4][5][6][7][8]. 5 The theoretical studies of rupture transition are focused mostly on regular rupture to super shear. The main 6 controlling parameters of the models are characteristic length of the initial rupture and critical value (i.e., 7 seismic ratio) regarding friction force [9][10][11]. Recently, a few numerical models have shown the evolution of 8 creeping faults or SSE into critical earthquakes [12][13]. These models use the coupling of friction laws with 9 heat and pore pressure. The assumption of releasing massive thermal energy is a vital part of the employed 10 coupled equations of the models. In these models, the weakening of the frictional-rock interface (i.e., fault) is 11 obviously due to rapid shear heating of pore fluids and does not accord with the aforementioned laboratory 12 friction experiments. Some other simpler models [14]-based on state-and-rate equations-capture essen-tials of rupture transitions in dry interfaces. However, these models are unable to predict all features of 14 the laboratory observations. Thus, in the latter models, the real mechanism of velocity weakening or 15 velocity strengthening in terms of friction evolution in micro and sub-micron scales are not known [6]. 16 In addition to these developments, recent precise laboratory measurements have presented new insights into 17 rupture events with micro-seconds resolution [6,15]. Among these findings, universal trends of appropri-  24 To support the aforementioned ideas, the authors used complex network techniques to extract "hid-25 den" information from recorded waveforms as well as reconstructed or recorded images through simple fric-26 tion-tests of glassy and rock samples [7]. We showed that functional networks constructed over sub-micron 27 events ("precursor or foreshocks" laboratory earthquakes) can unravel the possible regime of the rupture while 28 the changes of dynamic phases of network parameters are correlated with weak to very powerful events(i.e., 29 energy spectrum of events) [7]. Here, using the proposed functional friction networks, we explore the transi- 30 tion regimes of ruptures in Westerly granite-frictional interfaces. To this aim, we use several network parame- 31 ter spaces while the energy of the obtained networks (and then acoustic pulses) is analyzed using modularity 32 profiles. Interestingly, we find that our phenomenological formulation based on the functional friction net- 33 works presents a solution to dramatic rupture transitions in laboratory scale. Building on the evolutionary 34 phases of modularity's indexes, we also present a new way of estimating fracture energy where a glass transi-35 tion period is matched with a rapid evolutionary phase of module fast-growth. Our results from the trends of 36 fracture energy in different regimes of ruptures confirmed the recent theoretical and numerical studies regard- 37 ing the physics of fracture energy. Furthermore, a slip-weakening model, constructed upon the frictional re-38 sistivity of the interfaces, is proposed where the resistivity-slip rate phase space is related to our networks' 39 attributes. To complete our analysis, we developed a simple mean-field approach on the power law distribu-40 tion of centrality, which lead to a phase diagram with a fit-get-rich-phase while we change the control parame- 41 ter of the model. We show how the model can reach a gel-like or condensed state and interpret the "gel-like" 3 duration of 204.8 µs (recorded at 5 MHz) while the three main stick-slip events occurred. The experiment was 48 servo-controlled using an axial strain rate of 6

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Here we review the previously introduced algorithm. The algorithm includes the following five steps:

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(2) Each time series is divided according to maximum segmentation; we consider each recorded point in each 74 We put the length of each segment as unit. This essentiality considers the high temporal-resolution of the system's 76 evolution, smoothing the raw signals with 20-60 times window (for Lab.EQ1& 2 is equal to 1-3 s  ).

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(3) , ( )  Through this research, we use some of the main key networks' measures. We will use some of these 89 metrics to introduce phase-diagrams and further interpretation of obtained results. Each node is characterized 90 by its degree i k representing the number of links, and the betweenness centrality (B.C) [30]: , between h and j that passes through i. For scale free networks whose degree distribution follows a power law 94 ,the distribution of B.C (or load) of a node-also-follows a power law [54]. In the last section, we will use the

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The network's modularity characteristic is addressed as the quantity of densely connected nodes rela-100 tive to a null model (random model). Based on the role of a node in the network modules or communi-101 ties, each node is assigned to its within-module degree (Z-scores) and its participation coefficient (P).

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High values of Z indicate how well-connected a node is to other nodes in the same module, and P is a

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Furthermore, we will use clustering coefficient as the index of normalized triangles in each existing node. 119 The number of triangles surrounding a node in the constructed networks, denoted by i T , is used to define the 120 clustering coefficient:  Fig.2.b, we have compared the skewness versus the 202 mean modularity for the Lab.EQ2 events. An excellent collapsing of data in Q   shows an inverse correla-203 tion of the leftward asymmetric shape of the acoustic waveforms regarding the regime of ruptures. This is a 204 universal feature of "crackling noise" systems which has been allocated to the nature of the dissipation energy 205 such as eddy currents in Barkhusen noise (movement of magnetic domain wall) or threshold strengthening in 206 moment rate profiles in natural earthquakes [34][35]. We conclude that more deviation from symmetry is the 207 signature of ruptures with relatively high energy and critical ruptures, while approaching a less asymmetric 208 shape indicates ruptures with lower energy. Then, understanding the details of micro-seconds evolution of 209 Q-profiles helps to evaluate the general shape of crackling noises. 210 Next, to distinguish the role of main deformation phase I, we define a parameter of resistivity against 211 motion, denoted by resistivity: Fig.3a, we 212 have shown R-profiles from Lab.EQ1. In Fig.3c, we have also illustrated the collapsing of events from 213 Lab.EqQ1 in a normalized space : max . Next, we show that norm R Q   plane reflects a similar acceleration -deceleration phenomenon in fric-240 tional interfaces under high velocity (Fig.3.b) [36]. Here norm R is the normalized value of resistivity profile. To fit 241 a model on norm R Q   plane (Fig.3b and Fig.4), we use the fact that the duration of the second phase The rate of Q(t) per each generic phase in Q-profiles.

R
R-profiles as the reciprocal of Q-profiles. We use this value to emphasis on the first evolutionary phase as the nucleation and main deformation phase of micro-cracks.


Power exponent in

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Assuming independence of scaling exponents to each other and approaching 0   , R 3 class is van-350 ished ( Fig.5a- the phase III (Fig.1a). The interpretation is as follows: a fast-short time fracturing (phase I) induces a very 372 fast increase in the temperature of a tiny "process zone", which cools in typical time characteristics. The main 373 component of the theory is that the whole of the fracture energy is transferred to heat in the process 374 zone (Fig.8). This can be assumed as an effective temperature approach, well described in annealing-

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The increased temperature with respect to a reference temperature (room temperature) for 1D case 378 with an imminent released energy as the source is given by ,h is the thickness of the process zone in which the en-381 ergy rapidly dissipates, T D is the thermal diffusivity ,  is the energy released in phase I (or fracture energy), 382 and t is the cooling time. With first-order approximation, we estimate t as follows: The implication of the obtained relation is remarkable when we analyse the fracture energy variation while 387 considering the rupture velocity-regimes:

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The obtained result is comparable with the results of several investigations regarding the weak in-396 crease of  with rupture velocity [43][44]. We also can extend the concept of the rate of energy dissipation in a "slow class" regime (R3). In other words, how does fracture energy change in slow-slip regime? In this case, 398 0   and then decreasing T  yields decreasing  . In other words, far from the cross-over point (from the 399 R3 to the R2 class) and in the slow rupture class, the rate of energy dissipation is low. The latter conclusion 400 has been proved numerically in [45] and recently in [46] through spring-block models. In Fig.7

6) Mean-field Modeling of Friction-networks -
We consider a mean-field modeling approach 426 and use the power-law nature of the load distribution (or centrality- Fig.9). To this aim, we assume a nearly 427 constant exponent of the centrality distribution: ( ) , 3 P BC BC     (Fig.9a). To measure the power coeffi-428 cient of the centrality distribution, we used likelihood estimators for fitting the power-law distribution to data 429 (Fig.9b). Also, goodness-of-fit test using the Kolmogorov-Smirnov method was used, calculating p-value

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We now use some of the introduced scaling relations to simplify the relation. The first scaling law re-451 lates spatial mean centrality to the clustering coefficient (Fig.9c: proof rationalizes the interpretation of centrality distribution. , 2 1 A  where we considered an average value for node's degree and the expo-507 nent. Indeed the FGR state corresponds to the R2 or R1class. Also, we note the rapid decrease of  (or  ) 508 accelerates the growth of max k ,coinciding with the approach to the tail of the parameter space in Fig.9a. To 509 present a possible mechanical-interpretation of the condensed state of friction networks, we notice that reach-510 ing a possible condensed state is equal to approaching a less-collective motion of atoms (i.e., rupture). In par-511 ticular, the interface does not show collective local stick-lip motion, and the state of the motion of atoms (i.e., 512 particles or nodes) is controlled by a "single" atom or very condensed modules of atoms. This phenomenon is 513 the condition of incommensurate contacts; they can prevent collective atomic motion (or collective particle 514 motion), leading to a "superlubricity" (or super-rupture) state of friction [58-59].