Correlating the force network evolution and dynamics in slider experiments
CCorrelating the force network evolution and dynamics in slider experiments
Chao
Cheng , ∗ , Aghil
Abed Zadeh , ∗∗ , and Lou
Kondic , ∗∗∗ Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 07102 Department of Physics and Department of Neurobiology, Duke University, Durham, NC, 27708
Abstract.
The experiments involving a slider moving on top of granular media consisting of photoelastic parti-cles in two dimensions have uncovered elaborate dynamics that may vary from continuous motion to crackling,periodic motion, and stick-slip type of behavior. We establish that there is a clear correlation between the sliderdynamics and the response of the force network that spontaneously develop in the granular system. This correla-tion is established by application of the persistence homology that allows for formulation of objective measuresfor quantification of time-dependent force networks. We find that correlation between the slider dynamicsand the force network properties is particularly strong in the dynamical regime characterized by well-definedstick-slip type of dynamics.
A wide range of systems exhibit intermittent dynamics asthey are slowly loaded, with di ff erent dynamical regimesgoverning many industrial and natural phenomena. Inthese systems, the energy is loaded gradually with a stableconfiguration and then is dissipated in fast dynamics withmicroscopic and macroscopic rearrangements [1]. Exam-ples are fracture [2, 3], magnetization [4], and seismic ac-tivities [3, 5] such as earthquakes, in which the slowlyloaded energy relaxes via fast reconfiguration. This inter-mittent behavior has been observed in a number of granu-lar experiments and simulations [6–9]. In analyzing suchbehavior, a significant progress has been reached by study-ing the dynamics of a slider coupled with the boundary ofa granular system. A slider can exhibit a wide variety ofdynamics, including continuous flows and periodic or in-termittent stick-slip behavior [8, 10–12].While a significant amount of research on exploringintermittent dynamics of granular systems has been car-ried out, not much is known about the connection be-tween particle-scale response and the global dynamics,in particular for experimental systems. In slider exper-iments [8, 10], see also Fig. 1, it is possible to mea-sure particle scale response by using photoelastic tech-niques. These techniques allow for extracting dynamic in-formation about evolving particle interactions which typi-cally involve meso-scale force networks (so-called ‘forcechains’). Analysis of such time-dependent weighted net-works is not a simple task, and it has evolved throughlast decades in a variety of di ff erent directions, includingforce network ensemble [13, 14], statistics-based meth-ods [15, 16], and network type of analysis [17, 18]. In ∗ e-mail: [email protected] ∗∗ e-mail: [email protected] ∗∗∗ e-mail: [email protected] Figure 1: (a) Sketch of the experiment. A slider sits ontop of a 2D granular system with photoelastic disks, andis connected to a stage by a spring of constant k , pulledby a constant speed c . A force gouge measures the force f ; the granular medium is imaged with fast cameras. (b)Photoelastic response during loading [10]. Reprinted withpermission from [10].the present work, we will consider application of persis-tent homology (PH), which allows for formulating preciseand objective measures of static and dynamic propertiesof the force networks. This approach has been used exten-sively in analysis of the data obtained via discrete elementsimulations in the context of dry granular matter [19, 20]and suspensions [21], but its application to experimentaldata has so far been rather sparse [22, 23]. We show thatthis method allows to develop clear correlations betweenthe static and dynamic properties of the force networkson micro- and meso-scale and the macro-scale system dy-namics. a r X i v : . [ c ond - m a t . s o f t ] J a n Techniques
In our experiments, as shown in Fig. 1(a), a stage pulls a2D frictional slider with toothed bottom, of a fixed lengthof 25 cm and a mass of M =
85 g . The stage, which movesat constant speed c , is connected to the slider by a linearspring of sti ff ness K . The slider rests on a vertical bedof fixed depth L = . f experienced by the spring.We consider three experiments characterized by dif-ferent configurations of c and K : Exp. 1: K =
14 N / m, c = . / s; Exp. 2: K =
70 N / m, c = . / s, andExp. 3: K =
70 N / m, c = . / s. The total number ofanalyzed frames (images) is 30,000 for each experiment,corresponding to 250 seconds of physical time. The goal of the image processing in this study is to revealclear force signal and reduce noise e ff ects as much as pos-sible. As the fast imaging in our experiments constrainsthe resolution of images, we use brightness method to cap-ture force information, which works better that G methodfor the type of data collected, see [24]; similar approachwas used in [23]. We first remove background noise fromthe original images by applying a filter that removes pixelsof brightness below chosen threshold value so to removelow light area and particle textures. Multiple threshold val-ues were investigated, giving no quantitatively di ff erencein the results of the topological analysis that follow; wetypically use threshold value of 90 (the maximum bright-ness is 255), which is appropriate for capturing the rel-evant information. After thresholding, the image bright-ness is linearly mapped to 0-255 range. MATLAB built-in functions imerode and imdilate were applied to slightlydilate the bright regions so to fill the gaps between neigh-boring particles where in force chains are connected, andthen to erode away the unwanted excessive dilation to re-store the force networks with more accuracy. Fig. 2 showsan example of image processing; in our computations dis-cussed in what follows, we use grey scale version of thefigures such as Fig. 2 for the purpose of computing con-sidered topological measures. Persistence homology (PH) allows for formulating objec-tive measures describing force networks in both simula-tions and experiments. Analysis of experimental data,such as the ones considered here, presents some challengeswhich are discussed in some detail in [20].Each experimental image can be considered as an ar-ray of pixel brightness θ ∈ [0 , θ birth , θ death ) and each point rep-resents an object that could be either connected compo-nent (chain) or a loop (cycle). The lifespan of an objectis defined as θ birth − θ death , measuring how long the objectlasts as the threshold is varied. Total persistence (TP) ofa PD is defined as the sum of all lifespan of the points,TP(PD) = (cid:80) ( θ birth ,θ death ) ∈ PD ( θ birth − θ death ), which further re-duce the complexity of force networks to a single num-ber [25]. Note that TP is influenced by both how manycomponents there are, and by their lifespans.Another quantity related to PDs is the distance (or dif-ference) between them. The distance measures essentiallythe cost of mapping points in one PD to those in anotherPD; in the case of di ff erent number of points, the extraones are mapped to the diagonal. In particular, the degree-q Wasserstein distance between two persistence diagramsPD and PD’ is defined as d Wq ( PD , PD (cid:48) ) = inf γ :PD → PD (cid:48) (cid:88) p ∈ PD (cid:107) p − γ ( p ) (cid:107) q ∞ / q , where γ is a bijection between points from PD to PD’, γ : PD → PD (cid:48) . In the present work we use q = Figure 3 shows the calculated velocity of the slider and themeasured force, f , on the spring. This figure illustratesclearly the slider’s dynamics. We note that Exp. 1 exhibitscrackling stick-slip behavior as the driving rate is small.During a stick, the spring builds up the stress, while theslider is almost fixed, until the spring eventually yields,leading to a sharp velocity jump and drop of the force.igure 3: Sliders’ velocity and spring force for the consid-ered experiments: (a) Exp. 1, (b) Exp. 2, (c) Exp. 3.The system behaves more similarly to a continuous flowfor Exps. 2 and 3, as also discussed in [10].Figure 4: W q = x t , y t , with the Figure 5: Force on the slider and W2 distance (data fromFig. 3a) and 4a)).data ( x , x , · · · , x m ), and ( y , y , · · · , y m ). The cross-covariance is defined by c x y ( k ) = m m (cid:88) t = ( x t − ¯ x )( y t − k − ¯ y ) (1)where ¯ x = (cid:80) mi = x i / m , ¯ y = (cid:80) mi = y i / m , and k = , ± , ± , · · · is the chosen lag. When m is outside therange of y , y m =
0. Note that for the positive lag k , x t iscorrelating with y t − k (at earlier time), which means that wemay be able to use such correlation to predict the future x from earlier y . Finally, we define the sample standard de-viations of the series as • s x = √ c xx (0), where c xx (0) = Var ( x ). • s y = (cid:112) c yy (0), where c yy (0) = Var ( y ).The cross-correlation coe ffi cient is given by r x y ( k ) = c x y ( k ) s x s y , k = , ± , ± , · · · (2)Figure 6: Cross correlation coe ffi cient as a function of lag k (in seconds). r f ∗ and r v ∗ for Exp. 1, where ∗ stands eitherfor W2 distance (solid line), or for TP (dashed line).Figure 6 shows the cross-correlation coe ffi cients forExp. 1, where r f ∗ , r v ∗ correspond to the cross-correlationcoe ffi cient between the f or v and the measure of interest*, which can be w for W2 distance or t for TP. Focusingfirst at the results without lag, k =
0, we note that the r f ∗ orrelations are higher than the r v ∗ ones; we expect thatthis is due to the fact that the velocity data were obtainedby taking a discrete derivative of the slider position data,introducing further noise which may blur the actual signal.The r f ∗ results show that the correlation is higher for theTP data, which is not surprising since TP is expected toreflect the force on the slider, while W2 distance measuresthe temporal di ff erence in PDs.Considering next the results for non-zero lags, we notea di ff erent behavior of r f ∗ and r v ∗ curves, with r v ∗ curvesrising from negative to positive as lag is increased. Suchdi ff erence results from the fact that the structure of theforce and velocity profiles are rather di ff erent. The veloc-ity profile shows sharp transitions, while the force profileslowly builds up during the stick periods and drop dramat-ically at the events. More importantly, we note that withina reasonable range of positive lags, the r f w and r f t are stillsignificant in size, suggesting a potential for predictabil-ity. It should be noted however, that since the main part ofthe data is in a "stick" region, the correlation will naturallyweight more on these data points. A more insightful pro-cedure would involve correlating the measures just beforea slip event. Such an analysis would however require moredata points, and possibly also more detailed experimentalinput and therefore we leave if for the future work.Before closing, we note that for Exp. 2 and 3 we obtainconsistent results, however the correlations are weaker;e.g., the max of r f t goes down from 0.65 (Exp. 1), to 0.45(Exp. 2), and to 0.3 (Exp. 3). Despite the fact that allthree experiments fall into the same category of stick-slipin the dynamic phase diagram [10], clearly (see Fig. 3) theslip events are much stronger and better defined for Exp. 1,suggesting that in particular for such situation the persis-tence measures provide insightful information. We find that the tools of persistent homology (PH) allowfor correlating the dynamics of a slider and the photoelas-tic response of granular particles. In particular stick andslip regimes of the slider dynamics are well captured bythe PH measures. These results suggest that there is a po-tential for developing predictive capabilities by analyzingthe response of the force network to an external perturba-tion. One open question is how precise should the infor-mation about the forces between the granular particles beso to allow for further development of this potential. Wehope that our results set up a stage for this future work.
Acknowledgements
The authors acknowledge many insightful conversationswith R. Basak, M. Carlevaro, K. Daniels, M. Kramar, K.Mischaikow, J.Morris, L. Pugnaloni, A. Singh, J. Socolar,H. Zheng and J. Barés. CC and LK acknowledge supportby the ARO grant No. W911NF1810184.
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