Vibration can enhance stick-slip behavior for granular friction
NNoname manuscript No. (will be inserted by the editor)
Vibration can enhance stick-slip behavior forgranular friction
Abram H. Clark, Robert P. Behringer,and Jacqueline Krim
Received: date / Accepted: date
Abstract
We experimentally study the frictional behavior of a two-dimensionalslider pulled slowly over a granular substrate comprised of photoelastic disks.The slider is vibrated at frequencies ranging from 0 to 30 Hz in a direction par-allel to sliding. The applied vibrations have constant peak acceleration, whichresults in constant average friction levels. Surprisingly, we find that stick-slipbehavior, where stress slowly builds up and is released in intermittent slips, isenhanced as the frequency of vibration is increased. Our results suggest thatincreasing the frequency of vibration may help to combine many smaller rear-rangements into fewer, but larger, avalanche-like slips, a mechanism unique togranular systems. We also examine the manner in which the self-affine charac-ter of the force curves evolves with frequency, and we find additional supportfor this interpretation.
Keywords
Stick-slip · Granular friction · Vibration
A. H. ClarkNaval Postgraduate SchoolDepartment of PhysicsMonterey, CA 93943E-mail: [email protected]. P. BehringerDuke UniversityDepartment of PhysicsDurham, NC 27708J. KrimNorth Carolina State UniversityDepartment of PhysicsRaleigh, NC 27695E-mail: [email protected] a r X i v : . [ c ond - m a t . s o f t ] D ec Abram H. Clark, Robert P. Behringer, and Jacqueline Krim
The frictional response of granular material has broad relevance in earth sci-ence ( e.g. , fault mechanics [1,2,3], landslides [4], sediment erosion [5,6]) andindustry ( e.g. , food [7,8], pharmaceuticals [9], and detergents [10]). Moreover,yield and flow of granular materials in geologic or industrial settings is oftenaccompanied by latent vibrations in the system [11,12]. The interplay betweenvibrations and granular friction has important implications for dynamic earth-quake triggering [13,14] and other avalanche-like behavior [15].Friction in granular systems is not a simple material property: for example,the material friction coefficient µ is only weakly dependent on the grain-grainfriction coefficient [16,17]. Instead, an effective friction coefficient for granu-lar materials arises from the ability of the grains to form anisotropic forcenetworks [18,19], often called force chains. These structures can be correlatedover large distances and can thus slip and flow in collective and complex waysthat are difficult to predict [20,15,21,22,23]. Thus, near yield, the effect ofvibration is magnified, as marginally stable force chains can be disrupted andreorganize into either weaker or stronger configurations. Vibration can leadto compaction, dilation, and/or fluidization of the granular material, depend-ing on the magnitude of the acceleration that the grains experience. [24,25].Factors such as the peak velocity experienced by the vibrating grains and thedirection of vibrations also affect the dynamics, and the relative contributionof each is a matter of great current interest [26,27].Previous studies [28,27,24,29,26,30,31] have shown that increasing ampli-tude of vibration and increasing slider speed can each result in reduced stick-slip behavior and average granular friction levels. In this study, we demonstratethat, under certain conditions, increased frequency of vibration can surpris-ingly enhance stick-slip behavior. We present results for a slider pulled atconstant speed of 5 mm/s over a granular bed. The slider is vibrated in a di-rection parallel to its motion by means of an electromagnetic shaker. We vary f between 0 Hz and 30 Hz and set the displacement amplitude A of the shak-ing such that the dimensionless shaking acceleration Γ ≡ A (2 πf ) /g remainsapproximately constant at a low value of Γ ∼ .
01. Although the mean and thesize of the fluctuations in the friction coefficient both remain relatively con-stant with increasing f , the nature of the slips does change significantly with f . In particular, as f is increased, the pulling force is increasingly characterizedby steady, elastic-like increases and large avalanche-like stress drops. Addition-ally, the self-affine roughness of the force-versus-distance curves evolves withfrequency, with the Hurst exponent [32,33] increasing from approximately 0.5to 0.7 as f is increased from 0 to 30 Hz. To interpret this result, we suggest thatincreasing frequency enhances stick-slip via increased number of oscillationsduring shear. In this scenario, the grains become better organized and morecommensurate with the rough slider during rearrangements, converting manysmaller rearrangements into fewer larger ones. These conditions, particularlythe grain mobility induced by the mechanical vibrations, can increase staticfriction levels [34,35,36] and alter stick slip behavior. ibration can enhance stick-slip behavior for granular friction 3(a)(b) Fig. 1 (a) A schematic of the apparatus. (b) Dimensionless pulling force µ ≡ F/mg versusdistance for one experiment, where the slider is pulled across a granular bed at 5 mm/s andvibrated at 12 Hz. The data in (b) has been notch-filtered to suppress contributions fromthe 12 Hz signal itself.
Our experimental apparatus, depicted schematically in Fig. 1(a), has beendescribed in detail in an earlier publication [29]. It consists of a solid sliderthat is pulled at a constant speed over a two-dimensional granular material.The granular bed is roughly 1.5 meters long and 0.15 meters high. The grainsare bidisperse photoelastic disks with diameters 4 and 5 mm. The slider hasmass M = 0 .
17 kg. The bottom edge of the slider is patterned by half-roundcutouts of diameter comparable, but not equal to, the diameters of the grainsthat it is in contact with. The slider is coupled to a translational stage thatmoves at constant velocity of v = 5 mm/s via a spring with spring constant k = 160 N/m. A digital force sensor is used to record the spring force F asa function of distance traveled, which is then converted to a dimensionlessfriction coefficient µ = F/M g . Fig. 1(b) shows a typical plot of µ versusdistance.Magnets (not shown) are attached to the top of the slider and are positionedwithin two coils. We then drive oscillating current through the coils using afunction generator and an amplifier. This results in oscillatory electromagneticforces on the magnets, which causes the slider to vibrate in a direction parallelto its motion. We drive the shaker at vibration frequency f and fixed amplitudeof the time-varying voltage sent to the shaker. This results in an amplitude A of vibration that decreases as A ∝ f − , as shown in Fig. 2. We estimate A by taking the Fourier transform of the force data and examining the value Abram H. Clark, Robert P. Behringer, and Jacqueline Krim(a) -6 -4 -2 Fig. 2 (a) The amplitude of the applied vibration A plotted versus frequency f , with asolid line showing A ∝ f − . (b) The dimensionless shaking velocity Aω/v , where ω = 2 πf and v = 5 mm/s is the average speed of the slider, is plotted versus f . (c) The dimensionlessacceleration Γ ≡ Aω /g , where g is the gravitational acceleration, is plotted versus f . at the vibration frequency. Since the spring force is linearly proportional todisplacement, we assume the value A in the force signal is linearly proportionalto the displacement amplitude. For the data shown here, we vary f from 0to 30 Hz, and we estimate the dimensionless shaking amplitude to be Γ = A (2 πf ) /g ≈ . v ≈ . v ≈
100 mm/s), inertial or periodic oscillations were observed at the natu-ral frequency of the slider-spring system. At intermediate speeds, (e.g., v ≈
15 mm/s), they observed irregular behavior: not purely stick-slip but not yetdominated by periodic inertial oscillations. Our pulling speed of v = 5 mm/sputs us near the transition between stick-slip and irregular regimes, consistentwith prior reports by Krim, et al. [29] employing the same apparatus. In all experiments, µ begins at zero as the slider is still and the spring is notstretched. As we begin to pull the slider, µ rises during an initial transientphase, which persists over a short distance that is always less than 3 cm, thenreaches a “steady-state” phase, where µ fluctuates around a constant value. Weignore the initial transient phase and focus solely on the steady-state phase.Figure 3 displays typical data segments during the steady-state phase for µ and ∆µ , the discrete difference of µ , for no vibration and f = 28 Hz. We obtain ∆µ by subtracting neighboring pairs of data points in µ , such that positive valuesof ∆µ represent increasing µ and negative values of ∆µ represent decreasing µ (stress drops). Figure 4 shows a statistical characterization of the data shownin Fig. 3 for all frequencies studied. As can be seen in Fig. 4(a) the mean pulling ibration can enhance stick-slip behavior for granular friction 5(a) (b) Fig. 3
Plots of µ and ∆µ (defined as the difference in µ between each successive pair ofdata points) as a function of distance for slider velocity of 5 mm/s. Panel (a) shows datawith vibration at 0 Hz, and panel (b) shows data with vibration at 28 Hz.(a) (b) (c) Fig. 4 (a) The mean dimensionless pulling force (cid:104) µ (cid:105) , (b) the standard deviation σ µ of µ ,and (c) the skewness α of ∆µ are each plotted as a function of vibration frequency f . Solidblue lines are linear fits to the data, which show that (cid:104) µ (cid:105) and σ µ are both nearly independentof f , but α decreases strongly with f . force (cid:104) µ (cid:105) is virtually independent of f : increasing frequency clearly does notreduce the average granular friction for the regime studied here. Additionally,Fig. 4(b) shows that the standard deviation of the friction coefficient, σ µ , isalso relatively constant for increasing f .Despite the fact that the mean and the size of the fluctuations in µ both re-main relatively constant with increasing f , the nature of the slips does changesignificantly with f . This can be seen in Fig. 3, where the data in Fig. 3(c,d) Abram H. Clark, Robert P. Behringer, and Jacqueline Krim recorded at the higher frequency of 28 Hz are more asymmetric, with inter-mittent and larger stress drops. Such features are far less prominent for thedata recorded at 0 Hz (Fig. 3(a,b)). To quantify this asymmetry, we exam-ine the skewness α of the ∆µ data sets. This quantity is zero for symmetricdynamics (where stress rises and drops in the same way) and negative forstick-slip dynamics. Figure 4(c) shows that α is increasingly negative as f increases, ranging from α ≈ − . α ≈ − . ∆µ with large, intermittent stressdrops. Our results suggest a physical picture where, as vibration frequency increases,many small rearrangements are substituted for a fewer, larger avalanche-likeslips. In this scenario, the grains become better organized into more stableconfigurations, more compacted, and potentially more commensurate withthe rough slider during rearrangements. Each of these conditions can increasestatic friction levels [34,35,36], alter stick slip behaviors, and increase the skew-ness of the distribution of ∆µ . This picture is supported by measurements fromRef. [29] of the same slider being pulled over a compact, solid lattice (i.e., afixed row of grains), where there are no particle rearrangements and stick-slip dynamics are dominant. In particular, we analyzed data in Ref. [29] fromexperiments where the slider is pulled over the solid lattice at v = 5 mm/s,and we found the skewness of ∆µ to be -4.5, and increasing to α ≈ v is increased. Additionally, α was independent of whether or not we applied avibration at f = 11 Hz. We note that for the present study, as frequency isincreased, the values of α shown in Fig. 4(c) for sliding over a granular bed areapproaching the value associated with a compact, organized substrate. More-over, the frequency dependence of α is unique to the case of the granular bed,corroborating our assertion that it is associated with grain rearrangementsand not surface friction between grains.Under the physical picture that we suggest, the self-affine nature of theroughness profile [32,33] of the force curves would also be highly frequencydependent, as mobile grains would experience an increasing number of vibra-tions as the slider passed over them, readily contributing to rearrangement.Self-affine scaling of surface roughness is characterized by σ µ ( L ) ∝ L H , (1)where σ µ ( L ) is the standard deviation over a sample size L , and H is theHurst exponent, which varies between 0 and 1 and describes the manner inwhich the roughness scales with lateral extent. It can morevoer be directlymapped to power spectral analyses common in the literature and force powerspectra P f ( ω ) analyses common in the literature [31,38,30], since self-affineforce power spectral curves scale as P f ( ω ) ∝ ω − (2 H +1) . (2) ibration can enhance stick-slip behavior for granular friction 7(a) (b) (c) -3 -2 -2 -1 Fig. 5 (a) The standard deviation σ µ of µ is plotted as a function of sample length L for0 Hz and 28 Hz. The solid lines show fits of the form σ µ ∝ L H to the small- L portionof the data. (b) The self-affine exponent H increases with vibration frequency f . (c) Thecorrelation length ξ , defined as the value of L at which the solid fit line shown in (a) is 1.5times bigger than the data, is plotted versus f . Figure 5(a) shows the dependence of σ µ ( L ) on L for data recorded at 0 and28 Hz. Both data sets are described by Eq. (1) for small L and then plateau toa constant value at large L , characteristic of self-affine systems. The crossoverhappens at a correlation length ξ , which determines the length scale over whichthe structures tend to repeat. Increasing f is clearly associated with increasesin H (Fig.5(b)), which increases from approximately 0.55 at 0 Hz to 0.67 at 30Hz. This result is consistent with Fig. 3, where the high- f data for µ appears“smoother” than the low- f data, and is also close to the H values or 0.5-0.7 inferred from the power spectra data reported by Zadeh et al [31,38,30]for unvibrated granular systems. Figure 5(c) also shows that the correlationlength weakly decreases with f , and its value is roughly the same as a particlediameter. We have presented here a study of the frictional behavior of a two-dimensionalslider pulled slowly over a granular substrate comprised of photoelastic diskswhile being vibrated at frequencies ranging from 0 to 30 Hz in a directionparallel to sliding. Measurements were performed at speeds close to the tran-sition from stick-slip to steady sliding, where the impact of vibration is likelyto be magnified on account of the relative ease which which force chains can bedisrupted when close to transitions. We observe, surprisingly, that increasedfrequency of vibration can enhance stick-slip behavior for granular friction.Reports of reductions in friction when systems are mechanically vibrated arecommonplace in the literature. The apparent paradox is resolved by takingnote of the fact that vibrations can induce rearrangements in granular mate-rials that may not otherwise occur in rigid solid systems.In particular, the results can be explained by a physical picture where, asvibration frequency increases, many small rearrangements are substituted fora fewer, larger avalanche-like slips. At the low acceleration conditions underwhich the measurements were performed, the grains become better organized,
Abram H. Clark, Robert P. Behringer, and Jacqueline Krim more compacted and potentially more commensurate with the rough sliderduring rearrangements. These conditions, and particularly the grain mobilityinduced by the mechanical vibrations can increase static friction levels [34,35,36] and alter stick slip behaviors. The self-affine nature of the roughnessprofile [32,33] of the force curves would be highly frequency dependent in thisscenario, as mobile grains would experience an increasing number of vibrationsas the slider passed over them, readily contributing to rearrangement. Theforce curve profiles are consistent with this interpretation.
This work was supported by NSF DMR0906908 and NSF DMR0805204.
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