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Friction memory in the stick-slip of a sheared granular bed
Journal of Non-Crystalline Solids 357 (2011) 749–753
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Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
Friction memory in the stick-slip of a sheared granular bed Fabio Leoni a, Andrea Baldassarri b,c, Fergal Dalton d, Alberto Petri c,d,⁎, Giorgio Pontuale d, Stefano Zapperi e a
CEA, IRAMIS, SPEC, Grp. Ins¡tability & Turbulence, F-91191 Gif Sur Yvette, France Istituto dei Sistemi Complessi del CNR, via dei Taurini 19, I-00185, Roma, Italy c Dipartimento di Fisica, Sapienza Universitá, P.le A. Moro 5, I-00185 Roma, Italy d Istituto dei Sistemi Complessi del CNR, via Fosso del Cavaliere 100, I-00133, Roma, Italy e Istituto per l'Energetica e le Interfasi del CNR, Via R. Cozzi 53, I-20125, Milano, Italy b
a r t i c l e
i n f o
Article history: Received 14 July 2010 Available online 19 August 2010 Keywords: Granular media; Friction; Shear; Noise; Ageing
a b s t r a c t We investigate friction memory effects in the chaotic stick-slip motion of a plate shearing a granular bed. By analyzing separately trajectories' portions having increasing or decreasing instantaneous velocity, it is found that there are two quantitatively distinct granular friction–velocity curves for positive or negative acceleration, which cross one another in the weakening region. Interpreting acceleration dependence as an indirect consequence of contact ageing, we try to explain these effects by including rate-and-state (RS) friction equations in the stochastic model describing the plate motion. Preliminary results on a study case show that the main experimental features can be reproduced in this way, although quantitative agreement is partial. From the value obtained for the RS parameters we conclude that sliding friction decorrelation takes place at the length-scale of the solid-on-solid micro-contacts between grains and plate. In addition, the contemporary presence of noise and RS effects influences the average friction curve at large shear rate. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The physics of Granular Systems (GS) offers very rich opportunities to expand the understanding of many types of non-equilibrium and complex systems. At the simplest level a (dry) GS is composed of a large number of massive athermal particles interacting through elastic and frictional forces. As interactions are dissipative, energy is effectively removed from the system and one must therefore continuously inject additional energy to observe any kind of long term dynamics. These few ingredients are sufficient for the system to generate a wide variety of behaviours. As yet there exists no consistent physical framework representing the dynamic evolution of a GS from a general point of view such as the Navier–Stokes equation for fluids. The state of the grains themselves can sometimes be described as either gaseous, liquid or solid, depending on the system parameters and the applied excitation, and much existing research focuses on describing the granular material in a single experimental context though important progress have been made in recent years towards unifying different views (see e.g. [1]). One typical situation is that of a horizontal densely packed bed of granular medium subject to a horizontal shear force exerted by an upper plate. Besides being interesting both in itself and for industrial applications, this kind of system can also be considered to some extent ⁎ Corresponding author. Istituto dei Sistemi Complessi del CNR via Fosso del Cavaliere 100 I-00133 Rome Italy. E-mail address: [email protected] (A. Petri). 0022-3093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2010.07.046
a very simplified model for a low-pressure earthquake fault zone, where the granular material plays the role of the fault gouge (see e.g. [2,3] and refs. therein). One major item in this context concerns the response force that the medium exerts on the shearing plate. It has recently been shown [4] that the dynamics of the plate shearing the top of a granular channel can be well described by a stochastic equation of motion, where the reaction force of the medium to the shear stress is decomposed into two independent contributions: one random and one deterministic. In analogy with usual viscous forces the latter can be taken to be a function of velocity, while the former depends on the position of the plate. Comparison with experimental data [4,5] showed that this assumption allows a good quantitative description of the dynamics. Nevertheless it is known that GS can display hysteretic behaviour with respect to shear [6], which is not considered by the friction law adopted in [4]. Such behaviour can be ascribed to memory and/or ageing of the medium, and has been observed and investigated in both solid-on-solid experiments and gouge system friction [7–11] (see e.g. [2,3] and [12] for reviews). In the present article we investigate the effective stress response of a granular bed on a top shearing plate performing chaotic stick-slip motion, with large and intermittent velocity fluctuations. We find that the average effective stress dependence on the instantaneous shear velocity for positive and negative instantaneous acceleration is described by two distinct friction laws, which cross each other at low share rate. We ascribe this behaviour to memory and ageing and attempt to reproduce it by adopting a suitable rate-and-state friction law (instead of the simple velocity dependent law of ref. [4]) in the
F. Leoni et al. / Journal of Non-Crystalline Solids 357 (2011) 749–753
stochastic motion equation for the plate. To this end classic rate-andstate equations are suitably modified to allow for the Bagnold [13] and weakening–hardening [14–17] regimes characteristic of granular systems. However some discrepancies do remain and we shall discuss their possible origin. For the sake of brevity, henceforth the GS response to shear strain will be denoted “friction” without ambiguity. The rest of the paper is organized as follows: Section 2 describes the experiment and its equation of motion; Section 3 describes the rate-and-state approach to friction in solid-on-solid and in granular gouge experiments; in Section 4 results from the analysis of experimental data are presented, showing the dependence of friction on instantaneous acceleration; in Section 5 the classic rate-state equation is supplied with terms suitable to GS and is used in the equation of motion. Finally, results obtained by numerical simulation of the equation of motion are compared to experimental results and Section 6 is devoted to a short concluding summary. 2. Shearing plate dynamics on a granular channel In the experiments considered [4,15], the granular medium is composed of glass beads 2 mm ± 10% in diameter confined to a circular PVC Couette cell of height 7 cm and inner and outer diameters 25 and 40 cm. The shear force (energy injection) is provided by an overhead plate of weight 0.75 kg rotating in the channel on the granular surface, driven via a torsion spring so as to generate a stickslip motion (a simple sketch of the apparatus is shown in fig. 3 of Ref. [15]). The observed motion can be regular (e.g. periodic stick-slip, continuous sliding) depending on the system parameters (driving velocity, spring, inertia), however there exists a large parameter space in which the disorder inherently intrinsic to the granular medium gives rise to chaotic stick-slip motion. In previous work [4,5], it was shown that the dynamics of the top plate can be well approximated by a (stochastic) equation of motion of the kind :: Iθ = κðωD t−θÞ−F;
ð1Þ
where I is the moment of inertia of the top plate, θ its angular position, κ the torsional spring constant and ωD the driving angular velocity. F is the total effective frictional force exerted by the granular on the top plate, and in principle could be a function of many variables including ˙ ::: . However, plate dynamic variables and granular state F = F t; θ; θ; it was shown [4] that F is predominantly dependent on only two ˙ independent terms θ and θ: F θ;θ˙ = Fv θ˙ + Fr ðθÞ
such properties is a bounded random walk, which can be described by an Ornstein–Uhlenbeck process: dFr ðθÞ = ηðθÞ−aFr ðθÞ dθ
SðkÞ =
2D ; a2 + k2
(with experimental results as shown in the fig. 5b of Ref. [4]). The second friction term, Fv θ˙ was also obtained from experimental observations by averaging F as a function of the velocity θ˙ over a large number of slip events, as illustrated in Fig. 1. The resulting mean friction can be well approximated by a phenomenological curve [12]: h i ˙ ˙ Fv θ˙ = F0 + γ θ−2ω 0 ln 1 + θ = ω0
ð4Þ
where F0 is the mean torque at θ˙ = 0, γ is a damping term and ω0 indicates the (angular) velocity at the minimum torque. This model quantitatively reproduces the experimental slip distribution (and other observations) [4] for all driving velocities ωD considered in the experiments, and varying other parameters such as κ and I. In this article we consider a possible refinement of the model. In particular, we wish to transcend the empirical character of the friction law adopted, Eq. (4), and derive it from the properties of microscopic inter-grain interactions. To this end we considered a rate-and-state approach typical of friction models of solids [12], but that has been found to hold also for GS under controlled shear velocities [3]. In this case, the non-constant viscosity is ascribed to the coupled dynamics of the shearing surface and a state variable, taking into account the life-time of contacts between the grains and the surface. In such a context, the weakening behavior is not assumed “a priori” but is recovered as a dynamical effect. Moreover, many other more subtle dynamical effects arise [3,12].
ð2Þ
Here the first term is deterministic and contains the nonNewtonian viscous response of the granular medium, whereas the second term is a function expressing the random nature of the instantaneous response of the granular medium, due to the randomness of the configuration realised at every instant below the top plate. It is important to note that, unlike usual stochastic processes, the random component Fr in Eq. (2) is not a function of the time t but of the angle variable θ: during stick-slip motion when the top plate is stopped the granular medium does not change configuration, and therefore Fr must remain constant. If the top plate moves by some small angle, however, then the GS will slightly change configuration and the resulting frictional force will also change slightly and randomly, making Fr a correlated random function of θ. After some larger displacement, the GS will naturally become completely reorganised and bear no similarity to earlier, and so the frictional force will de-correlate w.r.t. its previous value. The simplest process with
ð3Þ
where η(θ) is an uncorrelated Gaussian white noise with zero mean and auto-covariance h η(θ)η(θ′) i = Dδ(θ − θ′). The second term on the r.h.s. sets both the correlation length and a bound on the random walk. The inverse correlation length a and the noise intensity D can be estimated experimentally from the power specrtum:
0.64
Frictional Torque (Nm)
750
0.62
0.6
0.58
0.56 0
0.05
0.1
0.15
0.2
0.25
Shear Angular Velocity (rad/s) Fig. 1. Instantaneous reaction force of the granular on the shearing plate as resulting from a series of chaotic slips (lines). By averaging at fixed instantaneous velocity θ˙ over the different slips a regular friction curve is obtained (circles), which can be fitted by phenomenological expressions like Eq. (4) (thick line).
F. Leoni et al. / Journal of Non-Crystalline Solids 357 (2011) 749–753
The most simple phenomenological expression for ϕ is that proposed by Dieterich [7]
3. Ageing and rejuvening in friction The understanding of solid-on-solid friction, albeit still incomplete, has advanced considerably in recent years. Whilst time has no role to play in classical da Vinci–Coulomb–Amontons friction laws, dynamic friction is observed to generally decrease as the interface velocity increases. Furthermore, when two bodies remain in static contact for a long period, the frictional force between them is observed to increase with the logarithm of the contact age. Bowden and Tabor [18] showed that the force of static friction between two sliding surfaces is strongly dependent on the real area of contact. This work led to the asperity contact theory of friction on which basis, in the 1970's and 80's, different authors established more refined properties and constitutive laws for the behaviour of frictional bodies, e.g. Dieterich [7], Rice [10] and Ruina [9,11] (see [2,3] and [12] for review). Although fundamentally different in nature, granular matter has been shown to share many similarities with solid surfaces as far as shear stress is concerned [3], with the addition of possible hardening when the top plate velocity becomes large enough [13–16]. It has been found in particular that the ageing of static friction with time and the weakening of dynamic friction with velocity in granular gouges can be described by the same phenomenological laws as solids [3,8,9], generally known as rate-and-state friction laws (RS). In the following we introduce rate and state variables in multicontact interfaces and reformulate the problem of friction following the rate-and-state approach so to allow the weakening and hardening observed, the latter proportional to the square shear rate characteristic of solid-granular interfaces. We will adopt the notation of Dieterich [7] and will not enter in the different interpretations of the Rice–Ruina and Dieterich approaches, since they ultimately lead to about the same phenomenology [3,19]. Multi-contact interfaces are generated when rough hard solids are brought into contact with an apparent pressures W / Σ well below the elastic moduli and the real area of contact Σr is a small fraction of the visible area of contact Σ (typically Σr / Σ ≃ 10 − 3 for solid-on-solid, that means that size of real contacts D0 is of the order of microns). Following Bowden and Tabor [18], the friction force between two solids F = μτn, where μ is the friction coefficient and τn the normal stress, can be expressed as F = σs ⋅Σr
ð5Þ
where σs is the shear strength of the interface. For a wide variety of materials, experiments (the so called stopand-go and slide-hold-slide experiments) [3,7,9,12] show that the static friction coefficient μs(tw) varies logarithmically over several decades of waiting time tw during which the system is at rest. The strengthening of μs is related to the slow growth of the real area of contact Σr by means of the following relation [12] t 2 + o ln Σr ðt Þ = Σr0 1 + m ln 1 + ; τ
751
ð6Þ
where m and τ are temperature dependent factors. Typically, at room temperature, m ≃ 10 − 2 and τ≲1s for solid-on-solid. A more complete expression for the evolution of the real contact area is obtained by considering the dynamical aspect of the contact aging. This is usually made by introducing the geometric age of the interface or “state” variable ϕ(t), which replaces t in Eq. (6). For non moving interfaces, ϕ(t) is simply proportional to the time which has been spent at rest at time t. For an interface sliding at the instantaneous velocity x˙ ðt Þ, ϕ˙ is time dependent. The typical microcontact size D0 characterises the interfacial configurational memory length.
˙ xϕ ϕ˙ = 1− : D0
ð7Þ
In conclusion Tabor's expression for the friction can be written as F = σs x˙ Σr ðϕÞ;
ð8Þ
in which memory (or time) dependent effects are contained within the real area of contact only, while the shear strength σs, depends only on the instantaneous sliding velocity. As friction laws are generally phenomenological in nature, experiments are fundamental to obtain expressions for the shear strength as a function of velocity (as a matter of fact, no ab initio calculation of friction is available at present). As far as rate-state equations are concerned, “velocity jump” experiments have emerged as a useful method for studying the shear strength of multicontact interfaces. In such experiments, the frictional response to a sudden jump of the driving velocity from Vi to Vf, after a long period of constant driving is studied. Experiments are performed with rigid driving stages such that the velocity jump be transmitted to the actual shear interface before the interface has moved more than the memory length D0. In this way, the geometric age does not vary significantly from its initial stationary value ϕi = D0 / Vi, and so the behaviour observed is fully attributable to the rate dependence of the shear strength σs. Despite their empirical nature, rate-and-state models can be easily recovered from quite simple assumptions, as a theory of linearized response to perturbations of steady-state sliding [20]. The form of the terms in Eq. (8) is recovered using physical considerations from the specific problem of friction in multi-contact interfarces. However, the dynamic evolution of the state equation expressed in Eq. (7) is quite general, being the simplest formulation of a “maturity” variable ϕ, increasing in time as long as the system does not move more than a typical distance D0. As we describe in Section 5, our modeling approach retains such elements of the rate-and-state theory, modifying only the specific term σs in order to account for the physics of granular media at high speed, which is markedly different from the solid-on-solid case. 4. Acceleration dependent shear response as a signature of rate-state friction In this Section we shall report the results of the analysis of data from the experiment described in Section 2, targetted at revealing memory effects in the deterministic part Fv θ˙ of the effective friction, Eq. (2). Then, in Section 5, we shall adopt the rate-and-state approach introduced for multi-contact interfaces and will derive an expression for Fv θ˙ , the force that encapsulates the memory effects while retaining the possibility of showing both weakening and hardening. In the present work we do not focus on “stop and go” or “velocity jump” measurements described in the previous section: rather we shall attempt to detect and characterize memory and state effects in the chaotic stick-slip motion performed by the plate (Fig. 1). This can be considered an important test for rate-state friction laws as it can confirm their validity in situations where the velocity changes continuously and randomly. In previous work [4] the friction force was assumed to be a function of the position and velocity of the disk only: F ð:::Þ = Fv θ˙ + Fr ðθÞ, where the fluctuating part Fr(θ) derives from the instantaneous “random” configuration of the disordered force chain network, and Fv θ˙ is a single valued function obtained by
752
F. Leoni et al. / Journal of Non-Crystalline Solids 357 (2011) 749–753
˙ an example is averaging F(...) over many slip events in intervals of θ; reported in Fig. 1. As seen above [3,6], many experimental observations show that the frictional force is determined to some extent also by the history of the motion or by some state variable, involving possibly the degree of fluidization in the case of GS [16,17]. In order to highlight these “memory” effects in the stick slip motion observed, the analysis of Ref. [4] has been refined by considering separately the force during periods of positive and of negative acceleration; the friction force may be written: Fv θ˙ = Fvþ θ˙ + Fv− θ˙ :
ð9Þ
We remark that Fv, Fv+ and Fv− are averages in intervals of θ˙ over all the observations. The two experimental friction curves obtained are shown in Fig. 2 as a function of disk velocity. It is seen in the figure that :: ˙ a clear dependence :: on θ exists: at low θ friction is larger for positive than for negative θ , whereas the two curves cross over each other for ˙ The main focus of our paper is now to reproduce this increasing θ. phenomenon within the framework of rate-and-state friction laws, and to seek to understand the origin of the cross-over in Fig. 2. 5. A rate-and-state friction law for solid-granular interface To test the rate-and-state formulation of friction force in the case of the stick-slip experiments described in Refs. [4,5], we employed the most simple expression for the shear strength σs x˙ that may account for the full fluidization and consequent hardening of the granular medium at high velocities, assuming a square dependence on velocity [13,14]. With this assumption and following Eq. (8), the deterministic component of the friction force Fv is taken to be described by the following equations: ˙ ˙ ϕ = 1 + χ θ˙ 2 A + A ln 1 + ϕ ;ϕ˙ = 1−θϕ ; ð10Þ FvF ≡Frs θ; 0 ϕ A0 τ θ0 where θ0 is a memory angle. For the fluctuating part, the relation Eq. (3) still holds. We note here that the explicit dependence on θ˙ of ˙ ϕ is monotonic, but it nevertheless allows for the expression Frs θ; both weakening and hardening of the average force, and will appear in fact as a dynamical state effect through the variable ϕ. Numerical simulations of the chaotic stick-slip motion of the plate on the granular bed have been performed according to Eq. (1), with F = Fr + Frs, Eqs. (3) and (10). In the simulation, inertia and spring
constant are set at the values chosen for the experiment. The parameters that appear in Eq. (10) have been estimated by running simulations with different parameters and choosing those which best fit the corresponding experimental curves Fv + and Fv − of Eq. (9). During stick events, the first part of Eq. (10) reduces to the static ϕ friction Fv = A0 + Aϕ ln 1 + implying A0 ≃ F0 from Eq. (4). In the τ rate-and-state formulation of solid-on-solid friction, τ can be experimentally estimated as the time at which real contact area growth begins to increase logarithmically. Ageing experiments that we are conducting show τ ≈ 1s, in agreement with many solid-onsolid and gouge friction results, and so our simulations have been performed around this value. From rate-and-state models it can also be shown that [12] Aϕ ≃ S / Y ≃ 10 − 3 ÷ 10 − 2, where S is the strain rate sensitivity and Y is the yield stress. For velocities in the range of steady ˙ sliding ( θ˙ ∼ N10−1 rad/s), the state variable approaches ϕ = ϕss = θ0 = θ. Therefore, up to a logarithmic term the friction torque for the rate-andstate model has a quadratic dependence on velocity [13–15] with the prefactor given by χ. The best results we have achieved by fitting data in Fig. 2 with simulations of Eqs. (1) and (10) are shown in Fig. 3. The corresponding values found for the parameters are s2
χ ( 2) rad 2.
A0 0.579
τ (s) 1.0
Aϕ 0.007
θ0 (rad) 5 10 − 4.
These results, although preliminary, indicate that the stochastic model with rate-and-state friction reproduces two main features of the Fv± curves, i.e. the cross-over at low velocities and the weakening_hardening behaviour. From the values obtained for the parameters some remarks can be drawn: 1. As the granular friction is dominated by force chains at low velocity, one might expect to have D0 ≃θ0 r, where r is the average radius of the channel, of the order of a few grains. The value found for D0 (≈ 160μm) indicates instead that the relevant lengthscale for the contacts is that of solid on solid friction, thus concerning intragrain friction. Mair et al. [19] found 4μm b D0 b 183μm for grains of diameter 50μm b d b 150μm. D0 therefore seems related to grain asperities even in this experiment, which was performed at much higher pressure where one might expect a stronger sensibility to granular texture. 2. The above result implies at least two relevant length scales for the process, since it had been found [4,5] that the friction itself has an autocorrelation length 1/ a of some grain diameters. The information about the granular structure is thus carried by just a, the force
0,61
0,6
Fv
Fv
0,61
0,59
0,58
0,6
0,59
0
0,05
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0,1
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dθ/dt Fig. 2. Circles: The friction dependence resulting by averaging at fixed instantaneous ˙ over many slips events (like those depicted in Fig. 1). The upper and lower velocity, θ, data sets results from separating positive (squares) and negative (diamonds) acceleration contributes. As expected the reaction force of the granular system is larger for positive acceleration, but it happens only at sufficiently low velocities. At larger values a cross-over takes place and decelerating events display higher friction.
0,58
0
0,05
0,1
0,15
0,2
0,25
dθ/dt Fig. 3. The friction curves reconstructed by adopting rate-state laws of the type Eq. (10) in the stochastic model Eq. (1).
F. Leoni et al. / Journal of Non-Crystalline Solids 357 (2011) 749–753
correlation length, not by D0. D0 also appears to rule the position of ffi pffiffiffiffiffiffiffiffiffi such a cross-over which scales as ðD0 Þ (Eq. (6)). 3. The value determined for χ is not simply the geometric value ˙ In fact, this is true derivable from the curvature of F θ˙ for large θ. if one suppresses the noise term in the model friction, Eq. (2), whereas we have that the presence of noise affects the friction curve, both its average value and at large ˙θ. This is an important fact which will be investigated further in the future. 4. The value of τ is in agreement with preliminary results from stopand-go experiments. Despite qualitative agreement, it must however be noted that the adopted model does not reproduce the experimental data accurately. In particular the Fv± curves appear less distinct and flatter. The maximum speed is lower than in experiment. These difficulties indicate that the adopted model is inherently incomplete. More work is needed to discover where these problems originate, i.e. in the RS or in the random part of the friction model adopted. With regard to the former it is known that RS still presents some fundamental problems: stop-go experiments have shown that parameters such as D0 can be velocity dependent (a clear problem as D0 should be a material parameter) [19]. For the latter, we know the correlation length 1/ a of the random friction force is not a fixed parameter [5] as assumed in our simulations. With regard to our analysis here, we must also note that general RS research is conducted using constant velocites much lower than those employed here ( 100 μm/s instead of 10 cm/s), and under much higher compressional stress ( MPa instead of 100 Pa). 6. Concluding remarks and summary We have implemented a rate-and-state friction law in a stochastic equation describing the stick-slip motion of a plate shearing a granular bed. The rate-and-state parameters have been determined by best fitting simulation results with experimental. Although the agreement is only partial, it nevertheless reproduces features not obtainable with the simpler friction law. In particular it is seen that friction depends not only on the velocity but also on the acceleration, at least through its sign.
753
The values obtained for the memory length indicate that grain– grain friction lies at the origin of the frictional memory effects. Work in progress is concerned with validating the obtained parameter values with different measurements and understanding the origin of discrepancies remaining between the model and the experiments.
Acknowledgements FD acknowledges the financial support from EU project TRIGS, contract no. 043386.
References [1] See e.g. Various Authors, EPJ E, 14, 341 (2004). [2] C.H. Scholz, The mechanics of earthquake and faulting, Cambridge University Press, Cambridge, 1990. [3] C. Marone, Annu. Rev. Earth Planet. Sci 26 (1998) 643. [4] A. Baldassarri, F. Dalton, A. Petri, S. Zapperi, G. Pontuale, L. Pietronero, Phys. Rev. Lett. 96 (2006) 118002. [5] A. Petri, A. Baldassarri, F. Dalton, G. Pontuale, L. Pietronero, S. Zapperi, The EPJ B, Condensed matter and complex systems 3 (2008) 531. [6] S. Nasuno, A. Kudrolli, J.P. Gollub, Phys. Rev. Lett. 79 (1997) 949. [7] J.H. Dieterich, J. Geophys. Res. 77 (1972) 3690. [8] J.H. Dieterich, Constitutive properties of faults with simulated gouge, in: N.L. Carter, M. Friedman, J.M. Logan, D.W. Sterns (Eds.), Mechanical Behavior of Crustal Rocks, Am. Geophys. Union Monogr., 24, 1981, p. 103, Am. Geophys. Union, Washington DC. [9] A. Ruina, J. Geophys. Res. 88 (1983) 10359�70. [10] J.R. Rice, Pure. Appl. Geophys. 121 (1983) 443. [11] J.R. Rice, A.L. Ruina, J. Appl. Mech. 50 (1983) 343. [12] T. Baumberger, C. Caroli, Adv. Phys. 55 (2006) 279. [13] R.A. Bagnold, Proc. R. Soc. A 295 (1966) 219. [14] S.B. Savage, M. Sayed, J. Fluid Mech. 142 (1984) 391. [15] F. Dalton, F. Farrelly, A. Petri, L. Pietronero, L. Pitolli, G. Pontuale, Phys. Rev. Lett. 95 (2005) 138001. [16] K. Lu, E.E. Brodski, H.P. Kavehpour, J. Fluid Mech. 587 (2007) 347 and refs. teherein. [17] P. Mills, P.G. Rognon, F. Chevoir, Europhys. Lett. 81 (2008) 64005 and refs. therein. [18] F.P. Bowden, D. Tabor, The friction and lubrication of solids I, Clarendon press, London, 1950. [19] K. Mair, C. Marone, J. Geoph. Res. 104 (1999) 28899. [20] J.R. Rice, N. Laposta, K. Ranjith, J. Mech, Phys. Solids 49 (2001) 1865–1898.
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
Friction memory in the stick-slip of a sheared granular bed Fabio Leoni a, Andrea Baldassarri b,c, Fergal Dalton d, Alberto Petri c,d,⁎, Giorgio Pontuale d, Stefano Zapperi e a
CEA, IRAMIS, SPEC, Grp. Ins¡tability & Turbulence, F-91191 Gif Sur Yvette, France Istituto dei Sistemi Complessi del CNR, via dei Taurini 19, I-00185, Roma, Italy c Dipartimento di Fisica, Sapienza Universitá, P.le A. Moro 5, I-00185 Roma, Italy d Istituto dei Sistemi Complessi del CNR, via Fosso del Cavaliere 100, I-00133, Roma, Italy e Istituto per l'Energetica e le Interfasi del CNR, Via R. Cozzi 53, I-20125, Milano, Italy b
a r t i c l e
i n f o
Article history: Received 14 July 2010 Available online 19 August 2010 Keywords: Granular media; Friction; Shear; Noise; Ageing
a b s t r a c t We investigate friction memory effects in the chaotic stick-slip motion of a plate shearing a granular bed. By analyzing separately trajectories' portions having increasing or decreasing instantaneous velocity, it is found that there are two quantitatively distinct granular friction–velocity curves for positive or negative acceleration, which cross one another in the weakening region. Interpreting acceleration dependence as an indirect consequence of contact ageing, we try to explain these effects by including rate-and-state (RS) friction equations in the stochastic model describing the plate motion. Preliminary results on a study case show that the main experimental features can be reproduced in this way, although quantitative agreement is partial. From the value obtained for the RS parameters we conclude that sliding friction decorrelation takes place at the length-scale of the solid-on-solid micro-contacts between grains and plate. In addition, the contemporary presence of noise and RS effects influences the average friction curve at large shear rate. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The physics of Granular Systems (GS) offers very rich opportunities to expand the understanding of many types of non-equilibrium and complex systems. At the simplest level a (dry) GS is composed of a large number of massive athermal particles interacting through elastic and frictional forces. As interactions are dissipative, energy is effectively removed from the system and one must therefore continuously inject additional energy to observe any kind of long term dynamics. These few ingredients are sufficient for the system to generate a wide variety of behaviours. As yet there exists no consistent physical framework representing the dynamic evolution of a GS from a general point of view such as the Navier–Stokes equation for fluids. The state of the grains themselves can sometimes be described as either gaseous, liquid or solid, depending on the system parameters and the applied excitation, and much existing research focuses on describing the granular material in a single experimental context though important progress have been made in recent years towards unifying different views (see e.g. [1]). One typical situation is that of a horizontal densely packed bed of granular medium subject to a horizontal shear force exerted by an upper plate. Besides being interesting both in itself and for industrial applications, this kind of system can also be considered to some extent ⁎ Corresponding author. Istituto dei Sistemi Complessi del CNR via Fosso del Cavaliere 100 I-00133 Rome Italy. E-mail address: [email protected] (A. Petri). 0022-3093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2010.07.046
a very simplified model for a low-pressure earthquake fault zone, where the granular material plays the role of the fault gouge (see e.g. [2,3] and refs. therein). One major item in this context concerns the response force that the medium exerts on the shearing plate. It has recently been shown [4] that the dynamics of the plate shearing the top of a granular channel can be well described by a stochastic equation of motion, where the reaction force of the medium to the shear stress is decomposed into two independent contributions: one random and one deterministic. In analogy with usual viscous forces the latter can be taken to be a function of velocity, while the former depends on the position of the plate. Comparison with experimental data [4,5] showed that this assumption allows a good quantitative description of the dynamics. Nevertheless it is known that GS can display hysteretic behaviour with respect to shear [6], which is not considered by the friction law adopted in [4]. Such behaviour can be ascribed to memory and/or ageing of the medium, and has been observed and investigated in both solid-on-solid experiments and gouge system friction [7–11] (see e.g. [2,3] and [12] for reviews). In the present article we investigate the effective stress response of a granular bed on a top shearing plate performing chaotic stick-slip motion, with large and intermittent velocity fluctuations. We find that the average effective stress dependence on the instantaneous shear velocity for positive and negative instantaneous acceleration is described by two distinct friction laws, which cross each other at low share rate. We ascribe this behaviour to memory and ageing and attempt to reproduce it by adopting a suitable rate-and-state friction law (instead of the simple velocity dependent law of ref. [4]) in the
F. Leoni et al. / Journal of Non-Crystalline Solids 357 (2011) 749–753
stochastic motion equation for the plate. To this end classic rate-andstate equations are suitably modified to allow for the Bagnold [13] and weakening–hardening [14–17] regimes characteristic of granular systems. However some discrepancies do remain and we shall discuss their possible origin. For the sake of brevity, henceforth the GS response to shear strain will be denoted “friction” without ambiguity. The rest of the paper is organized as follows: Section 2 describes the experiment and its equation of motion; Section 3 describes the rate-and-state approach to friction in solid-on-solid and in granular gouge experiments; in Section 4 results from the analysis of experimental data are presented, showing the dependence of friction on instantaneous acceleration; in Section 5 the classic rate-state equation is supplied with terms suitable to GS and is used in the equation of motion. Finally, results obtained by numerical simulation of the equation of motion are compared to experimental results and Section 6 is devoted to a short concluding summary. 2. Shearing plate dynamics on a granular channel In the experiments considered [4,15], the granular medium is composed of glass beads 2 mm ± 10% in diameter confined to a circular PVC Couette cell of height 7 cm and inner and outer diameters 25 and 40 cm. The shear force (energy injection) is provided by an overhead plate of weight 0.75 kg rotating in the channel on the granular surface, driven via a torsion spring so as to generate a stickslip motion (a simple sketch of the apparatus is shown in fig. 3 of Ref. [15]). The observed motion can be regular (e.g. periodic stick-slip, continuous sliding) depending on the system parameters (driving velocity, spring, inertia), however there exists a large parameter space in which the disorder inherently intrinsic to the granular medium gives rise to chaotic stick-slip motion. In previous work [4,5], it was shown that the dynamics of the top plate can be well approximated by a (stochastic) equation of motion of the kind :: Iθ = κðωD t−θÞ−F;
ð1Þ
where I is the moment of inertia of the top plate, θ its angular position, κ the torsional spring constant and ωD the driving angular velocity. F is the total effective frictional force exerted by the granular on the top plate, and in principle could be a function of many variables including ˙ ::: . However, plate dynamic variables and granular state F = F t; θ; θ; it was shown [4] that F is predominantly dependent on only two ˙ independent terms θ and θ: F θ;θ˙ = Fv θ˙ + Fr ðθÞ
such properties is a bounded random walk, which can be described by an Ornstein–Uhlenbeck process: dFr ðθÞ = ηðθÞ−aFr ðθÞ dθ
SðkÞ =
2D ; a2 + k2
(with experimental results as shown in the fig. 5b of Ref. [4]). The second friction term, Fv θ˙ was also obtained from experimental observations by averaging F as a function of the velocity θ˙ over a large number of slip events, as illustrated in Fig. 1. The resulting mean friction can be well approximated by a phenomenological curve [12]: h i ˙ ˙ Fv θ˙ = F0 + γ θ−2ω 0 ln 1 + θ = ω0
ð4Þ
where F0 is the mean torque at θ˙ = 0, γ is a damping term and ω0 indicates the (angular) velocity at the minimum torque. This model quantitatively reproduces the experimental slip distribution (and other observations) [4] for all driving velocities ωD considered in the experiments, and varying other parameters such as κ and I. In this article we consider a possible refinement of the model. In particular, we wish to transcend the empirical character of the friction law adopted, Eq. (4), and derive it from the properties of microscopic inter-grain interactions. To this end we considered a rate-and-state approach typical of friction models of solids [12], but that has been found to hold also for GS under controlled shear velocities [3]. In this case, the non-constant viscosity is ascribed to the coupled dynamics of the shearing surface and a state variable, taking into account the life-time of contacts between the grains and the surface. In such a context, the weakening behavior is not assumed “a priori” but is recovered as a dynamical effect. Moreover, many other more subtle dynamical effects arise [3,12].
ð2Þ
Here the first term is deterministic and contains the nonNewtonian viscous response of the granular medium, whereas the second term is a function expressing the random nature of the instantaneous response of the granular medium, due to the randomness of the configuration realised at every instant below the top plate. It is important to note that, unlike usual stochastic processes, the random component Fr in Eq. (2) is not a function of the time t but of the angle variable θ: during stick-slip motion when the top plate is stopped the granular medium does not change configuration, and therefore Fr must remain constant. If the top plate moves by some small angle, however, then the GS will slightly change configuration and the resulting frictional force will also change slightly and randomly, making Fr a correlated random function of θ. After some larger displacement, the GS will naturally become completely reorganised and bear no similarity to earlier, and so the frictional force will de-correlate w.r.t. its previous value. The simplest process with
ð3Þ
where η(θ) is an uncorrelated Gaussian white noise with zero mean and auto-covariance h η(θ)η(θ′) i = Dδ(θ − θ′). The second term on the r.h.s. sets both the correlation length and a bound on the random walk. The inverse correlation length a and the noise intensity D can be estimated experimentally from the power specrtum:
0.64
Frictional Torque (Nm)
750
0.62
0.6
0.58
0.56 0
0.05
0.1
0.15
0.2
0.25
Shear Angular Velocity (rad/s) Fig. 1. Instantaneous reaction force of the granular on the shearing plate as resulting from a series of chaotic slips (lines). By averaging at fixed instantaneous velocity θ˙ over the different slips a regular friction curve is obtained (circles), which can be fitted by phenomenological expressions like Eq. (4) (thick line).
F. Leoni et al. / Journal of Non-Crystalline Solids 357 (2011) 749–753
The most simple phenomenological expression for ϕ is that proposed by Dieterich [7]
3. Ageing and rejuvening in friction The understanding of solid-on-solid friction, albeit still incomplete, has advanced considerably in recent years. Whilst time has no role to play in classical da Vinci–Coulomb–Amontons friction laws, dynamic friction is observed to generally decrease as the interface velocity increases. Furthermore, when two bodies remain in static contact for a long period, the frictional force between them is observed to increase with the logarithm of the contact age. Bowden and Tabor [18] showed that the force of static friction between two sliding surfaces is strongly dependent on the real area of contact. This work led to the asperity contact theory of friction on which basis, in the 1970's and 80's, different authors established more refined properties and constitutive laws for the behaviour of frictional bodies, e.g. Dieterich [7], Rice [10] and Ruina [9,11] (see [2,3] and [12] for review). Although fundamentally different in nature, granular matter has been shown to share many similarities with solid surfaces as far as shear stress is concerned [3], with the addition of possible hardening when the top plate velocity becomes large enough [13–16]. It has been found in particular that the ageing of static friction with time and the weakening of dynamic friction with velocity in granular gouges can be described by the same phenomenological laws as solids [3,8,9], generally known as rate-and-state friction laws (RS). In the following we introduce rate and state variables in multicontact interfaces and reformulate the problem of friction following the rate-and-state approach so to allow the weakening and hardening observed, the latter proportional to the square shear rate characteristic of solid-granular interfaces. We will adopt the notation of Dieterich [7] and will not enter in the different interpretations of the Rice–Ruina and Dieterich approaches, since they ultimately lead to about the same phenomenology [3,19]. Multi-contact interfaces are generated when rough hard solids are brought into contact with an apparent pressures W / Σ well below the elastic moduli and the real area of contact Σr is a small fraction of the visible area of contact Σ (typically Σr / Σ ≃ 10 − 3 for solid-on-solid, that means that size of real contacts D0 is of the order of microns). Following Bowden and Tabor [18], the friction force between two solids F = μτn, where μ is the friction coefficient and τn the normal stress, can be expressed as F = σs ⋅Σr
ð5Þ
where σs is the shear strength of the interface. For a wide variety of materials, experiments (the so called stopand-go and slide-hold-slide experiments) [3,7,9,12] show that the static friction coefficient μs(tw) varies logarithmically over several decades of waiting time tw during which the system is at rest. The strengthening of μs is related to the slow growth of the real area of contact Σr by means of the following relation [12] t 2 + o ln Σr ðt Þ = Σr0 1 + m ln 1 + ; τ
751
ð6Þ
where m and τ are temperature dependent factors. Typically, at room temperature, m ≃ 10 − 2 and τ≲1s for solid-on-solid. A more complete expression for the evolution of the real contact area is obtained by considering the dynamical aspect of the contact aging. This is usually made by introducing the geometric age of the interface or “state” variable ϕ(t), which replaces t in Eq. (6). For non moving interfaces, ϕ(t) is simply proportional to the time which has been spent at rest at time t. For an interface sliding at the instantaneous velocity x˙ ðt Þ, ϕ˙ is time dependent. The typical microcontact size D0 characterises the interfacial configurational memory length.
˙ xϕ ϕ˙ = 1− : D0
ð7Þ
In conclusion Tabor's expression for the friction can be written as F = σs x˙ Σr ðϕÞ;
ð8Þ
in which memory (or time) dependent effects are contained within the real area of contact only, while the shear strength σs, depends only on the instantaneous sliding velocity. As friction laws are generally phenomenological in nature, experiments are fundamental to obtain expressions for the shear strength as a function of velocity (as a matter of fact, no ab initio calculation of friction is available at present). As far as rate-state equations are concerned, “velocity jump” experiments have emerged as a useful method for studying the shear strength of multicontact interfaces. In such experiments, the frictional response to a sudden jump of the driving velocity from Vi to Vf, after a long period of constant driving is studied. Experiments are performed with rigid driving stages such that the velocity jump be transmitted to the actual shear interface before the interface has moved more than the memory length D0. In this way, the geometric age does not vary significantly from its initial stationary value ϕi = D0 / Vi, and so the behaviour observed is fully attributable to the rate dependence of the shear strength σs. Despite their empirical nature, rate-and-state models can be easily recovered from quite simple assumptions, as a theory of linearized response to perturbations of steady-state sliding [20]. The form of the terms in Eq. (8) is recovered using physical considerations from the specific problem of friction in multi-contact interfarces. However, the dynamic evolution of the state equation expressed in Eq. (7) is quite general, being the simplest formulation of a “maturity” variable ϕ, increasing in time as long as the system does not move more than a typical distance D0. As we describe in Section 5, our modeling approach retains such elements of the rate-and-state theory, modifying only the specific term σs in order to account for the physics of granular media at high speed, which is markedly different from the solid-on-solid case. 4. Acceleration dependent shear response as a signature of rate-state friction In this Section we shall report the results of the analysis of data from the experiment described in Section 2, targetted at revealing memory effects in the deterministic part Fv θ˙ of the effective friction, Eq. (2). Then, in Section 5, we shall adopt the rate-and-state approach introduced for multi-contact interfaces and will derive an expression for Fv θ˙ , the force that encapsulates the memory effects while retaining the possibility of showing both weakening and hardening. In the present work we do not focus on “stop and go” or “velocity jump” measurements described in the previous section: rather we shall attempt to detect and characterize memory and state effects in the chaotic stick-slip motion performed by the plate (Fig. 1). This can be considered an important test for rate-state friction laws as it can confirm their validity in situations where the velocity changes continuously and randomly. In previous work [4] the friction force was assumed to be a function of the position and velocity of the disk only: F ð:::Þ = Fv θ˙ + Fr ðθÞ, where the fluctuating part Fr(θ) derives from the instantaneous “random” configuration of the disordered force chain network, and Fv θ˙ is a single valued function obtained by
752
F. Leoni et al. / Journal of Non-Crystalline Solids 357 (2011) 749–753
˙ an example is averaging F(...) over many slip events in intervals of θ; reported in Fig. 1. As seen above [3,6], many experimental observations show that the frictional force is determined to some extent also by the history of the motion or by some state variable, involving possibly the degree of fluidization in the case of GS [16,17]. In order to highlight these “memory” effects in the stick slip motion observed, the analysis of Ref. [4] has been refined by considering separately the force during periods of positive and of negative acceleration; the friction force may be written: Fv θ˙ = Fvþ θ˙ + Fv− θ˙ :
ð9Þ
We remark that Fv, Fv+ and Fv− are averages in intervals of θ˙ over all the observations. The two experimental friction curves obtained are shown in Fig. 2 as a function of disk velocity. It is seen in the figure that :: ˙ a clear dependence :: on θ exists: at low θ friction is larger for positive than for negative θ , whereas the two curves cross over each other for ˙ The main focus of our paper is now to reproduce this increasing θ. phenomenon within the framework of rate-and-state friction laws, and to seek to understand the origin of the cross-over in Fig. 2. 5. A rate-and-state friction law for solid-granular interface To test the rate-and-state formulation of friction force in the case of the stick-slip experiments described in Refs. [4,5], we employed the most simple expression for the shear strength σs x˙ that may account for the full fluidization and consequent hardening of the granular medium at high velocities, assuming a square dependence on velocity [13,14]. With this assumption and following Eq. (8), the deterministic component of the friction force Fv is taken to be described by the following equations: ˙ ˙ ϕ = 1 + χ θ˙ 2 A + A ln 1 + ϕ ;ϕ˙ = 1−θϕ ; ð10Þ FvF ≡Frs θ; 0 ϕ A0 τ θ0 where θ0 is a memory angle. For the fluctuating part, the relation Eq. (3) still holds. We note here that the explicit dependence on θ˙ of ˙ ϕ is monotonic, but it nevertheless allows for the expression Frs θ; both weakening and hardening of the average force, and will appear in fact as a dynamical state effect through the variable ϕ. Numerical simulations of the chaotic stick-slip motion of the plate on the granular bed have been performed according to Eq. (1), with F = Fr + Frs, Eqs. (3) and (10). In the simulation, inertia and spring
constant are set at the values chosen for the experiment. The parameters that appear in Eq. (10) have been estimated by running simulations with different parameters and choosing those which best fit the corresponding experimental curves Fv + and Fv − of Eq. (9). During stick events, the first part of Eq. (10) reduces to the static ϕ friction Fv = A0 + Aϕ ln 1 + implying A0 ≃ F0 from Eq. (4). In the τ rate-and-state formulation of solid-on-solid friction, τ can be experimentally estimated as the time at which real contact area growth begins to increase logarithmically. Ageing experiments that we are conducting show τ ≈ 1s, in agreement with many solid-onsolid and gouge friction results, and so our simulations have been performed around this value. From rate-and-state models it can also be shown that [12] Aϕ ≃ S / Y ≃ 10 − 3 ÷ 10 − 2, where S is the strain rate sensitivity and Y is the yield stress. For velocities in the range of steady ˙ sliding ( θ˙ ∼ N10−1 rad/s), the state variable approaches ϕ = ϕss = θ0 = θ. Therefore, up to a logarithmic term the friction torque for the rate-andstate model has a quadratic dependence on velocity [13–15] with the prefactor given by χ. The best results we have achieved by fitting data in Fig. 2 with simulations of Eqs. (1) and (10) are shown in Fig. 3. The corresponding values found for the parameters are s2
χ ( 2) rad 2.
A0 0.579
τ (s) 1.0
Aϕ 0.007
θ0 (rad) 5 10 − 4.
These results, although preliminary, indicate that the stochastic model with rate-and-state friction reproduces two main features of the Fv± curves, i.e. the cross-over at low velocities and the weakening_hardening behaviour. From the values obtained for the parameters some remarks can be drawn: 1. As the granular friction is dominated by force chains at low velocity, one might expect to have D0 ≃θ0 r, where r is the average radius of the channel, of the order of a few grains. The value found for D0 (≈ 160μm) indicates instead that the relevant lengthscale for the contacts is that of solid on solid friction, thus concerning intragrain friction. Mair et al. [19] found 4μm b D0 b 183μm for grains of diameter 50μm b d b 150μm. D0 therefore seems related to grain asperities even in this experiment, which was performed at much higher pressure where one might expect a stronger sensibility to granular texture. 2. The above result implies at least two relevant length scales for the process, since it had been found [4,5] that the friction itself has an autocorrelation length 1/ a of some grain diameters. The information about the granular structure is thus carried by just a, the force
0,61
0,6
Fv
Fv
0,61
0,59
0,58
0,6
0,59
0
0,05
0,15
0,1
0,2
0,25
dθ/dt Fig. 2. Circles: The friction dependence resulting by averaging at fixed instantaneous ˙ over many slips events (like those depicted in Fig. 1). The upper and lower velocity, θ, data sets results from separating positive (squares) and negative (diamonds) acceleration contributes. As expected the reaction force of the granular system is larger for positive acceleration, but it happens only at sufficiently low velocities. At larger values a cross-over takes place and decelerating events display higher friction.
0,58
0
0,05
0,1
0,15
0,2
0,25
dθ/dt Fig. 3. The friction curves reconstructed by adopting rate-state laws of the type Eq. (10) in the stochastic model Eq. (1).
F. Leoni et al. / Journal of Non-Crystalline Solids 357 (2011) 749–753
correlation length, not by D0. D0 also appears to rule the position of ffi pffiffiffiffiffiffiffiffiffi such a cross-over which scales as ðD0 Þ (Eq. (6)). 3. The value determined for χ is not simply the geometric value ˙ In fact, this is true derivable from the curvature of F θ˙ for large θ. if one suppresses the noise term in the model friction, Eq. (2), whereas we have that the presence of noise affects the friction curve, both its average value and at large ˙θ. This is an important fact which will be investigated further in the future. 4. The value of τ is in agreement with preliminary results from stopand-go experiments. Despite qualitative agreement, it must however be noted that the adopted model does not reproduce the experimental data accurately. In particular the Fv± curves appear less distinct and flatter. The maximum speed is lower than in experiment. These difficulties indicate that the adopted model is inherently incomplete. More work is needed to discover where these problems originate, i.e. in the RS or in the random part of the friction model adopted. With regard to the former it is known that RS still presents some fundamental problems: stop-go experiments have shown that parameters such as D0 can be velocity dependent (a clear problem as D0 should be a material parameter) [19]. For the latter, we know the correlation length 1/ a of the random friction force is not a fixed parameter [5] as assumed in our simulations. With regard to our analysis here, we must also note that general RS research is conducted using constant velocites much lower than those employed here ( 100 μm/s instead of 10 cm/s), and under much higher compressional stress ( MPa instead of 100 Pa). 6. Concluding remarks and summary We have implemented a rate-and-state friction law in a stochastic equation describing the stick-slip motion of a plate shearing a granular bed. The rate-and-state parameters have been determined by best fitting simulation results with experimental. Although the agreement is only partial, it nevertheless reproduces features not obtainable with the simpler friction law. In particular it is seen that friction depends not only on the velocity but also on the acceleration, at least through its sign.
753
The values obtained for the memory length indicate that grain– grain friction lies at the origin of the frictional memory effects. Work in progress is concerned with validating the obtained parameter values with different measurements and understanding the origin of discrepancies remaining between the model and the experiments.
Acknowledgements FD acknowledges the financial support from EU project TRIGS, contract no. 043386.
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