velocity weakening in a dynamical model of friction

PAGEOPH, Vol. 142, No. 3/4 ( 1 9 9 4 ) 0033-4553/94/040447-2051.50 + 0.20/0 �9 1994 Birkh/iuser Verlag, Basel

Velocity Weaken ing in a Dynamica l Mode l o f Frict ion

DMITRI PISARENKO 1 and PETER MORA 1

Abstract - -We introduce a discrete model for friction between rough elastic surfaces which is based on the microscopic description of contacts between asperities. Rough surfaces are modeled as spring-mass arrays with superposed asperities. The linear elastodynamics of the underlying surfaces is treated in the model separately from the nonlinear contact behavior of asperities. Unlike usual spring-block models, no a priori friction law is imposed in the model, which allows the frictional behavior corresponding to a chosen microscopic physics of contacts and topography of the rough surfaces to be simulated. We use the model to study the elastodynamical mechanism of friction related to the inertial response of the elastic medium to suddenly imposed tractions, and perturbations of contact properties due to the elastic waves propagating along the interface. The contribution of this mechanism to friction becomes important at high slip rates (above 1% of the wave speed in our simulations), where it results in the velocity weakening behavior. The mechanism of velocity weakening is first studied analytically on an isolated model element. The predicted behavior is then reproduced in numerical simulations with large surfaces. Simulations with stepping of the driving velocity demonstrate a difference between the frictional force measured directly on contacts, and at the loading point. The latter corresponds to laboratory measurements and includes the inertial response of both the loading mechanism and the elastic body to the variations of driving velocity. We speculate that a similar inertial response is present in certain experimental data.

Key words: Friction, velocity weakening, spring-block models.

I n t r o d u c t i o n

The friction between two rough surfaces is a p h e n o m e n o n of great impor tance

in m a n y different problems at scales ranging from that of microcracks in rocks up

to tectonic faults. I t is well k n o w n that the real area of contact of rough surfaces

represents only a small fraction of the total area of the surfaces (BOWDEN and

TABOR, 1950). Only those surface protrusions, or asperities, which are in contact,

cont r ibute to friction. 'The frictional force can thus be regarded as a statistical

quant i ty resulting from numerous local inl~eractions between the asperities. This

implies that the no t ion of friction may be adequate only with respect to sufficiently

large surfaces: below a certain scale this statistical sum is no longer representative

and, in the limit, one observes the contact of two asperities which canno t be

described in terms Of a friction law.

Institut de Physique du Globe de Paris, Drpartement de Sismologie, 4, place Jussieu 75252 Paris CEDEX 05, France.

448 D. Pisarenko and P. Mora PAGEOPH,

Many existing models of fault seismicity, such as the spring-block (BUR- RIDGE and KNOPOFF, 1967; CARLSON and LANGER, 1989) impose a friction law between the rough surfaces. They are therefore intrinsically macro-scale models: in order to produce the same friction law, the surfaces of the blocks, which represent the smallest scale in the model, should average out in the same way the individual features of the asperities, i.e., contain a large number of them. The spatial and temporal evolution one observes in this class of models is to a large extent the consequence of the particular form of the friction law used. Our present knowledge of the friction laws on real faults remains limited. Firstly, no direct measurement of friction on a fault, while it is in motion, can be imagined. Secondly, there is the fundamental problem of how results of laboratory experiments should be extrapolated to geological scales. WALSH (1971) demonstrated by a simple analysis that the loading stiffness in laboratory experiments on rock is typically 4 or 5 orders of magnitude higher than that corresponding to a large earthquake. Scaling of other parameters of the problem in a similar way may significantly change the proportion of competing mechanisms, and thus result in a behavior different from that is observed at laboratory scales. These considerations suggest that choosing a particular friction law as a basic ingredient for modeling fault dynamics may be restrictive.

The alternative approach that we follow in the present paper consists of constructing a model for the micro-scale interactions of rough surfaces. The description of contact of two rough surfaces is thus made at the smallest natural scale of the problem--that of asperities. The microscopic mechanisms, and hence the model based upon them, can be correctly scaled using scaling relations for the physical variables involved. The macroscopic characteristics of friction, such as friction law, obtained within such a model will therefore correspond to its microscopic physics.

Another advantage of the present approach is the possibility of separately examining the contributions of different microscopic mechanisms to friction, which is extremely difficult, if not unfeasible, in laboratory experiments.

In this work we are interested in the origin of velocity weakening--the decrease of friction with displacement velocity. In a number of models for fault dynamics, velocity weakening friction is the only nonlinear element, and hence a possible source of dynamical complexity. A continuous interest in the

spring-block models is in particular due to their ability to simulate stick-slip sliding with temporal and spatial patterns similar to those of real earthquakes. Considering that velocity weakening is a necessary condition for such unstable motion, it is important to improve our understanding of its origin. We shall demonstrate a new elastodynamical mechanism of velocity weakening, whose contribution to the rate dependence of friction may be important at high slip rates.

Vol. 142, 1994 Velocity Weakening in a Dynamical Model of Friction 449

Theoretical Background: Rate- and State-dependent Friction

The difference between static and kinematic friction was recognized since the works of Coulomb. Once the shear load on a frictional couple exceeds some critical value, slip occurs and further motion is generally resisted by a lower friction, i.e., /lk < #s- The kinematic friction is also observed to depend on the displacement velocity: in a stationary regime the frictional stress z tends to its steady-state value Zss which is a function of velocity V. No general form of friction law L~.(V) can be deduced, as observations show a large variability for different materials, surface roughnesses, conditions of experiment, etc. However, in numerous experimental studies (DIETERICH, 1979; TULLIS and WEEKS, 1986; KILGORE et al., 1993) friction was found to be a decreasing function of velocity

dzs,/dV <O, (1)

at least in some range of velocities. In the present paper we are interested in frictional behavior of this kind as it favors stick-slip instability.

Different microscopic mechanisms have been proposed to explain velocity weakening. One that is frequently invoked involves partial momentum transfer in a direction perpendicular to sliding. The faster the sliding, the less the penetration of asperities of one surface into cavities of the other, and thus, less energy is lost on the transverse movements of surfaces (early explanations of the origin of friction also used this "geometric" argument). Recent examples of this approach can be found in the works of LOMNITZ-ADLER (1991) and POSCHEL and HERRMANN (1993).

Other commonly accepted models are based on time-dependent contact strength: in laboratory experiments with metals (RABINOWIeZ, 1958) and rocks (DIETERICH, 1972) the shear strength of a contact of two surfaces at rest was observed to increase with time approximately as log t. This can be explained by plastic deformations of asperity tips. Such deformations increase the area of contact between asperities, and at the same time, lead to formation of new contacts because the separation between the rough surfaces reduces. The elasto-plastic model of contact (KRAGHELSKY, 1965) predicts that the increase of the total contact area with time is dominated by the formation of new contacts, while the surface of existing contacts grows quite slowly. When the rough surfaces slide past one another, the population of contacts is permanently renewed. Higher slip rate reduces the average age of contacts, which results in smaller real contact area, and consequently leads to weakening. This mechanism of rate dependence of friction was confirmed by DIETERICH (1993) in recent experiments with plexiglass blocks: a strong correlation between temporal variations of the frictional stress and those of the real contact area, measured by an optical method, was observed.

Experiments with metals (RABINOWICZ, 1958) and rocks (SCHOLZ and ENGELDER, 1976; DIETERICH, 1978, 1979; TULLIS and WEEKS, 1986) showed that


h(O)A /

�9

A

V1 V2

Uch

V1

SLIP

Figure I Evolution of the frictional stress in response to the loading velocity stepwlse variations. The steady-state behavior is described by the friction law z,,(V), while both the direct and the evolving effects are presented as convolution of the loading velocity time derivative with the function h(-), describing the

transitory process.

shear resistance of a dry frictional couple depends not only on the current displacement rate but also on the prehistory of slip, i.e., the system has a memory. Two opposite effects are observed in response to a sudden change in driving velocity (see Fig. 1). First, it causes an instantaneous change of shear stress of the same sign (the direct effect). Then the transition of stress to its new stationary value takes place over some characteristic slip (the evolving effect), provided the velocity is maintained constant. These observations lead to the concept of both rate- and state-dependent friction, the state of the surface being associated with the slip history. The memory effect can be related to the above discussed time-dependent contact strength: the change in velocity results in a gradual evolution of the average age of contacts towards its new stationary value, which is reached only when the entire population of contacts is reriewed. The slip distance over which this transition occurs is the characteristic of a given surface, and approximately equals the mean size of the contact junctions (RABINOWICZ, 1956; DIETERICH, 1979).

Using a single state variable framework, this can be summarized as follows

f_ . , t - t ' z(V, t) ='Ls(V) + ~ V(t ) h ( ~ ) dt', (2)

where h(t) is a rapidly decreasing positive function describing the transitional process, tch = uch / V is the characteristic time at a given velocity /I, and uch is the

Vol. I42, 1994 Velocity Weakening in a Dynamical Model of Friction 451

characteristic slip distance for the given surfaces. Note that the time scale of h(t) also depends on the current velocity V through tch, and that, in the present formulation, the direct effect is included in h(O, unlike equation (7) in RICE and RUINA (1983).

Experimental observations of friction carried out with centimeter size rock samples at driving velocities ranging from 10 - 3 to 10 3 microns per second are well described by the above rate- and state-dependent constitutive relation, though more than one state variable may be needed to better fit certain observed features (RUINA, 1983; TULLES and WEEKS, 1986). However, seismic rupture on a fault involves much higher slip velocities, which can be estimated as a few meters per second (HEATON, 1990). This may introduce additional mechanisms of friction, or give greater importance to those which are negligible at low slip rates. For instance, inertia should be taken into account at high slip velocities, i.e., deformation of rough surfaces must be described by the elastodynamic equations.

In this paper we introduce a model which is used to study the elastodynamic mechanism of friction in combination with topography of rough surfaces. Other mechanisms of rate dependence, such as the time-dependent strength of contacts, are not considered. This important simplification allows us to separately study the dynamical mechanism of interaction of rough surfaces and its contribution to rate and state dependence of friction at high slip velocities.

Elastodynamical Mechanism of Friction

Let us consider two asperities which are in contact. If a shear stress z is applied, the asperities and surrounding material will first deform. Once the stress reaches some critical value zcr, the asperities will slide past one another, releasing elastic energy stored during the loading phase, in the form of waves. The waves emanating from one asperity perturb positions of free asperities and stress on those which are locked, and thus modify the state of the surface. The interaction of asperities by means of the elastic waves propagating along the interface can be described in the following way. Consider the shear stress Txz at a point of the surface due to the tractions Tx in the x direction applied to asperities. Assuming a two-dimensional in-plane geometry of the problem (Fig. 2), it can be expressed as

Z~z(X,O=#~dt" f f (G .... ( x , t - t ' , x ' , O ) + G .... (x , t - t ' ,x ' ,O))Tx(x ' )dS, (3)

S

where p is the shear modulus, and G~ is the corresponding component of the Green's function for a half space. It is assumed that the surface S is free excepting the points of contact, which have small areas a (o, therefore the surface integral can be approximated by the sum over contacting asperities:

r~(x, t) ~ I~ Idt' ~ (G .... -4- Gx~,~,~T(~ . (4) , ) i

452

Z

D. Pisarenko and P, Mora PAGEOPH,

~'~::~ :*+~ :~:~'~:::~ ~ * ~ i : : ~ : :~::~ ::~:.":~,,~"~l ~:~:~: :v~*~.~ ~:: ' ~: ~ .~ ~ 5 ~ : ~ ~ g ~ ::~'::~':;~:g~:~!)~ ~ : : ~ ! ~ ; ~ i : ! i ~ : : g ~ ! : : ~ '

X

Figure 2 2-D in-plane geometry of the problem. The upper half space moves in the x direction which creates

tractions on the contacts between asperities.

The constitutive rate- and state-dependent relations assume that the state of contact is determined uniquely by the macroscopic movement of the surfaces. Though the perturbations of stress on asperities due to the waves propagating along the interface may be small compared to the absolute stress value, they can significantly modify the frictional behavior of the contact. LINKER and DIETERICH (1992) observed in the experiments with granite that the effect of sudden normal stress variations is similar to that of loading velocity stepping. They proposed therefore to include the dependence of state variable on normal stress in the constitutive law. The numerical simulations by MORA and PLACE (1994) suggest that normal stress variations occurring along the slip front can play an important role in the dynamical friction.

Tangential motions of particles at the interface due to the elastic waves can also have an impact on the frictional properties of contact. In the model introduced below we thoroughly consider the effect of such motions on rate dependence of

friction.

Description of the Model

The model introduced in this section tan be regarded as a tool for numerical study of the above described elastodynamical mechanism of friction, rather than its direct implementation. Therefore we state several important simplifying assump-

tions. We represent the rough surfaces as reference plane surfaces with superposed

asperities. This allows us to separately treat two basic mechanisms of the model: the linear elastodynamics of the underlying surfaces, and the nonlinear contact behavior of asperities. The stresses created by the interlocking of the asperities are assumed to be applied directly to the reference planes and concentrated at the


locations of asperities. The state of contact of two rough elastic surfaces is thus

represented by an equivalent distribution of stress on plane surfaces. This approxi- mation is validated if the typical size of asperities is small with respect to the surface dimensions, i.e., the surfaces are almost planar.

Though we assume for simplicity a linear stress-strain relation for the contact of asperities, the contact is nonlinear in the sense that it has a stress threshold, beyond which it breaks (the asperities slip past one another). Thus all the specific properties of asperity contacts, reflecting complex physical processes at these points (inelastic deformation, cohesions heating, etc.) are represented in the model by a single pa ramete r - - the shear strength of contact Fc,.. In order to exclusively observe the effect of the elastodynamical mechanism, we studied the model without time dependence of contact strength, i.e., for a given contact Fc~ = const. Under a given

normal stress the value of For is assumed to be a material property. The normal stress applied to the surfaces is represented in the model by the separation of the reference planes. The separation distance s and the vertical overlap Ah of the asperities of heights hj and h 2 are related as Ah = H(h~ + h 2 - S ) , therefore the contact strength F,.r depends only on the overlap Ah. For simplicity this dependence was chosen to be linear F~r = cAh, where c is constant. These assumptions reduce

the parametrization of rough surfaces in the model to their geometry and elastic constants.

The plane elastic surfaces are modeled by an array of particles of mass tn interconnected by springs of stiffness k~ (Fig. 3). All particles and springs are identical. The springs obey the Hooke's law, which corresponds to a crystal-like lattice with harmonic potentials. The waves radiated into the volume play a less important role in the imteraction of asperities than those propagating along the interface. We therefore discretize the medium only along the surface and not in the

k S _ y /

Figure 3 Representation of a rough surface in the model. The plane elastic surface is modeled by particles of mass m connected by springs k s. The loading stiffness of the elastic layer is modeled by leaf springs kt, which include viscous attenuation ct to account for the wave radiation. The topography (the heights of

asperities hl) controls the distribution of tractions applied to the particles.


depth direction: the particles are connected to a plane rigid support (the opposite surface of the elastic layer) by leaf springs of stiffness kt. This means that we take into account only the loading stiffness of the elastic layer and neglect the details of how stress and deformation are distributed throughout the volume.

A chain of harmonically coupled masses has a weak dispersion when the wavelength is considerably longer than the lattice constant. However, coupling to a rigid support (nonzero kt) introduces infinite phase velocity at infinite wavelength (BURRIDGE and KNOPOFF, 1967). This circumstance is sometimes misinterpreted as leading to a noncasual wave propagation along the chain. In fact it only means that at low frequencies (co < ~ ) the motion of particles is dominated by the waves reflected by the rigid boundary. The system has only 1-element thickness, thus the wave travel time through it simply equals the period of free oscillations 2nx /~ / k t. In order to retain a nonzero loading stiffness and, concurrently avoid the effect of multiple reflections (free oscillations) we include a viscous damping 0~ in the leaf springs. The amount of damping is adjusted empirically to better reproduce the response of an elastic half space (analytical solutions of the Lamb's problem). If necessary, this simplification can be removed from future models, by discretizing the medium in the third dimension, though at a high computational cost.

The last and vital element of the model is the contact of asperities. Whenever two asperities (shown as vertical bars atop particles on Fig. 3) overtake one another with a positive vertical overlap Ah, they are assumed to form a contact. For simplicity the contact is characterized by a linear stress-strain relation, i.e., it applies to the corresponding particles a force proportional to their relative horizontal displacement F = kcdx (see Fig. 4). Once this force exceeds the threshold value For, the contact breaks, and the asperities slip past one another. As stated above, the contact strength is proportional to the overlap of the asperities Fc~ = cab. In fact,

Ah

Ax Figure 4

Two asperities are assumed to form a contact when they slide past one another with a positive overlap Ah. The contact obeys the linear stress-strain relation F = k c Ax, where kc is the contact stiffness, and F is the force applied to the particles. The contact strength is limited by a critical force For which is

proportional to the vertical overlap For = cab.

Vol. I42, t994 Velocity Weakening in a Dynamical Model of Friction 455

once two asperities form a contact, they become connected to each other by a strength-limited spring of stiffness kc. We choose kc ~ ks to satisfy the assumption that stress created at contacts is transmitted to the reference planes.

It is important to ensure that the adopted assumptions allow us to correctly model the principal phenomena of concern and that the simplifications result only in second-order effects. In particular, viscous damping can only provide a qualitative representation of energy loss due to the wave radiation. It is also important to take into account that properties of an elastic continuum are significantly different from those of the corresponding discrete model at wavelengths comparable to the lattice constant:. To address these questions we shall consider the analogous continuum problem in a forthcoming work. Here we demonstrate the elastodynamic mechanism of velocity weakening friction predicted by the discrete model, nonetheless in our view this mechanism is general enough to be preserved in the continuous case.

Mechanism of Velocity Weakening in the Model

Before presenting the results of numerical simulations using the model described in the previous section, we would like to discuss more extensively the mechanism of interaction between the asperities in contact. In this section we demonstrate that even when all the elements of the model are linear (the springs obey Hooke's law) the presence of contacts with a limited strength introduces a nonlinear frictional behavior with respect to displacement velocity. Once the elementary contacts interact due to the propagation of elastic waves, this nonlinearity results in velocity weakening friction. First we demonstrate this effect by considering an isolated model element colliding with a rigid obstacle. We then reproduce the behavior predicted by a single element model in numerical simulations with rough profiles.

Contact Behavior of an Isolated Model Element

Consider two identical elastic half spaces sliding past one another at constant velocity V without friction, as shown in Figure 5. At some moment we fix two adjacent points on their surfaces together, which imitates the stress created by the locking of two asperities. What is the boundary condition imposed on the half spaces by such a contact? By choosing a mobile reference in which both half spaces move with speeds of V/2 in opposite directions, one can easily see by a symmetry argument, that the contact point remains at rest. This means that in the reference associated with the lower half space, the contact with the upper half space moving with velocity V imposes a traction boundary condition with the constant displacement rate of V/2. In other words, such a contact can be seen as a collision with an absolutely rigid obstacle moving at a constant velocity.


1

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~ ~i~;%/ii::iii/::i=:i=:::::::iii::iiiiil::::::!::!!iii!ii//iiiiiii~iiiiiiii::i

i~!~iiiiiiii#!~ ~'::::''. ........... !i::~i~t~ii :~,~I~ ~" ~ ::"' ".::::,~.:::r~:~:;;,

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Figure 5 The contact between the elastic half space moving with velocity V (top) and the identical fixed half space (bottom) moves with velocity V/2 (1). This results from the symmetry of the problem, when it is considered in the mobile reference, where both half spaces move "with V/2 in opposite

directions (2).

Let us now consider the behavior of an isolated model e lement- - the damped oscillator--subjected to a suddenly imposed constant rate displacement. A stepwise change in velocity imposed directly on a finite mass element would result in an infinite interaction force. This problem does not arise in the continuum elastodynamics, where abrupt changes of velocity can be used in boundary conditions or appear in solutions (RICE, 1993). The reason is that in an idealized continuous solid, the infinite instantaneous acceleration at the moment of the velocity change applies to an infinitesimal volume of matter with zero mass, and therefore does not create infinite force. However, when the medium is dis- cretized, the boundary condition with abruptly changing velocity should be smoothed, taking into account the elementary masses involved in motion. There- fore we apply the velocity stepwise change to an isolated model element through the strength limited contact with a finite stiffness, in the same way as we model

the contact of two asperities. The oscillator has the following parameters: mass m, stiffness k t and damping

~. The contact between the obstacle and the oscillator is characterized by the stiffness kc (kc >> kt) and the critical shear force Fc,. We calculate the force of interaction during the contact phase and its average value as a function of the oscillator parameters and the macroscopic velocity V. In this case, the dynamical equation of motion of the oscillator can be integrated analytically. If we take t = 0 as the beginning of the contact, the solution for the interaction force as a

function of time is given by

F( t ) = V t 1 + r + V A e -~/2( t -6) o,) s in~( t - f i ) - ~ c o s o g ( t - 6 ) + B , (5)


1 . 2 ,.." ,,.

5 . / "'"'" ,. ..,,-"

4 . . . . " T H R E S H O L D LEVEL .....

- - 0 , 4

0 . 2

,I 0 I I I [ 0 0.5 1 1,5 2 2.5 3 3.5

T IME

Figure 6 Interaction force between an isolated model element and a rigid obstacle moving with velocity V, as a function of time. The graphs are calculated using the analytic expression (5) for five different values of velocity. Time is normalized by the free oscillation period, force is normalized by the threshold value For (dashed line). Beyond the threshold value the contact breaks (symbols *), and the interaction force drops

to zero. Dotted lines continue the analytic solution after the contact is broken.

where the following nota t ion is used

kk~_ a 2 1 r=ktk~, c O = 4 m ( l + r ) - ~ - , 6 = - - a r c t a n - - ,

co 2 0

A ( l + r ) o ) ~ a 2 ' B = e (~/2)~ cos o ) 6 + c o s i n c o 6 .

Figure 6 shows the funct ion F(t) calculated for five different values o f velocity

V and with the typical oscillator parameters used in the simulations (time is

normalized by the free oscillation period, force is normalized by the threshold value

F~r). Two different components o f the interaction force during the contact phase can

be distinguished: linear increase due to the uniform loading o f the spring k~ (first term

in equat ion (5)) and exponentially decaying oscillations, produced by the collision

(second term in (5)). The average interaction force ( F ) can be estimated as

(f)-u- 1 F(x) dx = -T F(t) aft = Vu 3o'c F(t) dt, (6)

where the average is calculated over a fixed displacement u within which the contact

breaks, T = u~ V is the corresponding time interval given the velocity V, and t c is the dura t ion o f the contact . The last equality uses the fact that F(t) = 0 for t > t~.


0.6

0.5

w

r , r ~0 0.4

z

0 < 0.3

_Z

< m 0,2 uJ .,<

0.1

0 210 I I I 0 40 60 80 .... i 0 0

VELOCITY

Figure 7 The average interaction force (F) between an isolated model element and a rigid obstacle as a function of velocity V for four values of attenuation ~. The velocity is normalized by the reference value V* = u/T, where u is the displacement at which the contact breaks in the quasi-static limit, and T is the period of free osciIIations. The force is normalized by the threshold value For. For low velocities

(F) ~, 1/2 in accordance with (7).

In collisions at low velocities V, the oscillatory term of the interaction force will be mostly damped out by the end of the contact phase. In this case the duration of the contact, and thus the energy loss due to the collision, are determined by the intersection of the linearly increasing force component with the contact strength level For (graphs 1 through 3 in Fig. 6). The average force of interaction in this quasi-static limit does not depend on velocity V, and can be estimated as

(Fqs } ~ Fc~/2. (7)

An increase of the macroscopic velocity V leads to a higher slope of the linear term of the interaction force, but also increases the amplitude of the oscillating term in equation (5). At some critical velocity Vcr the maximum of the first oscillation reaches the threshold level, and the contact dynamically breaks much earlier than it would have in the quasi-static case (graph 4 in Fig. 6). Hence, for all velocities higher than V~r (e.g., graph 5 in Fig. 6) the average interaction force is substantially reduced, i.e., we observe velocity weakening.

In Figure 7 we show the average interaction force ( F ) as a function of velocity for the same parameters of the oscillator as in Fig. 6 and for four different values of damping ~. It was calculated by substituting the analytic expression for the interaction force (5) into (6), where u was taken equal to the displacement at which

Vol. I42, 1994 Velocity Weakening in a Dynamical Model of Friction 459

the contact breaks in the quasi-static limit. The force is normalized by the threshold

value, therefore for low velocities we obtain ( F ) ~ 1/2 in accordance with (7). The

graphs (F)(V) can be interpreted as effective friction laws for an isolated model element. The amount of damping which is empirically adjusted in the model in order to imitate the wave radiation, does not change the stepwise decreasing form of (F ) (V ) , which is essential for the mechanism of velocity weakening we are interested in.

The stepwise drop of the resistance of such a contact as a function of velocity is a phenomenon somewhat analogous to the rupture of materials at very high collision speeds. For example, a bullet can make a small hole in a glass without breaking it: the pressure on the glass in the contact area reaches its critical value, breaks the material locally and drops to zero before the elastic wave of a considerable amplitude is excited.

Numerical Simulations

In this section, we present a sequence of numerical simulations of the model.~ The numerical algorithm is based on the 4th-order Range-Kutta integration of the equation of motion of coupled damped oscillators

mJ2~ + a Z + krX, + k~ (2X,. - X,._ I - X,+~ ) = V,(t), (8)

where ~ is the i-th particle deviation from its equilibrium position, ks and k t - horizontal and leaf spring stiffnesses respectively, and a - -v i scous damping. The force terms Fi are calculated according to the contact stress-strain relation:

F i = { : c A x i kcAxi<F~." kc Axl > F~" (9)

where k~ is the contact stiffness, and Ax~ is the relative horizontal displacement of the asperities in contact. The force threshold F~" is calculated as

F~ r = cAh,, Ahi = H(h~ ') + h! 2) - S), (10)

where c is a material constant, h! ~) and h52~ are the heights of the contact asperities, and s is the separation between the reference planes. In all simulations F~ r was time independent.

Numerical Experiment with Realistic Profiles

We simulated friction experiments using realistic 1-D profiles with distributions of asperity locations and asperity heights similar to those measured on real rock

t Computations were performed on a Connection Machine CM-5 massivety paraltel computer at the French National Center for Parallel Computations in the Earth Sciences (C.N.C.P.S.T,), Paris.


surfaces. The rough surfaces were generated as random functions with given statistical parameters. To represent a profile as a random function, one needs essentially its two basic statistical characteristics: the one- and two-dimensional probability density functions (p.d.f.). The first contains information pertaining to the distribution of asperity heights independent of their mutual location. The second reflects the statistical dependence of topography values at different points. BROWN and SCHOLZ (1985a) found that the distributions of heights of many real rough surfaces are well described by two models: Gaussian and inverted chi-square. The latter generally gives better fit as it contains a parameter controlling the skewness of the p.d.f. The p.d.f.'s of heights of a number of rough surfaces are slightly skewed toward large values: frictional wear or surface preparation techniques result in flattened asperity tips and narrow valleys between them. In our numerical simulations however, we considered this effect to be of marginal importance, and used the Gaussian model. Besides the simplicity of numerical generating, it offers the following important advantage: a stationary Gaussian process is completely defined by its mean and correlation function (or its Fourier transform--power spectrum). Considering that spectral components of a Guassian process are independent Gaussian values, we can easily generate random topographies with Gaussian p.d.f. and given form of power spectrum. The corresponding numerical procedure consists of weighting the independent Gaussian spectral components by the spectrum shape and taking the inverse Fourier transform. The power spectra obtained from broad bandwidth study of natural rock profiles (BROWN and SCHOLZ, 1985b) have typically the form of "red noise": they linearly decrease in log-log scale with slopes ranging between - 2 and - 3 . In our numerical simulations we used random Gaussian topographies with k -2 spectra, where k is the spatial wavenumber.

The simulated experiment consisted of rubbing the upper sample, with velocity imposed at its top surface past the lower one, which was fixed. The two samples were put into contact at some initial position and with a given separation between their reference planes. The value of the vertical separation can be interpreted as the normal load applied. Figure 8 shows the frictional force as a function of slip in the velocity stepping experiment with two 1-D profiles. The upper and lower profiles contain 8000 and 10000 particles respectively, which allows the sliding of the upper profile within the limits of the lower one. The driving velocity has two stepwise changes with velocity ratio V2/V1 = 3. Note that the direct effect (sharp peaks) is observed as well as velocity weakening. Unlike laboratory experiments, the numerical simulation allows the frictional force to be measured directly at the locations of contacts. In the case of variable driving velocity, the force measured at the loading point (which corresponds to laboratory measurements) also includes the inertial response of both the elastic body and the driving mechanism. The comparison between the friction measured directly on contacts ~,i Fi (dashed line in Fig. 8) and at the loading point kt ~ iX/ (so l id line), shows that direct effect in the model is entirely due to the inertial response of the elastic body.

Vol. 142, 1994 Velocity Weakening in a Dynamical Model of Friction 46l

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v , I i v ,

o ' ' ' ' ;o 8'0 ' ' 2o 3o .o . ,oo ,,o SL IP

Figure 8 Frictional force as a function of displacement measured at the locations of contacts (dashed line) and at the loading point (solid line) in the simulated velocity stepping experiment. Displacement is normalized by the interparticle distance, stress is normalized by its steady-state value at V]. Both curves show the velocity weakening behavior. Insets show the direct effect pulses with a horizontal scale magnified by t0. The direct pulses are observed only in the loading point force because they are related to the inertial response of the sample. The velocities V~ and V~ equal 1% and 3% of the wave speed in the model, respectively, The transition between the steady states is very steep as the contact strength is time

independent.

Figure 9 shows a similar velocity stepping simulation (graph 2) compared to the experimental data (graph 1) reproduced from LINKER and D~ETeRICH (1992). The

simulation was performed with the same velocity ratio as used in the experiment

(V2/V] = 10), and the values of V1 and V2 were chosen in an almost velocity neutral part of the friction law of the model (discussed below), in order to facilitate the comparison with the experimental data.

On the simulated graph, the positive pulse is wider than the negative pulses. This effects the obvious explanation: the pulses have a characteristic du ra t i on - - t he

loading response t ime- -whi le the time scale of the graphs plotted versus slip is

inversely propotional to the displacement velocity. Thus during fast sliding (middle

interval) the time scale of the graph is expanded, and the positive peak on graph 2 is exactly 10 times wider than each of the two negative ones, as V2/V~ = 10. We argue that a similar effect is present in the experimental data: the positive pulse has a widened tip and the subsequent transition to the steady state clearly shows two different decay constants. This is not observed for the negative pulses. To illustrate a possible origin of such form of pulses we superpose the friction coefficient


P

0.1 ~ttrlt$ ] 1.0prnl6 t 0 1 p m l s

L

f L f

SLIP

Figure 9 Friction coefficient versus slip in the velocity stepping experiment with Westerly granite reproduced from LINKER and DIETEg~CH (1992) (graph 1), compared to the numerical simulation (graph 2). Solid line on graph 2 shows the loading point measurements, dashed line corresponds to the stress measured at the locations of contacts. The simulation does not reproduce the evolving effect as the contact strength is time independent, and the direct effect is purely inertial. The goal of the comparison is to underline the difference of form between the positive and the negative pulses. In the simulated curve 2 this difference is due to the different time scales of the graph during the high and low velocity intervals. Superposing the evolution law predicted by the constitutive relation of the form (2) (graph 3), which has the symmetric positive and negative pulses, and the inertial response (graph 4) with the negative pulses "compressed" by the factor V2/VI, gives the evolution (graph 5) qualitatively reproducing the experi-

mental data.

calculated using the rate- and state-dependent law of the form (2) with exponentially decaying evolving effect (graph 3), and a short impulse imitating the inertial response with a given characteristic time (graph 4). The positive and negative pulses described by the constitutive law are symmetric, while the inertial response pulses are "compressed" during the low velocity intervals. The result shown in graph 5 reproduces the form of the experimental data (graph 1).

The friction laws of the model T~s(V) obtained from simulations with different separation of the rough surfaces are shown in Figure 10. Velocity is normalized by the infinite wavelength wave speed in the model v = Ax/-~rn, where A is the interparticle spacing, and the friction coefficient is normalized by its value at V = l%v. The values of separation s given in Figure 10 are normalized by the topography roughness standard deviation (negative values correspond to an overlap of the topography mean levels). All friction laws show velocity weakening in the intermediate range of velocities studied (above 5% of v), and velocity neutral behavior (or weak velocity dependence) at low velocities. This confirms our


1.1

0.9

0.8

0.7

0.6

0,5

0.4

0.3

0.2

0,1

go;

O S= -20

+ S=-15

E} S= -10

X S= -5

t~ S= -2.5

S= 0

, , , , + , , L

0 . 1

VELOCITY 1

Figure I0 Friction laws of the model obtained from simulations with different separation of the rough surfaces. Velocity is normalized by the infinite wavelength wave speed in the model v, the friction coefficient is normalized by its value at V = 0.01 v. The values of separation s are normalized by the topography roughness standard deviation (negative values correspond to an overIap of the topography mean levels). The form of the friction taws suggests that the elastodynamical mechanism of interaction of the rough

surfaces may be responsible for velocity weakening behavior at relatively high slip rates.

expectation that the elastodynamical mechanism of friction may have importance only at relatively high slip velocities. The decrease of friction laws obtained with little penetration of the rough profiles (s = 0 through - 5 in Fig. 10) is monotone and starts practically at V = 0.01 v, whereas the curves corresponding to strong penetration (s = - 1 0 through -20 ) show a plateau or even velocity strengthening in the lower range of velocities studied. The latter situation corresponds to high normal loads when most asperities are in contact, and the topography controls only spatial variations of the contact strength.

Discussion

We have proposed a new elastodynamical mechanism of frictional interaction between rough surfaces. It is based on the inertial response of the elastic surfaces to suddenly imposed tractions and perturbations of the contact properties by the elastic waves propagating along the interface. We studied the influence of this mechanism on friction using the discrete numerical model which allows the microscopic physics of contacts to be specified, Hence we were able to observe the


frictional behavior of the model resulting uniquely from the elastodynamical mechanism under study in this paper. For this purpose other physical mechanisms which may be responsible for velocity weakening, such as time-dependent contact strength, were not included in the simulations. The form of friction laws produced by the model (Fig. 10) can be interpreted in the following way. Considering the effective friction laws of an isolated model element shown in Figure 6 can be approximated by a stepwise drop between two limit values, we can assume that a contact may contribute to the total frictional resistance by one of these values, depending on local sliding velocity. The local sliding velocity is determined by the loading velocity V and the perturbations due to the elastic waves produced by contact rupture and propagation along the interface. At low loading velocities (with respect to the velocity at which the resistance of contacts drops), most contacts have their maximal (quasi-static) resistance. At a higher velocity some fraction of contacts are broken dynamically, i.e., with low effective resistance. The proportion of dynamically breaking contacts is determined by the distribution of contact strength (controlled in the model by the topography), and by perturbations of local sliding velocity by waves. Finally, at high loading velocities most contacts contribute with their low dynamical resistance independently of their interaction through waves. This explains a steep decrease of the friction laws shown in Figure 10 at velocities above 50% of the wave speed. The rapidly decreasing high-speed part of the friction laws is probably of little relevance for seismological problems, as sliding at such velocities is not known to occur within faults, and moreover, other physical mechanisms (fracturing, heating) may be of greater importance than the elastodynamical mechanism considered here.

A closer inspection of experimental curves of the friction coefficient in velocity stepping experiments, like those reported by LINKER and DIETERICH (1992), and KILGORE et al. (1993), suggests that two distinct components are present in the evolving phase after the stepwise increase in the driving velocity. After the peak value, the initial rapid decrease of friction switches on half-way to a slower exponential-like decay. To satisfy this observation, constitutive relations with two state variables (TULLIS and WEEKS, 1986) or two different evolution laws with different characteristic distances (LINKER and DIETERICH, I992) have been proposed. We speculate that such form of positive direct pulses may result from the inertial response of the loading mechanism superposed on the frictional evolving effect. An argument which favors this hypothesis is that the measured positive and negative pulses of the frictional stress as a function of slip are not symmetric. This indicates the presence of a time-dependent response in the system in addition to the slip-dependent behavior, described by the constitutive laws.

Conclusions

1. The model for friction of rough elastic surfaces introduced in this paper is more general than the traditional spring-block models as it contains no a priori friction law.


2. The proposed elastodynamical mechanism of friction results in velocity

weakening. Its contr ibut ion to the rate dependence o f friction becomes impor tan t at

high slip velocities. 3. The simulations show a difference between the frictional stress measured at

the locations o f contacts and at the loading point. In particular, the inertial

response is observed only in the loading point stress.

4. The different fo rm of the experimentally observed frictional responses to

positive and negative stepwise variations o f the loading velocity may result f rom the

presence o f a mechanism with a characteristic time, such as the inertial response of

the loading system.

Ackn o wl e d g men t s

We thank Alain Cochard, Rafil Madar iaga and Jim Rice for many suggestions

and helpful comments on this work. Extensive critical reviews by Ian Main, Chris

Marone and John Rundle helped to improve the manuscript . Dmitr i Pisarenko

acknowledges the support f rom the Fondation des Treilles.

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(Received September 1, 1993, revised February 3, 1994, accepted February 17, 1994)

velocity weakening in a dynamical model of friction

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