History-dependent friction and slow slip from time-dependent microscopic junction laws studied in a statistical framework
To study how macroscopic friction phenomena originate from microscopic junction laws, we introduce a general statistical framework describing the collective behavior of a large number of individual micro-junctions forming a macroscopic frictional interface. Each micro-junction can switch in time between two states: A pinned state characterized by a displacement-dependent force, and a slipping state characterized by a time-dependent force. Instead of tracking each micro-junction individually, the state of the interface is described by two coupled distributions for (i) the stretching of pinned junctions and (ii) the time spent in the slipping state. This framework allows for a whole family of micro-junction behavior laws, and we show how it represents an overarching structure for many existing models found in the friction literature. We then use this framework to pinpoint the effects of the time-scale that controls the duration of the slipping state. First, we show that the model reproduces a series of friction phenomena already observed experimentally. The macroscopic steady-state friction force is velocity-dependent, either monotonic (strengthening or weakening) or non-monotonic (weakening-strengthening), depending on the microscopic behavior of individual junctions. In addition, slow slip, which has been reported in a wide variety of systems, spontaneously occurs in the model if the friction contribution from junctions in the slipping state is time-weakening. Next, we show that the model predicts a non-trivial history-dependence of the macroscopic static friction force. In particular, the static friction coefficient at the onset of sliding is shown to increase with increasing deceleration during the final phases of the preceding sliding event. We suggest that this form of history-dependence of static friction should be investigated in experiments, and we provide the acceleration range in which this effect is expected to be experimentally observable.
pacs:81.40.Pq, 46.55.+d, 81.40.Np, 62.20.mm
Solid friction is of considerable importance to a large number of fields, from geological Scholz (2002) to biological Scheibert et al. (2009a), engineering Zoback et al. (2012); Das and Zoback (2011) and materials Bartlett et al. (2012) sciences. It originates at the microscopic scales of the interface between two solids in contact. However, problems in friction often couple various time and length scales Persson (2000); Baumberger and Caroli (2006); Vanossi et al. (2013). To describe friction at large scales, upscaled/macroscopic friction laws are needed. Such laws are commonly formulated on length scales at which the local structure of the interface is assumed to be averaged out. The Amontons–Coulomb laws Coulomb (1821), the rate and state laws Rice and Ruina (1983); Bar Sinai et al. (2012) and other macroscopic friction laws parametrize the frictional response of the interface when submitted to external forces in terms of a handful of friction parameters, e.g. the static and kinetic friction coefficients. The microscopic origin of the friction forces does not explicitly enter in these descriptions, but is usually invoked to justify the basic features of the laws chosen for a given system, e.g. a proportionality between friction and normal forces.
The microscopic forces responsible for friction vary between systems. They can for example be associated with micro-contacts between asperities in rough interfaces Greenwood and Williamson (1966); Baumberger and Caroli (2006), pinned islands in boundary lubrication Persson (1995), or molecular bonds Schallamach (1963); Filippov et al. (2004); Srinivasan and Walcott (2009). We use the term micro-junction to refer to a single micro-contact, island or bond. To create a fundamental description of friction that takes its microscopic origins explicitly into account, two questions must be answered: (i) What is the behavior law for a given micro-junction? (ii) How can we upscale/integrate these laws to deduce the friction behavior at a larger length scale involving a large number of micro-junctions? The first question is addressed by the field of nanotribology (see e.g. Szlufarska et al. (2008)). Here we address the second question. In particular, we investigate the consequences that a time-dependent micro-junction behavior law has on the macroscopic friction force.
In principle, the state of a multi-junction interface could be monitored by following the individual state of each micro-junction. In practice, this task may not be possible for a series of reasons. First, the number of micro-junctions can be large, making it difficult to keep track of all the time evolutions of the parameters defining their individual states. Second, the properties of individual junctions (e.g. size, stiffness or threshold) are often known only in a statistical sense. Third, the external forces/stresses on the junctions are only known in average, through the total macroscopic applied loads on the whole interface.
A way around the above mentioned difficulties is the following: Instead of tracking individual junctions as they are loaded or start to slip under a small additional strain, the fraction of junctions that are loaded or start to slip is monitored. The idea of considering distributions rather than a finite set of micro-junctions will be used extensively here, and has also been used previously in various studies of friction (e.g. Schallamach (1963); Persson (1995); Farkas et al. (2005); Braun and Peyrard (2008, 2010)). Farkas et al. Farkas et al. (2005) studied the evolution of the junctions’ friction forces as a function of the displacement of a rigid slider. In particular, they showed how the macroscopic friction force depends on the distributions of both the shear stresses and strengths among the population of individual micro-junctions. Recently, Braun and Peyrard Braun and Peyrard (2008) showed that the evolution of the friction force as a function of the displacement of the slider can be solved with a differential equation – the master equation. Using this framework, they could study the relationship between the distribution of junction strengths and the occurence of either stick-slip motion or smooth sliding Braun and Peyrard (2010).
In these studies, the friction force was displacement dependent only. However, there is overwhelming experimental evidence that friction does not depend only on displacement. Among other phenomena: most interfaces have a velocity-dependent steady-state friction behavior (see e.g. Grosch (1963); Baumberger and Caroli (2006); Abdelounis et al. (2010)); most interfaces are aging, i.e. have a strength that increases with increasing time spent in contact before slip (see e.g. Dieterich (1978); Baumberger and Caroli (2006); Li et al. (2011)); the slip dynamics at short times after slip inception in polymethylmethacrylate is controlled by a time scale Ben-David et al. (2010a); the healing rate of seismic faults after an earthquake varies after a characteristic time scale Marone (1998); the friction force during reciprocating/oscillating motion depends on whether slip is accelerating or decelerating Bureau et al. (2001).
Motivated by these observations, a number of models have introduced time dependencies in the behavior of individual micro-junctions. Within the distribution approach, Schallamach introduced time rates for both the thermally activated bonding and de-bonding of molecules onto a surface to model the velocity-dependent friction of rubber Schallamach (1963); Persson, in a study of contacts with a lubrication film of molecular thickness (boundary lubrication), introduced a similar rate, but for the bonding of pinned adsorbate domains only Persson (1995); Braun and Peyrard also considered, in the master equation framework, the effects of a constant time delay for the repinning of micro-junctions and of an increase in the strength with the age of a pinned junction Braun and Peyrard (2011). Numerically, time-scales were also introduced for finite sets of micro-junctions put in parallel to model the friction between a surface and a slider. Filippov et al. used bonding and de-bonding time rates to model adhesive boundary lubricated surfaces and cold welding Filippov et al. (2004); in order to study micro-slip front propagation at a frictional interface, various models recently considered elastic sliders made of blocks connected by internal springs, each block being itself connected to the surface by a series of micro-junctions Braun et al. (2009); Capozza et al. (2011); Capozza and Urbakh (2012); Trømborg et al. (2014). Realistic results could be obtained using time delays between depinning and repinning of junctions.
Here, we present a general framework for models in which micro-junctions can switch between a time-dependent and a displacement-dependent state (Section II). The framework provides an explicit description of the distributions of individual junction states. It also allows for analytical continuum predictions that are useful to provide a systematic understanding of the effects of time-dependent junction laws. The framework can be applied to a whole family of behavior laws at the microscopic scale, and we show how previously studied models Farkas et al. (2005); Braun and Peyrard (2008, 2010, 2011); Braun et al. (2009); Capozza et al. (2011); Capozza and Urbakh (2012); Trømborg et al. (2014) are subsets of the general framework. We then explore the macroscopic consequences of microscopic variability in the transition time between the time-dependent and the displacement-dependent states. We first derive a general expression for the steady state friction force and show how it is directly related to the microscopic junction behavior. Depending on the microscopic laws used, the model can exhibit monotonic (strengthening or weakening) as well as non-monotonic velocity dependencies, all of which have been observed experimentally Chen et al. (2006); Bar-Sinai et al. (2013a); Dieterich (1978); Heslot et al. (1994); Baumberger and Caroli (2006) (Section III). In addition to this steady state phenomenology, the model gives insight into transient phenomena and static friction (Section IV). In particular, we show how the static friction coefficient is directly related to the distribution of shear forces on individual micro-junctions. We also predict a non-trivial history-dependence of the macroscopic static friction coefficient at the onset of sliding. More precisely, it is strongly influenced by the deceleration dynamics of the final phases of the preceding sliding event. We also show that slow slip, which has been reported in a wide variety of systems (see e.g. Ohnaka et al. (1987); Heslot et al. (1994); Hirose and Obara (2005); Peng and Gomberg (2010); Ben-David et al. (2010a); Yang et al. (2008)), spontaneously occurs in the framework if the friction force contribution from junctions in the slipping state is time-weakening. Section V contains the discussion and conclusion.
Ii Model description
We study the frictional behavior of a rigid slider (macroscopic block) that interacts with its substrate through a large number of micro-junctions (Figure 1). The junctions are assumed to be independent. They are all stretched by equal amounts when the slider moves. This assumption is valid if the lateral size of the slider is smaller than the elastic screening length, , so that the interface can be considered rigid Caroli and Nozières (1998); Braun et al. (2012). To study systems that are larger than , elastic interactions must be accounted for, for example by using spring–block models, with blocks of size , as in Braun et al. (2009); Trømborg et al. (2011); Amundsen et al. (2012); Capozza et al. (2011); Capozza and Urbakh (2012); Trømborg et al. (2014).
ii.1 The behaviour of individual junctions
We assume that individual junctions can exist in a displacement-dependent state, the pinned state; and in a time-dependent state, the slipping state. The junctions switch states as sketched in Figure 2. A junction remains in the pinned state until it is stretched beyond its breaking threshold force. It then enters the slipping state, where it stays for a random time (also called delay time), after which it is repinned or replaced by a different junction. In general, the force from each junction on the slider depends on the state of the junction. A pinned junction acts with a force , where is the stretching (the distance from the pinning point of the junction at the substrate to its attachment point on the moving slider). A slipping junction contributes a force that can depend on the time spent in the slipping state , . and are not necessarily displacement and time-dependent only; they can depend on other physical quantities, such as temperature and the velocity of the slider.
ii.2 A general framework for collective junction behaviour
To study the macroscopic friction force we need to know the collective behavior of a large number of junctions. In this section we introduce a general framework for the collective junction behavior, and show how various recent models are subsets of the framework. We then reduce the number of parameters and study the effect of disorder in the time at which the slipping-to-pinned transition occurs.
ii.2.1 Junction state distributions
When the number of junctions is large there is no need to keep track of the state of each individual junction. Instead, the collective state of the junctions can be described by two probability densities; one holding the information about pinned junctions and another holding the information about slipping junctions. Knowledge of these distributions can be used to determine the main variable of interest: The macroscopic friction force. In general, the instantaneous values of the distributions will depend on the past and present slip history of the slider.
Consider a system of junctions. Every time a junction leaves one of the states, the junction or its replacement enters the other state, so the total number stays unchanged.
This normalization condition can be written in a continuum formulation as
Equations (1) and (2) differ by a factor which will be absorbed into the force law. The stretching probability density holds the information about the stretching of pinned junctions. The slipping time probability density holds the information about the slipping time of slipping junctions. These distributions evolve with global time and with the motion of the slider, , so that and . The two time variables and evolve with the same increments, but serve different roles in the formalism. The global time is used to determine chronology and simultaneity, so that and are values taken at the same point in time. The slipping time , on the other hand, takes on different values for different junctions, or in the integral formulation, for different parts of ; because junctions enter the slipping state at different instants in time.
The macroscopic friction force on the slider is the sum of the forces from all junctions. The contribution to from the pinned junctions is a function of their stretching, , and the contribution from the slipping junctions in general depends on the slipping time, . We have that
The corresponding equation in the integral formulation is
where is the normal force on the slider and , . Two comments are in order at this point. First, we have absorbed the in the force law in order to have and normalized to . Second, the calculations that follow in the rest of the paper benefit from using a non-dimensional formulation of the force law and so we have divided by the normal force. Independently of wether the friction forces are proportional to the normal force, or have some other normal force dependence, this is the characteristic force level in the system. We will not study the effect of a varying normal force in this paper, so can always be recovered from by multiplying with .
ii.2.2 Evolution of S and A with time and displacement
The equations in the previous section apply when and are known. To use them, we also need to know how the distributions develop in time as the slider moves. Figure 3 shows the cycle during an infinitesimal time interval . The changes in the distributions have three contributions: 1) Rigid shift of the distribution. 2) Moving probability out of the distribution (junctions that break or repined/reform). 3) Probability received from the other distribution.
We start with the probability that moves from to . Recall that does not explicitly include a time dependence, but changes due to the changes in the position of the slider, . If the slider moved the distance during the infinitesimal time interval , the fraction of contacts that broke is
where is the probability for a contact with a stretching to break during a mesoscopic displacement . In Appendix A we show that this formulation is mathematically equivalent to having a distribution of junction strengths/thresholds, and how this function can be derived from .
In a formally equivalent way, the probability that moves from to (the fraction of junctions that repin/reform during a time interval ) is given by an integral over the distribution of slipping times multiplied by a probability function ,
Here we have written in a simple form where it only depends on the time since slipping was initiated , but it could also depend on other parameters such as the velocity of the slider, temperature, and so on. In this way more complicated rules for the evolution of slipping junctions can be modeled. In appendix B we show that is mathematically equivalent to having a distribution of delay times.
Note that while and have the normalization condition in equation (2), and can integrate to arbitrary values. For example, we will later use for all , whose integral over all diverges.
The portion of contacts that enter the pinned state will be assigned a stretching given by an initial distribution of stretchings . Combining the terms for contacts that leave and enter the pinned state with the rigid shift of , the evolution rule becomes
A formal equivalence for is achieved if contacts that break are assigned a non-zero initial slipping time from a distribution ,
should intuitively be a -function at , but we state it here because it gives symmetry to the equations and could in principle give more possibilities for the force law in the time-dependent state.
ii.3 Relating the general framework to previously published friction models
We are not the first to study friction models that fit in the framework defined above. In fact, our framework can be seen as a natural extension of existing results. In this section we give a few examples of previously studied models that are subsets of the general framework.
Farkas et al. Farkas et al. (2005) studied a model where junctions are represented by linear elastic springs. In the general framework this translates into , where is the shear stiffness of the junction. They investigated the influence of the distribution of junction stretchings on macroscopic friction force. The source of disorder in their model is that the junctions have different breaking thresholds. In the general framework this is encoded in , which can be mapped directly to a distribution of contact strengths (Appendix A). Whenever one junction reaches its breaking threshold (strengths), it is immediately replaced by an unloaded junction with the same properties. This is the special case of letting , , .
Braun and Peyrard (Braun and Peyrard, 2008, 2010) studied a model similar to the one of Farkas et al. They developed a master equation formulation of their model and used it to study the stick-slip and steady sliding regimes and the dependence of kinetic friction on sliding velocity. In Braun and Peyrard (2008, 2010) they assume that there is no time-dependent state of the junctions, that is, junctions that reach their breaking threshold are immediately replaced by new junctions pinned at a lower stretching. This is a subset of the general framework that is realized by letting , . Further, they introduce the forces in pinned springs as , that is, with an individual spring stiffness, but in the actual calculations they use , with the average spring stiffness. In place of they use a distribution of spring stretching thresholds ; Appendix A gives the mapping between these formulations. Their distribution of the initial spring stretchings is equivalent to in the general framework.
In Farkas et al. (2005); Braun and Peyrard (2008, 2010) the evolution of the junction states is controlled by a single variable, the position of the rigid slider. As a consequence of the disorder in breaking thresholds, these systems always approach a steady state when the block displacement is sufficiently large.
Braun and other co-workers introduced an enriched model in which junctions that reach their breaking threshold are removed and replaced by new unloaded junctions after a delay time Braun et al. (2009). As for and there is a mapping between and (it has the same mathematical form and can be found in appendix B). This model is the special case of , , and . There is also a viscous force term in their model, that acts directly on the slider. This could be added as an additional term in equation (5). The model formulated by Braun et al. (2009) has also been used by Capozza et al. Capozza et al. (2011); Capozza and Urbakh (2012). In Ref. Capozza et al. (2011) they also use a distribution of delay times, which would correspond to defining a .
The model used by us in Trømborg et al. (2014) can be formulated in the general framework by using the mapping between and a distribution of time that junctions will remain in the slipping state . The additional assumptions are , , , .
Following up on the idea of a delay time, Braun and Peyrard did a combined study of temperature activated breaking of pinned junctions, an increase of junction strength with time (ageing) and a delay time Braun and Peyrard (2011). The major component included in our framework, but not included in Braun and Peyrard (2011), is a disorder in the time spent in the slipping state. This is the focus of the present article. Although we have not included temperature effects and ageing in our presentation, the approach taken by Braun and Peyrard (2011) could be reused in our general framework: thermal breaking could be added as an additional time-and-temperature-dependent term in equations (6) and (8), and ageing could be included through a time-dependence in .
ii.4 Simplified model: Time-dependent junction behavior
To use the general framework we need to define the four underlying functions that are shown in Figure 3. and govern the flow of probability out of and , respectively. The other two, and , control the flow of probability into and , respectively. It is out of the scope of this paper to describe the effects of all four of them. Instead we focus on the effects of , which controls the transition from the slipping state to the pinned state, by reducing the other three functions to their simplest forms (Figure 4).
Braun and Peyrard have studied the effects of the strength distribution (equivalent to ) extensively. We recommend their papers to interested readers and take as our first simplification here that is collapsed to a delta function at a maximum stretching threshold ,
When all contacts have the same breaking threshold the portion of contacts that break during a time interval is reduced to
We define to be positive for a positive displacement of the slider. Second, we collapse the distribution of initial slipping times, , to a delta function at zero slipping time: . Third, we collapse the distribution of initial stretchings, , to a delta function at zero stretching: .
This leaves only one remaining underlying function: . The fraction of junctions that enter the pinned state during a time interval is
This probability should be removed from . Combining , and the rigid shift in we write
Note that for convenience we avoid -functions in by distributing the junctions entering uniformly in the interval . Also note that the junctions can break at both and . The corresponding equation for is slightly different (the equations are formally equivalent in the full framework, but here we have specified in equation (10) while remains unspecified).
Also note that the velocity of the slider is
which we will be useful in the next sections.
This concludes the description of the model we will use in the rest of the paper to study disorder in the time-dependent junction state. In sections III and IV we will discuss the implications this time-disorder has at the macroscopic scale. We start with the steady state, where the slider moves at constant velocity.
Iii Steady state
In this section we find expressions for the distributions and during steady sliding and derive the steady state friction coefficient as a function of velocity. The results are valid under the assumptions made on , and in the previous section. Additionally, we need to assume that a steady state can be found, which means that the block is sliding at constant velocity and that and are independent of the block’s position and of global time .
iii.1 Steady state distributions
The distance travelled by the block while a contact is in is
since we have already assumed that the distribution of initial stretchings is a delta function at zero stretching and that breaks all contacts at . The mean distance travelled while the contact is in is
where is the average time spent by a junction in the slipping state, i.e. the expected lifetime in the distribution. This is the expectation value of in the distribution of delay times (mapping from in Appendix B), and should not be confused with the expectation value of in , . The fraction of junctions in , , is then
Similarly, the fraction of contacts in , , is
In steady state, the exchange of probability between the distributions and is constant in time, otherwise the distributions would change. The same probability enters at and leaves at :
With constant this is only possible when the shape of is uniform. Since is nonzero only on we conclude that
The calculation for with a general is longer. It can be found in Appendix C, and we give the results for and here.
Note that the steady state distributions do not depend on or .
iii.2 Steady state friction coefficient
The steady state friction coefficient can be found using equation (5). Inserting for and we find
Equation (24) holds for any choice of , and , which results in a large variety of possible steady state friction laws. To find the steady state friction law for a particular system, one must specify , and based on the physical properties of the individual junctions. We will give a couple of examples of such single junction behavior laws and show that monotonous as well as non-monotonous velocity-dependent steady-state friction laws can be found within the model. We consider successively the cases of a velocity-independent and a velocity-dependent .
iii.2.1 No velocity dependence in , or
When is velocity-independent, the shapes of and are also velocity-independent, and so is . In these cases the velocity only controls the amount of probability in each of the distributions, i.e. the amplitudes and . The shape of , however, can still be different for different . We define the following velocity-independent expectation values:
We can write this in terms of the dimensionless parameters , and :
The velocity weakening, velocity strengthening and velocity independent solutions are apparent from this form. Increasing the velocity shifts probability from to , which results in a velocity weakening steady state friction coefficient if , a velocity strengthening steady state friction coefficient if , and a constant steady state friction coefficient if . For all three cases, the limit is . This is demonstrated in Figure 5.
If is velocity-dependent the shape of will also be velocity-dependent, and equation (24) can no longer be factorized as in the previous section. There are many possible choices for . Here we give an example motivated from a realistic microscopic picture of micro-contacts at the interface between rough solids, which results in a weakening-then-strengthening velocity-dependent steady state friction law.
Assume the following micro-contact behavior: The pinned state is linear elastic so that , where is an elastic constant with dimension 1/length. When a contact reaches the threshold , it enters the slipping state, in which it will slide a characteristic distance before the micro-contact vanishes (see Figure 6). We assume that, even in the absence of driving velocity , the contact, once in the slipping state, would slip with a small velocity , as observed experimentally in e.g. Heslot et al. (1994); Yang et al. (2008) and attributed to a creep-like process. The net velocity at the micro-contact interface in the slipping state is the sum of and of the macroscopic block velocity , giving
This is illustrated in Figure 6. We seek the time interval where the distance slipped equals the characteristic distance . Note that due to the creep , this occurs even if the slider velocity is zero (after a characteristic time ). In steady state, the velocity is constant, and we can solve for
To relate to we use a with
Of the many choices of that would work here, we choose the simplest one and assume that is not a function of . We get
which gives us
Inserting in equation (22) gives
To complete the friction law we need . Assume that the force from each micro-contact decreases with the time spent in the slipping state, for example due to friction induced heating. We take as our example
where is a positive constant with dimension 1/time. We can determine the steady state friction coefficient by inserting , and into equation (24). We get
This can be written in terms of the dimensionless quantities , , and :
Equation (38) is plotted in Figure 7 for different values of . Note that , which is directly related to the force threshold in the pinned state. In a physical system, the force in the slipping state is usually smaller than or equal to the threshold in the pinned state: . Depending on the value of , can be both velocity-weakening at low velocities and velocity-strengthening at high velocities, with a transition velocity given in appendix D. When , converges towards
which is different from . For the velocity-independent of the previous section, all junctions were at ( and ). In the present case, however, and converge to values different from these because when .
Weakening-then-strenthening velocity-dependent friction laws are often considered to be generic for frictional interfaces Baumberger and Caroli (2006); Bar-Sinai et al. (2013a). It was observed experimentally on interfaces as different as paper-on-paper Heslot et al. (1994), granite-on-granite Dieterich (1978) or PMMA-on-glass Bureau et al. (2002). It was used as a mesoscopic friction law in numerical models of the statistics of earthquakes at seismic faults Burridge and Knopoff (1967). It was also discussed theoretically in Bar Sinai et al. (2012); Bar-Sinai et al. (2013a, b). Note that Braun and Peyrard found an opposite behaviour, strengthening-then-weakening, when using ageing and delay times Braun and Peyrard (2011).
Iv History dependent friction
When the velocity is not constant, the slider will in general not be in the steady state. While steady state solutions are reasonable approximations in some limits, an understanding of transients is particularly important to the study of a large variety of frictional situations including oscillating contacts Rigaud et al. (2013), the onset of frictional sliding, be it quasi-static Prevost et al. (2013) or dynamic Rubinstein et al. (2004), the cessation of slip Ben-David et al. (2010a) and friction instabilities Rouzic et al. (2013).
In this section we will demonstrate two important consequences of a two-state junction law: slip history dependent static friction and slow slip. First, due to the coupling between the time-dependent state and the displacement dependent state, the distribution of junction stretchings, , depends on the velocity history of the slider. In turn, this distribution determines the static friction coefficient during onset of sliding, resulting in history dependent static friction. In particular, the static friction coefficient depends on the deceleration when the slider last came to rest. Second, if there is a weakening in the friction force in the slipping state that is distributed over a characteristic time (e.g. ), and if , then slow slip is predicted from the model.
To arrive at these promised results, we first need to develop a framework to make analytical predictions for transients. This is done in section IV.1. The applications are discussed in sections IV.2 and IV.3, which contain the main results of this paper.
iv.1 Analytical predictions for transients
In this section we introduce the necessary framework to make analytical predictions of the friction force during transients. We start with the distribution of junction stretchings, , after stopping. We then relate to the mesoscopic static friction coefficient. The distribution of junctions in the slipping state, , after onset of sliding can be found in Appendix E.
iv.1.1 Calculating after stopping
As we will see in Section IV.1.2, is of particular interest because it determines the development of the friction coefficient when sliding is initiated. In this section we will find after motion stops. Our strategy is to find as a function of time, and place it in according to the velocity profile.
Given an initial state , the probability in follows
where is the fraction of junctions, that had been slipping for a time at , that remain in after a time . This is found from solving equation (90) for the evolution of junctions in the slipping state. The cumulative probability that has left is
The instantaneous probability leaving is found by taking the derivative of equation (43) with respect to and multiplying this with .
Dividing by we get
Defining the rate of probability from to , , and using Leibniz’ integration rule on we find
We can find the stretching distribution after the slider has stopped (at time ), , if we assume that no junctions reach and break during the time , i.e. when the total displacement is smaller than . Recall from equation (13) that probability enters with amplitude (which results in a velocity dependence in after stopping) and combine this with the stretching that occurs when the probability enters and until to obtain
where is the displacement of the slider. Note that there is a time-dependence here, while is in general given as a function of stretching. To go from to , we need the velocity as the slider comes to rest. We show an example of such mapping in section IV.2. Equation (47) applies for any initial state where , and for any . Note that depends on the velocity the slider has as it comes to rest through the term , which has implications for the static friction coefficient assosiated the next onset of sliding.
iv.1.2 Macroscopic static friction coefficient
In this section we show how the macroscopic static friction coefficient depends on the distribution of forces acting on the junctions, . Consider a slider at rest where is known. At any instant in time, the friction force is
If we assume that breaking is fast compared to , the friction force as a function of displacement is
where the integral in the second term is the probability in , . should be replaced with an integral over from equation (108) or be solved numerically if the assumption that breaking is fast compared to does not hold. This would require knowledge of the velocity of the slider during breaking. In the following we assume that equation (49) is a good approximation. The macroscopic static friction coefficient is simply the maximum of :
This concludes the discussion of the framework for analytical predictions of transients. We will now use these results in a few examples.
iv.2 Deceleration dependent macroscopic static friction
Equation (50) predicts that the macroscopic static friction coefficient depends directly on . The highest possible static friction is found when is a delta-function so that all junctions contribute their maximum force simultaneously,
Increasing the width of will reduce because the first junctions break before the rest reach their maximum force contribution. When and is strictly increasing, the lowest static friction coefficient is obtained for uniform on ;
When , the ratio of maximum to minimum static friction (from the two limits of ) is . It is noteworthy that the value can vary this much even though all micro-junctions have the same breaking threshold. A similar result was already found by Farkas et al. Farkas et al. (2005). In our previous work Trømborg et al. (2014) we found the dependence of on the width of numerically. In appendix F we find the same result analytically for uniform by solving equation (50). Increasing the width of will in general reduce the static friction coefficient. However, the precise functional form of this dependence also depends on the shape of .
Combining equation (47) and equation (50), we see that the macroscopic static friction coefficient depends on the velocity history, because determines , which in turn determines . Even though the assumptions made when deriving equation (47) were quite restrictive, the dependence of friction on velocity history is general in our model. This velocity history dependence has interesting consequences: The macroscopic static friction coefficient depends on slip dynamics of the previous sliding event.
In this section we calculate for a slider that stops under constant deceleration. We then use this result to find the static friction coefficient of the next event as a function of the deceleration. Because the analytical results can only be found under quite strict assumptions, we will complement them with numerical solutions in which the restricting assumptions can be lifted. The implementation is straightforward using equation (13) and (14).
Assume that all probability is initially in ; . As initial condition, , for the analytical calculations we choose the steady state distribution of when , which is a good approximation to the steady state result for high velocities. Also assume that motion stops within the displacement . The calculations would be simplest for . However, results in distributions after stopping that are easier to interpret, and so we use this for clarity of presentation. Numerically we solve for three different including a constant .
Inserting in equation (47) we get
If we let in equation (23) we find
We have found , but the independent variable in the expression is time. To find as a function of we need to find and the correspondence between and . Numerically the inversion is trivial ( and come in indexed pairs and either is known as a function of the other). The analytical inversion requires a bit of bookkeeping. Assume that the slider stops under a constant deceleration, , so that
Because the block stops at and then remains at the inversion is not well defined for larger values of . This is handled by realizing that all the probability that leaves after the block has come to rest will enter at because . In the end we will therefore use
where is the probability that shifts from to in the time interval ,
For , when all the probability is in ,
is found by inserting
in equation (56). We obtain
Figure 8 (a) shows scaled with the characteristic time and length of the system: , , and