O M Braun

J Scheibert

Propagation length of self-healing slip pulses at the onset of sliding: a toy model

Keywords: Precursor to sliding, Self-healing slip pulse, Propagation length

Macroscopic sliding between two solids is triggered by the propagation of a micro-slip front along thefrictional interface. In certain conditions, sliding is preceded by the propagation of aborted fronts, spanning only part of the contact interface. The selection of the characteristic size spanned by those so-called precursors remains poorly understood. Here, we introduce a 1D toy model of precursors between a slider and a track in which the fronts are quasi-static self-healing slip pulses. When the slider's thickness is large compared to the elastic correlation length and when the interfacial stiffness is small compared with the bulk stiffness, we provide an analytical solution for the length of the first precursor, Λ, and the shear stress field associated with it. These quantities are given as a function of the bulk material parameters, the frictional properties of the interface and the macroscopic loading conditions. Analytical results are in quantitative agreement with the numerical solution of the model. In contrast with previous models, our model predicts that Λ does not depend on the frictional breaking threshold of the interface. Our results should be relevant to the various systems in which self-healing slip pulses have been observed.

Introduction

Despite the importance of friction [START_REF] Persson | Sliding Friction: Physical Principles and Applications[END_REF][START_REF] Braun | Nanotribology: Microscopic mechanisms of friction[END_REF], several fundamental aspects of the problem, such as a detailed description of the beginning of sliding motion, are still not fully understood. Recent experiments allowed visualization of the onset of sliding in a variety of experimental situations for sphere-on-plane [START_REF] Chateauminois | Friction and shear fracture of an adhesive contact under torsion[END_REF][START_REF] Prevost | Probing the micromechanics of a multi-contact interface at the onset of frictional sliding[END_REF] or planeon-plane [START_REF] Rubinstein | Detachment fronts and the onset of dynamic friction[END_REF][START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF][START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF][START_REF] Romero | Probing Locally the Onset of Slippage at a Model Multicontact Interface[END_REF] contacts, for randomly rough [START_REF] Rubinstein | Detachment fronts and the onset of dynamic friction[END_REF][START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF][START_REF] Chateauminois | Friction and shear fracture of an adhesive contact under torsion[END_REF][START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF][START_REF] Prevost | Probing the micromechanics of a multi-contact interface at the onset of frictional sliding[END_REF] or micro-structured [START_REF] Romero | Probing Locally the Onset of Slippage at a Model Multicontact Interface[END_REF] surfaces, for side-driven [START_REF] Rubinstein | Detachment fronts and the onset of dynamic friction[END_REF][START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF][START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF] or top driven [START_REF] Chateauminois | Friction and shear fracture of an adhesive contact under torsion[END_REF][START_REF] Prevost | Probing the micromechanics of a multi-contact interface at the onset of frictional sliding[END_REF][START_REF] Romero | Probing Locally the Onset of Slippage at a Model Multicontact Interface[END_REF] loading. These experiments have revealed that sliding initiation is always mediated by the propagation of micro-slip (rupture) fronts, separating a stuck region and a slipping region, along the contact interface. Propagation was found either quasi-static (controlled by the external loading) [START_REF] Chateauminois | Friction and shear fracture of an adhesive contact under torsion[END_REF][START_REF] Prevost | Probing the micromechanics of a multi-contact interface at the onset of frictional sliding[END_REF][START_REF] Romero | Probing Locally the Onset of Slippage at a Model Multicontact Interface[END_REF] or dynamic [START_REF] Rubinstein | Detachment fronts and the onset of dynamic friction[END_REF][START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF][START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF][START_REF] Romero | Probing Locally the Onset of Slippage at a Model Multicontact Interface[END_REF] with a variety of speeds from supersonic to anomalously slow [START_REF] Rubinstein | Detachment fronts and the onset of dynamic friction[END_REF][START_REF] Trømborg | Slow slip and the transition from fast to slow fronts in the rupture of frictional interfaces[END_REF]. Fronts spanning the whole contact precipitate macroscopic relative motion (sliding) of the bodies in contact. Before that, a series of aborted fronts spanning a portion of the contact only may be observed [START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF][START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF]. Such fronts are called precursors and will be the object of the present work.

In side-driven plane-on-plane contacts [START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF][START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF], all precursors were observed to nucleate near the trailing edge of the contact, where the loading is applied. The first precursor propagates through the shear and normal stress fields produced by the initial loading of the system. After the precursor has stopped, the mechanical state of the interface is modified all along the ruptured path, so that the next fronts will propagate through the stress field prepared by the previous ones [START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF]. In this context, successive precursors were observed to propagate longer and longer distances [START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF][START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF]. Here, we will address the problem of the selection of the characteristic length of the first precursor.

Several models have previously been used to investigate the series of precursors to sliding. Braun et al. [START_REF] Braun | Dynamics of Transition from Static to Kinetic Friction[END_REF] used a dynamic simulation in which the slider was modelled as a chain of blocks coupled by springs, while the interface was treated as a set of "frictional" springs with random breaking thresholds. They reproduced both a series of precursors of increasing length and a modification of the stress field by the precursors. These results stimulated the study of simplified models that would still reproduce series of precursors. In Refs. [START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF][START_REF] Scheibert | Role of friction-induced torque in stick-slip motion[END_REF][START_REF] Amundsen | 1D Model of Precursors to Frictional Stick-Slip Motion Allowing for Robust Comparison with Experiments[END_REF][START_REF] Trømborg | Transition from Static to Kinetic Friction: Insights from a 2D Model[END_REF], the interface was described by the phenomenological Amontons-Coulomb law of friction: the interface is pinned until the local ratio of shear to normal stress reaches the static friction coefficient µ s ; the interface is then locally slipping and characterized by a kinetic friction coefficient µ k < µ s . All these models produce series of growing precursors nucleated at the trailing edge. Maegawa et al. [START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF] further showed that an asymmetric normal loading influences the precursor length. Scheibert and Dysthe [START_REF] Scheibert | Role of friction-induced torque in stick-slip motion[END_REF] showed that a similar effect is obtained for a symmetric normal loading, as soon as the friction force is not applied exactly in the plane of the contact interface. Amundsen et al. [START_REF] Amundsen | 1D Model of Precursors to Frictional Stick-Slip Motion Allowing for Robust Comparison with Experiments[END_REF] showed that, to be realistic, 1D models should involve an intrinsic length scale for the interfacial stress variations, for instance through the introduction of the shear stiffness of the interface. Trømborg et al. [START_REF] Trømborg | Transition from Static to Kinetic Friction: Insights from a 2D Model[END_REF] showed that, in 2D models, such a length scale naturally arises when introducing the system's boundary conditions and bulk elasticity.

In all the above simplified models [START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF][START_REF] Scheibert | Role of friction-induced torque in stick-slip motion[END_REF][START_REF] Amundsen | 1D Model of Precursors to Frictional Stick-Slip Motion Allowing for Robust Comparison with Experiments[END_REF][START_REF] Trømborg | Transition from Static to Kinetic Friction: Insights from a 2D Model[END_REF], each broken contact point repins when its slipping velocity goes back to zero. As a consequence, the observed rupture mode is crack-like, i.e. the precursor front separates a stick region from a region which is slipping almost everywhere behind it. Recently however, Braun and coworkers have used a different friction law [START_REF] Braun | Collective Effects at Frictional Interfaces[END_REF][START_REF] Braun | Crack in the frictional interface as a solitary wave[END_REF]: the friction between two solids results from multiple individual micro-junctions which break under stress and immediately form again elsewhere. As a consequence, the rupture mode of the interface is of the self-healing slip pulse type, i.e. the slipping part of the interface is confined between the rupture front and a repinning front.

The scope of this work is to propose a toy model which incorporates this alternative friction law within a 1D elastic model of a solid slider, and use it to investigate the kinematics of the onset of sliding. We will show that, assuming quasi-static propagation, this toy model enables analytic predictions for the length of the first self-healing precursor and the shear stress field associated with it. We emphasize that the self-healing mode is not just a theoretical concept: it was observed in var-Fig. 1 (color online) Sketch of the toy model. Left: A 2D rectangular slider (gray) of dimensions L and H, of elastic constants E and ν, in contact with a track, is discretized into cubic blocks of size λ 3 c . Right: the slider is modeled using a 1D toy model as a bilayer. The blocks in the interfacial layer (IL) are connected through springs of stiffness K (Eq. ( 1)). In the upper part of the slider (US), each vertical slice of material is assumed to be a rigid block connected to its neighbours through springs of stiffness K L (Eq. ( 2)). The IL and the US are connected through transverse springs of stiffness K T (Eq. ( 3)). The IL is connected with the rigid track by "frictional" springs (or λ-contacts) of elastic constant k. The position of the leftmost block of the US is imposed and increased at a velocity v d .

ious experiments using widely different materials and loading conditions (see e.g. [START_REF] Lykotrafitis | Selfhealing pulse-like shear ruptures in the laboratory[END_REF][START_REF] Ronsin | Nucleation and Propagation of Quasi-Static Interfacial Slip Pulses[END_REF][START_REF] Romero | Probing Locally the Onset of Slippage at a Model Multicontact Interface[END_REF]). Comparatively, as far a the onset of sliding is concerned, it has received much less interest than the crack-like mode. This is why we believe that the present toy model is a valuable first step towards a better understanding of the full dynamics of self-healing slip pulses.

Toy model

We consider that macroscopic contacts are made of a large number of micro-junctions in parallel, with an average distance a c . Let an individual junction (microcontact, bridge, solid island, etc.) have an average radius r c and height h c . Considering it as a cylindrical flexible "rod", we may estimate its elastic constant as

k c = 3πEr 4 c /4h 3 c
, where E is the Young modulus of the material [START_REF] Braun | Collective Effects at Frictional Interfaces[END_REF]. Elastic theory introduces a characteristic size λ c (known as elastic correlation length) below which the frictional interface behaves rigidly [START_REF] Caroli | Hysteresis and elastic interactions of microasperities in dry friction[END_REF][START_REF] Braun | Collective Effects at Frictional Interfaces[END_REF]. Typical values of the correlation length lie in the µm scale. The set of N c = λ 2 c /a 2 c contacts within the area λ 2 c is considered as an effective contact (the λ-contact introduced in [START_REF] Braun | Collective Effects at Frictional Interfaces[END_REF][START_REF] Braun | Crack in the frictional interface as a solitary wave[END_REF]) with elastic constant k = N c k c .

At length scales larger than λ c , the slider's bulk is deformable and two distinct points along the interface can move by different amounts. To account for this elasticity, imagine that the slider is discretized into cubic blocks of size λ 3 c . The elastic coupling between adjacent blocks is modeled by springs connecting only the near-est neighbours, the stiffness of which can be worked out (see below) as functions of only two elastic parameters, Young's modulus E and Poisson's ratio ν. Because we aim at providing analytical solutions, we progressively reduce the dimensionality of the slider from 3D to 1D. A sketch of the final 1D model is shown in Fig. 1 (right). First the model is reduced to 2D (Fig. 1 left) by reducing the thickness of the slider to only one block's size, λ c . Then the height of the slider is modeled as a bilayer of blocks: The bottom-most layer (interfacial layer IL) is left as in the 2D model. Each block of the IL having a size λ c , it is connected to the track through a single λ-contact. In the upper part of the slider (US), each vertical slice is assumed to behave rigidly, with neighbouring slices being connected by internal springs. Note that a similar, though different, model was previously introduced in [START_REF] Brörmann | Friction on a Microstructured Elastomer Surface[END_REF].

Let us consider a slider of length L, height H and thickness W = λ c (by assumption). We assume that both L and H are much larger than λ c . Note that in this 1D model H should be interpreted as some effective height where the driving force is applied. The horizontal stiffnesses K and K L in the IL and US, respectively, can be derived to be:

K = Eλ c (1)
and

K L ≈ EH = (H/λ c )K . (2)
The total transverse stiffness of the slider is E/2(1+ ν)LW/H. We assume that we can ascribe the totality of this stiffness to the L/λ c transverse springs K T connecting the IL to the US. K T thus reads:

K T = Eλ c 2(1 + ν) λ c H = λ c 2(1 + ν)H K. (3)
Note that K T /K L ∼ (λ c /H) 2 << 1. This is a consequence of the rigid connection of the blocks constituting the US (from left to right in Fig. 1). We chose as an assumption to ascribe the totality of the transverse rigidity of the slider to only one plane between the IL and the US. Let us emphasize that, as a matter of fact, there is no exact way to capture 3D elasticity within a 1D model, even qualitatively. In particular, the stress profile in response to a side loading decays as a power law in 3D whereas it decays exponentially in 1D. The scope of our toy model is thus merely to help us investigate analytically the consequences of the self-healing nature of the rupture on the propagation length. We will see that an important parameter of the toy model is the ratio of interfacial to bulk stiffness k/K. The interface is denoted as stiff if k/K ∼ 1 and soft when k/K ≪ 1; we concentrate on the latter case which corresponds to most experimental situations. The system can be described by three variables: u t (x) describes the displacement field of the US, u(x) is the displacement field of the blocks in the IL, and u b (x) is the displacement of attachment points of the IL to the track.

The stress in the IL is then given by

σ c (x) = k [u(x) -u b (x)]/λ 2 c , (4)
while the stress which is driving the IL is equal to

σ d (x) = K T [u t (x) -u(x)]/λ 2 c . (5)
The whole system is driven by imposing the displacement of the left-most block of the US as u t (0, t) = v d t, with v d an arbitrarily small velocity.

Self-healing slip pulse-like rupture

The frictional behaviour of each junction connecting the IL to the track is the following. It acts as a spring of elastic constant k as long as its stretching remains below a critical value u s = f s /k but breaks and immediately repins with zero stretching when the local shear stress reaches σ s = f s /λ 2 c . When a loading is applied to the slider, it is transmitted differently to each λ-contact due to elasticity of the slider. The leftmost λ-contact is the first to reach its breaking threshold, slide and locally relax the interface. This causes an extra stress on the neighboring contacts, which tend to slide too, so that sliding events propagate as a kink, extending the initial relaxed domain. Such a scenario may be described as a solitary wave [START_REF] Braun | Collective Effects at Frictional Interfaces[END_REF][START_REF] Braun | Crack in the frictional interface as a solitary wave[END_REF]. The rupture mode is of the self-healing slip pulse type, with a pulse width equal to the lateral size of one contact, λ c , because a broken contact repins immediately, before the breaking of its neighbour. Note that, in the absence of the US, and for a displacement controlled loading of the left edge of the slider, no propagation would occur: the first block would not move after breaking of its λ-contact, and thus its rightneighbour would not be further loaded to its breaking threshold.

In order to simplify the analysis, we suppose the existence of a hierarchy of times. Namely, we assume that the pushing rate is adiabatically slow, v d → 0, while the sound speed (which determines the rate of propagation of the elastic stress) is very large, v R → ∞. Therefore, when a front propagates, it is accompanied by a quasistatic stress field defined by mechanical equilibrium of the forces acting on the interface layer (inertia effects are ignored). In the following, we show that there exist a typical length scale Λ which characterizes the stress accumulation and thus the precursor length.

Equations to be solved

In the discrete model we have x = iλ c , u t (x) → u t,i , u(x) → u i and u b (x) → u bi . In equilibrium, the displacements should satisfy

K L (u t,i+1 + u t,i-1 -2u t,i ) = K T (u t,i -u i ) , K (u i+1 + u i-1 -2u i ) = k (u i -u bi ) + K T (u i -u t,i ) .
In the continuum approximation, these equations reduce to the set of equations:

u ′′ t (x) = κ 2 T [u t (x) -u(x)] , (6)
u ′′ (x) = κ 2 [u(x) -w(x)] , (7)
w(x) = β u t (x) + (1 -β) u b (x) , (8)
where

κ T = (K T /K L ) 1/2 /λ c , κ = [(k + K T )/K] 1/2 /λ c and β = K T /(K T + k).
A key trick of our analytical approach is that the general solution of the equation

u ′′ (x) = κ 2 [u(x) - w(x)] is u(x) = D e ±κx -κ x dξ w(ξ) sinh[κ(x -ξ)] .
As we impose the position of the left-hand side (trailing edge) of the US we have

u t (0) = U t , (9)
while the left-hand side of the IL is free so that

Kλ c u ′ (0) + K T [U t -u(0)] = k [u(0) -u b (0)] . (10)
To simplify considerations, we ignore the right-hand side boundary conditions assuming that the chain is infinite,

L → ∞, so that u t (∞) = u(∞) = u b (∞) = 0.
Let initially, at t = 0, the system be completely relaxed:

u t (x) = u(x) = 0 and u b (x) = u (0) b (x) = 0.
We then start to push slowly the trailing edge of the US, so that U t = v d t with v d → 0. The solution of Eqs. ( 6)-( 10) is:

u t (x) = D 10 e -κ1x + D 20 e -κ2x , (11)
u(x) = α 1 D 10 e -κ1x + α 2 D 20 e -κ2x , (12)
where κ 2

1 2 = 1 2 (κ 2 +κ 2 T ± √ D), α1 2 = 1 2 [1-(κ 2 ± √ D/κ 2 T )], D = (κ 2 -κ 2 T ) 2 + 4βκ 2 κ 2 T , D 10 = (α 2 U t -u c )/(α 2 - α 1 ), D 20 = (u c -α 1 U t )/(α 2 -α 1 ) and u c ≡ u(0) = U t [λ c βκ 2 (κ 1 -κ 2 )+(K T /K) √ D]/[λ c κ 2 T (α 2 κ 2 -α 1 κ 1 )+ λ 2 c κ 2 √ D].
The positions U t and u c grow together until the trailing edge of the US reaches a position U t0 where the left end of the IL achieves the threshold value u s at some time t 1 = U t0 /v d . At t = t 1 the left-most λcontact breaks and attaches again with zero stretching,

u b (0) = u (1)
b (0) = u s , and the whole system relaxes, adjusting itself to a new configuration with u t (0) = U t0 , u b (0) = u s and u b (x ≥ λ c ) = 0. After relaxation, all contacts have shifted to the right, so that the IL stress σ c (x) for x ≥ λ c increases. As a consequence the stress on the second contact has grown above the threshold σ s so that it must break too. This domino-like process will continue until the stress at some (s 0 -th) contact will remain below the threshold. This first passage distance defines a characteristic length Λ = s 0 λ c which controls the kinematics of the onset of sliding.

We emphasize that the driving stress σ d (x), Eq. ( 5), changes self-consistently during front propagation, as u(x) changes. The moving self-healing crack leaves behind itself relaxed contacts. When the front reaches a position s, the current shapes of the displacement fields u t (x; s) and u(x; s) are given by solution of Eqs. ( 6)- [START_REF] Romero | Probing Locally the Onset of Slippage at a Model Multicontact Interface[END_REF]. The current displacement of the bottom surface of the IL, u b (x; s), is determined by the current propagation length of the front:

u b (x; s) = ũ(x + 0; x) for x < s, (13)
while ahead the moving front the contacts are still loaded,

u b (x; s) = u (0) b (x) for x > s. (14)
With these conditions, the functions u t (x; s) and u(x; s) take the following form. For the front tail (x < s):

u t (x; s) = [D 11 (s) sinh(κ 1 x) + D 12 (s) cosh(κ 1 x)] + [D 21 (s) sinh(κ 2 x) + D 22 (s) cosh(κ 2 x)] + ūt (0; x) , (15)
u(x; s) = α 1 [D 11 (s) sinh(κ 1 x) + D 12 (s) cosh(κ 1 x)] + α 2 [D 21 (s) sinh(κ 2 x) + D 22 (s) cosh(κ 2 x)] + ū(0; x) , (16)
where

ūt (x 0 ; x) = x x0 dξ u b (ξ) {b 1 sinh[κ 1 (x -ξ)] + b 2 sinh[κ 2 (x -ξ)]} , (17) ū
(x 0 ; x) = x x0 dξ u b (ξ) {α 1 b 1 sinh[κ 1 (x -ξ)] + α 2 b 2 sinh[κ 2 (x -ξ)]} , (18)
with

b 1 = (1 -β)κ 2 /κ 1 (α 2 -α 1 ) and b 2 = -b 1 κ 1 /κ 2 .
Ahead of the front (x > s):

u t (x; s) = D 1 (s) e -κ1(x-s) + D 2 (s) e -κ2(x-s) + ūt (s; x) , (19)
u(x; s)) = α 1 D 1 (s) e -κ1(x-s) + α 2 D 2 (s) e -κ2(x-s) + ū(s; x) . (20)
The coefficients D 1 (s), D 2 (s), D 11 (s), D 12 (s), D 21 (s) and D 22 (s) in Eqs. ( 15)-( 20) are functionals of the function u (0) b (x) and should be determined by the two lefthand-side boundary conditions ( 9) and ( 10) and the following four continuity conditions at x = s ± 0:

u t (s -0; s) = u t (s + 0; s) , (21)
u ′ t (s -0; s) = u ′ t (s + 0; s) , (22)
u(s -0; s) = u(s + 0; s) ,

u ′ (s -0; s) = u ′ (s + 0; s) , (23)
which finally lead to a set of integral equations that has to be solved self-consistently.

Numerical solution

A typical solution of the equations for the first front passage, for a completely relaxed IL initially (u

b (x) = 0), is shown in Fig. 2. We have assumed that front propagation is fast compared with the slow pushing, so that the left-most block of the US has no opportunity to move before the precursor arrests.

In these conditions, and when the slider's height is much larger than λ c , the displacement field in the US is approximately unchanged upon propagation of the precursor, even if the displacement in the IL almost double upon front propagation (Fig. 2(a)). Because the λ-contacts repin immediately after breaking, they also immediately start to load again as the next λ-contacts relax and the precursor propagates. As a consequence, the stress behind the front is increasing with the distance to the front (Fig. 2(b)). Ahead of the front, the relaxed λ-contacts induce an extra loading of the unbroken contacts ahead of the front, characterized by a stress peak at the front location (Fig. 2(b)). This extra stress ∆σ c is enough to bring initially less stressed contacts up to their threshold. However, because the initial stress is a decaying function of x, the extra stress is sufficient to bring the contacts above their threshold only over a finite length Λ (Fig. 2(c)).

Analytical solution

For the first precursor arising from a completely relaxed IL (u (0) b (x) = 0), we found some analytical results. The full proof is available in Appendix. Here, we will only provide the main results and the assumptions made to get them.

We assume that the elastic correlation length is much smaller than the slider's thickness (λ c ≪ H). Since λ c is typically in the micrometer range, this assumption is valid for a large range of macroscopic systems. We also assume that the interfacial stiffness is much smaller than the internal stiffness of the slider (k ≪ K). For rough interfaces, this assumption is also generally true [START_REF] Prevost | Probing the micromechanics of a multi-contact interface at the onset of frictional sliding[END_REF]. Based on the results of Fig. 2(a), we assume that the displacement field in the US is given by u t (x) ≈ U t0 e -κ2x and does not change during front propagation. Under these assumptions, we find an analytical expression for the length Λ of the first precursor (Eq. (66) in the Appendix). This expression approxi- mates, for Λ ≫ κ -1 (i.e. large H/λ c and/or k/K), to

Λ ≈ κ -1 2 ln {2/[(1 + β -2κ 2 /κ) Ψ 1 ]} , ( 25
)
where

Ψ 1 = 1 + λ c κ 2 /(1 + λ c κ).
The analytical results are shown in Fig. 3. Λ is found to increase quasi-linearly with the height of the slider (Fig. 3(a)) and to increase quasi-logarithmically with the stiffness of the interface (Fig. 3(b)). Comparison with the numerical results shows a quantitative agreement. We also checked that molecular dynamics simulations of the model agree with the analytical results (not shown).

Discussion

As for any toy model, we cannot expect our model to reproduce quantitatively experimental results. In this discussion, we will thus mainly list the consequences of the various simplifying assumptions that we made in order to allow for an analytical solution.

We have shown that in our model, the onset of sliding is characterized by precursory self-healing slip pulses. The characteristic length Λ of the first precursor is controlled by two parameters, H/λ c and k/K, determined by the slider and interface properties. Importantly, Λ does not depend on the threshold value σ s . This result is due to the fact that we use linear elasticity and immediate repinning of λ-contacts in a completely relaxed state. As a consequence, all curves Fig. 2(c) are unmodified if σ s is changed, and thus the length Λ is also unchanged.

This threshold-independence of Λ would mean that, if the interface would obey Amontons' law of friction (σ s = µ s σ n , with µ s the static friction coefficient and σ n the normal stress) locally (at the λ-contacts scale), Λ would not depend on µ s . This is a major difference with previous precursor models [START_REF] Scheibert | Role of friction-induced torque in stick-slip motion[END_REF][START_REF] Trømborg | Transition from Static to Kinetic Friction: Insights from a 2D Model[END_REF][START_REF] Amundsen | 1D Model of Precursors to Frictional Stick-Slip Motion Allowing for Robust Comparison with Experiments[END_REF], in which the length of the first (crack-like) precursor depends explicitly on µ s . The independence of Λ with µ s implies an independence with the normal load on the system, as long as the type of loading (pure side loading) is unchanged.

If an additional uniform shear stress σ 0 is applied to the top surface of the slider, then the threshold will decrease, σ s → σ s -σ 0 . In such conditions, the crack will nucleate at a lower macroscopic force and will propagate over a longer distance. Moreover, if σ 0 > σ f , where σ f is the Griffith threshold (see e.g. Eq. (35) in Ref. [START_REF] Braun | Collective Effects at Frictional Interfaces[END_REF] obtained for a simpler 1D model), then the crack will never stop, so that Λ → ∞.

We have considered a constant value of σ s along the interface. Let us consider a non-uniform normal load of the form σ n (x) = σ n0 +ǫx, as in top-driven systems with friction-induced torque [START_REF] Scheibert | Role of friction-induced torque in stick-slip motion[END_REF]. The thresholds thus also depend on x, σ s (x) = σ s0 + µ s ǫx. Λ will depend on the asymmetry parameter ǫ roughly as Λ(ǫ) ≈ Λ(0)(1 -ǫ).

We have assumed that broken contacts are immediately restored with zero stretching. Instead, one may assume that the contacts repin after some delay time τ d [START_REF] Braun | Dynamics of Transition from Static to Kinetic Friction[END_REF][START_REF] Braun | Collective Effects at Frictional Interfaces[END_REF][START_REF] Braun | Crack in the frictional interface as a solitary wave[END_REF][START_REF] Thøgersen | History-dependent friction and slow slip from time-dependent microscopic junction laws studied in a statistical framework[END_REF][START_REF] Trømborg | Slow slip and the transition from fast to slow fronts in the rupture of frictional interfaces[END_REF]. Assuming a constant τ d , the width of the self-healing crack will increase from λ to vτ d , with v the (minimal) propagation velocity of the front (see Refs. [START_REF] Braun | Collective Effects at Frictional Interfaces[END_REF][START_REF] Braun | Crack in the frictional interface as a solitary wave[END_REF]), and the length Λ will increase to Λ + vτ d . Note that experimentally, the repinning conditions is generally unraveled. In gels however, the interface was reported to re-stick when the slip velocity behind the rupture tip decreases back to a critical velocity depending of the gel composition [START_REF] Baumberger | Self-healing slip pulses and the friction of gelatin gels[END_REF][START_REF] Ronsin | Nucleation and Propagation of Quasi-Static Interfacial Slip Pulses[END_REF].

In the presence of inertia, which has been neglected in the present quasi-static model, the IL blocks will be able to continue increasing the load on their righthand neighbours even after repinning of their individual λ-contacts. This extra loading is expected to help breaking further contacts and thus increase the precursor length. In contrast, the front nucleation process will be unaffected because it originates from a static state of the system.

We have assumed that all λ-contacts behave as classical springs with a sawtooth-shape dependence u(σ c ). In real systems, they are composed of N c micro-junctions with a distribution of thresholds P c (f s ). If P c (f s ) is wide enough, the elastic instability will disappear, and the λ-contacts will smoothly slide with the pushing velocity v d after reaching the stretching u s [START_REF] Braun | Modeling friction on a mesoscale: Master equation for the earthquakelike model[END_REF][START_REF] Braun | Master equation approach to friction at the mesoscale[END_REF]. No discrete precursor will thus be observed, as a quasi-static front will continuously run through the interface.

Here we have considered the first precursor only. After its propagation, the system is left in the stress state shown in solid line in Fig. 2(b). If the system is further loaded, the left-most λ-contact will again be the first to reach its breaking threshold, and a second precursor will nucleate. It will propagate through the stress field left by the previous event, itself modified by the additional loading and this scenario will repeat itself. We have run numerical simulations for the series of precursors and found that all precursors propagate roughly over the same length Λ (between precursors, the stress peak left at the precursor extremity does propagate further quasi-statically due to the increased external loading). This is not in agreement with experimental data [START_REF] Rubinstein | Dynamics of precursors to frictional sliding[END_REF][START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF] and previous precursor models producing crack-like (rather than self-healing-like) rupture [START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF][START_REF] Scheibert | Role of friction-induced torque in stick-slip motion[END_REF][START_REF] Trømborg | Transition from Static to Kinetic Friction: Insights from a 2D Model[END_REF][START_REF] Amundsen | 1D Model of Precursors to Frictional Stick-Slip Motion Allowing for Robust Comparison with Experiments[END_REF], in which successive precursors propagate over increasing lengths.

This discrepancy disappears if aging of the λ-contacts -a newborn contact has a smaller size and thus a lower breaking threshold force f s -is taken into account [START_REF] Braun | Master equation approach to friction at the mesoscale[END_REF][START_REF] Braun | Role of aging in a minimal model of earthquakes[END_REF]. We checked this numerically by considering a timeincreasing f s . The previously broken region of the interface is characterized by "younger" contacts and therefore by smaller thresholds than the "old" unbroken part of the system. Hence, the next front will pass this region more easily and propagate deeper into the system. This process repeats itself until a precursor would span the whole interface and trigger macroscopic sliding of the interface. Note that in these numerical results, there is a competition between the aging and loading time scales, while the front propagation time scale is kept much smaller than the loading time scale.

The proposed toy model is important for all systems in which friction instabilities are central, from tribology to geophysics. It helps clarifying the problem of the selection of the size of the part of the interface which will slip as a function of the bulk material properties, interfacial parameters and loading conditions. It will be particularly relevant to systems in which self-healing slip-pulses have been observed (e.g. [START_REF] Lykotrafitis | Selfhealing pulse-like shear ruptures in the laboratory[END_REF][START_REF] Ronsin | Nucleation and Propagation of Quasi-Static Interfacial Slip Pulses[END_REF][START_REF] Romero | Probing Locally the Onset of Slippage at a Model Multicontact Interface[END_REF]). Our results emphasize the fact that the behaviour of precursors is intimately related to the repinning rule of the interface (at vanishing velocity for crack-like rupture; immediately in the present model). We thus advocate for an increased experimental effort to better constrain these repinning rules, which will be very useful to propose improved models for the onset of frictional sliding.

Appendix: Analytical derivation of Eq. ( 25) in the main text Let us introduce two dimensionless parameters h ≡ λ c /H and q ≡ k/K , (26)

so that K L = K/h, K T = Kh/2(1 + σ), β = 1/(1 + b), 1 -β = b/(1 + b), (λ c κ T ) 2 = hq/b, (λ c κ) 2 = q (1 + b)/b, ε ≡ κ 2 T κ 2 = h 1 + b , (27)
where b = 2(1 + σ)q/h, and consider the typical system with h, q ≪ 1. In this case ε ≪ 1, so that

κ 2 1 ≈ κ 2 (1 + βε) and κ 2 2 ≈ κ 2 ε (1 -β) = qε/λ 2 c , or (λ c κ 2 ) -1 ≈ [2(1 + σ) + h/q] 1/2 /h . (28)
In accordance with the numerics (see Fig. 2(a)), let us assume that in the case of h, q ≪ 1 the displacement field in the US is given by

u t (x) ≈ U t0 e -κ2x (29)
and does not change during front propagation. Before nucleation of the first precursor, the solution of Eq. [START_REF] Maegawa | Precursors of Global Slip in a Longitudinal Line Contact Under Non-Uniform Normal Loading[END_REF] in the main text is

u(x) = A 30 sinh(κx) + A 40 cosh(κx) -κβU x 0 dξ e -κ2ξ sinh[κ(x -ξ)] = 1 2 A 40 + A 30 - 1 2 βU κ κ + κ 2 e κx + 1 2 A 40 -A 30 -βU κ κ -κ 2 e -κx + βU κ 2 κ 2 -κ 2 2 e -κ2x . (30)
The right-hand-side boundary condition, u(x) → 0 at x → ∞, gives us

A 40 + A 30 = 1 2 βU κ κ + κ 2 , (31)
while the left-hand-side boundary condition (Eq. ( 10) in the main text) leads to the equation

(A 40 -A 30 )(1 + λ c κ)(κ + κ 2 ) = βU κ(1 + aκ + 2λ c κ 2 ) . (32)
Thus, before nucleation of the first precursor, the IL displacement field is

u(x) = βU κ 2 (κ 2 -κ 2 2 ) e -κ2x - κ 2 κ (1 + λ c κ 2 ) (1 + λ c κ) e -κx . (33)
Equation (33) allows us to couple the parameters U ≡ u t (0) and u c ≡ u(0):

U = u c 1 + κ 2 κ /(βΨ 1 ) , (34)
where

Ψ 1 = 1 + κ 2 κ λ c κ (1 + λ c κ) . (35)
When the displacement of the IL trailing edge reaches the threshold value u s at some U = U 0 = u s (1 + κ 2 /κ)/(βΨ 1 ), the front starts to propagate. In this case the solution of Eq. ( 7) in the main text, ahead of the propagating front, x > s, where u b (x) = 0 so that w(x) = βu t (x) = βU 0 e -κ2x , is given by

u(x; s) = A 3 (s) e -κ(x-s) + A 4 (s) e κ(x-s) -κβU 0 x s dξ e -κ2ξ sinh[κ(x -ξ)] = βU 0 κ 2 e -κ2x (κ 2 -κ 2 2 ) + A 3 (s) e -κ(x-s) + A 4 (s) e κ(x-s) - 1 2 βU 0 κ e -κ2s e κ(x-s) (κ + κ 2 ) + e -κ(x-s) (κ -κ 2 ) . (36)
The right-hand-side boundary condition gives us the coefficient A 4 (s),

A 4 (s) = 1 2 βU 0 κ e -κ2s (κ + κ 2 ) , (37)
so that Eq. (36) takes the form

u(x; s) = βU 0 κ 2 (κ 2 -κ 2 2 ) e -κ2x + A 3 (s) - 1 2 βU 0 κ e -κ2s (κ -κ 2 ) e -κ(x-s) . (38)
Behind the propagating front, x < s, where w

(x) = β u t (x)+(1-β) u b (x) and u b (x) = u(x+0; x) = A 3 (x)+ A 4 (x)
, the solution of Eq. ( 7) in the main text is given by

u(x; s) = A 1 (s) sinh(κx) + A 2 (s) cosh(κx) -κ (1 -β) x 0 dξ [A 3 (ξ) + A 4 (ξ)] sinh[κ(x -ξ)] -κβU 0 x 0 dξ e -κ2ξ sinh[κ(x -ξ)] = βU 0 F (x) + A 1 (s) sinh(κx) + A 2 (s) cosh(κx) -κ (1 -β) x 0 dξ A 3 (ξ) sinh[κ(x -ξ)] , (39)
where

F (x) = Ψ 2 κ 2 (κ 2 -κ 2 2 ) × κ 2 κ sinh(κx) -cosh(κx) + e -κ2x , (40)
F ′ (x) κ = Ψ 2 κ 2 (κ 2 -κ 2 2 ) × κ 2 κ cosh(κx) -sinh(κx) - κ 2 κ e -κ2x , (41)
Ψ 2 = (3 -β)κ + 2κ 2 2(κ + κ 2 ) , (42)
so that F (0) = 0 and F ′ (0) = 0. The coefficients A ... (s) in these equations are determined by the boundary and continuity conditions. The left-hand-side boundary condition (Eq. ( 10) in the main text) couples the coefficients A 1 (s) and A 2 (s). Using u b (0) = u s , u(0; s) = A 2 (s) and u ′ (0; s) = κA 1 (s), we obtain

A 2 (s) -(λ c κ) -1 A 1 (s) = Ψ 3 , Ψ 3 = βU 0 + (1 -β) u s . (43)
The continuity conditions (Eqs. ( 23) and ( 24

From Eq. ( 56) we obtain that at short distances, s ≪ κ -1 , A 3 (s) ≈ A 3 (0)(1 + γ 3 s), where

γ 3 = - κ 1 + λ c κ β + λ c κ - βU 0 A 3 (0) Ψ 2 Ψ ′ 6 (0) κ . ( 57
)
From Eqs. (37) and (56) it follows that u(s + 0; s) = A 3 (s)+A 4 (s) ≈ A 0 (1+γs) at short distances, s ≪ κ -1 , where

A 0 = A 3 (0) + A 4 (0) (58)
and

γ = [γ 3 A 3 (0) -κ 2 A 4 (0)]/A 0 , (59)
while for long distances, s ≫ κ -1 , u(s + 0; s) decays exponentially, u(s + 0; s) ≈ A e -κ2s ,

A = βU 0 κ 2(κ + κ 2 ) + Ψ 2 (1 + β -2κ 2 /κ) . ( 60
)
The function u(s + 0; s) may be approximated as u(s + 0; s) ≈ A 0 (1 + C) α e κ3s (e κ3s + C) α ,

where

α = 1 + κ 2 /κ 3 , (62)
C = (κ 2 + γ)/(κ 3 -γ) , (63)
and comparing Eqs. ( 59) and (61), we obtain a nonlinear equation, which defines the value κ 3 :

ln A A 0 = 1 + κ 2 κ 3 ln κ 2 + κ 3 κ 3 -γ . ( 64
)
Then, the IL stress ahead of the front is σ c (s) = k u(s + 0; s)/λ 2 c , and the equation σ c (Λ) = σ s defines the characteristic length Λ:

Λ ≈ κ -1 3 ln y , ( 65
)
where y is determined by the solution of the equation By = (y + C) α with B = (1 + C) α kA 0 /(σ s λ 2 c ).

Fig. 2 (

 2 Fig.2(a) Displacement fields ut(x; s) and u(x; s): when the front nucleates at t = t 1 (s = 0, dotted, ut(x, 0) ≈ U t0 e -κ2x ) and during front propagation for the distances s = 13 a (dashed) and s = 26 a (full precursor propagation length, solid curve). (b) Stress field in the IL during front propagation. (c) Dashed line: shear stress in the IL at front nucleation (t = t 1 -0). Solid line: shear stress in the IL, at the front tip, when it passes at location x (s = x) during its propagation. Dotted line: extra stress ∆σc(s). H/λc = 25, k/K = 0.03, ν = 0.3, L = 100λc.

Fig. 3 (

 3 Fig.3(color online) Dependence of the characteristic length Λ on model parameters. Symbols are for numerics, solid lines for analytics, dotted lines for approximate analytics (Eq. 66). (a) Λ versus H/λc for different values of the interface stiffness k/K = 0.03 (diamonds, red), 0.01 (squares, blue) and 0.003 (circles, black). (b) Λ versus k/K for fixed H/λc=10 (bottom, red), 25 (middle, blue, dots), 50 (top, black). ν = 0.3, L/λc = 100.

0 dξ A 3 0 dξ A 3 dξ A 3 0 dξ A 3 - κ (λ c κ 2 + κ 2 ( 1 +

 0303303221 ) in the main text) lead to two equationsκ (1 -β) s 0 dξ A 3 (ξ) sinh[κ(s -ξ)] + A 3 (s) = A 1 (s) sinh(κs) + A 2 (s) cosh(κs) + βU 0 Ψ 4 (s) (44) and κ (1 -β) s (ξ) cosh[κ(s -ξ)] -A 3 (s) = A 1 (s) cosh(κs) + A 2 (s) sinh(κs) + βU 0 Ψ 5 (s) , (45)whereΨ 4 (s) = F (s) -κ e -κ2s 2(κ + κ 2 ), and sum of Eqs. (44) and (45), we obtain two new equations:2A 3 (s) e κs -κ (1 -β) s 0 dξ A 3 (ξ) e κξ = A 2 (s) -A 1 (s) + βU 0 [Ψ 4 (s) -Ψ 5 (s)] e κs ,(48)κ (1 -β) s 0 dξ A 3 (ξ) e -κξ = A 2 (s) + A 1 (s) + βU 0 [Ψ 4 (s) + Ψ 5 (s)] e -κs .(49)Using Eq. (43), Eqs. (46) and (47) may be rewritten as2A 3 (s) e κs -κ (1 -β) s (ξ) e κξ = A 1 (s) (1 -λ c κ) λ c κ + Ψ 3 + βU 0 [Ψ 4 (s) -Ψ 5 (s)] e κs , (ξ) e -κξ = A 1 (s) (1 + λ c κ) λ c κ + Ψ 3 + βU 0 [Ψ 4 (s) + Ψ 5 (s)] e -κs .(51)Combining these equations, we finally come to the integral equation for the coefficient A 3 (s):A 3 (s)(1 + λ c κ) e κs -κ(1 -β) s (ξ) [cosh(κξ) + (λ c κ) sinh(κξ)] = λ c κΨ 3 + βU 0 Ψ 2 Ψ 6 (s) ,(52)whereΨ 6 (s) = κ (1 + λ c κ) 2(κ -κ 2 ) e (κ-κ2)s λ c κ) 2 e κs + (λ c κ 2 + κ 2 ) (κ -κ 2 ) e -κs κ e -κ2s .(54)From Eq. (52) we find thatA 3 (0) = [λ c κΨ 3 + βU 0 Ψ 2 Ψ 6 (0)]/(1 + λ c κ) .(55)Differentiating Eq. (52), we obtain a differential equation for A 3 (s):1 κ A ′ 3 (s) + A 3 (s) = A 3 (s) (1 -β) (1 + λ c κ)×[cosh(κs) + (λ c κ) sinh(κs)] e -κs + βU 0 Ψ 2 κ (1 + λ c κ) Ψ ′ 6 (s) e -κs .

Acknowledgements We wish to express our gratitude to M. Peyrard for numerous useful discussions. We thank J.K. Trømborg for a careful reading of the manuscript. This work was supported by the EGIDE/Dnipro grant No. 28225UH and by the People Programme (Marie Curie Actions) of the European Unions 7th Framework Programme (FP7/2007-2013) under Research Executive Agency Grant Agreement 303871. O.B . acknowledges a partial support from the NASU "RESURS" program.

Using Eq. (60), Λ may approximately be presented as

Equation ( 65) corresponds to the analytical solution for Λ, whereas Eq. (66) corresponds to the approximated analytical solution provided as Eq. (25) in the main text.


Bogdan Florin Cornea

Anne-Cécile Orgerie

Laurent Lefèvre
email: laurent.lefevre@inria.fr

Studying the energy consumption of data transfers in Clouds: the Ecofen approach

Keywords: energy consumption, Cloud data transfers, ethernet networks, simulation

Energy consumption is one of the main limiting factors for designing large scale Clouds. Evaluating the energy consumption of Clouds networking architectures and providing multi-level views required by providers and users, is a challenging issue. In this paper, we show how to evaluate and understand network choices (protocols, topologies) in terms of contributions to the energy consumption of the global Cloud infrastructures. By applying the ECOFEN model (Energy Consumption mOdel For End-to-end Networks) and the corresponding simulation framework, we profile and analyze the energy consumption of data transfers in Clouds.

I. INTRODUCTION

Cloud computing is increasingly becoming an essential component for Internet service provision, yet at the same time its energy consumption has become a key environmental and economic concern. While Cloud computing is relying on numerous servers and storage facilities, it also intensively uses telecommunication networks. The ever-growing appetite of new applications for network resources leads to an unprecedented electricity bill for these resources [START_REF] Bolla | Energy Efficiency in the Future Internet: A Survey of Existing Approaches and Trends in Energy-Aware Fixed Network Infrastructures[END_REF]. Moreover, for these bandwidth-hungry applications, networks can become a significant bottleneck [START_REF] Chowdhury | Managing data transfers in computer clusters with orchestra[END_REF].

In this context, Cloud computing actors need tools in order to design, to evaluate, and to improve their networking architectures. On one side, Cloud providers want to know the overall energy consumption of their networking infrastructures over time, depending on the incoming traffic. On the other end, Cloud users, like Platform-as-a-Service or Software-as-a-Service providers for instance, are interested in having a more detailed view of the energy consumption on a given equipment, or for a traffic pattern coming from a known application.

Evaluating the energy consumption of Cloud's networking architectures and providing these multi-level views required by providers and users, is a challenging issue. In particular, having precise evaluations to be able to compare different topologies or different network protocols requires practically huge monitoring infrastructures and carefully controlled environments. To overcome these technical issues, one can rely on simulation tools. While networking research community has an extended experience on network simulators, only few of them consider the energy dimension and the particular use-case of Cloud computing infrastructures. Moreover, the simulators including this energy dimension only consider it for wireless equipment, which is currently not suitable in the context of Cloud data centers.

In this paper, we propose to study the energy consumption of data transfers in Clouds from a simulation point of view. To achieve this goal, we use ECOFEN, an Energy Consumption mOdel For End-to-end Networks proposed in our previous paper [START_REF] Orgerie | Ecofen: An end-to-end energy cost model and simulator for evaluating power consumption in large-scale networks[END_REF]. This model has been implemented in NS-3 [START_REF]The NS-3 Network Simulator[END_REF], a discrete-event simulator for Internet systems, targeted primarily for research and educational use. This free software is widely used in the community, like its predecessor NS-2 [START_REF] Mccanne | NS Network Simulator[END_REF] which was used in our previous work [START_REF] Orgerie | Ecofen: An end-to-end energy cost model and simulator for evaluating power consumption in large-scale networks[END_REF].

The paper is organized as follows: Section II presents the related works. Section III introduces the ECOFEN model and simulator. Section IV provides preliminary results in order to evaluate the energy consumption of Cloud networks, and Section V shows results exploiting green levers in order to reduce the energy consumption of networking infrastructures. Section VI concludes this work and presents future work.

II. RELATED WORK

Energy efficiency is currently at the focus of attention of many research works in Cloud computing. Yet, telecommunication networks, which are an integral part of theses Cloud infrastructures, are often not considered because of their complexity, their heterogeneity and their large-scale properties. However, it has been shown that networking infrastructures play a non-negligible role in the electricity's consumption of data centers [START_REF] Bolla | Energy Efficiency in the Future Internet: A Survey of Existing Approaches and Trends in Energy-Aware Fixed Network Infrastructures[END_REF]. While the number of energy-aware components of large-scale architectures is increasing, the methods for evaluating the energy consumption are lagging behind. In many cases, direct measurements of the consumed energy are not possible due to lack of monitoring equipment, large-scale issues and the involvement of different providers.

In this context, simulation appears as an appealing solution allowing large-scale and fine-grained experiments. However, the community is lacking of such energy-aware simulators for wired networks. ECOFEN, an Energy Consumption mOdel For End-to-end Networks proposed in 2011 [START_REF] Orgerie | Ecofen: An end-to-end energy cost model and simulator for evaluating power consumption in large-scale networks[END_REF], has been implemented within the well-known network simulator NS-3 [START_REF]The NS-3 Network Simulator[END_REF].

The ECOFEN simulator provides energy-aware simulations based on packet-based and per-Byte energy consumption models based on experimental results from the literature [START_REF] Sivaraman | Profiling perpacket and per-byte energy consumption in the netfpga gigabit router[END_REF], [START_REF] Mahadevan | A power benchmarking framework for network devices[END_REF], [START_REF] Vishwanath | Estimating the energy consumption for packet processing, storage and switching in optical-ip routers[END_REF], [START_REF] Hähnel | ebond: energy saving in heterogeneous r.a.i.n[END_REF], [START_REF] Gupta | Dynamic ethernet link shutdown for energy conservation on ethernet links[END_REF].

Contrary to other work like the GreenCloud simulator based on NS-2 [START_REF] Kliazovich | GreenCloud: a packet-level simulator of energy-aware cloud computing data centers[END_REF], this simulator aims also at providing mod-els for energy-efficient network devices, i.e. able to dynamically adapt their link rate depending on the actual traffic [START_REF] Gunaratne | Ethernet Adaptive Link Rate (ALR): Analysis Of A Buffer Threshold Policy[END_REF], to switch on and off ports and entire devices [START_REF] Chiaraviglio | GreenCoop: Cooperative Green Routing with Energy-efficient Servers[END_REF], and to put them to sleep modes like in the 802.3az standard [START_REF] Christensen | IEEE 802.3az: the road to Energy Efficient Ethernet[END_REF].

III. ECOFEN, AN ENERGY-MODULE FOR THE NS-3

SIMULATOR

Proposed by Orgerie et al. [START_REF] Orgerie | Ecofen: An end-to-end energy cost model and simulator for evaluating power consumption in large-scale networks[END_REF], ECOFEN allows studying through simulation the energy consumption of wired network equipment. Initially, ECOFEN was developed as a module for NS-2 [START_REF] Mccanne | NS Network Simulator[END_REF], and was later on re-written for NS-3 [START_REF]The NS-3 Network Simulator[END_REF], which is mainly intended for scientists and academia. NS-3 is available by default in certain GNU/Linux distributions such as Fedora, thus allowing a simplified access to its functions, and to the energy module implicitly.

A. Characteristics of ECOFEN

The ECOFEN module brings to the NS-3 users community the possibility to analyze the energy profile of the network over time. Users can observe the global consumption of networks in a Cloud or can isolate and analyze a single network equipment.

With a precision of the order of milliwatts for the power and milliseconds for the time, ECOFEN provides several energy models and profiles for network interface cards (NIC), switches, routers, and interfaces (ports): basic, linear, and complete, all inspired by experiments from the literature. For calculating the NIC consumption, the basic energy model is used, this allowing users to set the power (in watts) in On and in Off states. For calculating the interface consumption, any of the three models can be used. The linear model allows replacing the On setting with two states -Idle and ByteEnergy -for calculating the energy per processed byte. For even more precision, the complete model allows replacing the ByteEnergy with four states indicating the energy consumed (i) per received byte, (ii) per sent byte, (iii) per received packet, and (iv) per sent packet.

The basic model provides coarser-grain energy modeling, compared to the complete one, whereas the linear model is in between. A different energy model may be used for each network equipment -NIC, switch, router, interface -the choice of model depending only on the expected granularity of the result. In addition, green functions (levers) can be applied to existing energy models. For instance, models are provided for calculating energy consumption during state transitions, i.e., from On (or Idle) to Off and vice versa or when switching between available link rates. These models are used when simulating power saving schemes, but they can also be used to simulate power failures.

B. Limitations

The energy consumption of complex infrastructures depends of several factors. Despite the complexity of the ECOFEN module, most limitations are due to the NS-3 implementation itself. For instance, simulation durations are driven by the packet-based nature of the simulator. Consequently, the time to simulate large scale systems with thousands of nodes, or millions of packets can be large. For some experiments (see Section IV-C), more than 5 hours of simulation are required to capture one minute of simulated network activity for a large topology (over 1000 nodes in 9 data-centers) as part of combined data transfers of 112GB. ECOFEN adds energy details on top of the existing network models. ECOFEN is thus dependent on the NS-3 models for network protocols. The availability of measured power consumptions for network devices is required for configuring the simulator, but some values can be found in the literature or on manufacturer's specifications.

IV. EVALUATING THE ENERGY CONSUMPTION OF CLOUD NETWORKS THROUGH SIMULATION

In Cloud systems, the demand for computing and networking resources varies in time. The computing resources are out of the scope of this paper and, therefore, we focus on profiling the energy consumption of network resources under various scenarios.

The experimental work described in this paper uses energy characteristics found in related work. These values are used to configure the NS-3 simulator. The magnitude of the energy used during data transfers, compared to idle operating mode, is not the focus of our experiments, but we present use cases and consumption analyses available through the ECOFEN/NS-3 energy module. Users are, however, given the freedom to input the power consumption for the devices of their choice, based on measurements taken or found in the literature.

A. Point-to-point VM migration

For a first scenario, the focus is on a type of data transfer which occurs frequently in Cloud systems: migrating Virtual Machines (VM). For this purpose, we have measured the size, duration and physical link conditions during a real migration, and compared the acquired information with results obtain by means of simulation (see Table I). A summary of consumption values used for configuring the simulator during this scenario can be found in Table II. Migrating VMs was done on two computing nodes from the Luxembourg site of the Grid'5000 platform [START_REF]Grid'5000[END_REF] (2 CPUs Intel@2.0GHz, 6 cores/CPU, 31GB RAM, 232GB disk). The measured size on disk for the VM is 1.5GB, with a virtual size of 10GB. The migration process relies on libvirt with qemu and kvm. Although libvirt imposes 32MB/s maximum bandwidth, we overcame this by configuring Debian backports, and upgrading libvirt to ver. 1.2.1. However, our experiments revealed that the command for setting the maximum migration bandwidth only proved to be reliable between 54 and 112Mbps. For this reason, we chose to take VM migration measurements, in turn, at 54, 80, 100, and 112 Mbps migration bandwidths. For configuring the simulator, interfaces were set to the smallest link rate that supports, in turn, each of these bandwidths. 2 Set or negotiated bandwidth. 3 A port on any of the following equipments: NIC, switch, router. 4 Extracted from [START_REF] Mahadevan | A power benchmarking framework for network devices[END_REF].

We are interested in the data that is transferred (migrated) between two computing nodes, hence we consider one switching device connecting both hosts (see Figure 1). Performing a VM migration provides information that helps configuring the simulator later: measurements of the migration duration and of the size of VM on disk, and traces of the network load during migration. We use this information to configure the ECOFEN module. The result can be seen in Figure 2, where the power profile at each second during the migration is showed. Since the power consumed during the data transfer appears in the upper half of Figure 2, a close-up is also showed in the lower half, were the 2mW variation becomes visible. With respect to the energy used by the two hosts employed in migrating the VM, ECOFEN can provide an insight on the impact of bandwidth on the overall network consumption. Figure 3 shows an analysis made with ECOFEN throughout the migration of a 1.5GB VM disk. First, ECOFEN allows a quick look into the behavior of network equipment and provides users with comparative results, globally or by network equipment. Second, for longer data transfers between two nodes in the Cloud, higher negotiated link rates mean lower energy consumption. Energy saving can be achieved at higher link rates once the transfer ends, as opposed to the slowest link rate that requires much more energy in time to finalize the migration. The experiment shows that only a small energy efficiency is observed for consecutive link rate, such as 100 and 1000Mbps, if the available bandwidth does not match the negotiated link rate, as in the case of 100 and of 112Mbps bandwidths. Time is listed in Table I.

We have shown how we can analyze the power and the energy in the case of a virtual machine migration, situation that often occurs in the Cloud. We move the focus on another aspect of the Cloud, i.e., the network activity at data-center level.

B. Data-center network

With respect to the network in data-centers, ECOFEN provides the following types of information: a global view, and a detailed view of a network equipment or of a group of network devices. The results obtained from experimenting with ECOFEN are discussed in this section.

Table III contains the relevant values used to configure the energy-aware network simulator. For NIC end hosts , and linecards on switches or routers, ECOFEN uses the basic energy model, which supports On/Idle and Off states. For the installed ports, on the other hand, ECOFEN has a complete model allowing a finer grained representation of states, as to account for the energy consumed per byte or per packet of transmitted data (inbound or outbound). Another relevant aspect in evaluating the energy consumption is the duration and the energy needed to switch a device (card or port) from one state to another. In addition to the values from Table III, we measured on Grid'5000 the intra-and interdata-center latency of 0.12ms and 13.8ms, respectively. The simulated energy consumption is read every 100ms, unless specified otherwise.

Assuming that a data-center in the Cloud can be described in a simplified way as in Figure 4, using NS-3 and the ECOFEN module, we can obtain two types of information, (i) an overview on the global network power consumption dynamics and (ii) a detailed view of one network equipment or of a group. This information is being expressed using three different metrics: (a) the power, in watts, (b) the energy, in joules, and (c) the power with respect to the communication volume, in watts and MB/s, respectively.

In the current scenario, a number of 200 point-to-point communications take place between end hosts A to E. The communicating pairs of end hosts and the size of each communication were randomly generated, but fixed during the [START_REF] Hähnel | ebond: energy saving in heterogeneous r.a.i.n[END_REF], for ports [START_REF] Hähnel | ebond: energy saving in heterogeneous r.a.i.n[END_REF], or for switches [START_REF] Mahadevan | A power benchmarking framework for network devices[END_REF], throughout the switching duration [START_REF] Gupta | Dynamic ethernet link shutdown for energy conservation on ethernet links[END_REF]. 3 References. * Equipment used: Gigabit PT EXPI9300PTLPBLK (for the NICend host ), unspecified edge LAN switch B at Idle Power (for the switch), unspecified edge LAN switch D at Idle Power (for the router), Gigabit PT EXPI9300PTLPBLK (for idle 1G port), 10 Gigabit X520-T2 (for idle 10G port), NetFPGA 1G rev. 2 (send/recv byte or packet energy at 1Gbps), unspecified Gigabit Ethernet (send/recv byte or packet energy at 10Gbps), Gigabit PT EXPI9300PTLPBLK (for switching On/Off NICend host ), unspecified edge LAN switch B (for switching On/Off a switch), Gigabit PT EXPI9300PTLPBLK (for switching On/Off ports on any device). Idle port values are the difference between consumption of interface when down (cable unplugged) and interface when up (no transfer). evaluation of different views and metrics. A reduced number of transfers is imposed for end host E, in order to observe the energy consumption when using energy saving techniques (green functions). Data size per communication is generated from the range 1024B to 10MB. The transport protocol in use is TCP and all network equipment is in On or Idle mode. For all interfaces installed on NIC end hosts and on switches, energy consumption also considers the number of transmitted bytes or packets.

1) Dynamic analysis of the overall network infrastructure: As indicated previously, with the ECOFEN energy module users can obtain a simulation of the overall network in a data-center. The bottom curve in Figure 5 shows the power consumption over the total communication time in the datacenter. At time t=0s and after time t=26s, the network has a constant consumption because it is in an Idle state. From The top curve of Figure 5 shows the communication volume over time, with the purpose of understanding the variations in power consumption. It can be seen that there is no network load during the first second, which corresponds to an Idle power consumption of 783.7W.

By the way ECOFEN was designed, it automatically supports energy consumption for any other transport protocol that NS-3 developers might consider adding to the simulator.

2) Dynamic analysis of end hosts NIC: A second type of information available through ECOFEN is a detailed view over the power and energy profile of any NIC end host . Figure 6 shows, for each end host, the data volume transferred per second (see upper figure), which shapes the power consumption profile (see the lower figure). This type of analysis result shows the Fig. 6. Detailed view of network traffic and power for NIC end hosts for the Cloud data-center considered for this scenario. Power is calculated every 100ms. Data volume and power for one NIC end host is drawn using the same color. contribution of each communicating NIC end host to the overall energy consumption of the network. It also allows identifying which end host A to E draws a higher power per second, and which communicates a longer time. Taking end host A as example, it has the highest number of simultaneous communications in the range of 5 to 6s, while end host E exchanges the most messages in the range 1.5 to 6s. On the other hand, end host D is part of the largest number of data transfers in this scenario, communicating during the 1 to 26s time interval.

C. Cloud Federation

For analyzing a large-scale Cloud system with ECOFEN, we consider the illustrative network topology of the Grid'5000 French platform (see Figure 7). This topology contains over one thousand computing nodes (multi-and many-core) geographically distributed on 9 sites, organized in 34 clusters. We With ECOFEN we can study and estimate the energy impact of one data-center switch, in case of a power failure or scheduled maintenance shutdown. Using values from Table III. for the data-center in Rennes, having one switch with 129 active 1 Gigabit and 1 active 10 Gigabit ports, we expect observing an energy saving of 145W if switching Off all ports, or 221.4W if switching Off the entire equipment. The next section details the use of green levers in ECOFEN.

V. GREEN LEVERS IN CLOUD NETWORKS

What is the impact of energy-efficient equipment in Cloud Networks? For this study, we enable green levers in the datacenter seen in Figure 4, then set TCP as transport protocol. Similarly to section IV, ECOFEN supplies the information about the consumption at network level or per equipment.

The green, or energy-aware, model of ECOFEN supports the On (Idle), and Off modes for every network equipment. In addition, the consumption of switching devices depends on (i) the number of transmitted bytes or packets, and (ii) the duration and the energy for switching between two states, e.g., from On to Off, or vice versa.

A. Analysis of energy-aware (green) devices -Global view Based on Figures 5 and6, we can apply energy-saving techniques on the NIC end host E, at any time after 6s. From this point, the end host network card is in Idle operating mode and switching it On or Off would not cause any data loss, as no communication is in progress. In Figure 9, ECOFEN provides an insight on the power profile for the overall network (see system power all ON). We study once more the power drawn by the network when the NIC end host E is switched Off for a duration of 5 seconds, starting at time t=10s (see end host E: card off ). Similarly, we analyze the power profile by modifying the number of active ports, hence the number of active links. Based on the simulation configuration for obtaining Figures 5 and6, we add a condition regarding one port on switch 4 that connects to the end host E, i.e., we test the shutting down of the port at time t=9s, for a duration of 5 seconds. This is allowed because during this time interval, there is no communication to/from end host E through this port of the switch (green color). Figure 9 shows the initial profile of the global network, as well as the power saving achievable by switching Off ports (1.12W) or NICs (1.82W). These three results are grouped together such that the power saving can be shown from a different perspective in Figure 10. In the lower figure is shown the network power when the two devices mentioned previously are switched Off as part of the green functions. These are later switched back On. One port on switch 4 is switched Off safely at time t=9s followed by a switching Off on the NIC end host E at time t=10s. This results in a total power saving of about 3W, until time t=14s when the switch port is turned back On, followed by the change of state On for the network card. For this scenario, each change in port state means a consumption or saving of 1.12W, whereas the power change per NIC end host is of about 1.82W.

B. Analysis of energy-aware (green) devices -at switch level

Based on the results shown in the previous figure, we isolate the energy consumption of switches and present a detailed view of their power profile. If one is interested in the energy behavior of switching equipment, a simple filter set on the simulator output produces the profiles shown in Figure 11. Two types of information are identified in this figure: power consumption groups together switches, and power savings are easy to observe. We observe the variation in power requirement for switches 0 through 4, as expected for the data-center network topology in Figure 4. There are three consumption levels in our data-center. The first groupswitches 0, 1, 4 -have a higher power consumption due to four active ports on each. Switch 3 has three active ports, while switch 2 has only two active ports. In this figure, ECOFEN provides additional information on the activity of switch 4, which turns Off one of its ports at 9 seconds to save energy. During this time, switch 4 will have three active ports, hence joining the group of switch 3. The port is then turned On at 14 seconds.

VI. CONCLUSION AND PERSPECTIVES

Energy consumption of networking devices must be taken into account in order to better compute energy consumption of Clouds infrastructures. This allows Clouds designers to propose realistic and precise models and algorithms. This paper presents the usage of the ECOFEN model (Energy Consumption mOdel For End-to-end Networks) and the corresponding simulation framework (designed as an NS-3 module). This approach permits global and complete (end to end) view of energy consumption of network devices. With ECOFEN, we can obtain two types of information, (i) an overview on the global network power consumption dynamics and (ii) a detailed view of one network equipment or of a group. ECOFEN allows the study of the impact of data transfers, in terms of static and dynamic energy consumption, in the Cloud. Moreover, with ECOFEN, network Cloud designers can evaluate the impact and manageability of Green leverages.

As future works, we are currently evaluating the precision of simulation results compared to measurements taken using wattmeters installed on the power supply of the corresponding physical network devices.

Fig. 1 .

 1 Fig. 1. Local transfer of a Virtual Machine between two computing nodes (from host A to host B) located on the same local area network.

Fig. 2 .

 2 Fig. 2. VM migration. Simulated power profile of the network during the migration. Link rate is set at 100Mbps. Energy consumption is calculated every 100ms.

Fig. 3 .

 3 Fig. 3. Energy profile when migrating a VM; labels indicate the energy. Time is listed in TableI.

Fig. 4 .

 4 Fig.4. Data-center configuration considered for this simulation scenario. The computing nodes are not considered, but only the network equipment of these end hosts, and the network infrastructure connecting them.

Fig. 5 .

 5 Fig. 5. Global view of the network traffic and power for this Cloud datacenter. Transport protocol in use: TCP. Energy consumption is calculated every 100ms; captured packet size (default NS-3): 578B time t=3s to time t=4s into the simulation, consumption reaches a peak value due to all end hosts engaged into communication.

Fig. 7 .

 7 Fig. 7. Grid'5000 network topology used for our Cloud platform network configuration. extend the previous experiment (see Section IV-B) to obtain the power profile for this network topology. The physical links connecting the data-centers have a transmission rate of 10Gbps. The link rates on each site are of 1 and 10Gbps. The communication protocol is TCP and 100 data transfers are simulated during this scenario. 80/100 transfers are of a random size of 10 to 20MB, and 20/100 transfers are of a random size of 1 to 10GB. A snapshot of one minute of the cloud network activity is shown in Figure 8. During this experiment, 100 transfers begin at random moments between 0 and 50s. Only some of the over 1000 computing nodes participate in data transfers and the figure shows the network equipment consumption, without taking into account cooling energy and computing nodes (except for their NICs).

  80 of 10-20MB each & 20 of 1-10GB each

Fig. 8 .

 8 Fig. 8. Global power profile of a Cloud Federation Network, based on Grid'5000 topology. Power is calculated every 100ms.

Fig. 9 .

 9 Fig. 9. Global power profile for the data-center network in Figure 4. Power is calculated every 100ms.

Fig. 10 .

 10 Fig.10. Global network profile for the considered data-center. One port then one card are switched Off, then back On. Power is calculated every 100ms.

Fig. 11 .

 11 Fig. 11. Power profile per switching device. One port on switch 4 is set to Off at 9s and to On at 14s. Power is calculated every 100ms.

TABLE I .

 I Real measurements taken on Petitprince cluster of Grid'5000 site in Luxembourg.2 Error between measured and simulated time. *Equipment used: NetFPGA 1G rev. 2 (for the NICend host), Nortel Ethernet Routing Switch 5510-24T (for the switch), unknown model of edge LAN switch B at 100Mbps and at 1Gbps port capacity (for idle port), 1 Gigabit Ethernet (for ByteEnergy).

	Disk	Bandwidth	Measured	Simulated time [s]	Error 2
	size	[M bps]	time 1	Ecofen module	Ecofen module	[%]
	[GB]		[s]	disabled	enabled
		54	221.7	249.8	249.8	12.6
	1.5	80 100	161 118	168.6 134.9	168.6 134.9	4.7 14.3
		112	123	120.4	120.4	2.1
	1

COMPARISON OF MEASURED VM MIGRATION DURATION AND SIMULATED TIMES, AT DIFFERENT BAND WIDTHS.

TABLE II .

 II VALUES USED IN SIMULATION; VM MIGRATION.

	Device	State	Value	Unit	Energy	Ref. 1	Bw. 2	Link
					model			rate
							[Mbps]
	NICend host	On;Idle Off	6.936 0	W W	basic basic	[6] -	-	-
	switch	On;Idle Off	66 0	W W	basic basic	-	-	-
		Idle ByteEnergy	0.11 3.423	W nJ	linear linear	[7] 4 [6]	54
	port 3	Idle ByteEnergy Idle ByteEnergy	0.11 3.423 0.11 3.423	W nJ W nJ	linear linear linear linear	[7] 4 [6] [7] 4 [6]	80 100	100
		Idle ByteEnergy	0.75 3.423	W nJ	linear linear	[7] 4 [6]	112	1000

1

References.

TABLE III .

 III VALUES USED IN SIMULATION; DATA-CENTER NETWORK AND CLOUD FEDERATION. One network port on end host NIC, switch or router. 2 Switching energy is calculated as the Power for NICend host

	Device	State	Value	Unit Energy	Ref. 3 Link
					model		rate
							[Gbps]
	NICend host	On;Idle Off	1.82 0.7	W W	basic basic	[9] [9]
	switch	On;Idle Off	150 0	W W	basic basic	[7] -	1 ; 10
	router	On;Idle Off	76.4 0	W W	basic basic	[7] -
		Idle	1.12	W	complete [9]
		Off Send/Recv byte	0 3.4	W nJ	complete -complete [6]	1
	port 1	Send/Recv packet Idle	197.2 0.53	nJ W	complete [6] complete [9]
		Off Send/Recv byte	0 14	W nJ	complete -complete [8]	10
		Send/Recv packet	1 504	nJ	complete [8]
	Device	Action	Value	Unit Energy	Ref. 3 Link
					model		rate
							[Gbps]
	NICend host	Switch On/Off	0.91*10 -3 0.5*10 -3	J s	complete [9] 2 complete [10]	1 ; 10
	switch	Switch On/Off	0.75 0.5*10 -3	J s	complete [7] 2 complete [10]	1 ; 10
	port 1	Switch On/Off	0.56*10 -3 0.5*10 -3	J s	complete [6] 2 complete [10]	1
	1

ACKNOWLEDGMENT

This work is funded by the INRIA HEMERA large-scale initiative. Experiments presented in this paper were carried out using the Grid'5000 testbed, supported by a scientific interest group hosted by Inria and including CNRS, RENATER and several Universities as well as other organizations (see https://www.grid5000.fr). We thank Ismael Cuadrado-Cordero at INRIA/Myriads (France) for providing us with part of the measurements (Section IV-A) for configuring our simulator.