Denis Delagnes
email: delagnes@enstimac.fr

Pascal Lamesle

François Delmas

Christophe Levaillant

N Mebarki

Relationship between microstructure and mechanical properties of a 5% Cr tempered martensitic tool steel

Keywords: Martensitic steel, Carbides, Fatigue, Second tempering

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Steels containing 5% Cr (AISI H11) were particularly developed for high temperature metal forming operations as pressure die casting for light alloys injection, forging dies, etc. [START_REF] Roberts | Tool Steels[END_REF]. Mechanical properties of such martensitic steels are strongly connected to their complex microstructure obtained after heat treatment that are generally performed in order to achieve a good hardness and/or tensile strength with sufficient ductility. However, as fatigue is one of the important causes of tool steel damage, it is of great importance to clearly understand the microstructural mechanisms controlling fatigue properties. Nevertheless, probably due to experimental difficulties, microstructural parameters (such as carbides at a nanometric scale and dislocations) giving suitable tensile and fatigue properties are never fully investigated. Notably, martensitic or bainitic steels have a typical fatigue behaviour as these steels present a strong softening till rupture (of about 25% of the stress amplitude measured during the first cycle for the AISI H11 steel, see Fig. 1 [2]).

The aim of the present work is to perform quantitative measurements of relevant microstructural parameters controlling the tempering and cyclic softening.

Experimental

The composition of the modified (low silicon) AISI H11 steel is shown in Table 1 and Table 2 shows the parameters of heat treatment and fatigue tests for all analysed specimens. Heat treatment includes austenitizing at 980 • C for 1 h followed by air cooling, first tempering at 550 • C for 2 h, second tempering for 2 h between 580 and 640 • C depending on the required hardness.

The microstructure of the steel was investigated by transmission electron microscopy (TEM) and energy-dispersive X-ray analysis (EDX). Selected area diffraction and EDX have been performed in order to determine crystal structure and chemical composition of the carbides extracted from the martensitic matrix. The distribution of carbide size is statistically approached from the analysis of TEM photos for nearly 300 particles per specimen.

In order to evaluate the weight fraction of carbides, carbides were extracted from the matrix by an electrolytic method described in Ref. [START_REF] Kim | [END_REF]. Crystallographic structure and chemical composition analysis were performed using X-ray diffraction (XRD). XRD experiments on the bulky material were also performed with peak profiles measurements in order to evaluate dislocation densities using the modified Williamson-Hall and the modified Warren-Averbach analysis [4][5][6].

Total strain controlled fatigue tests were carried out on a servohydraulic testing machine equipped with a resistance Tempered 550

• C + 580 • C - A5 Tempered 550 • C + 600 • C - A6 Tempered 550 • C + 620 • C - A7 Tempered 550 • C + 640 • C - A8 Tempered 550 • C + 620 • C ε t = 1.5% at 550 • C/230 MPa A9 Tempered 550 • C + 620 • C ε t = 2.0% at 550 • C/250 MPa
furnace. The axial strain was controlled with a 10-mm gauge length extensometer. The waveform of the cycle was triangular at a frequency of 1 Hz, the mean strain being zero.

Tests were performed at 550 • C with a total strain amplitude of 1.5% or 2%.

Results and discussion

Dislocation structure and density

Transmission electron microscopy

Low magnification bright-field transmission electron image of sample A4 is shown in Fig. 2. Laths are clearly separated by elongated iron carbides. In addition, observations of thin foils show a high density of intralath entangled dislocations even for high tempering temperatures. Consequently, the identification of individual dislocations (Burgers vector and glide plane) and density evaluation is very difficult using the classical TEM technique. Prior to fatigue testing, dislocation distribution is quite homogeneous on the whole even if, at a nanometric scale, a high density of dislocations was observed both near lath boundaries and carbides. To qualitatively compare dislocation structures obtained at different tempering temperatures, observations were performed in the same crystallographic orientation conditions ⃗ g = [ 1 1 0]. The recovery of the microstructure is shown by the clear decrease of the dislocation density as the second tempering temperature increases. This effect is strongly increased by the application of a cyclic strain (see Fig. 3).

X-ray diffraction; peak profile analysis

Coherently diffracting domains (CDD) and density of dislocations were evaluated using the theory and procedures introduced by Ungár et al. [4,5,[7][8][9]. Examples of conventional and modified Williamson-Hall plots are presented in Refs. [6,[START_REF] Mebarki | Proceedings of the 6th International Tooling Conference, The Use of Tool Steels: Experience and Research[END_REF]. The influence of the tempering temperature on CDD and density of dislocations are shown in Fig. 4. The highest density is measured for the as quenched sample (A2) in agreement with TEM observations. An increase of the second tempering temperature results in a strong decrease of the density of dislocations also in agreement with TEM observations described in Section 3.1.1. Density decreases by a factor 20, between the as quenched sample and the specimen tempered at 640 • C (A7). As dislocations contribute to limit the CDD size, the annihilation of dislocations results in an increase of CDD average size. As the consequence, the CDD size drastically increases with the temperature of the second tempering. The influence of fatigue loading is also shown in Fig. 4 for a specimen tempered at 620 • C (A6). After the fatigue test (A9), a decrease of the density of dislocations is observed. Conversely, no significant variation of CDD size oc- curs. Note that the density of dislocations after fatigue is however higher than that estimated for the sample A7. This latter result seems to be inconsistent with TEM observations. TEM observations led to the conclusion that the fatigue effect seems to be qualitatively stronger than the effect of an increase of the tempering temperature. Indeed, fatigue loading induces a new arrangement of dislocations increasing the heterogeneity inside martensitic laths rather than a strong decrease of the density of dislocations.

Precipitation and coarsening of carbides 3.2.1. Origin and sequence of the precipitation

For all tempering conditions, four types of carbides were identified depending on their morphology [6,[START_REF] Mebarki | Proceedings of the 6th International Tooling Conference, The Use of Tool Steels: Experience and Research[END_REF]: The TEM characterization of carbides before and after tempering confirms the precipitation of MC, M 3 C and M 7 C 3 carbides during tempering. Therefore, MC and M 7 C 3 are unambiguously the main carbides involved in the occurrence of the secondary hardening peak.

Influences of tempering temperature and fatigue loading on the precipitation and coalescence of carbides

The total weight fraction of carbides f m is the ratio between the mass of carbides collected after the dissolution and the initial mass of the sample. Results are shown in Fig. 5.

An increase of the weight fraction of carbides is observed for tempering temperatures above the secondary strengthening peak around 550 • C. Above 600 • C, the weight fraction levels off to a value of about 5.5%, very close to the maximum fraction calculated assuming that the whole carbon or alloying element has precipitated. Therefore, the saturation of f m can be explained by the absence of remaining carbon in solid solution and a stable distribution of carbon within the various carbides above 600 • C. In addition, the weight fraction is not influenced by a fatigue loading as indicated by the data for specimens A8 and A9 very similar to those of specimen A6. Two different populations of secondary carbides were identified after tempering:

(i) small carbides with an average size close to 6 nm. This first population is found for all tempering conditions and includes both chromium and vanadium carbides; (ii) carbides with an average size in the range 30-40 nm, only found for second tempering temperatures above 600 • C.

Evolution of the average size of carbides, including the two populations, is shown in Fig. 6. As both populations have nearly a constant average size for all the tempering conditions [6], the coarsening of carbides mainly results from the increased amount of the second population observed above 600 • C. As the weight and volume fraction remain constant, the softening observed during tempering is partly due to a mechanism of coalescence.

As shown in Fig. 6, the average size of carbides increases by nearly a factor two under the effect of isothermal fatigue performed at 550 • C (the temperature assumed to be reached near the surface of the tool for high pressure die casting operations). As no coarsening of carbides was observed during a second tempering at 580 • C (A3) as compared to a single tempering at 550 • C (A2) and because the duration of a fatigue test is shorter than a second tempering, it can obviously concluded that coalescence is only induced by fatigue. The driving force for the dynamic coalescence of carbides is probably the mechanical energy that complements the thermal energy not sufficient to provide coalescence alone. Such a mechanism of dynamic aging coalescence has been already observed in studies of the effect of cyclic strain in tempered martensitic steels [START_REF] Chang | [END_REF]12].

Conclusion

The decrease of the density of dislocations and the coalescence of secondary carbides are both involved in the decrease of the yield stress during tempering and fatigue.

Concerning the softening during tempering, the situation is summarised as follows:

• Between 550 and 600 • C, the precipitation is not fully completed as the distribution of carbon within the various precipitates is not stable or the transition from iron carbides to special carbides still occurs. An increase of the total weight fraction of carbides is observed that may results in an increase of the yield stress as no coalescence is observed. Nevertheless, the reduction of the density of dislocations as well as the reduction of the alloying elements content in the solid solution, lowering the solid solution hardening, induce a global softening of the tempered martensite. • Above 600 • C, both the coalescence of carbides and the decrease of the density of dislocations induce further softening of the material.

Fig. 1 .

 1 Fig. 1. Evolution of stress amplitude with the number of cycles for the AISI H11 steel.

Fig. 2 .

 2 Fig. 2. Bright-field TEM image of the dislocation structure for the sample A4.

Fig. 3 .

 3 Fig. 3. Bright-field TEM images of the dislocation structure, ⃗ g = [ 1 1 0]. (a) Before a fatigue test, sample A6, (b) after a fatigue test, sample A9 ( ε t = 2.0%).

Fig. 4 .

 4 Fig. 4. Influences of tempering temperature and fatigue on the average size of CDD and the density of dislocations.

  (i) facetted carbides: hexagonal Cr Fe carbides (M 7 C 3 type), (ii) large globular carbides (100-300 nm): face-centred cubic Cr Fe carbides (M 23 C 6 type), (iii) elongated carbides: orthorhombic Fe Cr carbides (M 3 C type), (iv) small globular carbides (<40 nm): face-centred cubic V carbides (MC type). XRD complementary results allow to determine the sequence of precipitation. The annealed steel contains M 2 C (M: Mo mainly), M 3 C (M: Fe mainly), M 23 C 6 (M: Fe and Cr mainly) and a small ratio of MC (M: V mainly). After quenching, only the vanadium carbide (MC) and small quantities of M 3 C and M 23 C 6 , probably not dissolved during the austenitization, were found. After the first and the second tempering, X-ray analysis has shown the presence of MC, M 3 C, M 7 C 3 , and traces of M 23 C 6 carbides.

Fig. 5 .

 5 Fig. 5. Influences of the heat treatment and isothermal fatigue on the weight fraction of carbides.

2 Fig. 6 .

 26 Fig.6. Influences of tempering temperature and fatigue loading on the average size of secondary carbides.

Table 1

 1

	Chemical composition (wt.%) of the modified AISI H11 steel
			C	C r	M o	V	S i	M n	P	S n	S b
	Weight percent	0.36	5.06	1.25	0.49	0.35	0.36	0.006	0.0022	0.0005
		900
		850			AISI H11 / ∆ε t =1%
		800
	/2 (MPa)	650 700 750		500˚C	400˚C	300˚C	200˚C
		600
		550	550˚C
		500
		0	3000	6000	9000	12000	15000
			Number of cycles

Table 2

 2 Conditions of heat treatment and fatigue testing of the various samples

	Sample Heat treatment conditions	Fatigue test/softening amplitude
	A1	Annealed	-
	A2	As air quenched	-
	A3	Tempered 550 • C	-
	A4

Acknowledgements

Authors gratefully acknowledge the Aubert&Duval Holding (ADh) company for providing samples and financial support. We would like to thank Prof. T. Ungár from the Eötvos University of Budapest for providing X-ray experiments and peak profile analysis and for so many precious advices. Special thanks are given to Mr. A. Grellier, Mr. P.E. Richy and Mrs. M.F. Gervais from the ADh company, Prof. A. Coujou and Dr. A. Couret from the Centre d'Elaboration des Matériaux et d'Etudes Structurales, CNRS for fruitful discussions.


Alain De Ryck

David Que

Gravity and Inertia Effects in Plate Coating

Keywords: fluid coating, thin liquid films, wetting

INTRODUCTION

Q Å eV 0 rge 3 /3h. [4] When a plate is drawn out of a wetting liquid bath (viscos-Introducing the latter expression in Eq. [3] and then Eq. ity h, surface tension g, and specific mass r) at a velocity [2] yields the surface velocity. In particular, it is easy to V, it comes out coated with a liquid film whose thickness e show that a stagnation point exists at the surface, at a thickwas first derived by Landau, Levich, and Derjaguin (1,2). ness h* given by At low velocities, the fabrication of the film takes place in a thin region (thickness on the order of e) called the dynamic meniscus, which joins the static meniscus to the film (Fig.

h* Å S 3 0 e 2 k 2
Ca D e,

[ 5 ] 1). In the dynamic meniscus, the interface (of profile h(x)) is slightly curved and the flow nearly vertical. So the lubrication approximation can be used and the Navier-Stokes equawhere k 01 Å q g/rg is the capillary length and Ca the capiltion reduces to lary number (Ca Å hV/g). The next step of the calculation consists of matching the dynamic meniscus with the static one, by expressing that hu yy Å p x / rg Å 0gh xxx / rg, [1] there is no pressure difference between them at the point where they match. But the location of this point is unknown, where u(y) is the velocity profile inside the dynamic menisso Landau, Levich, and Derjaguin performed an asymptotic cus, p the Laplace pressure, and g the acceleration of gravity.

matching, requiring the equality of the second derivatives The subscripts mean partial derivation.

of the profiles: As seen in Eq. [1], two different physical processes make the liquid flow in the dynamic meniscus: gravity (the plate is vertical) and capillary suction (the dynamic meniscus is

S d 2 h dx 2 D dyn. hr`Å S d 2 h dx 2 D static hr0 .
[6] curved). Equation [1] can be integrated using two boundary conditions: (i) no slippage at the wall: u(y Å 0) Å V; (ii) free liquid-air interface: u y (y Å h) Å 0. Hence a parabolic The static limit is the curvature at the top of the (static) profile for the velocity is obtained: meniscus, whose value is q 2k. To calculate the other limit, the curvature of the dynamic meniscus must be studied at 1 To whom correspondence should be addressed. large thicknesses. Dimensionless coordinates are introduced: AID JCIS 5444 / 6g42$$$$41 06-10-98 09:36:57 coida h Å eY

x Å e Ca 01 / 3 X.

[ 7 ]

Taking Eq. [4] into account, Eq. [3] becomes:

Y -Å 3 1 0 Y Y 3 / e 2 k 2 Ca Y 3 0 1 Y 3 . [8]
To determine the first limit in Eq. [6], the latter equation must be integrated once. The integration is difficult to achieve with all the terms; we are looking for a constant limit for Y 9, and Yobviously does not tend to zero as Y goes to infinity in Eq. [8] because of the gravity term.

It is first noted that two asymptotic solutions can be found according to the dominant mechanism causing the drainage Equation [9] can be integrated numerically once. The result is drawn in Fig. 2 and the sought-after limit is found:

THE VISCO-CAPILLARY REGIME

The LLD solution. If drainage by gravity can be ne-

S d 2 Y dX 2 D dyn.
Yr`Å 1.34, [10] glected, Eq. [8] reduces to a differential equation free of any parameter, which can be written, reintroducing h and x (see Eq. [7]), as

Y -Å 3 1 0 Y Y 3 , [9] S d 2 h dx 2 D dyn.
hr`Å 1.34 Ca 2 / 3 e .

[11] with the conditions Y r 1, Y * r 0, and Y 9 r 0 for X r `. The static limit in Eq. [6] being q 2k, the Landau-Levich-Derjaguin (or LLD) law is finally obtained: e Å 0.94 k 01 Ca 2 / 3 .

[12] Domain of validity. Gravity was neglected in Eq. [8]. If e is given by Eq. [START_REF] Koulago | Physical Hydrodynamics[END_REF], the approximation is justified if we have:

Ca 1 / 3 ! 1.
[13]

Hence, Eq. [START_REF] Koulago | Physical Hydrodynamics[END_REF] is practically valid for very small capillary numbers (Ca õ 10 03 ). Conversely, Ca Å 1 is expected to be the threshold above which drainage in the dynamic meniscus is mainly due to gravity. Length of the dynamic meniscus. Close to the flat film, Eq. [8] can be linearized. Setting Y Å 1 / e (with e ! 1)

The static meniscus is strained on a length l and a film of thickness e is entrained.

leads to AID JCIS 5444 / 6g42$$$$42 06-10-98 09:36:57 coida e-Å 03e S 1 0 e 2 k 2 Ca D .

[14]

Making k 2 Å 0 yields a linear equation of solution: e Å e s X , with s Å 03 1 / 3 (about 01.44). With the natural variables, the profile of the dynamic meniscus can be written h(x) Å e S 1 / exp S 0 x l DD , with [15] l É 0.69 e Ca 01 / 3 .

[16] l is the characteristic length of the dynamic meniscus pictured in Fig. 1. For very low capillary numbers (Eq. [13]), we have e ! l ! k 01 ; the extension of the dynamic meniscus is small compared with the size of the static meniscus but much larger than the film thickness, which justifies the lubrication approximation. Conversely, as Ca approaches unity, e and l both become on the order of the capillary length.

FIG. 3.

Profile of a liquid film on a solid drained by gravity. e is the

THE VISCO-GRAVITATIONAL REGIME

initial thickness of the film and l the region close to the top where the film is thinned.

A convenient way to derive the asymptotic entrainment law in the gravity regime involves considering Eq. [14]. Without the capillary term (e-Å 0), we immediately find top of the film (in this region, the flux of liquid is a constant given by Eq. [18]) and (ii) at the top of the film, Jeffreys has sought a power law solution for Eq. [19] ( h(x, t) Å e Å k 01 q Ca.

[17] Ax a t b ). If the origin of the x-axis is fixed at the top of the film, it is found that the profile is: This asymptotic regime was first predicted by Derjaguin (3) using dimensional arguments: when the withdrawal velocity is important, capillarity becomes negligible and the thickness h(x, t) Å r hx rgt .

[20] should no longer depend on the surface tension. Looking for a power law e Å k 01 Ca n independent of g gives n Å 1 2 , as found in [17].

This kind of slow thinning (h Ç t 01 / 2 ) is known as a Reyn-We can then be interested in the way the film gets thinner olds-type law. because of gravity after it is drawn. The thinning law of a Figure 3 summarizes these results. As time goes on, a vertical film of initial thickness e was first derived by Jefparabolic profile grows at the top of the film and matches freys (4). If the wall is immobile and the x-axis directed the lower part, which remains at the initial thickness. The downward, the flux (per unit width of the plate) is given by extension l of the thinned zone is determined by matching Eq. [3]: the solution of Eq. [20] with h Å e. It is thus found that l increases linearly with time:

Q Å rgh 3 3h , [18] l Å rg h e 2 t. [21]
where h is now the film thickness depending on time and position x. The film dries upstream, so it gets thinner from Returning to the coating problem, we finally conclude the top. Conservation of matter yields:

that two types of withdrawal can be described, as sketched in Fig. 4. If e õ k 01 q Ca , the thinned zone progresses more slowly than the film is made (

d l / dt õ V ), so that the Ìh Ìt Å 0 ÌQ Ìx Å 0 rg h h 2 Ìh Ìx .
[19] film is flat everywhere but at the top ( Fig. 4a ) . This case corresponds to the regime of entrainment limited by capillarity ( see Section 2 ) . If e Å k 01 q Ca , the thinned zone Equation [19] has two kinds of solutions: (i) a solution of constant thickness h Å e, which is valid far below the progresses at the same velocity as the plate. The entrained AID JCIS 5444 / 6g42$$$$42 06-10-98 09:36:57 coida film is thinned everywhere, with a parabolic profile ( Fig. 4b ) , and the Derjaguin equation ( Eq. [17 ] ) is only valid at the place where the film arises. The gravitational regime appears again as a borderline case; the film thickness cannot be larger than k 01 q Ca .

THE CROSSOVER BETWEEN THE TWO REGIMES

According to the mechanism causing the drainage in the dynamic meniscus, two asymptotic laws of liquid entrainment by a moving vertical plate have been found. At low capillary number (Ca õ 10 03 ), the entrainment is limited by capillarity and described by Eq. [START_REF] Koulago | Physical Hydrodynamics[END_REF]. At high capillary number, gravity is the principal cause of drainage and the entrainment law given by Eq. [17]. The latter regime is valid when Ca 1/3 is much larger than unity (see condition in Eq. [13]), and thus for Ca ú 10 3 . The crossover between the two regimes is especially broad since it concerns six orders of magnitude in capillary number. Therefore, it is worth studying the whole Eq. [8] with both the capillary and gravitational terms.

A first approach was performed by White and Tallmadge (5). Using Eq. [14], the linearized form of Eq. [8], they looked for an equivalent profile solution of the Landau problem. By this ingenious method, they finally obtained as a thickness dependence on the capillary number the simplest interpolation between the capillary and gravitational regimes:

Ca Å 1.09 e 3 / 2 k 3 / 2 / e 2 k 2 .
[22]

Here we propose a numerical solution of Eq. [8] and compare it with Eq. [22]. Following White and Tallmadge, the dimensionless quantity T Å ek/ q Ca is introduced and treated as a parameter. Then Eq. [8] becomes

Y -Å 1 0 Y Y 3 {3 0 T 2 (Y 2 / Y / 1)}. [23]
According to the value of T, we discuss the possibility for Eq. there is no solution becoming flat at infinity. For T Å 1, it Visco-capillary regime, which corresponds to the LLD law [START_REF] Koulago | Physical Hydrodynamics[END_REF]; the film can easily be seen in Eq. [23] that Y cannot be larger than is flat everywhere but at the top, where it is thinned. b) Visco-gravitational case, in which the film thickness is given locally by the Deryaguin law unity. Thus, the law e Å k 01 q Ca is found again as an asymp- For 0 õ T õ 1, the profile Y ( X ) can be deduced from the numerical integration of Eq. [23]. The procedure consists of starting from a point close to the flat film, whose coordinates contact line, rims indeed develop at the liquid front (and often break into vertical rivulets); that is the case for a film are calculated from the linearized Eq. [14]. Then, the profile is integrated step by step using a Runge-Kutta method. Such (bounded at its bottom by a contact line) flowing down a vertical plate (6). a profile is pictured in Fig. 5. It is very different from the Landau case (Fig. 2b). A bump appears at small X, which When going from large X (the film) to small X (the reservoir) in Fig. 5, the second derivative of the profile passes is clearly unphysical in our problem. In other cases where the film has not to match a reservoir, but is bounded by a through a maximum Y 9 M before changing its sign at the AID JCIS 5444 / 6g42$$$$43 06-10-98 09:36:57 coida inflection point I. We propose matching the two menisci at this point. Such a matching allows us again to find the Landau solution at the limit T r 0 (corresponding to Ca r 0 or to the absence of gravity). At this limit, Y 9 M tends towards 1.34 and Eq. [START_REF] Koulago | Physical Hydrodynamics[END_REF] is recovered. This choice for the matching possibly overestimates the film thickness, since the maximum value of Y 9 yields a maximum value for e. Equation [6] is then used to determine the film thickness e. (Of course, this point is questionable; as Ca tends towards unity, the dynamic meniscus sweeps into the static one and finally makes it disappear.) Thus we write, as in the Landau case:

e Å Y 9 M q 2 k 01 Ca 2 / 3 . [24]
The resulting T(Ca) dependence is drawn in Fig. 6, where it is compared with Eqs. the curvature we used is only valid when the slope of the interface is small. This condition is not satisfied at the match-For a given value of Ca, the above equation can be numeriing point. We introduce the exact curvature expression cally integrated following the same procedure as above.

Then, the maximum T (for Y as large as possible) is sought

C Å 0 h9 (1 / h* 2 ) 3 / 2 .
[25] by matching with the equation of the static meniscus, which can be written Then, Eq. [23] becomes

Y 9 0 q 2T Ca 01 / 6 ((1 / Ca 2 / 3 Y * 2 ) 1 / 2 / Ca 1 / 3 Y *) 1 / 2 1 (1 / Ca 2 / 3 Y * 2 ) 5 / 4 Å 0. [27] Y -Å 1 0 Y Y 3 (3 0 T 2 (Y 2 / Y / 1))
Results are drawn in Fig. 7. In Fig. 7a, an example of a

1 (1 / Ca 2 / 3 Y * 2 ) 3 / 2 / 3Ca 2 / 3 Y*Y 9 2 1 / Ca 2 / 3 Y * 2 .
[26] profile is displayed. It can be seen that the matching with a reservoir is more realistic. In Fig. 7b, these numerical results are shown to fit in a satisfactory way the data of Spiers et al.

(8) at low capillary numbers (0.01 õ Ca õ 0.2). Then, when approaching Ca Å 1, some discrepancies can be observed. We now discuss the main causes of these discrepancies.

THE VISCO-INERTIAL REGIME

Inertial effects may become nonnegligible at high withdrawal velocity. A Reynolds number associated with the flow inside the dynamic meniscus can be defined by taking as a characteristic length the thickness of the film (Re Å rVe/h). Considering that the thickness is given by the LLD law (Eq. [START_REF] Koulago | Physical Hydrodynamics[END_REF]), the Reynolds number can be written AID JCIS 5444 / 6g42$$$$43 06-10-98 09:36:57 coida which, as seen above, both Reynolds and capillary numbers are large.

(a) Inertial Effects at Low Capillary Number

The inertial term of the Navier-Stokes equation must be incorporated in the Landau theory, which leads to a modified equation for the profile. The principle of the calculation was first proposed by Esmail and Hummel (10) and developed recently by de Ryck (11) and Koulago (12): it consists of supposing that the flow inside the meniscus still has a parabolic profile, but that the effective pressure which provokes it is unknown. Writing the Navier-Stokes equation (with the usual boundary conditions) allows a more general expression for the profile of the dynamic meniscus to be found. Written with the dimensionless variables of Landau, it reads:

F Y 9 (1 / Ca 2 / 3 Y * 2 ) 3 / 2 G * Å 1 0 Y Y 3 {3 0 T 2 (Y 2 / Y / 1)} / 1 5 FCa 11 / 6 T S Y 2 0 2 3 (3 0 T 2 ) 2 D Y* Y 3 . [29]
The first part of the equation is Eq. [23], written with the whole expression for the curvature, given by Eq. [25]. The second term in the right member is the inertial contribution, which logically includes the number F. This term is not negligible in the case in which we are interested, since F was shown to be large for liquids of low viscosity.

Equation [29] can be integrated as explained above, which allows us to determine the maximum of the curvature. Then matching provides the film thickness, which can be ex- provokes a thickening of the film above a threshold in capillary number. The threshold Ca* depends on the nature of the liquid: the higher F, the smaller Ca*. A simple way to The prefactor F depends only on the nature of the fluid, and varies mainly with the liquid viscosity. For a very vis-understand this effect physically is the following: the plate puts the liquid close to it in motion, and the effect of inertia cous silicone oil (h Å 100 P), F is equal to 3 1 10 04 , and Re becomes larger than 1 if Ca is larger than 130 (which is indeed to project the liquid at the exit of the dynamic meniscus out of the reservoir, and thus to thicken the film. corresponds to V ¢ 25 cm/s). For water, F is much larger (F Å 2 1 10 5 ), and a Reynolds number of order one is This argument helps to understand the limitation of this effect: at still higher velocities, the problem may become un-obtained for a very small capillary number (Ca Å 7 1 10 04 , which corresponds to V Å 5 cm/s).

stationary. The plate does not spend enough time in the bath to put a thick layer of liquid in motion, so the film is limited Thus, for liquids of low viscosity, inertia must be considered even at a very low capillary number (which justifies in to the viscous boundary layer which develops in the bath.

In this case, the thickness decreases with the velocity. this case taking Eq. [START_REF] Koulago | Physical Hydrodynamics[END_REF] to define Re in Eq. [29]). We focus on this situation of practical importance (since water, Such effects were recently reported in fiber withdrawal experiments (13). The thickening effect was observed and with or without surfactants, is often used as a lubricant), which was not considered by previous authors. Soroka and found to be more spectacular than in Fig. 8 since gravity (a natural factor for limiting the thickness) could be neglected Tallmadge (9) and Esmail and Hummel (10) studied the effect of inertia in plate coating for viscous liquids, for (experiments were performed at small Bond numbers). Be-AID JCIS 5444 / 6g42$$$$44 06-10-98 09:36:57 coida and small capillary numbers), which is due to Tallmadge and Stella (14). Data on the coating of a plate with water are reported, and indeed a sharp increase of the film thickness above the LLD law is observed. A remarkable point (stressed by the authors) is the fact that the discrepancy is found at very low capillary numbers (thicknesses were reported for 10 04 õ Ca õ 10 03 ), in agreement with our description. Of course, new data would be useful to test our predictions quantitatively.

Note finally that the diverging behavior in Fig. 8 exhibits some inflection (and saturation). The last point for each curve may have no real significance; it corresponds to conditions for which the numerical integration of Eq. [29] becomes hard to achieve, since solutions have oscillations which make matching difficult. That is why our results stop at that point; it was impossible to produce a meaningful who have made attempts to describe this case, have shown sides, a scaling argument was proposed (and checked experithat the incorporation of the strain-rate component in the mentally), for predicting the location of the threshold: inertia equilibrium condition of the liquid-air interface leads to a provokes a large thickening when the dynamic pressure (of slight thickening of the film. On the other hand, Esmail and order rV 2 ) is equal to the Laplace pressure (of order g/r, Hummel (10) incorporated the same term in their inertial with r being the fiber radius). For a plate, we expect inertia theory of free coating. Their results, performed for F ranging to be important in a similar way when rV 2 becomes of the between 0 and 77, do not show a diverging behavior. same order as rgk 01 , the hydrostatic pressure in the meniscus, which gives as a threshold:

SUMMARY AND CONCLUDING REMARKS

Ca* É 1 q F . [30]
We proposed a numerical solution of the gravity-corrected Landau theory for plate coating, based on a new matching and on the consideration of the exact expression of the curva-Another way to derive this threshold consists of considerture in order to obtain realistic profiles. Then, we discussed ing the inertial term in Eq. [29], which must be considered an improvement of practical importance: if the withdrawal when the quantity FCa 11 / 6 T is of order one. Since we are at velocity becomes higher than a threshold, inertia must be low capillary number, the thickness (in the expression of T ) taken into account. We discussed the particular case of liqis given by the LLD law (e É Ca 2 / 3 ), which finally yields uids of low viscosity (aqueous solutions), for which the Eq. [30]. For water, we find Ca* Å 2 1 10 03 (or V É 16 capillary number can remain much smaller than unity even cm/s), in good agreement with the threshold observed in if the Reynolds number becomes of order one (or larger). curve 3 of Fig. 8. For F Å 10 4 , the difference from the curve In that case, a thickening of the film is predicted because of without inertia appears at around Ca* Å 0.01, as predicted by the influence of inertia, which ejects the fluid out of the Eq. [30]. Ca* may be written as reservoir. The observations of Tallmadge and Stella could be related to this effect, but new specific experiments would V * É q k 01 g,

[31] be useful to test it. The case where both the velocity and the capillary numbers are large is the most complicated and remains to be showing that inertia becomes important when the withdrawal velocity is close to the minimal velocity of capillaro-gravita-fully solved. In this case, the lubrication approximation is no longer justified (a two-dimensional flow must be taken tional waves (typically 10-30 cm/s).

To the best of our knowledge, there is only one experiment into account) and the matching with a static meniscus becomes questionable. corresponding with these features (large Reynolds numbers AID JCIS 5444 / 6g42$$$$44 06-10-98 09:36:57 coida 5. White, D. A., and Tallmadge, J. A., Chem. Eng. Sci. 20, 33 (1965).

FIG. 2 .

 2 FIG. 2. Profile of the dynamic meniscus in dimensionless coordinates defined in [7], obtained by numerical integration of [9]. Insert: curvature in the dynamic meniscus, capillary suction or gravity, respec-

FIG. 1 .

 1 FIG. 1.Withdrawal of a plate out of a quiescent wetting liquid bath.

FIG. 4 .

 4 FIG. 4. Coating of a plate vertically withdrawn out of a reservoir. a)

[

  17]; above this point, the whole film is thinned and has the parabolic totic regime. profile given by [20].

  FIG. 6. Film thickness scaled by the Derjaguin thickness (T Å et al. (8). Though the numerical calculation possibly overek/ q Ca) as a function of the capillary number. In the same figure are estimates the film thickness, the results are found to be below presented the LLD law [12], the Derjaguin law [17], the White and Tall- madge cross-over [22], and the numerical integration of [23]. The squares

FIG. 5 .

 5 FIG. 5.Profile of the dynamic meniscus obtained by a numerical integration of Eq. [23] for T Å 0.71. The main variation from Fig.2is the fact that the profile is not monotonous any longer, but presents a bump. The region beyond the inflection point I (dotted line) is unphysical in our

FIG. 7 .

 7 (a) Profile of the dynamic meniscus obtained from an integra-pressed as a function of the capillary number. The results tion of [26] with T Å 0.5 and Ca Å 0.044. The circle gives the location are displayed in Fig. 8 for F Å 2 1 10 5 (water) and F Å of the matching point (Equation [27] is verified). (b) For the same coordi-10 4 (long alkane). nates as above (T versus Ca), numerical results from [26] are compared A remarkable feature can be observed in Fig. 8: inertia with the data of Spiers et al. (8).

FIG. 8 .

 8 FIG. 8. Effect of inertia on the film thickness, obtained by integrating

ACKNOWLEDGMENTS 6. Huppert, H. E., Nature 300, 427 (1982). 7. Gutfinger, C., and Tallmadge, J. A., AIChE J. 11, 403 (1965). It is a pleasure to thank V. Shkadov and M. Velarde for very useful 8. Spiers, R. P., Subbaraman, C. V., and Wilkinson, W. L., Chem. Eng. discussions.

Sci. 29, 389 (1974). 9. Soroka, A. J., and Tallmadge, J. A., AIChE J. 17, 505 (1971).


A De Ryck

R Condotta

M Lubert

Interrupted shear of granular media

Keywords: 83, 80, Fg Granular solids -45, 70, -n Granular systems

The response of a granular material during a stop-and-go shear experiment is investigated using an annular shear cell and silicagel powders of different particle sizes. The experimental results are examined on the basis of the Dieterich-Rice-Ruina model for solid friction. In addition to making this analogy with solid friction, we describe a new instability that is observed when restarting shear, where the powder bed is found to slip and compact for short hold times but only dilates for long hold times. The minimum hold time to restore a non-slip behaviour has been investigated for different size particles and normal loadings. The observed dependencies show analogies between this behaviour and the sliding rearrangements seen above the stick-slip threshold.

Introduction

The mechanical resistance of bulk solids to flow and its temporal evolution is an important topic for the correct handling and storage of these materials. At low shear rates, and for non-cohesive powders, the behaviour observed is similar to solid friction where the shear stress τ necessary to make the powder move is proportional to the normal stress σ. However, the internal coefficient of friction, µ = τ /σ, is not single-valued since, the coefficient of friction to start the flow µ s is greater than the dynamic coefficient of friction µ d , necessary to maintain the sliding at constant velocity. Stop-and-go shear experiments at constant normal loading may demonstrate such behaviour [1]. These consist in interrupting the sliding for a hold time T before restarting shear. The shear stress then goes through a maximum, which increases logarithmically with the hold time.

In solid friction, the first physical explanation for this ageing is the plastic creep of the contact points between the two surfaces (due to the increase of the real area of contact and to the ageing of the adhesive junction at interface). Some recent experiments using polymer glasses at temperatures up to the glass transition temperature confirm these hypotheses [2,3]. A second origin is the condensation of humidity, which leads also to a logarithmic strengthening with time [4,5]. In this case, ageing should a e-mail: deryck@enstimac.fr vanish in dry air, which is what Dieterich and Conrad have observed in rock friction [6].

These two mechanisms also apply in granular friction. But, in this case, there is a third possibility where friction can be changed by small rearrangements between the grains during the creep phase. These rearrangements have been observed numerically in loading experiments [7,8] and shown experimentally by direct imaging in a 2D granular assembly [9].

Here, we present some experimental results of stopand-go shear experiments for non-cohesive silicagel powders, showing not only this strengthening but also a new temporal behaviour, not previously reported in the case of solid-solid friction. In Section 2, some results of the phenomenological model of solid friction developed by Dieterich, Rice and Ruina [10,11] are introduced in order to present the granular-friction results given in this paper. In Section 3, the experimental set-up and granular materials are presented. In Section 4, the mechanical response of these granular materials during steady-state shear experiments is detailed. Finally, Section 5 describes and discusses the stop-and-go shear experiments.

Theoretical background 2.1 The Dieterich-Rice-Ruina model

Logarithmic ageing is embedded in the phenomenological state-and-rate model formulated by Dieterich, Rice and Ruina to describe rock-rock friction. This model has been developed from the following experimental evidences: a) The steady-state sliding coefficient of friction decreases logarithmically with the sliding velocity V for numerous materials. b) The static coefficient of friction increases logarithmically with the contact time between the two solid bodies. c) When a jump is imposed in the driving velocity, the friction force relaxes to its new value over a characteristic distance D, which is of the order of a micrometer. The Dieterich-Rice-Ruina model can be written as [10,11]

µ = µ 0 + a ln V V 0 + b ln V 0 φ D , (1)
with dφ dt = 1 - V φ D , (2)
where µ 0 is the steady-state coefficient of friction at the sliding velocity V 0 . a and b are two numerical constants which depend on the material. φ is the age of the contact. Its evolution is given by equation ( 2): If there is no sliding, dφ/dt = 1 and φ = t. For steady-state sliding, we have

φ s = D/V and µ s = µ 0 + (a -b) ln(V /V 0 ).

Dynamics on an axisymmetric geometry

Disregarding inertia, the equilibrium of a frictional annulus of internal and external radii R i and R o entrained by an elastic motor at the rotational velocity ω reduces to 2πσ Ro Ri

µr 2 dr = Γ α , ( 3
)
where r is the radial co-ordinate. Γ α is the torque developed by the spring of which the angular deformation is α and the elastic constant is Γ . In practical cases, the elasticity comes from the elasticity of the drive motor and the elastic contacts between the two surfaces. This elasticity has a feedback on the friction force since the latter depends on the sliding velocity. We have

V = r(ω -dα/dt) . (4)

Onset of instability

If the sliding velocity is taken to be constant at the shear plane (R o /R i → 1), a linear stability analysis of the system (Eqs. (1) to ( 4)) shows that it becomes unstable in the case of a velocity weakening force (b > a) when the normal stress σ is higher than the threshold value:

σ 1 = D b -a Γ 3 π (R o + R i ) (R 3 o -R 3 i ) . ( 5
)
At this threshold, the system oscillates with a pulsation [12]. More generally, taking into account the radial dependency of the sliding velocity leads to a threshold value σ ℓ , which depends both on the ratio a/b and on the ratio R o /R i . Figure 1 gives the correction in percentage between σ ℓ and σ 1 and for the pulsation Ω ℓ . It can been concluded from this figure that the hypothesis of constant sliding velocity remains a good approximation for radii ratios R o /R i up to 2, or when the coefficient a is close to b. When not mentioned, the coefficient of friction will be considered as constant in the shear plane and evaluated from the expression 3Γ α/2πσ(R 3 o -R 3 i ).

Ω 1 = ω Ro+Ri 2D 1 -a/b

Creep relaxation

If the motor is stopped from a steady-state regime (α = α s and φ = φ s are constant) at t = 0, the angle α decreases to insure the continuity of the sliding velocity. We then have dα/dt(t = 0+) = -ω and dφ/dt(t = 0) = 0. (The continuity of the latter is due to Eq. (2).) The time derivative of equation (3):

2πσ Ro Ri a d 2 α dt 2 dα dt + b dφ dt φ r 2 dr = Γ dα dt (6)
reduces to

d 2 α dt 2 ∼ 3Γ 2πσ(R 3 o -R 3 i )a dα dt 2 (7)
at the beginning of the relaxation. Equation (7) means that the coefficient of friction decreases logarithmically like µµ s = -a ln(1 + t/t 0 ) ,

with

t 0 = 2πσ(R 3 o -R 3 i )a/3Γ ω.

Resuming shear

After a creep period of duration T , the shear driving motor is restarted. This leads to an increase of the shear force,

where Γ α max and Γ α s are, respectively, the maximum shear torque and the torque at steady state. Since the maximum occurs just at the beginning of the resuming, we have φ ≈ T and equation ( 9) leads to a logarithmic increase of the maximum friction coefficient with the duration of creep:

µ max -µ s ∝ b ln T . (10)
The coefficients a and b are then separately measurable by a series of stop-and-go shear experiments, as described in the following paragraph.

3 Experimental set-up

The shear cell

The experimental set-up is shown in Figure 2 and described in detail in [12]. The powder is poured inside an annulus of internal and external radii, respectively 5 and 10 cm and height H = 4 cm. A smaller cell was also used, whose dimensions are all reduced by a factor λ = 3/5. Both cells have a radii ratio of 2 and then, considering the conclusion of Section 1.2, the coefficient of friction will be considered constant on the shear plane.

Shear loading

This annulus is put into rotation by a motor at a rotational velocity ω = 3 • 10 -4 rad/s. A cover plate with 20 teeth (12 for the small cell) of 4 mm height, is placed on the powder surface. This cover plate is loaded by weights and held against rotation by two force transducers (at L = 12.5 cm from the axis of rotation), which measure the force F = Γ α/2L induced by the shear torque due to the friction between the particles. The friction measured is due to the fact that the powder gets trapped between the teeth of the cover plate, while the powder below is entrained by the rotation of the annulus. A gap between the cover and the annulus prevents friction between them but has the consequence of allowing extrusion of powder during the shear. The powder is also sheared in this gap, but the contribution to the measured torque is negligible, since, on the one hand, the surface times the lever arm is 15% lower than that developed by the base and, on the other hand, the normal loading on the side is expected to be lower than that exerted on the cover plate.

Normal loading

The cover plate is loaded by a weight M placed on a rod hooked on the cover plate. Since the cover plate and the rod are counter-weighted (not represented in Fig. 2 for clarity), the normal stress acting on the shear plane is

σ = (M + M p )g/π(R 2 o -R 2 i )
, where M p is the mass of the powder between the teeth (of order 50 g) and g is the acceleration due to gravity.

An inductive transducer measures the vertical displacement h of the cover plate caused by dilatancy of the powder sample and the extrusion. All the experiments are performed in ambient air.

The powders

The powders used are nanoporous silicagel (mean diameter of the pores: 60 Å) which have been ground and sieved (Fig. 3a). This material absorbs humidity well which ensures that the mechanical properties are insensitive to air humidity and no electrostatic effects are present when they are handled in ambient atmosphere. Two particle size cuts of similar size spread in logarithmic coordinates have been used: 5 to 20 µm and 60 to 200 µm as measured in ethanol suspension with a Malvern Mastersizer laser diffraction instrument (Fig. 3b). The finer powder is slightly cohesive. Thus, after a series of shear experiments, the powder remains as a block inside the annular cell which may be attributed to the fact that the van der Waals forces between particles are of the same order of magnitude as their weight. The coarser particle size cut is a free-flowing powder. Figure 4 shows the averaged shear stress in steady sliding versus the constant normal stress. The data, obtained at a rotation rate ω = 3 • 10 -4 rad/s, follow the same line for both cell sizes and particle sizes, which is fitted by µ = τ /σ = 0.68 ± 0.03.

Constant rate shear experiments

In permanent shearing, these powders present a stick-slip instability when the normal loading overcomes a threshold value. This instability has been extensively studied in [12] with the same silicagel sieved between 40 and 63 µm and also by Gollub and co-workers [13]. This is illustrated here by Figure 5 for the case of the 60-200 µm silicagel in the big cell. The onset of the stick-slip instability, with the occurrence of a small sinusoidal modulation of the shear stress, occurs for a critical loading mass M c close to 4 kg. When the loading mass is increased further, the amplitude of the oscillations increases. For a loading mass of M * = 6 kg or above, we observe a second instability where huge sliding occurs, leading to a reorganisation of the grains with a packing-down of several microns. This loss of height is recovered during the stick phase (Fig. 5).

The stick-slip instability and the sliding instability, presented in Figure 5, are also observed for different particle sizes and/or cell sizes, as described, respectively, in the next two sections.

Steady-state shearing and the onset of the stick-slip instability

Figure 6 shows the root mean-square deviation of the shear force versus the normal load M for the 5-20 and 60-200 µm silicagel (black and white points, respectively) and the small and big cells (circle and square points, respectively). Figure 7 shows the same data expressed as a percentage of the mean force.

At low normal load, the fluctuations of the shear force are proportional to the normal force (dashed lines in Fig. 6) and independent of the geometry. These fluctuations are greater for the coarser grains by a factor 3.3, which is close to the square root of the size ratio between the two size distributions. The origin of these fluctuations has not been studied and will be examined in a further investigation.

The onset M c is then determined at the departure of this linear regime. This threshold value is independent of the size of the particles, to within the precision of the measurements. However, it depends on the cell size: M c ≈ 4 kg for the big cell (open and solid squares in Fig. 6) and M c ≈ 12 kg for the small cell (open and solid circles). This size dependency is in accordance with equation (5), which predicts that the mass threshold scales with the inverse square of the size cell.

Sliding instability

On increasing the loading mass past the onset of the stickslip instability, we reach the second instability where huge rearrangements (sliding) occur. This behaviour is size dependent. In Figure 7, the experiments corresponding to the points above the dashed line present sliding. The mass thresholds M * to obtain this huge sliding corresponds to the same amplitude of variation δµ * ≈ 1.4 • 10 -3 of the coefficient of friction for both sizes of grain and both sizes of cell.

A qualitative explanation for this second instability, given in [12], is that the rearrangements occur as soon as the sliding velocity exceeds a critical value. This point has been checked by evaluating the maximum of the sliding rotational velocity ω * max at the threshold M = M * for each series of experiments. Assuming sinusoidal oscillations it can be expressed as Here Γ * and Ω * are the apparent elastic torque constant and the pulsation of the oscillations at threshold. Γ has been determined using the slope observed when restarting shear after a period of rest (as presented in the next paragraph). As in a previous study with a 40-63 µm silicagel [12], it is found that the sliding instability occurs when the sliding rotational velocity reaches a same critical value of 4.7 ± 0.6 • 10 -4 rad/s. This sliding instability is the ultimate permanent shearing regime observed. When the loading mass M is much higher than M * , the powder bed reacts as a regular succession of stick phases and sliding phases.

ω * max = ω + 2M * g R 3 o -R 3 i 3Γ * (R 2 o -R 2 i ) δµ * Ω * . (11)

Stop-and-go shear experiments

We now focus on the description of the force response to an interruption followed by a restarting of the shear. Figure 8 shows an experimental time recording of the shear stress τ and sample height h in the case of the coarse particles. In this experiment, the rotation rate is ω = 3 • 10 -4 rad/s for t < 0 s (region I), and is stopped between t = 0 s and t = 65 s (ω = 0 rad/s; region II) and set again to ω = 3 • 10 -4 rad/s for t 65 s (region III).

The region I is the permanent regime. The shear stress oscillates around 5.85 kPa. In this regime, the height of the sample decreases linearly with time by 0.12 ± 0.01 µm/s due to the extrusion of the particles past the cover.

During the hold time (region II), it can be observed that there is a creep response, and a peak in the shear stress curve when starting the shear rotation again (beginning of region III). These two effects are successively described in the two next paragraphs.

Creep during stop

During the hold time, the shear stress and the height of the powder bed decrease. This relaxation is logarithmic and the time and the coefficient of friction may be fitted by equation (8). Such a fit is shown in Figure 8 as a curve made of circles. It leads to a value of the coefficient a which is found to be of the order of 10 -2 , as in solid friction [1]. Furthermore this value appears to be slightly dependent on the loading mass, as can be observed in Figure 9, which includes the values extracted by the fit for different normal loads. The white points are for the fine particles (5-20 µm) and the black points for coarse particles (63-200 µm) for both cell sizes. The measurement of a for the fine powder (5-20 µm) is less easy since, as illustrated in Figure 10, the coefficient is less the denser is the powder. In this figure, the upper points have been obtained by increasing the loading mass and the lower points have been obtained by decreasing the loading mass after a 12-hour hold at M = 30 kg. The coefficient a measured in the dense state is similar to the one measured with the coarser powder (Fig. 9).

These experimental observations suggest that a, the coefficient of logarithmic dependence of the coefficient of friction with the sliding velocity, is a function of the packing fraction of the powder. For large particles, the particle bed quickly attains its state of maximal packing fraction and the relaxation response is unique. For the fine powder, the kinetics of rearrangements to attain the maximal packing fraction are slower and it is possible to observe the relaxation response of a loose sample.

The other point is that a depends on the loading mass M , and not on the normal stress σ. This feature is in accordance with the Da Vinci-Amontons observation that the friction force is independent of the apparent area of contact.

Resuming shear

The increase

When the rotation of the lower part at the angular velocity ω is restarted, we observe a linear increase in the shear torque (beginning of region III in Figure 8; a detailed view of that region is shown in Fig. 13a below). The slope gives the apparent elasticity Γ of the system which appears to be dependent on the loading mass. This is illustrated in Figure 11, which includes the experimental results for the two particle size cuts and the two cell sizes. This elasticity Squares: annular shear cell of 5 to 10 cm radii. Circles: smaller cell of 3 to 6 cm radii. The two bars scale, respectively, the λ 2 and λ 3 ratio of the surface and bulk term in equation ( 12), which should be respected, between the two cells.

has three origins and can be written as 1

Γ = 1 Γ apparatus + 1 Γ bulk + 1 Γ surface . (12)
Here the contribution of the apparatus does not depend on the loading mass.

Γ bulk = µ Lamé π(R 4 o -R 4 i )/2H
is the elastic contribution of the whole assembly of particles whose apparent shear modulus µ Lamé may depend on the loading. This term scales with λ 3 , where λ is the ratio between the linear size of the big and the small cell. λ = 3/5 in these experiments. Since the elastic torque is measured after a shear and relaxation experiment, we expect that a shear zone has been created, leading to a surface contribution for the elasticity:

Γ surface = kπ(R 2 o -R 2 i )
, where the stiffness k depends on the loading. This surface term scales with λ 2 . From Figure 11, we deduce (from the ratio between the measures with the big and small cells) that the surface term is predominant and grows with M 1/2 . This square-root dependency is different from the linear relationship predicted and observed in solid friction [14], different from the M 1/3 prediction for Hertzian contact, but similar to the shear-modulus-normal-force dependency observed in bulk materials [15]. Various interpretations for this M 1/2 have been proposed [16,17].

The peak

The linear increase of the shear force when restarting the flow ends with a peak of force before attaining the plateau value, as in solid friction. This maximum of force increases logarithmically with the time of creep T as in equation (10) (see insert in Fig. 12). It is in accordance with other experimental observations [18,[START_REF] Lubert | Handbook of Powder Technology[END_REF] which found that this logarithmic increase vanishes if the shear stress is not applied during the waiting time. The experimental coefficient b is plotted in Figure 12 versus the loading mass. We observe that b has the same mass dependency as does a and does not depend on the particle size of the silicagel.

Comparing Figures 10 and12 suggests that, for the 5-20 µm powder in the loose state, the coefficient a is greater than b, so there is no stick-slip. This is corroborated by the experimental observations. After filling the cell and beginning the shear at high normal load, there is a waiting time before the occurrence of the stick-slip oscillations. The powder bed compacts and a diminishes till the onset of instability given by equation ( 5) is attained.

With this last measurement, and using the stick-slip onset of instability (Eq. ( 5)), we find that the characteristic length of the Dieterich-Rice-Ruina model is D = 0.6 ± 0.3 µm for the 5-20 and 60-200 µm silicagel. This length is similar to that previously observed for a 40-63 µm silicagel [12] and similar to the D lengths reported for solid friction [1]. This indicates that the particles do not play the role of roughness in solid friction. There remains to investigate the behaviour of a sub-micronic silicagel powder, whose size is less than D.

The instability

When restarting the flow with these silicagel powders, we notice a new instability not observed in solid friction. The granular bed slips and compacts when the hold time is lower than a critical value T * . This phenomenon is illustrated by Figure 13a for a 60-200 µm silicagel powder sheared in the cell of 5 to 10 cm with M = 5 kg. For hold times T below 1 min, there is a slip which decreases the coefficient of friction by 0.15. This slip disappears for hold times greater than 1 min.

These slip events look like the sliding observed during permanent shearing at loading M > M * . They are associated with a settling of the powder bed (Fig. 13b), which is recovered after 5 s of shear (see upper curve in Fig. 8), that is after 100 ± 25 µm of tangential displacement. If the hold time T is greater than T * , the powder sample dilates during the resumption of shear and recovers the height lost during the creep.

This critical time T * increases with the normal loading and depends on the cell size and the particle sizes, as shown in Figure 14. For the same normal loading, T * is lower in the smaller cell for the two particle sizes. T * also depends on the size of the particles. In this case, the tendency is less clear since for the big cell (5 to 10 cm diameters) the smaller particles lead to smaller T * values, and the contrary for the small cell (3 to 6 cm diameters). However, Figure 14, which displays the mass dependency of T * , can be seen to be quite similar to Figure 2, which displays the mass dependency of δµ. This emphasises the fact that the hold time T * necessary to avoid sliding, increases sharply when the loading approaches this onset of sliding instability. To be more precise, the data of Figure 14 have been plotted in Figure 15 versus the reduced load mass M /M * . The data then collapse into two curves, depending on the particle size. Such a collapse does not occur when the load mass is rescaled by M c , the mass at the onset of shear instability. A link between the two sliding instabilities -one occurring when the loading mass overcomes a critical value (M > M * ), the other one when restarting a shear flow after a short period of rest (T < T * )-is thus demonstrated.

This correlation between the two sliding instabilities may come from the fact that both occur when the sliding velocity exceeds a critical value. In Figure 13a, the dashed line gives the slope of the temporal µ curve corresponding to the rotational sliding velocity of 4.7 • 10 -4 rad/s leading to the sliding instability in permanent shearing experiments. We observe in this figure that the longer is the hold time, the lower is the sliding velocity after the peak of friction. This behaviour, confirmed by the other experiments, awaits explanation since the numerical simulations of hold experiments using the Dieterich-Rice-Ruina model lead to the opposite conclusion.

The other remaining issue is to explain why T * diverges at a lower value of the reduced mass M/M * for the finer powder. A plausible hypothesis is that the texture acquired by the shear band is different for the two powders due to the inter-particles forces. These can be considered to be negligible for the coarser particles, but sufficient to induce rearrangements for the finer particles. When removing the cover plate after experiments with the fine powder, a sharp shear zone is observed between two blocks that act like a solid. It is not feasible to access the shear zone for the free-flowing coarser particles; investigation and confirmation of the hypothesis may best be made by numerical simulations.

Conclusion

The friction behaviour of silicagel powders of different particle sizes has been experimentally investigated in an annular shear cell, of inner and outer size ratio equal to 2, and thus small enough to allow this device to be considered as being analogue to a linear shear cell. The results may be arranged into two categories.

Size-independent behaviour

These results may be described on the basis of the Dieterich-Rice-Ruina model for solid friction. The coefficient a of logarithmic variation of the friction coefficient with the sliding velocity and the coefficient b of logarithmic variation of the friction coefficient with the ageing time are found to be of the same order of magnitude as in solid friction. They both have very a slight dependence on the normal loading force, which has not been reported for solid friction. Another difference with solid friction is that the creep response depends on the packing fraction of the granular media.

The other characteristic measured is the apparent shear elasticity of this granular media, which appears to be a surface and not a bulk shear elasticity, with a squareroot dependency on the normal load.

These results well fit the onset of stick-slip instability observed with these particles, for a renewal length of the population of contacts D of order a micron. This leads to the conclusion that this microscopic length scale is independent of the size of the particles, at least when their size is greater than D.

Size-dependent behaviour

Beyond these analogies, we have demonstrated a sliding instability which takes place after a short-time creep period. This behaviour is compared with the sliding instability observed in the permanent shearing regime for normal loading high enough to ensure the stick-slip instability. In both cases, the sliding occurs when the sliding velocity exceeds a critical value. Contrarily to the onset of instability, this behaviour is size dependent. More theoretical investigation is now needed to identify the origin of this rate-dependent instability and to build a granular friction model accounting for this behaviour.

Fig. 1 .

 1 Fig.1. Percentage relative difference between the normal stress at the onset of instability in an annular cell and the onset of instability in a linear geometry (continuous lines) for two ratios of outer to inner radii. Same percentage relative difference for the pulsation at the onset (dotted lines).

Fig. 2 .

 2 Fig. 2. Sketch of the experimental set-up.

Fig. 3 .

 3 Fig. 3. a) SEM images of the silicagels used. Left: 5-20 µm. Right: 60-200 µm. b) Cumulative size distribution of the silicagels used as obtained by laser diffraction in ethanol.

Fig. 4 .

 4 Fig. 4. Mean shear stress in steady sliding versus the normal stress. Open symbols: 60-200 µm silicagel. Solid symbols: 5-20 µm silicagel. Squares: annular shear cell of 5 to 10 cm radii. Circles: smaller cell of 3 to 6 cm radii.

Fig. 5 .

 5 Fig. 5. Shear at constant rotational velocity (ω = 3 • 10 -4 rad/s) of a 60-200 µm silicagel powder with a normal load of 4 and 6 kg in the annular shear cell of 5 to 10 cm radii. Two upper curves: shear stress versus time. Two lower curves: height of the granular bed versus time.

Fig. 6 .

 6 Fig. 6. Fluctuations of the shear force versus the loading mass. Open symbols: 60-200 µm silicagel. Solid symbols: 5-20 µm silicagel. Squares: annular shear cell of 5 to 10 cm radii. Circles: smaller cell of 3 to 6 cm radii. Dashed lines: indication of slope 1. The vertical arrows point to the onset of stick-slip instability (M c = 4 kg for the annular cell of 5 to 10 cm radii; M c = 12 kg for the annular cell of 3 to 6 cm radii). The horizontal arrow points to the limit of sensitivity of the force sensors.

Fig. 7 .

 7 Fig. 7. Amplitude of the fluctuations of the coefficient of friction expressed as a percentage, versus the loading mass. Open symbols: 60-200 µm silicagel. Solid symbols: 5-20 µm silicagel. Squares: annular shear cell of 5 to 10 cm radii. Circles: smaller cell of 3 to 6 cm radii. The dashed line indicates the slippage threshold. The vertical arrows point to the onset of stick-slip instability (M c = 4 kg for the annular cell of 5 to 10 cm radii; M c = 12 kg for the annular cell of 3 to 6 cm radii).

Fig. 8 .

 8 Fig. 8. Simultaneous recording of the shear stress (lower curve) and height of the granular bed (upper curve) during a stop-andgo experiment of hold duration T = 65 s. The experiment is performed with the 60-200 µm silicagel and a load mass of 5 kg (2.2 kPa). In regions I and III, ω = 3 • 10 -4 rad/s. In region II, ω = 0 rad/s. Circles: fit of the creep with equation (8). The slope indicated in region I is the extrusion observed during permanent shear at ω = 3 • 10 -4 rad/s.

Fig. 9 .

 9 Fig. 9. Coefficient a versus the load mass. Open symbols: 60-200 µm silicagel. Solid symbols: 5-20 µm silicagel. Squares: annular shear cell of 5 to 10 cm radii. Circles: smaller cell of 3 to 6 cm radii.

Fig. 10 .

 10 Fig.10. Coefficient a versus the load mass for the 5-20 µm silicagel. Crossed circles: experiments performed by increasing the load mass from 100 g to 30 kg. Solid symbols: same points as in Figure9, obtained by decreasing the load mass from 30 kg to 100 g.

Fig. 11 .

 11 Fig. 11. Elastic torque constant Γ versus the load mass. Open symbols: 60-200 µm silicagel. Solid symbols: 5-20 µm silicagel.Squares: annular shear cell of 5 to 10 cm radii. Circles: smaller cell of 3 to 6 cm radii. The two bars scale, respectively, the λ 2 and λ 3 ratio of the surface and bulk term in equation (12), which should be respected, between the two cells.

Fig. 12 .

 12 Fig. 12. Coefficient b versus the load mass, measured using the annular cell of 5 to 10 cm radii. Open symbols: 60-200 µm silicagel. Solid symbols: 5-20 µm silicagel. Insert: maximum of the coefficient of friction versus the hold time for the 60-200 µm silicagel with M = 500 g.

Fig. 13 .

 13 Fig. 13. a) Peak of friction after a period of rest ranging from 1 s to 18 h. The dashed line gives the slope corresponding to a sliding velocity of 4.7 • 10 -4 rad/s. b) Height of the powder bed for the experiments of Figure 5b. The zero level is set at the beginning of the resumption of shear.

Fig. 14 .

 14 Fig. 14. Minimal hold time to avoid the sliding instability when resuming the shear versus the loading mass. Open symbols: 60-200 µm silicagel. Solid symbols: 5-20 µm silicagel. Squares: annular shear cell of 5 to 10 cm radii. Circles: smaller cell of 3 to 6 cm radii.

Fig. 15 .

 15 Fig. 15. Minimal hold time to avoid the sliding instability when resuming the shear versus the loading mass M divided by M * . Open symbols: 60-200 µm silicagel. Solid symbols: 5-20 µm silicagel. Squares: annular shear cell of 5 to 10 cm radii. Circles: smaller cell of 3 to 6 cm radii.

We thank Julien Robert, student from the IUT of Nancy for his precious help in the experiments, John A. Dodds, Pierre Labourt-Ibarre and Rhodia for their constant interest and financial support.