R Ravaud

G Lemarquand
email: guy.lemarquand@ieee.org

V Lemarquand

Magnetic Pressure and Shape of Ferrofluid Seals in Cylindrical Structures

Keywords: ferrofluid seals, analytical approach, permanent magnets, magnetic pressure

This paper presents a three-dimensional analytical model for studying the shape and the pressure of ferrofluid seals in totally ironless structures. This three-dimensional analytical approach is based on the exact calculation of the magnetic field components created by ring permanent magnets whose polarizations are either radial or axial. We assume that the ferromagnetic particles of the ferrofluid are saturated.

Moreover, the permanent magnets (Neodymium Iron Boron) used in the considered applications create a magnetic field which is much higher than the magnetic field created by the magnetic particles in the ferrofluid so the latter is neglected. Nevertheless, the static behavior of the ferrofluid seal depends on both the magnetic field produced by the permanent magnets and the saturation magnetization of the ferrofluid particles. Furthermore, the accurate knowledge of the ferrofluid seal shape as well as the magnetic pressure inside the ferrofluid seal is very useful for the design of devices using both permanent magnets and ferrofluid seals. It is emphasized here that our structures are completely ironless and thus, there are no iron-base piston for these structures. Then, this paper makes a review of the main structures using ring permanent magnets and ferrofluid seals. For each ironless structure, the shape and the pressure of the ferrofluid seals are determined.

I. INTRODUCTION

The interest of using ferrofluid in the new technologies increases clearly with the understanding of its chemical and physical properties [START_REF] Choi | Simulation of hydrostatical equilibrium of ferrofluid subject to magneto-static field[END_REF]- [START_REF] Raikher | Orientational dynamics in magnetic fluids under strong coupling of external and internal relaxations[END_REF]. Ferrofluids are in fact a unique class of material that have been discovered in the 1960s. A ferrofluid is a stable colloidal suspension of sub-domain magnetic nano particles in a liquid carrier. These magnetic particles are coated with a stabilizing dispersing agent that prevents generally particle agglomeration. Such a property is very useful because in some applications (as the ones presented in this paper), the magnetic fields used are very intense. Besides, it is noted that we discuss the utility of using intense magnetic fields in this paper. Some industries use ferrofluids for realizing airtightness seals or in audio engineering to decrease the temperature of coils in loudspeakers.

As this choice allows us to improve greatly these devices, a care study of the shape of a ferrofluid seal must be carried out because this property is directly linked to its magnetic energy and its mechanical properties. Their properties prove to be useful and efficient in various engineering areas such as damping systems [START_REF] Bajkowski | A model for a magnetorheological damper[END_REF], heat transfers, motion control systems, sensors [START_REF] Rosensweig | Ferrohydrodynamics[END_REF] [START_REF] Doutres | On the use of a loudspeaker for measuring the viscoelastic properties of sound absorbing materials[END_REF]. They also have very promising medical applications [START_REF] Li | Cfd study on the magnetic fluid delivering in the vessel in high-gradient magnetic fields[END_REF] [START_REF] Park | New design of the magnetic fluid linear pump to reduce the discontinuities of the pumping forces[END_REF]. However, they are more commonly used in bearings and seals for rotating devices.

The pioneering work regarding the ferrofluid lubrication was done by Tarapov [START_REF] Tarapov | Movement of a magnetizable fluid in lubricating layer of a cylindrical bearing[END_REF]. Indeed, he considered a plain journal bearing lubricated by ferrofluid and submitted to a non-uniform magnetic field. Since then, numerous studies in the field of ferrofluid dynamic bearings have been carried out. Both static and dynamic characteristics of these bearings have been studied theoretically [START_REF] Walker | Ferrohydrodynamics thrust bearings[END_REF]- [START_REF] Cunha | A new boundary integral formulation to describe three-dimensional motions of interfaces between magnetic fluids[END_REF]. Moreover, recent trends in the ferrofluid lubrication applications are described and discussed [START_REF] Shah | Ferrofluid squeeze film in a long bearing[END_REF]- [START_REF] Meng | An analysis on the magnetic fluid seal capacity[END_REF]. Ferrofluids are also used in electrodynamic loudspeakers in which they fulfill several functions: they ensure the airtightness, they play a heat transfer part and they work as a radial bearing. Furthermore, a ferrofluid seal can replace the loudspeaker suspension and leads to a better linearity of the cone movement [START_REF] Rosensweig | Study of audio speakers containing ferrofluid[END_REF]- [START_REF] Ravaud | Mechanical properties of a ferrofluid seal: three-dimensional analytical study based on the coulombian model[END_REF]. This paper presents a three-dimensional analytical model for studying the shape and the static pressure of ferrofluid seals submitted to intense magnetic fields produced by ironless structures. We use the static magnetic pressure for studying the shape of the ferrofluid seals. Our three-dimensional analytical approach is based on the exact calculation of the magnetic field components created by ring permanent magnets whose polarizations are either radial or axial. Moreover, the approach taken is based on the coulombian model of permanent magnets. Consequently, it gives a great flexibility for the review of the different structures studied in this paper.

In addition, it must be emphasized here that our experimental conditions in which the ferrofluid is used differ from the other experiments proposed by the authors. Indeed, our applications (ironless loudspeakers, bearings) require very high magnetic fields. Consequently, we think that the most important parameter representing the pressure in the ferrofluid seal must be determined carefully. This term is often simplified in many studies because it is difficult to determine. However, this term is not only the most important parameter that must be determined carefully but this is also probably the only term that allows us to study the ferrofluid seal shape. The gravitational energy or the surface tension energy do not influence a ferrofluid seal shape used in these conditions. This is why we think that it is more useful to obtain an exact three-dimensional analytical expression of the magnetic pressure by only considering the preponderant term in the equation of equilibrium in the ferrofluid seal rather than using several simplifying terms whose influence is questionable in our applications.

As the ferrofluid considered in this paper (Ferrotec APG05) has a saturation magnetization of at least 32 kA/ m and the magnetic field is greater than 400kA/m, the value of this pressure is higher than 12800 N/m 2 . On the other hand, the surface tension exists. But when the values of both the surface tension coefficient, A, (A equals 0.0256kg/s 2 for the used ferrofluids) and the radius of curvature are considered, the effect of the surface tension can be omitted: this latter does not deform the free boundary surface.

II. THREE-DIMENSIONAL ANALYTICAL MODEL

First, we present the calculation tools to study the different configurations using one or several ring permanent magnets. As the goal of this paper is to determine the shape of the ferrofluid seals for different magnet configurations, the magnetic field created by the magnets must be calculated. Then, the way of obtaining the seal shape is discussed.

It is emphasized here that we do not use a two-dimensional model for calculating the magnetic field created by these arc-shaped permanent magnets because such an approach is not accurate in the near-field [START_REF] Ravaud | Analytical calculation of the magnetic field created by permanent-magnet rings[END_REF]. This is the reason why we use only a three-dimensional approach. 2. Arc-shaped permanent magnet whose polarization is axial: the upper surface is charged with the magnetic pole surface density +σ * and the lower surface is charged with the magnetic pole surface density -σ * , the inner radius is r in , the outer one is rout

A. Magnetic field created by the ring magnets

The magnetic field created by the ring magnets is determined by using the coulombian model. The magnets are assumed perfect (flawless) with a perfectly uniform magnetization. Consequently, each permanent magnet is represented by two charged surfaces. In the case of a permanent magnet whose polarization is radial, the magnetic poles are located on both curved surfaces of the ring and the magnetic pole surface density is denoted σ * (Fig. 1). The rings are assumed radially thin enough to neglect the magnetic pole volume density related to the magnetization divergence. In the case of a permanent magnet whose polarization is axial, the magnetic pole surface density σ * is located on the upper and lower faces of the ring (Fig. 2). The magnetic field components have been entirely determined in previous papers [START_REF] Ravaud | Analytical calculation of the magnetic field created by permanent-magnet rings[END_REF] [START_REF] Ravaud | Discussion about the analytical calculation of the magnetic field created by permanent magnets[END_REF]. Since the azimuthal component equals zero because of the cylindrical symmetry [START_REF] Ravaud | The three exact components of the magnetic field created by a radially magnetized tile permanent magnet[END_REF], only two components must be calculated. Therefore, the magnetic field depends on both radius r and altitude z.

B. Potential energy and magnetic pressure of the ferrofluid seal

The potential energy of the ferrofluid is very important parameter for the study of ferrofluid seals.

Its calculation requires several assumptions, generally admitted by the authors. First, the ferromagnetic particles of the ferrofluid are assumed to be small saturated spheres which can be freely oriented in all the directions of space. Thus, all the particles of the saturated ferrofluid are aligned with the permanent magnet orienting field. In addition, when submitted to these intense magnetic fields, the ferrofluid does not modify the field created by the permanent magnets and the field created by the ferrofluid itself is omitted. Furthermore, the aggregation in chains of the ferrofluid particles is omitted [START_REF] Ivanov | Ferrofluid aggregation in chains under the influence of a magnetic field[END_REF]. It is noted that, when the device is at laboratory temperature and at rest, the aggregation phenomenon is observed. In short, the magnetic pressure is expressed as follows:

p m (r, z) = µ 0 M s .H(r, z) = µ 0 M s H r (r, z) 2 + H z (r, z) 2 (1)
where both magnetic field components H r (r, z) and H z (r, z) are analytically calculated [START_REF] Ravaud | Analytical calculation of the magnetic field created by permanent-magnet rings[END_REF]. Thus, the potential energy of the ferrofluid seal, E m , can be deducted :

E m = - (Ω) H(r, z).(µ 0 M)dV = - (Ω) p m (r, z)dV (2)
where (Ω) is the ferrofluid seal volume. As a remark, the magnetic pressure is given in N/m 2 and the potential energy in J. The analytical expression of the magnetic pressure depends on the number of ring permanent magnets used in the ironless structure. For example, if we consider a structure made of two ring permanent magnets radially magnetized, with opposite magnetizations, the magnetic pressure p m (r, z)

inside the ferrofluid seal is given by [START_REF] Raikher | Orientational dynamics in magnetic fluids under strong coupling of external and internal relaxations[END_REF].

p m (r, z) = µ 0 M s H r,2 (r, z) 2 + H z,2 (r, z) 2 (3)
where M s is the intensity of magnetization of a magnetic particle. The radial component H r,2 (r, z) of the magnetic field created by the two permanent magnets is defined by ( 4).

H r,2 (r, z) = σ * 2πµ 0 (ς(u 1 ) -ς(u 2 )) (4)
with

102 ς(u) =    2i(1 + u) d(-1+u) c+e1+du (-(a 1 d + b 1 (c + e 1 ))) F * i sinh -1 [ √ -c+d-e1 √ c+e1+du ], c+d+e1 c-d+e1 d √ -c + d -e 1 e 1 d(1+u) c+e1+du √ 1 -u 2    +    2i(1 + u) d(-1+u) c+e1+du (b 1 c -a 1 d)Π * e1 c-d+e1 , i sinh -1 [ √ -c+d+e1 c+e1+du ], c+d+e1 c-d+e1 d √ -c + d -e 1 e 1 d(1+u) c+e1+du √ 1 -u 2    +    2i(1 + u) d(-1+u1) c+e2+du1 (-(a 2 d + b 2 (c + e 2 ))) F * i sinh -1 [ √ -c+d-e2 √ c+e2+du ], c+d+e2 c-d+e2 d √ -c + d -e 2 e 2 d(1+u) c+e2+du √ 1 -u 2    +    2i(1 + u) d(-1+u) c+e2+du (b 2 c -a 2 d)Π * e2 c-d+e2 , i sinh -1 [ √ -c+d+e2 c+e1+du ], c+d+e2 c-d+e2 d √ -c + d -e 2 e 2 d(1+u) c+e2+du √ 1 -u 2    -    2i(1 + u) d(-1+u) e3+du ((a 3 d -b 3 e 3 )) F * i sinh -1 [ √ -d-e3 √ e3+du ], -d-e3 d+e3 d √ -d -e 3 (-c + e 3 ) d(1+u) e3+du √ 1 -u 2    -    2i(1 + u) d(-1+u) e1+du (b 3 c -a 3 d)Π * -c+e3 d+e3 , i sinh -1 [ √ -d+e3 e3+du ], -d+e3 d+e3 d √ -d -e 3 (-c + e 3 ) d(1+u) e3+du √ 1 -u 2    -    2i(1 + u) d(-1+u) e4+du (a 4 d -b 4 e 4 ) F * i sinh -1 [ √ -d-e4 √ e4+du ], -d+e4 d+e4 d √ -d -e 4 (c + e 4 ) d(1+u) e4+du √ 1 -u 2    -    2i(1 + u) d(-1+u) e4+du (b 4 c -a 4 d)Π * -c+e4 d+e4 , i sinh -1 [ √ -d-e4 e4+du ], -d+e4 d+e4 d √ -d -e 4 (-c + e 4 ) d(1+u) e4+du √ 1 -u 2    (5)
The axial component of the magnetic field created by the two ring permanent magnets is given by [START_REF] Doutres | On the use of a loudspeaker for measuring the viscoelastic properties of sound absorbing materials[END_REF]. [START_REF] Rosensweig | Ferrohydrodynamics[END_REF] where K * [m] is given in terms of the incomplete elliptic integral of the first kind by ( 7)

103 H z,2 (r, z) = σ * πµ 0   -r in K * -4rrin (r-rin) 2 +z 2 (r -r in ) 2 + z 2 + r in K * - 4rrin (r-rin) 2 +(z-h) 2 (r -r in ) 2 + (z -h) 2   - σ * πµ 0   r in K * 4rrin (r-rin) 2 +z 2 (r -r in ) 2 + z 2 -r in K * 4rrin (r-rin) 2 +(z+h) 2 (r -r in ) 2 + (z + h) 2   (6) Parameters a 1 r in rz b 1 -r 2 in z c r 2 + r 2 in d -2rr in e 1 z 2 a 2 -r in r(z -h) b 2 r 2 in (z -h) e 2 (z -h) 2 a 3 r in rz b 3 -r 2 in z e 3 r 2 + r 2 in + z 2 a 4 r in r(-z -h) b 4 -r 2 in (-z -h) e 4 r 2 + r 2 in + (z + h) 2
K * [m] = F * [ π 2 , m] (7)
F * [φ, m] is given in terms of the elliptic integral of the first kind by ( 8):

F * [φ, m] = θ=φ θ=0 1 1 -m sin(θ) 2 dθ (8)
Π * [n, φ, m] is given in terms of the incomplete elliptic integral of the third kind by ( 9)

Π * [n, φ, m] = φ 0 1 (1 -n sin(θ) 2 ) 1 -m sin(θ) 2 dθ (9)
The parameters used in [START_REF] Rosensweig | Ferrohydrodynamics[END_REF] are defined in Table II-B. Moreover, when we input (4) in Mathematica, we have to take the real part of H r (r, z) because of the noise calculus. What's more, the parameter i used in ( 5) is the imaginary number (i 2 = -1).

C. Parameters used for describing the ironless structures

The further calculations are presented for magnets with 1T remanent magnetization in order to normalize the results. In fact, the magnets used in the prototypes are Neodymium Iron Boron ones for which the remanent magnetization can reach 1.5T. Furthermore, the interesting regions of space are the ones where the ferrofluid goes. With these 1T normalized magnets and for the proposed configurations, the magnetic field intensity there is greater than 400 kA/m. Of course, all the field values are proportional to remanent magnetization value. Then, we use commercial ferrofluids, either from the company Ferrotec or Ferrolabs

. Such ferrofluids have a saturation magnetization, M s , smaller than 32 kA/m and a particle concentration below 5,5 %. It is to be noted that for bearing or loudspeaker applications, a great bearing effect is seeked which requires high saturation magnetizations. Therefore, the magnetic field, H, created by the permanent magnets is far higher than the ferrofluid critical field [START_REF] Matthies | Numerical simulation of normal-field instability in the static and dynamic case[END_REF]. So, the ferrofluid is totally saturated and its magnetization is denoted M s .

Color plots of the magnetic pressure will be done in the further sections in order to visualize the characteristics of the studied configurations. The warmest color (red) is used for the regions of space where the magnetic pressure is the greatest and the coldest color (blue) for the ones where the magnetic pressure is the smallest. Moreover, the same color corresponds to the same value throughout this paper: the corresponding scale is given in Table II in which the magnetic pressure and the related magnetic field values are given.

III. SHAPE OF THE FERROFLUID SEAL IN VARIOUS MAGNETIC FIELDS

A. Description of the generic structure

Figure 3 shows the generic structure of all the devices studied. They consist of three outer stacked rings, of an inner non-magnetic piston and of ferrofluid seals. The piston is radially centered with the rings. The rings' inner radius, r in , equals 25 mm and their outer radius, r out , equals 28 mm. The rings can be either made out of permanent magnet -as here the middle ring-or out of non-magnetic material -like the upper and lower rings-. For each configuration, the ferrofluid seals are located in the air gap between the piston and the rings; their number, their position as well as the magnetization direction of r r out 0 Z R in Fig. 3. Device structure: three outer rings (permanent magnet or non-magnetic) radially centered with an inner non-magnetic piston and ferrofluid seals in the air gap between the rings and the piston. r in = 25mm, rout = 28mm.

the ring magnets will be discussed.

For each configuration, we have also represented the radial component of the magnetic field created by the ring permanent magnets: indeed, this parameter can also be optimized by changing the ring dimensions in electric machines.

It is assumed throughout this paper that the forming shapes of ferrofluid are considered to be proportional to the magnetic pressure. Such an assumption is possible because the magnetic field produced by the ring permanent magnets is very high.

The coordinate system (0, u r , u θ , u z ) is placed into the centre of the magnetic system in

B. Structures using one ring magnet

The first structure considered corresponds exactly to the configuration shown in Fig. 3. All the rings have the same square cross-section whose side equals 3 mm. The middle ring is a radially magnetized permanent magnet and the upper and lower rings are non-magnetic. The magnetic field created by the magnet in the air gap is calculated at a distance in length from the rings and along the Z axis which equals 0.1 mm. Moreover, its radial component H r shown in Fig. 4 is quite uniform in front of the magnet.

Two field gradients exist in front of the edges of the magnet. We can point out that the magnetic pressure calculated in the air gap between the rings and the piston is plotted on Fig. 4 as well. The ferrofluid seeks the regions of both intense field gradient and high magnetic energy. The color plot shows that, for seals thicker than 0.5 mm, the seal expands along the whole magnet height. A smaller amount of ferrofluid would give two separate seals, but they would be too thin to have interesting mechanical properties.

This result is linked to the magnet square section: were the magnet higher along Z, two separate seals would appear. It can be noted that this structure is the simplest one which can be used in electrodynamic loudspeakers. The second structure differs from the first one only in the magnetization direction which becomes axial. Figure 5 shows the structure, the calculated radial field component and the corresponding magnetic pressure. The radial component of the magnetic field is no longer uniform in front of the magnet; it shows instead a rather large gradient all over the magnet length and the non-magnetic rings. Nevertheless, the magnetic pressure deserves the same comments as in the first structure and is somewhat similar to the previous one. However, it is obvious that it differs in form from one structure to the other.

C. Structures using two ring magnets

The structures studied here derive from the preceding ones by stacking a second permanent magnet ring directly on the first one. Both ring magnets have the same dimensions and opposed magnetization directions. The radial field component and the magnetic pressure are shown in Fig. 7 when the ring magnets are radially magnetized and in Fig. 6 when the ring magnets are axially magnetized. The magnetic field in both cases results from the superposition of the single magnet fields. As a consequence, each radial magnet creates a region of uniform field in front of itself and the field directions are opposite. The field intensity in each uniform region is higher than in the single magnet structure because the leakage is decreased. In effect, three field gradients exist, and the one that appears in front of the magnets' interface is twice as high as those at the edges. From the gradient point of view, Fig. 7 can be compared with Fig. 5, and the former will prove more useful because the gradient is steeper. The axial double structure creates progressive field gradients with no peculiar interest. By contrast, the radial double structure is used to design two-coil loudspeakers. The repartition of the magnetic energy density in the structure using two ring magnets does not derive from the superposition of the ones in the structures using one ring magnet.

The main reason is due to the expression of the energy which depends on the square of the field. Both repartitions for radial and axial magnets seem alike at first sight, but the radial structure is in fact "more energetic" and the magnetic energy in it decreases slower at an increasing distance from the magnets.

Nevertheless, the maximum energy density is in front of the magnets' interface and the ferrofluid seal will be located there. We can point out that the axial length of the seal in structures using one magnet is smaller than the one in structures using two ring magnets. Furthermore, its energy density is approximately doubled for the radial magnets.

D. Structures using three ring magnets

All the structures considered now consist of three stacked ring permanent magnets whose magnetization directions undergo a 90 degrees rotation from one magnet to the neighboring one. Such configurations with a magnetization progressive rotation are related to Halbach cylinders [START_REF] Halbach | Design of permanent multiple magnets with oriented rec material[END_REF] and to their applications in the design of electrical motors [START_REF] Marinescu | Compensation of anisotropy effects in flux-confining permanent-magnet structures[END_REF], sensors [37][38] or of passive magnetic bearings [START_REF] Yonnet | Permanent magnet bearings and couplings[END_REF]. Contrary to electrical machines, the presented structures do not have a periodical magnet pattern.

Figure 8 shows the field radial component and the magnetic pressure for an assembly of a radially magnetized middle ring and two axially magnetized upper and lower rings. Moreover, the axial magnets have opposed magnetization directions. In Fig. 9, the middle ring is axially magnetized, both other rings are radially magnetized with opposed directions.

The energy density color plots show that two ferrofluid seals form in front of the magnets' interfaces and that those seals have a great magnetic energy. Consequently, they own good mechanical properties (great radial stiffness for example). The seals are rather similar in both structures. However, the magnetic field radial component shown in Fig. 8 is fairly uniform in the area in front of the middle magnet while it varies with no particularly interesting properties in Fig. 9. Therefore, the structure with the radial middle magnet is the useful one, especially for the loudspeaker design, and the field uniformity zone may be optimized.

The axial height of the middle magnet can be varied regarding the axial height of the upper and lower magnets. For example, the middle magnet is twice as high as each other magnet in Fig. 10 and half as small in Fig. 11. As a result, the length of the field uniformity area follows closely the middle magnet height. If the latter increases, the former increases too. We can say that the ratio middle of the magnet height over the upper and lower magnet height is important as well. Indeed, if it increases, the field uniformity region grows but the field intensity decreases, and inversely. It also influences the repartition of the magnetic pressure. Indeed, when the upper and lower magnets become too small, the ferrofluid can expand over their whole axial length and the seals can become fairly voluminous, depending on the ferrofluid available quantity. The field gradients are steep and the seal magnetic energy is quite high.

Consequently, the seals are well-fixed to the structure and have high mechanical performances. Inversely, when the middle magnet becomes too small, both seals gather to form a single one which expands over the whole height of the middle magnet. The energy in the seal is still high for radial thickness smaller than the ones presented in the previous case.

E. Discussion

So far, different magnet assemblies have been presented. Each assembly creates magnetic fields and magnetic field gradients so as to trap and fix ferrofluid to make seals. Their characteristics will be discussed and compared now.

The structures considered are either very simple or more complex. The complexity can be characterized by the number of separate parts necessary to build devices. We can say that from this point of view, the single magnet structures are the simplest ones. The complexity also lies in the magnets' magnetization direction. Indeed, an axial magnetization is easy to achieve for ring magnets whereas a radial one is technically more difficult. Thereby, ring magnets whose polarization is radial are more expensive than the ones whose polarization is axial. Nevertheless, some magnet manufacturers begin to deal with radial magnetization and to sell such products.

The utility of a structure depends on the intended applications. For example, a device can be designed solely to make ferrofluid seals with good mechanical properties. It can also be designed to create a radial magnetic field or to enable a ferrofluid seal to form as it is required in loudspeakers.

A high magnetic field gradient and a high magnetic pressure are required to achieve robust seals. From the preceding sections, we deduce that multi-magnet structures are more efficient than single magnet structures in creating high field gradients. Furthermore, the magnetic pressure is higher in the air gap near the magnets.

It can be emphasized here that the shape of the ferrofluid seal in the device depends on both the magnetic pressure and the ferrofluid volume. It can be noted that this latter one is related to the air gap size. The air gap exists because the inner part has to move, in rotation or in translation, inside the outer part without rubbing against it. The air gap size depends on mechanical considerations, such as the practical machining and the related possible tolerances. The color plots show clearly that the theoretical size of the air gap which allows an efficient ferrofluid seal depends on the magnetic structure dimensions.

For example, the structures shown in Fig. 11 or in Fig. 8 lead to seals thinner than the ones shown in Fig. 7 because the region of high magnetic pressure is radially thinner.

Ferrofluid seals are robust when their capacity is high; this means that they can resist to quite high axial pressures because they exert themselves an opposing force. We can say that this force is related to the energy variation of the ferrofluid seal. The higher the energy varies with the axial displacement, the more important the force is.

Moreover, a great force is attained with thin seals whose energy is high and varies extremely with axial displacements. Therefore, the smaller the air gap is, the more favorable the mechanical structure to build ferrofluid seals is.

The color plots can help optimizing both the number and the shape of ferrofluid seals for given ferrofluid volumes.

For example, the plots shown in Figs. 10 and 11 point out that two seals appear in these three magnet structures for a small volume of ferrofluid. Moreover, they show that the increase in the ferrofluid volume results in the growth of the seals which eventually join and can form a single big one.

The color plots can also be considered as tools to determine the adequate quantity of ferrofluid which will confer the intended mechanical properties on the seals. Furthermore, this study shows that some structures have greater performances than others for given magnet and ferrofluid volumes.

The requirements for the loudspeaker applications are different. Indeed, the magnetic structure is expected to fulfill several purposes, namely the creation of a radial uniform magnetic field as well as the creation of field gradients. Besides, a loudspeaker coil moves thanks to the magnetic field. The linearity performances are related to the field uniformity and the efficiency is proportional to the field level. Furthermore, the ferrofluid seal is used to ensure the airtightness, to transfer the heat from the moving part to the steady one, to work as a radial bearing and to replace the loudspeaker suspension.

Thereby, it contributes to the improvement of the loudspeaker linearity. The air gaps in loudspeakers are generally 0.1 to 0.5 mm wide: this explains why the presented magnetic field calculations have been carried out along a path at a distance in length which equals 0.1 mm from the ring magnets. Thinner air gaps would be more advantageous for ferrofluid seals but are not technically possible because of the machining tolerances or the recognition of the moving part thermal dilation. Therefore, the structures shown in Figs. 3, 7 and 10 are the ones that are the more useful for loudspeakers. The three magnet structures are the more complex, but they offer the possibility of a multi-criteria optimization. We can say that this optimization is enabled and simplified by the analytical formulations of both the magnetic field and the magnetic energy density. Eventually, such structures show very good performances.

IV. CONCLUSION

This paper has presented a three-dimensional analytical model for studying the shape of the static pressure of ferrofluid seals subject to intense magnetic fields produced by permanent magnet ironless structures. Our analytical approach has been applied to well-known structures composed of one, two or three stacked ring permanent magnets whose polarization is either radial or axial. Our analytical appraoch allows us to predict easily the shape and the pressure of ferrofluid seals. Then, we have discussed the interest of using simple or complex structures with ferrofluid seals. The single magnet structures are the simplest ones but the less efficient. The use of several magnets enhances both magnetic and mechanical properties: when the created magnetic field is doubled, the magnetic energy is quadrupled. Furthermore, three magnet structures enable to create one or two seals which have a high energy. The structures can also create a high uniform magnetic field over defined regions as well as efficient ferrofluid seals: this is useful for loudspeaker applications. The results obtained can be used for the design of cylindrical structures using both ring permanent magnets and ferrofluid seals. Fig. 1. Arc-shaped permanent magnet whose polarization is radial: the inner curved surface is charged with the magnetic pole surface density +σ * and the outer curved surface is charged with the magnetic pole surface density -σ * , the inner radius is r in , the outer one is r out . Fig. 2. Arc-shaped permanent magnet whose polarization is axial: the upper surface is charged with the magnetic pole surface density +σ * and the lower surface is charged with the magnetic pole surface density -σ * , the inner radius is r in , the outer one is r out .

Fig. 1 .

 1 Fig.1. Arc-shaped permanent magnet whose polarization is radial: the inner curved surface is charged with the magnetic pole surface density +σ * and the outer curved surface is charged with the magnetic pole surface density -σ * , the inner radius is r in , the outer one is rout

Fig. 4 .

 4 Fig.4. Right: upper and lower non-magnetic rings, a middle ring permanent magnet axially magnetized. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 5 .

 5 Fig.5. Right: upper and lower non-magnetic rings, a middle ring permanent magnet radially magnetized. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 6 .

 6 Fig.6. Right: Two ring permanent magnets with opposed axial magnetization. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 7 .

 7 Fig.7. Right: Two ring permanent magnets with opposed radial magnetization. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 8 .

 8 Fig.8. Right: axially magnetized upper and lower rings, radially magnetized middle ring. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 9 .

 9 Fig.9. Right: radially magnetized upper and lower rings, axially magnetized middle ring. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 10 .

 10 Fig.10. Right: axially magnetized upper and lower rings, radially magnetized middle ring, each axial magnet is half as small as the radial one. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 11 .

 11 Fig.11. Right: axially magnetized upper and lower rings, radially magnetized middle ring, each axial magnet is twice as high as the radial one. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 3 .

 3 Fig. 3. Device structure: three outer rings (permanent magnet or non-magnetic) radially centered with an inner non-magnetic piston and ferrofluid seals in the air gap between the rings and the piston. r in = 25mm, r out = 28mm.

Fig. 4 .

 4 Fig. 4. Right: upper and lower non-magnetic rings, a middle ring permanent magnet axially magnetized.Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 5 .

 5 Fig. 5. Right: upper and lower non-magnetic rings, a middle ring permanent magnet radially magnetized.Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 6 .

 6 Fig.6. Right: Two ring permanent magnets with opposed axial magnetization. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 7 .

 7 Fig. 7. Right: Two ring permanent magnets with opposed radial magnetization. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 8 .

 8 Fig. 8. Right: axially magnetized upper and lower rings, radially magnetized middle ring. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 9 .

 9 Fig. 9. Right: radially magnetized upper and lower rings, axially magnetized middle ring. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 10 .

 10 Fig.10. Right: axially magnetized upper and lower rings, radially magnetized middle ring, each axial magnet is half as small as the radial one. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

Fig. 11 .

 11 Fig.11. Right: axially magnetized upper and lower rings, radially magnetized middle ring, each axial magnet is twice as high as the radial one. Left: magnetic pressure in front of the rings. Bottom: magnetic field radial component at a 0.1 mm distance from the rings, along the Z axis.

TABLE I DEFINITION

 I OF THE PARAMETERS USED IN

TABLE II H

 II , MODULUS OF THE MAGNETIC FIELD (kA/m); pm, MAGNETIC PRESSURE (N.m -2 ); Ms = 32kA/m


Farhang Radjai

Emilien Azéma

Keywords: milieux granulaires, résistance au cisaillement, anisotropie de la texture, forces faibles et forts granular media, shear strength, fabric anisotropy, weak and strong forces. Revue

come

Shear strength of granular materials Farhang Radjai and Emilien Azéma

LMGC, CNRS-Université Montpellier 2 34095 Montpellier France.

RÉSUMÉ. La résitance des matériaux granulaires au cisaillement est généralement attribuée à l'anisotropie de la microstructure granulaire. La question de savoir comment l'anisotropie, et donc la résistance au cisaillement, dépend des propriétés des particules, reste ouverte. Dans cet article, nous proposons d'abord une synthèse sur le rôle des anisotropies de la texture et des forces vis-à-vis de la résistance au cisaillement dans l'état critique. Ensuite, un modèle des tats géométriques accessibles en termes de la connectivité des particules et de l'anisotropie de la texture sera présenté. Ce modèle intègre d'une manière très simple le fait que, en raison des exclusions stériques, les niveaux les plus élevés de la connectivité et de l'anisotropie ne peuvent pas être atteints simultanément, ce qui influence d'une manière signaificative les propriétés de résistance. Nous analysons également l'anisotropie des forces à la lumière du rôle spécifique des forces faibles par rapport aux chaînes de force, ce qui est à l'origine de l'anisotropie des forces. Enfin, nous discutons de l'effet de plusieurs paramètres tels que le frottement entre particules, la forme des particules et l'adhésion.

ABSTRACT. The shear strength properties of granular materials reflect their inherent force and fabric anisotropy. We analyze the role of fabric and force anisotropies with respect to the critical-state shear strength. Then, a model of accessible geometrical states in terms of particle connectivity and contact anisotropy is presented. This model incorporates in a simple way the fact that, due to steric exclusions, the highest levels of connectivity and anisotropy cannot be reached simultaneously, a property that affects seriously the shear strength. We also analyze the force anisotropy in the light of the specific role of weak forces in sustaining strong force chains and thus the main mechanism that underlies anisotropic force patterns. Finally, we briefly discuss the effect of interparticle friction, particle shape, and adhesion.

Introduction

Since the early work of Coulomb in 1773, the plastic yield behavior of granular materials has remained an active research field in close connection with soil mechanics and powder technology [START_REF] Mitchell | Fundamentals of Soil Behavior[END_REF][START_REF] Nedderman | Statics and kinematics of granular materials[END_REF]. According to the Mohr-Coulomb yield criterion, for normal and shear stresses σ and τ acting on a slip plane, the plastic threshold τ c is the sum of two terms :

τ c = c + σ tan ϕ, [1]
where c is a cohesive strength and ϕ is the internal angle of friction depending only on the nature of the granular material. This criterion expresses the pressure dependence of shear strength which is a distinctive feature of granular media. Given (1), the shear strength of cohesionless materials (c = 0) can be represented by the (dimensionless) stress ratio τ c /σ = µ c = tan ϕ. Since the angle ϕ is a bulk property, it can be expressed in terms of stress invariants. Let σ α (α = 1, 2, 3) be stress principal values. The average stress is p = (σ 1 + σ 2 )/2 in 2D and p = (σ 1 + σ 2 + σ 3 )/3 in 3D. We define the stress deviator by q = (σ 1 -σ 2 )/2 in 2D and q = (σ 1 -σ 3 )/3 in 3D under axisymmetric conditions (σ 2 = σ 3 ). With these notations, it can be shown that sin ϕ = q/p in 2D and sin ϕ = 3q/(2p + q) in 3D.

This picture of shear strength in granular media holds as a basic fact although the complex plastic behavior of granular media can not be reduced to a single strength parameter. In particular, the shear strength and plastic flow (dilatancy) depend on the granular structure and direction of loading, the latter reflecting the anisotropy of the structure. Since the shear strength is state-dependent, it cannot be considered as a material property unless attributed to a well-defined granular state. The internal angle of friction ϕ is often associated with the critical state (steady state or residual state) reached after long monotonous shearing ; see Fig. 1. This state is characterized by a solid fraction ρ c independent of the loading history and initial conditions [START_REF] Wood | Soil behaviour and critical state soil mechanics[END_REF].

The critical-state strength is below the peak shear stresses occurring for dense states with solid fraction ρ 0 > ρ c , but these states are metastable and often lead to strain localization [START_REF] Darve | Instabilities in granular materials and application to landslides[END_REF][START_REF] Vardoulakis | Bifurcation analysis in geomechanics[END_REF]. For loose states with ρ 0 < ρ c , the critical state is reached asymptotically following diffuse rearrangements. Hence, apart from these transients, which are governed by the evolution of internal variables pertaining to the microstructure and are important in formulating elastoplastic models, the critical-state shear strength represents a stable plastic threshold for a granular material.

In this paper, we are interested in the critical-state strength as a material property of cohesionless granular materials. The critical-state friction angle ϕ c can be described as a coarse-grained (or homogenized) friction angle between two granular layers sliding past each other. Nevertheless, the macroscopic status of ϕ c as a Coulomb friction angle, on the same grounds as those of dry friction between solid bodies, should not eclipse the fact that the granular friction angle is a bulk property to which ade-quate tensorial stress analysis should be applied (this was indeed the contribution of Mohr) and where the slip planes are not a priori defined, in contrast to solid friction which is a surface property at the macroscopic scale [START_REF] Radjai | Key features of granular plasticity[END_REF]. Depending on the boundary conditions, the critical state occurs either homogeneously in the whole volume of a granular sample or inside a thick layer of several particle diameters in the advent of strain localization [START_REF] Bardet | Shear-band analysis in idealized granular material[END_REF][START_REF] Herrmann | The deformation of rocks : fractals everywhere[END_REF][START_REF] Vermeer | The orientation of shear bands in biaxial tests[END_REF][START_REF] Moreau | Numerical Investigation of Shear Zones in Granular Materials[END_REF]. In both configurations, ϕ c stems from various granular phenomena such as friction between particles, anisotropy of the microstructure, organization of force networks and dissipation due to inelastic collisions. We consider below these effects and their respective roles in enhancing or restraining granular friction.

Effect of interparticle friction

While solid friction between particles underlies the frictional behavior of granular materials, it is not obvious how and through which physical mechanisms it comes into play. If shear deformation took place as a result of sliding between all contacts along a slip plane, the friction angle ϕ c would simply echo the friction between particles. An example of such a configuration is a regular pile of cubic blocs subjected to a vertical load. Horizontal shearing of this pile implies sliding between at least two rows so that the shear strength of the pile is a straightforward effect of friction between the blocs. However, discrete numerical simulations suggest that in sheared granular materials, rolling prevails over sliding [START_REF] Radjai | Bimodal character of stress transmission in granular packings[END_REF]. In quasistatic shear, sliding occurs at only ≃ 10% of contacts, and these sliding contacts belong essentially to weak contacts (see below) oriented on average along the minor principal stress directions [START_REF] Radjai | Contact forces in a granular packing[END_REF][START_REF] Staron | Friction versus texture at the approach of a granular avalanche[END_REF][START_REF] Staron | Multi-scale analysis of the stress state in a granular slope in transition to failure[END_REF]. Hence, the relationship between ϕ c and the local friction angle ϕ s involves the inhomogeneous distribution of forces and mobilization (or activation) of the friction force at rolling contacts. This relationship is far from linear as shown in Fig. 2. The critical-state coefficient µ c = tan ϕ c is above µ s = tan ϕ s at small values of the latter, and at larger values it tends to a plateau µ ∞ < µ s [START_REF] Corriveau | Internal friction angles : characterization using biaxial test simulations[END_REF][START_REF] Taboada | Additive decomposition of shear strength in cohesive granular media from grain-scale interactions[END_REF]. The transition from µ c -µ s < 0 to µ c -µ s > 0 occurs at µ c = µ s ≃ 0.5. Beyond µ s = 0.5, µ c is practically independent of µ s . The independence of ϕ c with respect to ϕ s at large values of the latter indicates that the role of interparticle friction is more subtle than expected from simple models. Moreover, the nonzero value of ϕ 0 shows clearly that the interparticle friction is not the only source of frictional behavior in the critical state [START_REF] Roux | Statistical approach to the mechanical behavior of granular media[END_REF].

The direct contribution of interparticle friction to shear strength, i.e. without interposition by the microstructure as will be analyzed below, may be evaluated from a decomposition of the shear stress. The stress tensor σ αβ in a control volume V can be expressed as [START_REF] Rothenburg | A micromechanical definition of the Cauchy stress tensor for particulate media[END_REF][START_REF] Christoffersen | A micromechanical description of granular material behavior[END_REF][START_REF] Moreau | Numerical Investigation of Shear Zones in Granular Materials[END_REF][START_REF] Bagi | Microstructural stress tensor of granular assemblies with volume forces[END_REF][START_REF] Staron | Multi-scale analysis of the stress state in a granular slope in transition to failure[END_REF])

σ αβ = n b ℓ i α f i β , [2]
where n b is the number density of bonds (contacts), ℓ i α is the α-component of the branch vector ℓ i joining the centers of particles at contact i and f i β is the β-component of the force vector f acting at the contact i between the two particles. The contribution of friction forces can be estimated by replacing in equation ( 2) f by f •t t, where t is the unit vector along the friction force. The contribution of normal forces is the complementary tensor obtained by replacing f by f • n n, where n is the unit vector perpendicular to the contact plane. The corresponding shear strengths q t and q n can then be calculated in the critical state. Numerical simulations show that the ratio q t /q is quite low (below 10%) [START_REF] Cambou | From global to local variables in granular materials[END_REF]. This counterintuitive finding underlines the role of interparticle friction as a parameter acting "behind the scenes" rather than a direct actor of shear strength. Our simulations show that, due to disorder and force/moment balance conditions as well as kinematic constraints such as rotation frustration, the friction forces inside a granular packing are strongly coupled with normal forces. For example, highly mobilized friction forces are rare events and the distribution of friction forces reflects for the most part that of normal forces. We consider below such effects in connection with granular microstructure.

Harmonic representation of the microstructure

The microscopic expression of the stress tensor in equation ( 2) is an arithmetic mean involving the branch vectors and contact forces. Hence, for analyzing the particle-scale origins of the shear strength, we need a statistical description of the granular microstructure and force transmission. Noticing that the shear stress corresponds to the deviation of stress components from the mean stress p = tr(σ)/d (for space dimension d) along different space directions, the useful information for this analysis is the density and average force of all contacts pointing in the same direction as a function of this direction. These functions can be expanded in Fourier series in 2D and in spherical harmonics in 3D [START_REF] Rothenburg | Analytical study of induced anisotropy in idealized granular materials[END_REF][START_REF] Ouadfel | Stress-force-fabric relationship for assemblies of ellipsoids[END_REF]. Since the contacts have no polarity, the period is π.

For illustration, we consider here only the 2D expansions truncated beyond the second term :

   P θ (θ) = 1 π {1 + a cos 2(θ -θ b )}, f n (θ) = f {1 + a n cos 2(θ -θ n )}, f t (θ) = f a t sin 2(θ -θ t ), [3]
where P θ is the probability density function of contact normals, and f n and f t are the force components along (radial) and perpendicular to (orthoradial) the branch vector, respectively. The parameters a, a n and a t are the anisotropies of branch vectors, radial forces and orthoradial forces, respectively, θ b , θ n and θ t being the corresponding privileged directions. The sine function for the expansion of the orthoradial component f t is imposed by the requirement that the mean orthoradial force is zero to satisfy the balance of force moments over particles whereas the mean radial force f is positive (repulsive). We also note that for circular and spherical particles the radial and orthoradial force components coincide with normal and tangential forces, respectively. This harmonic representation with only three anisotropy parameters provides a good approximation for numerical data. Using the functions (3), the stress components σ αβ can be written as an integral over space directions :

σ αβ = n b ℓ π 0 { f n (θ)n α (θ) + f t (θ)t β (θ)} P θ (θ) dθ, [4]
where n x = cos(θ) and n y = sin(θ), t x = -sin(θ) and t y = cos(θ). It has been also assumed that the branch vector lengths ℓ are not correlated with forces.

Equation ( 4) together with the harmonic approximation expressed in equation (3) yield the following expression for the normalized stress deviator [START_REF] Radjai | Key features of granular plasticity[END_REF] :

q p ≃ 1 2 {a cos 2(θ σ -θ b ) + a n cos 2(θ σ -θ n ) + a t cos 2(θ σ -θ t )} , [5]
where θ σ is the major principal direction of the stress tensor. In deriving equation ( 5), the cross products among the anisotropies have been neglected. In the critical state, the privileged directions coincide, i.e. θ b ≃ θ n ≃ θ t ≃ θ σ , so that [START_REF] Rothenburg | Analytical study of induced anisotropy in idealized granular materials[END_REF][START_REF] Ouadfel | Stress-force-fabric relationship for assemblies of ellipsoids[END_REF])

q c p ≃ 1 2 {a c + a nc + a tc } , [6]
where the anisotropy parameters refer to the critical state. In 3D, a similar relation can be established by means of spherical harmonics [START_REF] Azéma | Quasistatic rheology and force transmission in a packing of polyhedral particles[END_REF] :

q c p ≃ 2 5 {a c + a nc + a tc } [7]
These relations exhibit two microscopic sources of the shear strength in a granular packing : 1) fabric anisotropy, represented by the parameter a and 2) force anisotropy, captured into the parameters a n and a t . Hence, the material parameters influence the shear strength via fabric and force anisotropies. For example, the saturation of ϕ c for ϕ s > 0.5 (section 2 means that, increasing the interparticle friction beyond this limit does not enhance anisotropy.

Accessible geometrical states

In this section, we focus on the fabric anisotropy a which represents the excess and loss of contacts along different space directions with respect to the average contact density. The latter is commonly represented by the coordination number z (mean number of contacts per particle). In a granular material, z is bounded between two limits z min and z max . The lower bound z min is dictated by the force balance requirement. For example, stable particles often involve more than three contacts in 2D and more than four contacts in 3D. On the other hand, the upper bound z max is constrained by steric exclusions [START_REF] Troadec | Model for granular texture with steric exclusion[END_REF]. For example, in 2D for a system of monodisperse particles, a particle can not have more than 6 contacts. In practice, this limit is reduced to 4 as a result of disorder.

Within the harmonic approximation, the geometrical state of a granular system is defined by its position in the space of coordinates (z, a). We define two limit states : 1) the loosest isotropic state, characterized by (z = z min , a = 0), and 2) the densest isotropic state, characterized by (z = z max , a = 0). These states can be reached only by complex loading. For example, it is generally difficult to bring a granular system towards a dense isotropic state via isotropic compaction. The reason is that the rearrangements occur mainly in the presence of shearing, and the latter leads to fabric anisotropy.

It is natural to assume that all accessible geometrical states are enclosed between the two isotropic limit states. In order to represent the geometrical states, it is useful to define the state function

E(θ) = zP θ (θ) = z π {1 + a cos 2(θ -θ b )}. [8]
The two limit isotropic states are E min = z min /π and E max = z max /π. The assumption that the geometrical states are constrained to stay between the two isotropic limit states, implies that the anisotropy a can not exceed a maximum a max depending on the value of z. With harmonic approximation ( 8), we obtain This function is shown in Fig. 3. By construction, a max (z min ) = a max (z max ) = 0. The largest anisotropy is

a max (z) = min 2 1 - z min z , 2 z max z -1 . [9]
a c = a max (z c ) = 2 a max -a min a max + a min , [10]
with z c = (z min + z max )/2. According to equation ( 10), a max increases with z for z < z c , and it declines with z for z < z c . When a = a c is reached along a monotonic path, neither anisotropy nor coordination number evolve since both contact gain and contact loss are saturated. In this picture, the critical state corresponds to the intersection between the two regimes with z = z c and a = a c . In 2D with weakly polydisperse circular particles and µ s > 0.5, a good fit is provided by assuming z min = 3 and z max = 4. This yields z c = 3.5 and a c = 2/7. For lower values of µ s , a c declines.

Fig. 4 shows the evolution of a with z in simulated biaxial compression of two initially isotropic samples with initial coordination numbers z 0 = 3.1 and z 0 = 3.7. In both simulations, z tends to the same critical-state value z c ≃ 3.35 with a c ≃ 0.24. Remarkably, the anisotropy of the dense packing reaches and then follows closely the limit states. Equation ( 9) provides here an excellent fit to the data with only one fitting parameter z max . In the loose case, the trajectory remains entirely inside the domain of accessible states and the limit states are reached only at the critical state Equation ( 10) predicts that the critical state anisotropy a c increases with z maxz min . The shape, size and frictional characteristics of the particles may therefore influence a c via z min and z max . For example, increasing the sliding friction between the particles allows for lower values of z min (stable configurations with less contacts) without changing z max (which depends only on steric exclusions) and leads to larger values of a c .

One interesting aspect of the model of accessible states presented in this section is to show that the largest values of a and z can not be reached simultaneously. The critical value a c is not obtained with z max but with z c which is below z max . But higher levels of force anisotropies a nc and a tc can be achieved with higher values of z.

Weak and strong force networks

According to equation ( 10), the shear strength is proportional to force anisotropies a nc and a tc in the critical state. As for fabric anisotropy a, which was discussed in the last section, we would like to analyze here the mechanisms that underly force anisotropies. A basic feature of force distribution in granular media is the occurrence of numerous weak forces together with a subnetwork of strong forces appearing often sequentially (force chains). The probability density function (pdf) P n (f n ) of normal forces in a macroscopically homogeneous system in the critical state is such that more than 58% of contact forces are below the mean force f n and they have a nearly uniform distribution [START_REF] Radjai | Nonsmoothness, Indeterminacy, and Friction in Two-Dimensional Arrays of Rigid Particle[END_REF][START_REF] Mueth | Force distribution in a granular medium[END_REF][START_REF] Tsoungui | Partial pressures" supported by granulometric classes in polydisperse granular media[END_REF]. These weak forces contribute only ≃ 29% to the mean stress p. The pdf of strong forces (above the mean normal force f n ) decays exponentially Radjai1996a, Cop-persmith1996a, Radjai1999, Majmudar2005, Metzger2004. The very large number of weak forces, reflecting the arching effect, is a source of weakness for the system. Weak regions inside a packing correspond to locally weak pressures and they are more susceptible to fail. A quantitative analysis of grain rearrangements indicates that during a quasistatic evolution those weak regions undergo local rearrangements, and nearly all sliding contacts are localized in weak regions [START_REF] Staron | Preavalanche instabilities in a granular pile[END_REF][START_REF] Staron | Multi-scale analysis of the stress state in a granular slope in transition to failure[END_REF][START_REF] Nicot | From microscopic to macroscopic second-order work in granular assemblies[END_REF].

Let S(ξ) be the set of contacts with a normal force f n < ξ f n . The set S(∞) is the whole contact set. The weak and strong sets are S(1) and S(∞) -S(1), respectively. The partial shear stress q(ξ)/p and the fabric and force anisotropies a(ξ), a n (ξ) and a t (ξ) can be calculated as a function of ξ [START_REF] Radjai | Bimodal character of stress transmission in granular packings[END_REF]. Our simulations show that q(ξ) ≃ 0 ; see Fig. 5. This means that nearly the whole stress deviator is carried by the strong contact network, the weak contacts contributing only to the mean stress. Hence, the total stress tensor σ is a sum of two terms :

σ = p w I + σ s , [ 11
]
where I is the unit tensor, p w is the weak pressure, and σ s represents the strong stress tensor. Hence, from the stress transmission viewpoint, the weak contact set is a "liquidlike" phase whereas the strong contact set appears as a "solidlike" backbone transmitting shear stresses. The weak and strong networks are shown in Fig. 6 in thickness of segments joining particle centers for an assembly of 4000 particles subjected to biaxial compression. The zero shear stress in the weak network implies that, according to equation ( 6), at least one of the corresponding anisotropies is negative. Since the critical-state angles are assumed to be equal (θ b ≃ θ n ≃ θ t ≃ θ σ ), a negative value corresponds to a rotation π/2 of the principal axes. Indeed, our numerical data show that the privileged direction of weak contacts is perpendicular to the major principal stress direction [START_REF] Radjai | Bimodal character of stress transmission in granular packings[END_REF]. The strong forces occur at contacts that are, on average, aligned with the major principal direction of the stress tensor. Lateral weak forces prop the particles against deviations from alignment at strong contacts. In other words, the weak contacts play the same stabilizing role with respect to the particles sustaining strong forces as the counterforts with respect to an architectural arch. This bimodal transmission of shear stresses corresponds thus to a statistical description of arching effect in granular media.

This stress-fabric correlation can be interpreted as a way for a granular system to optimize the shear strength. Indeed, the stress deviator q increases if a larger number of strong forces occur at contacts aligned with the major principal direction, implying thus a surplus of weak contacts in the perpendicular direction. This weakening of forces at contacts pointing in one direction has the same effect for force anisotropy as the loss of contacts in the same direction for fabric anisotropy. As a result, force weakening in the weak network is all the more efficient as it leads to lower amount of contact loss. This condition can, for example, be achieved for higher level of connectivity, i.e. larger values of z in the critical state.

Effect of material parameters

In this section, we briefly discuss the effect of several material parameters with respect to the mechanisms that underly shear strength in granular media. More details will be given elsewhere.

There are several shape parameters that may lead to enhanced shear strength through force anisotropy or fabric anisotropy. We consider here polygonal particles as compared to circular particles [START_REF] Azéma | Force transmission in a packing of pentagonal particles[END_REF]. The first sample, denoted S1, is composed of 14400 regular pentagons of three different diameters : 50% of diameter 2.5 cm, 34% of diameter 3.75 cm and 16% of diameter 5 cm. The second sample, denoted S2, is composed of 10000 disks with the same polydispersity. The coefficient of friction is 0.4 between particles and 0 with the walls. At equilibrium, both numerical samples are in isotropic stress state. The solid fraction is 0.80 for S1 and 0.82 for S2. The isotropic samples are subjected to vertical compression by downward displacement of the top wall.

Figure 7 displays the evolution of a as a function of the cumulative shear strain ε q in both packings. In both cases, a increases from 0 to its largest value in the critical state. Surprisingly, the fabric anisotropy is quite weak in the pentagon packing whereas the disk packing is marked by a much larger anisotropy (≃ 0.3). Fig. 8 shows the evolution of a n and a t . We see that, in contrast to fabric anisotropies, the force

0 0.1 0.2 0.3 0.4 ε q 0 0.1 0.2 0.3 0.4 a S1 S2
Figure 7. Evolution of the anisotropy a with cumulative shear strain ε q for a packing of pentagons (S1) and a packing of disks (S2).

anisotropies in the pentagon packing are always above those in the disk packing. This means that the aptitude of the pentagon packing to develop large force anisotropy and strong force chains is more dependent on particle shape than on the buildup of an anisotropic structure.

According to equation ( 6), in spite of the weak fabric anisotropy a, the larger force anisotropies a n and a t allow the pentagon packing to achieve higher levels of shear strength compared to the disk packing, as shown in Fig. 9. Our numerical data show that the strong force anisotropy of the polygon packing results from the edge-to-edge contacts that capture most strong force chains, whereas vertex-to-edge contacts belong mostly to the weak network. The pentagons provide thus an interesting example where the role of fabric anisotropy in shear strength is marginal. Similar conclusions hold for polyhedral particles in 3D [START_REF] Azéma | Quasistatic rheology and force transmission in a packing of polyhedral particles[END_REF].

The effect of the coefficient of friction µ s between particles on the shear strength was discussed in section 2. The saturation of the critical-state friction angle ϕ c with increasing µ s is related to the fact that, due to disorder, particle equilibria are fundamentally controlled by normal forces. Ideal situations where friction needs to be fully mobilized over a large number of contacts exist but are marginal. For example, a column of particles each with two contacts may in principal exist, but is of practically zero chance to occur within a disordered granular material. The effect of µ s over a c manifests itself through z min which decreases with µ s . On the other hand, larger values of µ s allow for reinforced stabilizing effect of weak contacts, increasing thereby force anisotropies and thus shear strength.

The effect of adhesion is to allow for tensile forces mainly in the direction of extension between the particles. We find that the tensile forces between particles play the same stabilizing role with respect to the strong compressive forces as the weak network [START_REF] Radjai | Cohesive granular texture[END_REF]. Remark that the privileged direction of weak compressive forces coincides with that of tensile forces. As a result, the main contribution to the shear strength comes from force anisotropy. The fabric anisotropy is generally low and partially inhibited by the presence of adhesion. Note also that adhesion between particles involves a force scale so that its contribution to the shear strength is mainly expressed through the Coulomb cohesion c (equation ( 1)), but it can also influence the internal angle of friction ϕ c through fabric anisotropy.

Conclusion

In this paper, we presented a brief account of physical mechanisms that underly the critical-state shear strength of granular materials. The short-comings of the picture of granular friction in direct analogy with solid friction was discussed. Recalling the expansion of the stress tensor in force and fabric anisotropies, a model was presented for the accessible geometrical states within a harmonic representation of the microstructure. This model, consistent with numerical simulations, relates the critical-state fabric anisotropy to two isotropic limit states corresponding to the lowest and highest contact densities of a granular packing. The force anisotropy was analyzed in the light of the bimodal character of force transmission. It was shown that the shear strength is mainly sustained by the strong force network so that force anisotropy is mainly related to the aptitude of a granular assembly to build up strong force chains. Finally, the effect of material parameters with respect to fabric and force anisotropies was discussed.

Figure 1 .

 1 Figure 1. Normalized shear stress as a function of cumulative shear strain in a 2D simple shear simulation by the contact dynamics method for two different values of the initial solid fraction.

Figure 2 .

 2 Figure 2. The critical-state friction coefficient µ c as a function of sliding friction coefficient µ s between particles in biaxial shearing of a sample of 5000 particles.

Figure 3 .

 3 Figure 3. Domain of accessible geometrical states based on the harmonic representation of granular microstructure.

Figure 4 .

 4 Figure 4. Evolution of the geometrical state of a sheared packing for two different initial states simulated by the contact dynamics method.

Figure 5 .

 5 Figure 5. The partial shear stress, normalized by the mean stress, as a function of force threshold ξ.

Figure 6 .

 6 Figure 6. Weak and strong normal forces represented in two different grey levels (color online). Line thickness is proportional to the normal force.

Figure 8 .Figure 9 .

 89 Figure 8. Evolution of force anisotropies a n (a) and a t (b) as a function of cumulative shear strain ε q in samples S1 and S2.

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Revue, Volume X -n˚x/année

Remerciements

N. Estrada and A. Taboada are acknowledged for figure 2 as well as interesting discussions about granular friction. I present my special thanks to S. Roux for inspiring ideas he shared with me about the plasticity of granular media and its microscopic origins.

ANNEXE POUR LE SERVICE FABRICATION


Eunice Phd Cascant
email: eunice.cascant@univ-lyon3.fr

How entrepreneurial ecosystem embeddedness reduces institutional bottlenecks faced by women refugees' social entrepreneurs in France

Keywords:

How entrepreneurial ecosystem embeddedness reduces institutional bottlenecks faced by women refugees' social entrepreneurs in France

(a) Their migration trajectory differs from that of voluntary migrants and has an impact on their social networks (Backman et al., 2021). (b) They are not able to fully benefit from the host country's entrepreneurial ecosystems due to blocked mobility and disconnections [START_REF] Duan | Understanding immigrant entrepreneurship : A home-country entrepreneurial ecosystem perspective[END_REF]. (c) They are prone to precarious conditions due to labor market discrimination, sociocultural, gendered and information barriers hindering them from accessing entrepreneurial opportunities [START_REF] Kainat | Sociocultural barriers to information and integration of women refugees[END_REF][START_REF] Malach Pines | Gender differences in entrepreneurship : Equality, diversity and inclusion in times of global crisis. Equality, Diversity and Inclusion[END_REF][START_REF] Miller | Underdog entrepreneurs : A model of challengebased entrepreneurship[END_REF][START_REF] Naudé | Migration, entrepreneurship and development : Critical questions[END_REF]. This is particularly true for women refugees usually left out when considering broader discussions surrounding diversity and complexity of gender in entrepreneurship as previously argued in [START_REF] Marlow | Annual review article : Is it time to rethink the gender agenda in entrepreneurship research[END_REF] critical and thought-provoking analysis of gender within entrepreneurship research [START_REF] Marlow | Annual review article : Is it time to rethink the gender agenda in entrepreneurship research[END_REF]. Nevertheless, WRSE view entrepreneurship as a way of self-empowerment, integration, creation of social value, and a pathway out of poverty [START_REF] Santos | Entrepreneurship as a pathway into and out of poverty : A configuration perspective[END_REF]. Although impacted by non-equitable ecosystems and non-double crossed system policies. Our research aims at opening this black box and bridging both research and empirical gaps by investigating "How does embedding in host country's entrepreneurial ecosystems reduces institutional bottlenecks encountered by WRSE? We draw upon the mixed embeddedness theory as "it has been proven to be a powerful concept to analyse the processes and consequences of entrepreneurial activities of those suffering from little accessibility to entrepreneurial resources" (Zhu et al., 2019 : 392). It is also suitable for understanding the interactions of WRSE through their agency in navigating contextual ecosystems [START_REF] Granovetter | Economic Action and Social Structure : The Problem of Embeddedness[END_REF][START_REF] Langevang | Bounded Entrepreneurial Vitality : The Mixed Embeddedness of Female Entrepreneurship[END_REF]. In addition to their engagement with a wide array of actor in public and private support structures [START_REF] Audretsch | Entrepreneurial ecosystems in cities : Establishing the framework conditions[END_REF][START_REF] Neumeyer | Sustainable business models, venture typologies, and entrepreneurial ecosystems : A social network perspective[END_REF].

Design/Methodology/Approach: We conducted a qualitative study to gain a deeper understanding of our subject matter (Eisenhardt, 1989). We carried out 35 semi-structured interviews with WRSE and service providers, as well as 10 interviews with women native entrepreneurs in France from April-July 2021 and January-September 2022. We aimed at having a comparative analysis as to whether both groups encounter the same institutional challenges during their venturing processes. This process was followed by content analysis which led us to seven thematic categories [START_REF] Gioia | Seeking qualitative rigor in inductive research : Notes on the Gioia methodology[END_REF]. Notably: (1) Ecosystem institutional service provider legal support. ( 2 Findings: Our research responds to suggestions towards wider and non-discriminatory perspectives on what constitutes entrepreneurship whilst leading to better theory and more insights in embracing entrepreneurial diversity [START_REF] Welter | Everyday Entrepreneurship-A Call for Entrepreneurship Research to Embrace Entrepreneurial Diversity[END_REF]. We highlight the roles played by non-state actors in reducing barriers to entry by offering active support on the restrictive legal and bureaucratic start-up regulations. How findings highlight how WRSE are able to access ecosystem funding through the different actors which are key pillars in sustaining their ventures [START_REF] Landström | Advanced Introduction to Entrepreneurial Finance[END_REF]. Our findings unveil how women refugees benefit from inter and intra-ecosystem networks that are crucial in their social venture sustainability.

Originality and value:

Our work builds on new realities by addressing diversity in women entrepreneurship with regard to structural contexts, processes, and individuals in the French context. WRSE turn societal problems into opportunities through the development of products and services that generate social value. Yet entrepreneurship research has not yet acted as "a window into and a tool for shaping social and economic equity constructed to include not only issues of structural inequality but also empowerment and emancipation more broadly" (Welter et al., 2017, p. 317). To the entrepreneurial ecosystem research, we integrate a new term of sustainable inclusive entrepreneurial ecosystems to cater for disadvantaged entrepreneurs such as refugees. We recommend building equitable ecosystems that require consciousness in raising and double-crossing systems' policies to accommodate women refugee social entrepreneurs [START_REF] Orser | Building back better through feminist entrepreneurship policy[END_REF] We advance the framework of mixed embeddedness by providing an improved conceptualization encompassing the entrepreneurial ecosystem elements within the formal and informal institutions that were not originally included [START_REF] Kloosterman | Mixed Embeddedness : (In)formal Economic Activities and Immigrant Businesses in the Netherlands[END_REF]. Our policy implications are two-fold: 1) Ecosystem enablers and actors, there is a need to create sustainable inclusive policies and implement programs that cater for a demographically diverse set of entrepreneurs. 2) We urge policymakers to measure sustainable entrepreneurial ecosystem indicators with inclusive entrepreneurial support. We posit that female refugee social entrepreneurs do not face the same institutional hurdles as their native counterparts. We recommend policymakers to promote diversity and equality in entrepreneurship as women empowerment is a cornerstone of the 2030 Sustainable development goals agenda (OECD & European Commission, 2021).

Keywords: Gender, women refugee social entrepreneurship, embeddedness, ecosystems.

  ) Financial training and accessibility. (3) Intra and internetwork accessibility. 4) Management training and coaching for supporting start-ups. (5) Pioneering technologies through innovation support programs. (6) Economic empowerment and (7) Measuring of intended social impact of the women refugee social ventures.