Xavier Chateau
email: xavier.chateau@lcpc.fr

Guillaume Ovarlez

Kien Luu

Homogenization approach to the behavior of suspensions of noncolloidal particles in yield stress fluids

 and to experimental data found in the literature.

I Introduction

Heterogeneous systems consisting of particles suspended in a fluid medium make up in a wide variety of materials of practical interest, both natural (slurries, debris flows, lavas, ...) or manmade (concretes, food pastes, paints, cosmetics, ...). This abundance explains why the behavior of these materials have been extensively studied from both a theoretical and an experimental point of view.

In the framework of man-made materials, it does not seem to exist a well established method to obtain a material with given rheological characteristics from components having known properties despite the fact that is is a problem encountered in numerous industrial processes. The set-up of such a methodology requires the ability to predict the overall behavior of the material from that of its constituents. A general answer for this problem has yet to be found.

The main difficulty in modeling the behavior of suspensions comes from the fact that the material is multiscale and contains many interacting constituents. Experimental data show that the overall rheological properties of a suspension depend upon the shape and size of the particles, the interaction between particles (colloidal or noncolloidal), the interaction between the particles and the suspending fluid (hydrodynamic), the properties of the suspending fluid (Newtonian or not) and the type of flow the suspension is subjected to.

In this paper, we focus on suspensions made up of noncolloidal particles dispersed in a yield stress fluid such as suspensions made up of a coarse fraction and a colloidal fraction. If the colloidal particles are much smaller than the coarse ones, the latter interact with the other components of the suspension only through hydrodynamic interactions: they see a homogeneous phase fluid which behavior is that of the colloidal suspension [Sengun and Probstein (1989a,b); [START_REF] Ancey | Yield stress for particle suspensions within a clay dispersion[END_REF]].

Like for any other suspensions, the rheological properties of non-Newtonian suspensions depend upon the shape, the surface texture and the size distribution of the coarse particles.

In addition to experimental approaches, modeling the behavior of non-Newtonian suspensions using their constituents characteristics in the framework of numerical methods appears promising because of the ability of these methods to track the particles localization in the suspension and to account for several interparticle interactions. Solving the problem for noncolloidal particles immersed in a Newtonian fluid is feasible using modern computers and provides useful results [START_REF] Brady | Computer simulation of viscous suspensions[END_REF]]. The most serious drawback of these methods to their generalization to non-Newtonian suspending fluid is the fact that the Newtonian hydrodynamic force applied to the particle is evaluated by means of close form equations depending on the macroscopic strain rate and on the particle velocity relative to the bulk fluid [START_REF] Bossis | Dynamic simulation of sheared suspension. i. general method[END_REF]; [START_REF] Brady | Stokesian dynamics[END_REF]]. Thanks to these relations, it is not necessary to solve the continuous Stokes equation in the bulk fluid in order to simulate the behavior of the suspension. This enables to simulate the flow of representative elementary volume of a suspension containing numerous particles at a reasonable computational cost. To our knowledge, such an equation does not exist for particles immersed in a yield stress fluid, which prevents a generalization of the method to be easily proposed.

Trying to evaluate the velocity field in the bulk yield stress fluid by means of a finite element method is such a time consuming task that no situation of practical interest can be solved using these tools today [START_REF] Johnson | 3d simulation of fluid-particle interactions with the number of particles reaching 100[END_REF]; [START_REF] Roquet | An adaptive finite element method for bingham fluid flows around a cylinder[END_REF]; [START_REF] Yu | A fictitious domain method for dynamic simulation of particle sedimentation in bingham fluids[END_REF]].

To obtain estimates for the overall characteristics of non-Newtonian suspensions, the change of scale method appears then to be a powerful tool. As a reminder homogenization technique aims at identifying the macroscopic properties of a material modeled as a continuous medium from those of their constituents. The first results in this field were obtained by [START_REF] Einstein | Eine neue bestimmung der moleküldimensionen[END_REF] for the viscosity of a dilute suspension. Since, different problems have been studied such as the viscosity of multimodal suspensions [START_REF] Farris | Prediction of the viscosity of multimodal suspensions from unimodal viscosity data[END_REF]], the viscosity of concentrated suspensions [START_REF] Krieger | A mechanism for non-newtonian flow in suspensions of rigid spheres[END_REF]; [START_REF] Frankel | On the viscosity of a concentrated suspension of solid spheres[END_REF]], the effect of the shape and orientation of particles on the behavior of Newtonian suspensions [START_REF] Batchelor | The stress generated in a non-dilute suspension of particles by pure straining motion[END_REF]], the effect of interparticle interactions [START_REF] Batchelor | The determination of the bulk stress in a suspension of spherical particles to order c 2[END_REF]] or the effect of Brownian motions on the overall behavior [START_REF] Batchelor | The effect of brownian motion on the bulk stress in a suspension of spherical particles[END_REF]; [START_REF] Russel | Colloidal Dispersions[END_REF]].

It is worth noticing that it has not been possible to obtain exact solutions for most of the problems cited above; generally, only estimates of the overall characteristics of the suspension have been obtained. This situation is very similar to the one prevailing in the field of homogenization approaches to the behavior of solid materials [START_REF] Zaoui | Continuum micromechanics: Survey[END_REF]]. This is not surprising since the two problems are very similar as it has been recognized by [START_REF] Batchelor | The determination of the bulk stress in a suspension of spherical particles to order c 2[END_REF]. The main difference comes from the fact that the morphology of the heterogeneities within the representative elementary volume is a given for solid materials whereas it is an unknown for particle suspensions since the flow of the representative elementary volume of suspension and the morphology of the particles are coupled Interestingly, novel developments have taken place in nonlinear continuum micromechanics in the last twenty years so that both estimates and variational bounds are now available [see [START_REF] Ponte Castañeda | Nonlinear composites[END_REF] for a review]. Because of the similarities between the solid problem with the liquid one, it is quite natural to address the modeling of the behavior of non-Newtonian suspensions within this framework.

The aim of this paper is to provide a first approach to the overall behavior of a suspension of non-Brownian and noncolloidal particles immersed in an incompressible yield stress fluid. First, the main features of the homogenization approach to the behavior of non-Newtonian suspensions are recalled. Then, the secant-method of [START_REF] Castañeda | The effective mechanical properties of nonlinear isotropic composites[END_REF] and [START_REF] Suquet | Overall potentials and extremal surfaces of power-law or ideally plastic materials[END_REF] is applied in order to estimate the overall behavior of the suspension. The estimates are compared to new experimental results obtained by [START_REF] Mahaut | Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids[END_REF] and to experimental results of the literature in the next part. Finally, the validity of the theoretical approach is discussed before some general conclusions are drawn.

II Homogenized behavior

We examine the overall behavior of a suspension, the liquid phase of which is homogeneous and nonlinear. We restrict ourselves to the situation where the constitutive behavior of the fluid is characterized by an energy function w (d) where d = √ 2d : d denotes the second invariant of the Eulerian strain rate tensor d. The Cauchy stress σ is obtained by differentiation of the potential w with respect to the strain rate tensor d if w is differentiable. Otherwise, the derivative should be interpreted as the subdifferential of convex analysis.

The condition of naught strain for the particles can also be written in term of a dissipation potential with w defined by

w (d) = 0 if d = 0 w (d) = ∞ if d = 0 (1)
Thus, the suspension is made of a heterogeneous medium, the behavior of which is described by

σ = ∂w ∂d (d, x) (2)
w being a strict convex function. The location in the representative elementary volume is defined by the position vector x.

Ω ℓ Ω s u (x) = D • x Ω ∂Ω Figure 1:
The representative elementary volume of the suspension submitted to a macroscopic strain rate loading (Hashin boundary condition).

It is assumed that it is possible to define a representative elementary volume of the suspension occupying a domain Ω with boundary ∂Ω such that it is large enough to be of typical composition and its overall properties do not depend on the way it is loaded at the macroscopic scale. Ω s and Ω ℓ denote the solid and the liquid domain respectively. The volume fraction of particles ϕ is the ratio of the volume fraction of Ω s in Ω. For simplicity, it is assumed that the boundary of Ω is located in the liquid domain as depicted in Fig. 1. At the microscopic scale, Ω is considered as a structure. < a > (resp. < a > α with α = s, ℓ) denotes the average of a over Ω (resp. Ω α ). The liquid phase is homogeneous. We adopt an Eulerian description of the movement and we restrict our attention to the situations where the evolutions of the system are quasistatics (i.e. inertial effects are negligible) and all the long range forces other than the hydrodynamic ones are negligible.

As shown by [START_REF] Hill | Elastic properties of reinforced solids: some theoretical principles[END_REF], the overall behavior of the suspension reads:

Σ = ∂W ∂D (D) with W (D) = min d∈C(D) w (d) (3)
where C (D) denotes the set of Eulerian strain rate fields kinematically admissible with D. It is recalled that a strain rate field d defined over Ω is said to be kinematically admissible with D if exists a velocity field u defined over Ω, complying with the Hashin condition

(∀ x ∈ ∂Ω) u (x) = D • x (4) such as: (∀ x ∈ Ω) d (x) = 1 2 grad u (x) + t grad u (x) (5)
When the fluid obeys a Herschel-Bulkley law with a yield stress τ c , a consistency η and a power law exponent n > 0, the dissipation potential reads:

w (d) = τ c d + η n + 1 dn+1 if trd = 0 w (d) = ∞ if trd = 0 (6)
Then the fluid's state equation reads

d = 0 if s : s/2 < τ c s = τ c + η dn d d trd = 0 if s : s/2 ≥ τ c (7)
where p = -trσ/3 is the hydrostatic pressure, s = σ+pδ the deviator of σ and δ the second order unit tensor. As it is common for incompressible materials, the pressure p is not determined by the state law. Considering that the particles are rigid and that the bearing fluid is homogeneous and incompressible, it is easily shown from Eq. ( 3) that the macroscopic potential is also defined by

W (D) = min d∈G(D) (1 -ϕ) τ c d ℓ + η n + 1 dn+1 ℓ if trD = 0 (8)
where G (D) denotes the subset of C (D) which elements comply with the naught strain rate constraint over the domain occupied by the particles and the incompressible constraint over the fluid domain. Of course, the set G (D) is only defined for macroscopic strain rate complying with the condition trD = 0. Eq. ( 8) is completed by the condition W (D) = ∞ if trD = 0, which enforced the incompressible constraint at the macroscopic level.

The set G (D) being convex, the minimization problem (8) admits only one solution, which ensures the validity of the method. The identification of the macroscopic behavior of the suspension from Eq. (3) or Eq. ( 8) requires the resolution of a continuous convex minimization problem. This problem of minimization has to be solved for each morphology of the suspension defined by the shape of the particles and the distribution of the particles within the suspension. Of course, it is not possible to solve this problem in most situations of practical interest.

To remedy this difficulty, various estimation techniques of the macroscopic behavior have been proposed, in particular, by [START_REF] Castañeda | The effective mechanical properties of nonlinear isotropic composites[END_REF][START_REF] Castañeda | Exact second-order estimates for the effective mechanical properties of nonlinear composite materials[END_REF][START_REF] Castañeda | On the homogenized behaviour of reinforced and other bingham composites[END_REF] and [START_REF] Suquet | Overall potentials and extremal surfaces of power-law or ideally plastic materials[END_REF].

The key feature in these methods is the use of a rigorous variational principle (Eq. 3 for the yield stress fluid suspensions) to determine the best possible choice of a "linear comparison composite" to estimate the effective behavior of the nonlinear one. A detailed description of these methods is beyond the scope of the present paper. The reader is referred to [START_REF] Ponte Castañeda | Nonlinear composites[END_REF]] for a more detailed review.

In the following, we obtained estimates relevant to our problem in the framework of a simple approach which does make use explicitly of the variational Eq. 3. The main features of the method used to obtain these estimates are recalled in the following section (largely inspired by the presentation of [START_REF] Suquet | Continuum micromechanics[END_REF]) for completeness of the paper.

III Secant estimate of the behavior

It is assumed that the solid particles are isotropically distributed over the representative elementary volume. The macroscopic behavior of the suspension is therefore also isotropic.

It is possible to write the behavior of the nonlinear fluid in the following form

σ = 2µ sct ( d)d -pδ (9)
where µ sct ( d) denotes the secant modulus of the fluid phase. The secant modulus is no more than the apparent viscosity of the fluid, as depicted in Fig. 2. Using Eq. ( 9) for the state equation of the suspending fluid, it is possible to replace the original nonlinear problem of homogenization by a linear homogenization problem for a suspension of rigid particles immersed in a heterogeneous incompressible isotropic fluid with apparent viscosity η (x) = µ sct x, d (x) .

d s ij ( d) d µ sct ( d) 1 s ij
Then, the difficulty which remains to be solved is to calculate, or to estimate, the field d over the domain occupied by the fluid phase in the representative elementary volume. Since it is impossible to analytically determine the local response of the nonlinear fluid, it is impossible to compute the secant modulus field over the representative elementary volume. Then an approximation has to be introduced to make analytic calculations feasible. The approximation for secant methods consists of replacing the secant modulus field over the representative elementary volume by a modulus which is uniform over subdomains of the representative elementary volume. In this paper, we consider only one domain for the liquid phase to simplify. Then the secant modulus is uniform over the fluid phase. This estimate reads:

(∀ x ∈ Ω ℓ ) µ sct x, d (x) ≃ µ sct ℓ ( deff ℓ ) (10)
where deff ℓ is an equivalent effective strain rate which remains to be defined as a function of the mean value of the field d in the fluid phase, and thus, of the value of the macroscopic strain rate. The replacement of the field of heterogeneous secant modulus defined over the fluid phase by a homogeneous field simplifies considerably the resolution of the problem. Consequently, the estimate technique of the nonlinear macroscopic behavior includes three steps:

1. First of all, it is necessary to solve a linear homogenization problem for a suspension of particles immersed in an isotropic homogeneous fluid of viscosity µ ℓ . If the suspension is isotropic at the macroscopic scale, the macroscopic behavior is characterized by a macroscopic viscosity proportional to the viscosity of the fluid, the coefficient of proportionality depending on the morphology of the particles. As this problem has been the subject of many works, numerous results and estimates are available in the literature dealing with the rheology of suspensions or the homogenization to the behavior of heterogeneous linear materials. Let g(ϕ) denotes the coefficient of proportionality linking the macroscopic viscosity to the microscopic viscosity.

µ hom = µ ℓ g(ϕ) (11)
2. Then, one must choose a measure of the effective strain rate for the fluid phase. As the considered material is isotropic at the microscopic scale, the strain rate is characterized by the second order moment of the quantity d defined by:

deff ℓ = d2 ℓ (12)
This choice corresponds to the modified approach described by [START_REF] Suquet | Continuum micromechanics[END_REF]. [START_REF] Castañeda | The effective mechanical properties of nonlinear isotropic composites[END_REF] demonstrated that this choice is optimal in the framework of a variational approach to the solution of the nonlinear homogenization problem under consideration. It would have been simpler to choose deff ℓ = < d > ℓ , a quantity easily computed from the egality [START_REF] Suquet | Continuum micromechanics[END_REF] showed that estimates of the overall properties of the heterogeneous material obtained using this effective liquid strain rate are less accurate than those obtained using Eq. ( 12).

d ℓ = D/(1 -ϕ). Unfortunately,

Finally, the nonlinear character of the problem is taken into account by integrating into

the relation of homogenization ( 11) linking µ hom to the viscosity of the liquid phase, the fact that the value of the fluid viscosity depends on deff ℓ . As deff ℓ depends on the value of the macroscopic strain rate D, the value of the macroscopic secant modulus also depends on the value of D. The only difficulty implementing this step is the calculation of deff ℓ as a function of D for the particular homogenization scheme used. It has been shown by [START_REF] Kreher | Residual stresses and stored elastic energy of composites and polycristals[END_REF] that:

d2 ℓ = 1 1 -ϕ ∂µ hom ∂µ ℓ D2 = 1 1 -ϕ g(ϕ) D2 (13)
By remplacing µ ℓ by µ sct in the relation of linear homogenization Eq. ( 11) and then by combining the obtained equation with the localization Eq. ( 13), one obtains the following estimate for the macroscopic secant modulus (i.e. apparent viscosity) of any nonlinear materials with any isotropic microstructure:

µ hom ϕ, D = g(ϕ) * µ sct d with d = D g(ϕ) 1 -ϕ (14)
Using notations of Sengun and Probstein (1989b), the apparent viscosity Eqs. ( 14) reads

η(ϕ, γ) = η cr (ϕ) * η fr ( γeff ) with γeff = γ η cr (ϕ) 1 -ϕ (15)
which is much more general than Eqs (2.8), (2.9), (4.1) and (4.2) of Sengun and Probstein (1989b) as they allow to take into account any estimate g(ϕ) (i.e. η cr (ϕ)) of the relative viscosity of a Newtonian suspension. It is worth noting that Eqs. ( 14) and ( 15) are valid for any particles shape and dispersity. Interestingly, even if the estimate g(ϕ) does not rely on a morphological model, Eq. ( 13) allows to estimate the localization factor associated with the relative viscosity function g(ϕ).

Coming back to the Herschel-Bulkley suspension problem, the secant modulus of the suspending fluid reads:

µ sct x, d (x) = τ c d (x) + η d (x) n-1 (16)
Then, putting Eq. ( 16) into Eq. ( 14) yields

µ hom (ϕ, D) = τ hom c D + η hom Dn-1 (17)
with:

τ hom c = τ c (1 -ϕ)g(ϕ) (18)
and:

η hom = ηg(ϕ) g(ϕ) 1 -ϕ n-1 2 (19)
It is therefore predicted that, at the macroscopic scale, the suspension behaves as a Herschel-Bulkley fluid with same exponent as that of the suspending fluid. Overall yield stress and macroscopic consistency are defined by Eqs. ( 18) and ( 19). This result does not depend on the scheme (i.e. g(ϕ)) used to link the viscosity of the bearing fluid to the overall viscosity of the suspension. The quality of the prediction depends thus only on the validity of the assumption that the field µ sct ℓ x, d (x) can be estimated by the quantity µ sct ℓ ( deff ℓ ).

IV Experimental validation

An experimental procedure has been designed by [START_REF] Mahaut | Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids[END_REF] which complies with the assumptions made to obtain the theoretical results presented above. As this procedure is described in detail in the paper entitled Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids, no further details concerning the experimental work are given in this paper; we restrict ourselves to comparisons between experimental data and theoretical predictions.

A Elastic modulus vs. yield stress

The accuracy of the estimates obtained in the framework of the theoretical approach presented above for the overall properties of the yield stress suspension depends on the assumption made to take into account the nonlinear behavior of the suspending fluid and on the scheme used to estimate the overall linear behavior of the suspension. It is possible to check experimentally the validity of the assumption made on the heterogeneities of the secant modulus over the liquid domain irrespective of the errors induced by the choice of a particular homogenization scheme. For this, it is enough to remark that Eq. ( 11) enables to calculate the macroscopic elastic modulus G hom of an isotropic suspension of particles dispersed in an isotropic incompressible linear elastic matrix whose shear modulus is equal to G (both problems pose exactly in the same way provided that d and µ ℓ be identified with the infinitesimal strain tensor and the elastic shear modulus).

As a consequence, it is possible to obtain a general relationship between the dimensionless elastic modulus and the dimensionless yield stress of a suspension of rigid particles dispersed in a yield stress fluid that is true whatever the scheme as long as the particle distribution is isotropic and using an uniform secant modulus estimate is relevant. Combining Eqs. ( 11),( 18) and ( 19) yields the relations:

τ hom c /τ c = (1 -ϕ)G hom /G (20)
and

η hom /η = (G hom /G) n+1 (1 -ϕ) n-1 (21)
Moreover, the yield stress and the consistency not being independent of one another, it is also possible to determine the consistency from the yield stress and the concentration:

η hom /η = τ hom c /τ c n+1 (1 -ϕ) n (22)
In Fig. 3, we have plotted the dimensionless yield stress τ c /τ c as a function of the dimensionless quantity (1 -ϕ)G hom /G for all the systems studied by [START_REF] Mahaut | Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids[END_REF]. It is recalled that yield stress fluids have a solid linear viscoelastic behavior below the yield stress, so that the macroscopic elastic modulus of the suspensions could be experimentally measured through oscillatory shear measurements. A good agreement between the experimental results -2 and the micromechanical estimation (20) (which is plotted as a straight line y = x in these coordinates) is observed. These results show that the data are consistent with the assumption that an uniform estimate of the secant modulus over the fluid domain allows to accurately estimate the overall properties of the suspension in the studied situations.

-1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 log τ hom c /τ c -1 log (1 -ϕ)G hom /G -1 y = x

B Elastic modulus

In this section, we summarize the results of the elastic modulus measurements performed on all the materials.

The evolution of the dimensionless modulus G hom /G as a function of the volume fraction ϕ of noncolloidal particles for all the studied materials are summarized in Fig. 4. It is observed that the experimental data are very well fitted to the Krieger-Dougherty law [START_REF] Krieger | A mechanism for non-newtonian flow in suspensions of rigid spheres[END_REF]): [START_REF] Mahaut | Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids[END_REF]. The open square symbols are suspensions of polystyrene and glass beads in bentonite, the open circle symbols are suspensions of glass beads in carbopol while the solid diamond symbols are suspensions of polystyrene and glass beads in emulsion. The solid line is the Krieger-Dougherty Eq. ( 23) with ϕ m = 0.57. MRI techniques. The small discrepancy between the two values of ϕ m comes certainly from the anisotropy induced by the flow in the experiments of [START_REF] Ovarlez | Local determination of the constitutive law of dense suspension of non colloidal particles through magnetic resonance imaging[END_REF] as it was also observed in the experiments performed by [START_REF] Parsi | Fore-and-aft asymmetry in a concentrated suspension of solid spheres[END_REF].

G hom G = g(ϕ) = 1 - ϕ ϕ m -2.5ϕm (

C Yield stress

We now present the results of the yield stress measurements.

The yield stress τ c was measured with a method which avoids destroying the homogeneity and the isotropy of the suspension. The evolution of the dimensionless yield stress τ hom c /τ c as a function of the solid volume fraction ϕ of noncolloidal particles is depicted in Fig. 5 for the studied materials.

Fig. 5 clearly shows that the yield stress of the suspensions reads as the product of the suspending fluid yield stress times a function f (ϕ). When both the dimensionless yield stresses and the dimensionless elastic moduli are drawn on the same diagram (see Fig. 6), it is obvious that the yield stress function f (ϕ) is different from the elastic modulus function g(ϕ).

It is possible to directly use Eqs. ( 17), ( 18) and ( 19) to evaluate the function f (ϕ).

For the dilute suspensions, the Einstein relation g DL (ϕ) = 1 + 5/2ϕ is exact to the first order of ϕ. Introducing this relation in Eqs. ( 18) and ( 19) and keeping only the first order terms yield the dilute estimates:

τ DL c = τ c (1 + 3/4ϕ) ( 24
)
η DL = η 1 + 7n + 3 4 ϕ (25)
The coefficients of growth of these two estimates with the solid volume fraction are different from that of the Einstein law. According to the experimental results depicted in Fig. 6 the change of scale method also predicts that the two functions g(ϕ) and f (ϕ) are different. The only approximation performed to obtain Eqs. ( 24) and ( 25) is to assume that a uniform estimate of the secant modulus over the fluid domain enables to accurately estimate the overall properties of the suspension. Thus, estimates ( 24) and ( 25) are not rigorously exact in contrast with the Einstein law. It is worth noticing that for a suspension of particles in a Bingham fluid, the overall consistency of the suspension is given by the classical Einstein function 1 + 5/2ϕ.

For larger values of the solid volume fraction, the viscosity of a Newtonian suspension is classically estimated using the Krieger-Dougherty equation [START_REF] Krieger | A mechanism for non-newtonian flow in suspensions of rigid spheres[END_REF]; [START_REF] Quemada | Phenomenological rheology of concentrated dispersions -i clustering effects and the structure-dependent packing fraction[END_REF]]. Putting the second equality of Eq. ( 23) into Eqs. ( 18) and ( 19) yields the Krieger-Dougherty estimates for the effective yield stress and the effective consistency of the suspension:

τ KD c = τ c (1 -ϕ)(1 -ϕ/ϕ m ) -2.5ϕm
(26)

η KD = η (1 -ϕ) 1-n 2 (1 -ϕ/ϕ m ) -1.25(n+1)ϕm (27)
Both the macroscopic yield stress and the macroscopic consistency diverge when ϕ tends towards ϕ m . The values obtained with Eq. ( 26) are plotted in Fig. 5, taking ϕ m = 0.57. Good accordance is observed between theoretical and experimental data, accounting for the validity of Eq. ( 26). Note that no attempt to fit the value of ϕ m has been performed.

D Experimental results from the literature

In this paragraph, the theoretical estimates are compared with experimental data of [START_REF] Ancey | Yield stress for particle suspensions within a clay dispersion[END_REF], [START_REF] Erdogan | Determination of aggregate shape properties using X-ray tomographic methods and the effect of shape on concrete rheology[END_REF] and [START_REF] Geiker | On the effect of coarse aggregate fraction and shape on the rheological properties of self-compacting concrete[END_REF].

The results for the yield stress are depicted in Fig. 7. The experimental data of [START_REF] Geiker | On the effect of coarse aggregate fraction and shape on the rheological properties of self-compacting concrete[END_REF] have not been drawn in Fig. 7 because they report to measure yield stress ten times larger than other researchers. 26) with ϕ m = 0.635 and the straight dashed line is the dilute overall yield stress (24). The dashed curves are the theoretical Eq. ( 26) with ϕ m = 0.67 (dashed curve below the solid curve) and ϕ m = 0.6 (dashed curve above the solid curve).

The theoretical curve in Fig. 7 is calculated using ϕ m = 0.635, as reported by [START_REF] Ancey | Yield stress for particle suspensions within a clay dispersion[END_REF]. We also plotted the curve using ϕ m = 0.67 (dashed curve below the solid curve) and ϕ m = 0.6 (dashed curve above the solid curve). The experimental data of [START_REF] Ancey | Yield stress for particle suspensions within a clay dispersion[END_REF] are roughly well fitted by the theoretical estimate (26) with ϕ m = 0.635 even if the theoretical model always predicts yield stress lower than the measured ones. These discrepancies can not be impart to the experimental procedure. The suspensions was prepared by adding monodisperse glass beads to a clay dispersion and the yield stress was measured by means of a slump test, an experimental procedure which ensures that the materials remain isotropic and homogeneous. It is believed that the discrepancy between the experimental data of [START_REF] Ancey | Yield stress for particle suspensions within a clay dispersion[END_REF] and our theoretical predictions could come from a wrong evaluation of the experimental maximum packing fraction. To support this opinion, it is worth noting that the estimate ϕ m = 0.635 of [START_REF] Ancey | Yield stress for particle suspensions within a clay dispersion[END_REF] was neither measured nor estimated from experimental data. Moreover, this value 0.635 is greater than the values reported in the literature for a dense suspension of non colloidal suspension of monodisperse spherical particles [START_REF] Ovarlez | Local determination of the constitutive law of dense suspension of non colloidal particles through magnetic resonance imaging[END_REF]]. It can be seen in Fig. 7 that the experimental data are well fitted by Eq. ( 26) with ϕ m = 0.6. [START_REF] Erdogan | Determination of aggregate shape properties using X-ray tomographic methods and the effect of shape on concrete rheology[END_REF] used a Couette-vane rheometer and applied a strong preshear to the samples before to measure the overall properties of the concrete suspension. Nevertheless, his results compare rather well to the theoretical ones, even if the lack of experimental data for values of the solid fraction larger than 0.4 does not allow to draw a definitve conclusion.

We now pay attention to the results of the consistency measurements. Dimensionless consistency η hom /η measured by [START_REF] Erdogan | Determination of aggregate shape properties using X-ray tomographic methods and the effect of shape on concrete rheology[END_REF] and [START_REF] Geiker | On the effect of coarse aggregate fraction and shape on the rheological properties of self-compacting concrete[END_REF] are plotted in Fig. 8 as a function of the normalized solid volume fraction ϕ/ϕ m . Four types of aggregates were used by [START_REF] Geiker | On the effect of coarse aggregate fraction and shape on the rheological properties of self-compacting concrete[END_REF] : glass beads, sea dredged, crushed, and a mix of 30% sea dredged and 70% crushed aggregates. None of these coarse aggregates are monodisperse (the ratio of the bigger particle size to the smaller one is close to 4 for the all the aggregates). Moreover, only the beads are spherical. Then, most of the suspensions studied by [START_REF] Geiker | On the effect of coarse aggregate fraction and shape on the rheological properties of self-compacting concrete[END_REF] do not comply with the assumptions made to compute Eqs. ( 26) and ( 27). Consequently, it is not surprising that the data of [START_REF] Erdogan | Determination of aggregate shape properties using X-ray tomographic methods and the effect of shape on concrete rheology[END_REF] are relatively well fitted by the theoretical prediction while the fit between most of the experimental data of [START_REF] Geiker | On the effect of coarse aggregate fraction and shape on the rheological properties of self-compacting concrete[END_REF] and the theoretical estimate is less satisfactory. More precisely, while it seems that Eq. ( 27) is able to roughly estimate the consistency of suspensions of spherical particles (glass beads) even if they are not monodisperse, the discrepancy between experimental and theoretical data is no more acceptable for the other aggregates. As depicted in Fig. 8, tuning the value of the maximum solid fraction does not allow to improve the quality of the fit.

It is believed that this discrepancy comes from the fact that the sea dredged, crushed and mixed aggregates are not spherical (an aspect ratio equal to 2 has been measured for all these particles). As the Krieger-Dougherty Eq. ( 23) applies only to suspensions of spherical particles, Eq. ( 27) cannot fit the properties of suspensions of non-spherical particles. One concludes that using a function g(ϕ) able to accurately estimate the linear properties of a suspension of non spherical particles into Eqs. ( 18) and ( 19) will improve the quality of the estimates. This opinion is supported by the fact [START_REF] Geiker | On the effect of coarse aggregate fraction and shape on the rheological properties of self-compacting concrete[END_REF] observed that the relative consistency of nonspherical particles suspensions is greater than that of the glass beads suspensions when measured for the same solid volume fraction (see Fig. 8), a trend classically observed for Newtonian suspensions.

V Validity of the theoretical model

For all the results, the experimental data are more dispersed when the solid volume fraction increases and the discrepancy between the experimental and the theoretical results increases when the solid volume fraction tends towards that of the maximum packing fraction. It could be asserted that this discrepancy comes simply from the dispersion of the experimental results. Nevertheless, it is believed that heterogeneities of the secant modulus over the liquid domain are not any more negligible for the larger values of ϕ. When the solid volume fraction value tends towards the maximum packing fraction, the fluid located near the closest points of two adjacent particles experiences a much larger shear rate than the fluid located far from these points (see Fig. 9). In this situation, it is necessary to take into account the heterogeneities of the secant Particles High shear rate Low shear rate Figure 9: Heterogeneities of the strain rate field for a concentrated suspension. modulus in the framework of more complex models in order to obtain more accurate estimates.

To use an uniform estimate of the secant modulus induces a second drawback of the model. As no difference is made between all the points located in the fluid phase, such a model is unable to account for the fact that the fluid does not start to flow everywhere at the same time (or more precisely that the yield stress is not reached everywhere at the same time in the fluid domain).

For the Bingham suspension, [START_REF] Castañeda | On the homogenized behaviour of reinforced and other bingham composites[END_REF] has shown that beyond the yield stress, the behavior of the suspension is first nonlinear and then asymptotes to a purely linear behavior for higher strain rate values. As this phenomenon is induced by the presence of strong fluctuations of the strain rate field in the fluid phase at the onset of yield which induce fluctuations of the secant modulus, the simplified model described above is not able to account for this nonlinearity.

Nevertheless, it is possible to show that the simplified approach performed in this paper furnishes useful estimates.

Thanks to the properties of the minimization problem (8) it is easily established that the potential W (D) admits the following lower bound:

W (D) ≥ Φ 1 (D) + Φ 2 (D) (28) with Φ 1 (D) = (1 -ϕ) τ c min d∈G(D) d ℓ (29) and Φ 2 (D) = (1 -ϕ) η n + 1 min d∈G(D) dn+1 ℓ (30)
The two quantities Φ 1 and Φ 2 define the effective behavior of two suspensions with the same microstructure as the yield stress suspension. Φ 1 defines the constitutive law of a suspension of particles embedded in a plastic material obeying the standard flow rule for a von Mises criterion with yield shear stress τ c . Φ 2 is the overall potential of a suspension of rigid particles dispersed in a fluid whose behavior is described by the incompressible viscous power-law:

σ = 2η dn-1 d -pδ trd = 0 (31)
Thanks to the properties of the function dm ℓ for m ≥ 1 and of the set G (D), it can be shown that the material defined by the potential Φ 1 is an isotropic rigid plastic material and the overall potential Φ 2 is that of an isotropic incompressible viscous material whose overall behavior is described by an homogeneous function of degree n. Moreover, if the suspending fluid obeys the Bingham law (n = 1), the effective potential Φ 2 is that of a Newtonian fluid.

Then, we can state that the macroscopic behavior of the isotropic suspension is energetically "underestimated" by a yield stress state law with same exponent as that of the bearing fluid. Both the overall yield criterion and the viscous behavior can be determined through the solution of two homogenization problems allowing to explicitly calculate the lower bound (28). Of course, the overall potential of the yield stress suspension is not the sum of two simple potentials. Nevertheless, it is obvious that the first term of the lower bound (28) prevails for the lower values of the macroscopic strain rate D, while for the larger values of the strain rate, the second term must be taken into account. This property allows to show that, although the macroscopic behavior of the suspension is not Herschel-Bulkley, it tends toward that of a yield stress fluid with yield criterion defined by Φ 1 and viscous flow rule described by Φ 2 for respectively the lower values of the macroscopic strain rate (near the yield stress) and the larger values of the macroscopic strain rate (far from the yield stress). These results are summarized in Fig. 10 for a suspension of particles in a Bingham fluid. On the left part of Fig. 10 are drawn the overall potential W and both potentials Φ 1 and Φ 2 as a function of the second invariant of the macroscopic strain rate. For clarity, the lower bound Φ 1 + Φ 2 is not drawn. The overall behaviors of the suspension and of the materials associated with the potentials Φ 1 , Φ 2 and the lower bound Φ 1 + Φ 2 are depicted on the right part of Fig. 10. This diagram is very similar to the Fig. 3 of [START_REF] Castañeda | On the homogenized behaviour of reinforced and other bingham composites[END_REF]. Σ denotes the second invariant of the macroscopic deviatoric stress tensor.

Interestingly, using the method described in section III, it can be shown that Eqs (26) and ( 27) allow to estimate the value of the yield stress defined by Φ 1 and the value of the consistency defined by Φ 2 . Then, even if the overall behavior of the suspension is not exactly Herschel-Bulkley, the estimates ( 26) and ( 27) can account for well defined properties of the suspension. Nevertheless it must be kept in mind that these estimates are not associated to lower bounds of the potentials Φ 1 and Φ 2 . A more accurate estimate for the overall behavior of a two-dimensional Bingham suspension, which is of the form suggested by the solid curve in the right part of Fig. 10, has been proposed by [START_REF] Castañeda | On the homogenized behaviour of reinforced and other bingham composites[END_REF].

D W W Φ 2 Φ 1 Φ 1 Φ 2 D Σ Yield stress
It is clear from Fig. 10 that it is not possible to deduce the value of the macroscopic yield stress Σc from the measurements of the hight limit strain rate behavior of the overall material. Indeed, if one tries to evaluate the overall yield stress of the suspension from the high-limit strain rate behavior as depicted in Fig 10, the obtained estimate, denoted Σ∞ c , is higher than the true yield stress Σc one must apply to make the suspension flowing. It seems that [START_REF] Geiker | On the effect of coarse aggregate fraction and shape on the rheological properties of self-compacting concrete[END_REF] used such a procedure to measure the overall yield stress of the concretes they studied. This result could explain why they measured yield stress values much higher (ten times) than the ones measured by others authors.

The theory predicts that for the higher values of the macroscopic shear rate, the particles do not change the power-law index n. This result is in agreement with previous theoretical and experimental approaches to the rheological properties of concentrated suspensions in non-Newtonian fluid [START_REF] Polinski | Rheological behavior of filled polymeric systems i. yield stress and shear-thinning effects[END_REF]; [START_REF] Castañeda | On the homogenized behaviour of reinforced and other bingham composites[END_REF]].

VI Conclusion

Theoretical estimates for the overall rheological properties of a suspension of noncolloidal and non-Brownian particles immersed in a nonlinear fluid have been proposed. To our knowledge, this approach and the associated estimates are original in the field of rheology. As the problem to be solved is nonlinear, it was necessary to make some approximations in order to compute simple analytical estimates. Here, the estimates are valid provided that the heterogeneities of the secant modulus can be neglected over the domain filled by the fluid phase. Then, the overall properties of the nonlinear suspension are estimated from that of a fictitious linear suspension having the same microstructure.

In the case where the suspending fluid obeys a Herschel-Bulkley law, the estimates for the yield stress and the consistency read τ hom c /τ c = (1 -ϕ)g(ϕ) and η hom /η = g(ϕ) n+1 /(1 -ϕ) n-1 where g(ϕ) denotes the ratio of the macroscopic to the microscopic properties of the fictitious linear material.

Comparison of the theoretical results with experimental data for suspensions of monodisperse rigid noncolloidal particles dispersed in yield stress fluids have also been presented. The experimental procedure used to obtained these data was designed by [START_REF] Mahaut | Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids[END_REF] to evaluate the purely mechanical contribution of the particles to the paste behavior, independently of the physicochemical properties of the materials.

It was found from both the theoretical and the experimental approaches that the dimension-less yield stress and the dimensionless consistency depend on the bead volume fraction only. It has been shown that the yield stress/solid volume fraction and the consistency/solid volume fraction relationships are well fitted to the very simple laws τ hom c /τ c = (1 -ϕ) 1/2 (1 -ϕ/ϕ m ) -1.25ϕm and η hom /η = (1 -ϕ) (1-n)/2 (1 -ϕ/ϕ m ) -1.25(n+1)ϕm respectively.

As long as the coarse particles are spherical, the theoretical estimates agree quite well with experimental data found in the literature even if experimental procedures used by the authors do not exactly comply with the assumptions made to obtain the theoretical results. This is all the more satisfactory that only four numbers are necessary to calculate the estimates of the overall properties of the suspension as a function of the solid volume fraction: the close packing density of the particles and three suspending fluid properties (yield stress, consistency and power law index).

We now plan to study the case of polydisperse systems and that of anisotropic particle distributions. To improve the theoretical estimates, it will also be necessary to separate the fluid domain in several geometrical domains in order to better describe the heterogeneities over the liquid domain. In particular this situation is expected to be founded for higher values of the solid volume fraction or for polydisperse suspensions.

Figure 2 :

 2 Figure 2: Secant modulus formulation of nonlinear incompressible isotropic fluid.

Figure 3 :

 3 Figure 3: Dimensionless yield stress τ hom c /τ c as a function of (1 -ϕ)G hom /G for all the systems studied by Mahaut et al. (2007). The open square symbols are suspensions of polystyrene and glass beads in bentonite, the open circle symbols are suspensions of glass beads in carbopol while the solid diamond symbols are suspensions of polystyrene and glass beads in emulsion. The figure's coordinates were chosen so that the y = x line represents the theoretical relation (20).

Figure 4 :

 4 Figure4: Dimensionless elastic modulus G hom /G vs the beads volume fraction ϕ for all the systems studied by[START_REF] Mahaut | Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids[END_REF]. The open square symbols are suspensions of polystyrene and glass beads in bentonite, the open circle symbols are suspensions of glass beads in carbopol while the solid diamond symbols are suspensions of polystyrene and glass beads in emulsion. The solid line is the Krieger-Dougherty Eq. (23) with ϕ m = 0.57.

Figure 5 :Figure 6 :

 56 Figure 5: Dimensionless yield stress τ hom c/τ c vs the beads volume fraction ϕ for all the systems studied by[START_REF] Mahaut | Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids[END_REF]. The open square symbols are suspensions of polystyrene and glass beads in bentonite, the open circle symbols are suspensions of glass beads in carbopol while the solid diamond symbols are suspensions of polystyrene and glass beads in emulsion. The solid line is the theoretical prediction (26) with ϕ m = 0.57. The dashed curve is the dilute estimate (24) for the yield stress.

Figure 7 :

 7 Figure 7: Dimensionless yield stress τ hom c /τ c vs beads volume fraction ϕ for suspensions of Ancey and Jorrot (2001) (empty diamond-shaped) and Erdogan (2005) (solid diamond-shaped). The solid line is the theoretical prediction (26) with ϕ m = 0.635 and the straight dashed line is the dilute overall yield stress (24). The dashed curves are the theoretical Eq. (26) with ϕ m = 0.67 (dashed curve below the solid curve) and ϕ m = 0.6 (dashed curve above the solid curve).

Figure 8 :

 8 Figure 8: Dimensionless consistency η hom /η vs. the solid volume fraction ϕ for suspensions of Erdogan (2005) (solid diamond-shaped) and suspensions of Geiker et al. (2002) (empty circle for glass beads, other empty symbols for sea dredged, crushed and mixed aggregates). The solid line is the Krieger-Dougherty estimate for the consistency (n = 1) with ϕ m = 0.65. The dashed lines are the same estimate with ϕ m = 0.55 (above the solid curve) and ϕ m = 0.75 (below the solid curve).

Figure 10 :

 10 Figure 10: One-dimensional sketch of the functions W , Φ 1 and Φ 2 as a function of the second invariant of D (left). The associated state laws (fictitious materials, lower bound and overall material) are depicted on the right part of the figure. The suspending fluid is Bingham (n = 1).


Olivier Cahuc

Jean-Yves K'nevez •

Alain Gérard

Philippe Darnis

Gaëtan Albert

Claudiu F Bisu

Céline Gérard

Keywords: self-excited vibrations, experimental model, turning, torsor measurement, central axis of torsor 2

This work aims at analysing the moment effects at the tool tip point and at the central axis, in the framework of a turning process. A testing device in turning, including a six-component dynamometer, is used to measure the complete torsor of the cutting actions in the case of self-excited vibrations. Many results are obtained regarding the mechanical actions torsor. A confrontation of the moment components at the tool tip and at the central axis is carried out. It clearly appears that analysing moments at the central axis avoids the disturbances induced by the transport of the moment of the mechanical actions resultant at the tool tip point. For instance, the order relation between the components of the forces is single. Furthermore, the order relation between the moments components expressed at the tool tip point is also unic and the same one. But at the central axis, two different order relations regarding moments are conceivable. A modification in the rolling moment localization in the (y, z) tool plan is associated to these two order relations. Thus, the moments components at the central axis are particularly sensitive at the disturbances of machining, here the self-excited vibrations.

Introduction

In the three-dimensional cutting configuration, the mechanical actions torsor (forces and moments), is often truncated: the moments part of this torsor is neglected due to the lack of adapted metrology [START_REF] Mehdi | Dynamic behavior of thin wall cylindrical workpiece during the turning process, Part 2: Experimental approach and validation[END_REF], [START_REF] Yaldiz | Design, development and testing of a turning dynamometer for cutting force measurement[END_REF], [START_REF] Yaldiz | A dynamometer design for measurement the cutting forces on turning[END_REF], [START_REF] Lian | Self-organizing fuzzy control of constant cutting force in turning[END_REF], [START_REF] Marui | Chatter vibration of the lathe tools. Part 2 : on the mechanism of exciting energy supply[END_REF]. Unfortunately, until now the results on the cutting forces are almost still validated using platforms of forces (dynamometers) measuring the forces three components. However, forces and pure moments (or torque) can be measured [START_REF] Couétard | Capteurs de forces à deux voies et application à la mesure d'un torseur de forces[END_REF]. Recently, an application consisting in sixcomponent measurements of the actions torsor in cutting process was carried out in the case of high speed milling [START_REF] Couétard | Mesure des 6 actions de coupe en fraisage grande vitesse[END_REF], drilling [START_REF] Laporte | Phenomenological model for drilling operation[END_REF], [START_REF] Yaldiz | Design, development and testing of a four-component milling dynamometer for the measurement of cutting force and torque[END_REF], etc. Cahuc et al., in [START_REF] Cahuc | Experimental and analytical balance sheet in turning applications[END_REF], present another use of this six-component dynamometer in an experimental study: the taking into account of the cutting moments allows a better machine tool power consumption evaluation. It allows a better approach of the cut [START_REF] Bisu | Nouvelle analyse des phénomènes vibratoires en tournage[END_REF], [START_REF] Couétard | Caractérisation et étalonnage des dynamomètres à six composantes pour torseur associé à un système de forces[END_REF], [START_REF] Laporte | A parametric model of drill edge angles using grinding parameters[END_REF] and thus allows to reach new properties of the vibrations of the chip-tool-workpiece system in the dynamic case.

Moreover, the tool torsor has the advantage of being transportable in any space point and, especially, at the tool tip, here called O point. The following study is carried out in several stages, including two major ones. The first one is related to the analysis of forces. The second one is dedicated to determining of the central axis and a first moments analysis at the central axis during the cut.

In section 2, we present first the experimental device used and associated measurement elements.

Section 3 is devoted to the measurement of the cutting process torsor. An analysis of the forces exerted during the cuttig process is performed. It allows to establish experimentaly several properties of the cutting actions resultant. The case of the moments at the tool tip point is also accurately examinated. The central axis of the torsor is required (section 4.1). The existence of central axes highlighted from the multiple tests confirms noticeably the presence of moments at the tool tip point. In section 4.2 we carry out more particularly the analysis of the moments at the central axis by studying the case the most sensitive to vibrations (ap=5 mm, f = 0.1 mm/rev). Before concluding, this study gives a certain number of properties and drives to some innovative reflexions.

Experimental device

The experimental device presented in Fig. 1 is a conventional lathe (Ernault HN 400). The machining system dynamic behaviour is identified using a three-direction Fig. 1 Experimental device and associated measurement elements accelerometer fixed on the tool. Two unidirectional accelerometers are positioned on the lathe and on the front bearing of the spindle, in order to identify the influence of this one during the cutting process. All the cutting actions (forces and torques at the tool tip point) are measured by a six-component dynamometer [START_REF] Couétard | Capteurs de forces à deux voies et application à la mesure d'un torseur de forces[END_REF]. The instantaneous spindle speed is continuously controlled (with an accuracy of 1%) by a rotary encoder directly coupled with the workpiece. The connection is carried out by a rigid steel wire. The test workpieces are cylindrical and have a diameter of 120 mm and a length of 30 mm. The geometry of the support part holder was optimized by a finite element analysis (using SAMCEF R software) in order to confer a maximum rigidity to the unit. This procedure is described in [START_REF] Bisu | Etude des vibrations auto-entretenues en coupe tridimensionnelle: nouvelle modélisation appliquée au tournage[END_REF]. Thus, under a load P = 1,000 N, for material having a Young modulus E = 21×10 5 N/mm 2 , the workpiece dimensions selected are: D 1 = 60 mm (diameter), L 1 = 180 mm (length), for a bending stiffness of 7×10 7 N/m (Fig. 2) [START_REF] Bisu | New method to characterize a machining system: application in turning[END_REF]. These values are within the higher limit of the stiffness zone acceptable for a conventional lathe [START_REF] Ispas | Vibrations des systèmes technologiques[END_REF], [START_REF] Koenigsberger | Machine Tools Structures[END_REF], [START_REF] Konig | Zweikomponenten schnittkraftmesser[END_REF]. The insert tool used is a TNMA 16 04 12 type carbide not coated, without chip breaker. The machined material is a chrome molybdenum 42CrMo24 type alloy. Moreover, the tool geometry (Fig. 3) is characterized by the cutting angle γ, the clearance angle α, the inclination angle of edge γ s , the direct angle κ r , the nozzle radius r ǫ and the sharpness radius R [START_REF] Laheurte | Application de la théorie du second gradient à la coupe des matériaux[END_REF]. In order to limit the influence of the tool wear on measurements, the tool insert is examined after each test, and is changed if necessary (Vb ≤ 0.2 mm ISO 3685). Nevertheless, the present study does not strictly speaking deal with fretting and tool wear. The tool parameters are detailed in the Table 1. -6 0 6 0 -6 0 91 0 1, 2mm 0, 02mm Table 1 Geometrical characteristics of the tool 3 Cutting torsor actions

Tests

The experiments are performed within a framework similar to the one described in Cahuc et al., [START_REF] Cahuc | Experimental and analytical balance sheet in turning applications[END_REF]. For each test, the complete torsor of the mechanical actions is measured using the six-component dynamometer according to the method initiated in Toulouse [START_REF] Toulouse | Contribution à la modélisation et à la métrologie de la coupe dans le cas d'un usinage tridimensionnel[END_REF], developed by Couétard [START_REF] Couétard | Caractérisation et étalonnage des dynamomètres à six composantes pour torseur associé à un système de forces[END_REF] and used in several occasions [START_REF] Couétard | Mesure des 6 actions de coupe en fraisage grande vitesse[END_REF], [START_REF] Cahuc | Experimental and analytical balance sheet in turning applications[END_REF], [START_REF] Darnis | Energy balance with mechanical actions measurement during turning process[END_REF], [START_REF] Laheurte | Metrological devices in cutting process[END_REF], [START_REF] Laheurte | Evaluation de l'énergie mise en jeu et du comportement des outils de coupe dans l'usinage[END_REF]. These mechanical actions are evaluated for two depths of cut: ap = 2 and 5 mm; and for four feed rates: f = 0.05; 0.0625; 0.075 and 0.1 mm/rev. The sixcomponent dynamometer gives the instantaneous values of all the torque cutting components in the threedimensional space (x, y, z) related to the machine tool (Fig. 1). Measurements are performed in O', which is the center of the six-component dynamometer transducer. Then, they are transported to the tool point O via the moment transport classical relations. Measurement uncertainties of the six-component dynamometer are: ±4% for the forces components and ±6% for the moment components.

To ensure the results repetitivity, for each configuration specified above, 4 series of measurements were carried out. For each series of measures, we analyzed the results over 15 seconds of recording. With 44 points of measurements recorded per spin round of the piece, the median values discussed thereafter correspond finally to the average of 7,590 measurements.

Resultant of the cutting actions analysis

For the four values feed rate f indicated above, two examples of forces resultant measurements applied to the tool tip point are presented: one of these for the stable case (quasi non-existent vibrations) ap = 2 mm (Fig. 4), and another one for the unstable case (self-excited vibrations), ap = 5 mm, f = 0.1 mm/rev (Fig. 5). In the stable case (ap = 2 mm, f = 0.1 mm/rev), it appears that the force component amplitudes remain almost independent of the time parameter. Thus, the amplitude variation is limited to 1 or 2 N around their rated values, starting with 200 N for (F x ) 400 N for (F z ) and 600 N for (F y ). These variations are quite negligible and lower than uncertainties of measurements. Indeed, when the rated stress is reached, the component noticed as the lowest value is (F x ), while the highest in the absolute value is (F y ). Taking as reference the absolute value of (F x ), the following relation between these three components comes:

|F x | = |F z | 2 = |F y | 3 . (1)
In the unstable case, we observe that the force component on the cutting axis (F y ) has the most important average amplitude (1,500 N). It is also the most disturbed (±1, 000 N), with oscillations between -2,450 N and -450 N. The force along the feed rate axis (F z ) has also an important average amplitude (1, 000 N), but the oscillations are smaller: ±200 N in absolute value and ±20% in relative value. The efforts along the radial direction (F x ) are the weakest in terms of average amplitude (200 N), but also the most disturbed regarding the relative value (±240 N). These important oscillations are the tangible consequence of the contact tool/workpiece frequent ruptures and, thus, demonstrate the vibration and dynamical behaviour of the system workpiece-toolmachine WTM.

Finally, taking as reference the absolute value of (F x ), the following relation between the absolute values of these three components comes:

|F x | = |F z | 5 = |F y | 7.5 . (2)
In other words, whatever the depth of cut ap, we have the following order relation between the absolute values of the cutting actions components:

|F x| ≤ |F z| ≤ |F y|. ( 3
)
More precisely, in the case of turning studied here, for the depth of cut ap = 2 mm (stable case) or ap = 5 mm (unstable case) we have, with the errors of experiments close the following relation between the absolute values of the cutting actions components:

|F x | = |F z | ap = |F y | 1.5ap . ( 4
)
We find that the cutting actions are proportional to the depth of cut. This result is in agreement with [START_REF] Benardos | Prediction of workpiece elastic deflections under cutting forces in turning[END_REF], [START_REF] Sr | The impact of cutting conditions on cutting forces and vibration signals in turning with plane face geometry inserts[END_REF]. In the unstable case (ap = 5mm, f = 0.1 mm/rev) the force resultant component detailed highlights a plan in which a variable cutting forces F v moves around a rated value F n [START_REF] Bisu | Self-excited vibrations in turning: forces torsor analysis[END_REF]. This variable force is an oscillating action (Fig. 6) that generates tool tip displacements and maintains the vibrations of elastic system block-tool BT. Thus, the cutting force variable and the self-excited vibrations of elastic WTM system are interactive, in agreement with research work [START_REF] Lian | Self-organizing fuzzy control of constant cutting force in turning[END_REF], [START_REF] Koenigsberger | Machine Tools Structures[END_REF], [START_REF] Sr | The impact of cutting conditions on cutting forces and vibration signals in turning with plane face geometry inserts[END_REF], [START_REF] Ball | A treatise on the theory of screws[END_REF].

Study of the moments to the tool point

Like for the torsor resultant, we give here two examples of statements of the moment components at the tool point in the machine frame (Fig. 1). The stable case is first presented, see Fig. 7 (without vibrations ap = 2 mm). Then the unstable case is shown in Fig. 8 (with vibrations ap = 5 mm). point. However, we can note that the moment components at the tool tip point are more disturbed than their equivalents regarding the resultant. Thus, the moment components seem more sensitive to the self-excited vibrations than the force components applied. Thus, the follow-up of those could be a good mean of precociously detecting the existence of regenerative vibrations.

The analysis of the results shown on

|M oz | = |M oy | 1.5 = |M ox | 2 , (ap = 2mm). ( 5
)
The analysis of the results show on Fig. 8 allows to establish that among the three moment components, the average value of the component along the cutting axis is the lowest (M oy = -100 dN.m). It is also the most disturbed (±280 dN.m i.e. variations of about 280%!). The component of the torque along the radial axis (x) is continuously positive M ox = 170 dN.m (±150 dN.m); it is thus the most raised moment component and, proportionally, the least disturbed (±90 %). The M oz moment component along the feed rate direction is continuously negative, with an average value about -135 dN.m in absolute value, the highest, but less disturbed than M oy (with oscillations of ±140 dN.m i.e. only ±105 %). Fi-nally, for this depth of cut, we have the following relation between the absolute values of the moment components:

|M oy | = |M oz | 1.4 = |M ox | 1.7 , (ap = 5mm), (6)
with a module of average torque at the tool tip point about |M o | = 275 dN.m. The comparison of Eq. ( 2) and Eq. ( 6), shows that the role of x and y axes is reversed for ap = 5 mm. This allows to think that the transport of the moments at the tool tip point has a major effect. Moreover, whatever the value of ap (= 2 mm; 5 mm), M ox modulus keeps the highest value in the turning cases here considered. It is also the only positive and slightly decreasing component when the depth of cut increases. Furthermore, one notes that the components M oy and M oz are continuously negative when the depth of cut increases. One notes also that the component M oy is decreasing while the component M oz is

The modulus of the moments component at the tool tip point, increases with the depth of cut, like the one of the resultant.

In the same way, for ap = 5 mm, the general order relation (3) for the resultant is replaced for the moment at the tool tip point by:

|M oy | ≤ |M oz | ≤ |M ox | . (7)
The x and y axes positions are also reversed regarding the resultant and moment at the tool tip point components. This can be allotted to the moments transport at the tool tip point.

It is always important to know the influence of the vibrations on the behavior of the moments at the tool tip point. So, we remain in the case of the strong vibrations (ap = 5 mm) to examine their influence when the feed rate increases. An example of the behavior of the moment components average values at the tool tip point is described in figure 9. The following conclusions are obtained (ap = 5 mm, f variable):

-the validity of the order relation of ( 7) is extended to the case of a variable feed rate, -the component M oz which consumes the mechanical power goes through a light extremum around feed rate f = 0.0625 mm/rev, -the component M oy varies like the component M oz , but in more accentuated way, and also goes through a extremum more marked around f = 0.0625 mm/rev, -the component M ox is quasi-constant and continuously negative, -finally the results indicated above are more accentuated in the tool reference system than in the machine tool reference system.

These results show that it is better to follow the moments evolution in the tool reference system than this one of the machine tool. Let us see now in the following section the influence of the vibrations on the central axis of the mechanical actions torsor. 4 Moments analysis at central axis

Central axis

It is well-known that, it is possible to associate a central axis, to any torsor (except the torsor of pure moment), which is the single object calculated starting from the six torsor components [START_REF] Brousse | Cours de mécanique (1 er cycle et classe prépa[END_REF]. A torsor [A] O in a point O is composed of the resultant forces R and the resulting moment M O :

[A] O = R, M O . ( 8
)
The central axis is the straight line classically defined by:

OA = R ∧ M O |R 2 | + λR, ( 9
)
where O is the point where the mechanical actions torsor was moved (here, the tool tip) and A the current point describing the central axis. Thus, OA is the vector associated with the bi-point [O, A] (Fig. 10).

This line (Fig. 10a) corresponds to the geometric points where the mechanical actions moment torsor is minimal. The central axis calculation consists in determining the points assembly (a line) where the torsor can be expressed along a slide block (straight line direction) and the pure moment (or torque) [START_REF] Brousse | Cours de mécanique (1 er cycle et classe prépa[END_REF].

The central axis is also the point where the resultant cutting force is colinear with the minimum mechanical moment (pure torque). The test results enable us to check, for each measurement point, if there is colinearity between the resultant cutting force R and moment M A calculated at a point of the central axis (Fig. 10b). The meticulous examination of the six mechanical action torsor components shows that the forces and the moment average values are not null. For each point of measurements, the central axis is calculated in the stable (Fig. 11a) and unstable modes (Fig. 11b). In any rigour, the case ap = 2 mm and f = 0.1 mm/rev should be described as quasi-stable movement because the vibrations exist, but their amplitudes are very small -of the order of micrometers-thus, quasi null compared to the other studied cases. Considering the cutting depth value ap = 5 mm and f = 0.0625 mm/rev, the recorded amplitude was 10 times more important. The central axes beam is larger in unstable case, and is also more tilted compared with the normal axis on the plane (x, y). The self-excited vibrations, due to the variable moment generation, can explain this central axis dispersion.

Moments analysis

The minimum moment (pure torque) M A is obtained by transporting the moment from tool tip at the central axis. Constant and variable parts are deduced from the moment values at the central axis. Like for the efforts, the variable part is due to the self-excited vibrations, as revealed below.

The moments contribution on the areas of contact tool-workpiece-chip is expressed using this decomposition. The observations resulting from the analysis show that the tool vibrations generate rotations; cause variations of contact; and, thus, generate variable moments, confirming the efforts analysis detailed in Section 3. Let us express the moments along the three axes of the machine tool: swivel moment in the y direction and the two rolling moments along x and z directions.

For the unstable case, characterized by ap = 5 mm and f = 0.1 mm/rev, the analysis of the cutting torque components average values at the central axis (subscripted by the letter a) allows to establish that among the three moment components, the average value of the component along the radial axis is (in absolute value) the lowest (M ax = -4 dN.m). It is very disturbed in relative value (±24 dN.m i.e. 600%). The absolute value of the component along the cutting axis is the highest (M ay = -34 dN.m) with a weaker disturbance in absolute value (±160 dN.m i. e. 470%). The component along the feed rate axis is medium (M az = 8 dN.m). It is the most disturbed in relative value (±110 dN.m i. e. 1,300%). Thus, one has, by the errors of measurement, the following relation between the absolute values of the three torque components:

|M ax | = |M az | 2 = |M ay | 8 , (ap = 5mm). (10)
The confrontation of Eq ( 6) and [START_REF] Cahuc | Experimental and analytical balance sheet in turning applications[END_REF] shows the interest of transporting the moment from the tool tip point to central axis. It is also interesting to note that the role of pivot of z axis is preserved. The role of x and y axes is reversed here. Thus, the order relation ( 7) is replaced by:

|M ax | ≤ |M az | ≤ |M ay | . (11)
At the central axis, the modulus of the torque components and those of the forces are always similar. Thus, the torque along the feed rate axis z is framed by the component on the radial axis x which is the weakest. The strongest is on the cutting axis y. It is as also interesting to note as the modulus of the average torque of cut at the central axis is worth to |M a | = 90 dN.m, either 3 times less than at the tool point.

This result encourages us to study again the influence of the vibrations on the moments evolution at the central axis when the feed rate varies. Let us consider the case of the strong vibrations (ap = 5 mm) to examine their influence when the feed rate increases. An example of the behavior of the average values of the moment components at the central axis in the tool reference system is given at figure 12. We note that the evolution of the moments components at the central axis is very different from those recorded at the tool tip figure 9.

The examination of the moment components at the central axis is enriching. We obtain the following conclusions (ap = 5 mm, f variable): and quasi-constant till f = 0.0625 mm/rev, then strongly decreasing after and becomes negative after the value f = 0.075 mm/rev. -Finally the results indicated above are accentuated in the tool reference system than in the machine tool one, -For variable feed rate (ap = 5 mm), one notes that the torque modulus is increasing with feed rate as for the resultant cutting force, on the central axis. -The single order relation [START_REF] Bisu | Nouvelle analyse des phénomènes vibratoires en tournage[END_REF] is replaced by the two following:

|M ax | ≤ |M ay | ≤ |M az | (if, f ≤ 0.0625mm/rev), ( 12
)
|M ax | ≤ |M az | ≤ |M ay | (if, f > 0.0625mm/rev). ( 13
) -These two relations are associated to a modification of the localization of the rolling moment in the (y, z) tool plan (and in agreement with figure 12). This one passes from the fourth quadrant of the plan (y, z) for f ≤ 0.0625 mm/rev to the second quadrant for f > 0.0625 mm/rev. This change of the bearing moment orientation is associated to a modification in the ejection of the chip. This one is evacuated towards the left of the x axis (f ≤ 0.0625 mm/rev) then towards the right of the x axis (f > 0.0625 mm/rev). This analysis of the results presented at figure 12 is confirmed by the examination of the corresponding recording videos.

The results show that it is better to study the moments evolution in the tool reference system than in the machine tool system. It is also obvious that the analysis of the mechanical actions moment must be performed at the central axis.

Conclusion

Experimental procedures here developed allowed to determine the elements required for a rigorous analysis of the influence of tool geometry, its displacement and the evolution of contacts tool/workpiece and tool/chip on the machined surface. These experimental results allowed to establish a vectorial decomposition of actions analysing the resultant of applied actions torsor during a turning process. Thus, it is demonstrated that the cutting force evolves around a constant rated value. This variable force moves in a plan tilted compared to the machine spindle.

Furthermore, a really innovative study of the moments at the tool tip point, and at the central axis of the mechanical action torsor, is presented. For the resultant like for the moment, at the tool tip point or the central axis, the modulus of these elements is increasing with the feed rate. If one compares the average values of the components modulus action with those of the moments at the tool tip point an order relation clearly appears. In particular one notes that the roles of x and y axes are inversed, and z axis plays the role of "pivot". On the other hand, at the central axis there does not exist a unic single order relation but two. These two relations corresponds to a modification of the localization of the rolling moment in the (y, z) tool plan. This one goes from the fourth quadrant of the plan (y, z) for f ≤ 0.0625 mm/rev to the second quadrant for f > 0.0625 mm/rev. This change of the rolling moment orientation is associated to a modification in the ejection of the chip. This one is evacuated towards the left of the x axis (f ≤ 0.0625 mm/rev) then towards the right of the x axis (f > 0.0625 mm/rev). This analysis of the results presented in figure 12 and is confirmed by the examination of the corresponding recording videos.

Furthermore, this review allows to better understand the results obtained in ( [START_REF] Bisu | Self-excited vibrations in turning: forces torsor analysis[END_REF], Fig. 13). Indeed, it was noticed that the place of moments has a shape of buttonhole mainly inscribed in a plan. Its tips come slightly out of this plan along a small segment. This particular buttoncoole shape reveals that the moment is passing from the second to the fourth quadrant (see also [START_REF] Bisu | Self-excited vibrations in turning: forces torsor analysis[END_REF], Fig. 16).

Only the moment modulus along the radial axis remains the weakest when the feed rate increases. This property is common with the component modulus of the same row of the resultant. For the modulus of the other components of the moments at the central axis, their relative position depends on the feed rate value. Moreover, one notes that the moment components at the central axis are subjected to oscillations much more important than these same components at the tool tip point. This shows that the moments components at the central axis are particularly sensitive to the disturbances of machining, here the self-excited vibrations. Thus, the follow-up of these moments evolution at the central axis allow to follow, a priori, the surface quality of the machined piece. The more important the amplitudes of vibrations of tools are, the poorer the surface quality of the piece manufactured is. This suggests one should look further into the study of the cutting moments, especially, at the central axis. More precisely, all these remarks show that it is particularly important to be placed under cutting conditions such that the central axis of the torsor passes by the cutting line (or so not with more close possible of the cutting line). Indeed, such a configuration corresponds to the cutting moments minimum. This situation leads then to the minimum friction conditions supporting of the minimal rises in temperature, and thus also a minimum tool wear, since all these elements are dependant and increasing with the applied forces [START_REF] Toh | Static and dynamic cutting force analysis when high speed rough milling hardened steel[END_REF], [START_REF] Lalwani | Experimental investigations of cutting parameters influence on cutting forces and surface roughness in finish hard turning of MND250 steel[END_REF].

Fig. 2

 2 Fig. 2 Geometry and dimension of the support part holder and the workpiece

Fig. 3

 3 Fig. 3 Tool geometry

Fig. 4

 4 Fig. 4 Signals related to the resultant components of cutting forces following the three (x, y, z) cutting directions; test case using parameters ap = 2 mm, f = 0.1 mm/rev, and N = 690 rpm

Fig. 5

 5 Fig. 5 Signals related to the resultant components of cutting forces following the three (x, y, z) cutting directions; test case using parameters ap = 5mm, f = 0.1 mm/rev, and N = 690 rpm

Fig. 6

 6 Fig. 6 Cutting force Fv evolution around the nominal value Fn.

Fig. 7

 7 Fig. 7 The moment components time signals that act on the tool tip following the three directions of space machine in the stable case ap = 2 mm, f = 0.1 mm/rev and N = 690 rpm

Fig. 8

 8 Fig.8The moment components time signals that act on the tool tip following the three directions of space machine in the unstable case ap = 5 mm, f = 0.1 mm/rev and N = 690 rpm

  Fig. 7 allows to establish that, among the three moment components, the average value of the component along the axis feed rate is the lowest (M oz = -40 dN.m). It is the most disturbed in relative value (±20 dN.m i.e. 50%). The absolute value of the component along the axis of cut plays the role of pivot (M oy = 60 dN.m) with a weaker disturbance in absolute value (±16 dN.m i.e. 30%). The highest component is along the radial axis (M ox = 80 dN.m) with a weaker disturbance in absolute value (±20 dN.m i.e. 25%). Taking as reference the average values of the component M oz , it comes :

Fig. 9

 9 Fig. 9 Evolution of the moment components at the tool tip point in the tool tip reference system for ap = 5 mm

Fig. 10

 10 Fig. 10 Central axis representation (a) and of the colinearity between vector sum R and minimum moment MA on central axis (b)

Fig. 11

 11 Fig. 11 Central axes representation obtained for 68 spin rounds of the workpiece feed rate f = 0.0625 mm/rev; a) stable process ap = 2 mm; b) unstable process ap = 5 mm

Fig. 12

 12 Fig. 12 Evolution of the moments components at the central axis in the tool tip reference system for ap = 5mm.

Acknowledgements The authors would like to thank the CNRS (Centre National de la Recherche Scientifique UMR 5469) for the financial support of this project.