R Penas

A Gaudin

A Kreis

E Balmes

Dissipation in hysteretic rubber mount models

Rubber mounts are elements of extreme importance in automotive suspension, and accurate modeling is crucial for comfort design. While mount characterization is typically done using cycles, actual performance is often associated with transients. The paper thus focuses on the impact of power dissipation on suspension models during transients. For scalar or 0D hysteretic models that respect Madelung rules and Masing's law, a method is introduced to compute instant dissipation even for models where it is not explicitly available. It is shown that stiffness and dissipation depend only on the last turning point and this dependence should be the core aspect for identifying and modeling hysteretic dissipation. The second section introduces two different 0D models having the same full cycle dissipation and force amplitude, thus the same storage and loss moduli in a first harmonic approximation of the mount behavior, though having different instantaneous dissipation. The case of a transient starting torque soliciting a suspended powertrain is finally considered. The different suspension models are shown to have different instant dissipation which might deeply modify the conclusions drawn from the dynamic simulation.

INTRODUCTION

Rubber mounts are widely used in automotive industry due to their unique combination of moderate stiffness and high dissipation. These properties are particularly interesting for softening and filtering vibrations in articulations between moving parts. Despite the intense utilization of rubbery materials, their behavior is still subject of active research, given that current modeling strategies cannot provide fully predictive design.

One of the main modeling demands for this type of articulations comes from multibody simulations where a low computational cost is a modeling constraint. For this reason, scalar or 0D models are the most commonly used for emulating rubber articulations. These models can describe and accumulate different nonlinear effects, while keeping a relatively low complexity. Models of this kind are often found in the specific literature such as [START_REF] Coveney | Rate-dependent modeling of a highly filled vulcanizate[END_REF], [START_REF] Sjöberg | On dynamic properties of rubber isolators[END_REF], [START_REF] Bourgeteau | Modélisation numérique des articulations en caoutchouc de la liaison au sol automobile en simulation multi-corps transitoire[END_REF].

To identify this complex behavior, the most used technique is to impose cyclic solicitations and fit the force response with scalar models. Cyclic identification does not provide information on dissipation during transients. The first section describes hysteretic models which are the focus of this work. It is in particular shown how instant dissipation, even if is not accessible experimentally, can be computed and used as a valuable asset for analysis, by giving an estimation of when and where the dissipation is concentrated. The second section describes two models equivalent from the point of view of cyclic dissipation but leading to very different behavior in an engine start simulation.

In the scope of this work, hysteresis is taken as a rate-independent behavior that respects the Madelung rules and the Masing's law (see [START_REF] Brokate | Hysteresis and phase transitions[END_REF]). The rate-independence means that viscoelastic contribution is not taken in account and should be object of further work.

The Madelung rules state that :

• Any curve Γ 1 emanating from a turning point A of the force/displacement graph (figure 1) is uniquely determined by the coordinates of A.

• If any point B on the curve Γ 1 becomes a new turning point, then the curve Γ 2 originating at B leads back to the point A.

• If the curve Γ 1 is continued beyond the point A, then it coincides with the continuation of the curve Γ which led to the point A before the Γ 1 -Γ 2 cycle was traversed. Since this is true for any cycle length, the tangent stiffness is an even function of the displacement, that is K + (e) = K -(-e). Thus in figure 2 rather than displaying the classical cycle on the top, one chooses to represent the tangent stiffness.

Other hysteretic model main characteristic is the fact that the first loading curve should be homothetic to the first unloading curve, with a scale factor of two, known as Masing's law [START_REF] Brokate | Hysteresis and phase transitions[END_REF]. Combining both laws, illustrated in figure 2, one can represent K(e) as a function of distance to turning point with a factor 2 on e when starting from 0.

Concerning dissipation, the parity property is quite useful, since dissipation can be split equally between

E d = e B e A F dis de = 1 2 F de = 1 2 ( e B e A F + de + e A e B F -de) = e B e A F (e)de -F (e B -e A )(e B -e A ) (2)
which can be used to compute the instantaneous value of dissipated energy. These rules are applicable when neglecting viscoelastic and nonlinear elastic effect. Figure 3 illustrates tests with triangular enforced displacement. A low, constant speed was used to minimize viscoelastic effects which should remain negligible. Nonlinear elastic effects, from both geometric and material natures are also negligible due to the use of deformations below 10%. The results are shown in the force/displacement plane in figure 3. This plot suggests that the stated hypotheses are plausible, but a simpler verification is to show, as in figure 4, the instant stiffness as function of the distance to the last turning point. In this representation, the verification of hypotheses is confirmed by the superposition of all curves. Since the K(e -e turn ) curve completely defines an hysteretic model, a non-parametric representation of the behavior is simply achieved though its discretization. [START_REF] Segalman | A four-parameter iwan model for laptype joints[END_REF] in his work proposes a model based on a rheologic representation, shown in figure 5. This model allows the reproduction of the curve respecting both laws with a continuous force evolution and control on the accuracy. An alternate hysteretic model that respects the stated hypotheses is the STS model, introduced and detailed by [START_REF] Coveney | Rate-dependent modeling of a highly filled vulcanizate[END_REF]. This model is based on an spring in parallel of an assembly of identical Jenkins cells in series, shown in figure 7. By making the springs stiffness tend to infinity and the friction forces tend to zero, with their product constant (k t F f = C t ), one reaches to a continuous model, defined by the equation 3. (3)

e 0 k 0 k 1 F f 1 e 1 k n F fn e n
k 0 e 0 e 1 F f e 2 F f e k k t F f k t k pa

NONLINEAR ELASTICITY AND HYSTERETIC DAMPING

In order to apply the the models on section 2, identification tests were made on three mounts that compose a powertrain suspension. These tests were made with enforced displacement at constant and low speed, and the results are illustrated by figure 8. The curly shape of test data suggests the presence of a non linear elastic component and classical dissipative models are not capable of representing this kind of behavior. A nonlinear elastic model was built using the mean test load for each displacement and is shown in figure 9 left.

The remaining forces, shown in figure 9 right, are supposed to be the generated by the dissipative model in parallel of the nonlinear elastic model. The left mount (and on a smaller scale the anti torque mount) presents significantly less dissipation around null displacement. It is probably because a significant part of the rubber is not excited within small displacements, and after a certain level, it is solicited by a self-contact. Tracing the stiffness as function of distance to the turning point without the nonlinear elastic contribution, in figure 10, as it was suggested in section 2, one can see that its evolution for small distances to the turning point is quite similar to the ones provided by hysteretic models. The absence of dissipation around null displacement creates bumps in the curves between 6 and 10 mm, more visible for the left mount. The end of the curve shows strong negative stiffness, as for a form of symmetry, which is probably due to the removal of a nonlinear elastic part. To fit the dissipation in parallel of the non-linear elasticity, two different models were used: one is the STS model and the other is a simple viscous dissipation. Both models were tuned to dissipate the same amount of energy for a harmonic solicitation with frequency close to the powertrain rolling frequency (around 10 Hz), with the same amplitude as the identification tests. Figure 11 shows the instant dissipation for both models of the right mount under the stated load, and even if the total dissipation is the same, their instant dissipation is different.

APPLICATION CASE

With the mounts identified, a model of the whole powertrain suspension was developed. The powertrain unit modeled by its mass and inertia matrix, while the three mounts are attached on one side to the powertrain unit, and the other to the chassis, considered as rigid as shown in figure 12. The model input is the engine torque, applied in the direction of the Figure 14 shows the experimental results and simulated forces for both models in the direction of the vehicle motion.

With forces and displacements available for post treatment, viscous dissipation may be explicitly computed, and hysteretic dissipation may be computed by equation 2. Figure 15 illustrates the evolution of dissipated energy for both computed models, and it highlights the fact that for this transient simulation, the dissipated energy differs both in total dissipated energy and timing of when the energy is dissipated. This demonstrate that the original equivalence shown in 3 is not sufficient to provide a proper transient simulation. To analyze the implications of this difference into the system level, a modal analysis was carried around the resting position of the engine. As expected, the six powertrain modes are three translational modes and three rotational modes, listed in table 1. Modal amplitudes are the decomposition of the displacement on the directions of the eigenvectors (Bianchi, Balmes, Vermot Des Roches, & Bobillot 2010). This decomposition of the displacement vec-tor q is given by

q j = φ j T M {q} (4)
with φ the mass normalized eigenvectors and M the mass matrix, and q j the amplitude associated to the mode j. The time derivative of those amplitudes yield the modal speeds which are also necessary for computing modal energies using

2E j = q2 j + ω 2 j q 2 j (5)
with E j the energy and ω j the frequency associated to mode j. This information is useful to know where the energy flows before being dissipated. Modal energy evolution through time is shown in figure 16, for both hysteretic and viscous models. Modes 4 and 5 take most of the imposed energy on the system, which is natural, since they represent the rotations along the direction of the imposed torque. The different dissipation models impact strongly on how the energy is transmitted to modes, leading to conflicting information of which mode is more important to address.

CONCLUSION

This work focused on a specific range of models whose objective is to emulate the rate-independent nonlinear component of rubber mounts behavior. Models of this type used in industry are often identified with full-cycle dissipation equivalence. This equivalence does not take into account all the necessary aspects for a predictive transient simulation.

To sustain this argument, an application case where a starting powertrain unit was simulated with two suspension models that differ only on the dissipation instants and the simulations would lead to different design choices.

The need for identification strategies accounting for a conservative nonlinear elastic part, rate-dependent dissipation (viscoelastic) and rate-independent (hysteretic) dissipation has only been illustrated here and should be an important aspect of future work. For industrial application, evolution of dissipation with non-linear elasticity may also represent an important component of the model for many mounts.

Figure 1 :

 1 Figure 1: Illustration of Madelung rules

Figure 2 :

 2 Figure 2: Illustration of Masing's law. a) classical cycle, tangent stiffness as function time (b) and distance to turning point (c)

Figure 3 :

 3 Figure 3: Imposed displacement and force-displacement response

Figure 4 :

 4 Figure 4: Tangent stiffness as function of distance to last turning point

Figure 5 :Figure 6 :

 56 Figure 5: Rheologic scheme for Iwan's modelDespite needing a high number of Jenkins cells for accurate representation, as seen on figure6, the low computational effort associated to each cell makes it an interesting model for application purposes.

Figure 7 :

 7 Figure 7: Rheologic scheme for the STS model

Figure 8 :

 8 Figure 8: Force/displacement response for the three tested mounts

Figure 9 :

 9 Figure 9: Non-linear elastic response (left) and dissipative forces (right) for the tested mounts

Figure 10 :

 10 Figure 10: Stiffness in function of the distance to the last turning point without nonlinear elastic component

Figure 11 :

 11 Figure 11: Instantaneous dissipation for a STS model and a viscous model under harmonic solicitation

Figure 12 :

 12 Figure 12: Assembled powertrain unit and its suspension

Figure 13 :

 13 Figure 13: Measured torque for a starting engine

Figure 14 :Figure 15 :

 1415 Figure 14: Measured and computed forces at the mounts. Top: Left mount; Middle: Right mount; Bottom: Anti-torque mount

Figure 16 :

 16 Figure 16: Modal energies for each model. Top: Viscous model; Bottom: Hysteretic model

Table 1

 1

	: Powertrain suspension modes
	Mode description	Frequency
	Translation perpendicular to vehicle direction 4.00Hz
	Translation in vehicle direction	6.04Hz
	Translation in vertical direction	7.13Hz
	Pitching powertrain	9.18Hz
	Yawing powertrain	11.66Hz
	Rolling powertrain	13.38Hz


Pierre Gilormini
email: pierre.gilormini@ensam.eu

Francisco Chinesta
email: francisco.chinesta@ensam.eu

Viscous drag and rod orientation kinematics in an orthotropic fluid

Keywords:

A model is proposed, where the evolution of the orientation of each fiber is coupled to the orientations of the surrounding fibers in the flow of a fiber-filled fluid and includes the effects of the fiber volume fraction and aspect ratio. This is performed by accounting for the effective behavior of the fiber-filled fluid, which is anisotropic although the fibers are embedded in an isotropic Newtonian fluid. The rotation of a fiber in these conditions is predicted by a dumbbell model, which allows an extension of Jeffery's equation to anisotropic cases. This involves the numerical evaluation of the drag force applied on a sphere in an orthotropic incompressible fluid, which is evaluated by finite element simulations. A simple fit is proposed for the practical use of the coupled model, which is applied finally to the orientation kinematics of a population of fibers in a simple shear flow, and the results are compared with the ones given by the standard uncoupled approach.

Introduction

In our former works we addressed the kinematics of rods, simulating short rigid straight fibers involved in reinforced plastics, by assimilating them to dumbbells with hydrodynamic forces applied on the dumbbell beads [START_REF] Chinesta | From single-scale to two-scale kinetic theory description of rods suspensions[END_REF] . Such a modeling framework allows recovering the Jeffery equation for a prolate spheroid with infinite aspect ratio [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF] , as shown below. This rationale has been extended to study rigid [START_REF] Abisset-Chavanne | On the multiscale description of dilute suspensions of non-brownian clusters composed of rods[END_REF] and deformable [START_REF] Perez | On the multi-scale description of micro-structured fluids composed of aggregating rods[END_REF] clusters composed of rods, to address confinement effects [START_REF] Perez | A multi-scale description of orientation in simple shear flows of confined rod suspensions[END_REF][START_REF] Scheuer | Second-gradient modelling of orientation development and rheology of dilute confined suspensions[END_REF] , to include higher-order gradients [START_REF] Abisset-Chavanne | A second-gradient theory of dilute suspensions of flexible rods in a newtonian fluid[END_REF] , to consider viscoelastic surrounding fluids [START_REF] Borziacchiello | Orientation kinematics of short fibres in a second-order viscoelastic fluid[END_REF] , or to account for rods inertia [START_REF] Scheuer | Microscopic modelling of orientation kinematics of non-spherical particles suspended in confined flows using unilateral mechanics[END_REF] . Moreover, in [START_REF] Abisset-Chavanne | On the multiscale description of dilute suspensions of non-brownian clusters composed of rods[END_REF] the authors proved that the Jeffery equation for general ellipsoids is recovered by using a rigid tri-dumbbell analog. All these developments leading to the equivalence between ellipsoids and dumbbells were performed under the assumption that the particle is immersed in an isotropic fluid. When the particles volume fraction increases, the so-called semi-dilute regime is reached, where direct contacts between fibers can still be neglected but hydrodynamic interactions come into play as referred in the papers above (and the abondant references therein) or as addressed within a thermodynamic setting in [START_REF] Grmela | One and two-fiber orientation kinetic theories of fiber suspensions[END_REF] . This induces an anisotropic effective behavior of the fluid surrounding a particle and should alter its orientation kinematics, which is expected to deviate from the standard Jeffery prediction.

To our knowledge, the effects of this anisotropy on the rod kinematics has never been addressed, and all theories and models giving the orientation kinematics rely on an isotropic environment, in opposition with what is expected from self-consistent approaches, for instance. Therefore, the present work aims at developing an approximate model for the rotation of rigid spheroids with large aspect ratios (representing slender rods and fibers accurately) that takes into account the effective fluid behavior resulting from a population of fibers with a non-isotropic distribution of orientations. Since no analytical solution is available in the anisotropic case for the problem solved by Jeffery in isotropic conditions, a slender spheroid will first be assumed equivalent to a dumbbell, the orientation kinematics will be solved for the latter in the anisotropic case, and a direct comparison with a finite element simulation will validate the equivalence with a slender spheroid. Finally, the model will be applied to the orientation kinematics of a whole population of fibers suspended in a fluid submitted to a simple shear in order to emphasize the effects of taking anisotropy into account in the flow of fiber-filled plastics.

Rod kinematics in an anisotropic medium

To derive the dumbbell-based model, we start by revisiting the kinematics of a slender rod of length 2 L immersed in an isotropic Newtonian fluid with viscosity 𝜂, where the kinematics described by the velocity field v ( x ) is assumed unperturbed by the presence of the rod, whatever its orientation. The latter is defined by the unit vector p , and inertia effects are neglected in the sequel. If we assume the rod can be represented by the dumbbell depicted in Fig. 1 and that hydrodynamic forces apply on the two (identical) beads only, each being proportional to the difference between the velocity of the fluid 𝐯 0 + ∇ 𝐯 𝐩 𝐿 (with v 0 the velocity of the fluid at the dumbbell centroid G ) and the velocity of the bead 𝐯 𝐺 + ṗ 𝐿 (with v G the velocity of the dumbbell centroid), the force acting on the bead at p L (the origin of coordinates is taken at G ) reads

𝐅 ( 𝐩

𝐿 ) = 𝜁 ( 𝐯 0 + ∇ 𝐯 𝐩 𝐿 -𝐯 𝐺 -ṗ 𝐿 ) , ( 1
)
where 𝜁 is the drag coefficient in the isotropic fluid, and the force on the opposite bead at -𝐩 𝐿 is

𝐅 (-𝐩 𝐿 ) = 𝜁 ( 𝐯 0 -∇ 𝐯 𝐩 𝐿 -𝐯 𝐺 + ṗ 𝐿 ) . ( 2
)
Adding these two forces and enforcing the linear momentum balance 𝐅 ( 𝐩 𝐿 ) + 𝐅 (-𝐩 𝐿 ) = 𝟎 leads to 𝐯 0 = 𝐯 𝐺 , that is, the dumbbell centroid moves with the unperturbed fluid velocity. As the resulting torque must also vanish, the only possibility is that the forces act along the direction of p , that is 𝐅 ( 𝐩 𝐿 ) = 𝜆 𝐩 , with 𝜆 a real number:

𝜆 𝐩 = 𝜁 𝐿 (∇ 𝐯 𝐩 -ṗ ) . ( 3
)
Taking the scalar product of both sides with p provides the 𝜆 value:

𝜆 = 𝜁 𝐿 𝐩 𝑇 ∇ 𝐯 𝐩 , (4)
since 𝐩 𝑇 𝐩 = 1 and consequently 𝐩 𝑇 ṗ = 0 , and (3) gives

𝜁 𝐿 ( 𝐩 𝑇 ∇ 𝐯 𝐩 ) 𝐩 = 𝜁 𝐿 (∇ 𝐯 𝐩 -ṗ ) (5)
leading finally to the rotation rate of the dumbbell:

ṗ = ∇ 𝐯 𝐩 - ( 𝐩 𝑇 ∇ 𝐯 𝐩 ) 𝐩 (6)
which coincides with the well known Jeffery's equation for a rigid spheroid with an infinite aspect ratio. Therefore, the rotation of a slender fiber in an isotropic Newtonian fluid is equal to the rotation of a rigid dumbbell, and this remarkable coincidence, which was unrecognized in the early papers where rigid dumbbells were studied (see [START_REF] Bird | Kinetic theory and rheology of dumbbell suspensions with brownian motion[END_REF] , for instance), has been stressed in [START_REF] Chinesta | From single-scale to two-scale kinetic theory description of rods suspensions[END_REF] .

It may be noticed that the drag coefficient is absent in [START_REF] Scheuer | Second-gradient modelling of orientation development and rheology of dilute confined suspensions[END_REF] , but this does not apply when the fluid is anisotropic, as we show now. In this case, the drag force is not necessarily aligned with the velocity of the bead with respect to the fluid, unless possibly along a symmetry axis of the fluid behavior, and an anisotropic drag tensor 𝜻 is introduced, similarly to what has been done in [START_REF] Curtiss | A kinetic theory for polymer melts. i. the equation for the single-link orientational distribution function[END_REF] . This second-order tensor is positive definite, so that the drag force does dissipate energy, and therefore it has an inverse. The hydrodynamic force acting on the bead at p L reads now

𝐅 ( 𝐩 𝐿 ) = 𝜻( 𝐯 0 + ∇ 𝐯 𝐩 𝐿 -𝐯 𝐺 -ṗ 𝐿 ) , ( 7
)
and, taking into account the positivity of 𝜻, 𝐯 0 = 𝐯 𝐺 yields again from the linear momentum balance. The angular momentum balance implies

𝜻(∇ 𝐯 𝐩 𝐿 -ṗ 𝐿 ) = 𝜆 𝐩 , ( 8
)
which, using the invertibility of 𝜻, can also be written as

∇ 𝐯 𝐩 -ṗ = 𝜆 𝐿 𝜻 -1 𝐩 . ( 9
)
Taking the scalar product with p gives

𝐩 𝑇 ∇ 𝐯 𝐩 = 𝜆 𝐿 𝐩 𝑇 𝜻 -1 𝐩 , ( 10
)
which leads to the value of 𝜆:

𝜆 = 𝐿 𝐩 𝑇 ∇ 𝐯 𝐩 𝐩 𝑇 𝜻 -1 𝐩 (11)
that can be introduced into (9) and gives finally

ṗ = ∇ 𝐯 𝐩 - 𝐩 𝑇 ∇ 𝐯 𝐩 𝐩 𝑇 𝜻 -1 𝐩 𝜻 -1 𝐩 . ( 12
)
Since 𝜻 -1 appears at both the numerator and denominator, it can be replaced by a normalized inverse drag tensor 𝝃 where 𝜉 11 + 𝜉 22 + 𝜉 33 = 1 , so that

ṗ = ∇ 𝐯 𝐩 - 𝐩 𝑇 ∇ 𝐯 𝐩 𝐩 𝑇 𝝃 𝐩 𝝃 𝐩 (13)
generalizes [START_REF] Scheuer | Second-gradient modelling of orientation development and rheology of dilute confined suspensions[END_REF] . This expression for the rotation of a rigid dumbbell in an anisotropic fluid will be assumed below to govern also the rotation of a prolate spheroid with infinite aspect ratio. In this sense, this extends Jeffery's result to the anisotropic case.

Of course, ( 6) is recovered in the isotropic case, where the normalized inverse drag tensor reduces to 𝝃 = 𝜹∕3 ( 𝜹 denotes the second-order identity tensor), because the drag tensor 𝜻 simplifies to 𝜁 𝜹. By taking the scalar product with p , it may also be checked that [START_REF] Advani | The use of tensors to describe and predict fiber orientation in short fiber composites[END_REF] does satisfy the condition 𝐩 𝑇 ṗ = 0 to preserve the unit length of p . Before comparing the kinematics ( 13) and ( 6) for a population of fibers, it is necessary to address three main issues: the effective behavior of the fiber-filled fluid must be detailed, the associated second-order 𝝃 tensor must be computed, and the extension of (13) to a fiber must be validated. This is performed in the following Sections.

Orthotropic behaviors considered

The effective viscous behavior of a Newtonian fluid filled with fibers is deduced here by applying the Advani and Tucker [START_REF] Advani | The use of tensors to describe and predict fiber orientation in short fiber composites[END_REF] approach. This two-step procedure combines a model for the effective behavior when all fibers are aligned, and a closure relation applied to the symmetric second-order orientation tensor a otherwise. The latter is given by

𝐚 = ∫ (0 , 1)
𝐩 ⊗ 𝐩 𝜓( 𝐩 ) d 𝐩 [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] where the orientation distribution function 𝜓( p ), defined over the surface (0 , 1) of the unit ball, gives the fraction of fibers that are parallel to p and verifies the normality condition:

∫ (0 , 1) 𝜓( 𝐩 ) d 𝐩 = 1 (15)
from which 𝑎 11 + 𝑎 22 + 𝑎 33 = 1 results.

In the first step of the Advani-Tucker procedure, the Mori-Tanaka model [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] , as reformulated by Benveniste [START_REF] Benveniste | A new approach to the application of mori-tanaka's theory in composite materials[END_REF] , is used here for the case where all fibers are aligned and represented as very elongated prolate spheroids. This model was developed originally for composite materials, and was preferred by Tucker and Liang [START_REF] Tucker | Stiffness predictions for unidirectional short-fiber composites: review and evaluation[END_REF] at the end of their review of various models that predict the stiffness of short-fiber composites. Taking advantage of the formal equivalence between the equations governing the deformation of an elastic solid at small strain and those governing the flow at a given time in a linear viscous fluid at low Reynolds number (see [START_REF] Beaussart | Constitutive relationships for anisotropic viscous materials[END_REF] , for instance), this model is extended readily to fiber-filled fluids. For a volume fraction f of aligned rigid fibers in an incompressible fluid of viscosity 𝜂, the effective behavior is transversely isotropic and defined by the following three material constants: an elongational viscosity

𝜂 𝐸 = 3 ( 1 + 2 3 𝑓 1 -𝑓 1 -𝑤 2 ℎ ( 𝑤 ) -2 𝑤 2 + 2 ℎ ( 𝑤 ) 𝑤 2 ) 𝜂 ( 16
)
a longitudinal shear viscosity

𝜂 𝐿 = ( 1 + 2 𝑓 1 -𝑓 1 -𝑤 2 (1 + 𝑤 2 )(2 -3 ℎ ( 𝑤 )) ) 𝜂 ( 17
)
and a transverse shear viscosity

𝜂 𝑇 = ( 1 + 4 𝑓 1 -𝑓 1 -𝑤 2 3 ℎ ( 𝑤 ) -2 𝑤 2 ) 𝜂 ( 18
)
where w > 1 denotes the fiber aspect ratio, and

ℎ ( 𝑤 ) = 𝑤 ( 𝑤 2 -1) 3∕2 ( 𝑤 √ 𝑤 2 -1 -arccosh 𝑤 ) . ( 19
)
When w → ∞, it may be noted that h → 1 (with already h ≥ 0.99 when w ≥ 16), whereas 𝜂 E → ∞ but both 𝜂 L and 𝜂 T tend to (1 + 𝑓 ) 𝜂∕(1 -𝑓 ) .

In the second step of the Advani-Tucker procedure, non-aligned fibers are considered, and an evaluation of the anisotropic fluid behavior is obtained by combining the above unidirectional case and information about the orientation distribution function. The latter is reduced to the second-order orientation tensor a , provided that a suitable closure relation is applied, which defines a fourth-order orientation tensor related to a . Here, the IBOF closure proposed by Chung and Kwon [START_REF] Chung | Invariant-based optimal fitting closure ap-proximation for the numerical prediction of flow-induced fiber orientation[END_REF] is used, which has been developed by fitting over a large set of flow simulations and a values. As a result, the effective behavior is orthotropic, with its three material symmetry planes being normal to the principal axes of a . In the latter system of axes, six material parameters are defined for an incompressible orthotropic fluid behavior: three elongational viscosities 𝜂 11 , 𝜂 22 and 𝜂 33 , and three shear viscosities 𝜂 12 , 𝜂 23 and 𝜂 31 . These 6 parameters are deduced from the volume fraction f and aspect ratio w of the fibers (to get the reference unidirectional case), and from the a tensor (to include the orientation distribution) through the expressions given in [START_REF] Chung | Invariant-based optimal fitting closure ap-proximation for the numerical prediction of flow-induced fiber orientation[END_REF] . The linear relation between the deformation rate D and the stress deviator s (or extra-stress, such that the stress tensor is given by 𝝈 = 𝐬 -𝑝 𝜹, where p is a pressure) writes as follows in the principal axes of orthotropy (the reverse expressions for the components of s in terms of D are more involved):

𝐷 11 = 𝑠 11 𝜂 11 + ( 1 𝜂 33 - 1 𝜂 11 - 1 𝜂 22 ) 𝑠 22 2 + ( 1 𝜂 22 - 1 𝜂 33 - 1 𝜂 11 ) 𝑠 33 2 𝐷 22 = ( 1 𝜂 33 - 1 𝜂 11 - 1 𝜂 22 ) 𝑠 11 2 + 𝑠 22 𝜂 22 + ( 1 𝜂 11 - 1 𝜂 22 - 1 𝜂 33 ) 𝑠 33 2 𝐷 33 = ( 1 𝜂 22 - 1 𝜂 33 - 1 𝜂 11 ) 𝑠 11 2 + ( 1 𝜂 11 - 1 𝜂 22 - 1 𝜂 33 ) 𝑠 22 2 + 𝑠 33 𝜂 33 (20)
and 𝐷 12 = 𝑠 12 ∕(2 𝜂 12 ) , 𝐷 23 = 𝑠 23 ∕(2 𝜂 23 ) , 𝐷 31 = 𝑠 31 ∕(2 𝜂 31 ) for the shear components, in agreement with Beaussart et al. [START_REF] Beaussart | Constitutive relationships for anisotropic viscous materials[END_REF] using different notations. These relations are complemented by an objective evolution law for the a tensor, which includes the rotation of the orthotropy axes and induces an updating of the six viscosities through the Advani-Tucker model.

The special case of transverse isotropy about axis 1 that applies when all fibers are parallel is recovered when 𝜂 11 = 𝜂 𝐸 , 𝜂 12 = 31 = 𝜂 𝐿 , 𝜂 23 = 𝜂 𝑇 , 𝜂 22 = 𝜂 33 = 4 𝜂 𝐸 𝜂 𝑇 ∕( 𝜂 𝐸 + 𝜂 𝑇 ) . In the limit isotropic case, where 𝑎 11 = 𝑎 22 = 𝑎 33 = 1∕3 , one has 𝜂 11 = 𝜂 22 = 𝜂 33 = 3 𝜂 12 = 3 𝜂 23 = 3 𝜂 31 . The Abaqus [19] finite element code is employed in the simulations reported below, using again the equivalence between elastic solids and linear fluids, which requires the addition of three Poisson's ratios to the material constants. They were defined such that a Poisson's ratio of 0.5 minus 10 -6 results in the isotropic case, which means that the incompressibility condition is approached extremely closely.

Viscous drag in orthotropic fluids

Preliminary validation tests

The drag of a sphere translating in an unbounded isotropic incompressible viscous fluid has been calculated in 1851 by Stokes in a famous paper [START_REF] Stokes | On the effect of the internal friction of fluids on the motion of pendulums[END_REF] , but an extension to the case of a finite domain has been performed by Lin et al. [START_REF] Lin | Force-displacement relationships for spherical inclusions in finite elastic media[END_REF] . These authors have shown that when a sphere of radius R moves with velocity v at the center of a spherical fluid domain of radius qR , with zero velocity at the boundary, the drag force can be expressed in closed form:

𝐹 = 24 𝑞 5 + 𝑞 4 + 𝑞 3 + 𝑞 2 + 𝑞 4 𝑞 5 -5 𝑞 4 -5 𝑞 3 + 5 𝑞 2 + 5 𝑞 -4 𝜋𝑅𝜂𝑣 ( 21
)
which recovers the well-known relation 𝐹 = 6 𝜋𝑅𝜂𝑣 derived by Stokes when q → ∞. This result, which has been established originally for a spherical inclusion displaced in a finite spherical elastic medium [START_REF] Lin | Force-displacement relationships for spherical inclusions in finite elastic media[END_REF] , implies that the drag force is 8% over Stokes' value when 𝑞 = 30 , over about 5% when 𝑞 = 50 , and still over 2% when 𝑞 = 100 . This is an indirect consequence of the slow r / R decrease of the velocity field in Stokes' solution, where r denotes the radial coordinate. Since a precise evaluation of the drag force in an unbounded orthotropic fluid is desired in this study, a mesh extending from the moving sphere up to 100 times its radius has been defined for the numerical simulations. More precisely, one fourth of the inner and outer spheres has been considered, which is bounded by symmetry planes (see Fig. 2 a). The mesh has been generated with the Netgen mesh generator [START_REF] Schöberl | NETGEN an advancing front 2d/3d-mesh generator based on abstract rules[END_REF] , it contains 607, 852 nodes, 493, 384 quadratic 10-node tetrahedral elements using a hybrid formulation (i.e., where the displacements and the pressure are degrees of freedom), which results in 2, 262, 943 unknowns. As shown in Fig. 2 b, the mesh has been refined at the two points where a singularity is expected, i.e., where a streamline ends onto the moving sphere, for symmetry reasons.

A unit displacement of the central spherical surface has been prescribed along the Y axis, which corresponds to a unit velocity in terms of the elasticity-viscosity equivalence, whereas the surface of the outer sphere was fixed, and the resulting force has been computed with Abaqus. When the fluid is isotropic, with a unit 𝜂 value, the computed force on a sphere of unit radius is 19.2827, whereas a value of 19.2831 is given by [START_REF] Lin | Force-displacement relationships for spherical inclusions in finite elastic media[END_REF] , i.e., 1.023 times Stokes' value. This very good agreement validates the mesh and the finite element approach for computing the drag force. Moreover, this suggests also that an error of about 2% might be expected when the drag of a sphere in an unbounded anisotropic fluid will be evaluated from simulations with a mesh of finite size extending up to 100 times the sphere radius. Nevertheless, the results for the components of the normalized inverse drag tensor may be even more precise, since ratios between drag forces in different directions are involved and errors may cancel out. For instance, the exact value of 1/3 Fig. 3. Component of the normalized inverse drag tensor 𝝃 parallel to the main symmetry axis of an incompressible transversely isotropic fluid where 𝜂 𝐸 = 3 𝜂 𝑇 . Finite element results (symbols) and analytical solution of Gómez-González and Álamo [START_REF] Gómez-González | Flow of a viscous nematic fluid around a sphere[END_REF] (solid curve). is obtained for each component in the isotropic case, although the drag force is overestimated by 2% in each direction.

Another validation of the finite element modeling is provided by comparing to the remarkable solution obtained by Gómez-González and del Álamo [START_REF] Gómez-González | Flow of a viscous nematic fluid around a sphere[END_REF] for the viscous drag of a sphere in a transversely isotropic fluid. Actually, the results of Gómez-González and Álamo [START_REF] Gómez-González | Flow of a viscous nematic fluid around a sphere[END_REF] have been obtained in the more general case of a nematic fluid, but the case of incompressible transverse isotropy is obtained for a special choice of the material variables: 𝛼 1 = 4( 𝜂 𝑇 -𝜂 𝐿 ) , 𝜂 𝑎 = 𝜂 𝑇 and 𝜂 𝑏 = 𝜂 𝑐 = 𝜂 𝐿 in the notations of Gómez-González and Álamo [START_REF] Gómez-González | Flow of a viscous nematic fluid around a sphere[END_REF] . This corresponds to an incompressible transversely isotropic fluid where 𝜂 𝐸 = 3 𝜂 𝑇 , with 𝜂 L / 𝜂 T left as a single anisotropy parameter, the isotropic case being recovered when 𝜂 𝐿 ∕ 𝜂 𝑇 = 1 . The quite complex expressions obtained for the drag force when the sphere is displaced either parallel ( 𝜁 ∥ ) or perpendicular ( 𝜁 ⊥ ) to the direction of transverse isotropy are not repeated here, but they can be found in [START_REF] Gómez-González | Flow of a viscous nematic fluid around a sphere[END_REF] . They allow to compute the component of the normalized inverse drag tensor along the axis of transverse isotropy:

𝜉 11 = 𝜁 ⟂ 𝜁 ⟂ + 2 𝜁 ∥ ( 22
)
with 𝜉 22 = 𝜉 33 = (1 -𝜉 11 )∕2 , so that 𝜉 11 + 𝜉 22 + 𝜉 33 = 1 . When the 𝜂 L / 𝜂 T ratio is varied from 0.1 to 10, the curve shown in Fig. 3 is obtained for 𝜉 11 . The agreement between the finite element results (for a finite fluid domain) and the analytical solution (for an unbounded fluid domain) is quite good, and the discrepancy with respect to the analytical values is at most -1 . 3 %.

Application to orthotropic fluids

The evaluation of the viscous drag of a sphere in an incompressible orthotropic fluid has already been addressed in the context of the dumbbell kinetic theory at the scale of the polymer molecule. Among ealy papers, Curtiss and Bird [START_REF] Curtiss | A kinetic theory for polymer melts. i. the equation for the single-link orientational distribution function[END_REF] introduced this concept, and Bird and Wiest [START_REF] Bird | Anisotropic effects in dumbbell kinetic theory[END_REF] reviewed various empirical formulae. More recently, Azaiez [START_REF] Azaiez | Constitutive equations for fiber suspensions in viscoelastic media[END_REF] proposed a linear combination of the identity tensor (which applies to the isotropic case) and a , and the associated polymer-fiber interaction parameter has been fitted to experimental results by Guo et al. [START_REF] Guo | Rheology of fiber filled polymer melts: role of fiber-fiber interactions and polymer-fiber coupling[END_REF] . The concern here is the viscous drag at the upper scale of a fiber, modelled as a dumbbell, and a quantitative evaluation is sought by using finite element simulations.

In order to sample all possible a tensors for given fiber volume fraction and aspect ratio, a set of 45 points is defined in the classical triangle used by Cintra and Tucker [START_REF] Cintra | Orthotropic closure approximations for the flow-induced fiber orientation[END_REF] , for instance, where a 11 ≥ a 22 ≥ a 33 (see Fig. 4 ). This ordering of the principal values of a and the associated definition of axes (1,2,3) is kept from now on in this paper. Points A and B correspond to isotropic ( 𝑎 11 = 𝑎 22 = 1∕3 ) and unidirectional ( 𝑎 11 = 1 ) distributions of fiber orientations, respectively, lines AB and AC include the axisymmetric orientation distributions ( 𝑎 22 = 𝑎 33 along AB, 𝑎 11 = 𝑎 22 along AC), and line BC is defined by 𝑎 33 = 0 . The 44 non-trivial (i.e., not corresponding to the isotropic case) points considered are aligned on a 0.05 × 0.05 grid, as shown in Fig. 4 . Consequently, 132 finite element simulations are required for given fiber volume fraction and aspect ratio: one per translation of the sphere along each of the 3 principal directions of the orthotropic behavior associated with each of the 44 non-trivial points. Once the drag force ( F 1 , F 2 , or F 3 ) has been evaluated along each of the principal axes of the a tensor, where the inverse drag tensor 𝝃 also is diagonal, the normalized components of the latter are given by

𝜉 11 = 1 1 + 𝐹 1 𝐹 2 + 𝐹 1 𝐹 3 ( 23
)
with a similar relation for 𝜉 22 , and 𝜉 33 = 1 -𝜉 11 -𝜉 33 . Since 5 volume fractions ( 𝑓 = 0 . 1 , 0.15, 0.2, 0.25, and 0.3) were considered, and 5 aspect ratios ( 𝑤 = 20 , 35, 50, 75, and 100) were used for each volume fraction, a total of 3, 300 finite element simulations were performed.

No systematic simple trend could be found among the results, like 𝜉 11 ≤ 𝜉 22 ≤ 𝜉 33 for instance, or a component always increasing or decreasing when f or w increases. The expression proposed by Azaiez [START_REF] Azaiez | Constitutive equations for fiber suspensions in viscoelastic media[END_REF] for the inverse drag tensor, after normalization and using slightly different notations, namely

𝝃 = 𝜎 3 𝜹 + (1 -𝜎) 𝐚 (24)
which recovers 𝝃 = 𝜹∕3 in the isotropic case ( 𝐚 = 𝜹∕3 ) whatever the value of 𝜎, or for any a tensor if 𝜎 = 1 , has been fitted to the set of results pertaining to fixed f and w values. In each case, 132 ( a 11 , 𝜉 11 ), ( a 22 , 𝜉 22 ), or ( a 33 , 𝜉 33 ) pairs (one set of three for each of the 44 non-trivial a tensors) were considered for determining the optimal 𝜎 value. Fig. 5 a illustrates that the linear law ( 24) is quite acceptable in the favorable case of a low volume fraction ( 𝑓 = 0 . 10 ) of moderately elongated fibers ( 𝑤 = 20 ), but the agreement with the numerical results is less satisfactory in the more extreme case of a high volume fraction ( 𝑓 = 0 . 30 ) of very elongated fibers ( 𝑤 = 50 ) shown in Fig. 5 b. Since a simple, though crude, law may nevertheless be useful in practice, the 𝜎 values obtained for the 25 combinations of f and w considered here are given in Table 1 , which can be used to interpolate for intermediate fiber volume fractions f and aspect ratios w . It may be noted that 𝜎 increases when either f or w increases.

Rotation of a prolate spheroid

When a rigid prolate spheroid is free to translate and rotate in an unbounded isotropic viscous fluid submitted to simple shear at infinity, the solution detailed by Jeffery [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF] gives the rate of rotation of its long axis. The purpose of this Section is to evaluate this rotation when the fluid is orthotropic, since no analytical result is available, and to compare with the dumbbell model proposed in Section 2 . In order to test the validity of this model, a few cases only are treated because the problem is defined by 9 parameters: the aspect ratio of the spheroid, its orientation with respect to the plane and direction of simple shear, the orientation of the principal axes of the second-order orientation tensor, two independent principal values of the latter, and the fiber volume fraction in the fluid surrounding the fiber considered (assuming all fibers have the same aspect ratios). A mesh has been prepared, where a prolate rigid spheroid with a length of 20 and a diameter of 1 ( 𝑤 = 20 ) is embedded at the center of a cube of side 60. The mesh has been refined in the vicinity of the sharp ends of the spheroid, it has 427, 163 nodes, 311, 584 10-node quadratic tetrahedral hybrid elements, with a total of 1, 593, 076 unknowns. The sides of the cube are normal to the Cartesian axes and parallel to the symmetry planes of the fluid orthotropy, the long axis of the fiber is parallel to the (1,1,1) direction in order to avoid favoring an orthotropy axis ( Fig. 6 ), and simple shear is applied in the X -direction parallel to the XZ -plane. The origin of the coordinate system is taken at the center of the cube and a velocity 𝑣 𝑋 = ̇𝛾𝑌 , 𝑣 𝑌 = 𝑣 𝑍 = 0 is applied on all cube faces. In these conditions, the analytical solution of Jeffery [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF] for an unbounded isotropic fluid domain gives the rotation rate of the unit vector parallel to the long axis p of the spheroid as ṗ = 𝛀 𝐩 + 𝑘 𝐃 𝐩 -𝑘 ( 𝐩 𝑇 𝐃 𝐩 ) 𝐩 [START_REF] Azaiez | Constitutive equations for fiber suspensions in viscoelastic media[END_REF] where 𝛀 and D are the antisymmetric and symmetric parts of the velocity gradient ∇ v , respectively, and 𝑘 = ( 𝑤 2 -1)∕( 𝑤 2 + 1) , with (6) being recovered when w → ∞ (extremely elongated spheroids). Eq. ( 25) leads to the following expressions for the components of ṗ in the case considered here:

ṗ 𝑋 = ̇𝛾 √ 3 ( 1 2 + 𝑘 6 ) , ṗ 𝑌 = ̇𝛾 √ 3 ( - 1 2 + 𝑘 6 ) , ṗ 𝑍 = -̇𝛾 √ 3 𝑘 3 ( 26
)
which allows a validation of the finite element simulation. With the mesh described above, where 𝑘 = 0 . 995 because 𝑤 = 20 , the three values of ( 26) have been recovered extremely precisely, with relative differences of less than 0.03%. Because of the large value of k involved in the finite element model, the obtained rate of rotation is close to the value for w → ∞, where

ṗ 𝑋 = -2 ṗ 𝑌 = -2 ṗ 𝑍 = 2 ̇𝛾∕(3 √
3 ) . Therefore, the components of ṗ obtained in the numerical simulations with an orthotropic fluid can be compared to the values given explicitly by the approximate dumbbell model ( 13) using the inverse drag tensor, namely:

ṗ 𝑋 = ̇𝛾 √ 3 (1 -𝜉 𝑋𝑋 ) , ṗ 𝑌 = -̇𝛾 √ 3 𝜉 𝑌 𝑌 , ṗ 𝑍 = -̇𝛾 √ 3 𝜉 𝑍𝑍 ( 27
)
for a slender fiber tilted to the (1,1,1) direction with respect to the Cartesian axes. Note that in [START_REF] Cintra | Orthotropic closure approximations for the flow-induced fiber orientation[END_REF] the ( X, Y, Z ) axes are aligned with the principal axes of a (and, therefore, of 𝝃), but three cases arise when either ( 𝑋, 𝑌 , 𝑍) = (1 , 2 , 3) , or ( 𝑋, 𝑌 , 𝑍) = (3 , 1 , 2) , or ( 𝑋, 𝑌 , 𝑍) = (2 , 3 , 1) , leading to three different rotations of the fiber. For instance, when the fluid surrounding the fiber contains a volume fraction 𝑓 = 0 . 3 of fibers with an aspect ratio 𝑤 = 20 , two cases are considered, which correspond to points B and C in Fig. 4 , i.e., where either 𝑎 11 = 1 (aligned fibers) or 𝑎 11 = 𝑎 22 = 1∕2 . The following

Table 2

Components of ṗ given by the dumbbell model and by the finite element simulations for a spheroid with its long axis ( 𝑤 = 20 ) parallel to the (1,1,1) direction and immersed in an orthotropic fluid (where 𝑓 = 0 . 3 and 𝑤 = 20 , 𝑎 11 = 1 in case B, 𝑎 11 = 𝑎 22 = 1∕2 in case C) submitted to a simple shear. Three orientations of the fluid orthotropy with respect to the Cartesian axes are considered in each case.

Case B

Case C 2 when the dumbbell model is applied. These results are compared in Table 2 with the rotations of the prolate spheroid given by the finite element simulations; they suggest that the dumbbell model evaluates the components of ṗ to within ± 0.01. This validates the assumption that the solution to the Jeffery problem in the anisotropic case can be approximated reasonably well by a dumbbell model. The rotation rate predicted by Jeffery's equation for a slender fiber is recovered when 𝜉 𝑋𝑋 = 1∕3 in [START_REF] Cintra | Orthotropic closure approximations for the flow-induced fiber orientation[END_REF] , and therefore a rotation towards the alignment of all other fibers (component ṗ 𝑋𝑋 ) is obtained faster or slower than predicted by Jeffery's equation when 𝜉 XX is smaller or larger than 1/3, respectively. The values of ṗ 𝑋𝑋 predicted by [START_REF] Cintra | Orthotropic closure approximations for the flow-induced fiber orientation[END_REF] when using values of 𝜉 XX < 1/3 as obtained in Section 4.2 are confirmed by the finite element simulations shown above ( Table 2 ), but they do not agree with the predictions of Koch and Shaqfeh [START_REF] Koch | The average rotation rate of a fiber in the linear flow of a semidilute suspension[END_REF] . The latter authors obtained ṗ 𝑋𝑋 values lower than given by Jeffery's equation by using a direct evaluation and summation of hydrodynamic effects to evaluate the rotation of a slender fiber when all other fibers immersed in the fluid are aligned in a given direction. Since our finite element simulations of a rotating fiber do not use the drag factor but merely rely on the preliminary evaluation of the effective behavior of the fluid surrounding the rotating fiber, we note that an agreement with Koch and Shaqfeh [START_REF] Koch | The average rotation rate of a fiber in the linear flow of a semidilute suspension[END_REF] might be obtained, at least qualitatively, with a different evaluation of the effective behavior that would lead to a slower rotation in the finite element simulations and to 𝜉 XX > 1/3 in Section 4.2 . At the present stage, we may just observe that these two approaches, which are based on different evaluations of the effects of surrounding fibers in semi-dilute suspensions, lead to opposite trends. It may also be recalled incidentally that [START_REF] Koch | The average rotation rate of a fiber in the linear flow of a semidilute suspension[END_REF] is limited to aligned fibers, whereas our approach is more general.

( X, Y, Z ) Model ṗ 𝑋 ∕ ̇𝛾 ṗ 𝑌 ∕ ̇𝛾 ṗ 𝑍 ∕ ̇𝛾 ṗ 𝑋 ∕ ̇𝛾 ṗ 𝑌 ∕ ̇𝛾 ṗ 𝑍 ∕ ̇𝛾 (1,

Simulating the orientation kinematics of a population of rods

In this Section, the orientation kinematics of a population of initially randomly oriented rods is computed when the surrounding Newtonian fluid is submitted to a simple shear ( 𝑣 𝑋 = ̇𝛾𝑌 , 𝑣 𝑌 = 𝑣 𝑍 = 0 , like in the previous Section). Two cases are considered, where the coupling of the rod orientation kinematics with the orientations of the surrounding rods is either ignored or taken into account. In the former no-coupling case, each rod evolves independently from the others, and its orientation history is obtained readily by integrating the standard Jeffery's Eq. ( 6) . In the coupled case, the generalized Jeffery's Eq. ( 13) is used for evolving each orientation and, by contrast, the whole population of orientations must be updated at every time step.

At any time t in the integration procedure, the N rods involved in the numerical simulation of the suspension are described by their orientations 𝐩 𝑡 𝑖 (with 𝑖 = 1 , ⋯ , 𝑁), from which the second-order orientation tensor is computed as a discrete counterpart of ( 14) :

𝐚 𝑡 = 1 𝑁 𝑁 ∑ 𝑖 =1 𝐩 𝑡 𝑖 ⊗ 𝐩 𝑡 𝑖 . ( 28
)
This tensor can then be used to compute the inverse drag tensor 𝝃 𝑡 at time t through [START_REF] Bird | Anisotropic effects in dumbbell kinetic theory[END_REF] , and finally ( 13) is applied to each rod in the population to obtain its rate of rotation ṗ 𝑡 𝑖 in order to update the orientations with a mere explicit Euler scheme: 𝐩 𝑡 +Δ𝑡 𝑖 = 𝐩 𝑡 𝑖 + Δ𝑡 ṗ 𝑡 𝑖 for 𝑖 = 1 , ⋯ , 𝑁. The next time step can then be handled. This procedure is noticeably simplified by the use of the appropriate, preliminarily fitted, and constant 𝜎 value in [START_REF] Bird | Anisotropic effects in dumbbell kinetic theory[END_REF] , which can be read from Table 1 when the aspect ratio w of the fibers and their volume fraction f have been defined. Since the rods considered in this discrete approach do not have a finite diameter (hence an infinite aspect ratio and a zero volume), the fiber volume fraction and aspect ratio are present through the value of 𝜎 only. This explains why the simulations described below use the same set of N rods when different f and w values are considered. Neither f nor w affects the predictions of the uncoupled model, of course, since they are not present in [START_REF] Scheuer | Second-gradient modelling of orientation development and rheology of dilute confined suspensions[END_REF] .

A population of 𝑁 = 5 , 000 rods is considered, with orientations which are initially distributed randomly and uniformly on the unit sphere, so that an almost isotropic orientation distribution is depicted. Since the rods are expected to orient along the shear direction ( X -axis) because both ( 6) and ( 13) predict ṗ = 𝟎 if all fibers are aligned along this direction, the associated a XX component of the orientation tensor should approach the value of 1 at long times. This quantity is plotted in Fig. 7 (where ̇𝛾 = 1 s -1 has been used) as predicted by the standard Jeffery model and by the coupled model that takes into account the anisotropic effective medium. The predictions of the latter model depend on the fiber volume fraction and aspect ratio, and three cases are considered: 𝑓 = 0 . 10 and 𝑤 = 20 , where 𝜎 = 1 . 182 applies according to Table 1 , 𝑓 = 0 . 20 and 𝑤 = 20 ( 𝜎 = 1 . 242 ) to demonstrate the effect of the volume fraction, 𝑓 = 0 . 20 and 𝑤 = 50 ( 𝜎 = 1 . 376 ), to illustrate the influence of the aspect ratio. Of course, the orientation distribution is the same ( 𝑎 𝑋𝑋 = 1∕3 ) in all cases initially and tends to be the same ( 𝑎 𝑋𝑋 = 1 ) at large times, but Fig. 7 shows that significant differences can be observed at intermediate times, although all curves are still very close to each other at small times. At any time, the orientation is more pronounced (less isotropic) with the coupled model than with the standard Jeffery prediction. Fig. 7 also shows that an increase in the fiber volume fraction tends to increase a XX , but this effect is moderate (the curve for 𝑓 = 0 . 30 and 𝑤 = 20 , not shown for clarity, would be very close to the 𝑓 = 0 . 20 and 𝑤 = 20 curve), whereas an increase of the fiber aspect ratio increases a XX more strongly. Thus, for shear flows of fiber-Fig. 7. Time evolution of the a XX component of the orientation tensor when simple shear is applied along the X -axis (with ̇𝛾 = 1 s -1 ) and the effective anisotropic fluid is either accounted for (solid lines) or ignored (standard Jeffery prediction, dashed line). Three combinations of fiber volume fractions f and aspect ratios w are considered, as indicated. filled polymers, noticeable differences can be obtained in the final orientation state if the flow implies a significant anisotropy and if the total applied shear is not enough for reaching the fully aligned terminal state.

Concluding remarks

A model has been proposed, where the evolution of the orientation of each fiber is coupled to the orientations of the surrounding fibers in the flow of a fiber-filled fluid and includes the effects of the fiber volume fraction and aspect ratio. This has been performed by accounting for the effective behavior of the fiber-filled fluid, which is anisotropic although the fibers are embedded in an isotropic Newtonian fluid. The rotation of a fiber in these conditions could be predicted by using a dumbbell model, which allowed an extension of Jeffery's equation to anisotropic cases. This involved the numerical evaluation of the drag force applied on a sphere in an orthotropic incompressible fluid, and a simple fit has been proposed for its practical use in the coupled model. The assumed equivalence between the rotations of a fiber and of a dumbbell in a anisotropic fluid has been validated directly by finite element simulations. This work calls for the following remarks:

(i) The proposed model relies on some important assumptions. First, an effect of the fibers surrounding a given fiber is accounted for, but merely through the anisotropy they induce, not by the direct interactions they may have with the fiber considered. In other words, fiber-to-fiber contacts are neglected here, and this may be acceptable in two cases: when the fiber volume fraction is not too large, and when the fibers are sufficiently aligned (a decrease of the number of contacts has been observed by Mezher et al. [START_REF] Mezher | Analysis of the folgar & tucker model for concentrated fibre suspensions in unconfined and confined shear flows via direct numerical simulation[END_REF] , for instance). The first condition is closely related to the limit where the Mori-Tanaka model used here is reliable, so that large fiber volume fractions should be considered cautiously, and other homogenization models may be preferred in such cases.

The second circumstance is increasingly likely to occur when the orientation distribution moves away from the isotropic case. Fortunately, this is rapidly obtained in simple shear with the coupled model. (ii) The use of the fitted coefficient 𝜎, as proposed in Section 4.2 , is convenient in practice, because it leads to a simple and fast application of the coupled model, but it may give a crude approximation of the drag tensor in some circumstances, as illustrated in Fig. 5 b. A direct use of the whole set of computed values of the drag tensor (obtained with some uncertainty through a numerical simulation, though) might be more satisfactory, but this would lead to a more complex scheme. In particular, the drag tensor would have to be updated at each time step, because the orientation tensor evolves, whereas 𝜎 is constant. This might be performed with a suitable interpolation procedure using the results of the finite element simulations. (iii) For practical reasons, the numerical simulation of the injection of fiber-filled polymers relies on the second-order orientation tensor to condense the orientation distribution function. As mentioned in Section 3 , this has the consequence that the effective behavior of the fiber-filled fluid is orthotropic, which simplifies the evaluation of the drag tensor (reduced to 3 finite element simulations).

In general, complex, flows of fiber-filled fluids, the orientation distribution function may not have three orthogonal symmetry planes, and the effective behavior would not be orthotropic, but of a more general anisotropy type instead. In these conditions, the numerical evaluation and tabulation of the drag tensor would be a formidable task. (iv) The fiber aspect ratio appears at two places in the present coupled model. First, it has a strong effect on the effective anisotropic behavior of the fiber-filled fluid, as shown in Section 3 , especially through the elongational viscosity of the reference unidirectional case. The effects induced on the drag tensor have been taken into account in this study, and they are reflected by the variations of 𝜎 in each column of Table 1 . The aspect ratio comes also into play through coefficient k in the general Jeffery equation ( 25) , but it is missing in [START_REF] Advani | The use of tensors to describe and predict fiber orientation in short fiber composites[END_REF] , where an infinite aspect ratio is assumed. This is justified by the negligible effect of w in [START_REF] Azaiez | Constitutive equations for fiber suspensions in viscoelastic media[END_REF] when reasonably large values are used, as shown in Section 5 , and by the next point below. (v) A tri-dumbbell model has been proposed in [START_REF] Abisset-Chavanne | On the multiscale description of dilute suspensions of non-brownian clusters composed of rods[END_REF] , which reproduces Jeffery's equation for a general ellipsoid in an isotropic fluid, and we explored its extension to the present context of an anisotropic fluid like we did for the simple dumbbell in Section 2 , in order to account for a finite aspect ratio. This involved extremely complex analytical expressions (not reported here) with, finally, numerical values for reasonably large aspect ratios very close to the predictions of the considerably simpler mono-dumbbell model used in the present paper. (vi) There is no two-dimensional analog to the effect of anisotropy described in this paper. In such a simplified case, one would consider a distribution of orientations along the unit circle rather than on the unit sphere. This might appear as an ideal simple test case, but taking into account the anisotropy of the fiber-filled fluid would have no effect at all, unfortunately. The fundamental reason is that the viscosities along the two principal axes of the orientation tensor would be equal, although the two principal values of this tensor can have different values (with their sum being equal to 1). This property of an incompressible twodimensional fluid has been shown by Fletcher [START_REF] Fletcher | Deformable, rigid, and inviscid elliptical inclusions in a homogeneous incompressible anisotropic viscous fluid[END_REF] and, as a result, the drag tensor (and its inverse) has its two principal values equal. In other words, the drag is isotropic in two dimensions. Actually, the equivalent to the Jeffery solution is available for a two-dimensional anisotropic problem: the rotation of a rigid ellipse is merely independent of the surrounding anisotropy, as shown in [START_REF] Fletcher | Deformable, rigid, and inviscid elliptical inclusions in a homogeneous incompressible anisotropic viscous fluid[END_REF] .

Fig. 1 .

 1 Fig. 1. Hydrodynamic forces applied on a dumbbell immersed in a Newtonian fluid.

Fig. 2 .

 2 Fig. 2. (a) Mesh used in the finite element simulations to the viscous drag of a sphere moving in an orthotropic fluid. The sphere (in red), with a velocity parallel to the Y -axis (horizontal), is embedded in a 100 times larger fluid domain (in green), and one-fourth of the geometry is considered for symmetry reasons. (b) Enlarged view in the vicinity of the rigid moving sphere; to avoid meshing its volume, the velocities of all nodes at its surface are set equal and parallel to the Y -axis. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4 .

 4 Fig. 4. Set of 45 ( a 11 , a 22 ) pairs used for the evaluation of the inverse drag tensor.

Fig. 5 .

 5 Fig. 5. Linear fit (straight line) obtained with the set of ( a 11 , 𝜉 11 ) (red circles), ( a 22 , 𝜉 22 ) (blue squares), and ( a 33 , 𝜉 33 ) (yellow triangles) pairs obtained for 𝑓 = 0 . 10 and 𝑤 = 20 (a), or for 𝑓 = 0 . 30 and 𝑤 = 50 (b). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6 .

 6 Fig. 6. The cubic volume (shown translucent) meshed for the study of the rotation of a tilted prolate spheroid (shown in light color). A simple shear is applied, which translates the upper and lower faces of the cube in opposite directions.

Table 1

 1 Values of the 𝜎 parameter obtained for various combinations of fiber volume fraction f and aspect ratio w .

		𝑓 = 0 . 10	𝑓 = 0 . 15	𝑓 = 0 . 20	𝑓 = 0 . 25	𝑓 = 0 . 30
	𝑤 = 20	1.182	1.218	1.242	1.260	1.274
	𝑤 = 35	1.273	1.306	1.328	1.343	1.355
	𝑤 = 50	1.327	1.357	1.376	1.389	1.399
	𝑤 = 75	1.382	1.407	1.422	1.433	1.441
	𝑤 = 100	1.415	1.437	1.450	1.459	1.466

  𝜉 11 = 0 . 1704 and 𝜉 22 = 𝜉 33 = 0 . 4148 in case B, 𝜉 11 = 𝜉 22 = 0 . 2720 and 𝜉 33 = 0 . 4560 in case C. They lead to the values reported in Table

	2,3)	Spheroid	0.4888	-0 . 2448	-0 . 2440	0.4180	-0 . 1593	-0 . 2587
		Dumbbell	0.4790	-0 . 2395	-0 . 2395	0.4203	-0 . 1570	-0 . 2633
		Difference	-0 . 0098	0.0053	0.0045	0.0023	0.0023	-0 . 0046
	(3,1,2)	Spheroid	0.3326	-0 . 0886	-0 . 2440	0.3188	-0 . 1612	-0 . 1576
		Dumbbell	0.3379	-0 . 0984	-0 . 2395	0.3141	-0 . 1570	-0 . 1570
		Difference	0.0053	-0 . 0098	0.0045	-0 . 0047	0.0042	0.0006
	(2,3,1)	Spheroid	0.3318	-0 . 2455	-0 . 0863	0.4161	-0 . 2586	-0 . 1576
		Dumbbell	0.3379	-0 . 2395	-0 . 0984	0.4203	-0 . 2633	-0 . 1570
		Difference	0.0061	0.0060	-0 . 0121	0.0042	-0 . 0047	0.0006
	values have been obtained with the finite element simulations reported
	in Section 4.2 for the components of the normalized inverse drag ten-
	sor:

Acknowledgments

The authors gratefully acknowledge the ANR (Agence Nationale de la Recherche, France) for its support through project AAPG2018 DataBEST .

Supplementary material

Suppleme ntary material associated with this article can be found, in the online version, at doi: 10.1016/j.jnnfm.2019.07.006 .


Sylvain Rode
email: sylvain.rode@univ-perp.fr

Mathilde Gralepois
email: mathilde.gralepois@univ-tours.fr

E Daniel-Lacombe

Éric Daniel-Lacombe

Les transactions entre la ville et l'inondation pour un urbanisme plus résilient

Keywords:

Les projets d'aménagement urbain en zone inondable se présentent comme des configurations -à la fois spatiales et actorielles -lors desquelles sont mises en oeuvre des transactions entre la ville et l'eau afin d'intégrer le risque d'inondation et aboutir à une forme d'urbanisme résilient. L'étude transversale de quatre projets d'aménagement urbain développés en zone inondable constructible en France -le quartier Rives de Maine à Angers, le quartier Berges de la Robine à Narbonne, le quartier de l'Eure St-Nicolas au Havre et le quartier Matra à Romorantin -permet de faire ressortir les grandes catégories de transactions nécessaires à un urbanisme résilient. Nous identifions tout d'abord trois grands types d'espaces génériques de transaction ville/risque d'inondation, qui sont des solutions d'organisation de l'espace adaptées au risque d'inondation et constituent donc des réponses à l'enjeu de l'urbanisation résiliente des zones inondables : les espaces surélevés, les transparences hydrauliques, l'aménagement paysager de l'interface ville/eau. Nous montrons ensuite que la prise en compte de l'inondation dans un projet d'aménagement urbain tient à la capacité des acteurs du projet à articuler registre technique et registre sensible, mais

aussi à leur capacité à articuler logique réglementaire et logique de projet. Il s'agit de rendre possible la confrontation des différents acteurs et l'invention de solutions d'organisation de l'espace qui soient le résultat de négociations inventives. Mots-clefs :

I INTRODUCTION

Le projet urbain semble désormais envisagé par la littérature scientifique sur l'urbanisme résilient comme « une ouverture du champ des possibles » [START_REF] Gralepois | Ville et inondation : une cohabitation délicate[END_REF] potentiellement favorable à une meilleure intégration des risques à la conception urbaine et à l'aménagement. Nous partons du postulat que les projets d'aménagement urbain en zone inondable se présentent comme des configurations (spatiales et actorielles) à l'occasion desquelles doivent être mises en oeuvre des transactions nouvelles entre la ville et l'eau afin d'intégrer pleinement le risque d'inondation à la fabrique urbaine et ainsi espérer aboutir à une forme d'urbanisme résilient. En informatique, la transaction est une suite d'opérations qui fait passer d'un état antérieur à la transaction à un état postérieur. En droit, « la transaction est un contrat par lequel les parties, par des concessions réciproques, terminent une contestation née, ou préviennent une contestation à naître » (article 2044 du Code civil). En sciences sociales, la notion de transaction sociale insiste sur l'idée que la décision finale est le résultat d'une transformation des termes de l'échange et de la modification des priorités initiales [START_REF] Rémy | La vie quotidienne et les transactions sociales : perspectives micro ou macrosociologiques[END_REF] des différentes parties pour aboutir à une décision. Le terme de transaction sociale implique que le cadre de la négociation n'est pas seulement technique et matériel, mais exprime également « un jeu de pouvoir et de contre-pouvoir autour d'un problème à résoudre » [START_REF] Rémy | Négociations et transaction sociale[END_REF]. Nous parlons ici de « transactions ville/risque d'inondation » pour définir l'ensemble des modes d'articulation du développement urbain et de la prévention des risques qui doivent être mis en oeuvre à l'occasion des projets d'aménagement urbain en zone inondable constructible. Les transactions ville/risque d'inondation se déclinent ainsi en deux registres :

• un registre spatial. Quelles formes particulières d'organisation de l'espace et du bâti sont mises en oeuvre aux différentes échelles afin d'adapter les formes architecturales et urbaines à l'inondation dans les projets d'aménagement urbain ? • un registre actoriel. Les transactions ville/risque d'inondation ne se réduisent pas à la mise en oeuvre de solutions concrètes et fonctionnelles d'organisation de l'espace urbain. Le projet d'aménagement urbain est aussi un espace-temps où s'opèrent des transactions entre différentes catégories d'acteurs. Or, ces jeux d'acteurs -singuliers au sein du projet d'aménagement urbain en zone inondable [Rode, Gralepois, 2017] -jouent un rôle important dans la réussite ou l'échec de ces transactions ville/risque et, in fine, dans la possibilité d'un urbanisme résilient. L'objectif de l'article est de mettre en évidence la nécessité, pour espérer parvenir à une forme d'urbanisme résilient, de mettre en oeuvre des transactions ville/risque d'inondation qui relient étroitement la gouvernance du projet (rôles et interactions entre les différents acteurs, jeux de négociation) et les solutions d'organisation de l'espace de projet retenues pour intégrer le risque. Car si la littérature opérationnelle et académique sépare souvent les deux aspects, d'une part les solutions spatiales [START_REF] Gilsoul | La ville engloutie[END_REF] et d'autre part les solutions actorielles [START_REF] Guevara | L'adaptation aux risques d'inondation façonnée par les métiers de la ville[END_REF], cet article vise à poser les bases d'une réflexion praticienne et académique sur les modalités de convergence de ces deux registres d'action, qui constituent les deux faces indissociables des transactions ville/risque d'inondation pour un urbanisme résilient. Les résultats présentés dans cet article sont issus du travail d'analyse réalisé sur trois quartiers de villes moyennes projetés récemment en zone inondable (Angers / quartier Rives de Maine, Narbonne / quartier Berges de la Robine et Le Havre / quartier de l'Eure St-Nicolas) dans le cadre du projet de recherche PRÉCIEU 1 . Nous y ajoutons un quatrième cas d'étude, le quartier Matra à Romorantin, développé récemment en zone inondable et inondé en juin 2016, à peine achevé 2 . Les méthodes employées sont fondées sur la collecte et l'analyse des documents de planification des territoires concernés (Plan de Prévention des Risques d'Inondation -PPRI -et Plan Local d'Urbanisme -PLU) ainsi que des documents d'urbanisme opérationnel à l'échelle du projet d'aménagement urbain (dossiers de création et de réalisation des Zones d'Aménagement Concerté -ZAC), une série d'entretiens semi-directifs auprès des acteurs de ces projets (élus, professionnels de l'architecture, de l'urbanisme et de l'aménagement paysager qui interviennent en zone inondable), des visites de terrain, deux comités d'expertise de confrontation des résultats et un séminaire final d'étude. À travers une analyse transversale de nos quatre cas d'étude, nous identifions trois grands types d'espaces génériques de transaction ville/risque d'inondation : les espaces surélevés, les transparences hydrauliques, les aménagements paysagers (II). L'analyse des transactions spatiales et 1 PRojet d'Étude sur la Contrainte d'Inondation dans les Espaces Urbanisés (PRÉCIEU) financé par le programme Risques, Décision, Territoires (RDT) du Ministère de l'Environnement, du Développement Durable et de l'Énergie (MEDDE). La direction scientifique de ce projet a été assurée par Mathilde Gralepois (Université de Tours) et Sylvain Rode (Université de Perpignan). 2 Éric Daniel-Lacombe, l'architecte qui a conçu ce quartier, était en effet un membre de l'équipe pluridisciplinaire du projet de recherche PRÉCIEU.

actorielles à l'oeuvre dans les projets d'aménagement urbain en zone inondable permet de mettre en exergue un certain nombre de conditions pour intégrer pleinement l'inondation à la fabrique urbaine et ainsi espérer aboutir à une forme d'urbanisme résilient (III).

II TROIS GRANDS TYPES D'ESPACES GÉNÉRIQUES DE TRANSACTION VILLE/RISQUE D'INONDATION

L'analyse de nos cas d'étude permet d'identifier trois grands types d'espaces génériques de transaction ville/risque d'inondation : les espaces surélevés (1), les transparences hydrauliques (2), les aménagements paysagers (3). Déclinés à des échelles différentes, ces types d'espaces de transaction font appel à des expertises professionnelles spécifiques et sont souvent combinés au sein d'un même projet d'aménagement urbain afin d'en assurer la résilience.

II.1 Les espaces surélevés

La surélévation constitue probablement la réponse la plus classique de l'urbanisme résilient au défi de l'urbanisation en zone inondable. Elle est identifiée aussi bien par la littérature technique [CEPRI, 2015] que par la littérature scientifique [Rode, Gralepois, 2017]. Elle se décline sous plusieurs formes : remblai, bâtiments sur pilotis, bâtiments surélevés sur parkings. Le point commun à ces solutions de surélévation est de porter hors d'eau les espaces fonctionnels sensibles (logements toujours, locaux commerciaux parfois, équipements vulnérables mais aussi voirie ou espaces de circulation). Ce principe de surélévation se retrouve dans les quatre projets étudiés. La solution du remblai est mise en oeuvre à Narbonne, où il s'agit de fabriquer un nouveau sol en rajoutant 50 cm de terres sur le terrain naturel existant afin de surélever le quartier. Néanmoins ce recours au remblai doit s'accompagner de déblais dans d'autres zones du périmètre de projet afin de compenser le volume remblayé, le principe mis en avant par la réglementation étant de parvenir à un équilibre entre remblai et déblai. La surélévation des bâtiments sur parkings a été mise en oeuvre à Angers dans le quartier Saint-Serge : parkings en rez-de-chaussée et dalle surélevée. À Angers toujours, dans le quartier Thiers Boisnet, ce sont les bâtiments et les espaces de circulation piétons qui ont été surélevés. Au Havre, dans le quartier de l'Eure-Saint Nicolas, les bâtiments également sont surélevés, ainsi que les socles des cours à l'échelle de l'îlot. À Romorantin, dans le quartier Matra, la surélévation combine l'usage du remblai et des pilotis, utilisés pour rehausser les immeubles collectifs comme les logements individuels (Fig. 1). Un enjeu important est alors de veiller à ce que les espaces situés sous les constructions sur pilotis demeurent libres et ne soient pas colonisés au bout de quelques années.

Photographie : Sylvain Rode Photographies : Eric Daniel-Lacombe Architecte

II.2 Les transparences hydrauliques

Si une première exigence de l'urbanisme résilient est la mise hors d'eau des bâtiments et des espaces de circulation afin de permettre la continuité de la vie urbaine en cas d'inondation, le second enjeu est de faire en sorte que les nouveaux quartiers développés en zone inondable ne fassent pas obstacle au libre écoulement de l'eau de la rivière en crue. Pour cela, les architectesurbanistes qui conçoivent le parti de composition urbaine de ces quartiers doivent y ménager des transparences hydrauliques, c'est-à-dire des espaces libres de constructions qui pourront accueillir l'eau dans le quartier, la guider, la stocker, etc. L'orientation des bâtiments et des voiries surélevées doit être pensée en fonction de paramètres hydrauliques. Le travail de composition urbaine se fait ainsi en intégrant des données issues de modélisations hydrauliques. À Romorantin comme à Narbonne, les formes urbaines proposées à l'échelle du quartier mettent en oeuvre ce principe des transparences hydrauliques (Fig. 2).

Source : Eric Daniel-Lacombe Architecte

Figure 1 :

 1 Figure 1 : Le quartier Matra à Romorantin : différents types de bâtiments sur pilotis

Figure 2 :

 2 Figure 2 : Le quartier Matra à Romorantin : multiplier les transparences pour permettre la circulation de l'inondation

  2004). -L'activité de projet dans l'aménagement urbain. Processus d'élaboration et modes de pilotage. Le cas de la ligne B du tramway strasbourgeois et d'Odysseum à Montpellier, Thèse de doctorat, Ecole Nationale des Ponts et Chaussées, Aménagement et urbanisme. Bernateau D. (2001). -« Le transfert de risque entre maître d'ouvrage et maître d'oeuvre. Cas de projets complexes », in Bonnet M., Claude V. et Rubinstein M. (éd.), La commande... de l'architecture à la ville, La Défense : Plan Urbanisme construction architecture, 121-134. Bonnet F. (2016). -Atout risques, des territoires exposés se réinventent, Marseille, Parenthèses. CEPRI (Centre Européen pour la Prévention du Risque d'Inondation) (2015). -« Comment saisir les opérations de renouvellement urbain pour réduire la vulnérabilité des territoires inondables face au risque d'inondation ? -Principes techniques d'aménagement ». Champy F. (2000). -« Les architectes, les urbanistes et les paysagistes », in Paquot T., Lussault M., Body-Gendrot S. (dir.), La Ville et l'Urbain, l'état des savoirs, Paris, La Découverte, 215-224. Claude V. (2006). -Faire la ville. Les métiers de l'urbanisme au XX e siècle, Marseille, Parenthèses.

Le rôle central du jeu d'acteurs dans les transactions ville/risque d'inondation

  Ces trois grands types d'espaces génériques de transactions ville/risque d'inondation ont donc pour but, en les combinant au sein d'un espace de projet, de rendre possible l'urbanisation en zone inondable par la production de formes architecturales et urbaines résilientes à l'inondation. Mais, par delà leur apparente simplicité, ces solutions techniques et fonctionnelles d'organisation de l'espace résultent en réalité de transactions entre les différents acteurs des projets d'aménagement urbain. L'intégration pleine et entière de l'inondation à la fabrique urbaine en faveur d'un urbanisme résilient se joue ainsi largement dans les transactions actorielles singulières à l'oeuvre dans le projet d'aménagement urbain en zone inondable. Dans la droite ligne du paradigme de la transaction, qui se représente « la vie sociale comme étant une confrontation d'une pluralité d'acteurs en relation partiellement conflictuelle et en négociation pour déterminer des zones d'accord en fonction de leur capacité de pression respective »[START_REF] Rémy | Produire ou reproduire ? Une sociologie de la vie quotidienne[END_REF], le projet d'aménagement urbain en zone inondable peut être conçu comme un véritable espace de transactions socio-professionnelles entre acteurs(discussions, échanges, controverses, oppositions, divergences, négociations, compromis, invention, etc.) leur permettant ainsi d'inventer des transactions spatiales (c'est-à-dire qui traduisent dans l'espace urbain la place donnée au risque d'inondation) à la fois techniques et sensibles. Véhiculant « une posture d'action Aujourd'hui on le ferait, mais à leur époque cela n'a pas été fait. On a géré la contrainte de l'eau » (entretien Agence d'Urbanisme de la Région Angevine, 07/04/2014). De fait, les transactions ville/risque d'inondation ont toutes les chances de ne pas s'avérer pleinement efficaces si elles sont réduites à leur seule dimension technique et réglementaire. En étant une réponse « sectorielle », elles peuvent être efficaces d'un point de vue de la protection contre l'inondation, mais pas du fonctionnement urbain dans son ensemble, de son urbanité. En activant le registre sensible, le travail paysager de mise en visibilité de l'eau aide à développer la compréhension des enjeux liés à la prévention des risques d'inondation. « La notion de projet urbain renvoie aussi à une multiplicité de techniques, parce qu'il se rapporte à plusieurs compétences d'aménagement, de construction, d'écologie. Cette multiplicité de techniques n'a de sens que si elle a une légitimation globale (dans la conception même des choses et dans les moyens d'articuler toutes les techniques) de nature politique. Le projet urbain […] s'identifie avec un ensemble d'actions inscrites dans la durée et légitimées par le pouvoir politique »[START_REF] Ingallina | Le projet urbain[END_REF]. Ainsi, pour que les transactions ville/risque d'inondation au sein d'un projet d'aménagement urbain puissent être fertiles, un certain nombre de caractéristiques du jeu d'acteurs doivent également être réunies. Une alchimie entre un maître d'ouvrage et les services de l'État (1) ainsi qu'entre les différents professionnels de l'équipe de maîtrise d'oeuvre (2) est tout particulièrement nécessaire à l'invention d'un urbanisme résilient. Une certaine continuité politique peut être bienvenue pour éviter le risque de remise en cause des orientations du projet au moment d'alternances politiques. Le cas de Narbonne, où les trois dernières élections municipales ont conduit à des changements de maires et de majorités municipales, illustre de manière patente les conséquences directes qu'une forme d'instabilité politique peut avoir en déstabilisant profondément un projet, en revenant sur les choix d'un prédécesseur, produisant ainsi une forme de paralysie de l'action publique.Le rôle des services de l'État apparaît essentiel dans un projet d'aménagement urbain en zone inondable. Ils ont en effet pour mission d'élaborer la règle (à travers l'élaboration des Plans de Prévention des Risques d'Inondation) et de s'assurer de son respect. Mais ils doivent aussi faire preuve d'une capacité à intégrer la logique de projet comme un espace-temps propice à une vraie réflexion sur l'intégration du risque d'inondation dans un projet d'aménagement global, c'est-à-dire une opportunité pour inventer des solutions en faveur d'une ville plus résiliente plutôt que comme un simple lieu d'application d'une règle pensée comme intangible. Pour cela, il semble souhaitable que les services de l'État intègrent l'espace de transaction du projet comme un acteur à part entière 'espace de transaction ville/risque d'inondation peut alors devenir un lieu de « rencontre institutionnelle [censée être féconde] entre les services de l'État -qui dépassent leur fonction régalienne pour apporter une expertise aiguisée -et les collectivités locales, dont on stimule les capacités d'établir une vision partagée, une vision politique fédératrice »[Bonnet, 2016].III.2.2 Le rôle des professionnels du projet urbainLe rôle et la place des différents professionnels du projet urbain réunis au sein des équipes de maîtrise d'oeuvre se trouvent redéfinis dans le projet d'aménagement urbain en zone inondable. Si la « transversalité des métiers »[Bonnet, 2016] est capitale dans les transactions ville/risque d'inondation, il est néanmoins possible d'identifier des enjeux spécifiques selon les métiers. Les professionnels de la conception urbaine que sont les architectes-urbanistes découvrent souvent la problématique de l'inondation et les enjeux hydrauliques une fois qu'ils sont impliqués dans le projet, c'est du moins le cas dans les premiers projets d'aménagement urbain développés en zone inondable constructible dans la perspective d'inventer un urbanisme résilient. Dans ce cas, ces architectes-urbanistes en charge de la conception du quartier ne sont pas spécialistes de l'urbanisme résilient et ils n'ont donc pas de recette toute faite à mobiliser pour apporter des réponses architecturales et urbaines aux problématiques hydrauliques. L'essentiel n'est-il pas, alors, d'avoir à faire à des professionnels de la conception qui prennent au sérieux la question du risque dans la conception du projet et en fassent un paramètre à part entière de la conception urbaine ? C'est ce qui s'est passé pour le quartier Matra à Romorantin, où l'architecte-urbaniste en charge de la conception du projet a « découvert » la problématique de l'inondation une fois le projet déjà assez avancé. Dans ce cas, les professionnels de la conception urbaine ont accepté de jouer le jeu de l'inondation. Il a fallu apprendre au fur et à mesure de l'avancement du projet, s'acculturer à de nouvelles notions, intégrer de nouveaux paramètres, faire avec la présence incontournable de nouveaux acteurs, mobiliser de nouvelles expertises, et inventer des solutions adaptées au territoire et à son inondabilité, pour finalement en faire un paramètre à part entière de la conception. De fait, ce rapport subjectif aux éléments naturels et fluctuants du territoire dont se saisit le paysagiste apparaît comme incontournable dans un projet d'aménagement urbain en zone inondable. La mise en scène paysagère, souvent esthétisée, de l'eau conduit à affirmer la présence sensible de l'eau dans la ville, dont on a souligné l'importance afin de ne pas se cantonner à une approche technique de l'urbanisme résilient. Le paysagiste participe donc de la double dimension technique et sensible des transactions ville/risque d'inondation. Toutefois, si son intervention est utile et recherchée, le travail du paysagiste ne suffit pas à résoudre certains problèmes posés par l'urbanisme résilient en termes de production d'urbanité[Rode, Gralepois, 2017] et a finalement « une portée limitée dans le renouvellement des pratiques de l'urbanisme »[START_REF] Guevara | L'adaptation aux risques d'inondation façonnée par les métiers de la ville[END_REF]. 'invention de transactions ville/risque d'inondation nouvelles et efficaces afin de parvenir à un urbanisme résilient ne doit donc pas se limiter au seul registre technique, au risque de produire des formes architecturales et urbaines stéréotypées, peu satisfaisantes en terme de fonctionnement ordinaire et quotidien. Comme dans tout projet mais peut-être plus encore pour les projets en zone inondable, la capacité des différents acteurs à coopérer pour inventer ensemble des solutions adaptées et situées apparaît déterminante. De fait, la réussite dans la prise en compte de l'inondation par un projet d'aménagement urbain tient largement à la capacité des acteurs de ce projet à créer un espace de transactions qui permette la confrontation des points de vue et contraintes des différents acteurs et l'invention de solutions d'organisation de l'espace qui soient le résultat de négociations inventives et créatives. Lorsque les dispositifs spatiaux retenus comme réponse à la problématique de l'inondabilité ne sont pas le fruit d'une proposition collective des acteurs du projet d'aménagement urbain, et qu'ils ne sont pas appropriés et portés par un collectif d'acteurs, bien souvent ils ne produisent pas la résilience ou l'urbanité recherchée. L'enjeu est aussi de s'assurer du caractère durable de ces solutions d'urbanisme résilient : au-delà de l'invention d'un urbanisme résilient durant la phase de conception urbaine d'un projet de quartier nouveau, il est également nécessaire de faire en sorte que l'urbanisme conserve sa résilience à l'épreuve du temps et de la vie quotidienne des quartiers. L'enjeu n'est-il pas finalement, pour l'ensemble des acteurs du projet d'aménagement urbain en zone inondable, de parvenir à mieux articuler la logique réglementaire (ce qui est autorisé versus ce qui est interdit) et la logique de projet (l'invention collective de solutions adaptées aux territoires et à leurs enjeux singuliers) ?

	III.2 Ces IV CONCLUSION
	professionnels de l'urbanisme ont accepté une forme de déstabilisation de leur cadre cognitif et
	professionnel d'intervention, les obligeant ainsi à délaisser les solutions toutes faites et
	possiblement reproductibles d'un projet à un autre au profit d'un registre d'invention créative
	permettant de dégager des réponses propres à un territoire, et adaptées à ses problématiques
	spécifiques.
	Ce type de positionnement de l'architecte-urbaniste intervenant en zone inondable, devant
	composer avec l'incertitude et inventer des réponses nouvelles et adaptées à l'inondabilité, ne
	contribue-t-il pas, somme toute, à réaffirmer le caractère prudentiel de sa pratique
	professionnelle [Champy, 2000], c'est-à-dire la mobilisation de savoirs dans des situations
	d'incertitude, dans lesquelles on ne peut se contenter d'appliquer un savoir normalisé [Vézinat, exprimant les 2010] ? Mais dans le même temps, l'intervention des professionnels de la modélisation hydraulique, caractéristiques du milieu naturel, doit provoquer des « effets de révélation qui déclenchent la en cadrant fortement le travail de composition urbaine et en ayant tendance à lui imposer des réflexion » (Bachoc, Lyon, 2015). C'est le parti adopté dans le quartier Matra à Romorantin, où « la réponses techniques parfois stéréotypées à la problématique de l'inondabilité, ne limite-t-elle pas ce montée des eaux s'impose à l'imagination » [Daniel-Lacombe, Paquot, 2016] grâce à un ensemble caractère prudentiel ? de signes et à un traitement sensible de l'espace du projet. Par exemple, « le bassin de rétention des eaux de pluie dans le parc public est tout autant un catalyseur de l'imaginaire qu'un objet sdans laquelle l'architecte 'décidait' de la forme urbaine d'un quartier » [Ingallina, 2001].
	technique. […] Les présages qu'il porte ne sont pas faciles à interpréter, mais ils attirent l'attention Les paysagistes, « nouveaux venus dans le domaine de l'aménagement de l'espace » [Claude,
	de tous les habitants et font l'objet de conversations et de débats d'opinion à propos du climat, de la 2006], jouent eux aussi un rôle majeur dans ces transactions ville/risque d'inondation dans la
	rivière, et de ses incidences sur le quartier » [Daniel-Lacombe, Paquot, 2016]. Il s'agit bien ainsi de mesure où leur travail va participer directement à la conception de certaines transactions spatiales,
	contribuer à la culture du risque des habitants des quartiers résilients. A contrario, le cas d'Angers notamment à travers les aménagements paysagers. Présentant la profession de paysagiste
	montre la nécessité d'aller au-delà d'une simple réponse technique à la prise en compte du risque concepteur, la Fédération Française du Paysage souligne : « Son savoir-faire n'est pas seulement

III POUR INTÉGRER PLEINEMENT L'INONDATION À LA FABRIQUE URBAINE spécifique, celle du dialogue, du partenariat, de la négociation »

[Arab, 2004]

, le projet est censé le permettre. Néanmoins, les rapports de force à l'oeuvre entre acteurs, leurs logiques politiques et/ou professionnelles potentiellement discordantes peuvent compliquer les choses. Les transactions ville/risque d'inondation se jouent alors dans une articulation entre registre technique et registre sensible (1) ainsi que dans une articulation des interventions des différents acteurs du projet (2).

III.1 Articuler registre technique et registre sensible

Un enjeu majeur des transactions ville/risque d'inondation pour un urbanisme résilient est que les espaces de transaction parviennent à articuler mise hors d'eau des bâtiments et culture sensible du risque en inscrivant la présence sensible de l'eau dans l'espace du projet. C'est ce qu'exprimait André Bachoc lors du séminaire Risques, Décision, Territoires à mi-parcours qui s'est tenu à Lyon en novembre 2015 en évoquant l'importance de « raconter l'histoire de l'eau » à travers une écriture architecturale, urbaine et paysagère spécifique aux nouveaux quartiers développés en zone inondable. L'enjeu est ainsi de parvenir à rendre l'eau visible sur le territoire, alors même qu'elle n'y est pas présente en permanence. Cette écriture sensible, dans l'aménagement urbain. Certains acteurs des projets Saint-Serge et Thiers Boisnet en ont conscience a posteriori, estimant que la prise en compte des inondations a été « peut-être une réponse exclusivement technique : il faut surélever. Il n'y a pas eu d'approche sensible comme sur Rives Nouvelles, de dire on va prendre en compte la contrainte, mais que cette donnée soit un espace de projet.

III.2.1 La maîtrise d'ouvrage et les services de l'État

Le rôle des élus locaux, et d'une manière plus générale de la maîtrise d'ouvrage dans ses niveaux régalien et opérationnel

[Bernateau, 2001]

, ne doit pas être sous-estimé. Si ce ne sont pas eux qui dessinent le projet et décident des solutions techniques et sensibles propres à traduire la contrainte hydraulique dans l'aménagement, leur rôle en termes d'élaboration du programme et de conduite du projet n'est pas sans influence sur le cours du projet et ses orientations. Les élus locaux doivent informer le projet en portant une vision pour le territoire, une vision du projet, en fixant un cap et en contribuant à ce qu'il soit tenu tout en faisant preuve de souplesse pour s'adapter aux imprévus. et non pas qu'ils se considèrent uniquement comme un acteur extérieur, surplombant, celui qui dit la règle. Ils ne peuvent se contenter, comme c'est souvent le cas dans les projets hors zone à risque, d'un rôle « d'observateur vigilant »

[Claude, 2006]

. En participant pleinement à la dynamique du projet, il devient alors possible pour les services de l'État de se placer dans une logique constructive d'échanges avec les acteurs politiques et professionnels du projet, afin inventer avec eux des solutions d'urbanisme résilient.

Ltechnique, puisqu'il sait intégrer la dimension subjective, culturelle et sociale du paysage, et mettre en oeuvre sa propre créativité ; ni seulement architectural, puisqu'il connaît le végétal, le sol, l'eau, et aussi la complexité, l'évolution et l'incertitude d'un milieu vivant. » 3 . L

http://www.f-f-p.org/fr/paysagiste-concepteur/definition/