Erwin Gander

Victorita Dolean

Martin J Gander

Erwin Veneros

Schwarz Methods for Second Order Maxwell Equations in 3D with Coefficient Jumps

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Classical Schwarz methods need in general overlap to converge, but in the case of hyperbolic problems, they can also be convergent without overlap, see [START_REF] Dolean | Why Classical Schwarz Methods Applied to Hyperbolic Systems Can Converge even Without Overlap[END_REF]. For the first order formulation of Maxwell equations, we have proved however in [START_REF] Dolean | Optimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients[END_REF] that the classical Schwarz method without overlap does not converge in most cases in the presence of coefficient jumps aligned with interfaces.

Optimized Schwarz methods have been developed for Maxwell equations in first order form without conductivity in [START_REF] Dolean | Optimized Schwarz methods for Maxwell equations[END_REF], and with conductivity in [START_REF] Dolean | Optimized Schwarz methods for Maxwell's equations with non-zero electric conductivity[END_REF][START_REF] El Bouajaji | Optimized Schwarz methods for the time-harmonic Maxwell equations with damping[END_REF]. These methods use modified transmission conditions, and often converge much faster than classical Schwarz methods. For DG discretizations of Maxwell equations, optimized Schwarz methods can be found in [START_REF] Dolean | A domain decomposition method for solving the threedimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods[END_REF][START_REF] Dolean | Optimized Schwarz algorithms for solving timeharmonic Maxwell's equations discretized by a discontinuous Galerkin method[END_REF][START_REF] Dolean | Domain decomposition methods for electromagnetic wave propagation problems in heterogeneous media and complex domains[END_REF]. Optimized Schwarz methods were also developed for the second order formulation of Maxwell equations, see [START_REF] Alonso-Rodriguez | New nonoverlapping domain decomposition methods for the harmonic Maxwell system[END_REF], and [START_REF] Peng | Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics[END_REF][START_REF] Peng | One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems[END_REF] for scattering problems with applications.

While usually coefficient jumps hamper the convergence of domain decomposition methods, this is very different for optimized Schwarz methods. For diffusive problems, it was shown in [START_REF] Dubois | Optimized Schwarz methods for the advection-diffusion equation and for problems with discontinuous coefficients[END_REF] that jumps in the coefficients can actually lead to faster iterations, when they are taken into account correctly in the transmission conditions: optimized Schwarz methods benefit from jumps in the coefficients at interfaces. We had shown in [START_REF] Dolean | Optimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients[END_REF] that this also holds for the special case of transverse magnetic modes (TMz) in the two dimensional first order Maxwell equations. We show in this short paper that these results for the TMz modes (and the corresponding ones for the transverse electric modes (TEz)) can be used to formulate optimized Schwarz methods for the 3D second order Maxwell equations which then in some cases converge faster, the bigger the coefficient jumps are. The time dependent Maxwell equations in their second order formulation are

ε∂ 2 t E + ∇ × (µ -1 ∇ × E ) = ∂ t J , (1)
where E = (E 1 , E 2 , E 3 ) T is the electric field, ε is the electric permittivity, µ is the magnetic permeability, and J is the applied current density. We assume that the applied current density is divergence free, divJ = 0. There is a similar system also for the magnetic field

H = (H 1 , H 2 , H 3 ) T , µ∂ 2 t H + ∇ × (ε -1 ∇ × H ) = ∇ × ε -1 J , (2)
but we will only consider the equations (1) for the electric field in this short paper.

The time dependent Maxwell equations (1) form a system of hyperbolic partial differential equations [START_REF] Dolean | Optimized Schwarz methods for Maxwell equations[END_REF]. Imposing incoming characteristics is equivalent to imposing the impedance condition

B n j (E m,n ) = 1 µ m (∇ × E m,n × n j ) × n j + iω Z m (E m,n × n j ) = s, (3)
where Z m = µ m ε m . We are interested here in the time-harmonic Maxwell equations, which are obtained by supposing that E (x,t) = e iωt E(x) for a fixed frequency ω. After some simplifications, we obtain from equation (1) the time harmonic second order Maxwell equation

εω 2 E -∇ × (µ -1 ∇ × E) = -iωJ. ( 4
)
We are interested here in the heterogeneous case, where the domain Ω of interest consists of two non-overlapping subdomains Ω 1 and Ω 2 with interface Γ , and piecewise constant parameters ε j and µ j in Ω j , j = 1, 2. We want to solve such problems using the Schwarz algorithm

ε 1 ω 2 E 1,n -∇ × (µ -1 1 ∇ × E 1,n ) = -iωJ, in Ω 1 , T n 1 (E 1,n ) = T n 1 (E 2,n-1 ) on Γ , ε 2 ω 2 E 2,n -∇ × (µ -1 2 ∇ × E 2,n ) = -iωJ, in Ω 2 , T n 2 (E 2,n ) = T n 2 (E 1,n-1 ) on Γ , (5)
with the transmission condition

T n j (E m,n ) = (Id -A j )( 1 µ m n j × ∇ × E m,n ) - iω j µ j (Id + A j )(n j × (E m,n × n j )). ( 6
)
with

ω j = ω √ ε j µ j , j = 1, 2.
The classical Schwarz algorithm is obtained for the choice A j = 0, for j = 1, 2. We see that the classical Schwarz algorithm is exchanging characteristic information at the interfaces between subdomains, i.e.

T n j (E m,n ) = B n j (E m,n ) where B is defined in (3).
In [START_REF] Dolean | Optimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients[END_REF], we studied the classical Schwarz algorithm for the first order Maxwell equations on the domain 2 and interface Γ = {0} × R 2 and the Silver-Müller radiation condition. We showed that the convergence factor of the classical Schwarz algorithm in 3D is ρ cla = max{ρ Ecla , ρ Mcla }, where ρ Ecla and ρ Mcla are the convergence factors of the TEz and TMz cases in 2D. We then proved that if there are coefficient jumps along interface Γ , i.e. µ 1 = µ 2 and/or

Ω = R 3 , with subdomains Ω 1 = (-∞, 0] × R 2 and Ω 2 = [0, ∞) × R
ε 1 = ε 2 , the classical Schwarz algorithm is diver- gent in 3D if µ 1 ε 2 = µ 2 ε 1 . If µ 1 ε 2 = µ 2 ε 1 , we obtained ρ Ecla = ρ Mcla , and ρ cla < 1 for the propagative modes, |k| < ω j , j = 1, 2, but ρ cla (|k|) = 1 for the evanescent modes, |k| > ω j , j = 1, 2,
so the algorithm is stagnating for all evanescent modes. It is thus never convergent in 3D. We then investigated in [START_REF] Dolean | Optimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients[END_REF] the 2D case of TMz modes in more detail, and found that the classical Schwarz algorithm in the presence of coefficient jumps is convergent in certain situations, depending on the jumps in ε and µ.

These results also hold for the second order Maxwell equations when the Schwarz algorithm [START_REF] Dolean | Optimized Schwarz methods for Maxwell's equations with non-zero electric conductivity[END_REF][START_REF] Dolean | Domain decomposition methods for electromagnetic wave propagation problems in heterogeneous media and complex domains[END_REF] with classical transmission conditions is applied, and for the convergent cases from [START_REF] Dolean | Optimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients[END_REF] in 2D, we have the following new contraction estimate:

Theorem 1 (Classical Schwarz in 2D). If the classical Schwarz algorithm [START_REF] Dolean | Optimized Schwarz methods for Maxwell's equations with non-zero electric conductivity[END_REF][START_REF] Dolean | Domain decomposition methods for electromagnetic wave propagation problems in heterogeneous media and complex domains[END_REF] in 2D converges, then we have the asymptotic convergence factor estimate

ρ Mcla (k, ω 1 , ω 2 , Z) = ρ Ecla (k, ω 1 , ω 2 , Z) = 1 -O(h 2 ) with Z = µ 1 ε 2 µ 2 ε
1 and h the uniform mesh size.

Proof. As in [START_REF] Dolean | Optimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients[END_REF], we can write the convergence factors for the TMz case as

ρ Mcla (k, ω 1 , ω 2 , Z) = k 2 -ω 2 1 -iω 1 Z k 2 -ω 2 2 -iω 2 /Z k 2 -ω 2 1 + iω 1 k 2 -ω 2 2 + iω 2 1 2 , (7)
and for evanescent modes (k > ω 1 , ω 2 ), equation ( 7) is equal to

ρ Mcla (k, ω 1 , ω 2 , Z) = 1 + (Z 2 -1)ω 2 1 Z 2 k 2 Z 2 -Y 2 - (Z 2 -1)ω 2 2 k 2 , (8)
with Y = ω 2 ω 1 . From equation [START_REF] Dolean | Optimized Schwarz methods for Maxwell equations[END_REF] we see that lim k→∞ ρ Mcla = 1. If the classical Schwarz algorithm is convergent then ρ Mcla < 1, ∀k, the previous remark permits us to conclude that the maximum over all the frequencies must be at k = k max = c max h , the largest frequency supported by the numerical grid, where h is the mesh size and c max is a constant depending on the geometry. To conclude the proof, we just insert k = c max /h into (8) and the result follows by expansion. The proof for the TEz case is similar.

Optimized Schwarz for Second Order Maxwell Equations

Since the classical Schwarz method is not an effective solver for Maxwell equations in the presence of coefficient jumps, we introduce now more effective transmission conditions which take the coefficient jumps into account. We consider algorithm [START_REF] Dolean | Optimized Schwarz methods for Maxwell's equations with non-zero electric conductivity[END_REF][START_REF] Dolean | Domain decomposition methods for electromagnetic wave propagation problems in heterogeneous media and complex domains[END_REF] with the particular choice

A j := γ jM S T M + γ jE S T E , S T M = ∇ τ ∇ τ •, S T E = ∇ τ × ∇ τ ×,
where τ is the tangential direction to the interface. We note that S T M -S T E = ∆ τ I, where ∆ τ is the Laplace-Beltrami operator in the tangential plane (for example , ∆ τ = ∂ yy +∂ zz when n = (1, 0, 0)). The constants γ 1E , γ 2E and γ 1M , γ 2M can be chosen in order to optimize the algorithm.

Performing a Fourier transform in the yz plane, we find after a lengthy calculation the iteration matrix of the optimized Schwarz algorithm to be

IT = C E 0 0 C M ( 9
)
with the coefficients

C E = ((λ 1 -iω 1 /Z) -γ 2M |k| 2 (λ 1 + iω 1 /Z))((λ 2 -iω 2 Z) -γ 1M |k| 2 (λ 2 + iω 2 Z)) (2ω 1 -i(λ 1 -iω 1 )(1 -γ 1M |k| 2 ))(2ω 2 -i(λ 2 -iω 2 )(1 -γ 2M |k| 2 )) , C M = ((λ 1 -iω 1 Z) -γ 2E |k| 2 (λ 1 + iω 1 Z))((λ 2 -iω 2 /Z) -γ 1E |k| 2 (λ 2 + iω 2 /Z)) ((1 -γ 1E |k| 2 )(λ 1 -iω 1 ) + 2iω 1 )((1 -γ 2E |k| 2 )(λ 2 -iω 2 ) + 2iω 2 ) , (10)
with λ j = |k| 2 -ω 2 j , j = 1, 2. If we choose for the parameters the values

γ 1M = λ 2 -iω 2 Z |k| 2 (λ 2 + iω 2 Z) , γ 1E = λ 2 -iω 2 /Z |k| 2 (λ 2 + iω 2 /Z) , γ 2E = λ 1 -iω 1 Z |k| 2 (λ 1 + iω 1 Z) , γ 2M = λ 1 -iω 1 /Z |k| 2 (λ 1 + iω 1 /Z) , (11)
then the iteration matrix IT in (9) vanishes and we have convergence in two iterations. The corresponding transmission conditions are called transparent conditions, and are optimal, since they lead to a direct solver. But the operators corresponding to the symbols in [START_REF] Dubois | Optimized Schwarz methods for the advection-diffusion equation and for problems with discontinuous coefficients[END_REF] are non local and thus costly to use. We therefore propose to replace λ 1 and λ 2 in (11) by zeroth order approximations s 1E , s 1M , s 2E and s 2E . The convergence factor of the method is then the maximum of the spectral radius of ( 9) over all Fourier frequencies. We obtain

ρ opt = max{ρ Eopt , ρ Mopt }, (12)
with

ρ Eopt (|k|, ω, ε 1 , ε 2 , µ 1 , µ 2 , s 1M , s 2M ) = (λ 2 -s 2M )(λ 1 -s 1M ) (λ 2 +s 1M ε 2 /ε 1 )(λ 1 +s 2M ε 1 /ε 2 ) 1/2 , ρ Mopt (|k|, ω, ε 1 , ε 2 , µ 1 , µ 2 , s 1E , s 2E ) = (λ 1 -s 1E )(λ 2 -s 2E ) (λ 2 +s 1E µ 2 /µ 1 )(λ 1 +s 2E µ 1 /µ 2 ) 1/2 . ( 13
)
These factors can be optimized separately and they are once again the convergence factors of the TMz and TEz cases in 2D. In order to optimize we have to choose s jE , s jM , j = 1, 2 such that ρ opt is as small as possible for all numerically relevant frequencies k ∈ K := [k min , k max ]. Here k min is the smallest frequency relevant to the subdomain, and k max = c max h is the largest frequency supported by the numerical grid, h being the mesh size, see for example [START_REF] Gander | Optimized Schwarz methods[END_REF]. We search for s jE and s jM of the form s jE = c jE (1 + i), s jM = c jM (1 + i) such that s jE , s jM , j = 1, 2 will be the solutions of the min-max problems min

s 1E ,s 2E ∈C max k∈K ρ Mopt (|k|, ω, ε 1 , ε 2 , µ 1 , µ 2 , s 1E , s 2E ), (14)
min

s 1M ,s 2M ∈C max k∈K ρ Eopt (|k|, ω, ε 1 , ε 2 , µ 1 , µ 2 , s 1M , s 2M ). ( 15
)
Since the optimization can be performed independently, we can use our results from [START_REF] Dolean | Optimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients[END_REF] and obtain

Corollary 1 (2D asymptotically optimized contraction factor). For TMz, the solution of ( 14) for Y = 1 gives the asymptotic convergence factor

ρ * Mopt =        1 -O(h 1/4 ) if Z = Y , µ min µ max + O(h) if Z ≤ Y < √ 2Z or Y ≤ Z < √ 2Y , 4 1 2 + O(h) if Z < √ 2Y or Y > √ 2Z. (16)
If Z = 1 and Y = 1, we obtain after excluding the resonance frequency [START_REF] Dolean | Optimized Schwarz methods for Maxwell equations[END_REF] ρ * Mopt =

µ min µ max + O(h).
For the TEz case, the same conclusion holds if we replace Y by Y -1 and µ by ε.

The results in 3D follow now by a systematic consideration of both cases together:

Theorem 2 (3D asymptotically optimized contraction factor, Case A). If Z = Y,Y -1 and Y = 1, the optimized convergence factor ρ * opt in [START_REF] El Bouajaji | Optimized Schwarz methods for the time-harmonic Maxwell equations with damping[END_REF] has the asymptotic behavior:

1. If min max{(ZY ) -1 , ZY }, max{Z/Y,Y /Z} > √ 2, then ρ * opt = 4 1/2 + O(h). ( 17
) 2. If min max{(ZY ) -1 , ZY }, max{Z/Y,Y /Z} = max{Z/Y,Y /Z} ≤ √ 2, then ρ * opt = µ min µ max + O(h). (18)

If min max{(ZY

) -1 , ZY }, max{Z/Y,Y /Z} = max{(Y Z) -1 ,Y Z} ≤ √ 2, then ρ * opt = ε min ε max + O(h). (19)
Proof. To prove 1. we use twice Corollary 1. If max{Z/Y,Y /Z} > √ 2, we use the third result in [START_REF] Peng | Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics[END_REF] for the TMz case. Similarly if max{ZY, (ZY ) -1 } > √ 2 we use also the third result in ( 16) but for the TEz case. From equation [START_REF] El Bouajaji | Optimized Schwarz methods for the time-harmonic Maxwell equations with damping[END_REF] we know that ρ opt is the maximum of ρ Eopt and ρ Mopt , and if both of them have the asymptotic behaviour 4 [START_REF] Dolean | Optimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients[END_REF].

Finally, for 3., one can proceed as for 2 to obtain (19).

Theorem 3 (3D asymptotically optimized contraction factor, Case B).

If Z = Y or Z = Y -1
, then the optimized convergence factor ρ * opt in [START_REF] El Bouajaji | Optimized Schwarz methods for the time-harmonic Maxwell equations with damping[END_REF] satisfies

ρ * opt = 1 -O(h 1/4 ). (20)
Proof. We use the first result in [START_REF] Peng | Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics[END_REF] of Corollary 1 and proceed as in Theorem 2.

Theorem 4 (3D asymptotically optimized contraction factor, Case C). If Y = 1 and Z = Y , then the optimized convergence factor ρ * opt in [START_REF] El Bouajaji | Optimized Schwarz methods for the time-harmonic Maxwell equations with damping[END_REF] satisfies

ρ * opt = µ min µ max + O( √ h). ( 21
)
Proof. After excluding the resonance frequency, we apply the second part of Corollary 1. Note that in this case ρ Mopt = ρ Eopt .

Theorem 2 and 4 contain the important result that in the presence of jumps in the coefficients, the convergence of the optimized Schwarz method for Maxwell equations gets faster when the jump increases, the method benefits from the jumps! In the first part of Theorem 2, the convergence is independent of the jump in the coefficients, and in all these cases the nonoverlapping method converges independently of the mesh parameter, also unusual for optimized Schwarz methods without jumps in the coefficients. In the case of Z = Y or Z = Y -1 (µ 1 = µ 2 or ε 1 = ε 2 ) in Theorem 3 however, the convergence factor depends on h and deteriorates as h goes to zero, as in the case without jumps presented in [START_REF] Dolean | Optimized Schwarz methods for Maxwell equations[END_REF]. We now illustrate graphically the improvement of the optimized Schwarz method over the classical one in 2D. We show in Figure 1 in red the divergence regions and in blue the convergence regions for different values of Z and Y . In the left graphic the white part is still an open problem. In the right the light blue line have convergence dependant of the mesh size h, the light blue region have convergence dependent on the coefficients µ ′ s and the dark blue region have convergence independent of the mesh size h and the coefficients µ ′ s, the red line is the zone of resonance corrected with theorem 1. We clearly see that the optimization of the transmission conditions transforms an algorithm that fails for a large range of problems into one that works in all cases.

C 0 C -1 0 1 2 3 4 5 1 2 3 4 5
ε 1 = ε 2 ω 1 = ω 2 µ 1 = µ 2 Y Z 1 2 3 4 5 1 2 4 5 Y Z ω 1 = ω 2 µ 1 = µ 2 µ 1 /µ 2 = √ 2 µ 2 /µ 1 = √ 2

Conclusions

Classical Schwarz methods applied to 3D Maxwell equations with jumps in the coefficients aligned with the interfaces do not converge, and this is also the case for the second order formulation of Maxwell equations. Using however optimized transmission conditions, we showed that one can obtain Schwarz methods for the 3D Maxwell equations that converge independently of the mesh parameter in some cases, and even become faster as the jumps get larger at the interfaces. These methods directly benefit from the jumps in the coefficients. We presented precise asymptotic convergence factor estimates for the many different cases of coefficient jumps, and are currently working on the numerical implementation of these methods in the full 3D setting.

Fig. 1

 1 Fig. 1 Convergence regions in blue and divergence regions in red for classical Schwarz (left) and optimized Schwarz (right)

  1/2 + O(h), we get (17) as required. For 2. we know that max{Z/Y,Y /Z} ≤ √ 2, which means that we can use the second result in (16), i.e. ρ Mopt = Eopt = 4 1/2 + O(h), and we clearly see ρ Mopt > ρ Eopt . If max{(ZY ) -1 , ZY } ≤ √ 2, we obtain by hypothesis the inequality max{Z/Y,Y /Z} ≤ max{(ZY ) -1 , ZY } ≤ √ 2, and this implies µ min µ max ≥ ε min ε max . Then we obtain ρ Mopt ≥ ρ Eopt and thus

	and ZY = ε 2 ε 1 which implies 1 ≥	µ min µ max + O(h). We note that Z/Y = µ 1 µ 2 µ max ≥ 4 1 µ min 2 . If max{(ZY ) -1 , ZY } > √ 2, by
	Corollary 1 we have ρ


Tristan Guillot

Douglas N C Lin

Mathieu Pierre Morel

Mathieu Havel

Vivien Parmentier

Evolution of Exoplanets and their Parent Stars

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The discovery of 51 Peg b (Mayor & Queloz 1995) heralded the birth of a new field: exoplanetology. The study of exoplanets is important in itself, but they also offer us a new window to study their parent stars, the interactions between stars and planets and the processes that led to their formation.

This paper stems from a presentation given in October 2013 in Roscoff, France, at the school "The ages of stars". We present a few key elements based on the authors' work to understand the physical and dynamical evolution of exoplanets and their parent stars. This is by no means a proper review of this extremely rich field.

We first examine the physical mechanisms that govern the evolution of fluid planets, brown dwarfs and stars. In Section 3, we show that the global composition of exoplanets may be derived from evolution models and linked to the composition of their parent star. In Section 4, we discuss how star-planet interactions shape the population of close-in exoplanets and brown dwarfs, leading some to be swallowed by their star.

Thermal evolution of exoplanets

The wide range of exoplanets discovered include objects from a fraction of an Earth-mass to giant planets many times the mass of our Jupiter. It also includes objects orbiting only a few stellar radii away from their parent stars to objects at orbital distances beyond 100 AU. We focus on objects for which both a mass and a radius may be determined, corresponding to the ones discovered transiting in front of their parent star. We also choose to restrict ourselves to giant planets and brown dwarfs, i.e. fluid objects mostly made of hydrogen and helium. Because of their large masses (more than about 100 times the mass of the Earth), these objects inherit a large amount of internal energy from their gravitational contraction. Because of a non-negligible thermal expansion coefficient, the progressive loss of this initial energy results in both a cooling and a contraction of the planets. It can indeed be shown that the energy loss per unit time (i.e. the intrinsic luminosity) follows a modified Kelvin-Helmholtz relation:

L ≈ η GM 2 Rτ , (2.1)
where τ is the age, and η is a factor that hides most of the complex physics. In the approximation that Coulomb and exchange terms can be neglected, η ≈ θ/(θ + 1) where θ is the electron degeneracy factor. The poor compressibility of giant planets in their mature evolution stages imply that η ≪ 1 (η ∼ 0.03 for Jupiter): the luminosity is not obtained from the entire gravitational potential, but from the much more limited reservoir constituted by the thermal internal energy (Guillot 2005). The evolution of giant planets and brown dwarfs is thus akin to the premain sequence evolution of stars (see also Burrows et al. 1997;Chabrier & Baraffe 2000). Giant planets thus gradually cool and contract but with a timescale that depends also on the amount of irradiation from their parent star (e.g. Hubbard 1977). Giant planets and brown dwarfs are thought to be mostly convective but planets very close to their star (such as 51 Peg b) develop an outer radiative zone that progressively extends to the deeper levels (Guillot et al. 1996;Guillot & Showman 2002). This radiative zone and the atmosphere to which it is linked (see Parmentier & Guillot 2014) govern the rate of cooling and contraction. The black line is applicable to the evolution of solar composition planets, brown dwarfs and stars, when isolated or nearly isolated (as Jupiter, Saturn, Uranus and Neptune, defined by diamonds and their respective symbols), after 5 Ga of evolution. The dotted line shows the effect of a 15M⊕ core on the mass-radius relation. Orange and yellow curves represent the mass-radius relations for heavily irradiated planets with equilibrium temperatures of 1000 and 2000 K, respectively, and assuming that 0.5% of the incoming stellar luminosity is dissipated at the center. For each irradiation level, two cases are considered: a solar-composition planet with no core (top curve), and one with a 100M⊕ central core (bottom curve).

Circles with error bars correspond to known planets, brown dwarfs and low-mass stars, color-coded as a function of their equilibrium temperature(below 750, 1500, 2250 K and above 2250 K, respectively, from darkest to lightest). [From (Guillot & Gautier 2014)]

Another factor affecting the sizes of exoplanets is whether they contain more or less heavy elements1 : Everything else being equal, planets with a higher fraction of heavy elements tend to be smaller.

Figure 1 compares a mass-radius diagram of known exoplanets, brown dwarfs and low-mass stars against some theoretical relations. After about 5 Ga of evolution, the mass-radius curve for non-irradiated hydrogen-helium planets has a maximum for a mass of 4 M Jup and a radius slightly above that of Jupiter. For planets of larger masses, degeneracy effects in the equation of state begin to dominate, thermal effects become less important and cannot oppose as efficiently the increased compression: the radius decreases with mass until in the stellar domain nuclear reactions set in. However, for heavily-irradiated hydrogen-helium planets (i.e. for "hot Jupiters"), the radius tends to decrease with mass: for smaller masses the warm atmosphere is less bounded to the planet because of the weaker gravity (see Guillot 2005). The presence of a massive core removes that effect. The variety of irradiation level and compositions is responsible for an ensemble of known giant exoplanet that is "trumpet-shaped" in that diagram.

A link between stellar and exoplanetary compositions

The possibility to measure planetary masses, radii, irradiation levels and to constrain the age through the stellar age thus offers the possibility to determine the global compositions of giant exoplanets. However, a problem remains: a significant fraction of hot Jupiters is found to have radii above that theoretically predicted for hydrogen-helium planets (Bodenheimer et al. 2001;Guillot & Showman 2002). A number of explanations have been proposed (e.g. Laughlin et al. 2011). For energetic considerations, the most plausible class of models appear to be those which invoke the dissipation of a small fraction (of order 1% or less) of the stellar flux relatively deep in the planetary interior as a mean to slow the planetary cooling and contraction (Guillot & Showman 2002;Laine et al. 2008;Batygin & Stevenson 2010;Arras & Socrates 2010). A definitive test of these models would be an accurate measurement of stellar ages coupled to the determination of precise planetary masses and radii for a statistically significant number of objects.

Under the assumption that this class of models prevails, it is relatively straightforward to parametrize this 'missing physics' through the dissipation of a fixed fraction of the irradiation energy into the planetary interior. This fraction may be chosen so that model radii for hydrogen-helium planets are always (or almost always) above the observational constraint. The global amount of heavy elements present in the planet may then be inferred from evolution models (Guillot et al. 2006).

Figure 2 shows the result of such an exercice when applied to hot Jupiters with known masses and radii. In that plot, the mass of heavy elements is plotted against the metallicity of the parent star. For some objects, the chosen (small) value of the dissipation parameter imply an unphysical negative mass of heavy elements. This is for example the case of the still unexplained large radius of CoRoT-2b (see Guillot & Havel 2011). However, two robust results are the requirement of very large masses in heavy elements (above 100 M ⊕ ) for some objects, and the correlation between the mass of heavy elements inferred in the planets and that in the parent star (Guillot et al. 2006;Burrows et al. 2007;Guillot 2008;Laughlin et al. 2011;Moutou et al. 2013). Interestingly, these two results are also observed when one uses a subset of the ensemble of planets that is weakly irradiated and does not require any extra heat source to explain their radius (Miller & Fortney The evolution model assumes that 0.5% of the incoming irradiation flux is dissipated at the planet's center. Circles which are labeled 1 to 23 correspond to the CoRoT giant planets. Gray symbols correspond to a subset of known transiting systems Guillot (2008); Laughlin et al. (2011). Unphysical negative values for MZ correspond to insufficient heat sources leading to a radius that is larger than observed. [From Moutou et al. (2013).] Bottom panel: Same plot for weakly irradiated planets. [From Miller & Fortney (2011).] In both panels, a curve shows a linear relation between log MZ and [Fe/H] fitted to the results for weakly irradiated planets.

2011).

The large amount of heavy elements shows that solids were efficiently stored and delivered into the planets. This must have occurred in the circumstellar disk at the time of the formation of the parent star and its planets. Furthermore, the correlation between planet and star 'metallicity' shows that the storage/accretion/delivery of solids was strongly favored by an increased metallicity. This favor scenarios in which giant planets are formed by the accretion of a central core followed by the capture of a hydrogen-helium envelope which may be polluted in variable amounts by dust and planetesimals.

Dynamical interactions between stars and close-in exoplanets

Another link between the properties of stars and of their close-in exoplanets is a direct consequence of tidal interactions between them. Figure 3 shows the masses of known exoplanets as a function of the effective temperature of their parent star. The symbol sizes are inversely proportional to the planets' orbital periods which effectively helps focusing on short period planets. Obvious observational biases are indicated in the figure: stars with low effective temperature are faint which makes it difficult to observe planets around them with either radial velocimetry or photometry in the visible. Similarly, at high effective temperatures two effects combine against the discovery of small planets: the fast rotation of F-dwarfs makes the spectral lines broader which reduces the sensitivity of radial velocity surveys and the stars become larger which also acts against the discovery of transiting planets by photometry. A paucity of close-in companions is obvious for masses above about 5 M Jup and around stars with effective temperatures between about 5000 and 6000 K. This cannot be explained by an observational bias: if present, massive companions around G-dwarfs would be at least as easily detected as massive companions to F-dwarfs.

Given the small different in mass between F-and G-dwarfs, the different planet populations must be due to a different dynamical evolution caused by different dissipation regimes. This may be shown by defining the inward migration timescale (see Barker & Ogilvie 2009):

τ mig = 12 Ma Q * 10 6 M p 1 M Jup -1 M * 1 M ⊙ 8/3 R * 1 R ⊙ -5 × P orb 1 day 13/3 1 - P orb P spin -1 (4.1)
where M * and R * are the stellar mass and radius, respectively, M p is the companion's mass, P orb its orbital period and P spin the stellar spin period. Q ′ * ≡ Q * /k 2 is the equivalent stellar tidal dissipation factor defined as the ratio between the tidal dissipation factor and the second Love number. The value of τ mig for known brown dwarfs and exoplanetary companions for an assumed fixed Q ′ * = 10 6 ranges from 10 6 yrs and more for G-dwarfs and only about 10 5 yrs and more for F-dwarfs (Guillot, Lin & Morel, in preparation). We interpret this difference as arising from the engulfment of massive planets and brown dwarfs around G-dwarfs and their preservation around F-dwarfs. This has two reasons: (i) F-dwarfs are known to be rapid rotators which have a weaker magnetic braking due to their small outer convective zone. They are hence less efficient at extracting angular momentum. On the other hand in systems in which a G-dwarf is spun-up by a close-in companion, stellar winds and magnetic fields yield a rapid loss of angular momentum from the system and a fast runaway migration of the companion onto the central star. (ii) G-dwarfs have a radiative center while F-dwarfs have a central convective zone. This implies that gravity waves propagating in the inner radiative zone may break and dissipate their energy in G-dwarfs but not in F-dwarfs (see Barker & Ogilvie 2010).

Figure 4 shows the result of a dynamical model that includes tidal interactions between stars and their companion (Barker & Ogilvie 2009), stellar evolution (Morel & Lebreton 2008), the magnetic braking of stars (Bouvier et al. 1997), and a consistent calculation of tidal dissipation (Q ′ * ) by gravity waves (Barker & Ogilvie 2010). The latter is calculated consistently by including the evolution of the stellar interior which enter the calculation of Q ′ * (see also Barker 2011). The figure shows for stars of 0.8 to 1.4 M ⊙ and companions of 0.2 to 200 M Jup on an initial 3-day orbital period the fraction of the star's main-sequence lifetime on which the companion is able to survive. We find that massive companions around G-dwarfs generate tides that break at the star center, leading to a strong dissipation, their inward migration and eventual demise. The process is much less efficient for small-mass companions essentially because the waves that they exert have a too small amplitude to break except at late ages when the star stars leaving the main sequence. For massive companions, the inward migration is slow because of the large initial angular momentum. Around F-dwarfs, very little migration occurs both because internal gravity waves cannot reach the center and because of their weak magnetic braking.

Massive close-in planets and brown dwarfs are therefore engulfed preferentially around G-dwarfs in qualitative agreement with the observations. Actually, while this goes in the right direction, it may be argued that the observations show an even stronger deficit of massive companions around G-dwarfs than found by the models. This clearly requires further work.

Conclusion

A proper understanding of the properties, formation and evolution of planetary systems requires combining observations and theoretical studies of both the planets and their parent stars. This clearly demonstrated by the two examples that we discussed: (i) the determination of the global composition of giant exoplanets which appears to be directly linked to the metallicity of the parent star; (ii) the fate of close-in massive exoplanets and brown dwarfs which depends both on the planetary and stellar properties and the magnitude of their tidal interactions. This is of course not restricted to these two examples. The prospect of discovering thousands of transiting planets and measuring at the same time both the stellar and planetary properties very accurately thanks to a number of ground-based and space-based projects (including CHEOPS, TESS, PLATO) is extremely promising.

Fig. 1 .

 1 Fig.1. Theoretical and observed mass-radius relations. The black line is applicable to the evolution of solar composition planets, brown dwarfs and stars, when isolated or nearly isolated (as Jupiter, Saturn, Uranus and Neptune, defined by diamonds and their respective symbols), after 5 Ga of evolution. The dotted line shows the effect of a 15M⊕ core on the mass-radius relation. Orange and yellow curves represent the mass-radius relations for heavily irradiated planets with equilibrium temperatures of 1000 and 2000 K, respectively, and assuming that 0.5% of the incoming stellar luminosity is dissipated at the center. For each irradiation level, two cases are considered: a solar-composition planet with no core (top curve), and one with a 100M⊕ central core (bottom curve). Circles with error bars correspond to known planets, brown dwarfs and low-mass stars, color-coded as a function of their equilibrium temperature(below 750, 1500, 2250 K and above 2250 K, respectively, from darkest to lightest). [From(Guillot & Gautier 2014)]

Fig. 2 .

 2 Fig. 2. Mass fraction of heavy elements in the planets as a function of the metallicity of their parent star. Top panel: Result for hot Jupiters with masses smaller than 2 MJup. The evolution model assumes that 0.5% of the incoming irradiation flux is dissipated at the planet's center. Circles which are labeled 1 to 23 correspond to the CoRoT giant planets. Gray symbols correspond to a subset of known transiting systems Guillot (2008);Laughlin et al. (2011). Unphysical negative values for MZ correspond to insufficient heat sources leading to a radius that is larger than observed.[From Moutou et al. (2013).] Bottom panel: Same plot for weakly irradiated planets.[From Miller & Fortney (2011).] In both panels, a curve shows a linear relation between log MZ and [Fe/H] fitted to the results for weakly irradiated planets.

Fig. 3 .

 3 Fig. 3. Mass (times the sine of the orbital the inclination) of known exoplanets and brown dwarfs as a function of effective temperature of the parent stars. Some of the key systems are labelled. The size of the symbols is inversely proportional to their orbital period. (Large symbols correspond to important tidal interactions between the star and the companion.) [Figure adapted from Bouchy et al. (2011).]

Fig. 4 .

 4 Fig. 4. Contour plot showing the ratio of the lifetime age to the main sequence lifetime of a companion of mass M pla orbiting around a star of mass Mstar initially on a 3 day orbit and assuming [Fe/H]=0.0. The model used is the full internal gravity wave model [see text]. The points correspond to the observed low-metallicity population, with -0.15 ≤ [Fe/H] < 0.15. [Figure from Guillot, Lin & Morel (in preparation).]

Here, anything other than hydrogen and helium is dubbed "heavy element".


E Orgéas

V Bertrand

C Deplano

Geindreau

L Bailly
email: lucie.bailly@irphe.univ-mrs.fr

M Toungara

L Orgéas

E Bertrand

V Deplano

C Geindreau

In-plane mechanics of soft architectured fibre-reinforced silicone rubber membranes

Keywords: Micro-mechanical model, Tensile tests, Biomimetism, Fibre-reinforced membrane, Anisotropy, Material design, Silicone rubber

Silicone rubber membranes reinforced with architectured fibre networks were processed with a dedicated apparatus, allowing a control of the fibre content and orientation. The membranes were subjected to tensile loadings combined with continuous and discrete kinematical field measurements (DIC and particle tracking). These tests show that the mechanical behaviour of the membranes is hyperelastic at the first order. They highlight the influence of the fibre content and orientation on both the membrane in-plane deformation and stress levels. They also prove that for the considered fibrous architectures and mechanical loadings, the motion and deformation of fibres is an affine function of the macroscale transformation. These trends are fairly well described by the micromechanical model proposed recently in Bailly et al. (JMBBM, 2012). This result proves that these materials are very good candidates for new biomimetic membranes, e.g. to improve aortic analogues used for in vitro experiments, or existing textiles used for vascular (endo)prostheses.

Introduction

Human abdominal aortic tissue (AA) is a complex soft sandwich structure, arranged in three different layers. Within these layers, the distribution of all components displays a double-helix architecture of collagen fibres, elastic fibres, smooth muscle cells and other molecules of the extracellular matrix [START_REF] Holzapfel | Collagen in arterial walls: Biomechanical aspects[END_REF]. The fibrous microstructure is characterised by a particular wavy arrangement under stress-free conditions [START_REF] Freed | Elastic response of crimped collagen fibrils[END_REF][START_REF] Holzapfel | Collagen in arterial walls: Biomechanical aspects[END_REF][START_REF] Rezakhaniha | Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy[END_REF] and by distinctive preferred orientations when subjected to a mechanical loading [START_REF] Holzapfel | Determination of material models for arterial walls from uniaxial extension tests and histological structure[END_REF][START_REF] Horny | Orientations of collagen fibers in aortic histological section[END_REF]. These histological features provide the abdominal aortic tissue a highly non-linear, anisotropic and essentially hyperelastic mechanical behaviour (Vande Geest et al., 2006). Over the past decades, several synthetic materials aiming at mimicking the microstructural and/or mechanical specificities of aortic tissue were developed (i) to perform in vitro experiments, or (ii) for bioengineering issues:

(i) In vitro experiments were carried out to characterise biomechanical factors involved in the evolution of arterial diseases such as Abdominal Aortic Aneurysm (AAA). For that purpose, several deformable AAA analogues were processed. Most of them were placed into vascular flow simulators to investigate endovascular repair [START_REF] Schurink | Endoleakage after stent-graft treatment of abdominal[END_REF][START_REF] Flora | Aneurysm wall stress and tendency to rupture are fea-tures of physical wall properties: An experimental study[END_REF][START_REF] Chong | Flow patterns in an endovascular evg for abdominal aortic aneurysm repair[END_REF][START_REF] Gawenda | Endotension is influenced by wall compliance in a latex aneurysm model[END_REF]. Few were utilised to quantify biological fluid-structure interactions within AAAs [START_REF] Ene | In vitro evaluation of the effects of intraluminal thrombus on abdominal aortic aneurysm wall dynamics[END_REF][START_REF] Deplano | New insights into the understanding of flow dynamics in an in vitro model for abdominal aortic aneurysms[END_REF][START_REF] Deplano | Flow of a blood analogue fluid in a compliant abdominal aortic aneurysm model: experimental modelling[END_REF]. Inflation tests were conducted by increasing internal pressure in order to identify AAAs deformation or rupture locations [START_REF] Morris | Experimental assessment of stress patterns in abdominal aortic aneurysms using the photoelastic method[END_REF][START_REF] Doyle | Experimental modelling of aortic aneurysms: Novel applications of silicone rubbers[END_REF][START_REF] Doyle | Identification of rupture locations in patient-specific abdominal aortic aneurysms using experimental and computational techniques[END_REF][START_REF] Corbett | Engineering silicone rubbers for in vitro studies: Creating aaa models and ilt analogues with physiological properties[END_REF][START_REF] Meyer | Stereoscopically observed deformations of a compliant abdominal aortic aneurysm model[END_REF]. However, all these materials were made up of homogeneous elastomers with isotropic material properties, which are very far from the anisotropic hyperelastic behaviour of aortic tissue and from its fibrous microstructure.

(ii) In parallel, in the field of vascular tissue engineering, numerous implants were designed so as to be mechanically as close as possible to the host tissue macroscale mechanical behaviour. To this end, polymeric fibrous composites were largely designed as bioinspired scaffolds [START_REF] Li | Biomaterials for Tissue Engineering Applications: A Review of the Past and Future Trends[END_REF]. Various manufacturing processes were also developed to elaborate fibrous networks [START_REF] Ratcliffe | Tissue engineering of vascular grafts[END_REF][START_REF] Chen | Biomimetic materials processing for tissueengineering processes[END_REF]. Currently, a specific interest is provided to fibre electrospinning and textile processing techniques:

(a) Electrospinning technology is favoured to manufacture nano-structured biomaterials made up of filaments close to collagen/elastin fibrils dimensions [START_REF] Matthews | Electrospinning of collagen nanofibers[END_REF][START_REF] Boland | Electrospinning collagen and elastin : preliminary vascular tissue engineering[END_REF][START_REF] Agarwal | Use of electrospinning technique for biomedical applications[END_REF][START_REF] Clure | A three-layered electrospun matrix to mimic native arterial architecture using polycaprolactone, elastin, and collagen: A preliminary study[END_REF]. Recent advances allow a better control of the electrospun fibres morphology, strength and size by adjusting set-up conditions or polymer solution parameters [START_REF] Kowalczyk | Electrospinning of bovine serum albumin. optimization and the use for production of biosensors[END_REF]Soliman et al., 2011). However, the control of the fibres position and orientation is still a challenging task to generate the desired macroscopic mechanical anisotropy [START_REF] Bu | Continuously tunable and oriented nanofiber direct-written by mechano-electrospinning[END_REF].

(b) Textile processing techniques are also widely used to develop vascular (endo)prostheses [START_REF] Jacobs | Mechanical failure of prosthetic human implants: A 10-year experience with aortic stent graft devices[END_REF][START_REF] Chakfe | Influence of the textile structure on the degradation of explanted aortic endoprostheses[END_REF]VanLieshout et al., 2006). Knitted, braided or woven fabrics are characterised by a fine regular arrangement of the fibres, therefore allowing a better control of the macroscopic anisotropic properties (Xu et al., 2010;[START_REF] Schreiber | Improving the mechanical properties of braided shape memory polymer stents by heat setting[END_REF]Yang et al., 2012). However, due to their inappropriate fibrous microstructures, the current commercial textile prostheses do not exhibit suitable deformability (Surovtsova, 2005;[START_REF] Sarkar | The mechanical properties of infrainguinal vascular bypass grafts: Their role in influencing patency[END_REF][START_REF] Demanget | Computational comparison of the bending behaviour of aortic stent-grafts[END_REF][START_REF] Demanget | Finite element analysis of the mechanical performances of 8 marketed aortic stent-grafts[END_REF][START_REF] Lemercier | Comparison between the mechanical behaviour of the human healthy aa and commercial prostheses under various mechanical loadings[END_REF].

Therefore, the global objective of this work is to design and process new artificial materials, able to mimic the macroscopic anisotropic properties of AA and AAA tissues, their hyperelastic J-shape mechanical response, and their main histological features (fibrous microstructures with distinctive fibre orientations). In the short run, these materials represent appropriate candidates to perform in vitro experiments similar to those carried out in [START_REF] Deplano | Flow of a blood analogue fluid in a compliant abdominal aortic aneurysm model: experimental modelling[END_REF], with more relevant models of AAs or towards more efficient mechanical performances. To this end, optimal elastomeric fibrous composites were identified in a previous theoretical work using a multi-scale homogenisation process combined with microstructure optimisation [START_REF] Bailly | Towards a biomimetism of abdominal healthy and aneurismal arterial tissues[END_REF]. The optimised architectured materials are composed of a thin and soft hyperelastic membrane reinforced by periodic fibrous lattices displaying four preferred orientations and hyperelastic fibres. The present study aims at (i) providing experimental advances concerning the manufacturing and mechanical characterisation of such architectured soft fibre-reinforced membranes, and (ii) validating the assumptions stated to build the micro-mechanical model in [START_REF] Bailly | Towards a biomimetism of abdominal healthy and aneurismal arterial tissues[END_REF]. Thus, a summary of the theoretical background is first reported in Section 2. The manufacturing processes of the elastomeric fibrous membranes are presented in Section 3, along with the method used to characterise their mechanical behaviour. Experimental results are detailed in Section 4. A comparison between these results and the prediction of the micromechanical model is proposed and discussed in Section 5.

Theoretical background

To mimic the main histological features of the arterial tissues, various fibrous architectures were investigated with the micro-mechanical model proposed in [START_REF] Bailly | Towards a biomimetism of abdominal healthy and aneurismal arterial tissues[END_REF], i.e. single or bi-layers lattices of fibres. As a first step in the validation of the model assumptions, the present study focuses on the one-layer fibrous structure, as illustrated in Figure 1(a). This structure comprises a single lattice of crossed, identical continuous and straight fibres of initial cross-section S 0 , embedded into a thin and soft elastomer matrix. Each fibrous lattice can be seen as a repetition of a Representative Elementary Cell (REC) lying in the (e x , e y ) plane, as (i) The REC geometry is characterised by two periodic vectors P 1 = A 1 A 4 = 0 E 1 and P 2 = A 1 A 2 = 0 E 2 (resp. p 1 = a 1 a 4 = 1 e 1 and p 2 = a 1 a 2 = 2 e 2 ) joining the extremities A 1 , A 2 and A 4 (resp. a 1 , a 2 and a 4 ) of the REC in the initial C 0 (resp. deformed C) configuration. The angles (e x , P 1 ) and (e x , P 2 ) [resp.

(e x , p 1 ) and (e x , p 2 )] are noted θ 01 and θ 02 (resp. θ 1 and θ 2 ) in C 0 (resp. in C).

Initial REC configuration is defined so that: θ 02 = -θ 01 = θ 0 .

(ii) Fibres are supposed to be linked at the REC extremities A 1 , A 2 , A 3 and A 4 , in such a way that the relative displacement of fibres at these points is zero.

Between these points, the initial and deformed lengths of fibre i (i = 1 and 2) are expressed as f 0i = ||P i || = 0 and f i = ||p i || = i , respectively. Furthermore, its elongation λ f i and its tension t f i are expressed as λ f i = i / 0 and t f i = t f i e i (no summation on the index i). Fibres are assumed to behave as hyperelastic bars, i.e. the tensions t f i only depend on the fibre elongations λ f i via a fibre strain energy function W f . (iii) The mechanical interactions between the fibres lattice and the homogeneous elastomer membrane, and between the fibres are negligible.

The macroscale mechanical behaviour of such periodic fibrous lattice was entirely determined analytically, by using the homogenisation of periodic discrete structures [START_REF] Caillerie | Discrete homogenization in graphene sheet modeling[END_REF][START_REF] Bailly | Towards a biomimetism of abdominal healthy and aneurismal arterial tissues[END_REF]. It was shown that the overall macroscopic Cauchy tension tensor τ c of the one-layer fibrous membrane can be written as:

τ c = τ m + 1 ||p 1 × p 2 || 2 i=1 (t f i ⊗ p i ), (1)
where τ m corresponds to the Cauchy tension tensor of the elastomer membrane contribution. Thus, knowing the mechanical behaviour of the elastomeric membrane, that of the fibres, and the geometry of the REC, the mechanical behaviour of the considered architectured and fibre-reinforced membranes is fully estimated: no constitutive parameter has to be fitted a posteriori.

3 Experimental procedure

Materials

Matrix -A biocompatible high strength RTV Silicone elastomer dispersion (PN 40021, Applied Silicone Corporation) was selected as matrix material, due to its high deformability and specific physical properties. It allowed an easy impregnation of the fibrous network and cured at room-temperature. To minimise the scattering of the experimental results, a particular attention was brought to subject the processed membranes to the same curing time (24h) and to keep them at the same room temperature (290K) and relative humidity (50%) until mechanical testing.

The silicone in-plane mechanical response was investigated by means of a uniaxial tensile-testing device (Bose Corporation, Eden Prairie, Minnesota) equipped with a load cell of capacity of 450±0.01 N. Displacement-controlled tests were performed at room temperature on 8 rectangular samples of mean initial gauge L 0 = 21.21 mm (STD = 1.07 mm), width W 0 = 10.13 mm (STD = 0.47 mm) and thickness T 0 = 1.05 mm (STD = 0.22mm), STD being the data standard deviation. Note that these dimensions were determined by photographing the undeformed samples with an accuracy of 0.02 mm. Each sample was subjected to 10 load-unload cycles, with a maximal tensile elongation of 1.30 and an elongation rate of 10 -2 s -1 , representative of AA tissue's physiological loadings [START_REF] Bailly | New experimental protocols for tensile testing of abdominal aortic analogues[END_REF]. During the tests, pictures of the sample's gauge zone were recorded with a JAI CCD camera (1624 × 1236 pixels, 25 Hz). Before, speckles were drawn on the surface of each strip so that strain field measurements could be carried out using 2D Digital Image Correlation (DIC), as illustrated in Figure 1(d). Elongations of the specimen along the stretch e y and the transversal e x directions, i.e. respectively λ yy and λ xx , were calculated using a DIC software (7D) developed by Vacher et al. (1999). Longitudinal force F y was also measured during the tests and corresponding component of the first Piola-Kirchhoff stress tensor was defined as Π yy = F y /W 0 T 0 .

Typical experimental data are displayed in Figure 1(d), showing both the elongation path (λ xx , λ yy ) and the stress-elongation response (Π yy , λ yy ). A low hysteresis was measured during the load/unload sequences. Therefore, as a first reasonable approximation, it can be inferred that the considered silicone rubber behaved nearly as a hyperelastic material. Besides, within the investigated elongation range, its mechanical behaviour could be fairly well modelled with a classical neo-Hookean model (Treloar, 1943), thus characterised by a volumetric strain-energy function

W m = c m (I 1 -3)/2.
The constant c m is a positive material parameter, and I 1 = tr(C) denotes the first principal invariant of the right Cauchy-Green tensor C. This model was adjusted on both curves, as illustrated in Figure 1(d). For all samples, the average material parameter was found around c m = 0.24 MPa (STD = 0.01 MPa).

Fibre -A 50 μm diameter polyamid monofilament was selected as a fibre candidate.

This fibre is commonly used as fishing wire (Feeling 36605, Sensas -0.8 lb, 0.350 kg).

Uniaxial tensile tests were performed to characterise the mechanical behaviour of 16 fibre samples cut from this filament on a dedicated testing machine (Instron 3367, 100 ± 0.01 N). Tests were conducted until failure, using an elongation rate of 10 -2 s -1 , and an initial fibre length of 200 mm at rest. The force within each fibre sample t f was measured during the tests and the corresponding first Piola-Kirchhoff stress was estimated as Π f = t f /S 0 , S 0 being the fibre unloaded cross-section. (1998) for Shape Memory Alloys (see Figure 1(e)), so that the mean tension within the fibre is expressed as:

t f = S 0 λ f ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ μ 1 2 f + sgn { s -[ f + sgn s ] 2 + α 2 1 } + sgn σ 0 +μ 2 f + μ 3 2 f + sgn { [ f -sgn c ] 2 + α 2 2 -2 c + α 2 2 } ,
(2)

with s = α 2 1 -(2σ 0 /μ 1 ) 2 4σ 0 /μ 1 and sgn = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 if f ≥ 0 -1 if f < 0.
This tension is a function of the logarithmic strain, f = ln λ f , and of 7 constitutive

parameters: μ 1 , μ 2 , μ 3 , σ 0 , c ,

Architectured soft fibre-reinforced membranes

Processing route -The as-described fibre and matrix materials were used to manufacture the target fibre-reinforced membranes. For that purpose, a specific device was designed to manufacture lattices of crossed and identical straight fibres, impregnated inside rectangular membranes (160 mm × 50 mm), as shown in Figure 2(a).

The device consists of a pantographic deployable structure and a Teflon TM rectangular mould. Each arm of the pantograph is equipped with a 1 mm-tooth comb.

Three parameters of the device were adjusted before each composite membrane's elaboration:

(i) the number of teeth involved in the manufacturing (see Figure 2 (iii) the angle β 0 as defined in Figure 2(c), adjustable by rotation of the mould. This parameter allows to control the orientation of the fibre network in the (e x , e y ) plane. Two cases were investigated:

β 0 = {0 • , 45 • }.
At a chosen configuration ( 0 , θ 0 , β 0 ), a first unidirectional fibre network was built using the same fibre. The fibre was guided alternatively by two combs of the pantograph facing to each other. In the same way, a second unidirectional network was built using the other combs of the device, and superimposed on the first one. So as to create a 1 mm-thick membrane, the lattice was then embedded by a 20 m -layer of silicone casted into the mould. Note that the depth of the mould was designed Samples -Eight rectangular strips (S i , i ∈ [1, . . . , 8]) were cut from the composite membranes. Their geometrical dimensions were determined so as to preserve the same separation scale from a given sample to another one, by considering 6 RECs along the width and 9 RECs along the length. Strip's initial dimensions were optically measured and summarised in Table 1. In the following, for the sake of clarity, the composite strip S i , characterised by the triplet of parameters ( 0 , θ 0 , β 0 ), will be denoted as S i ( 0 , θ 0 , β 0 ). In practice, the values of ( 0 , θ 0 ) slightly differed from the target ones. Thus, optical measurements were processed using a particle-tracking method applied on strips' images registered at rest. The image processing was based on the tracking of experimental REC extremities, as illustrated in Figure 3 upon one REC. For each strip, the measured values of ( 0 , θ 0 ) were averaged on all the surface's RECs and noted ( 0 , θ 0 ). Values of ( 0 , θ 0 ) are detailed in Table 1. The accuracy of the tracking method was assessed at (0.05 mm, 0.2 • ). Finally, the target values of ( 0 , θ 0 ) were achieved within a mean precision of (0.07 ± 0.05 mm, 0.4 ± 0.2 • ) over the whole database.

Tensile tests -The in-plane mechanical response of the S i samples was investigated using the tensile-testing device and the protocol previously described for the homogeneous membrane. The stretch direction was defined by the vector e y . Thus, for β 0 = 0 • , the stretch direction matches the REC diagonal (A 4 A 2 direction). For β 0 = 45 • , the stretch direction matches the fibres one, as illustrated in Figure 3.

Each test was repeated twice in order to ensure its reproducibility, and coupled with image recording of the deformed strip. In the following, focus is given to the first load of each test only. The discrete kinematical field measured at the fibre scale was derived from the tracking of experimental REC nodes A i on each image, as displayed in Figure 3. The image processing was repeated three times on four centered RECs for each composite sample. This yielded to average values of the microscopic longitudinal and transversal elongations, i.e. λ yy and λ xx respectively.

The local value of the half-angle between the fibres was also derived. This angle was noted θ 0 and its evolution was calculated along the test. In summary, for each strip S i , Table 2 compares (i) the target value of angle θ 0 , (ii) the angle θ 0 averaged over all surface's RECs, and (iii) the angle θ 0 averaged over the four RECs associate to the measured elongations (λ xx , λ yy ).

Finally, some samples' surfaces were additionally covered by speckles to perform 2D DIC measurements. In that case, the continuous kinematical field applied on the composite membrane was derived in average over a whole REC. Therefore, kinematics measured at the composite scale (global, DIC measurements) and at the fibre scale (local, particle tracking measurements) could be compared.

Experimental results on composite membranes

General trends

Typical measurements are reported in Figure 3, showing the local loading paths (λ xx , λ yy ), the derived local half-angle between the fibres θ as a function of the elongation λ yy , and the corresponding tension-elongation response in the composite membrane (τ yy , λ yy ). Two tests repeated on samples S 1 (1, 45, 0) and S 2 (1,45,45) are illustrated, highlighting important general features:

• whatever the loading angle β 0 , the repeatability of each test is ensured.

• the local kinematical fields are optically measurable, with an accuracy on angle θ being of 0.2 • . During the tests, a typical quasi-linear θ-variation is observed for β 0 = 0 • , against zero θ-variation for β 0 = 45 • (see second row in Figure 3).

This angle variation comes up to 15 • at the applied maximal strain when β 0 = 0 • . Note that for each sample, a slight change in the value of initial angle θ 0 (± 1/2 • ) is observed between the first test and the second one: this is probably due to possible fibrous rearrangement (e.g. slight rotation of the fibres due to the low hysteresis measured after loading cycles) or local variations in the sample's architecture (see Table 2).

• the mechanical effects of the fibrous reinforcement are highly dependent on the loading direction. For β 0 = 0 • , the fibres progressively align with the stretch direction (see pictures in Figure 3 The lattice contribution also comes to the tensile response of individual fibre, weighted by the volume fraction of the fibres parallel to the loading direction, v f . This volume fraction is defined by the ratio of the volume of fibres parallel to the loading direction within one REC in configuration C 0 (πd 2 0 0 /4), over the volume of the REC (T 0 2 0 sin(2θ 0 )). This ratio yields to v f = πd 2 0 4T 0 0 sin(2θ 0 ) , i.e. 0.2% for S 2 (1,45,45). ), and the one determined by particle tracking (λ t xx , λ t yy ) was calculated using:

Comparison between global and local kinematics

e r = [(λ DIC xx -λ t xx ) 2 + (λ DIC yy -λ t yy ) 2 ] 0.5 [(λ t xx -1) 2 + (λ t yy -1) 2 ] 0.5
(3)

Regarding data series with β 0 = 45 • , the average value, e r , comes to 13.8% for λ yy ∈ [1 : 1.14], and drops down to 6% for λ yy > 1.05. Regarding data series with Note that this quantitative agreement is nearly unchanged when calculating the global kinematical field in average over a surface of 3×4 RECs (not shown). From these results, the overall membrane and the REC were considered to be subjected to the same macroscopic transformation gradient (see the discussion below). In the following, the discussed loading paths (λ xx , λ yy ) refer exclusively to the elongations calculated at the fibre scale.

β 0 = 0 • , e

Influence of the loading angle β 0

The graphs of Figure 5 Regarding the elongation paths shown in Figure 5(b), the influence of the initial fibre length is noticeable when the loading direction is not aligned with the fibres ones.

Indeed, for β 0 = 0 • , the contraction of the sample initial width (15% for λ yy = 1.25) is twice lower for 0 = 3 mm than that observed for 0 = 1 mm. By contrast, for β 0 = 45 • , the impact of fibre length 0 on the sample contraction is negligible.

Regarding the tension-elongation responses shown in Figure 5(b), all composites exhibit a global stiffening as compared to the silicone membrane, except for case S 3 (3,45,0). The similarity between the responses of S 3 (3, 45, 0) and the silicone is mainly ascribed to the differences in thickness samples (ratio of values being 0.76). This similarity disappears when tension data are converted into first Piola-Kirchhoff stresses (not shown). For a given stretch direction β 0 , the samples with higher volume fraction of fibres (i.e. for 0 = 1 mm) are stiffer. In particular, the ratio of the tensions achieved for S 4 (3,45,45) by the ones obtained for S 2 (1,45,45) varies between 0.20 and 0.41, around a mean value of 0.36. Beyond low elongations (λ yy > 1.03), this ratio rises up to 0.39 in average. This value comes close to the ratio of the volume fractions of the fibres parallel to the loading direction within each sample (i.e. 0.40).

Influence of the initial half-angle between fibres θ 0

The graphs plotted in Figure 5 Note that above results are consistent with those obtained by comparing samples for S 8 (3, 30, 0) is linked with the small thickness of the sample as compared to the matrix (ratio of values being 0.67, i.e. the lowest in the database).

5 Validation of the micro-mechanical model

Relevance of some of the model assumptions

As detailed in sections 2 and 3, some major microscale geometrical and mechanical assumptions were formulated in order to apply the micro-mechanical model. Several experimental observations gained from results shown in section 4 ensure the relevance of these assumptions:

(i) For the tensile loadings under study, the REC geometry is assumed to be diamond-shaped in the initial (resp. actual) configuration C 0 (resp. C). Besides, the displacements of nodes A i are assumed to be affine functions of the macroscale gradient of the geometrical transformation the samples are subjected to.

The tracking of the experimental RECs extremities showed that their diamond pattern remained topologically unaffected by the transformation (see Figure 3).

Besides, Figure 4 also proved that the motion of nodes A i were identical to the displacement field recorded at the macroscale with the DIC technique. These observations thus validate above assumptions.

(ii) The fibre mechanical behaviour is predicted by a hyperelastic model (Eq. 2), adjusted on experimental tensile tests performed on fibres (Figure 1(d)). This model assumes a symmetrical behaviour under compressive loading.

It was shown that the fibres are subjected to compressive loading for cases with β 0 = 45 • . No buckling effect was observed during the considered tests (because the compressive strains remained small and because compressed fibres are confined within the matrix). Thus, the above assumption is reasonable for the considered configurations.

(iii) Finally, the mechanical interactions between the fibres and the matrix, and between the fibres forming the lattice are assumed negligible. Therefore, at the first order, the tension decomposition in Eq. ( 1) can be considered as a reasonable approximation of the membrane mechanics.

The relevance of this last assumption is detailed in the following.

Comparison between experimental data and theoretical predictions

Theoretical predictions derived from Eq. ( 1) were confronted to the experimental results presented in section 4, and displayed along in Figures 5 and6. All these predictions were calculated by prescribing the experimental elongation paths (λ xx , λ yy ) as input loadings, instead of using an additional assumption of simple tensile test conditions. Note that each plotted experimental elongation path was associated to a local value of the initial half-angle between fibres, θ 0 (see bold values in Table 2).

Therefore, in the following, the comparison between experiments and predictions is focused on the Cauchy tension -elongation curves (τ yy , λ yy ).

Over the whole database, whatever the considered triplet of parameters ( 0 , θ 0 , β 0 ), Figures 5 and 6 demonstrate that the micro-mechanical model predictions fairly well fit the experimental data (cases of discrepancies observed in the database are detailed thereafter). This result is obtained despite the model simplicity and bearing in mind that none arbitrary macroscale adjustment was made a posteriori, i.e. all the model data were acquired at the microscale (the REC scale). In particular, the model is able to predict very precisely the mechanical behaviour of composites S 1 (1, 45, 0), S 3 (1, 45, 0) and S 6 (1, 30, 0). The average relative difference between data and predicted tensions is equal to 5.8%, 9.5% and 9.7% respectively. This mean difference remains below 20% for most other composites, with a value of 11.7% for S 2 (1, 45, 45), 16.4% for S 4 (3, 45, 45), 18.7% for S 7 (3, 60, 0) and 15.8% for S 8 (3,30,0). Note that this quantitative agreement between predictions and experimental data is similar when considering the second local value of θ 0 in Table 2 (not shown).

Such a good accordance validates model assumption (iii).

However, some discrepancies can be observed in two particular cases:

• the first case is pointed out in Figure 7(a). Two tests were performed on sample S 1 (1, 45, 0), with a good repeatability in both loading path and tensionelongation curves (see Figure 3(a)). The unique input difference between both test conditions relied on the applied experimental path and the associate value of initial angle θ 0 , as measured to determine the local kinematical field. The local angle θ 0 was assessed to 47.5 • in one case, and to 46.4 • in the other (see Table 2). The impact of the weak difference between both θ 0 -values on the macroscopic tension τ yy is illustrated in Figure 7(a), along with the model predictions (solid lines). The load t f predicted within REC fibre i (i = 1 and 2) in each case is plotted in Figure 7(b). The elongations λ f predicted at the fibre scale are also plotted as function of the macroscopic loading λ yy in Figure 7(c).

Thus, it is shown that a very small change in the lattice fibrous architecture can induce very different tensile loadings at the fibre scale: in case θ 0 = 46.4 • , the fibre elongation λ f stayed close to 1 up to λ yy ∈ [1 : 1.05]. This implies that the fibres in the REC were subjected to tension later than the ones of the sample with θ 0 = 47.5 • . Such a delayed stretching altered the overall mechanical behaviour of the composite in a noticeable manner. From this point of view, the initial half-angle between fibres appears as a critical parameter in case of loading direction far from the fibre orientation.

• the second important discrepancy of the database is highlighted in Figure 7(d).

Two tests were performed on sample S 5 (1, 60, 0), with a good repeatability in both loading path (not shown) and tension-elongation curves. Again, the unique difference between both tests relied on the measurement of the local angle θ 0 , i.e. 61.9 • in one case, and 62.7 • in the other. The impact predicted on the macroscopic tension τ yy , the fibre load t f and the fibre elongation λ f is displayed in Figures 7(d), 7(e) and 7(f) respectively. In that lattice configuration, both theoretical predictions are rather close to each other despite the 1-2 • of difference in θ 0 assessment. The REC fibres were subjected to tension from the beginning of the test (λ f > 1 once λ yy > 1 in each case). Thus, for comparable macroscopic loadings (e.g. λ yy = 1.18), the load achieved within the REC fibres is predicted four times greater than in previous lattice configuration ( θ 0 ≈ 45 • ), as shown when comparing Figures 7(e) and 7(b). However, the predictions largely deviate from the experimental tensions (mean relative difference being 86.7%). These discrepancies could be induced either by local stress heterogeneities generated within the composite, or by a misalignment of the sample's microstructure with respect to the load direction.

Conclusion

The process of artificial membranes able to mimic the mechanical and histological specificities of aortic tissue is needed to perform in vitro experiments as carried out in [START_REF] Deplano | Flow of a blood analogue fluid in a compliant abdominal aortic aneurysm model: experimental modelling[END_REF], using more relevant models of aortic wall. This will help improving the current knowledge on AAA evolution, and can also contribute to bioengineering challenges (e.g. building aligned and orderly fibrous structures with a controlled positioning, improving the deformability of future prostheses).

In this global context, the present study proposed an original technical device to design architectured soft-fibre reinforced membranes. With non-linear, anisotropic mechanical properties at the macroscale, and favoured fibrous orientations at the microscale, these membranes represent appropriate candidates for biomimetic goals.

The control of a macroscale mechanical anisotropy was allowed only by changing two microstructural parameters: the initial half-angle between the fibres θ 0 , and the fibre length 0 . The particular case of a single lattice of crossed, identical and straight monofilaments of 50 μm in diameter, embedded into a 1 mm-thick soft matrix in silicone was investigated in this work. Eight fibrous membranes of different microstructures were manufactured. The target values of ( 0 , θ 0 ) were achieved within a mean precision of (0.07 ± 0.05 mm, 0.4 ± 0.2 • ) over the whole database.

The fibrous membranes were then characterised using a tensile testing device, for two loading directions β 0 . Optical recordings of displacement fields were realised at both macro-and microscale. These experiments allowed the validation of the assumptions of the micro-mechanical model presented in a previous work [START_REF] Bailly | Towards a biomimetism of abdominal healthy and aneurismal arterial tissues[END_REF], and based on the homogenisation of periodic discrete structures. This study demonstrates that the model is able to describe the experimental database in a good quantitative agreement, whatever the triplet of parameters ( 0 , θ 0 , β 0 ).

The present experimental work must be completed by new biaxial tensile loadings, in order to assess the micro-mechanical model for loadings closer to the ones that arterial tissues are subjected to. Further tests will be coupled with 3D DIC measurements, in order to capture possible out-of-plane displacement fields of the deformed membranes. Finally, manufacturing of wavy substructures in the fibrous lattice would allow the biomimetism of the membranes, as expected in the target solution identified in [START_REF] Bailly | Towards a biomimetism of abdominal healthy and aneurismal arterial tissues[END_REF]. ߚ Differences between the target value of the initial half-angle between fibres θ 0 , the mean value measured on the whole sample surface θ 0 , and the value measured over 4 RECs to determine the local kinematical field θ 0 ( † bold values are the ones used for theoretical predictions illustrated in Figures 5 and6).

ൌ Ͳι ϭ Ϯ ϯ ϰ ௬ ௫ ϭ Ϯ ϯ ϰ ɉ ௫௫ ൌ ϭ ɉ ௬௬ ൌ ͳ ɉ ௫௫ ൌ ͲǤͺ ɉ ௬௬ ൌ ͳǤͳͷ ɉ ௫௫ ൌ ͲǤ ɉ ௬௬ ൌ ͳǤʹ ߚ ൌ Ͷͷι ɉ ௫௫ ൌ Ϭ͘ϵϳ ɉ ௬௬ ൌ ͳ͘Ϭϰ ɉ ௫௫ ൌ Ϭ͘ϵϳ ɉ ௬௬ ൌ ͳ͘ϭϬ ɉ ௫௫ ൌ ϭ ɉ ௬௬ ൌ ͳ
Samples θ 0 ( • ) θ 0 ( • ) θ 0 ( • ) S i ( 0 , θ 0 , β 0 ) mean (min,

  sketched in Figure 1(b) in the initial configuration C 0 . The microscale geometrical and mechanical assumptions related to the REC are summarised as follows:

  Figure 1(e)presents the mean stress value Π f , as function of fibre elongation λ f , measured in average over the 16 samples. The obtained fibre mechanical behaviour is repeatable and non-linear. Failure loads varied from 1.93 to 2.35 N (mean value 2.21 N), and failure stretches ranged between 1.20 and 1.30 (mean value 1.26). Finally, the experimental data could be described by a hyperelastic model proposed inOrgéas et al.

  α 1 and α 2[START_REF] Orgéas | Etude expérimentale et numérique du comportement thermomécanique d'un alliage à mémoire de forme industriel NiTi[END_REF]. The model parameters were identified from the experimental data given in Figure1(e) over the stretch range λ f ∈ [1 : 1.15]. The latter covers the range of tensile loadings predicted within the fibres in the experiments below. The identification yielded to μ 1 = 2444.94 MPa, μ 2 = 223.16 MPa, μ 3 = 38652.62 MPa, σ 0 = 42.18 MPa, c = 16.25 × 10 -2 , α 1 = 9.64 × 10 -3 and α 2 = 5.20 × 10 -2 .

  (c)). This parameter allows the control of the initial fibre length of the lattice, 0 , and thus, the fibre content. Length 0 can vary by step of 1 mm. (ii) the angulation of the pantograph, adjustable by translation of the arm's crossjunctions as illustrated in Figure 2(b). This parameter allows the control of the initial half-angle between fibres, θ 0 . It can be varied between 20 • and 70 • by step of 5 • .

  (i) to account for the average mass loss (58%) measured after the silicone curing due to solvent evaporation and shrinkage of the silicone during curing, and (ii) to spatially equilibrate the microstructure along the membrane's thickness. Finally, several composite membranes were processed, characterised by six microstructural arrangements ( 0 , θ 0 ) as follows: (1 mm, 45 • ), (3 mm, 45 • ), (1 mm, 30 • ), (3 mm, 30 • ), (1 mm, 60 • ), (3 mm, 60 • ). These target values of both parameters were chosen in line with previous predictions found in[START_REF] Bailly | Towards a biomimetism of abdominal healthy and aneurismal arterial tissues[END_REF]. Typical illustrations of manufactured microstructures are presented in Figure2(d).

  (a)). Hence, the lattice mechanical contribution combines the rotations of the fibres with their tensile response. These mechanisms typically yield to a drastic contraction of the sample along the transverse direction (up to λ xx = 0.7), and low values of the mechanical tension in the overall membrane at λ yy = 1.3 (maximum value of τ yy below 0.5 N/mm). On the contrary, for β 0 = 45 • , the fibres arranged perpendicular to the stretch direction are slightly compressed. They strongly restrain the membrane width's contraction along the transverse direction (λ xx > 0.97 during the test). In that case, only the fibres aligned with the loading direction contribute to the overall mechanical tension measured in the composite membrane. Therefore, as the silicone rubber matrix displays soft mechanical properties, the lattice mechanical contribution at the macroscale is nearly equivalent to the composite's one.

Figure 4

 4 Figure 4(a) confronts the global kinematics measured at the composite scale by 2D DIC, with the local kinematics measured at the fibre scale by tracking the displacements of the REC nodes A i . SamplesS 3 (3, 45, 0) and S 4 (3, 45, 45) were chosen for illustration. Whatever the loading direction, a fairly good quantitative agreement was measured between global and local kinematics, as shown by the superimposition of the loading paths derived from both techniques in Figure4(a). In particular, the relative difference e r between a couple of stretches determined by DIC (λ DIC xx , λ DIC yy ), and the one determined by particle tracking (λ t xx , λ t yy ) was calculated using:

  r comes to 17.7% for λ yy ∈ [1 : 1.23]. It falls below 15% for λ yy > 1.05.

  (a) compare the loading paths and the Cauchy tensionelongation responses measured on three samples: the isotropic homogeneous silicone membrane, samples S 1(1, 45, 0) and S 2(1, 45, 45). All samples are characterised by a similar initial thickness (relative STD being 6.7 %), which enables a comparison relying on tension curves.Both the mechanical structural reinforcement and the anisotropy induced by the fibrous lattice are emphasised in the left graph of Figure5(a). For the silicone membrane, the shrinkage of the sample width is about 10% at λ yy = 1.25. For composite S 1(1, 45, 0), the progressive alignment of the fibres along the loading direction favours the sample contraction along the x-direction, and yields to a width-shrinkage of about 30% at λ yy = 1.25. On the contrary, for sample S 2(1, 45, 45), the width's decrease is restrained to about 2% beyond λ yy = 1.05.The anisotropy is also highlighted in the right graph of Figure5(a), showing a global stiffening of the mechanical behaviour induced by the fibrous lattice. The more the fibres are aligned along the stretch direction, the higher the stiffness of the sample and stress levels. A tangent modulus was assessed at a tension value typically measured in aortic tissue for tensile loadings(Vande Geest et al., 2006;[START_REF] Bailly | Towards a biomimetism of abdominal healthy and aneurismal arterial tissues[END_REF], i.e. τ yy = 0.1 N/mm. The stiffening was characterised by a modulus of 1.13 N/mm for S 1(1, 45, 0) [resp. 19.76 N/mm for S 2(1, 45, 45)], thus increased by a ratio 1.3 (resp. 22.5) in comparison to the silicone case (0.88 N/mm). At last, the mechanical responses of the silicone and the sample S 1 (1, 45, 0) were initially superimposed. This suggests that, for this fibrous architecture, i.e. for β 0 = 0 • and for values of θ 0 below 45 • , the contribution of the lattice is negligible, at least for λ yy < 1.05.4.4 Influence of the initial fibre length 0The graphs of Figure5(b) compare the loading paths and the tension-elongation responses measured on five samples of comparable initial thickness (relative STD being 13.7 %): the silicone membrane, samples S 1 (1, 45, 0), S 2(1, 45, 45), S 3 (3, 45, 0) and S 4(3, 45, 45). The influence of initial fibre length 0 can be assessed by comparing the mechanical response of S 1 (1, 45, 0) and S 3(3, 45, 0), and that of S 2(1, 45, 45) and S 4(3, 45, 45).

  (c) compare the loading paths and the tensionelongation responses measured on five samples of similar initial thickness (relative STD being 8.2 %) for various values of θ 0 . All composite samples are characterised by an identical initial fibre length ( 0 = 1 mm) and the same loading direction (β 0 = 0 • ). For θ 0 = 30 • , the decrease of the sample's width is very close to the one of the matrix alone, as shown in the left graph of Figure 5(c). Tension-elongation responses are also rather similar. These results imply that, for such lattice architectures and such elongation range, the fibres are not loaded and they weakly disturb the deformation of the silicone.By contrast, for θ 0 = 60 • , the fibres tend to align along the loading direction as soon as λ yy > 1. Therefore, the reduction of the sample width is much higher than in the previous case (θ 0 = 30 • ): λ xx = 0.68 versus λ xx = 0.90 for λ yy = 1.25. Furthermore, the elongation paths being similar for cases θ 0 = 45 • and θ 0 = 60 • , the influence of the fibres' initial orientation is clearly demonstrated on the corresponding tensionelongation responses. The more the fibres are aligned along the stretch direction, the higher the stiffness of the sample.

Figure 1 .

 1 Figure 1. (a) Rhombic lattice and corresponding periodic Representative Elementary Cell of the one-layer fibrous structure: (b) initial undeformed configuration C 0 , (c) actual configuration C. (d) Mechanical behaviour of the silicone matrix: typical elongation path (λ xx , λ yy ) and corresponding nominal stress Π yy versus elongation λ yy . The continuous line gives the prediction of the neo-Hookean model with c m = 0.24 MPa. (e) Mechanical behaviour of the fibre: mean nominal stress Π f versus fibre elongation λ f (average value over 16 samples). Error bars represent the data standard deviation.

Figure 2 .

 2 Figure 2. (a) 3D-scheme of the device designed to manufacture the composite membranes. (b) 2D-view of the device's motion possibilities. The yellow part represents the lower part of the mould dedicated to the membrane casting (superior part of the device being not shown here). (c) Schematic zoom of a 2D-view of the pantographic structure for the lattice elaboration. (d) Pictures of some resulting composite membranes. Composites with perpendicular fibres could be created by two different manufacturing processes (see first and last rows).

Figure 3 .

 3 Figure 3. Typical post-processing of the measured data, using the tracking of the experimental REC nodes A i . Illustration for two distinct loading directions β 0 , with S 1 (1, 45, 0) (a) and S 2 (1, 45, 45) (b).

Figure 4 .

 4 Figure 4. Comparison between global kinematics at the macroscale determined by 2D DIC and local kinematics at the microscale determined by tracking of experimental REC extremities. Illustration for two distinct loading directions, with samples S 3 (3, 45, 0) and S 4 (3, 45, 45). Same legend for (a) and (b).

Figure 5 .

 5 Figure 5. Influence of (a) the loading direction β 0 , (b) the initial fibre length 0 and (c) the initial half-angle between fibres θ 0 on the mechanics of the composite membranes. Comparison between the predictions of the micro-mechanical model (solid lines) and experimental data (symbols).

Figure 6 .Figure 7 .

 67 Figure 6. Same as in Figure 5 for composites with 0 = 3 mm and β 0 = 0 • . Influence of the initial half-angle between fibres, θ 0 .

Table 1

 1 Dimensions and microstructural parameters of composite samples S i ( 0 , θ 0 , β 0 ). † Values measured over all RECs of the strip's surface: mean value (minimum -maximum).

	Samples	0 (mm) †	θ 0 ( • ) †	L 0 (mm) W 0 (mm) T 0 (mm)
		±0.05	±0.2	±0.02	±0.02	±0.02
	S 1 (1, 45, 0) 0.96 (0.86 -1.08) 45.3 (43.0 -47.6)	11.22	7.75	1.06
	S 2 (1, 45, 45) 0.96 (0.87 -1.10) 45.2 (43.8 -46.4)	10.29	5.99	0.98
	S 3 (3, 45, 0) 2.93 (2.60 -3.08) 45.8 (43.8 -48.1)	28.31	21.12	0.85
	S 4 (3, 45, 45) 2.85 (2.76 -2.96) 45.1 (44.2 -46.5)	22.28	15.34	0.81
	S 5 (1, 60, 0) 0.95 (0.79 -1.10) 60.4 (57.6 -63.5)	18.04	6.39	0.93
	S 6 (1, 30, 0) 0.97 (0.86 -1.07) 30.2 (28.6 -31.8)	20.82	10.31	1.11
	S 7 (3, 60, 0) 2.90 (2.68 -3.07) 60.2 (57.8 -62.6)	47.96	17.54	1.05
	S 8 (3, 30, 0) 2.92 (2.61 -3.16) 31.1 (29.1 -33.3)	37.91	25.29	0.77

  max) 1 st test, 2 nd test †

			±0.2	±0.2
	S 1 (1, 45, 0)	45	45.3 (43, 47.6)	47.5, 46.4
	S 2 (1, 45, 45)	45	45.2 (43.8, 46.4)	44.2, 45.1
	S 3 (3, 45, 0)	45	45.8 (43.8 -48.1)	45.6, 45.6
	S 4 (3, 45, 45)	45	45.1 (44.2 -46.5)	44.2, 43.7
	S 5 (1, 60, 0)	60	60.4 (57.6, 63.5)	61.9, 62.7
	S 6 (1, 30, 0)	30	30.2 (28.6, 31.8)	31.3, 32.2
	S 7 (3, 60, 0)	60	60.2 (57.8, 62.6)	60.5, 60.3
	S 8 (3, 30, 0)	30	31.1 (29.1, 33.3)	31.3, 30.9

Acknowledgements

-The authors would like to thank A. Lemercier for her contribution to the experimental work. They gratefully acknowledge the CNRS, the Univ. Joseph Fourier and the Institute Carnot PolyNat for their financial support. The laboratory 3SR is part of the LabEx Tec 21 (Investissements d'Avenir -grant agreement ANR-11-LABX-0030). IRPHE institute is part of the LabEx MEC.

of Vascular Surgery 28(2), 234-41. Soliman, S., Sant, S., Nichol, J., Khabiry, M., Traversa, E. and Khademhosseini, A. (2011) Yang, H., Zhu, G., Zhang, Z., Wang, Z., Fang, J. and Xu, W. (2012), 'Influence of weft-knitted tubular fabric on radial mechanical property of coaxial three-layer small-diameter vascular graft', J Biomed Mater Res 100B, 342-349.