Rinaldo M Colombo
email: rinaldo.colombo@unibs.it

Vincent Perrollaz
email: vincent.perrollaz@univ-tours.fr

Abraham Sylla
email: abraham.sylla@unimib.it

Peculiarities of Space Dependent Conservation Laws: Inverse Design and Asymptotics

. More precisely, characterizations of attainable sets and the set of initial data evolving at a prescribed time into a prescribed profile were obtained. Here, we present an explicit example that underlines deep differences between the 𝑥-dependent and 𝑥-independent cases. Moreover, we add a detailed analysis of the time asymptotic solution of this example, again underlining differences with the 𝑥-independent case.

Introduction

Consider the scalar one dimensional conservation law 𝜕 𝑡 𝑢 + 𝜕 𝑥 (𝐻 (𝑥, 𝑢)) = 0 (𝑡, 𝑥) ∈ ]0, +∞[ × R 𝑢(0, 𝑥) = 𝑢 𝑜 (𝑥) 𝑥 ∈ R .

(CL)

Denote by

𝑆 𝐶 𝐿 : R + × L ∞ (R; R) → L ∞ (R; R) (1)
the semigroup whose orbits are the entropy solutions to (CL). For any positive 𝑇 and for any assigned profile 𝑤 ∈ L ∞ (R; R) the inverse design is the set of initial data evolving into this profile at time 𝑇, i.e.,

𝐼 𝐶 𝐿 𝑇 (𝑤) 𝑢 𝑜 ∈ L ∞ (R; R) : 𝑆 𝐶 𝐿 𝑇 𝑢 𝑜 = 𝑤 . (2)
In the homogeneous case, a general characterization of 𝐼 𝐶 𝐿 𝑇 (𝑤) is given in [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF]. Other more specific results in this setting are [START_REF] Liard | Initial data identification for the one-dimensional Burgers equation[END_REF], devoted to Burgers' equation; [START_REF] Ancona | On the attainable set for scalar nonlinear conservation laws with boundary control[END_REF], specific to the attainable set for boundary value problems arising in the modeling of vehicular traffic. However, the homogeneous case significantly differs from the present non-homogeneous one and less results in the literature are available. A first step in this direction, limited to the study of the attainable set, is [START_REF] Ancona | Attainable profiles for conservation laws with flux function spatially discontinuous at a single point[END_REF], see also the related preprint [START_REF] Adimurthi | Exact and optimal controllability for scalar conservation laws with discontinuous flux[END_REF], where 𝐻 in (CL) consists of an expression for 𝑥 > 0 and another expression for 𝑥 < 0. A specific inverse design problem related to conservation laws, e.g., the minimization of a sonic boom, is considered in [START_REF] Gosse | Filtered gradient algorithms for inverse design problems of onedimensional Burgers equation[END_REF], while [START_REF] Esteve-Yagüe | Reachable set for Hamilton-Jacobi equations with non-smooth Hamiltonian and scalar conservation laws[END_REF] is devoted also to Hamilton-Jacobi equation.

In [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF], the characterizations obtained in [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF] is extended to the non-homogeneous case. The analytic techniques developed in [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF] take advantage of the connection 𝑢 = 𝜕 𝑥 𝑈 between (CL) and the Hamilton-Jacobi equation

𝜕 𝑡 𝑈 + 𝐻 (𝑥, 𝜕 𝑥 𝑈) = 0 (𝑡, 𝑥) ∈ ]0, +∞[ × R 𝑈 (0, 𝑥) = 𝑈 𝑜 (𝑥) 𝑥 ∈ R . (HJ)
We know, on the basis of [START_REF] Colombo | Conservation laws and Hamilton-Jacobi equations with space inhomogeneity[END_REF], that both Cauchy problems for (CL) and (HJ) are globally well posed under the same set of assumptions, namely:

Smoothness : 𝐻 = 𝐻 (𝑥, 𝑢) ∈ C 3 (R 2 , R). (C3) Compact NonHomogeneity : ∃𝑋 > 0, ∀(𝑥, 𝑢) ∈ R 2 |𝑥| > 𝑋 =⇒ 𝜕 𝑥 𝐻 (𝑥, 𝑢) = 0. (CNH)
Strong Convexity :

∀𝑥 ∈ R, 𝑢 ↦ → 𝜕 𝑢 𝐻 (𝑥, 𝑢) is an increasing C 1 -diffeomorphism of R onto itself. (CVX)
Assumption (CVX) implies that 𝐻 is strictly convex with respect to the second variable. Clearly, the strongly concave case is entirely analogous. Let us also mention that for well-posedness, (CVX) is relaxed to a uniform coercivity assumption coupled to a genuine nonlinearity assumption in [6, (UC)-(WGNL)]. Rather than tackling directly the characterization of the inverse design for (CL), [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF] deals first with (HJ) and then uses the correspondence to get back to (CL). Below, we construct an explicit example showing that when 𝐻 depends on 𝑥, the inverse design 𝐼 𝐶 𝐿 𝑇 (𝑤) may have properties, in a sense, opposite to those that hold in the homogeneous case, according to [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF]. Indeed, in the 𝑥-independent case, the presence of a shock in 𝑤 implies that 𝐼 𝐶 𝐿 𝑇 (𝑤) is a cone with infinite dimensional extremal faces. In our 𝑥-dependent example, in spite of 𝑤 displaying a shock, 𝐼 𝐶 𝐿 𝑇 (𝑤) is a singleton. On the contrary, in the homogeneous case, 𝐼 𝐶 𝐿 𝑇 (𝑤) is a singleton if and only if 𝑤 is continuous. Furthermore, in this example, the time asymptotic solution is neither constant nor a rescaling of a solution to a Riemann Problem. On the contrary, as 𝑡 → +∞, the solution converges locally uniformly to a compactly supported profile displaying a single singularity, which is a stationary entropic shock wave. This suggests various questions about the characterization of time asymptotic profiles to non homogeneous conservation laws and, in particular, about their stability properties.

Notations and Results

Recall the classical definition of entropy solution [11, Definition 1], as tweaked in [START_REF] Colombo | Conservation laws and Hamilton-Jacobi equations with space inhomogeneity[END_REF].

Definition 1 Fix 𝑢 𝑜 ∈ L ∞ (R; R). A bounded function 𝑢 ∈ L ∞ (R + × R; R) is a solution to (CL) if for all test functions 𝜑 ∈ C ∞ 𝑐 (R + × R; R + )
and for all scalar 𝑘 ∈ R: which we consider equipped either with initial or with final conditions. For all

∫ +∞ 0 ∫ R |𝑢(𝑡, 𝑥) -𝑘 | 𝜕 𝑡 𝜑(𝑡, 𝑥) d𝑥 d𝑡 + ∫ +∞ 0 ∫ R sgn (𝑢(𝑡, 𝑥) -𝑘) (𝐻 (𝑥, 𝑢(𝑡, 𝑥)) -𝐻 (𝑥, 𝑘)) 𝜕 𝑥 𝜑(𝑡, 𝑥) d𝑥 d𝑡 - ∫ +∞ 0 ∫ R sgn (𝑢(𝑡, 𝑥) -𝑘) 𝜕 𝑥 𝐻 (𝑥, 𝑘) 𝜑(𝑡, 𝑥) d𝑥 d𝑡 + ∫ R |𝑢 𝑜 (𝑥) -𝑘 | 𝜑(0, 𝑥) d𝑥 ≥ 0.
𝑤 ∈ L ∞ (R; R) such that 𝐼 𝐶 𝐿 𝑇 (𝑤) ≠ ∅, define 𝜋 𝑤 : R -→ R 𝑥 ↦ -→ 𝑞(0)
where (𝑞, 𝑝) solves (HS) with

𝑞(𝑇) = 𝑥 𝑝(𝑇) = 𝑤(𝑥) . (4)
We also introduce the set

R 𝑇 𝑞 ∈ C 1 ( [0, 𝑇]; R) : ∃ 𝑝 ∈ C 1 ( [0, 𝑇]; R) such that (𝑞, 𝑝) solves (HS) .
whose elements we call Hamiltonian rays. The map 𝜋 𝑤 assigns to 𝑥 ∈ R the intersection of the minimal backward characteristics emanating from (𝑇, 𝑥), see [ A characterization of the attainable set for (CL) can be obtained by that for (HJ) provided in [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF]Theorem 3.2]. The latter theorem extends to the 𝑥-dependent case some of the properties known to hold in the 𝑥-independent case, see [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF]. However, the extension to the 𝑥-dependent case can not be merely reduced to the rise of technical difficulties. Indeed, some properties are irremediably lost and new phenomena arise.

The most apparent difference between the two situations is described in Figure 1, with reference to extremal backward generalized characteristics (thicker curves), whose behaviors in the two cases are quite different. In the 𝑥-independent case, extremal backward characteristics define a non uniqueness gap. On the contrary, in the 𝑥-dependent case, extremal backward characteristics may well intersect at the initial time, so that the non uniqueness gap disappears. Furthermore, in the 𝑥independent case, an isentropic solution, see [6, Theorem 3.1], can be constructed filling the non uniqueness gap with Hamiltonian rays, that is solutions to (HS), emanating from 𝑞(𝑇) = 0, 𝑝(𝑇) = (1 -𝜆)𝑤(0-) + 𝜆𝑤(0+), for 𝜆 ∈ [0, 1]. On the contrary, the same procedure might not work in the 𝑥-dependent case, see also [START_REF] Barron | Regularity of Hamilton-Jacobi equations when forward is backward[END_REF] for a multi-d study. The numerical integrations in Figure 1 𝑜 is one sided Lipschitz continuous, but also that the solution ũ to (CL) with initial datum 𝑢 * 𝑜 evolving into 𝑤 is Lipschitz continuous on any compact subset of ]0, 𝑇 [ × R. Thus, ũ satisfies the inequality in Definition 1 with an equality, i.e., it is an isentropic solution and also reversible in time. All this is no longer true in the heterogeneous case, as highlighted by the following theorem. + 𝑔(𝑥) where 𝑔(𝑥)

1 -(1 -𝑥 2 ) 4 if |𝑥| ≤ 1 1 otherwise. ( 5
)
Then, (C3), (CNH) and (CVX) hold. Moreover, there exists 𝑤 ∈ L ∞ (R; R) such that

(i) For all 𝑇 > 0, 𝐼 𝐶 𝐿 𝑇 (𝑤) is a singleton. (ii) For all 𝑇 > 𝜋/(2 √
2) the solution 𝑢 to (CL) with 𝑢(0) being the vertex of 𝐼 𝐶 𝐿 𝑇 (𝑤) displays an entropic shock sited at 𝑥 = 0, whose size grows in time, that arises at time 𝑡 = 𝜋/(2

√
2).

Elements of proof are presented in Section 3. As we mentioned before, the contrary to (ii) holds when 𝐻 does not depend on 𝑥. In the homogeneous case, the contrary to (i) also holds true, see [5, (G1) in Proposition 5.2]. The evolution of the numerical solution to (CL)-( 5) with initial data in 𝐼 𝐶 𝐿 𝑇 (𝑤) (which is a singleton) computed with a standard finite volume scheme, is represented in Figure 2, see also Figure 4.

Therefore, there exist an 𝑥-dependent Hamiltonian 𝐻, a profile 𝑤 and a time 𝑇 > 0 such that 𝐼 𝐶 𝐿 𝑇 (𝑤) ≠ ∅ but in any solution evolving from an initial datum in 𝐼 𝐶 𝐿 𝑇 (𝑤), shocks arise before time 𝑇, so that no reversible solution is possible, see Figure 4. In other words, the profile 𝑤 can be reached exclusively producing a sufficient amount of entropy and no isentropic solution evolves into 𝑤. Each of these facts requires 𝐻 to depend on 𝑥 and is false in an 𝑥-independent setting. As a consequence, no direct definition of 𝑢 * 𝑜 is available, as it was in the 𝑥-independent case, explaining why in [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF] the authors had to resort to (HJ) for its construction.

A further difference between the homogeneous and the non homogeneous cases is in the long time behavior.

Theorem 3 Let 𝐻 be as in [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF], choose

𝑢 𝑜 (𝑥) -2 if 𝑥 < 0 2 if 𝑥 > 0 , ( 6
)
and consider the corresponding entropy solution 𝑢 to (CL). Then, for all 𝑥 ∈ R\{0},

lim 𝑡→+∞ 𝑢(𝑡, 𝑥) = -sgn 𝑥 √ 2(1 -𝑔(𝑥)) = - √ 2 sgn 𝑥 (1 -𝑥 2 ) 2 |𝑥| ≤ 1 , 0 |𝑥| > 1 . ( 7
)
The proof is sketched in Section 4, refer to Figure 2, bottom right, for the graph of [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF].

2) a shock is formed (bottom left) and in subsequent times the size of the shock grows (bottom right). The latter graph also portrays the asymptotic profile [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF].

In particular, note that a stationary shock is present at 𝑥 = 0 in the time asymptotic limit. Outside this shock, the solution is classical.

Theorem 2 -Sketch of the Proof

Note first that 𝐻, as defined in [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF], satisfies (C3), (CNH) with 𝑋 = 1 and (CVX).

Here follows a sequence of lemmas describing the behaviors of the solutions to (HS)- [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF]. The proofs, relying on standard ODE arguments, can be found in [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF]. We refer to Figure 3 for an illustration of these behaviors.

Lemma 1 [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF]Lemma 5.11] Let 𝐻 be as in [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF] and 𝑢 𝑜 be as in [START_REF] Colombo | Conservation laws and Hamilton-Jacobi equations with space inhomogeneity[END_REF]. Fix 𝑞 𝑜 ≥ 0. Denote by (𝑞, 𝑝) the solution to (HS)-( 5) with initial datum

(𝑞 𝑜 , 𝑢 𝑜 (𝑞 𝑜 +)) = (𝑞 𝑜 , 2). Then, 𝑞 is increasing on [0, +∞[ and 𝑞(𝑡) -→ 𝑡→+∞ + ∞.
Lemma 2 [7, Lemma 5.12] Let 𝐻 be as in [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF] and 𝑢 𝑜 be as in [START_REF] Colombo | Conservation laws and Hamilton-Jacobi equations with space inhomogeneity[END_REF]. Fix 0 ≤ 𝑞 𝑜 < 𝑞 𝑜 and denote by (𝑞, 𝑝), respectively ( 𝑞, 𝑝), the global solution to (HS)-( 5) with initial datum (𝑞 𝑜 , 𝑢 𝑜 (𝑞 𝑜 +)) = (𝑞 𝑜 , 2), respectively ( 𝑞 𝑜 , 𝑢 𝑜 ( 𝑞 𝑜 +)) = ( 𝑞 𝑜 , 2). Then, 𝑞(𝑡) < 𝑞(𝑡), for all 𝑡 ≥ 0.

Lemma 3 [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF]] Let 𝐻 be as in [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF] and 𝑢 𝑜 be as in [START_REF] Colombo | Conservation laws and Hamilton-Jacobi equations with space inhomogeneity[END_REF]. Fix 𝑝 𝑜 ∈ ]0, 2[. Denote by (𝑞, 𝑝) the global solution to (HS)-( 5) with initial datum (0, 𝑝 𝑜 ).

(i) If 𝑝 𝑜 ∈ ] √ 2, 2[, then 𝑞 is increasing on [0, +∞[ and 𝑞(𝑡) -→ 𝑡→+∞ + ∞. (ii) If 𝑝 𝑜 = √ 2, then 𝑞 is increasing on [0, +∞[ and 𝑞(𝑡) -→ 𝑡→+∞ 1. (iii) If 𝑝 𝑜 ∈ ]0, √ 2[
, then 𝑞 is periodic and the map

T : ]0, √ 2[ -→ ]0, +∞[ 𝑝 𝑜 ↦ -→ the smallest period of 𝑞 (8)
is increasing, continuous and lim

𝑝 𝑜 → √ 2 T ( 𝑝 𝑜 ) = +∞. Moreover, 𝑞(𝑡) > 0 for 𝑡 ∈ ]0, T ( 𝑝 𝑜 )/2[ and 𝑞(𝑡) < 0 for 𝑡 ∈ ]T ( 𝑝 𝑜 )/2, T ( 𝑝 𝑜 ) [.
Lemma 4 [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF]Lemma 5.16] Let 𝐻 be as in [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF] and 𝑢 𝑜 be as in [START_REF] Colombo | Conservation laws and Hamilton-Jacobi equations with space inhomogeneity[END_REF]. Fix 0 < 𝑝 𝑜 < 𝑝 𝑜 < 2 and denote by (𝑞, 𝑝), respectively ( 𝑞, 𝑝), the global solution to (HS)-( 5) with initial datum (0, 𝑝 𝑜 ), respectively (0, 𝑝 𝑜 ). Then,

(𝑖) 𝑝 𝑜 ∈]0, √ 2[ =⇒ ∀𝑡 ∈ ]0, T ( 𝑝 𝑜 )/2], 𝑞(𝑡) < 𝑞(𝑡) (𝑖𝑖) 𝑝 𝑜 ∈ [ √ 2, 2[ =⇒ ∀𝑡 ∈]0, +∞[, 𝑞(𝑡) < 𝑞(𝑡).
Recall that (C3)-(CNH)-(CVX) ensure that for all (𝑞 𝑜 , 𝑝 𝑜 ) ∈ R 2 , calling (𝑞, 𝑝) the solution to (HS)- [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF] with datum (𝑞 𝑜 , 𝑝 𝑜 ) at time 0, the map

F : R 3 -→ R 2 (𝑡, 𝑞 𝑜 , 𝑝 𝑜 ) ↦ -→ (𝑞(𝑡), 𝑝(𝑡)) (9)
and its two projections F 𝑞 , F 𝑝 are of class C 2 , see [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF]Lemma 5.2].

Lemma 5 [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF]Lemma 5.17] Let 𝐻 be as in [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF] and 𝑢 𝑜 be as in [START_REF] Colombo | Conservation laws and Hamilton-Jacobi equations with space inhomogeneity[END_REF]. Then, there exists a unique map

Δ : ]0, +∞[ 2 -→ ( [0, +∞[ × {2}) ∪ ({0} × ]0, 2]) (𝑡, 𝑥) ↦ -→ (𝑞 𝑜 , 𝑝 𝑜 ) (10)
such that

F 𝑞 (𝑡, 𝑞 𝑜 , 𝑝 𝑜 ) = 𝑥 and ∀ 𝑠 ∈ ]0, 𝑡 [, F 𝑞 (𝑠, 𝑞 𝑜 , 𝑝 𝑜 ) > 0 . ( 11
)
Moreover, (i) Δ is continuous.

(ii) Δ is monotone, in the sense that setting Δ(𝑡 𝑜 , 𝑥 𝑜 ) = (0, 𝑝 𝑜 ) and Δ(𝑡 𝑜 , 𝑥 ′ 𝑜 ) = The construction of the solution to (CL)-( 6) now follows from the next proposition.

Proposition 1 [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF]Proposition 5.18] Let 𝐻 be as in [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF] and 𝑢 𝑜 as in [START_REF] Colombo | Conservation laws and Hamilton-Jacobi equations with space inhomogeneity[END_REF]. Then,

𝑢 : ]0, +∞[ × (R \ {0}) -→ R (𝑡, 𝑥) ↦ -→ F 𝑝 (𝑡, Δ(𝑡, 𝑥)) if 𝑥 > 0 -F 𝑝 (𝑡, Δ(𝑡, -𝑥)) if 𝑥 < 0. ( 12
)
is the solution to (CL) in the sense of Definition 1 and it is a classical solution outside 𝑥 = 0.

We refer to [7, § 5.4] for the concluding remarks proving the properties of 𝑢.

Proof of Theorem 3

Since 𝑢 is odd in the space variable, we consider only the case 𝑥 > 0. Using the notation (9), introduce the maps Indeed, note first that by (14), 𝑝 𝑜 (𝑡, 𝑥) > 0 and the Implicit Function Theorem, 𝑝 𝑜 is sufficiently regular. Item (ii) in Lemma 5 implies that 𝜕 𝑥 𝑝 𝑜 (𝑡, 𝑥) ≥ 0, so that 𝜕 𝑡 𝑢(𝑡, 𝑥) ≤ 0. Hence, (15) and ( 16) allow to complete the proof of [START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF].

As a final remark, we note that as 𝑡 → +∞, 𝑢(𝑡) converges uniformly compact subsets of R\ {0}, thanks to pointwise convergence and monotonicity, by Dini's Theorem [START_REF] Rudin | Principles of mathematical analysis[END_REF]Theorem 7.13]. A further technical argument, based on (ii) of Theorem 2, allows to extend the local uniform convergence on all of R.

( 3 )

 3 Definition 1, taken from by [6, Definition 2.1] is, apparently, weaker than the classical Kružkov definition since it does not require the "trace at 0 condition" [11, Formula (2.2)]. Nevertheless, under Assumption (C3), Definition 1 ensures uniqueness and uniform L 1 loc -continuity in time of the solution, as proved in [6, Theorem 2.6]. As usual, in connection with (CL) we use the system of ordinary differential equations 𝑞 = 𝜕 𝑝 𝐻 (𝑞, 𝑝) 𝑝 = -𝜕 𝑞 𝐻 (𝑞, 𝑝), (HS)

  , right, referred to (HS) with Hamiltonian (5), show that extremal backward characteristics still do not intersect in ]0, 𝑇 [ × R, but the intermediate Hamiltonian rays may well cross each other and even exit the region bounded by the extremal characteristics.

Fig. 1

 1 Fig.1Left, in the 𝑥-independent case, the Hamiltonian rays fill the non uniqueness gap. Right, in the 𝑥-dependent case defined by the Hamiltonian (5), extremal characteristics still do not intersect, but Hamiltonian rays do and may well exit the non uniqueness gap or also intersect.

Theorem 2 [ 7 ,

 27 Theorem 4.1] Define 𝐻 (𝑥, 𝑝) 𝑝 2 2

Fig. 2

 2 Fig. 2 Numerically computed solution to (CL) with 𝐻 as in (5) and initial datum (6), as described in Theorem 2. The Riemann initial datum (6) first evolves similarly to a rarefaction (top left); then gets non monotone (top right); at time 𝑡 = 𝜋/(2

Fig. 3

 3 Fig.3On the horizontal axis, the 𝑞 component of solutions to (HS)-(5), while time 𝑡 is on the vertical axis. Dotted curves are those considered in Lemma 1 and in Lemma 2; dashed-dotted curves are those considered in (i) of Lemma 3; solid curves refer to (iii) of Lemma 3. The two thicker curves depict solutions corresponding to the initial data (0, √2) and (0, 2).

( 0 ,

 0 𝑝 ′ 𝑜 ), if 0 < 𝑥 𝑜 < 𝑥 ′𝑜 , then 𝑝 𝑜 < 𝑝 ′ 𝑜 . (iii) For all 𝑥 ∈ ]0, +∞[, lim 𝑡→0+ Δ(𝑡, 𝑥) = (𝑥, 2).

Fig. 4 2 . 2 = 2 ( 1 - 2 ( 1 -

 4222121 Fig. 4 Illustration of Proposition 1: evolution in time of the solution to (CL) in connection with the orbits of (HS)-(5).

  8, Defi-nition 3.1, Theorems 3.2-3.3], with the axis 𝑡 = 0. Remark that in the 𝑥-independent case, all Hamiltonian rays are straight lines, as also any extremal characteristics, a key simplification exploited in [5, Formula (2.3)]. 𝐼 𝐶 𝐿 𝑇 (𝑤) ≠ ∅ if and only if 𝜋 𝑤 admits a non decreasing representative. In this case, 𝐼 𝐶 𝐿 𝑇 (𝑤) is a closed convex cone with a unique vertex 𝑢 * 𝑜 .

Theorem 1

[START_REF] Colombo | Initial Data Identification in Space Dependent Conservation Laws and Hamilton-Jacobi Equations[END_REF] Corollary 3.5]

Let 𝐻 satisfy (C3), (CNH) and (CVX). Fix 𝑇 > 0 and 𝑤 ∈ L ∞ (R; R). Then,

Acknowledgements

The first author was partly supported by the GNAMPA 2022 project Evolution Equations, Well Posedness, Control and Applications. This research was funded, in whole or in part, by l'Agence Nationale de la Recherche (ANR), project ANR-22-CE40-0010. For the purpose of open access, the second author has applied a CC-BY public copyright license to any Author Accepted Manuscript (AAM) version arising from this submission.


Frédéric Dubois
email: frederic.dubois@unilim.fr

Jérôme Dopeux
email: jerome.dopeux@unilim.fr

Octavian Pop
email: ion-octavian.pop@unilim.fr

Mickaël Metrope
email: mickael.metrope@unilim.fr

Long-term creep behavior of timber columns: Experimental and numerical protocols

Keywords: Compression creep, Viscoelastic behavior, Spectrum decomposition, Creep extrapolation, Eurocodes

This work deals with an experimental and numerical approach to determining the differed behavior of compressed wood elements subjected to creep loading. An experimental protocol is associated with a mechanical model, whose purpose is to isolate the effects of creep from those of shrinkage-swelling, while allowing for a long duration test to be conducted. The experimental protocol is based on a compression creep test applied to five wooden columns disposed in a climatic room for one year. The numerical model is based on the use of a Kelvin-Voigt rheological model associated with a shrinkage-swelling element. An optimization study is proposed in order to interpolate the model parameters with respect to the experimental measurements. In the context of a linear viscoelastic behavior, the result of this interpolation allows to propose a model extrapolation for long times. Within the framework of the European code, Eurocode 5, this project aims to identify the long-term creep coefficient (kdef) adapted to compressed elements, in addition to evaluating the partial creep coefficients as a function of loading duration class.

Introduction

Today, wood is one of the materials of the future in civil engineering because it is natural and it requires low consumption of fossil resources. Moreover, timber constructions store carbon dioxide which is the major cause of climate change. Finally, wood is often a local material that enhances short circuits by limiting the environmental impact in terms of transport. Thus, many countries and regions, with their abundant forest resources, attach considerable importance to the use of this resource in construction by considering wood as an economical and efficient structural material. Today, many technological and scientific challenges are proposed to realize timber road bridges or high-rise buildings beyond ten floors. These new structures require the completion of regulatory texts in order to adapt them to these new constructions.

In recent years, several experimental and numerical studies have been carried out to investigate the use of structural wood elements in the construction of high-rise buildings or large-span structure. In order to increase the resistance of these construction elements, most of these studies propose reinforcement solutions. Concerning large-span structures Xu et al. [START_REF] Xu | Compressive performance of steel-timber composite L-shaped columns under concentric loading[END_REF] propose an interesting study, using the L shaped column fabricated using Steel-Timber Composites (STC). The study demonstrated that for axial compression the L-STC columns stiffness and loadcarrying capacity are enhanced in comparison with CLT or Glue Laminated elements. Kang et al. [START_REF] Kang | Behaviour of glulam timber frames with different beamcolumn connections and braces under reversed cyclic loads[END_REF] have also studied the strength capacity, stiffness, ductility, energy dissipation and equivalent damping ratio glulam timber frames under reversed cyclic loading conditions. The study showed that the seismic behavior is significantly enhanced by using top-and-seat steel-angle connections and diagonal braces. O'Callaghan et al. [START_REF] O'callaghan | Experimental investigation of the compressive behaviour of GFRP wrapped spruce-pine-fir square timber columns[END_REF] show that in the case of square timber columns wrapped with fiber-reinforced polymer (FRP) composites under compression loading, a proper thickness of FRP wrapping can prevent undesirable failures and reduce the influence of naturally defects of wood. For Cross-Laminated Timber (CLT) elements subjected to blast loads, Lopez-Molina and Doudak [START_REF] Lopez-Molina | Retrofit techniques for Cross-Laminated Timber (CLT) elements subjected to blast loads[END_REF] show that reinforcing the panels with steel straps had minimal effect on the ultimate strength, but significantly affected the post-peak resistance and ductility. The whole of these works relates more particularly to the strength capacity of the carrying elements.

In high-rise buildings, vertical supports such as columns are typically subjected to compression or compound bending loads. If, today, some works have been done on the axial and transverse compressive strength [START_REF] Zheng | Experimental investigation on the rheological behavior of timber in longitudinal and transverse compression[END_REF], few studies have focused on the differed deformation behavior. In this context, Jockwer et al. [START_REF] Jockwer | Long-term deformation behaviour of timber columns : monitoring of a tall timber building in Switzerland[END_REF] proposed an experimental monitoring of the short and long term deformation behavior of highly loaded components. This work is based on a monitoring approach of the vertical carriers. It highlights the difficulty of separating the different effects induced by a complex mechanical behavior coupling viscoelasticity, shrinkage and swelling caused by moisture variations and mechanosorptive effects. In a similar way to the consideration of deferred effects on bent structures, the deferred deformations can be parameterized by a creep coefficient balanced by a service class related to the external environment. But moreover, the effects of shrinkage-swelling must be controlled, which is not the case today for bent elements.

Considering the effects of creep on the long-term behavior of wood structures is nowadays a necessity for both bending and compression elements. The European regulation Eurocode 5 [START_REF]Eurocode 5, Design of timber structures -Part 1-1 : General and rules for buildings[END_REF] integrates this notion only in the case of the calculation of the deflection of bending elements and in the long-term stiffness of fasteners. This consideration has been reflected by an amplification of the deflections through a coefficient kdef modulated by the duration of the given loading and service class. In the context of developing high-rise structures, the vertical carriers play a predominant mechanical role. At the serviceability limit state, the cumulative compressive deformation generates vertical displacements that can no longer be ignored or neglected. Currently, design engineers use creep coefficients determined on the basis of bending due to a lack of guidance from the Eurocode.

In addition, structural elements are subject to dimensional variations as a result of drying and wetting kinetics.

While in bending parts, these effects do not significantly contribute to the variations and changes in deflection, such is no longer the case for compressed elements whose effects are proportional to the height of vertical supports.

Therefore, with respect to high-rise buildings beyond ten floors, these effects can no longer be neglected.

However, the compressed load-bearing elements are dimensioned with large cross-sections (1m x 1m for instance).

In contrast to bent elements, the compressive stress is distributed evenly over the entire cross-sectional area. Under these conditions, the creep effects are evenly distributed over the entire resisting section. Moreover, the size of transversal sections induces important inertia effects in terms of moisture transfer. If it is important to consider moisture content variations between the manufacturing area of the elements, their storage places and their final destination in the building, the humidity variations induced by a service class 1 (in heated or air-conditioned interior), will not generate representative hydric disturbances on the whole of the cross sections. For this reason, this work focuses on the characterization of the creep behavior in a climatic chamber at constant temperature and relative humidity.

At the local scale, in assuming a linear viscoelastic behavior combined with swelling-shrinkage effects, mechanical behavior is defined as the relationship between stresses ( ) t  and strains ( ) t  . According to a unidimensional approach, this mechanical behavior is defined by the following expression [8] [9] [10]:

( ) ( ) ( ) ( ) ( ) t o t o t J t d w t w t        = - +  -   (1)
where ( ) Jt is the creep function. According to a uniaxial approach, the effects of shrinkage-swelling are taken into account by considering the differential variation between moisture content ( ) To proceed, a first section will detail the materials and methods by presenting the creep experimentation that entails a creep test bench adapted to compressed elements.

wt
Since compressed columns are placed in a climate-controlled chamber after several days of conditioning, the prediction of moisture content evolution needs to be evaluated. A numerical approach, optimized with experimental measurements, will be presented in the second section in order to separate the swelling-shrinkage effects from the creep response.

As such, to correct the first instants of creep from the shrinkage-swelling effects, a creep recovery protocol will be developed in the third section.

The last section will discuss the final analysis of the experimental data in the aim of determining the long-term behavior of the compressed elements by means of extrapolating the experimental results to 50 years of creep response. This extrapolation allows setting the new value of kdef and offering an assessment of the results obtained.

Materials and methods

2.1.Experimental set-up

The experimental equipment used herein has been based on an experimental creep device capable of handling five glue laminated spruce columns. To limit the risk of instability, specimen height is limited to 75cm and its crosssection is set at 100 x 200mm² corresponding to a slenderness ratio  of 26 in the week direction. The cross section is composed of 5 lamellae whose growth rings are also visible on the Fig. 1. Based on the mechanical strength class, the stress state is limited to 30% of the ultimate limit state compression value of 7.2MPa, which corresponds to an axial force loading of 144kN for each specimen. This force loading is adjusted by the system's three screws. The device design has been enhanced to obtain an acceptable stiffness relative to the experimental specimens, Fig. 3.

Fig. 3 Experimental creep device

Given that a creep state necessitates a controlled and homogeneous stress distribution, a distribution plate is placed between a force sensor and the column head. The loading device is presented in Fig. 4. In searching for a homogeneous distribution of internal moisture, as opposed to the work conducted on CLT creep proposed by Gräfe et al. [START_REF] Gräfe | CLT under in-plane loads : Investigation on stress distribution and creep[END_REF] [12], we have opted to lay the columns out horizontally so as to generate identical thermal and hydric convective exchanges on all four faces. Moreover, the enclosure has been equipped with fans that allow stirring the air and controlling exchange conditions on the surface. Fig. 4 Loading system principle In accordance with service class no. 1, the experimental device is to be placed in a climate-controlled room, where the climate is controlled in terms of both temperature (20° +/-2°C) and relative humidity (65% +/-5% RH).

Initial moisture content distribution

Since the creep response is influenced by the temporal evolution of the moisture content, the experimental protocol requires the integration of its control. Between the moisture state during the glue laminated design of the column and subsequent installation in the climate-controlled room, it is necessary to measure the humidity state. In respecting this consideration, during specimen machining, a 2-cm thick plate was isolated, with 15 cubes 2.5 cm to a side being extracted in order to measure the local moisture constant after drying in an oven at 103°C, Fig. 5.

The moisture content values are listed in The projection of local moisture content measurements onto a finite element support allows representing a map of the moisture content distribution, as shown in Fig. 6. The moisture highlights a drying phase between the column

S11 S21 S31 S41 S51 S12 S22 S32 S42 S52 S13 S23 S33 S43 S53
design time and the conditioning phase before experimental testing with a maximum deviation of 3% between the core and periphery of the specimens.

Fig. 6 Moisture content mapping in the cross-section

Creep and recovery experimentation

According to the moisture content map shown above and the updated environmental climate imposed in the climate-controlled chamber, the creep response will be made noisy by a moisture content evolution starting at its initial value ini w and extending until reaching stability equ w . Since a decoupling between moisture content and creep evolution is required, we have opted for a specific loading that includes both an initial creep loading during the moisture content stabilization phase and a recovery phase in a state of constant moisture content (Fig. 7). In such a context, we have chosen specimen no.

Moisture content corrections

Mass transfer simulation

The evaluation of recovery time r t is primarily based on a mass transfer simulation by applying a nonlinear orthotropic Fick's Law through the cross-section, as described by the following equation:

( ) ( )

w div grad w t  =  D (4)
where D denotes the second-order diffusion tensor. In the main orthotropic frame of reference, this tensor is represented by a diagonal matrix. By limiting the simulation to the cross-section characterized by the radial and tangential directions, respectively R and T , the diffusion tensor can be characterized by the two following nonlinear radial and tangential diffusion coefficients:

( ) ( )

exp o R R R D w D k w =   and
( ) ( ) Table 2 Diffusion properties for the spruce species

exp o T T T D w D k w =   (5)
In terms of boundary conditions, the water exchanges taking place between the environmental area and the specimen surfaces are expressed via the sorption isotherm curve found in Merakeb's formulation [START_REF] Merakeb | Modeling of the sorption hysteresis for wood[END_REF], i.e.:

( ) ( ) ( )

ln ln exp equ RH s w RH RH wT   =       (6)
 and  are two thermodynamic parameters characterizing the sorption isotherm. In combining Eq. ( 6), [START_REF]Eurocode 5, Design of timber structures -Part 1-1 : General and rules for buildings[END_REF] and the temperature and relative humidity evolutions, Fig. 8 serves to predict the average balanced moisture content at around 13.9%. é q w = 13,9%

é q w = 13,9%

é q w = 13,9%

é q w = R T T R
The initial moisture content map has been introduced as the interpolated distribution shown in Fig. 7. Moreover, the effects of convective exchanges are to be considered as an instantaneous exchange.

Moisture content evolution

The predicted time evolution of the average moisture content is plotted in Fig. 11. In terms of moisture content mapping evolution, let's note a global increase with a homogenization taking place from one day to the next. The average moisture content evolution shows stability from right around 200 days. According to the force loading principle (Fig. 10), it can be concluded that the recovery time must be set at between 200 days and one year upon completion of the force loading process. The experimental strategy employed herein focuses on a time recovery of 200 days for column no. 3. The total experimental time equals one year. At the end of the creep and recovery experimental test, the moisture content evolution was controlled by cutting three 2-cm thick plates, from which 15 cubes 2 cm to a side were placed in an oven for drying and evaluating the final moisture content map.

Fig. 10 Time evolution of the average moisture content and various MS maps

Lastly, the final moisture content was defined for 50 cubes, with an average value of 13.85% and a standard deviation of 0.4%, thus confirming the accuracy prediction along with a homogeneous state.

Creep end recovery testing

Based on specimen no. 3, this experimental test is composed of a creep phase with, in parallel, a phase of constant moisture equilibrium. This experimental protocol induces several phenomena, such as swelling effects, an impact on the axial elastic modulus and an increased creep response time. The subsequent protocol has been designed to control and isolate all these effects. For specimen no. 3, the external loading is shown in Fig. 11 for an axial stress value of 7.3 MPa. The force evolution highlights the effects of swelling at the start of loading, resulting in a slight increase in force during the first 120 days, despite the correction introduced by the experimenter.

Instantaneous elastic response

The elastic modulus is evaluated by conducting a linear regression modulated by loading and unloading time, respectively. Table 4 provides the elastic modulus values for all specimen evaluated at the loading phase (at an average moisture content of 11.6%) and at the unloading time (at a moisture content of 13.9%).

Creep and recovery behavior under an increasing moisture content

By applying the principle of superposition, legitimized by the assumption of a linear behavior, after unloading at the time r t , the recovery deformation admits the form deduced from the expression (1) :

( ) ( ) ( ) ( ) ( ) ( ) ( ) o o r o t J t t J t t w t w t   =  - - - +  - (8)
The creep function ( )

Jt is normally represented by a creep spectrum, according to the following Prony series, such that:

( ) Finally, by regrouping the Eq. ( 8), ( 9) and [START_REF] Dubois | Modeling of the viscoelastic, mechano-sorptive behavior in wood[END_REF], an optimization algorithm is used to determine the parameters m  and the axial shrinkage swelling coefficient  . The convergence criterion of the optimization is based on a least squares minimization of the difference between the experimental measurements and the model data over the entire experimental time range (360 days). The model parameters are given in Table 5 and the axial shrinkage swelling coefficient converges at the value of ( ) This fact explains the small differences between model and experiment. However, for all samples, the differences disappear after one year of creep, which will allow, in the continuation of this study, to have a good extrapolation until 50 years. In this context, we note the importance, in the future, to integrate a recovery phase after 6 months to better interpret the effects of swelling process.

1 1 1 1 exp t M m m L m Jt E   - =     =  +  -        (
1 day  ( ) 2 day  ( ) 3 day  1  2  3  1.

Numerical extrapolation

Since long-term timber behavior must be predicted for 50 years, let's use an extrapolation method based on a property of the linear viscoelastic behavior, whereby after the first primary creep phase, the time-dependent strain adopts the following form [START_REF] Dubois | Modeling of the viscoelastic, mechano-sorptive behavior in wood[END_REF], as illustrated in Fig. 15:

( ) ( ) ( ) ln ln t a t b  =  + (11)
Fig. 15 Logarithmic linearity during secondary creep For all columns, the linear correlation yields coefficients a and b, as listed in Table 8.

Column

Differed strain

All experimental results allow for a determination of the creep strain under a constant axial stress loading. These results, completed by the approximation process, are indicated in Table 6.

Columns

Spectral decomposition of creep functions

The spectral decomposition of creep allows separating the temporal spectrum the times defined in Eurocode 5, i.e., one week, six months, ten and fifty years. Under these conditions, the creep function [START_REF] Dubois | Modeling of the viscoelastic, mechano-sorptive behavior in wood[END_REF] can be rewritten as follows:

( ) For each loading duration, the ) [START_REF]Eurocode 1, Basis of design and actions on structures and National, Application document -Part 1 : Basis of design (ENV 1991-1)[END_REF]. By combining the time evolution of the creep function (Eq. ( 11)) with the definition of differed displacement at the column head, kdef and 2,m  can be defined as follows: These results are completed by the values assumed in Eurocode 5 [START_REF]Eurocode 5, Design of timber structures -Part 1-1 : General and rules for buildings[END_REF] and Eurocode 1 [START_REF]Eurocode 1, Basis of design and actions on structures and National, Application document -Part 1 : Basis of design (ENV 1991-1)[END_REF]. Results have been given herein for service class no. 1; they reveal an atypical behavior of column no. 1, which may be due to either a problem during the initial loading or a misinterpretation of the increase in moisture during the first 150 days (corresponding to the moisture equilibrium). For this reason, we propose discarding column no. 1 in the statistical processing of results.

Two major remarks can thus be deduced from this study. First, let's note that the def k of the compressed elements is half that set for the flexural elements (i.e., 0.3 vs. 0.6). This first conclusion prompts us to consider a complement for Eurocode 5. Secondly, regarding the values of 2,m  , it is concluded that the values assigned in Eurocode 1 can be applied directly to the combinations of actions targeting compressed elements.

Conclusion and perspectives

This work has made it possible to define the long-term behavior of wood elements subjected to compressive loading. First, an experimental device was created to establish a mechanical state of creep by limiting the stresses to a service limit state, as defined in Eurocode 5. An experimental protocol was then proposed to isolate the effects of shrinkage and swelling that contribute to the mechanical response of to a state. During a second step, a spectral decomposition of the deferred behavior allowed accurately identifying the creep kinetics over an entire year of loading. Lastly, an extrapolation strategy was used to determine the creep function over 50 years, which correspond to the deferred effects of a permanent load. This work has allowed drawing a conclusion on the kdef coefficient to be considered for compressed elements by highlighting a value half that typically used for elements in bending.

In parallel, the intermediate creep effects led to the conclusion that the partial creep coefficients corresponding to each loading duration class were indeed well matched. The work program was exclusively limited to service class 1. Moreover, this work was limited to correcting the creep coefficient kdef. However, creep effects also cause an apparent decrease in mechanical strength through the coefficient kmod at the ultimate limit state. A short-term perspective calls for developing this study in order to determine the kmod coefficient, which moreover should exert a lesser impact on both the short-and long-term strength of compressed elements.

In conclusion, this study has highlighted the effects of shrinkage-swelling in combination with the effects of creep.

Over the longer term, this effort will require deep reflection on the incorporation of hygrothermal conditions with respect to the displacement and stress response of elements in compression.

Fig. 1 Fig

 1 Fig. 1 Specimen geometryGiven that the definition of creep requires measuring the axial strain, a 500mm long zone is identified in bypassing the set of boundary condition problems. In this case, the local strain is calculated with the relative displacement u  , as measured in this area in accordance with the following expression:

Fig. 5

 5 Fig. 5 Cutting of the cross section into 15 cubes

Fig. 7

 7 Fig. 7 Creep end recovery loadings vs. average evolution of moisture content over time Columns are loaded at time

D

  are the anhydrous diffusion coefficients for the radial and tangential directions.

  typically observed in wood materials. According to Manfoumbi's findings[START_REF] Manfoumbi | Découplage expérimental des déformations d'éléments bois sous chargements hydrique et mécanique variables[END_REF], spruce diffusion properties are given in Table2.

sw

  designates the equilibrium moisture content reached for 100% RH; its value is temperature-dependent in accordance with the following expression[START_REF] Varnier | Influence of temperature in a mass transfer simulation: application to wood[END_REF]:

Fig. 8 Fig. 9

 89 Fig. 8 One-year temperature and relative humidity variations in the climate-controlled chamber The mass transfer algorithm implemented in the finite element software Castem is based on the mesh shown below, in considering the orthotropic reference cylinder in the radial-tangential plane with its heterogeneity being induced by the various lamellas. The isoperimetric finite element size (2 mm) and time increment (1 day) are fixed in order to ensure spatial and temporal convergence. The boundary conditions are based on an imposed and constant moisture content of 13.9% on the surface, Fig. 9.

Fig. 11

 11 Fig. 11 Creep and recovery experiment for specimen no. 3The recovery phase starts after 220 days. As shown in Fig.12, the experimental protocol makes it possible to define the following four phases:

Fig. 12

 12 Fig. 12 Creep and recovery response

9 ) m  3 M

 9m3 denotes the creep coefficient, in accordance with the strain kinetics coupled with characteristic times m  , in response to an exponential distribution. For our specific case, in considering a total experimental time of one year, let's consider the three characteristic times of one week, six months and one year ( = ). In these conditions, characteristic times m  are calculated to have a 99% creep limit à time

  Fig. 13 illustrates a good fit between the experimental and model creep strains.

Fig. 13

 13 Fig. 13 Experimental and model creep strains, column n°3 Fig. 14 shows the synthetic results for the other 4 specimens. Note that the correction of the swelling effects was taken from the column n°3.

Fig. 14 5 - 3

 1453 Fig. 14 Experimental and model results for samples 1, 2, 4 and 5

t

  Fig. 17 Results of creep optimization (Column no. 3)

  By applying these expressions to the experimental values in

Table .

 .

	S11	S21	S31	S41	S51
	10.8%	12.1%	11.9%	11.5%	10.6%
	S12	S22	S32	S42	S52
	11.3%	12.7	12.7%	12.6%	11.2%
	S13	S23	S33	S43	S53
	10.3%	11.3%	11.3%	10.9%	11.0%

Table 1

 1

Initial moisture content distribution in the cross-section

Table 3

 3 indicates the isotherm parameter values for spruce.

			RH	o s w	L	C	pv	anh C
	0.82	1.435	35.4%	11 K -- 2500kJ kg	11 K -- 4180kJ kg	11 K -- 2160kJ kg

Table 3

 3 Sorption isotherm parameters for spruce

Table 4

 4 Elastic modulus measured by linear correlation during loading and unloading phases

		Instantaneous strain at 11.6%	Creep response + swelling induced increase
						Instantaneous strain at 13.9%
						Recovery response at
						constant moisture content
	-1.0E-04
	2.0E-04
	60	120	180	240	300	360

Table 5

 5 Creep spectrum properties

	52	39.1	62.1	3 1.93 10 -	2 7.60 10 -	2 9.48 10 -

Table 6

 6 Deferred creep strains

												1	2	3	4	5
					(	MPa	)	)				7.30	7.25	7.30	7.23	7.22
				( inst m m /	)			6 544 10 -	6 487 10 -	6 522 10 -	6 499 10 -	6 482 10 -
		( six months mm /	)	6 545 10 -	6 488 10 -	6 523 10 -	6 499 10 -	6 483 10 -
		( ten years mm /	)		6 588 10 -	6 542 10 -	631 10 -	6	578		6	6 543 10 -
		( fifty years mm /	)	6 632 10 -	6 609 10 -	6 662 10 -	6 635 10 -	6 622 10 -

  denotes the longitudinal elastic modulus defined for the equilibrium moisture content of 13.9%.

	7.0E-04 where L E m  are
	the characteristic times calculated based on load duration times; they have been calculated according to the
	following expression:
			4.0E-04
			Long-term strain extrapolated
			2.0E-04
			1.0E-04
			0.0E+00
			Time (days)
	Jt	=	4 m = 1   1 1 1 exp t m m L E   -   +  -      	(12)

  Table 7 specifies the various optimization parameters for each column.

	5.2 def k	interpretation
	By analogy with the bending creep of beams, the deferred displacement of the column head	u	can be expressed
																							fin
	as follows:
	u	fin	=	G u	, inst	( 1  +	k	def	)	+	4 	Qn u	, inst	( 1  +		2,	n		k	def	)	(14)
													n	1 =
	where:	G u	, inst	denotes the instantaneous displacement induced by the permanent loads;	Q u	inst ,	the
																							n
	instantaneous displacement induced by the operating loadings characterized by their loading time	t ;	k	the
																							n	def
	long-term creep amplification; and parameter 2,n  a moderation of the creep effects, as a function of the loading
	duration characterized by	n t (	n 	 1,..., 4	
																		Experimentation/Extrapolation	Optimization
					9.5E-05
					9.0E-05
					8.5E-05
			J(t) (Mpa-1)	7.5E-05 8.0E-05
					7.0E-05
					6.5E-05
					6.0E-05
									0							3000				6000	9000	12000	15000	18000
																							Time (days)
						m											1				2	3	4
						t											7 days			6 months	10 years	50 years
						m
				 (days)										2.34				60.1	1,200	6,010
					m
		m  (Column no. 1)						2 2.46 10 -		2 3.76 10 -	3.65 10	5.98 10 -
		m  (Column no. 2)						2 5.64 10 -		2 9.24 10 -	2 6.93 10 -	1 1.24 10 -
		m  (Column no. 3)						3 5.03 10 -		2 9.32 10 -	2 6.32 10 -	1 1.13 10 -
		m  (Column no. 4)						3 5.03 10 -		2 8.36 10 -	2 6.95 10 -	1 1.24 10 -
																2 6.95 10 -		2 6.79 10 -	2 7.32 10 -	1 1.32 10 -
																				Table 7 Spectral decomposition results

m

 (Column no.

5)

  Table for the five tested columns, the long-term properties presented in Table 8.

									Average value /
				Column 1	Column 2	Column 3	Column 4	Column 5		Eurocodes 5 and 1
									Standard deviation
	k	def	0.16	0.29	0.27	0.28	0.28	0.28 /1%	0.6
			0.16	0.01	0.02	0.02	0.02	0.01/1%	0.0
		2,2	0.39	0.32	0.36	0.31	0.27	0.31/ 4%	0.3
		2,3	0.62	0.57	0.59	0.56	0.53	0.56 / 4%	0.6
		2,4	1.0	1.0	1.0	1.0	1.0	1.0 / 0%	1.0

Table 8

 8 Long-term experimental properties vs. Eurocode 5 and Eurocode 1 (Service class no. 1)

The authors would like to thank the AdivBois association 'Association for the Development of Timber Living Buildings' for funding this study. This work was carried out within the framework of the study on high-rise timber buildings.


Boucaud-Maitre Denis
email: denis.boucaud@gmail.com

Health trajectories of older persons living in French

Keywords: Senior housing, mortality, nursing home, health trajectories Senior housing, mortality, nursing home, health trajectories

Senior housing for older adults could be an alternative or a transitional care model between home care and nursing home care. Using two longitudinal cohorts of community dwellers aged 65 years or older, we compared risks of mortality and of nursing homes admission between older adults who did or did not move to senior housing over time. In the 3C study (n=2104, 17 years of follow-up), 143 (6.8%) participants moved into a senior housing during the follow-up. This move was associated with a lower risk of mortality (hazard ratio (HR): 0.64; 95% confidence interval (CI) 0.46 to 0.77) and a higher risk of nursing home admissions (HR: 1.54 (1.10-2.15)). The risks of hospitalizations (HR: 0.54 (0.40-0.73)) and falls (HR: 0.63 (0.50 to 0.79)) were lower. In the PAQUID study (n=3777, 27 years of follow-up), 161 (4.3%) participants moved into a senior housing. This move was also associated with a lower mortality risk (HR: 0.72 (0.58 to 0.88)) and a higher risk of nursing home admissions (HR:

1.39 (1.05-1.86)). Our results showing lower risks of mortality suggest that senior housing may be a relevant model for vulnerable older adults.

Introduction

The aging of the population in developed countries poses significant challenges for health policy worldwide. Finding the appropriate care option for dependent or vulnerable older adults remains difficult [START_REF] Araujo De Carvalho | Organizing integrated health-care services to meet older people's needs[END_REF]. The WHO, within the framework of Decade of Healthy Ageing, promotes the integration of care and services that meet the individual needs of older people and access to long-term care [START_REF] Cesari | Implementing care for healthy ageing[END_REF]. To date, the most widespread social model revolves around home care and nursing home carefor people with or at risk of unmet home care needs. It has been previously reported that the majority of people aged 50 and over would like to age in their current home (85% according to a French national survey institute (3)), regardless of their functional status. Several services promote aging in place for this purpose (remote monitoring, telemedicine, home care services) [START_REF] Tappenden | The clinical effectiveness and costeffectiveness of home-based, nurse-led health promotion for older people: a systematic review[END_REF][START_REF] Coster | What is the impact of professional nursing on patients' outcomes globally? An overview of research evidence[END_REF]. Nonetheless, social isolation, social vulnerability, and loneliness among older people living at home [START_REF] Freedman | Social isolation and loneliness: the new geriatric giants: Approach for primary care[END_REF] increase the risk of adverse health events; they are respectively associated with a 29%, 26%, and 32% increase in mortality risks [START_REF] Holt-Lunstad | Loneliness and social isolation as risk factors for mortality: a meta-analytic review[END_REF]. On the other hand, voluntary or involuntary nursing home admission is often the result of a deterioration in health caused mainly by falls, cognitive impairment, dependence in daily activities, or strokes [START_REF] Vejux | Repérage des facteurs influençant le risque d'admission en établissement médico-social [Screening risk and protective factors of nursing home admission[END_REF][START_REF] Luppa | Prediction of institutionalization in the elderly. A systematic review[END_REF]. Nursing homes are the last place of residence as well as the place of death for most institutionalized older people, after an average length of stay of three years and four months in France [START_REF] Muller | Direction de la Recherche[END_REF].

The transition from one's home to a nursing home usually disrupts residents' social life and lifestyle [START_REF] Groenvynck | Interventions to Improve the Transition From Home to a Nursing Home: A Scoping Review[END_REF]. Institutionalized older adults experience greater cognitive decline than community-dwelling individuals [START_REF] Harmand | Cognitive decline after entering a nursing home: a 22-year follow-up study of institutionalized and noninstitutionalized elderly people[END_REF], a decreased in quality of life [START_REF] Villeneuve | Trajectory of Quality of Life Before and After Entering a Nursing Home: A Longitudinal Study[END_REF] and increased psychotropic medication use [START_REF] Atramont | Health status and drug use 1 year before and 1 year after skilled nursing home admission during the first quarter of 2013 in France: a study based on the French National Health Insurance Information System[END_REF] after moving into nursing home. For these reasons, we need to study alternative and transitional care models between home care and nursing home care. Among these models, "senior housing" (also called "residential care facilities ", "independent living communities", "assisted living facilities", or "continual care retirement communities") have been developed over the last decades. Usually, each resident has a private apartment and access to common areas and services. Senior housing differ in size, type (residence or village), services offered, and costs. These facilities aim to promote the autonomy of older people and help them maintain an active social life [START_REF] Jolanki | Senior Housing as a Living Environment That Supports Well-Being in Old Age[END_REF]. In the United States, between 980,000 and 2 million people live in these community-based residential care settings, depending on the definition we use [START_REF] Toth | US Department of Health and Human Services. Understanding the characteristics of older adults in different residential settings: data sources and trends[END_REF]. In France, there are about 2,300 residences of this type (representing approximately 120,000 accommodation places), called "Residences for autonomous seniors" accommodating an average of 48 residents each. Two-thirds are public, 30% are private nonprofit and 4% are commercial. These structures have a social vocation, they receive funding and the rents are moderate. External service providers (private doctors, home nursing services, private nurses) most often provide medical care and routine care for residents. In some senior housing, there are medical staff on site, such as care assistants or nurses. Staffing levels are lower in senior housing than in nursing homes, with a staff ratio of 13 per 100 beds compared to 64 per 100 beds in nursing homes [START_REF] Letenneur | Independent Living Communities: A Privileged but Neglected Place for Secondary Prevention in Ageing[END_REF]. The mean age at admission is 81 years, and the mean length of stay is five years [START_REF] Toth | US Department of Health and Human Services. Understanding the characteristics of older adults in different residential settings: data sources and trends[END_REF]. Most residents have no disability, but more than half meet the criteria for frailty (53.7%) and suffer from depressive disorders (53.5%) [START_REF] Kerimel | Seniors living in residential homes: a target population to implement ICOPE (Integrated care for older people) program in primary care[END_REF].

Few studies have assessed the effectiveness of senior housing in preventing adverse health events (deaths or hospitalizations) compared to home care. Available studies [START_REF] Inacio | Mortality in the first year of aged care services in Australia[END_REF][START_REF] Thomas | State Variability in the Prevalence and Healthcare Utilization of Assisted Living Residents with Dementia[END_REF][START_REF] Bartley | Health Outcomes From Assisted Living Facilities: A Cohort Study of a Primary Care Practice[END_REF][START_REF] Mccann | A Census-based longitudinal study of variations in survival amongst residents of nursing and residential homes in Northern Ireland[END_REF] suggest a higher risk of mortality in senior housing compared with home care or community. However, these studies are of short duration (1-5 years of follow-up) and the baseline medical characteristics of the older adults were not comparable between these two populations. Regarding hospitalization, one longitudinal study observed a decrease for heavy hospital use over time for older people living in senior housing compared with older people remaining at home [START_REF] Park | The Role of Senior Housing in Hospitalizations Among Vulnerable Older Adults With Multiple Chronic Conditions: A Longitudinal Perspective[END_REF].

The choice to invest in a senior housing is an important issue for a village or a town. Thus, assessing the relevance of this investment is crucial for the community. Research is necessary to determine whether this type of geriatric facility can be an effective alternative to home care and promote higher survival rates. The present study uses data from two population-based cohorts (3C and PAQUID) of older people living at home initially and followed over time, contrary to previous studies [START_REF] Kerimel | Seniors living in residential homes: a target population to implement ICOPE (Integrated care for older people) program in primary care[END_REF][START_REF] Inacio | Mortality in the first year of aged care services in Australia[END_REF][START_REF] Thomas | State Variability in the Prevalence and Healthcare Utilization of Assisted Living Residents with Dementia[END_REF][START_REF] Bartley | Health Outcomes From Assisted Living Facilities: A Cohort Study of a Primary Care Practice[END_REF]. Its primary objective was to compare mortality and risks of nursing home admission between older people who moved to a senior housing during follow-up and older people who did not moved into senior housing. Secondary objectives were to compare the risk of hospitalization, fall, and frailty that might explain a potential difference in survival, using data from the 3C cohort.

Materials and Methods

The 3C study

The 3C Study is an on-going multicenter prospective cohort study conducted in France. Its primary objective was to assess the relationship between vascular factors and dementia in people aged 65 years and over. The detailed study protocol has been published elsewhere [START_REF] Lenoir | Depression history, depressive symptoms, and incident dementia: the 3C Study[END_REF]. Briefly, noninstitutionalized men and women aged 65 years or over were randomly selected from electoral rolls of three French cities. The acceptance rate was 37%, resulting in 9294 participants (4931 from Dijon, 2104 from Bordeaux, and 2259 from Montpellier). Participants were followed-up every two to three years (7 follow-ups) for 17 years, even if they entered a senior housing or a nursing home. The Bordeaux sample provided information on vital status at baseline (1999)(2000) and at each follow-up (n=2104).

The number of participants lost to follow-up in this sample was 277 at 17 years. By the end of the last follow-up, 1257 participants had died.

Data from the 3C Study

Trained nurses and psychologists collected the data during face-to-face interviews using standardized questionnaires. For the present study, the following variables were extracted at baseline and each follow-up: sociodemographic characteristics (gender, age, and educational level), transition to a nursing home or a senior housing, vital status, hospitalizations, falls, global cognitive performance, Frailty was assessed using the five components of the phenotype proposed by Fried et al. [START_REF] Fried | Frailty in older adults: evidence for a phenotype[END_REF]. Weight loss was assessed in a self-reported manner (recent and unintentional weight loss of 3 kg or more or a body mass index (BMI, calculated from anthropometrical measurements) lower than 21 kg/m2).

Participants assessed their level of exhaustion with two questions also used in the Cardiovascular

Health Study: ''I felt that everything I did was an effort'' and ''I could not get going.''. Slowness was defined as the lowest quintile on a timed 6-meter walking test, at usual pace, adjusted for gender and height, among participants of this study. Participants answering ''yes'' to the question ''Do you have difficulty rising from a chair?'' were categorized as frail for weakness. Low physical activity corresponded to performing daily leisure activities (such as walking, gardening, or exercising) less than once a week. For the present study, participants meeting three or more criteria were considered frail and the others non-frail.

Validation analysis in the PAQUID study

The PAQUID cohort started in 1988-1989 with a representative sample of 3777 participants aged 65 years and over and living at home in two French departments (Gironde and Dordogne) [START_REF] Dartigues | The Paquid epidemiological program on brain ageing[END_REF]. The selection was stratified by sex, age, and urban unit's size. Face-to-face interviews were conducted at home every 2-3 years by specially trained neuropsychologists. Sociodemographic, environmental, and health-related data,including entry into a nursing home or a senior housing, were collected prospectively at each wave with the participant or a proxy, when self-assessment was impossible or invalid. The PAQUID program included a systematic and regular record and check of deaths with the administrative data from death certificates obtained from general practitioners and the city administrative department (date and cause of death). Data on hospitalizations, falls, and frailty were unavailable for this cohort. The number of older people lost to follow-up in this cohort was 98. By the end of the study, 3665 participants had died.

Statistical analysis

Descriptive analyses were conducted using frequencies and percentages for categorical variables and means and standard deviations (SD) for continuous variables.

In our analyses, nursing home admission and death were considered as competing risks in a semiparametric illness-death model accounting for left-truncation and right censoring. We used a semiparametric approach with M-splines approximation of baseline intensities in order to obtain smooth estimates of the hazard functions [START_REF] Joly | A penalized likelihood approach for an illnessdeath model with interval-censored data: application to age-specific incidence of dementia[END_REF]. This model allowed us to make predictions and obtaining the following probabilities:

 The probability of being alive without entering a nursing home  The probability of being alive and entering a nursing home

 The probability of dying

We also presented the predictions in two subgroups: people who never moved to a senior housing and people who did so during follow-up. We performed univariate and multivariate models taking age as the baseline time and adjusted a priori for gender, educational level, and IADL disability (at baseline).

The exponential of each regression coefficient can be interpreted as a hazard ratio (HR) like in Cox models. All regression coefficients were estimated simultaneously using the Smooth Hazard R package [START_REF] Touraine | SmoothHazard: An R Package for Fitting Regression Models to Interval-Censored Observations of Illness-Death Models[END_REF].

Cox proportional hazard models with delayed entry were performed to estimate the risks of hospitalizations, falls, and frailty, using data from the 3C Study. Models were adjusted for gender, educational level, and IADL disability (at baseline).

Ethics declarations: The Ethical Committee of the University Hospital of Kremlin-Bicêtre (Paris) has approved the protocol of the 3C study. Each participant signed a consent form. The ethical committee of the Bordeaux University Teaching Hospital has approved the PAQUID study according to the principles embodied in the declaration of Helsinki in1988. The study investigators informed all participants and their proxies of the ongoing research activity. They were free to accept or refuse to participate. Participants provided written informed consent.

Results

Characteristics of the 3C study sample

Table 1 shows the baseline characteristics of the 3C sample. The average age of the 2104 participants was 74.6 years old (SD: 5.3), and 61.2% were women. During the 17-year follow-up, 143 (6.8%) participants moved into a senior housing.

At the time of admission into a senior housing, the mean age of the 143 participants was 82.8 years old and most of them were women (80.4%). Their mean MMSE score was 25.9 and 48.6% were dependent in at least one IADL (table 1).

Effects of moving into a senior housing on the risks of death and nursing home admission

In all three models (1. univariate, 2. adjusted for sex and education level, and 3. adjusted for sex, education level, and IADL score), moving into a senior housing was associated with a decreased mortality risk over time compared to older adults who did not move into a senior housing (table 2 and figure 1).

Of the 143 participants who moved into a senior housing during follow-up, 46 (32.2%) subsequently moved into a nursing home. They were 255 (13.0%) among those who never lived in senior housing.

In all three models, having lived in a senior housing was associated with an increased risk of nursing home admission (table 2).

Having lived in senior housing did not affect the risk of death after nursing home entry in all three models (Table 2). Figure 1 shows the probabilities of death and nursing home admission at each follow-up according to whether the participants moved into a senior housing during follow-up. The probability of dying was higher for older persons who did not move into a senior housing. In contrast, the probability of being still alive without entering a nursing home was higher for those who went through a senior housing. Risks of hospitalization, falls and frailty.

Among the 143 participants who moved into senior housing, 88 (61.5%) were hospitalized during follow-up, compared to 1389 (70.8%) among those who never moved into senior housing. In all three models, having moved into a senior housing was associated with a decreased risk of hospitalization and falls (Table 3).

For non-frail older adults, having moved to a senior housing was associated with a decreased risk of frailty incidence only in the univariate model (table 3). The HR estimates were similar but no more significant when we controlled for gender and educational level. Of the 161 participants who moved into a senior housing during follow-up, 51 (31.7%) subsequently entered a nursing home. In all three models (1. univariate, 2. adjusted for sex and education level, 3.

adjusted for sex, education level, and IADL score), moving into a senior housing was associated with a decreased risk of death and an increased risk of nursing home admission (table 5) compared older adults who did not use senior housing. Having lived in a senior housing did not affect the risk of death once individuals entered a nursing home (table 5).

Discussion

With ageing of the population, the demand for long-term services and support for older people is rising. When staying at home becomes difficult (due to deteriorating health, loss of autonomy, absence or exhaustion of caregivers, loneliness, unsuitable housing…), it may be necessary to look for and assess alternatives, such as senior housing or foster families [START_REF] Boucaud-Maitre | Foster families to support older people with dependency: a neglected strategy[END_REF]. In these two independent cohorts investigated in this study, we observed that mortality was reduced by about 30% in older adults who had moved into senior housing compared to their counterparts who had not. Moreover, the risk of hospitalization was also reduced by half, as previously described in a study by Park et al [START_REF] Park | The Role of Senior Housing in Hospitalizations Among Vulnerable Older Adults With Multiple Chronic Conditions: A Longitudinal Perspective[END_REF] (even though to a lesser extent). Finally, we observed a decreased risk of falling that is associated with a higher risk of death in the scientific literature [START_REF] Salari | Global prevalence of falls in the older adults: a comprehensive systematic review and meta-analysis[END_REF].

Previous cohort studies have suggested an increased risk of mortality in senior housing compared with communities in US [START_REF] Thomas | State Variability in the Prevalence and Healthcare Utilization of Assisted Living Residents with Dementia[END_REF][START_REF] Bartley | Health Outcomes From Assisted Living Facilities: A Cohort Study of a Primary Care Practice[END_REF], Australia [START_REF] Inacio | Mortality in the first year of aged care services in Australia[END_REF] or Ireland [START_REF] Mccann | A Census-based longitudinal study of variations in survival amongst residents of nursing and residential homes in Northern Ireland[END_REF]. Nevertheless, the longitudinal design of our study, with older people initially in their homes, provides a better knowledge on the consequences of entry into senior housing in a life-course perspective than these other short-term studies (1-5 years of follow-up). We noted that the clinical characteristics of older adults living in French senior housing are probably different compared to other countries. In the United States or Canada, the characteristics of older people in senior housing are closer to those of nursing home residents, since a majority of older adults suffers from dementia (58% in the Canadian study by Maxwell et al. (31) or 68% of individuals in the American study by Watson et al. (32)). In our cohorts, the mean age of older adults entering a senior housing was 82.8 years at admission and the mean MMSE score was 25.2 ± 3.3.

Therefore, whereas senior housing is often considered in other countries as an alternative to nursing homes, our results suggest that senior housing in France is rather a stepping-stone between home care and nursing home care. Indeed, compared to older adults who did not move into a senior housing, the health trajectory of older adults who lived at home and then moved to a senior housing puts them at increased risk of nursing home admission. Indeed, about 30% of the residents of senior housing moved to a nursing home, likely due to cognitive or functional decline.

In France, the target population for senior housing is rather older people who are rather frail (or at risk of frailty) and for whom their housing is no longer suitable [START_REF] Letenneur | Independent Living Communities: A Privileged but Neglected Place for Secondary Prevention in Ageing[END_REF]. A senior housing offers many potential advantages: safe housing (no stairs, adapted lightning, especially at night, bathrooms and toilets designed to avoid the risk of falling), personal support, and health care, along with the facilitation of social interactions. Senior housing could promote autonomy of older adultsthe primary concern in this populationand help manage the health trajectory of vulnerable older people.

We have observed that senior housing reduce mortality risks, often the primary concerns of families and health professionals [START_REF] Kane | Finding the right level of posthospital care: "We didn't realize there was any other option for him[END_REF]. For practical and logistical reasons, senior housing may be a more appropriate setting for secondary prevention than the general population, where frail older people are scattered. The array of medical and support services in senior housing may help adapt to the constraints and needs of older adults to promote lifestyle-based group interventions such as diet, physical activity, and social interactions [START_REF] Mihalko | Promoting physical activity in independent living communities[END_REF]. Finally, staffing levels in these facilities, particularly nursing staff, are thought to play a pivotal role in preventing adverse health events. Stearns et al. [START_REF] Stearns | Determinants and effects of nurse staffing intensity and skill mix in residential care/assisted living settings[END_REF] found an association between reduced rates of hospital admissions and a higher number of licensed staff in senior housing.

In this study, several limitations are noteworthy. The sample size was small for older adults who used senior housing. Nevertheless, the results are consistent across the two independent cohorts we investigated. The demographic characteristics of the participants were similar to those of a recent French cohort living in senior housing [START_REF] Kerimel | Seniors living in residential homes: a target population to implement ICOPE (Integrated care for older people) program in primary care[END_REF]. As stated above, the clinical characteristics of older people in the French senior housing are different from those observed in other countries, which limits comparisons with the scientific literature. We hypothesize that housing security, improved care, and support management may explain the lower mortality observed in the senior housing group.

Nevertheless, other factors could have been studied, such as medication use, marital status or cognitive decline [START_REF] Van Patten | Rates of Cognitive and Functional Impairments in Older Adults Residing in a Continuing Care Senior Housing Community[END_REF]. Moreover, our cohorts do not include systematically collected information on patient characteristics, such as previous hospitalizations and comorbidities, which are often the main causes of admission into an institution and are significant predictors of mortality. Further research would be necessary to understand the most appropriate profile of senior housing residents, mainly in terms of cognitive functioning, and to determine the factors leading to nursing home admission in people living in senior housing [START_REF] Van Patten | Rates of Cognitive and Functional Impairments in Older Adults Residing in a Continuing Care Senior Housing Community[END_REF]. This transition may happen too late, as senior housing and nursing homes still suffer from a pejorative representation in the general population. Finally, from a public health perspective, medico-economic studies should assess the cost-efficiency of senior housing.

Conclusion

Our results support the view that senior housing as appropriate transitional care option for older people when living at home becomes challenging. In the two independent cohorts investigated in this study, we observed lower risks of mortality and hospitalization in older adults who moved into senior housing compared to their counterparts who did not. We need to know more of the benefits of these facilities and the characteristics of older adults who might benefit most from them.

  and functional status. The Mini-Mental State Examination (MMSE) score, ranging from 0 to 30, measured global cognitive functioning. The Instrumental Activities of Daily Living (IADL) scale (Lawton's IADL scale) and the Activities of Daily Living (ADL) scale (Katz's scale) assessed functional status. Depression was assessed with the CES-D (Center for Epidemiologic Studies-Depression) scale. Falls were self-reported.

Figure 1 :

 1 Figure 1: Estimated probabilities of death, nursing home admission, and being alive according to residential status (no senior housing in blue versus senior housing in red)

Baseline characteristics of the participants of 3C study Characteristics of the residents at the time of entry into senior housing

	Variables	n	Mean (SD) or n (%)	N Total	Mean (SD) or N (%)
	Age (years)	2104	74.6 (5.1)	143	82.8 (5.9)
	Gender (women)	2104	1288 (61.2%)	143	115 (80.4%)
	High educational level	2099	1837 (87.5%)	143	119 (83.2%)
	MMSE score	2081	27.2 (2.4)	139	25.9 (3.3)
	Instrumental
	Activities of Daily Living (at least >	2095	296 (14.1%)	142	69 (48.6%)
	1IADL)
	Activities of Daily
	Living (at least > 1ADL)	2102	30 (1.4%)	143	14 (9.8%)
	CESD score	2054	7.5. (7.7)	121	9.8 (8.6)

Table 1 : Baseline characteristics of participants of 3C study and characteristics of the residents at the time of entry into senior housing

 1

Table 2 : Estimated effects of moving into senior housing on the risks of mortality and nursing home admission, illness- death model. 3C study (n=2095).

 2

				Model 2 (Adjusted by	Model 3 (Adjusted by
		Model 1 (Univariate)	gende rand education	gender, education level
				level)		and IADL score)
	Transition	HR (CI 95%)	p	HR (CI 95%)	p	HR (CI 95%)	p
	Death	0.59 (0.46-0.77) <0.001 0.65 (0.50-0.85)	0.001	0.64 (0.49-0.83) <0.001
	Transition to nursing home	1.63 (1.17-2.27)	0.004	1.58 (1.13-2.20)	0.007	1.54 (1.10-2.15)	0.011
	Death after nursing home admission	0.94 (0.63-1.38)	0.739	0.96 (0.65-0.1.43)	0.843	0.96 (0.65-1.43)	0.845

Table 3 : Estimation of the effects of a moving into a senior housing on the risks of hospitalizations, falls, and frailty, Cox model. 3C study (n=2095).

 3

				Model 2 (Adjusted by	Model 3 (Adjusted by
		Model 1 (Univariate)	gender and education	gender, education level and
				level)		IADL score)
	Variables	HR (CI 95%)	p	HR (CI 95%)	p	HR (CI 95%)	p
	Hospitalization	0.52 (0.39-0.70)	<0.001 0.53 (0.39-0.71)	<0.001	0.54 (0.40-0.73)	<0.001
	Fall	0.64 (0.51-0.80)	<0.001 0.64 (0.51-0.80)	<0.001	0.63 (0.50-0.79)	<0.001
	Frailty	0.60 (0.42-0.86)	0.005	0.65 (0.40-1.06)	0.085	0.63 (0.39-1.05)	0.074

Validation analysis in the PAQUID study

Characteristics of the PAQUID study sample Table

1

presents the baseline characteristics of the study sample from the PAQUID study population.

The mean age of the 3777 participants was 75.5 years old at baseline (SD: 6.9), and 58.2% of them were women. During the 30-year follow-up, 161 (4.3%) participants moved into a senior housing. At the time of entry into senior housing, the participants were 82.6 (SD: 6.6) years old, predominantly female (78.9%), and 63.8% were dependent in at least one IADL (table

4

). On average, residents remained in senior housing for 3.38 ± 4.55 years.

Baseline

characteristics of the participants of PAQUID study Characteristics of the residents at the time of entry into senior housing

	Variables	n	Mean (SD) or n (%)	N Total	Mean (SD) or N (%)
	Age (years)	3777	75.5 (6.9)	161	82.6 (6.6)
	Gender (women)	3777	2200 (58.2%)	161	127 (78.9%)
	High educational level	3777	2435 (64.5%)	161	93 (57.8%)
	MMSE score	3697	25.5 (4.1)	157	24.5 (4.5)
	Instrumental
	Activities of Daily Living (at least >	3770	1148 (30.5%)	160	102 (63.8%)
	1IADL)
	Activities of Daily
	Living (at least > 1ADL)	3772	182 (4.8%)	161	18 (11.2%)
	CESD score	3628	10,0 (9.2)	149	12.8 (10.7)

Table 4 : Baseline characteristics of participants of PAQUID study and characteristics of the residents at the time of entry into senior housing

 4

Table 5 :

 5 Estimated effects of the move into senior housing on the risks of mortality and nursing home admission, illnessdeath model. PAQUID cohort (n=3710).

				Model 2 (Adjusted by	Model 3 (Adjusted by
		Model 1 (Univariate)	gender and education	gender, education level
				level)		and IADL score)
	Transition	HR (CI 95%)	p	HR (CI 95%)	p	HR (CI 95%)	P
	Death	0.63 (0.51-0.77) <0.001 0.68 (0.56-0.84)	<0.001	0.72 (0.58-0.88)	0.002
	Transition to nursing home	1.46 (1.10-1.94)	0.009	1.34 (1.00-1.78)	0.047	1.39 (1.05-1.86)	0.023
	Death after entry in nursing home	0.94 (0.70-1.27)	0.693	0.98 (0.69-1.40)	0.943	1.05 (0.66-1.54)	0.980

Data availability:

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Author contributions: JFD conceived the study. JFD, HA, LL, MTT and DBM designed the study. CM extracted and analyzed data. DBM wrote the first draft, which all authors revised for critical content. All authors approved the final manuscript. DBM is the guarantor. The corresponding author attests that all listed authors meet authorship criteria and that no others meeting the criteria have been omitted.

Competing interests: No conflicts of interest.

All authors declare no support from any organization for the submitted work; no financial relationships with any organizations that might have an interest in the submitted work in the previous three years; no other relationships or activities that could have influenced the submitted work.

Funding source: None


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  Modulation of gut microbiota by antibiotics did not affect anhedonia in a high-fat diet-induced model of depression in male mice Magali Monnoye, Pauline Flauss, Catherine Philippe, Nathalie Castanon, Sylvie Rabot, Sylvie Vancassel, Laurent Naudon To cite this version: Magali Monnoye, Pauline Flauss, Catherine Philippe, Nathalie Castanon, Sylvie Rabot, et al.. Modulation of gut microbiota by antibiotics did not affect anhedonia in a high-fat diet-induced model of depression in male mice. FENS Forum of Neuroscience, Jul 2022, Paris, France. hal-04093340

POSTER SESSION 03 -SECTION: GUT-BRAIN AXIS AND THE MODULATION OF APPETITE

Magali Monnoye 1 , Pauline Flauss 1 , Catherine Philippe 1 , Nathalie Castanon 2 , Sylvie Rabot 1 , Sylvie Vancassel 2 , Laurent Naudon 1 1 INRAE, Micalis, Jouy en Josas, France, 2 INRAE, Nutrineuro, Bordeaux, France Long-term consumption of a high-fat diet (HFD) causes obesity and is a risk factor for depression. HFD has a significant impact on the gut microbiota, and dysbiosis of the microbiota is now associated with certain psychiatric disorders such as anxiety and depression. We aimed to investigate whether modulation by antibiotic treatment of the composition of the gut microbiota in diet-induced obese (DIO) mice has an impact on depressive-like behavior. In this study, we analyzed the effects of a 15 weeks HFD consumption by male mice on their depressive-like behavior assessed in the forced swim and sucrose preference tests. Two weeks before beginning the behavioral tests, a group of DIO mice were given a combination of two non-absorbable antibiotics, neomycin and polymyxin B. HFD induced anhedonia, as revealed by the sucrose preference test, and significant changes in gut microbiota composition at the phyla and family levels. On the other hand, there was no significant effect of HFD on the peripheral and brain inflammatory profiles. In DIO mice treated with the antibiotics, an even more pronounced alteration in the composition of the gut microbiota occurred, without any change in anhedonia behavior. Only four families of bacteria were not affected in their abundance by the antibiotic treatment, the Rikenellaceae, Streptococcaceae, Erysipelotrichaceae and Bifidobacteriaceae. This stability concomitant with that of anhedonia may suggest that these families were involved in this depression-like behavior.


G Brezeanu

P Godignon

E Dimitrova

C Raynaud

D Planson

A Mihaila

F Udrea

J Millan

G Amaratunga

C Boianceanu

Towards the Fabrication and Measurement of High Sensitivity SiC-UV Detectors with Oxide Ramp Termination

Keywords: UV detector, Oxide ramp termination, OBIC measurements, Medici simulation

The paper is focused on the simulation, fabrication and measurement of the high UV detection performances 6H-SiC pn photodiodes based on an efficient planar termination. The oxide ramp technology used for these photo-detectors is extensively presented. The OBIC technique was used for optical measurements of the samples. The impact of diode's bias as well as their photo-response on some incident light power values has been revealed by measurements and numerical simulations. The measurement shows a stronger diode's bias effect on its optical properties than the simulation data.

Introduction

Wide bandgap semiconductors are desirable for UV optical detection applications because of their insensitivity to longer wavelengths and their very small dark current levels, even at elevated temperatures. Low dark current enables SiC photodiodes to exceed Si UV detector sensitivities by 4 orders of magnitude [1,2]. Many of the applications for UV detection involve hostile environments such as in situ combustion monitoring and satellite based missile plume detection, where the ruggedness of SiC is an important advantage. Other applications capitalize on the sensitivity of wide bandgap semiconductor detectors, such as air quality monitoring, gas sensing and personal UV exposure dosimeter. Among the wide bandgap semiconductors, SiC is once again leading the way, being the first to reach the marketplace [1,2]. The photo-detectors are, usually, Schottky or pn junction diodes reverse biased. For SiC diodes we had previously proposed a simple planar edge termination based on a field plate overlapping on oxide ramp at the periphery of the contact. We have proved experimentally the use of this termination on 4H-SiC and 6H-SiC for high voltage Schottky barrier diodes and Schottky UV photo-detectors, respectively [3]. In this paper we use the oxide ramp edge termination for pn shallow junction barrier photodiodes. Extensive measurements based on OBIC technique have proved the high efficiency of SiC photo-diodes with oxide ramp profile. Medici simulations have also been carried out to demonstrate the excellent UV detection sensitivity of the fabricated SiC samples.

Structure

The pn junction photo-detectors, with oxide ramp termination, were realized on nn + 6H-SiC wafer. The n + substrate has a typical resistivity of 0.03Ωcm. The parameters of n epitaxial layer are x e =8µm and N d =8.5×10 15 cm -3 . The main technological challenge of the targeted SiC diode is fabrication of an efficient oxide ramp profile [3]. A field plate termination is achieved by anode metal overlapping of the oxide ramp region. In order to obtain different oxide ramp angles various combinations of oxide layers have been deposited on test samples. Two oxide ramp profiles, shown in Fig. 1, have been obtained. The first one is a typical ramp oxide termination, (Fig. 1a) while in the second one, the oxide profile, has two ramps (Fig. 1b) and corresponds to a two level field plate configuration. This profile, which has been obtained using only a photolithographic process, is also very attractive termination. For pn photo-detectors two insulating layers comprised of an undoped oxide layer and an 8% phosphorus doped oxide layer, with a total thickness of 1.1µm, were grown onto the epilayer. Then two steps wet etching were used: an initial etching in a standard oxide etch solution followed by an over-etching in a P-etch solution. An oxide ramp termination as in Fig. 1a was achieved. The oxide ramp angle is smaller than 4°. A layout and a cross section of pn detectors with oxide ramp termination are shown in Fig. 2. After oxidation, Ni e-gun evaporation is performed on the heavily doped backside substrate and annealed at 1000 ° C to obtain a good ohmic contact (M3 in Fig. 2.). The p + layer with a surface doping concentration of about 10 19 cm -3 and a thickness of 0.15 µm, was formed by Al implantation. 140Å platinum was deposited and patterned by lift-off as the anode ohmic contact on p + layer. Platinum (M1 in Fig. 2) is a transparency metal for the photon flux. In order to achieve the same surface potential two layers of Ti (1000 Å) and Al (1µm) were deposited on front-side (M2 in Fig. 2). An antireflection coating (which covers M1) is used to reduce the absorption losses.

Measurements

The SiC photo-detectors, with oxide ramp termination, have been optical tested using the OBIC technique. For these measurements, an Ar ++ laser, which generates a UV fascicle with a wavelength in the range from 300 nm to 363.8 nm, is focused on the device and is modulated by a chopper of 70 Hz. The measurements have been performed at room temperature at several wavelengths, from the above mention interval, which ought to correspond to a light penetration depth in the 4H-SiC a little bigger the 11 µm. A synchronic sensor detects the local variations of the photocurrent induced by the incident flux, which has a diameter of 4 µm, on the device. A computer controls 2 motors, which assure the left to right and up to down movement (with a 3 µm step) of the test sample under the laser beam, and registers the measured current as a function of the position of the UV fascicle, respectively. The output power of the laser beam was 10mW (which corresponds at an incident power density of 0.76 W/cm 2 ) or 20 mW.

Results. Discussion

Fig. 3 presents the measured photocurrent dependence as a function of UV laser beam position on the sample. A current peak appears on the two sides of the sample at any bias and wavelength, respectively. The right peak is always higher than the left peak. This asymmetry can be due to a difference in the surface planarity of the sample. When the wavelength is increased, the left peak becomes higher. The peak also rises with the reverse bias of the photo-detector and with incident laser beam power (Fig. 3a.). Normally, the current peak is multiplied by a factor of 2, for an increase of the incident light power from 10 to 20mW (Fig. 3b.).

JBD-DC light-300K V(K)=500V V(K)=100V V(K)= 50V V(K)= 10V V(K)= 5V Photocurrent Density, J ph (A/cm 2 )
Wavelength,λ (nm) (b) Fig. 4

(a) Measured and (b) simulated photocurrent vs. wavelength for different reverse biases

A comparison of measured and Medici simulated data [3] of the photocurrent density at different light wavelengths and biases are shown in Figs. 4. We can observe a significant increase of the photocurrent density at high voltage detector bias. Note, that the measurement data shows a higher increase than the one predicted by the simulations. The differences between the measured and Medici simulated data for photocurrent density are due to different power densities use in measurements and simulations, respectively. Also, note that the simulated data (Fig. 4b) corresponds to a uniform light illumination of the sample.

Conclusions

A simple termination of the planar pn junction structure was proposed and experimented on 6H-SiC photodiodes. This technique, which is based on oxide etching under small angles around anode contact window, assured a field plate surrounding the anode contact window. Extensive investigation have shown that an oxide with the thickness more than 1µm and ramp oxide profile smaller than 4 o are recommended for a efficient edge termination. We demonstrated that this termination is highly suitable for UV photo-detectors. The response at a pulse light excitation has been investigation. Measurements based on OBIC method have been performed at different biases of the device and various wavelengths and power of the incident laser.

The photocurrent density increases with reverse bias and light power level and is almost flat over the spectral range from 300 to 360nm. The measured reverse voltage dependence is higher than the one predicted by the simulations. This photocurrent behavior was predicted by Medici simulations.

The transient response to a very short pulse light excitation has also been investigated. Owing to high photocurrents and high speed pn SiC photodiodes are a strong candidate for transient UV detection.

Fig. 1

 1 Fig. 1 Two types of oxide ramp termination: (a) with a single ramp and (b) with two ramps.

Fig. 2

 2 Fig. 2 Layout and cross-section of Schottky photo-detector structure.

Fig. 3 J

 3 Fig. 3 J ph measured on pn detectors at 363.8 nm: (a) bias and (b) incident power effect

  Figs. 5 show simulated transient response of two photo-detectors for various pulse light durations (from 40 to 400ps). During the pulse light time the photocurrent increases exponentially. It subsequently decreases when the recombination process becomes predominant. At lower doping of n-epilayer the photocurrent density increases and the bias influence is reduced (Fig 5b). The effect of the photodiode bias voltage is more important in this case (Figs. 5) than for a uniform light illumination (Fig.4b). The differences between the simulated photocurrent density (Figs. 5) and the measured ones (Fig.4a) can be explained by the very short pulse time width used in the Medici investigation.