Clotilde Fermanian Kammerer

Patrick Gerard
email: patrick.gerard@math.u-psud.fr

Caroline Lasser
email: classer@ma.tum.de

Clotilde Fermanian-Kammerer

AND Patrick G Érard

Wigner Measure Propagation and Conical Singularity for General Initial Data

Introduction

We consider the Schrödinger equation

(1.1) iε∂ t ψ ε = -ε 2 2 ∆ψ ε + V (x)ψ ε , (t, x) ∈ R × R d , ψ ε |t=0 = ψ ε 0 .
where the potential V (x) displays a conical singularity: there exist two scalarvalued functions w, V 0 ∈ C ∞ (R d , R) and a vector-valued function g ∈ C ∞ (R d , R p ), 0 < p ≤ d such that g = 0 is a system of equations of a codimension p submanifold of R d and

(1.2) ∀x ∈ R d , V (x) = w(x)|g(x)| + V 0 (x).

We suppose that V satisfies Kato conditions (see [START_REF] Gérard | Homogenization limits and Wigner transforms[END_REF]) so that the Schrödinger operator -ε 2 2 ∆ + V (x) is essentially self-adjoint. Moreover, we are concerned with the effects of conical singularities in the potential; therefore, we assume that V 0 and w are smooth. This smoothness assumption can be slightly relaxed as discussed in Remarks 4.1 and 5.1 below.

Assuming that (ψ ε 0 ) ε>0 is uniformly bounded in L 2 (R d ), the families (ψ ε (t)) ε>0 are uniformly bounded in L 2 (R d ) for all t ∈ R and we study the time evolution of their Wigner transforms defined for (x, ξ) ∈ R 2d by

W ε (ψ ε (t))(x, ξ) = (2π) -d R d e iξ•v ψ ε t, x -ε v 2 ψ ε t, x + ε v 2 dv,
and of their Wigner measures µ t , which are the weak limits of W ε (ψ ε (t)) in the space of distributions (see [START_REF] Hörmander | The analysis of linear partial differential operators III. Pseudo-differential operators[END_REF], [START_REF] Kammerer | Wigner measures and codimension two crossings[END_REF] or the book [START_REF] Lions | Sur les mesures de Wigner[END_REF]).

Conical singularities naturally appear for smooth matrix-valued potentials in the context of eigenvalue crossings. In this case, evolution laws were derived for Wigner measures in [7, ?, 8, 6, 17, 9] and normal forms have been obtained in [?, ?]. For single equations displaying conical singularities, transport equations were established in [START_REF] Gérard | Mesures semi-classiques et ondes de Bloch, Séminaire sur les Équations aux Dérivées Partielles[END_REF] in the context of acoustic waves with constant coefficients (see [START_REF] Kato | Schrödinger operators with singular potentials[END_REF] for a review). Here, we are interested in a situation with variable coefficients, where the propagation phenomenon may hit the conical singularity.

Recently, Ambrosio and Figalli [START_REF] Ambrosio | A Figalli : Almost everywhere well-posedness of continuity equations with measure initial data[END_REF] have proposed a new approach to deal with singular potentials more general than ours. The main concept of [START_REF] Ambrosio | A Figalli : Almost everywhere well-posedness of continuity equations with measure initial data[END_REF] is a regular Lagrangian flow on the space of probability measures, which allows to prove a propagation result of Wigner measures along classical trajectories in average with respect to the initial data, see [START_REF] Ambrosio | Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data[END_REF]. A related result was given in [START_REF] Kammerer | A Landau-Zener formula for non-degenerated involutive codimension three crossings[END_REF], where the authors consider mixed states. On the contrary, our aim here is to consider pure states.

In this situation, we need to keep the classical point of view for singularities of the form (1.2). Using the fact that classical trajectories exist and are unique we prove the propagation result for every individual initial data. In particular, we study the case of initial data with Wigner measures which concentrate on these singularities.

Wigner measures have nice geometric properties : they are measures on the cotangent space to R d , that is, on R 2d , and they propagate along classical trajectories as we now recall. Let µ t be a Wigner measure of (ψ ε (t)) ε>0 and define

S = (x, ξ) ∈ R 2d , g(x) = 0 . Since 1 2 |ξ| 2 + V (x) ∈ C ∞ (R 2d \ S)
, it is well-known (see [START_REF] Kammerer | Propagation through generic level crossings: a surface hopping semigroup[END_REF] or [START_REF] Figalli | Semiclassical limit for mixed states with singular and rough potentials to appear in Indiana University Mathematics Journal[END_REF]), that outside S the Wigner measure satisfies the transport equation

(1.3) ∂ t µ t + ∇ x • (ξ µ t ) -∇ ξ • (∇V (x)µ t ) = 0 in D (S c ).
The classical trajectories associated with (1.1) are the Hamiltonian trajectories of the function 1 2 |ξ| 2 + V (x), i.e. the solution curves of the ODE system (1.4) ẋt (x 0 , ξ 0 ) = ξ t (x 0 , ξ 0 ), ξt (x 0 , ξ 0 ) = -∇V (x t (x 0 , ξ 0 )) , subject to the initial conditions

x |t=0 = x 0 , ξ |t=0 = ξ 0 .
For (x 0 , ξ 0 ) / ∈ S, the smoothness of V near x 0 implies the existence and the uniqueness of a local solution of (1.4). We denote by Φ t the flow induced by these trajectories Φ t (x, ξ) = (x t (x, ξ), ξ t (x, ξ)) , for points (x, ξ) / ∈ S and t small enough so that Φ t (x, ξ) / ∈ S. Then, equation (1.3) says, that outside S the measure µ t propagates along the classical trajectories as long as they do not hit S. The transport equation (1.3) comes from the analysis of the Wigner transform and from an Egorov type theorem (see [START_REF] Figalli | Semiclassical limit for mixed states with singular and rough potentials to appear in Indiana University Mathematics Journal[END_REF] or [START_REF] Lions | Sur les mesures de Wigner[END_REF]): for

a ∈ C ∞ 0 (R 2d ) and t ∈ R such that the support of a • Φ -s does not intersect S for all s ∈ [0, t], then (1.5) a, W ε (ψ ε (t)) = a, W ε (ψ ε 0 ) • Φ -t + o(1)
, where the error o(1) turns out to be O(ε 2 ) in this context of smooth coefficients. In this article, we study what happens when classical trajectories attain the set

S * = S ∩ {dg(x)ξ = 0},
and we extend to these points the equations (1.3) and (1.5).

We first prove that the transport equation (1.3) still holds outside S \ S * . Theorem 1.1. There exists a continuous map t → µ t such that µ t is a semiclassical measure of the family (ψ ε (t)) ε>0 . Moreover, µ t (S * ) = 0 for almost every t ∈ R and

(1.6) ∂ t µ t + ∇ x • (ξµ t ) -∇ ξ • (∇V (x)µ t ) = 0 in D (R × (S \ S * ) c ) .
Note that in view of µ t (S * ) = 0 for almost every t ∈ R, for all j ∈ {1, • • • , d}, ∂ xj V (x)µ t is well defined as a measure on R × (S \ S * ) c . Remark 1.2. More precisely, we prove in Section 2 (see Remark 2.2) that the measures µ t satisfy

∂ t µ t + ∇ x • (ξµ t ) -∇ ξ • (∇V (x)µ t ) = ρ where ρ is a distribution supported on R t × (S \ S * ) such that ∃N ∈ N, ∃C > 0, ∀a ∈ C ∞ 0 (R 1+2d t,x,ξ ), | a, ρ | ≤ C sup Rt×(S\S * ) sup |α|≤N w(x)∂ α ξ a .
The main tool at this stage of the proof is provided by two-microlocal Wigner measures, as introduced in [START_REF] Lasser | Propagation through conical crossings: an asymptotic semigroup[END_REF], [START_REF] De Verdière | The level crossing problem in semi-classical analysis I. The symmetric case[END_REF] and [START_REF] De Verdière | The level crossing problem in semi-classical analysis II. The hermitian case[END_REF]. It was already the case in [START_REF] Gérard | Mesures semi-classiques et ondes de Bloch, Séminaire sur les Équations aux Dérivées Partielles[END_REF] in the context of accoustic waves with constant coefficients and in [7, ?, 8, 6, 17] when dealing with matrix-valued potentials presenting eigenvalue crossings.

The points of S * have good properties: for every point (x 0 , ξ 0 ) ∈ S * , there exists a unique classical trajectory passing through it. This fact relies on the observation that if a solution of (1.4)

satisifies (x t , ξ t ) -→ t→0 (x 0 , ξ 0 ) ∈ S * , then g(x t ) |g(x t )| -→ t→0 ± ± dg(x 0 )ξ 0 |dg(x 0 )ξ 0 | =: ω 0 .
These broken trajectories are continuous but they are not C 1 . They allow to uniquely extend the flow Φ t to a continuous map on open sets Ω which intersects S inside S * . This generalized flow is smooth in the variable ξ as stated in the following proposition.

Proposition 1.3. For (x 0 , ξ 0 ) ∈ S * there exists τ 0 > 0 and a unique Lipschitz

continuous map t → (x t (x 0 , ξ 0 ), ξ t (x 0 , ξ 0 )) , t ∈ [-τ 0 , τ 0 ] satisfying (1.4) for t = 0 such that x 0 (x 0 , ξ 0 ) = x 0 , ξ 0 (x 0 , ξ 0 ) = ξ 0 and ẋt (x 0 , ξ 0 ) -→ t→0 ξ 0 , ξt (x 0 , ξ 0 ) -→ t→0 ± -∇V 0 (x 0 ) ∓ w(x 0 ) t dg(x 0 )ω 0 .
Besides, there exists a neighborhood Ω of (x 0 , ξ 0 ) such that Ω ∩ S ⊂ S * and two smooth maps

[-τ 0 , τ 0 ] × (Ω ∩ S * ) : t → Φ t ± (x, ξ) such that ∀(x, ξ) ∈ Ω ∩ S * , Φ t (x, ξ) ±t∈[0,τ0] = Φ t ± (x, ξ) ±t∈[0,τ0] .
Therefore, the flow Φ t extends to a continuous map

t → Φ t (x, ξ), t ∈ [-τ 0 , τ 0 ], (x, ξ) ∈ Ω.
Moreover, for |t| < τ 0 and α ∈ N d , the maps (x, ξ) → ∂ α ξ Φ t (x, ξ) are continuous maps on Ω with bounded locally integrable time derivatives ∂ t ∂ α ξ Φ t (x, ξ).

Remark 1.4. For points of S which are not in S * , one may lose the uniqueness of the trajectory as the example V (x) = -|x| shows: the curves x t = ω t 2 2 and ξ t = ωt satisfy (1.4) for all t and pass through (0, 0) at time t = 0 independently of the choice of the vector ω ∈ S d-1 . It is likely, that these non-unique trajectories induce new phenomena: in particular, the problem could become ill-posed in terms of semi-classical measures, as suggested by the example V (x) = -|x| 3/2 proposed in [START_REF] Athanassoulis | Strong and weak semiclassical limits for some rough Hamiltonians[END_REF]. Proposition 1.3 is proved in Section 6.1 (note that the existence of the broken trajectories was already proved in Proposition 1 of [START_REF] Kammerer | Semiclassical analysis of generic codimension 3 crossings[END_REF]). Note also that the flow Φ t preserves the Liouville measure close to points of S * ; however, besides, as a consequence of Theorem 1.1 we obtain the following Theorem.

Theorem 1.5. If the initial data (ψ ε 0 ) ε>0 has a unique Wigner measure µ 0 and if there exists τ 0 such that for t ∈ [0, τ 0 ] the trajectories Φ t issued from points of the support of µ 0 do not reach S \ S * , then (ψ ε (t)) ε>0 has a unique measure

µ t = (Φ t ) * µ 0 for t ∈ [0, τ 0 ].
Theorem 1.5 is proved in Section 7. Note that the fact that Φ t (x, ξ) is not smooth in (x, ξ) close to S * makes the proof of Theorem 1.5 nontrivial. Besides, we emphasize that Theorem 1.5 holds for initial data µ 0 which can see S * .

Let us now introduce the set A consisting of functions a = a(x, ξ) on R 2d such that, for every α with |α| ≤ d + 1, the function ∂ α ξ a is continuous and

∂ α ξ a(x, ξ) (1 + |ξ|) d+1 -→ (x,ξ)→∞ 0 ,
endowed with the norm

(1.7) M (a) := max |α|≤d+1 sup (x,ξ) ∂ α ξ a(x, ξ) (1 + |ξ|) d+1 .
Notice that this space is a variant of the space introduced by Lions-Paul in [START_REF] Hörmander | The analysis of linear partial differential operators III. Pseudo-differential operators[END_REF]. Then a • Φ t ∈ A for a ∈ A is compactly supported with supp(a) ∩ S ⊂ S * , and one can consider the action of W ε (ψ ε 0 ) on a • Φ t . Since the Wigner transform is convergent for the weak star topology in the dual space of A, Theorem 1.1 implies a weaker version of Egorov's theorem (1.5): For a ∈ C ∞ 0 (R 2d ) and t ∈ R such that the support of a • Φ -s does not intersect S \ S * for all s ∈ [0, t], we have

a, W ε (ψ ε (t)) = a, W ε (ψ ε 0 ) • Φ -t + o(1).
However, we are not able to estimate the convergence rate in full generality. This issue, which is interesting for numerical purpose, will be the subject of further works.

Organization of the paper: The scheme of the proof of Theorem 1.1 is explained in the next Section 2. Then, Section 3 is devoted to the analysis of the time-continuity of the measure µ t , and the transport equation is established in Section 4; the proof of a technical lemma is the subject of Section 5. The analysis of the generalized flow is made in Section 6 where we prove Proposition 1.3 and the computation of the measure µ t stated in Theorem 1.5 is done in Section 7.

Scheme of the proof of Theorem 1.1

Wigner transforms are closely related to pseudodifferential operators via the formula :

∀a ∈ C ∞ 0 (R 2d ), ∀f ∈ L 2 (R d ), a, W ε (f ) = (op ε (a(x, ξ))f, f ) L 2 (R d )
, where the operator op ε (a) is the semi-classical Weyl-quantized pseudodifferential operator of symbol a defined by : ∀f

∈ L 2 (R d ), (2.1) op ε (a(x, ξ))f (x) = (2π) -d R 2d a x + x 2 , ε ξ e iξ•(x-x ) f (x )dx dξ,
see [START_REF] Lions | Sur les mesures de Wigner[END_REF] for example. Besides, by a simple adaptation of Lemma 1.1 in [START_REF] Kammerer | Propagation through generic level crossings: a surface hopping semigroup[END_REF] (see also Lemma 3.1 below and Remark 3.2), one can prove that the operator op ε (a) is uniformly bounded in L 2 (R d ): there exists a constant C > 0 such that for any

a ∈ L 1 loc (R 2d ), we have (2.2) op ε (a) L(L 2 (R d )) ≤ C M (a),
where M (a) has been defined in (1.7). The proof of the Theorem 1.1 consists in three steps.

2.1. First step, existence of the measure. Let T > 0, we prove the existence of a sequence (ε k ), ε k → 0 as k → +∞, and of a continuous map t → µ t from [0, T ] into the set of positive Radon measures such that for all compactly supported

a ∈ A ∀t ∈ [0, T ], op ε k (a)ψ ε k (t) , ψ ε k (t) -→ k→+∞ a(x, ξ)dµ t (x, ξ).
This comes from the fact (proved in Section 3) that there exists a constant C > 0 such that for all a ∈ C ∞ 0 (R 2d ),

(2.3) d dt (op ε (a)ψ ε (t) , ψ ε (t)) ≤ C.
Then, by diagonal extraction, considering a dense family of C ∞ 0 (R 2d ) and using Ascoli's Theorem, we obtain the existence of the sequence (ε k ) and of the associated family of measures µ t . Finally, we extend the convergence to compactly supported symbols a ∈ A by approaching them by

a n ∈ C ∞ 0 (R 2d ) with M (a -a n ) -→ n→+∞ 0.
2.2. Second step, the transport equation. We derive the following equation satisfied by

µ t for t ∈ [0, T ]. Proposition 2.1. There exists a distribution ρ on [0, T ] × S × R d ξ such that (2.4) ∂ t µ t + ∇ x • (ξµ t ) -∇ ξ • ∇V (x)1 g(x) =0 µ t = ρ(t, x, ξ).
Besides, there exists

N ∈ N * and C > 0 such that for all a ∈ C ∞ 0 ([0, T ] × R 2d ), (2.5) | a(t, x, ξ) , ρ | ≤ C sup (t,x,ξ)∈[0,T ]×S×R d sup |α|≤N w(x)∂ α ξ a(t, x, ξ)
where the function w is defined in (1.2). Moreover, if Ω is an open set with µ t 1 S∩Ω = 0, then for all a compactly supported on Ω, a, ρ = 0.

Proposition 2.1 is proved in Section 4; the distribution ρ is defined by use of two-microlocal Wigner measures in the spirit of [START_REF] Lasser | Propagation through conical crossings: an asymptotic semigroup[END_REF], [START_REF] De Verdière | The level crossing problem in semi-classical analysis I. The symmetric case[END_REF] and [START_REF] De Verdière | The level crossing problem in semi-classical analysis II. The hermitian case[END_REF].

2.3. Third step, the measure above the singularity. We now prove (2.6) µ t 1 S * = 0 for almost every t ∈ R. We consider the test function a δ (t, x, ξ) depending on the small parameter δ ∈]0, 1[,

a δ (t, x, ξ) = δ Φ g(x) δ θ(t)b(x, ξ) where b ∈ C ∞ 0 (R 2d \ {dg(x)ξ = 0}), b ≥ 0, θ ∈ C ∞ 0 ([0, T ]), θ ≥ 0 and Φ ∈ C ∞ (R d ) satisfies ∃c 0 > 0, ∀ξ ∈ Supp b, ∀x ∈ S, ∇Φ(0) • (dg(x)ξ) > c 0 .
Then, in view of (2.5), testing a δ against ρ(t, x, ξ) and letting δ go to 0, we obtain

(2.7) a δ , ρ -→ δ→0 0.
On the other hand, using (2.4), we obtain

a δ , ρ = a δ , ∂ t µ t + ξ • ∇ x µ t -∇V (x) • ∇ ξ µ t 1 g(x) =0 µ t = (dg(x) • ξ) • ∇Φ g(x) δ θ(t)b(x, ξ) , µ t + O(δ)
where we have used that µ t is a measure. Therefore, we obtain

(2.8) a δ , ρ -→ δ→0 R 2d+1 (dg(x)ξ) • ∇Φ(0)θ(t)b(x, ξ)dµ t (x, ξ)1 S dt.
In view of (2.7) and (2.8), we have

R 2d+1 (dg(x)ξ) • ∇Φ(0)θ(t)b(x, ξ)dµ t (x, ξ)1 S dt = 0.
This identity implies (2.6).

Conclusion.

We can now conclude the proof of Theorem 1.1. By (2.6) and the last point of Proposition 2.1, a, ρ = 0 for all a ∈ C ∞ 0 (R 2d+1 ) with supp(a)∩S ⊂ S * . Then, (2.4) writes (1.6) outside S \ S * , which finishes the proof. Remark 2.2. Note that we have proved that the distribution ρ is supported above R × (S \ S * ); therefore Remark 1.2 is a consequence of this observation and of Proposition 2.1.

Existence of the measure

Let us prove (2.3). We observe that

(3.1) d dt (op ε (a)ψ ε (t) , ψ ε (t)) = i ε - ε 2 2 ∆ + V (x) , op ε (a) ψ ε (t) , ψ ε (t) .
By using integration by parts, one easily obtain

(3.2) i ε - ε 2 2 ∆ , op ε (a) = op ε (ξ • ∇ x a)
and this family of operators is uniformly bounded. Set

L 0 = i ε [V (x) , op ε (a)] ,
we are going to prove that this family is also uniformly bounded in ε, even though V has a singularity on S. We use the following lemma to control the norm of the considered operators.

Lemma 3.1. Consider L ε an operator of kernel K ε (x, y) of the form

(3.3) K ε (x, y) = 1 (2πε) d k x + y 2 , x -y ε ,
such that the function k satisfies

(3.4) N (k) := v∈R d sup Y ∈R d |k(Y, v)| dv < +∞. Then L ε is uniformly bounded in L(L 2 (R d ))
and there exists C > 0 such that

L ε L(L 2 (R d )) ≤ C N (k).
Remark 3.2. Note that this lemma yields the uniform boundedness of the operator op ε (a) for a ∈ C ∞ 0 (R 2d ) and more generally for symbols a compactly supported such that ∂ β ξ a is bounded and locally integrable for all |β| ≤ d + 1.

Lemma 3.1 implies the boundedness of

L 0 on L 2 (R d ). Indeed, L 0 has a kernel K ε (x, y) of the form (3.3) with k ε (X, v) = i ε (V (X + εv/2) -V (X -εv/2)) a(X, ξ)e iξ•v dξ.
We write

V (X + εv/2) -V (X -εv/2) = ε G(X, εv) • v,
and the boundedness of ∇ (|g(x)|) on compact subsets of R d implies the existence of a constant C > 0 such that |G(X, εv)| ≤ C. Writing, thanks to an integration by parts,

k ε (X, v) = -e iξ•v ∇ ξ a(X, ξ) • G(X, εv)dξ
and using that a is smooth and compactly supported in ξ, we obtain (again by integration by parts)

∀N ∈ N, v 2N k(X, v) = -G(X, εv) • ∇ ξ i∇ ξ 2N a(X, ξ)e iξ•v dξ.
Therefore, we have

(3.5) ∀N ∈ N, ∃C N > 0, sup Y,v v 2N |k ε (Y, v)| ≤ C N
and the condition (3.4) is satisfied. Let us now prove Lemma 3.1.

Proof. We observe

sup x |K ε (x, y)| dy = 1 (2πε) d sup x k x + y 2 , x -y ε dy = 1 (2π) d sup x |k (x -εv/2, v)| dv ≤ C sup Y |k(Y, v)| dv.
Similarly,

sup y |K ε (x, y)| dx = 1 (2πε) d sup y k x + y 2 , x -y ε dx = 1 (2π) d sup y |k (y + εv/2, v)| dv ≤ C sup Y |k(Y, v)| dv.
Therefore, by Schur lemma, the condition (3.5) is enough to yield the boundedness of the operator L ε .

The transport equation

Let us now prove Proposition 2.1. We choose ε = ε k , the subsequence of Section 2.1. In view of (3.1), we need to pass to the limit in the term

L 0 = i ε [V (x) , op ε (a)].
L 1 = i ε [V 0 (x) , op ε (a)]. The kernel of L 1 is of the form (3.3) with k ε (X, v) = i ε (V 0 (X + εv/2) -V 0 (X -εv/2)) a(X, ξ)e iξ•v dξ = -∇ ξ a(X, ξ) • ∇V 0 (X)e iξ•v dξ + εr ε (X, v)
where r ε satisfies

v 2N r ε (X, v) = v 2N i ε a(X, ξ)e iξ•v V 0 (X + εv/2) -V 0 (X -εv/2) -ε∇ x V 0 (X) • v dξ = ε i∇ ξ 2N a(X, ξ)Θ ε (X, v)dξ with |Θ ε (X, v)| ≤ C|v| 2 . Therefore, |r ε (X, v)| ≤ C v 2N -2
, and by Lemma 3.1, the operator op ε (r ε ) is uniformly bounded in ε by choosing N large enough. As a conclusion, we get

(4.2) (L 1 ψ ε (t) , ψ ε (t)) -→ ε→0 -< ∇ ξ a • ∇ x V 0 , µ t > .
Remark 4.1. The previous argument only uses the following property of V 0 :

(4.3) |V 0 (X + v) -V 0 (X) -∇ x V 0 (X)v| ≤ C|v| 2 .
Therefore, it is enough to suppose that V 0 is differentiable and satisfies (4.3). (The fact that ∇ ξ a • ∇ x V 0 ∈ A is compactly supported allows us to pass to the limit in (4.2).)

It remains to study the contribution of the singular part of the potential. We first discuss the case where g(x) = (x 1 , • • • , x p ), then we reduce to this situation by a local change of coordinates. The analysis relies on a second microlocalisation on the singular set S = {x 1 = • • • = x p = 0} in the spirit of [START_REF] De Verdière | The level crossing problem in semi-classical analysis II. The hermitian case[END_REF]: we explain this point in the next subsection. 4.2. Two microlocal Wigner measures. These measures are used to describe the concentration of the family ψ ε (t) above S = {x 1 = • • • = x p = 0} (see [START_REF] De Verdière | The level crossing problem in semi-classical analysis II. The hermitian case[END_REF] for more details). We set

x = (x 1 , • • • , x p ) and x = (x , x ).
We consider two-microlocal test symbols b(t, x, ξ, y) ∈ C ∞ (R 2d+p+1 ) satisfying

• there exists a compact K ⊂ R 2d+1 such that for all y ∈ R p , the function

(t, x, ξ) → b(t, x, ξ, y) is compactly supported in K, • there exists R 0 > 0 and b ∞ (t, x, ξ, ω) ∈ C ∞ (R 2d+1 × S p-1 ) such that for |y| > R 0 , b(t, x, ξ, y) = b ∞ t, x, ξ, y
|y| , and we analyze the action of the operator op ε (b(t, x, ξ, x /ε)) as ε goes to 0.

(op ε (b(t, x, ξ, x /ε)ψ ε (t) , ψ ε (t)) -→ ε→0 x =0 b ∞ t, x, ξ, x |x | dµ t (x, ξ)dt + b ∞ (t, (0, x ), ξ, ω)dν(t, x , ξ, ω) + tr b W (t, (0, x ), (D y , ξ ), y)dM (t, x , ξ )
where, for all (x , ξ ) ∈ R 2(d-p) , we denote by b W (t, (0, x ), (D y , ξ ), y) the operator obtained by the Weyl-quantization of the symbol (y, η) → b(t, (0, x ), (η, ξ ), y).

This result (which is proved in [START_REF] De Verdière | The level crossing problem in semi-classical analysis II. The hermitian case[END_REF]) calls for several remarks. First, we point out that for any open set Ω ⊂ R 2d the mass of the measure µ t above S ∩ Ω can be expressed in terms of the mass of ν and of the trace of M according to

S∩Ω µ t (dx, dξ) = π x ,ξ (S∩Ω)×S p-1 ν(t, dx , dξ, dω) (4.4) +tr π x ,ξ (S∩Ω) M (t, dx , dξ )
where π x ,ξ and π x ,ξ denotes the canonical projection (x, ξ) → (x , ξ) and (x, ξ) → (x , ξ ) respectively. As a consequence, M and ν are measures absolutely continuous with respect to the Lebesgue measure dt. Note also that for any test function a(t, x , ξ ), the operator a , M is a positive trace-class operator on L 2 (R p y ) so that tr a, M ≥ 0; therefore, each term of the sum (4.4) is positive. As a consequence, we have the following result: Then, up to a subsequence in ε, we have

limsup δ→0 limsup R→∞ lim ε→0 op ε b t, x, ξ, x ε 1 -χ x Rε χ x δ ψ ε (t) , ψ ε (t) = b ∞ (t, (0, x ), ξ, ω)dν(t, x, ξ, ω), limsup R→∞ lim ε→0 op ε b t, x, ξ, x ε χ x Rε ψ ε (t) , ψ ε (t)
= tr b W (t, (0, x ), (D y , ξ ), y)dM (t, x , ξ ).

Remark 4.5. The family (Φ ε (t)) ε>0 with

(4.6) Φ ε (t, y, x ) = ε p/2 ψ ε (t, εy, x ) is uniformly bounded in L 2 (R d-p
x , H) where H = L 2 (R p y ). Besides, we have for b compactly supported in all the variables,

op ε b t, x, ξ, x ε ψ ε (t) , ψ ε (t) L 2 (R d ) = (op ε (A ε (x , ξ )) Φ ε (t) , Φ ε (t)) L 2 (R d-p ,H)
where A ε (x , ξ ) = b W (t, εy, x , ξ , D y ) where for c(y, η) ∈ C ∞ 0 (R 2p ), the operator c W (y, D y ) is the pseudodifferential operator of Weyl symbol c(y, η) c W (y, D y ) = op 1 (c(y, η)).

The operator A ε (x , ξ ) is a semiclassical symbol valued in the set of compact operators on H, since b(t, x , x , η, ξ , y) is compactly supported in (y, η). Therefore, the measure M is a semi-classical measure of the uniformly bounded family (Φ ε (t)) ε>0 of L 2 (R d-p , H).

In the following subsection, we use these measures ν and M to obtain a transport equation on the measure µ t . 4.3. Concentration on a vector space. We use the cut-off function χ of (4.5) and write for R > 0

(4.7) i ε [|x |w(x) , op ε (a)] = L 2 + L 3 with L 2 = i ε |x |w(x) , op ε a(x, ξ)χ x εR .
We study separately the operators L 2 and L 3 .

Analysis of L 2 . We observe

(L 2 ψ ε (t) , ψ ε (t)) = L 2 Φ ε (t) , Φ ε (t)
with L 2 = i |y|w(εy, x ) , op 1 a(εy, εx , η, εξ )χ y R and Φ ε (t) defined in (4.6). The operator L 2 is a semiclassical operator of symbol

A ε (x , ξ ) = i |y|w(εy, x ) , a W (εy, x , D y , ξ )χ y R
valued in the set of operators on H (with the notations of Remark 4.5). Besides, if χ is a cut-off function such that χ = 1 on the support of χ, we can write

A ε (x , ξ ) = Ãε,R + O(1/R) in operator norm with Ãε,R = i χ y R |y|w(εy, x ) , a W (εy, x , D y , ξ )χ y R ,
which is a compact operator. By Remark 4.5, for all test functions θ

limsup R→+∞ limsup ε→0 θ(t) (L 2 ψ ε (t), ψ ε (t)) dt (4.8) = tr θ(t)i |y|w(0, x ), a W (0, x , D y , ξ ) dM (t, x , ξ ) .
Analysis of L 3 . The following lemma relates L 3 with the two-microlocal test symbols of subsection 4.2.

Lemma 4.6. There exists ε 0 > 0, N 0 ∈ N and C > 0 such that for all a ∈ A, ε ∈]0, ε 0 [ and R > 1,

L 3 + op ε ∇ x (|x |w(x)) • ∂ ξ a(x, ξ) 1 -χ x Rε L(L 2 (R d )) ≤ CM N0 (a) R -3 + ε where ∀N ∈ N, M N (a) = max |α|≤N sup (x,ξ) ∂ α ξ a(x, ξ) (1 + |ξ|) d+1 .
We postpone the proof to Section 5. By Lemma 4.6, we are left with the operator

op ε ∇ x (|x |w(x)) • ∇ ξ a(x, ξ) 1 -χ x Rε .
Notice that the function

(x, ξ) → ∇ x (|x |w(x)) • ∇ ξ a(x, ξ) 1 -χ x Rε
is smooth. We decompose this function in three parts :

∇ x (|x |w(x)) • ∇ ξ a(x, ξ) 1 -χ x Rε = b 1 x, ξ, x ε + b 2 x, ξ, x ε + c ε,δ (x, ξ) with b 1 (x, ξ, y) = w(x) y |y| • ∇ ξ a(x, ξ) 1 -χ y R , b 2 (x, ξ, y) = |x |∇w(x) • ∇ ξ a(x, ξ) 1 -χ y R 1 -χ x δ , c ε,δ (x, ξ) = |x |∇w(x) • ∇ ξ a(x, ξ) 1 -χ x Rε χ x δ .
• The symbols b 1 (x, ξ, y) and b 2 (x, ξ, y) are smooth two microlocal symbols. Therefore by Proposition 4.2 (see also Remark 4.4), we obtain that, up to a subsequence ε k , for all test functions θ and all j ∈ {1, 2},

limsup δ→0 limsup R→+∞ limsup ε k →0 θ(t) op ε b j x, ξ, x ε ψ ε (t) , ψ ε (t) dt = θ(t)(b j ) ∞ (x, ξ, ω) , ρ with ρ(t, x, ξ, ω) = µ t (x, ξ)1 x =0 ⊗ δ ω - x |x | + δ(x ) ⊗ ν(t, x , ξ, ω), (b 1 ) ∞ (x, ξ, ω) = w(x)ω • ∇ ξ a(x, ξ), (b 2 ) ∞ (x, ξ, ω) = |x |∇w(x) • ∇ ξ a(x, ξ).
• Let us now consider the symbol c ε,δ . The operator op ε (c ε,δ (x, ξ)) has a kernel of the form (2πε

) -d k ε x+y 2 , x-y ε with k ε (X, v) = 1 -χ X Rε χ X δ |X | ∇w(X) • ∇ ξ a(X, ξ)e iξ•v dξ.
Therefore, using integration by parts in ξ, we obtain

∀N ∈ N, ∃C N > 0, v N |k ε (X, v)| ≤ C N δ, which yields limsup δ→0 limsup R→+∞ limsup ε→0 op ε (c ε,δ ) L(L 2 ) = 0.
Finally, we obtain

limsup R→+∞ limsup ε→0 θ(t) (L 3 ψ ε (t) , ψ ε (t)) dt (4.9) = -θ(t) (∇ x (|x |w(x)) • ∇ ξ a(x, ξ)) 1 x =0 dµ t (x, ξ)dt -θ(t)w(x)ω • ∇ ξ a((0, x ), ξ)dν(t, x , ξ, ω)
where we have used (b 1 ) ∞ x, ξ,

x |x | + (b 2 ) ∞ x, ξ, x |x | = ∇ x (|x |w(x)) • ∇ ξ a(x, ξ).
As a conclusion, in view of (4.8) and (4.9), we have

θ(t) ((L 2 + L 3 )ψ ε (t) , ψ ε (t)) dt (4.10) -→ ε→0 -θ(t) (∇ x (|x |w(x)) • ∇ ξ a(x, ξ)) 1 x =0 dµ t (x, ξ)dt -θ(t)w(x)ω • ∇ ξ a((0, x ), ξ)dν(t, x , ξ, ω) +tr θ(t)i |y|w(0, x ), a W (0, x , D y , ξ ) dM (t, x , ξ ) .
Let us now conclude the proof of Proposition 2.1.

Proof. We have

d dt (op ε (a)ψ ε (t) , ψ ε (t)) = i ε - ε 2 2 ∆ + V (x) , op ε (a) ψ ε (t) , ψ ε (t) = ((op ε (ξ • ∇ x a) + L 1 + L 2 + L 3 ) ψ ε (t) , ψ ε (t)) , see (3.
2), (4.1) and (4.7). By usual Weyl calculus (see for example [START_REF] Gérard | Ergodic properties of eigenfunctions for the Dirichlet problem[END_REF]Theorem 18.5.4.]) the commutator resulting from L 2 can be written as

i |y|w(0, x ), a W (0, x , D y , ξ ) = - y |y| w(0, x )(∂ η a) W (0, x , D y , ξ ) + r,
where the symbol of r depends on products of w and η-derivatives of a. Passing to the limit ε → 0, we obtain

∂ t µ t = -ξ • ∇ x µ t + 1 x =0 (∇ x V 0 • ∇ ξ µ t + ∇ x (|x |w(x)) • ∇ ξ µ t ) + ρ from (4.
2) and (4.10) for some distribution ρ(t, x, ξ) satisfying the estimate (2.5). This is the transport equation (2.4) in the case S = {x = 0}. The observation of Remark 4.3 concludes the proof.

4.4. More general submanifolds. We now suppose that S is not necessarily a vector space. We work locally close to a point

x 0 ∈ S in local coordinates x = ϕ(z) with z = (z , z ) ∈ R d = R p ×R d-p such that z = g(x)
. We consider v ε = ψ ε •ϕ. By Egorov's theorem (see [START_REF] Kammerer | Propagation through generic level crossings: a surface hopping semigroup[END_REF]Lemma 1.10] for example), the semi-classical measures µ and μ of v ε and ψ ε , respectively, are linked by

µ(z, ζ) = µ(ϕ(z), t dϕ(z) -1 ζ).
Besides, v ε solves locally, close to x 0 ,

iε∂ t v ε = op ε ( 1 2 | t dϕ(z) -1 ζ| 2 )v ε + (|z | w(z) + V 0 (z))v ε
where w and V 0 are smooth. It is not difficult to check that the arguments of the preceding sections also apply to this equation with a modified Laplacian. We leave the details to the reader.

Proof of Lemma 4.6

We write L 3 = T ε L 3 T * ε where T ε is the scaling operator defined by

∀f ∈ L 2 (R d ), T ε f (x) = ε d/2 f (εx).
The we have

L 3 = 1 i op 1 a(εx, ξ) 1 -χ x R , |x |w(εx) . The kernel of L 3 is of the form (2π) -d K ε x+y 2 , x -y with K ε (X, v) = 1 i e iξ•v a(εX, ξ) 1 -χ X R × X - v 2 w εX -ε v 2 -X + v 2 w εX + ε v 2 dξ.
We set

A ε (X, v) := X - v 2 w εX -ε v 2 -X + v 2 w εX + ε v 2
Therefore, the operator L 3

is

L (1) 3 = -op 1 ∇ ξ a(εx, ξ) • ∇ x (|x |w(εx)) 1 -χ x R = -T * ε op ε ∇ ξ a(x, ξ) • ∇ x (|x |w(x)) 1 -χ x Rε T ε . (5.6) • For j = 2, we write X - v 2 -X + v 2 + X |X | • v = (5.7) X • v - 2 |X + v /2| + |X -v /2| + 1 |X | . Since (5.8) 1 |X + v /2| + |X -v /2| - 1 2|X | = 2|X | -|X + v /2| -|X -v /2| 2|X | (|X + v /2| + |X -v /2|) , we observe |X | X + v 2 + X - v 2 -2|X | = |X | X + v 2 + |X | X - v 2 -2|X | 2 ≤ 1 2 X + v 2 2 + X - v 2 2 + 2|X | 2 -2|X | 2 ,
where we have used ab ≤ 1 2 (a 2 + b 2 ). Expanding the terms

X ± v 2 2
, we obtain

|X | X + v 2 + X - v 2 -2|X | ≤ 1 2 |v | 2 2 + 4|X | 2 -2|X | 2 = |v | 2 4 .
Plugging the latter inequality in (5.8) and (5.7), we obtain

X - v 2 -X + v 2 + X |X | • v = X |X | • v G(X , v ) with (5.9) |G(X , v )| ≤ C |X | -3 |v | 2 ,
where we have used that, by the triangle inequality,

X + v 2 + X - v 2 -2|X | ≥ 0
Therefore, by (5.2),

A (2) ε (X, v) = w(εX) X |X | • v G(X , v ),
and integrating by parts, we have

v 2N K (2) ε (X, v) = G(X , v ) i∇ ξ 2N w(εX) X |X | • ∇ ξ a(εX, ξ) 1 -χ X R e iξ•v dξ.
Using (5.9) and the fact that a is a smooth compactly supported function of ξ, we obtain that for all N ∈ N, there exists a constant C N such that sup

X,v v 2N K (2) ε (X, v) ≤ C N R -3 .
We then conclude by Lemma 3.1 that there exists P ∈ N such that (5.10)

L (2) 3 L(L 2 (R d ))
≤ CM P (a)R -3 .

• For j = 3, we transform A

ε (X, v). We write

1 2 X - v 2 + X + v 2 -2|X | = v 4 • 2X + v /2 |X + v /2| + |X | - 2X -v /2 |X -v /2| + |X | = v • G(X , v )
with G(X , v ) a bounded function. Therefore,

A (3) ε (X, v) = -ε∇w(εX) • v G(X , v ) • v + εR ε (X, v) with |R ε (X, v)| ≤ C v 3 for some constant C > 0, if εX is in the support of a.
Finally, we obtain

K (3) ε (X, v) = -ε 1 -χ X R ∇w(εX) • ∇ ξ G(X , v ) • ∇ ξ a(εX, ξ) -εR ε (X, v)a(εX, ξ) e iξ•v dξ,
whence, by integration by parts, for all

N ∈ N v 2N |K (3) ε (X, v)| ≤ C N ε sup k≤2N +4
e iv•ξ i∇ ξ k a(εX, ξ)dξ ≤ CM P (a)ε for some P ∈ N. We then conclude by Lemma 3.1

(5.11) L

(L 2 (R d )) ≤ CM P (a) ε. (3) 3 L
We can now conclude the proof of Lemma 4.6 which comes from (5.5) and from equations (5.6), (5.10) and (5.11).

The generalized flow

In the next two subsections, we prove Proposition 1.3 in two steps: we first prove existence and uniqueness of the trajectories, then we focus on their regularity. 6.1. Existence and uniqueness of the trajectories. We work with initial data (x 0 , ξ 0 ) ∈ S * and prove local existence and uniqueness of a Lipschitz map t → (x t , ξ t ) satisfying ẋt = ξ t , ξt = -∇V (x t ) for t = 0. Then, we have

(6.1) 1 t g(x t ) -→ t→0 ± dg(x)ξ,
where dg(x) = (∂ j g i (x)) i,j is a p × d matrix and ξ 0 is thought as a column (a d × 1 matrix); similarly, g(x) is a p × 1 matrix. Therefore, we have

ξt = -∇V 0 (x t ) -|g(x t )|∇w(x t ) -w(x t ) t dg(x t ) g(x t ) |g(x t )| -→ t→0 ± -∇V 0 (x) ∓ w(x) t dg(x) dg(x)ξ |dg(x)ξ| .
For (x, ξ) ∈ S * , we introduce the systems

ẋ± t = ξ ± t , x ± 0 = x, ξ± t = -∇V 0 (x ± t ) ∓ sgn(t)|g(x ± t )|w(x ± t ) ∓ sgn(t) t dg(x ± t ) g(x ± t ) |g(x ± t )| , ξ ± 0 = ξ.
We note that, under existence condition, we have

∀(x, ξ) ∈ R 2d , Φ t (x, ξ) = Φ t ± (x, ξ) := (x ± t , ξ ± t ) if ± t > 0.
Let us prove the existence of a solution Φ t + , the proof for Φ t -is similar. We note that if such a map exists, then t -1 g(x + t ) -→ t→0 ± dg(x)ξ and we set

y t = 1 t g(x + t ) -dg(x)ξ. Since sgn(t) g(x + t ) |g(x + t )| = t -1 g(x + t ) |t -1 g(x + t )| = dg(x)ξ + y t |dg(x)ξ + y t | ,
we are left with the system    d dt (ty t ) = dg(x + t )ξ + t -dg(x)ξ, y 0 = 0 ẋ+ t = ξ + t , x + 0 = x ξ+ t = B(t, x + t , y t ), ξ + 0 = ξ where

B(t, X, Y ) := -∇V 0 (X) -t|dg(x)ξ + Y | -t dg(X) dg(x)ξ + Y |dg(x)ξ + Y |
Note that we can write

y t = 1 t t 0 dg(x + s )ξ + s -dg(x)ξ ds = 1 0 dg(x + tθ )ξ + tθ -dg(x)ξ dθ.
Besides, since the function B is smooth near (t, X, Y ) = (0, x, 0) for (x, ξ) ∈ S * , we can apply a fixed point argument to the function

F x,ξ : (y t , x + t , ξ + t ) → 1 0 dg(x + tθ )ξ + tθ -dg(x)ξ dθ, x + t 0 ξ + s ds, ξ + t 0 B(s, x + s , y s )ds
in the set B t0,δ defined for δ, t 0 > 0 small enough by

B t0,δ = sup |t|<t0 |x + t -x| + |ξ + t -ξ| + |y t | < δ, y 0 = 0, x + 0 = x, ξ + 0 = ξ .
In this manner, we construct the smooth trajectory t → Φ t + (x, ξ) for (x, ξ) ∈ S * , which defines Φ t (x, ξ) = Φ t + (x, ξ) for t > 0. The proof is similar for t < 0 by using Φ t -. Note also that the smoothness of B with respect to x and ξ implies that the fixed point of F x,ξ depends smoothly on the parameter (x, ξ) ∈ S * . These trajectories allow to uniquely extend the flow t → Φ t (x, ξ), t ∈ [-τ 0 , τ 0 ] for (x, ξ) ∈ Ω with Ω ∩ S ⊂ S * . Besides, ∂ t Φ t is bounded and locally integrable. 6.2. Regularity of the trajectories. We now prove that (x, ξ) → ∂ α ξ Φ t (x, ξ) are continuous on Ω for every multi-index α by solving the system satisfied by ∂ α ξ Φ t (x, ξ). For this, we argue by induction on |α|.

Let us consider ∂ α ξ Φ t for |α| = 1 and let us first suppose t > 0 (the argument for t < 0 is similar). We denote by 1 j the vector of N d with 1 on the j-th coordinate and 0 elsewhere. We have

d dt ∂ ξj Φ t (x, ξ) = M (x t )∂ ξj Φ t (x, ξ), ∂ ξj Φ 0 (x, ξ) = (0, 1 j ) with M (x) = 0 Id -B(x) 0
and for all δx ∈ R d ,

B(x)δx = d 2 V (x)δx = d 2 V 0 (x)δx + |g(x)|d 2 x w(x)δx + (∇w(x) • δx) t dg(x) g(x) |g(x)| + w(x)d 2 g(x) δx, g(x) |g(x)| + w(x) |g(x)| t dg(x) dg(x)δx - g(x) |g(x)| • (dg(x)δx) t dg(x) g(x) |g(x)| .
Due to (6.1) there exists a d × d matrix B 0 such that

B(x t ) = B 1 (x, ξ) t + B 0 (x, ξ) + O(t) for t > 0 with B 1 (x, ξ)δx = w(x) |g(x)| t dg(x) dg(x)δx - dg(x)ξ |dg(x)ξ| • (dg(x)δx) t dg(x) dg(x)ξ | dg(x)ξ| .
for all δx ∈ R d . For solving the system, we take advantage of the fact that the initial condition is such that

(6.2) 0 0 -B 1 (x, ξ) 0 ∂ ξj Φ 0 (x, ξ) = 0.
We set

Z(t) = 1 t ∂ ξj Φ t -∂ ξj Φ 0 = 1 t ∂ ξj x t ∂ ξj ξ t -1 j .
We have

d dt (tZ(t)) = M (x t ) tZ(t) + 0 1 j , t d dt Z(t) + Z(t) + Q 0 Z(t) = tP (t)Z(t) + F (t) with Q 0 = 0 0 B 1 0
and t → P (t) and t → F (t) smooth maps for t ≥ 0. Note that there exists a unique vector Z(0) such that

Z(0) + Q 0 Z(0) = F (0).
We set Z(t) = Z(t) -Z(0) and we have

t d dt Z(t) + Z(t) + Q 0 Z(t) = tP (t) Z(t) + t F (t)
with t → F (t) smooth. We obtain

d dt te Q0lnt Z(t) = e Q0lnt tP (t) Z(t) + t F (t) ,
where the function t → e Q0lnt is absolutely continuous on R + , whence

te Q0lnt Z(t) = t 0 e Q0lnσ σP (σ) Z(σ) + σ F (σ) dσ Z(t) = t 1 0 e Q0lnθ θP (tθ) Z(tθ) + θ F (tθ) dθ,
which is solved by a fixed point argument. At this first step of the induction, we have obtained that the quantity

∂ ξj Φ t (x, ξ) = tZ(t) + 0 1 j = t(Z(0) + Z(t)) + 0 1 j is a continuous map on t ≥ 0. Besides ∂ t ∂ ξj Φ t (x, ξ) = Z(t) + Z(0) + t d dt Z(t)
have finite limits when t goes to 0 + . Arguing similarly for t ≤ 0, we build a continuous map t → ∂ ξj Φ t (x, ξ) with a locally integrable bounded derivative ∂ t ∂ ξj Φ t (x, ξ).

We now proceed to the last step of the induction: we suppose that the functions t → ∂ β ξ Φ t (x, ξ) are well-defined for all β ∈ N d such that |β| ≤ n with

∂ β ξ Φ t (x, ξ) = O(t) if |β| > 1 and ∂ ξj Φ t = (0, 1 j ) + O(t), for t close to 0. Therefore, 0 0 -B 1 (x, ξ) 0 ∂ β ξ Φ 0 (x, ξ) = 0. Let us consider ∂ α ξ Φ t for |α| = n + 1. The function ∂ α ξ Φ t satisfies an ODE system of the form d dt ∂ α ξ Φ t = M (x t )∂ α ξ Φ t + F (∂ β ξ Φ t ) where the arguments of F are all associated with multi-indices β such that |β| ≤ n. Besides, F (∂ β ξ Φ t ) is the sum of terms of the form ∂ γ x,ξ M (x t )∂ α1 ξ x t • • • ∂ αp ξ x t with γ ∈ N 2d , α j ∈ N d and |α 1 | + • • • + |α p | = |γ| + 1, |γ| ≤ n. It is easy to check that ∀α ∈ N 2d , ∂ α x,ξ B(x t ) = O(t -|α|-1 ). Since ∂ αj x t = O(t), we obtain ∂ γ x B(x t )∂ α1 x t • • • ∂ αp x t = O(t |α1|+•••+|αp|-|γ|-1 ) = O(1), γ ∈ N d . Therefore, the map t → F (∂ β ξ Φ t
) is continuous when t goes to 0. We are left with a system of the same form as in the first step of the induction, since the initial data Finally, note that Ω f ∩ V (t f ) ⊂ V out (t f ) and Ω i ∩ V (t i ) ⊂ V in (t i ) are submanifolds of dimension 2d -p. Note also that the vector field

(7.2) H(x, ξ) = ξ • ∇ x -∇V (x) • ∇ ξ ,
is smooth close to points (x, ξ) of V in ∪ V out and, by the definition of V in and V out , it is tangent to these submanifolds. Therefore H is a vector field of V in and of

V out . φ Ω Ω t t S x S i f f (S) t t i Figure 1
With each point (x, ξ) of the projection on T * R d of V , one can associate the time τ (x, ξ) where the trajectory issued from (x, ξ) passes through S: we have Φ τ (x,ξ) (x, ξ) ∈ S. If (x, ξ) is in the projection of V in , we have τ (x, ξ) > 0 and if (x, ξ) is in the projection of V out , we have τ (x, ξ) < 0. If t 0 ∈ [t i , t f ], we can define a map χ t0 from V (t 0 ) to S as

χ t0 : (x, ξ) → t 0 + τ (x, ξ), Φ τ (x,ξ) (x, ξ) ∈ S.
Note that χ t0 is a homeomorphism from V (t 0 ) onto S.

Set µ := µ t dt as a measure of (t, x, ξ). We define the traces of µ on S as the measures

µ in = (χ ti ) * µ ti 1 Ωi∩V (ti) , µ out = (χ t f ) * µ t f 1 Ω f ∩V (t f ) .
Since µ satisfies the transport equation (1.6) on V in/out -where H is a smooth vector field-and since it does not see the set S, it is given on V by the formula

(7.3) µ1 V = Φ t-t f * (µ t f 1 t>τ +t f 1 Ω f ∩V (t f ) ) dt + Φ t-ti *
(µ ti 1 t<τ +ti 1 Ωi∩V (ti) ) dt .

On the other hand, µ and µ1 V c satisfy (1.6), so µ1 V does. This implies µ in = µ out . Indeed, the following lemma holds.

Lemma 7.2. The measure µ1 V satisfies the equation

∂ t (µ1 V ) + ∇ x • (ξµ1 V ) -∇ ξ • (∇V (x)µ1 V ) = 1 S (µ out -µ in ) .
Before proving Lemma 7.2, we observe that the relation µ out -µ in = 0 implies

µ t f 1 Ω f ∩V (t f ) = (χ t f ) * µ out = (χ t f ) * µ in = (χ t f ) * (χ ti ) * (µ ti 1 Ωi∩V (ti) ) = Φ t f -ti * (µ ti 1 Ωi∩V (ti) ) ,
as announced. Let us now prove Lemma 7.2.

Proof. In order to compute (∂ t + H)(µ1 V ), introduce a nondecreasing function ϕ ∈ C ∞ (R) such that ϕ(s) = 0 for s ≤ 1 , ϕ(s) = 1 for s 2 .

Then

µ1 V = lim δ→0 + Φ t-t f * µ t f 1 Ω f ϕ t -t f -τ δ dt + Φ t-ti * µ ti 1 Ωi ϕ t i + τ -t δ dt .
Notice that the right hand side is supported into V out ∪ V in , where H is smooth, so that we can make easy computations. We obtain, in the set of distributions,

(∂ t + H)(µ1 V ) = lim δ→0 + Φ t-t f * µ t f 1 Ω f 1 δ ϕ t -t f -τ δ dt -Φ t-ti * µ ti 1 Ωi 1 δ ϕ t i + τ -t δ dt .
Therefore, given a = a(t, x, ξ) ∈ C ∞ 0 (M ),

(∂ t + H)(µ1 V ), a = lim δ→0 + R Ω f a t, Φ t-t f (x, ξ) 1 δ ϕ t -t f -τ (x, ξ) δ dµ t f (x, ξ) dt - R Ωi
a t, Φ t-ti (x, ξ) 1 δ ϕ t i + τ (x, ξ) -t δ dµ ti (x, ξ) dt .

Passing to the limit in the integral as δ tends to 0 + , we conclude

(∂ t + H)(µ1 V ), a = Ω f
a(χ t f (x, ξ)) dµ t f (x, ξ) -Ωi a(χ ti (x, ξ)) dµ ti (x, ξ)

where we have used the definition of χ t and the fact that ϕ (u)du = 1. Lemma 7.2 follows by the definition of µ ti and µ t f . 7.3. End of the proof of Theorem 1.5. We first focus on µ t above S * . Since t → µ t and t → Φ t are continuous, we only need to prove that for a ∈ C ∞ 0 (R 2d ) such that supp (a) ∩ S * = ∅, we have Besides, since the support of a•Φ -t does not intersect Φ t (S * ), we have by using (7.1) where we have used the Fubini theorem. We observe that a1 Φ -t ((S * ) c ) = a -a1 Φ -t (S * )

µ t 1 (S * ) c = µ t 1 (S *
where, for every (x, ξ), Φ t (x, ξ) ∈ S * for at most one value of t. Therefore T 0 a1 Φ -t (S * ) dt = 0 identically, and we obtain (7.4).

To conclude the proof, it remains to calculate µ t above Φ t (S * ). For this, we work in a small neighborhood ω of a point (x t , ξ t ) ∈ Φ t (S * ). Since the flow is transverse to S * , by restricting ω if necessary, we can find θ < 0 such that the assumptions of Theorem 1.5 holds on [θ, τ 0 ] and such that Φ θ-t (ω) ∩ S * = ∅. We now argue between the times θ and 0 on one hand, and between the times θ and t, on the other hand. The previous analysis gives µ 0 = (Φ -θ ) * µ θ on Φ -t (ω) and µ t = (Φ t-θ ) * µ θ = (Φ t ) * µ 0 on ω. This completes the proof.

4 . 1 .

 41 The smooth part. Let us consider the smooth part of the potential and set (4.1)

Proposition 4 . 2 .

 42 There exists a positive Radon measure ν on R 2d-p+1 t,x ,ξ × S p-1 ω and a positive measure M on R 2(d-p)+1 t,x ,ξ valued in the set of trace-class operators on L 2 (R p y ) such that, up to a subsequence,

Remark 4 . 3 .

 43 If µ t 1 S∩Ω = 0, then, by(4.4) and because of the positivity of ν and M , we obtain ν = 0 and M = 0 above π x ,ξ (S ∩ Ω) and π x ,ξ (S ∩ Ω), respectively. Moreover, we have the following characterization of the measures ν and M : Remark 4.4. Let b(t, x, ξ, y) be a two-microlocal test symbol and χ ∈ C ∞ 0 (R p ) a cut-off function such that (4.5) χ(y) = 1 for |y| ≤ 1 and χ(y) = 0 for |y| ≥ 2 with 0 ≤ χ ≤ 1.

(7. 4 )

 4 ∀T ∈ [0, τ 0 ], T 0 a • Φ -t , µ t dt = T 0 a, µ 0 dt. Since µ([0, T ] × S * ) = 0, we can write T 0 a • Φ -t , µ t dt = T 0 a • Φ -t , µ t 1 (S * ) c dt.

0 a 1 Φ

 01 ) c ∩(Φ t (S * )) c = (Φ t ) * µ 0 1 (S * ) c on supp a • Φ -t . Therefore, T 0 a • Φ -t , µ t dt = T 0 R 2d a(x, ξ)1 Φ -t ((S * ) c ) (x, ξ) dµ 0 (x, ξ)dt = T -t ((S * ) c ) dt, µ 0

and write

with

For j ∈ {1, 2, 3}, we set (5.4)

We denote by L (j)

3 the operators of kernel (2π) -d K (j) ε x+y 2 , x -y , so that we have (5.5)

.

We now study successively each of these operators.

Remark 5.1. Here as for V 0 one can relax the C 2 regularity: assuming that w is differentiable and satisfies (4.3), an argument similar to the one of the beginning of section 4 allows to perform the proof (see Remark 4.1).

• For j = 1, we obtain

have an analogous property to (6.2), and one can argue similarly, which concludes the proof of Proposition 1.3.

7.

Propagation of the measure 7.1. Preliminaries. Before proving Theorem 1.5, we begin with a crucial remark.

Remark 7.1. Because of (6.1), the quantity (dg(x)ξ) • g(x) changes of sign close to t = 0 on trajectories passing through S * at time t = 0: (dg(x)ξ) • g(x) > 0 on the outgoing branches and (dg(x)ξ) • g(x) < 0 on the incoming ones.

Let us now prove Theorem 1.5. Note that it is enough to prove the corollary under the assumption that between times t = 0 and t = τ 0 the trajectories Φ t (x, ξ) issued from points of the support of µ 0 cross S * at most once. We proceed in two steps: under the assumptions of Theorem 1.5, we first calculate µ t near points which are not in S * ∪ Φ t (S * ), then we deal with the general case. Before starting the proof, let us introduce the following notation: given a subset A of R 2d+1 and t ∈ R, we set A(t) := {(x, ξ) ∈ R 2d : (t, x, ξ) ∈ A} .

7.2.

The measure away from the singularity. In this section, we prove (7.1)

We consider the set M consisting of the points (t, Φ t f -t (x, ξ)) t∈[ti,t f ] for all (x, ξ) ∈ Ω f . We have a partition of M , M = V ∪ V c , where

The set V c is an open subset of R 2d+1 which is invariant by Φ t and we have

In particular, we have

). We will use latter that the measure µ1 V c also is a solution of the transport equation (1.6).

Let us now focus on V . In view of Remark 7.1, by reducing Ω f and t i if necessary, we can assume that the quantity (dg(x)ξ) • g(x) vanishes in V only at points of S * . Then, the set S := (R×S)∩M -which is a subset of [t i , t f ]×S * and a submanifold of dimension (2d -p) -separates V into two sets:

• the incoming region V in , where (dg(x)ξ) • g(x) < 0, which contains trajectories entering into S, • the outgoing region V out , where (dg(x)ξ) • g(x) > 0, which contains trajectories which are issued from S, and we have V = S ∪ V in ∪ V out . Note that by the characterization through the function (dg(x)ξ) • g(x), the sets V out and V in have disjoint projections on R 2d . Because of the links between Φ t and Φ t ± , the sets V in and V out are submanifolds of dimension 2d -p + 1

V in = {(t, x, ξ) ∈ M, ∃(s, y, η) ∈ [t, t f ] × S, (x, ξ) = Φ t-s -(y, η)}, V out = {(t, x, ξ) ∈ M, ∃(s, y, η) ∈ [t i , t] × S, (x, ξ) = Φ t-s + (y, η)}.


Valter Carvelli

Giulia Gramellini

Stepan V Lomov

Alexander E Bogdanovich

Dmitri D Mungalov

Ignaas Verpoest

FATIGUE BEHAVIOUR OF NON-CRIMP 3D ORTHOGONAL WEAVE AND MULTI-LAYER PLAIN WEAVE E-GLASS REINFORCED COMPOSITES

Keywords: (A) Textile composites, (B) Mechanical Properties, Fatigue

The paper studies tension-tension fatigue behavior of a single-ply non-crimp 3D orthogonal weave Eglass composite and of a laminated composite reinforced with four plies of a standard plain weave fabric.

Both composites have same total thickness and very close fiber volume fraction. The paper presents the description of the materials, the results of quasi-static tensile and of tension-tension fatigue tests, including the damage development during fatigue tensile loading. The non-crimp 3D woven fabric composite, loaded in both principal in-plane directions (warp and fill), shows the best quasi-static tensile properties and, when loaded in the fill direction, exhibits much longer fatigue life than its laminated plain weave counterpart. During both quasi-static and fatigue loading, the latest damage initiation is observed for the 3D woven composite in both in-plane directions. The PW laminate develops delamination between the plies for each maximum stress in the cycle considered. Contrary to that, the 3D composite is not affected by delamination neither under quasi-static nor under fatigue loading conditions.

INTRODUCTION

A key factor driving the increased applications of composites over recent years is the occurrence of new advanced reinforcements developed for fiber reinforced polymer composites, new resin systems and * Corresponding author: tel.: +39 02 2399 4354 fax: +39 02 2399 4369 e-mail: valter.carvelli@polimi.it 2 / 24 innovative manufacturing methods. A major driving force has been the development of manufacture and design methods for textile preforms, which permit the production of more sophisticated composites next to the more traditional two-dimensional laminated structures [START_REF] Tong | 3D Fibre Reinforced Polymer Composites[END_REF][START_REF] Bogdanovich | Three-Dimensional Reinforcements for Composites[END_REF].

Relatively thick, single layer non-crimp 3D orthogonal woven preforms are gaining fast growing interest in the composites industry. Composites reinforced with single-layer non-crimp 3D woven preforms are nowadays used in a broad range of applications: from personnel and ground vehicle composite armor, to load bearing structures such as bridge decks or windmill blades, marine structures like boats and yachts, and aircraft engines [START_REF] Bogdanovich | Advancements in manufacturing and applications of 3D woven preforms and composites[END_REF][START_REF] Mohamed | A new generation of 3D woven fabric preforms and composites[END_REF][START_REF] Leblanc | Shock loading of three-dimensional woven composite materials[END_REF][START_REF] Grogan | Ballistic resistance of 2D and 3D woven sandwich composites[END_REF][START_REF] Tekalur | Shock loading response of sandwich panels with 3-D woven E-glass composites skins and stitched foam core[END_REF]. Efficient modern methods of low-cost manufacturing of non-crimp 3D single-layer preforms, see [START_REF] Bogdanovich | Three-Dimensional Reinforcements for Composites[END_REF][START_REF] Mohamed | A new generation of 3D woven fabric preforms and composites[END_REF][START_REF] Mohamed | Comparative analysis of different 3D weaving processes, machines and products[END_REF], increased the interest in the research activities to exploit the full potential of these materials. It is well understood and generally appreciated that 3D woven non-crimp reinforcements provide efficient delamination suppression, markedly improved fracture toughness [START_REF] Tamuzs | Delamination properties of translaminar-reinforced composites[END_REF], significantly increased damage tolerance as well as exceptional impact, ballistic and blast performances [START_REF] Leblanc | Shock loading of three-dimensional woven composite materials[END_REF][START_REF] Grogan | Ballistic resistance of 2D and 3D woven sandwich composites[END_REF][START_REF] Tekalur | Shock loading response of sandwich panels with 3-D woven E-glass composites skins and stitched foam core[END_REF]. In addition, as shown in [START_REF] Brandt | Mechanical performance of composites based on various threedimensional woven-fibre performs[END_REF][START_REF] Quinn | Examination of failure of 3D woven composites[END_REF][START_REF] Lomov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 1: materials, methods and principal results[END_REF][START_REF] Ivanov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 2: comprehensive experimental results[END_REF], this kind of 3D reinforcement provides significantly higher in-plane strength properties than 2D woven fabric laminates. This is a very distinct positive factor, knowing that 3D interlock woven fabric composites suffer from substantial inherent crimp of the performs and, consequently, show significantly reduced in-plane stiffness and strength even in comparison with 2D plain weave composites [START_REF] Tong | 3D Fibre Reinforced Polymer Composites[END_REF][START_REF] Brandt | Mechanical performance of composites based on various threedimensional woven-fibre performs[END_REF]. The difference between composites reinforced with 3D non-crimp orthogonal weaves and 3D interlock weaves is caused, primarily, by the absence of crimp in all inner warp and fill yarn layers. Only minimal deviation from straightness is observed in the surface fill yarn layers due to their interaction with through thickness (Z) yarns. The absence of other types of fiber waviness and misalignment is due to the specialty 3D orthogonal weaving process and machines used for manufacturing this type of non-crimp fabric, see further information in [START_REF] Bogdanovich | Three-Dimensional Reinforcements for Composites[END_REF][START_REF] Mohamed | A new generation of 3D woven fabric preforms and composites[END_REF][START_REF] Mohamed | Comparative analysis of different 3D weaving processes, machines and products[END_REF].

The tensile properties and failure mechanisms of 3D woven composites have been investigated since the mid-1980s. Numerous works have compared the tensile properties of different kind 3D woven composites against two-dimensional plain weave laminates which were, in most of the cases, not equivalent, due to the principal parameters (such as fiber volume fraction and total thickness) were rarely matched, thus not allowing for a meaningful comparison. The exceptions can be found in [START_REF] Brandt | Mechanical performance of composites based on various threedimensional woven-fibre performs[END_REF] and in the 3 / 24 recent publications [START_REF] Lomov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 1: materials, methods and principal results[END_REF][START_REF] Ivanov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 2: comprehensive experimental results[END_REF]. The latter ones present a consistent experimental investigation of the quasistatic in-plane mechanical properties and damage development of 3D orthogonal weave composites. The authors have compared the quasi-static tensile properties and progression of damage of non-crimp 3D orthogonal weave composites to respective properties of the plain weave laminates having nearly equal total thickness and fiber volume fraction. Also importantly, same constituent materials, same resin system, same composite fabrication method and equipment were used in the production of both types of composites.

Fatigue data for non-crimp 3D woven composites are on a high demand by industry. Fatigue of crossply prepreg tape laminates and 2D woven fabric laminates, have been extensively studied (see recent review works [START_REF] Talreja | Damage and fatigue in composites -A personal account[END_REF][START_REF] Harris | Fatigue in composites. Science and technology of the fatigue response of fibrereinforced plastics[END_REF]). At the same time, this topic has been barely touched even for the 3D interlock weave composites, see [START_REF] Rudov-Clark | Tensile fatigue properties of a 3D orthogonal woven composite[END_REF][START_REF] Mouritz | Tensile fatigue properties of 3D composites with through-thickness reinforcement[END_REF][START_REF] Gude | Damage evolution of novel 3D textile-reinforced composites under fatigue loading conditions[END_REF] among few publications. To the best of these authors' knowledge, no fatigue test data are available in literature for non-crimp 3D orthogonal weave composites for either inplane or out-of-plane loading cases. Therefore, this work is viewed as initial effort to generate fatigue test data and get knowledge of fatigue behavior of this class of composites. For such initial effort, in-plane tension-tension fatigue loading in both principal reinforcement directions was chosen. The other fatigue loading cases, like in-plane tension-compression, flexure, in-plane and out-of-plane shear, are of a practical importance and have to be addressed in future work. Fatigue properties of composites reinforced with carbon, S-2 glass and ceramic fiber non-crimp 3D orthogonal woven performs are of a great practical interest too.

The present work aims to provide a contribution to scientific understanding of fatigue behavior of noncrimp 3D woven E-glass composites. One specific material of this kind has been manufactured and studied, side-by-side, with an equivalent thickness and fiber volume fraction plain weave reinforced laminate. The paper presents the characteristics of the two materials, the description of the quasi-static mechanical properties, the fatigue response to tensile-tensile cyclic loading and the transmitting light pictures observation aimed at comparing the damage patterns and their progression caused by cyclic loading at different applied stress amplitudes.

The materials investigated and compared in this study are (i) composite reinforced with a single-ply non-crimp 3D orthogonal weave E-glass fabric commercialized under trademark 3WEAVE ® (as illustrated in Figure 1a) and (ii) laminated composite reinforced with four plies of plain weave E-glass fabric (shown in Figure 1b). These two materials are termed in the rest of the paper as "3D" and "PW" fabrics and composites, respectively. The 3D composite is nominally identical to one of the materials (reinforced with "96 oz" fabric) which was characterized in [START_REF] Lomov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 1: materials, methods and principal results[END_REF][START_REF] Ivanov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 2: comprehensive experimental results[END_REF] for quasi-static loading, but it belongs to a different batch of production. Due to the possible minor differences between the production batches and the different fiber volume fractions, the quasi-static tensile testing, similar to the reported in [START_REF] Lomov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 1: materials, methods and principal results[END_REF][START_REF] Ivanov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 2: comprehensive experimental results[END_REF], has been performed for the material studied here. The results of that testing are summarized in the following section.

The non-crimp 3D single-ply orthogonal weave preform has the properties listed in Table 1a. The fiber architecture has 3 warp and 4 fill (a.k.a. "weft") layers. The fabric construction results in ~49%/~49%/~2% ratio of the fiber amounts (by volume) in the warp, fill and Z fiber directions.

The properties of the plain weave preform given in Table 1b show slightly unbalanced reinforcement architecture with respect to the total fiber volume content in the warp and weft directions. In order to reach the overall fiber balance in the composite laminate, the fabric layer lay-up (0/90/90/0º) was used when making the four-ply composite. Due to the balanced warp-and weft-directional fiber content in the resulting composite laminate, it was assumed that its in-plane properties are identical, and its mechanical characterization has been performed in one direction only.

The 3D woven fabric was produced by 3TEX Inc. on their proprietary 3D weaving machine, while the 2D woven fabric was produced on a regular Dornier weaving machine. The fiber material in both cases was PPG Hybon 2022 E-glass. Sample panels of both composites were fabricated in identical laboratory conditions at 3TEX Inc. using Dow Derakane 8084 Epoxy-Vinyl Ester resin. Both single-ply 3D and 4ply PW composites were fabricated in a vacuum bag using Vacuum Assisted Resin Transfer Molding (VARTM) method at room temperature. The 3D and PW composites have (see Table 2) almost identical areal density, very close sample thickness and fiber volume fraction.

QUASI-STATIC TENSILE TESTS

The experimental characterization of the composites included quasi-static in-plane tensile loading up to failure with acoustic emission (AE) registration, full-field surface strain mapping (SM), and direct damage observation on backlit samples (AE and SM devices are detailed in [START_REF] Lomov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 1: materials, methods and principal results[END_REF][START_REF] Lomov | Experimental methodology of study of damage initiation and development in textile composites in uniaxial tensile test[END_REF]).

The tensile tests were performed using an INSTRON 4505 machine (load cell 100kN) setting the crosshead speed of 2 mm/min. The prismatic samples had width 25 mm and total length 250 mm.

Aluminum tabs (1.5 mm thick) were glued in the grip zones with a free length of 150 mm. Test samples were cut out in two principal reinforcement directions determined by the textile preform architecture, i.e. in the warp and fill directions of the 3D composite and in the warp direction of the surface layers in the PW laminate.

The results of the quasi-static tensile tests are summarized in Table 3 for both composite materials and for each loading direction in terms of the Young's modulus (E), ultimate stress (σ ult ) and ultimate strain (ε ult ). Each average value and standard deviation in Table 3 was obtained from six tested samples.

The 3D composite exhibits very close Young's moduli in both the warp and fill directions. The difference between elastic modulus of the PW composite and the 3D composite is slightly greater than the respective standard deviations, i.e. it is statistically meaningful. At the same time, as Table 2 shows, the total fiber volume fraction is slightly higher in the case of the PW laminate. We suggest that absence of crimp in the 3D composite vs. the PW reinforced laminate not only compensates for a slightly lower fiber volume fraction in the former one, but results in considerably higher in-plane Young's moduli in both principal directions.

When comparing the ultimate stress and strain values presented in Table 3, a significant difference, much beyond the scatter of experimental data, is observed for the 3D composite in fill direction vs. PW laminate. The average ultimate stress is 26% higher and the average ultimate strain is 19% higher than respective values for the PW laminate. It is also seen in Table 3 that the average ultimate stress of the 3D composite in fill direction is 22% higher, and the average ultimate strain is 21% higher than respective values for the same material in warp direction. The 3D composite shows slightly higher average ultimate stress and nearly the same average ultimate strain in warp direction than the PW counterpart, although the differences are within statistical scatter.

The trends seen in Table 3 are practically the same as the ones revealed in [START_REF] Lomov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 1: materials, methods and principal results[END_REF], where same kinds of composites were compared; this makes it unnecessary to repeat the explanations and interpretations presented in the above work. The most important conclusion from both studies, which is essential for this work, is that the studied non-crimp 3D orthogonal weave composite has significantly higher in-plane strength and failure strain in fill direction than in warp direction, and than respective in-plane properties of the studied PW laminate. As suggested in [START_REF] Lomov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 1: materials, methods and principal results[END_REF], the main reasons for this may be in: (i) the absence of yarn crimp in the 3D composite and (ii) the lowest fiber damage caused by weaving in the fill direction of 3D preform, as discussed in [START_REF] Bogdanovich | Three-Dimensional Reinforcements for Composites[END_REF][START_REF] Mohamed | Comparative analysis of different 3D weaving processes, machines and products[END_REF].

TENSILE-TENSILE FATIGUE TESTS

The response to fatigue loading of the two composite materials was investigated by means of tensiontension cyclic tests. The geometry of the samples is the same as adopted in the quasi-static tensile tests described in the previous section.

Experimental Parameters

Two hydraulic MTS testing machines (load cell 100kN) were adopted for the fatigue investigation.

Tests were performed under constant stress amplitude, sinusoidal wave-form tensile-tensile loading and assuming the ratio R = 0.1 (ratio of the minimum to the maximum stress in the cycle). The stress was evaluated by using the sample cross-sectional area averaged from three measurements. Eight different maximum stress levels were considered, in order to get a complete characterization, from 350 MPa (close to the static failure stress) to the stress level at which complete failure did not occur at least before 5 million cycles (σ 5m ). Complete failure means separation in two parts of the sample during cyclic loading.

To avoid the influence of tab zones and grip pressure, a cyclic test was considered "valid" if the sample broke at a distance greater than 2 cm from the tabs. The tests were conducted till three samples showed valid failure for each considered stress level.

To understand the influence of the loading frequency and the temperature on the mechanical response of the two composites in both principal in-plane directions, preliminary DMA (Dynamic Mechanical Analysis) bending tests were performed assuming frequency in the range 0.1÷80 Hz and temperature from the room up to 80 o C (note that Tg temperature of Derakane 8084 resin used here is 115 o C). These tests were done with the same number of cycles for all frequencies. The main conclusion of that preliminary study was that below 40 o C the mechanical response of the materials does not vary for loading frequency between 1 and 30 Hz. Based on this conclusion, further fatigue characterization was performed for several loading frequencies in the range 1÷10 Hz, depending on the applied maximum stress and the correct response of the loading device, as detailed in Table 4. The temperature of the samples, measured by a laser thermometer, was not higher than 34 o C during cyclic loading which in any case is much lower than the resin's Tg.

Results

The response to cyclic tensile loading at different stress levels allows one to deduce the stress level σ 5m . According to the obtained experimental results, the σ 5m value has been determined as 60 MPa for both cases of the 3D composite in fill direction and the PW laminate, while it was determined as 55 MPa for the 3D composite in warp direction.

Principal information on the behavior of the composite materials during the cyclic loading is extracted from the observed variation of the shape of the fatigue stress-displacement cycles (further called a "cycle curve" for brevity) during the test. The plot in Figure 2a shows the comparison of some cycle curves of a 3D sample loaded in fill direction with a maximum stress of 60 MPa (e.g. σ 5m ); this is in the range of "high cycle" fatigue. The slope of the cycle curve segment passing through the points of maximum and minimum stress (further called a "cycle slope" for brevity) continuously decreases up to two million cycles (see Figure 2a). Above two million cycles, the slope remains nearly constant. These observations are further confirmed by observing the evolution of the maximum and minimum displacements under increasing number of cycles (see Figure 2b). The reduction of the cycle slope within the initial two million cycles suggests that there is some stiffness degradation of the material caused, in turn, by the damage accumulation during cycling.

Differently to what has been observed at σ 5m maximum stress level, the evolution of the damage reflects in a continuous decrease of the material stiffness if the maximum stress is in the range of low cycle fatigue. In Figure 3a, some cycles of a 3D sample fatigued in fill direction with a maximum stress of 300 MPa (note that average static strength is 540 MPa) show a continuous decrease of the cycle slope (respectively, a continuous decrease of the stiffness). The increasing hysteresis area of the cycles and the increase of the recorded maximum and minimum displacements (as seen in Figure 3b) demonstrate a gradual damage development, leading to the ultimate failure of the sample, as illustrated in Figure 4.

The observations described above are valid for the fatigue tests in warp direction of the 3D composite and for the PW laminate.

The fatigue tests performed in a wide stress range enable for depicting a Wöhler-type diagram for both the 3D and PW composites. In Figure 5a, twenty five valid tests in fill direction of the 3D composite (conducted at eight stress levels) are characterized by the variations of maximum stress in the cycle (σ max ) vs. the number of cycles to failure N. The results of twenty seven valid tests in warp direction of the 3D composite (conducted at nine stress levels) and the results of twenty seven valid tests of the PW composite (conducted at eight stress levels) are shown in Figure 5b and Figure 5c, respectively. In each diagram of Figure 5, the respective average static tensile strength (given in Table 3) is also indicated in its correspondence to the extrapolated lowest number of cycles, N=1.

A comparison of the fatigue behavior is detailed in Figure 6 and Table 5. Figure 6 shows the actual "fatigue life" (e.g. number of cycles to failure) for different stress levels. Table 5 presents a comprehensive statistical analysis for each stress level in term of average, standard deviation and covariance of the cycles to failure. The fatigue performance for each stress level of the 3D composite in warp and fill directions is found in exact correspondence with the trends observed in the static tests (see Section 3). Firstly, the PW composite shows, except for σ 5m , significantly worse fatigue performance than the 3D composite tested in fill direction. Secondly, the fill-directional cyclic loading of the 3D composite vs. the PW composite results in the average number of cycles ~51% higher at 350 MPa stress, ~38% higher at 150 MPa stress, and ~31% higher at 70 MPa stress (see Table 5). Further on, the fatigue life of the 3D composite is much longer in fill direction than in warp for all of the applied stress levels. Specifically, the average number of cycles to ultimate failure is ~39% higher at the stress 350 MPa, ~50% higher at the stress 150 MPa, and ~239% higher at the stress 70MPa (Table 5). The PW composite shows a shorter fatigue life vs. the 3D composite in warp direction for stress level 350 MPa (the number of cycles is ~7% higher for 3D warp). However, for the stress levels 150 MPa and 70 MPa the average numbers of cycles till ultimate failure are higher in the case of the PW composite (by ~9% and ~160%, respectively, see Figure 6 and Table 5). reliable predictions of the fatigue life for the other stress levels, which were not investigated in the present experimental study. Such a fitting is not aimed at determining the best possible fitting of the obtained experimental data. The diagrams in Figure 5 show the fitting of experimental results by means of three linear segments (for the 3D composite in fill direction the two segments coincide along the same straight line) in three maximum stress ranges: (I) σ 5m ÷100 MPa, (II) 100÷350 MPa, (III) 350 MPa÷ σ ult . This separation into the three distinct stress ranges correlates with different dominating damage types imparted during the cyclic loading (see further details in section 5 for two stress levels in range (I) and (II)) and has an analogy with the separation of the modes of fatigue damage in [START_REF] Talreja | Damage and fatigue in composites -A personal account[END_REF].

Here, the following fitting by a semi-logarithmic function is used:

c N m + = σ log max (1)
In eqn. (1), 'm' and 'c' are parameters to be defined by the least squares algorithm. The proposed fitting of the fatigue life over the entire stress range is depicted in Figure 7. The quality of the fitting is related to the coefficient of correlation R 2 [START_REF] Bain | Introduction to probability and mathematical statistics[END_REF]. Values of R 2 close to 1 confirm the reliability of the fitting. All curves in Figure 7 have the coefficient of correlation in the range 0.95÷1.

A comparison of all of the obtained fatigue curves for both composites is presented in Figure 7. The 3D composite fatigued in warp direction exhibits considerably lower fatigue performance in comparison to the PW counterpart for the lower stress levels; both show comparable performance for the intermediate stress levels; the 3D composite shows slightly better performance for the higher stress levels. The difference in the fatigue performance of the 3D composite in the warp and fill directions can be attributed to several reasons. Firstly, there is more of a fiber damage imparted to the warp yarns than to the fill ones during weaving. As hypothesized in [START_REF] Lomov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 1: materials, methods and principal results[END_REF], this may also be the major reason causing significant difference in the static strength values (and also observed in the static testing of the present work, see Table 3). The significance of this effect has been recently confirmed by a comprehensive experimental study performed at 3TEX for all of the dry E-glass yarns used in the "96 oz" 3D non-crimp orthogonal woven preform.

Results of that study will be published elsewhere. Secondly, as illustrated by Figure 1a, in a 3D orthogonal weave composite the presence of Z yarns creates multiple local "pockets" of pure matrix. It has been discovered earlier for various types of 3D reinforced composites (stitched and Z-pinned [21, 22, and 200 MPa (in range II of the Wöhler diagram, see Figure 7). Samples fatigued at the stress level σ 5m have been monitored up to 3 million cycles, while for samples loaded at 200 MPa stress the damage evolution has been monitored till complete failure.

To highlight important peculiarities of the observed local damage patterns and explain the evolution of the local damage events for the 3D and PW composites and different loading directions, pictures of the backlit samples have been taken during loading after different numbers of cycles and were compared in Figure 8 and Figure 9 for maximum stress σ 5m and 200 MPa, respectively (the loading direction is horizontal; vertical direction corresponds to the sample width ). These pictures show a frame of width 25 mm and length 30 mm focused on the center of the sample.

First interesting observation is that certain damage was imparted during the first stage of fatigue loading (see Figure 8 after 10,000 cycles). No significant damage has developed if σ 5m level stress was applied quasi-statically, as indicated by the AE registrations. The comparison of pictures for a fixed number of cycles shows that in the PW laminate there is a higher density of initiated transverse cracks (see Figure 8 after 10,000 cycles). This shows that, similarly to the case of the quasi-static tensile loading, the 3D composite is less prone to transverse cracking within the yarns or at the yarn-matrix interfaces as compared to the PW composite counterpart.

When increasing the number of cycles, longitudinal cracks appear in the PW composite after approximately 100,000 cycles, while in the 3D composite only transverse cracks with increasing length and density are observed at that number of cycles. Longitudinal cracks in both loading directions of the 3D composite are still not visible below 3 million cycles. This observation confirms that, similar to the quasi-static loading case, under the cyclic loading the 3D woven composite is less prone to longitudinal cracking.

The main differences observed for the two loading cases of the 3D composite under maximum stress σ 5m are:

• for warp-directional loading the existing transverse cracks increase in their length and more significantly in their width;

• for fill-directional loading the transverse cracks increase in number and mutually interconnect.

of the transverse cracks produced during the fill-loading.

Pictures in Figure 8 show that for the 3D composite, after 1 million cycles, the damage patterns do not change significantly in regard of transverse cracks density; only few cracks continue to increase in width and length. This observation matches the conclusion deduced in section 4, where nearly constant cycle slope was observed after approximately same number of cycles has been reached. The PW laminate shows a stable pattern of transverse cracks developed after 1 million cycles. A continuous development of longitudinal cracks gradually increasing their length takes place.

The damage evolution during fatigue tests under maximum stress 200 MPa is illustrated in Figure 9.

The damage pattern after the first cycle is similar to that seen in a quasi-static tensile test under the same stress level. In fill-directional fatigue loading of the 3D composite, the transverse cracks saturate after approximately 100 cycles; at that number of cycles the longitudinal cracks and the cracks on Z yarn surfaces in fill direction are initiated. Further cycling induces a gradual growth and multiplication of the aforementioned longitudinal cracks; the immediate predecessor of ultimate failure is an intensive formation of transverse and longitudinal cracks inside the yarns, which leads to a macro-splitting of 3D woven composite, subsequent fibre breakage in the loading direction, and separation of the sample into parts.

Under warp-directional fatigue loading of the 3D composite, the transversal cracks increase in length and in width faster than it occurs under fill-directional loading. The predecessors of ultimate failure are longitudinal cracks and local debonds around Z yarns (see Figure 9 after 1,000 cycles). The cycling of the PW laminate causes delamination associated with intensive development of the longitudinal cracks.

Moreover, in the PW laminate the longitudinal cracking begins right after the first cycle, while in the 3D composite only few longitudinal cracks had appeared even after 100 cycles of the fatigue loading.

The main conclusions from the damage monitoring during fatigue tests are as follows.

The different slopes of the fatigue curve in range (I) and (II) (see Figure 7) could be related to the different predominant damage mechanism: transverse cracks in range (I) and longitudinal cracks in range (II).

Inherent crimp of the PW fabric and irregularities of the fabric ply nesting during lamination induce stress concentrations and facilitate damage development during cyclic loading. The damage is initiated earlier during fatigue loading in the PW laminate than in the 3D woven composite.

The absence of crimp and nesting provides better fatigue performances of the 3D composite (in terms of delayed damage initiation) when tested in both fill and warp directions.

A worse fatigue performance of the 3D composite under warp-directional loading could be attributed to the type of damage and the rate of damage development during fatigue loading. The fabric architecture of the 3D woven preform (see Figure 1a) explains this phenomena: when the load is applied in warp direction (which is, at the same time, the direction of the Z yarn crowns), both the warp and Z yarns are loaded in tension. As the consequence, the stress concentration induced by the Z yarn in its crossover area with the fill yarn becomes significant. This is confirmed by the damage pattern observed during cyclic warp-directional loading of the 3D composite. Indeed, longitudinal cracks appear in the Z yarn crossovers (see pictures in Figure 9 at failure), and local debonds start from the crossovers due to relative displacements between adjacent warp and Z yarns.

It appears that the influence of Z-crowns depends more on the number of cycles than on the applied stress level. In fact, the static strength of the 3D composite in warp direction is higher than that of the PW laminate, yet the fatigue performance of the 3D composite in warp direction at relatively low stress levels appears to be worse than that of the PW laminate. This behaviour could be the consequence of the frictional forces acting cyclically between warp yarns on one side, and through thickness segments of Z yarns on the other, as described above.

CONCLUSIONS

The experimental study presented in this work addressed the fatigue tensile mechanical response of a single-ply non-crimp 3D orthogonal weave E-glass composite and of a laminated composite (reinforced with four plies of a standard plain weave fabric). The tensile-tensile fatigue tests have shown that:

1) The fatigue performance of the PW composite is significantly worse than that of the 3D composite tested in fill direction for all the studied stress levels, except for the σ 5m stress level.

14 / 24

2) The 3D composite has significantly longer fatigue life when tested in fill direction than when tested in warp direction for all of the applied stress levels.

3) The 3D composite tested in warp direction has slightly longer fatigue life for the low-cycle fatigue stress range, ≥350 MPa, but significantly shorter fatigue life for the high-cycle fatigue stress range, ≤150 MPa, than the PW laminate.

The reliability of these three conclusions, in term of fatigue life, can be assessed by the confidence level index [START_REF] Bain | Introduction to probability and mathematical statistics[END_REF]. The confidence levels in Table 6 show that, in strict statistics terms, the hypothesis "the 3D composite in fill direction has a longer fatigue life than the PW laminate" is valid with confidence level 95÷99% (except for σ 5m ); the hypothesis "the 3D composite in fill direction has a longer fatigue life than the 3D composite in warp direction" is valid with confidence level 98÷99%; the hypothesis "the 3D composite in warp direction has a shorter fatigue life than the PW laminate for stress levels ≤ 150MPa" is valid with confidence level 98÷99%.

The damage observations made during tensile-tensile fatigue loading suggest that:

1) When cycling at the stress level σ 5m , longitudinal cracks appear in the PW laminate, while in the 3D composite only transverse cracks develop.

2) The 3D composite exposed to cyclic maximum stress σ 5m shows transverse cracks increasing in length and width under warp-directional loading but increasing in number and interconnections under fill-directional loading.

3) Fatigue loading at stress level 200 MPa facilitates in the 3D composite a faster increase of the length and width of transversal cracks under warp-directional loading than under fill-directional loading.

LIST OF FIGURE CAPTIONS

LIST OF TABLE CAPTIONS

Table 1. Properties of the preforms: (a) 3D and (b) PW. Table 2. Properties of 3D and PW composites (± means standard deviation). Table 3. Average values and standard deviations of the quasi-static mechanical properties. Table 4. Frequency of the cyclic loading adopted for the fatigue tests at different stress levels. Table 5. Statistical parameters of the fatigue life at the different stress levels. Table 6. Confidence levels for three hypotheses ('>' means longer fatigue life, '<' means shorter fatigue life).

Fabric plies 1

Areal

4 )

 4 In the PW laminate the longitudinal cracking begins after the first loading cycle with maximum stress of 200 MPa, while in the 3D composite only few longitudinal cracks appear even after first hundred cycles. In general, inherent crimp of the PW fabric and irregularities of the fabric ply nesting during lamination induce local stress concentrations and facilitate earlier damage initiation and development in the PW laminate during cyclic loading in comparison with the 3D composite. The absence of crimp and ply nesting makes, obviously, a strong positive effect on the fatigue behaviour of the non-crimp 3D woven fabric composite in the fill-directional loading case. 27. Mouritz AP. Compression properties of z-pinned composite laminates. Composites Science and Technology 2007; 67: 3110-3120. 28. Chang P, Mouritz AP, Cox BN. Flexural properties of z-pinned laminates. Composites Part A 2007; 38: 244-251.

Figure 1 .

 1 Figure 1. Internal architecture of the: (a) non-crimp 3D orthogonal weave single-ply preform (fill yarns on the surface, warp yarns underneath them, and Z-crowns oriented in warp direction over fill yarns); (b) 2D plain weave fabric.Figure 2. 3D composite loaded in fill direction, max stress 60 MPa: (a) some stress vs. displacement curves; (b) maximum and minimum stress and displacement vs. number of cycles.Figure 3. 3D composite loaded in fill direction, max stress 300 MPa: (a) some stress vs. displacement curves; (b) maximum and minimum stress and displacement vs. number of cycles.Figure 4. 3D composite loaded in fill direction, max stress 300 MPa: fatigue failure of a sample.Figure 5. Maximum stress σµαξ vs. number of cycles to failure N for: (a) 3D composite loaded in fill direction; (b) 3D composite loaded in warp direction; (c) PW laminate. ('→' means no failure).Figure 6. Comparison of the fatigue life at different maximum stress levels: (a) 350 MPa, (b) 150 MPa and (c) 70 MPa. Vertical lines indicate average fatigue life.Figure 7. Fitting and comparison of the fatigue life curves.Figure 8. Damage development during fatigue loading for σµαξ=σ5µ.Figure 9. Damage development during fatigue loading for σµαξ=200 MPa.

Figure 2 .

 2 Figure 1. Internal architecture of the: (a) non-crimp 3D orthogonal weave single-ply preform (fill yarns on the surface, warp yarns underneath them, and Z-crowns oriented in warp direction over fill yarns); (b) 2D plain weave fabric.Figure 2. 3D composite loaded in fill direction, max stress 60 MPa: (a) some stress vs. displacement curves; (b) maximum and minimum stress and displacement vs. number of cycles.Figure 3. 3D composite loaded in fill direction, max stress 300 MPa: (a) some stress vs. displacement curves; (b) maximum and minimum stress and displacement vs. number of cycles.Figure 4. 3D composite loaded in fill direction, max stress 300 MPa: fatigue failure of a sample.Figure 5. Maximum stress σµαξ vs. number of cycles to failure N for: (a) 3D composite loaded in fill direction; (b) 3D composite loaded in warp direction; (c) PW laminate. ('→' means no failure).Figure 6. Comparison of the fatigue life at different maximum stress levels: (a) 350 MPa, (b) 150 MPa and (c) 70 MPa. Vertical lines indicate average fatigue life.Figure 7. Fitting and comparison of the fatigue life curves.Figure 8. Damage development during fatigue loading for σµαξ=σ5µ.Figure 9. Damage development during fatigue loading for σµαξ=200 MPa.

Figure 3 .

 3 Figure 1. Internal architecture of the: (a) non-crimp 3D orthogonal weave single-ply preform (fill yarns on the surface, warp yarns underneath them, and Z-crowns oriented in warp direction over fill yarns); (b) 2D plain weave fabric.Figure 2. 3D composite loaded in fill direction, max stress 60 MPa: (a) some stress vs. displacement curves; (b) maximum and minimum stress and displacement vs. number of cycles.Figure 3. 3D composite loaded in fill direction, max stress 300 MPa: (a) some stress vs. displacement curves; (b) maximum and minimum stress and displacement vs. number of cycles.Figure 4. 3D composite loaded in fill direction, max stress 300 MPa: fatigue failure of a sample.Figure 5. Maximum stress σµαξ vs. number of cycles to failure N for: (a) 3D composite loaded in fill direction; (b) 3D composite loaded in warp direction; (c) PW laminate. ('→' means no failure).Figure 6. Comparison of the fatigue life at different maximum stress levels: (a) 350 MPa, (b) 150 MPa and (c) 70 MPa. Vertical lines indicate average fatigue life.Figure 7. Fitting and comparison of the fatigue life curves.Figure 8. Damage development during fatigue loading for σµαξ=σ5µ.Figure 9. Damage development during fatigue loading for σµαξ=200 MPa.

Figure 4 .Figure 5 .

 45 Figure 1. Internal architecture of the: (a) non-crimp 3D orthogonal weave single-ply preform (fill yarns on the surface, warp yarns underneath them, and Z-crowns oriented in warp direction over fill yarns); (b) 2D plain weave fabric.Figure 2. 3D composite loaded in fill direction, max stress 60 MPa: (a) some stress vs. displacement curves; (b) maximum and minimum stress and displacement vs. number of cycles.Figure 3. 3D composite loaded in fill direction, max stress 300 MPa: (a) some stress vs. displacement curves; (b) maximum and minimum stress and displacement vs. number of cycles.Figure 4. 3D composite loaded in fill direction, max stress 300 MPa: fatigue failure of a sample.Figure 5. Maximum stress σµαξ vs. number of cycles to failure N for: (a) 3D composite loaded in fill direction; (b) 3D composite loaded in warp direction; (c) PW laminate. ('→' means no failure).Figure 6. Comparison of the fatigue life at different maximum stress levels: (a) 350 MPa, (b) 150 MPa and (c) 70 MPa. Vertical lines indicate average fatigue life.Figure 7. Fitting and comparison of the fatigue life curves.Figure 8. Damage development during fatigue loading for σµαξ=σ5µ.Figure 9. Damage development during fatigue loading for σµαξ=200 MPa.

Figure 9 .

 9 Figure 1. Internal architecture of the: (a) non-crimp 3D orthogonal weave single-ply preform (fill yarns on the surface, warp yarns underneath them, and Z-crowns oriented in warp direction over fill yarns); (b) 2D plain weave fabric.Figure 2. 3D composite loaded in fill direction, max stress 60 MPa: (a) some stress vs. displacement curves; (b) maximum and minimum stress and displacement vs. number of cycles.Figure 3. 3D composite loaded in fill direction, max stress 300 MPa: (a) some stress vs. displacement curves; (b) maximum and minimum stress and displacement vs. number of cycles.Figure 4. 3D composite loaded in fill direction, max stress 300 MPa: fatigue failure of a sample.Figure 5. Maximum stress σµαξ vs. number of cycles to failure N for: (a) 3D composite loaded in fill direction; (b) 3D composite loaded in warp direction; (c) PW laminate. ('→' means no failure).Figure 6. Comparison of the fatigue life at different maximum stress levels: (a) 350 MPa, (b) 150 MPa and (c) 70 MPa. Vertical lines indicate average fatigue life.Figure 7. Fitting and comparison of the fatigue life curves.Figure 8. Damage development during fatigue loading for σµαξ=σ5µ.Figure 9. Damage development during fatigue loading for σµαξ=200 MPa.

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Internal architecture of the: (a) non-crimp 3D orthogonal weave single-ply preform (fill yarns on the surface, warp yarns underneath them, and Z-crowns oriented in warp direction over fill yarns); (b) 2D plain weave fabric.

Figure 4 .Figure 5 .Figure 6 .Figure 7 .

 4567 Figure 4. 3D composite loaded in fill direction, max stress 300 MPa: fatigue failure of a sample.

Figure 8 .Figure 9 .

 89 Figure 8. Damage development during fatigue loading for σ max =σ 5m .

Table 1 . Properties of the preforms: (a) 3D and (b) PW.

 1

	3D non-crimp woven preform
	density [g/m 2 ]	3255
	Warp Insertion density [ends/cm]	2.76
	layer Top and bottom layer yarns [tex]	2275	Plain weave preform
	Middle layer yarns [tex]	1100	Fabric plies		4
	Weft Insertion density [picks/cm]	2.64	Areal density [g/m 2 ]	3260
	layer Yarns [tex]	1470	Ply insertion density: Warp [ends/cm]	1.95
	Z-yarns Insertion density [ends/cm]	2.76		Weft [picks/cm]	1.6
	Yarns [tex]	1800	Warp and Weft Yarns [tex]	2275
	(a)			(b)
			3D	PW
	Panels thickness [mm]		2.58 ± 0.05	2.57 ± 0.09
	Fiber volume fraction [%]	53.22 ± 0.63 54.42 ± 0.65

Table 2 . Properties of 3D and PW composites (± means standard deviation).

 2

		E	σ ult	ε ult
		[GPa]	[MPa]	[%]
	3D-Fill	26,34 ± 0.63	540 ± 20	2,92 ± 0.05
	3D-Warp	26,39 ± 0.76	441 ± 26	2,41 ± 0.13
	PW	24,68 ± 1.51	427 ± 23	2,45 ± 0.18

Table 3 . Average values and standard deviations of the quasi-static mechanical properties.

 3

	Maximum Stress [MPa]	350	300	200	150	≤95
	F r e q u e n c y [ H z ]	1	2	4	8	1 0

Table 4 . Frequency of the cyclic loading adopted for the fatigue tests at different stress levels.

 4

	Stress level		3D-Fill			3D-Warp			PW
	[MPa]	Average Stand. Dev. Covariance Average Stand. Dev. Covariance Average Stand. Dev. Covariance
	55				5,000,000	n/a	n/a
	60	5,000,000	n/a	n/a	2,393,181	33,563	1.4%	5,000,000	n/a	n/a
	70	2,263,028 104,676	4.6%	666,110	93,163	14.0%	1,731,774 162,194	9.4%
	80	771,452	184,724	23.9%	153,056	38,828	25.4%	433,181	190,699	44.0%
	95	192,313	34,368	17.9%	55,018	17,772	32.3%	105,785	37,825	35.8%
	150	13,595	995	7.3%	9,083	164	1.8%	9,883	233	2.4%
	200	3,372	118	3.5%	2,506	268	10.7%	2,723	292	10.7%
	300	344	6	1.7%	256	22	8.4%	299	11	3.6%
	350	118	13	11.1%	84	15	17.2%	78	32	41.3%

Table 5 . Statistical parameters of the fatigue life at the different stress levels.

 5

	Stress level	Confidence Level for a hypothesis
	[MPa]	3D-Fill > PW	3D-Fill > 3D-Warp 3D-Warp < PW
	70	99.75%	100.00%	99.99%
	80	96.08%	99.97%	98.72%
	95	99.28%	99.99%	98.81%
	150	99.93%	99.97%	99.77%
	200	99.20%	99.89%	82.80%
	300	99.93%	99.95%	98.64%
	350	95.24%	98.49%	61.27%

Table 6 . Confidence levels for three hypotheses ('>' means longer fatigue life, '<' means shorter fatigue life).

 6

Acknowledgements

The research reported here has been carried out in the framework of the Erasmus Master program of Giulia Gramellini in K.U. Leuven and Politecnico di Milano. The help of laboratory staff of the Department MTM -Bart Pelgrims, Kris van der Staey and Johan Vanhuist -is gratefully acknowledged.

 [START_REF] Steeves | Mechanics of failure in composite structures[END_REF][START_REF] Steeves | In-plane properties of composite laminates with through-thickness pin reinforcement[END_REF][START_REF] Partridge | Manufacture and performance of z-pinned composites[END_REF][START_REF] Chang | Properties and failure mechanisms of z-pinned laminates in monotonic and cyclic tension[END_REF]27,28]) that through-thickness matrix channels increase local stress concentration and reduce resistance of a composite to initial damage formation. It was also shown experimentally that the presence of such channels may significantly, up to 30%, reduce the in-plain static strength of stitched and Z-pinned composites under tensile and compressive loads. However, there is important difference in our case: as is seen in Figure 1a, there are no through-thickness matrix channels between adjacent fill yarns; instead, there are only relatively thin matrix pockets limited in their thickness to the thickness of the fill yarn layer. Yet, as experimental data show, even such small matrix pockets play significant role in the crack initiation process of 3D woven composites.

It is also worth noting that in the warp-directional loading case Z yarns are directly subjected to the inplane load, while in the fill-directional loading case they are not loaded directly. Possible effect caused by this aspect is, that the cyclic frictional contact between warp and Z yarns differs between the two loading cases. Under tensile loading in warp direction, the tensile stress is applied to warp and Z yarns, while the reduction of the sample width takes place due to Poisson's effect. The latter one results in a respective reduction of the distance between adjacent Z and warp yarns. These yarns could get in a contact and, due to both of them are exposed to cyclic tensile stress, their mutual friction effect may become sufficient to significantly reduce the overall fatigue life of the sample. This hypothesis will be the subject of a more indepth investigation in the future work.

FATIGUE DAMAGE DEVELOPMENT

Damage initiation and development during tensile-tensile fatigue tests have been investigated by means of taking pictures of backlit samples subjected to cyclic loading. The same technique was used in [START_REF] Ivanov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 2: comprehensive experimental results[END_REF], together with microscopical investigation of the cross sections of the damaged samples, and it was shown that for the studied material the backlit damage characterisation is an accurate tool. For fatigue loading the backlit technique has a great advantage: it allows continuous registration of damage development for the same sample, which cannot be done via micrography. Tensile-tensile fatigue tests with accompanied damage observation have been performed for the 3D and PW composite materials loaded in the same principal in-plane directions as described before. The following two maximum stress levels were used in the cycle: σ 5m (stress level contained in range I of the Wöhler diagram, see Figure 7)