Cédric Nicolas Jacon

C Édric Lecouvey

CRYSTAL ISOMORPHISMS AND WALL CROSSING MAPS FOR RATIONAL CHEREDNIK ALGEBRAS

Crystal isomorphisms and wall-crossing maps for rational Cherednik algebras

Introduction

The rational Cherednik algebra associated to a complex reflection group W was introduced by Etingof and Ginzburg in [START_REF] Etingof | Ginzburg Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism[END_REF] as a particular symplectic reflection algebra. In this paper, we will focus on the case where W is the complex reflection group G(l, 1, n) := (Z/lZ) n ⋊S n . The rational Cherednik algebra then depends on the choice of a certain parameter s in a (l + 1)dimensional C-vector space. There is a distinguished category of modules over these algebras, the category O, which may be constructed in the same spirit as the BGG category for a reductive Lie algebra. This category has been intensively studied during the last decade because of its interesting structure and also its connection with other important mathematical objects such as cyclotomic Hecke algebras or cyclotomic q-Schur algebras.

As the simple modules of the associated complex reflection groups, the simple modules in this category are labelled by the set of multipartitions. A natural and important question is then to understand how the set of simple modules in the category O for different choices of the parameter s are related. In [START_REF] Losev | Supports of simple modules in cyclotomic Cherednik categories O[END_REF], Losev has defined a collection of hyperplanes in the space of parameters called "essential walls". Then, he has shown the existence of a perverse and derived equivalence between the categories O associated to distinct parameters s and s ′ with integral difference separated by a single wall. In particular, this equivalence induces a bijection between the simples of these categories. These bijections, called "wall crossing maps", are of great interest because they commute with both actions of the Heisenberg algebra and the affine type A algebra g. In particular, they can be used to compute the support (as coherent sheaves on the reflection representation of W ) of the simple modules in the category O and to obtain a classification of the finite-dimensional irreducible representations. Indeed, by the works of Losev [START_REF] Losev | Supports of simple modules in cyclotomic Cherednik categories O[END_REF], in the case where the parameter s satisfies a certain "asymptotic" condition, the support can be quite easily computed. Now, one can always go from a category O parametrized by a parameter s to a category O parametrized by an asymptotic parameter by using a composition of these wall-crossing maps. As, in addition, they leave the support invariant, these wall-crossing maps thus allow its computation in general.

The goal of the present paper is to show that, despite their very abstract definition, there is a simple combinatorial procedure to compute the wall crossing bijections.

Because of the above properties, these bijections can also be interpreted as crystal isomorphisms for certain integrable g-modules: the Fock spaces. On the other hand, in [START_REF] Jacon | Crystal isomorphisms for irreducible highest weight Uv( sle)-modules of higher level Algebras and Representation[END_REF], we described distinguished crystal isomorphisms between such Fock spaces and presented a simple procedure on multipartitions of fixed rank to compute them. The aim of this paper is to show how these two isomorphisms are related. To do this, we first review the Uglov g-module structure of the Fock space and explain how it can be extended. We then construct and explicitly describe crystal isomorphisms corresponding to this generalized structure. The last section explores the connections of these isomorphisms with Losev's wall crossing bijections. We explain how the crystals appearing in the context of Cherednik algebras are related with the usual crystals of Fock spaces (in the sense of Uglov). Finally, the main result gives a simple combinatorial procedure on l-partitions of fixed rank for computing an arbitrary wall crossing bijection without referring to any crystal structure Acknowledgement. The authors thank Ivan Losev for useful discussions on his work [START_REF] Losev | Supports of simple modules in cyclotomic Cherednik categories O[END_REF][START_REF] Losev | Rational Cherednik algebras and categorification[END_REF]. The authors also thank the referee for helpful comments and suggestions. Both Authors are supported by "Projet ANR ACORT : ANR-12-JS01-0003".

Colored graphs and Fock spaces of JMMO type

We first describe structures of colored oriented graphs on the sets of lpartitions (that we define below), then focus on a particular case which is connected to the action of the quantum affine group U q ( sl e ).

Combinatorics of l-partitions.

Definition 2.1. Let n ∈ N and l ∈ N:

• A partition λ of rank n is a sequence

(λ 1 , . . . , λ r ),
of decreasing non negative integers such that

1≤i≤r λ i = n.
By convention, we identify two partitions which differ by parts equal to 0.

• A l-partition (or multipartition) λ of rank n is an l-tuple of partitions (λ 1 , . . . , λ l ) such that the sum of the rank of the partitions λ i for 1 ≤ i ≤ l is n. We denote by Π l (n) the set of all l-partitions of rank n and if λ ∈ Π l (n), we sometimes write λ ⊢ l n. The empty l-partition (which is the l-tuple of empty partitions) is denoted by ∅.

Let κ ∈ Q + and s = (s 1 , s 2 , . . . , s l ) ∈ Q l . We set s := (κ, s).

One can associate to each λ ⊢ l n its Young diagram:

[λ] = {(a, b, c) a ≥ 1, 1 ≤ c ≤ l, 1 ≤ b ≤ λ c a }.
This diagram sometimes will be identified with the l-partition itself. We define the s-content of a node γ = (a, b, c) ∈ [λ] as follows:

cont(γ) = b -a + s c ∈ Q,
and the residue of γ is by definition the content of the node in Q/κ -1 Z when κ = 0 and equals the residue when κ = 0. Definition 2.2. Let I s be the subset of Q/κ -1 Z formed by the classes x + s j + κ -1 Z where x ∈ Z and j ∈ {1, . . . , l} when κ = 0 and the subset of Q formed by the rationals of the form x + s j where x ∈ Z and j ∈ {1, . . . , l} when κ = 0.

We say that γ is a z-node of λ when res(γ) = z + κ -1 Z. Finally, we say that γ is removable when γ = (a, b, c) ∈ λ and λ\{γ} is an l-partition. Similarly γ is addable when γ = (a, b, c) / ∈ λ and λ ∪ {γ} is an l-partition.

Fix z ∈ I s . We assume that we have a total order ≤ on the set of z-nodes of an arbitrary l-partition. We then define two operators depending on z as follows. We consider the set of addable and removable z-nodes of our l-partition. Let w z (λ) be the word obtained first by writing the addable and removable z-nodes of λ in increasing order with respect to ≤ next by encoding each addable z-node by the letter A and each removable i-node by the letter R. Write w z (λ) = A p R q for the word derived from w z by deleting as many subwords of type RA as possible. The word w z (λ) is called the zword of λ and w z (λ) the reduced z-word of λ. The addable z-nodes in w z (λ) are called the normal addable z-nodes. The removable z-nodes in w z (λ) are called the normal removable z-nodes. If p > 0, let γ be the rightmost addable z-node in w z . The node γ is called the good addable z-node. If q > 0, the leftmost removable i-node in w z is called the good removable z-node.

We then define e ≤ z µ = λ and f ≤ z λ = µ if and only if µ is obtained from λ by adding to λ a good addable z-node, or equivalently, λ is obtained from µ by removing a good removable z-node. If µ as no good removable z-node then we set e ≤ z µ = 0 and if λ has no good addable z-node we set f ≤ z λ = 0. 2.2. Extended JMMO structure associated to s. We can define from s and ≤ a colored oriented graph G s,≤ as follows:

• vertices : the l-partitions λ ⊢ l n with n ∈ Z ≥0

• the arrows are colored by elements in I s and we have:

λ i → µ for i ∈ Q/κ -1 Z if and only if e ≤ i µ = λ, or equivalently f ≤ i λ = µ.
The l-partitions such that e z µ = 0 for all z will be called highest weight vertices. The set I s of elements in Q/κ -1 Z coloring the arrows is called the indexing set of the graph. Observe the graph G s,≤ is not an affine A

(1) e-1crystal graph in general notably because its indexing set can be distinct from Z/eZ.

Finally two graphs G s 1 ,≤ 1 and G s 1 ,≤ 2 on l-partitions with indexing sets I 1 and I 2 are isomorphic if there exists a bijection [START_REF] Etingof | Ginzburg Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism[END_REF] Ψ :

Π l (n) → Π l (n),
and a bijection:

(2) ψ :

I 1 → I 2 , such that • λ is a highest weight vertex in G s 1 ,≤ 1 if and only if Ψ(λ) is a highest weight vertex in G s 2 ,≤ 2 ,
• we have an arrow

λ i → µ in G s 1 ,≤ 1 if
ψ(i) → Ψ(µ) in G s 2 ,≤ 2 .
In addition, if Ψ is the identity, we will say that the two graphs are equivalent (in particular, the graph structures coincide up to their coloring). We will see that, for a good choice of the orders ≤ 1 and ≤ 2 , the graphs G s 1 ,≤ 1 and G s 2 ,≤ 2 have the structure of a Kashiwara crystal graph. In this case, a crystal isomorphism between G s 1 ,≤ 1 and G s 2 ,≤ 2 is a graph isomorphism such that ψ = id. In particular, each crystal isomorphism yields a graph isomorphism but the converse is false in general.

2.3.

Extended JMMO Fock space structure. We now define a Fock space structure that, as far as we know, first appeared in the work of Gerber [START_REF] Gerber | Generalised canonical basic sets for Ariki-Koike algebras[END_REF]. This structure generalizes the ones defined by Uglov [START_REF] Uglov | Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials[END_REF] and Jimbo-Misra-Miwa-Okado (JMMO) [START_REF] Jimbo | Combinatorics of representations of Uq( sl(n)) at q = 0[END_REF].

Condition 2.3. We assume in this section that s = (s 1 , . . . , s l ) ∈ Z l and κ = 1/e where e ∈ N >1 ⊔ {∞} (with κ = 0 when e = ∞).

Let g e := U q ( sl e ) be the quantum group of affine type A

(1) e-1 if e is finite, otherwise we set g ∞ := U q (sl ∞ ). The associative Q(q)-algebra g e has generators e i , f i , t i , t -1 i (for i = 0, ..., e -1) and ∂. We refer the reader to [12, §2.1] for the relations they satisfy (we do not use them in the sequel.) We denote by U ′ q ( sl e ) the subalgebra generated by e i , f i , t i , t -1 i (for i = 0, ..., e -1). We write Λ i , i = 0, ..., e -1 for the fundamental weights of g e .

The Fock space F is the Q(q)-vector space defined as follows:

F = n∈Z ≥0 λ⊢ l n Q(q)λ.
Set I = Z if e = ∞ and I = Z/eZ otherwise.

For any e ∈ N >1 ⊔ {∞}, there is an action of g e on the Fock space F. This action can be regarded as a generalization of Uglov's construction [12, §2.1]. It is defined by using an order on the nodes on the l-partitions with the same residue modulo e. This order depends on the choice of an l-tuple of rational numbers m = (m 1 , . . . , m l ). If e is finite, for 1 ≤ i, j ≤ l with i = j and N ∈ Z, let us define

m s,e i,j,N := {(m 1 , . . . , m l ) ∈ Q l |s i -m i -(s j -m j ) = N.e}. If e = ∞, we define for 1 ≤ i, j ≤ l m s,∞ i,j := {(m 1 , . . . , m l ) ∈ Q l |s i -m i -(s j -m j ) = 0}. Let M s,
e be the union of the hyperplanes m s,e i,j,N for all 1 ≤ i, j ≤ l and N ∈ Z if e is finite and m s,∞ i,j for all 1 ≤ i, j ≤ l if e = ∞. Now consider m / ∈ M s,e . Let γ, γ ′ be two removable or addable i-nodes of λ for i ∈ I s . We denote

(3) γ m γ ′ def ⇐⇒ b -a + m c < b ′ -a ′ + m c ′ .
Thanks to our assumption m / ∈ M s,e , it is easy to verify that the above definition indeed defines a total order on the set of i-nodes of any l-partition. This thus yields a graph G e,m,s as in §2.2. It was also proved in [START_REF] Gerber | Generalised canonical basic sets for Ariki-Koike algebras[END_REF] that one can mimic Uglov's Fock space construction and define a g e -action on the Fock space F from any order m . This gives an integrable g e -module that we denote by F e,m,s . The submodule generated by the empty l-partition is then an irreducible highest weight module of weight

Λ s = Λ s 1 + • • • + Λ s l .
Remark 2.4. Note that the Fock space F e,m,s depends on the choice of m (because of the order m ) and on the choice of s modulo e (because of the definition of the residue of a node). Also in the case where s = (s 1 , . . . , s l ) and s ′ = (s ′ 1 , . . . , s ′ l ) satisfy s j ≡ s ′ j (mod e) for j = 1, . . . , l, the associated Fock spaces can be identified and we can write F e,m,s = F e,m,s ′ . Remark 2.5. The inverse order - m of m also yields the structure of an integrable g e -module on F we denote by F - e,m,s . 2.4. Relations with JMMO Fock space structure [START_REF] Etingof | Ginzburg Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism[END_REF]. The operators e m z and f m z defined from the order m as in §2.2 coincide in fact with the Kashiwara crystal operators and G e,m,s or G ∞,m,s are the crystal graphs corresponding to the g e -module structure on our Fock space. To recover the crystal structure used by Uglov (or the crystal structure introduced by JMMO), it suffices to choose m such that

m c = s c + δ c , c = 1, . . . , l,
where e > δ 1 > • • • > δ l ≥ 0. We will simply denote by G e,s this JMMO structure.

Now let us consider m

′ / ∈ M s,e . For c = 1, . . . , l, define δ ′ c ∈ Q as the unique rational number such that 0 ≤ δ ′ c < e which is equivalent to m ′ c -s c modulo e. Thus there exists (s ′ 1 , . . . s ′ l ) ∈ Z l such that s j ≡ s ′ j (mod e) and m ′ c = s ′ c + δ ′ c , c = 1, . . . , l. We have e > δ ′ σ(1) > • • • > δ ′ σ(l) ≥ 0, for a permutation σ ∈ S l . Then the map (4) F e,m ′ ,s = F e,m ′ ,s ′ σ → F e,σ(m ′ ),σ(s) (λ 1 , . . . , λ l ) → (λ σ(1) , . . . , λ σ(l) )
is an isomorphism of g e -modules. It also defines a crystal isomorphism between the crystal G e,m ′ ,s and the JMMO crystal G e,σ(s) .

This implies that for any m 1 / ∈ M s,e and m 2 / ∈ M s,e , the Fock spaces F e,m 1 ,s and F e,m 2 ,s are isomorphic. The crystals G e,m 1 ,s and G e,m 2 ,s are then also isomorphic as crystals. This means they are isomorphic in the sense of § 2.2 with indexing set Z/eZ and ψ = id. The modules F e,m 1 ,s and F e,m 2 ,s are reducible in general, so such an isomorphism is not unique. However, its restriction to the connected component of G e,m 1 ,s with empty highest weight vertex yields the connected component of G e,m 2 ,s with empty highest weight vertex.

Description of the canonical crystal isomorphisms

In this section we assume that s = (κ, s) with κ = 1 e and s ∈ Z l . Then G e,m,s is a Kashiwara crystal for any m / ∈ M s,e .

3.1. Crystal isomorphisms. The hyperplanes m s,e i,j,N divide R l into chambers. We first show that the orders m are the same for all the parameters m in the same (open) chambers. We also show that one can restrict to a finite sets of chambers in order to understand our crystal isomorphisms. Proposition 3.1. Assume that m 1 and m 2 are both in the same chamber with respect to the decomposition in §2.3, then the orders m 1 and m 2 on the i-nodes of an arbitrary l-partition coincide.

Proof. Consider γ = (a, b, c) and γ ′ = (a ′ , b ′ , c ′ ) two distinct i-nodes and assume we have γ ≺ m 1 γ ′ but γ ′ ≺ m 2 γ. This means that: b -a + m 1,c < b ′ -a ′ + m 1,c ′ , b -a + m 2,c > b ′ -a ′ + m 2,c ′ and b -a + s c = b ′ -a ′ + s c ′ + ke with k ∈ Z. We get b ′ -a ′ = b -a + s c -s c ′ -ke. By replacing b ′ -a ′ by
its expression in terms of a and b in the first above inequality we obtain:

(5) (s c -m 1,c ) -(s c ′ -m 1,c ′ ) > ke,
whereas the second inequality yields:

(6) (s c -m 2,c ) -(s c ′ -m 2,c ′ ) < ke.
This means that m 1 and m 2 are separated by the affine hyperplane with equation (s c -m c )-(s c ′ -m c ′ ) = ke, so we get the desired contradiction.

Proposition 3.2. Consider a wall m s,e i,j,N such that:

|N.e + (s j -s i )| ≥ n,
and pick two parameters m 1 and m 2 separated by this wall (and only it) then the order m 1 and m 2 are the same on Π l (n).

Proof. Consider γ = (a, b, c) and γ ′ = (a ′ , b ′ , c ′ ) two distinct z-nodes in a given l-partition of n such that γ ≺ m 1 γ ′ but γ ′ ≺ m 2 γ.
Since we know that m 1 and m 2 are separated by the unique wall m s,e i,j,N , one can assume (with the notation of the previous proof) that c = i and c ′ = j and k = N . We obtain:

b -a + s i = b ′ -a ′ + s j + N.e, but as we have |(b -a) -(b ′ -a ′ )| < n,
this leads to a contradiction.

The above propositions shows that if we choose m 1 and m 2 such that :

• these parameters belong to the same chamber with respect to the decomposition in §2.3, • or satisfy the condition of Proposition 3.2, then the associated crystal structures are not simply isomorphic but equal. To investigate our canonical isomorphims, we thus have a finite set M s,e n of hyperplanes and chambers to consider (depending on n), namely M s,e n is the union of the m s,e i,j,N for 1 ≤ i < j ≤ l and |N.e + (s js i )| ≤ n. Indeed, the canonical isomorphisms we aim to characterize will be equal to the identity if we stay inside one chamber. So it just remains to understand what happens when we move from one chamber to another. This is equivalent to describing the crystal isomorphisms corresponding to the crossings of the walls m s,e i,j,N with 1 ≤ i < j ≤ l and |N.e + (s js i )| ≤ n.

3.2.

A combinatorial procedure. We first study the case of e = ∞, define and describe our canonical crystal isomorphisms. The walls are then defined as:

m s,∞ i,j := {(m 1 , . . . , m l ) ∈ Q l |s i -m i -(s j -m j ) = 0}, for each 1 ≤ i, j ≤ l.
Let us first consider the case where l = 2. Set s = (s 1 , s 2 ). We wish to describe a crystal isomorphism when we cross a wall:

m (s 1 ,s 2 ),∞ 1,2 := {(m 1 , m 2 ) ∈ Q 2 |s 1 -m 1 -(s 2 -m 2 ) = 0}.
Let (λ 1 , λ 2 ) be a bipartition of n. We define the minimal integer

d ≥ |s 1 -s 2 | such that λ 1 d+1+s 1 -s 2 = λ 2 d+1 = 0 if s 2 ≥ s 1 , and otherwise the minimal integer d ≥ |s 1 -s 2 | such that λ 2 d+1+s 2 -s 1 = λ 1 d+1 = 0. To (λ 1 , λ 2
), we associate its symbol. This is the two row tableau:

S(λ 1 , λ 2 ) = s 2 -d + λ 2 d+1 • • • • • • • • • s 2 -1 + λ 2 2 s 2 + λ 2 1 s 2 -d + λ 1 d+1+s1-s2 • • • s 1 + λ 1 1 when s 2 ≥ s 1 S(λ 1 , λ 2 ) = s 1 -d + λ 2 d+1+s2-s1 • • • s 2 + λ 2 1 s 1 -d + λ 1 d+1 • • • • • • • • • s 1 -1 + λ 1 2 s 1 + λ 1 1 when s 2 < s 1 .
We will write S(λ 1 , λ 2 ) = L 2 L 1 where the top row (resp. the bottom row) corresponds to λ 2 (resp. λ 1 ). By definition of d, the entry in the leftmost column of S(λ 1 , λ 2 ) is equal to s 2d (resp. s 1d) when s 2 ≥ s 1 (resp. s 2 < s 1 ). So from S(λ 1 , λ 2 ) it is easy to obtain s 1 and s 2 since |s 1s 2 | is the difference between the lengths of L 1 and L 2 . Once we have S(λ 1 , λ 2 ) and (s 1 , s 2 ), we can recover the bipartition (λ 1 , λ 2 ).

We now recall a combinatorial procedure used in [START_REF] Jacon | Crystal isomorphisms for irreducible highest weight Uv( sle)-modules of higher level Algebras and Representation[END_REF] to describe the combinatorial R-matrix

G ∞,(s 1 ,s 2 ) = G ∞,s 1 ⊗ G ∞,s 2 ≃ G ∞,s 2 ⊗ G ∞,s 1 = G ∞,(s 2 ,s 1 ) .
On the set of bipartitions, the above R-matrix can be written

R ∞ (s 1 ,s 2 ) : Π 2 (n) → Π 2 (n) (λ 1 , λ 2 ) → ( λ 2 , λ 1 ).
We proved in [START_REF] Jacon | Crystal isomorphisms for irreducible highest weight Uv( sle)-modules of higher level Algebras and Representation[END_REF] it is a rank preserving crystal isomorphism. Also R ∞ (s 1 ,s 2 ) is the unique g ∞ -crystal isomorphism between the two Fock spaces G ∞,(s 1 ,s 2 ) and G ∞,(s 2 ,s 1 ) because there is no multiplicity into their decomposition in irreducible components (only in level 2). We also have R

∞ (s 1 ,s 2 ) • R ∞ (s 2 ,s 1 ) = R ∞ (s 2 ,s 1 ) • R ∞ (s 1 ,s 2 ) = id.
The bipartition ( λ 2 , λ 1 ) can be defined from its symbol

L 1 L 2 as follows. Suppose first s 2 ≥ s 1 . Consider x 1 = min{t ∈ L 1 }. We associate to x 1 the integer y 1 ∈ L 2 such that (7) y 1 = max{z ∈ L 2 | z ≤ x 1 } if min{z ∈ L 2 } ≤ x 1 , max{z ∈ L 2 } otherwise.
We repeat the same procedure to the lines L 2 -{y 1 } and L 1 -{x 1 }. By induction this yields a sequence {y 1 , ..., y d+1+s 1 -s 2 } ⊂ L 2 . Then we define L 1 as the line obtained by reordering the integers of {y 1 , ..., y d+1+s 2 -s 1 } and L 2 as the line obtained by reordering the integers of L 2 -{y 1 , ..., y d+1+s 1 -s 2 } + L 1 (i.e. by reordering the set obtained by replacing in L 2 the entries y 1 , ..., y d+1+s 1 -s 2 by those of L 1 ). Now, suppose s 2 < s 1 . Consider x 1 = min{t ∈ L 2 }. We associate to x 1 the integer y 1 ∈ L 1 such that ( 8)

y 1 = min{z ∈ L 1 | x 1 ≤ z} if max{z ∈ L 1 } ≥ x 1 , min{z ∈ L 1 } otherwise.
We repeat the same procedure to the lines L 1 -{y 1 } and L 2 -{x 1 } and obtain a sequence {y 1 , ..., y d+1+s 1 -s 2 } ⊂ L 1 . Then we define L 2 as the line obtained by reordering the integers of {y 1 , ..., y d+1+s 2 -s 1 } and L 1 as the line obtained by reordering the integers of L 1 -{y 1 , ..., y d+1+s 2 -s 1 } + L 2 .

Example 3.3. Assume (s 1 , s 2 ) = (0, 3) and consider the bipartition of 38 given by (λ 1 , λ 2 ) = (6.5.5.4, 5.5.3.3.2). Then d = 7 and:

S(λ 1 , λ 2 ) = -4 + 0 -3 + 0 -2 + 0 -1 + 2 0 + 3 1 + 3 2 + 5 3 + 5 -4 + 0 -3 + 4 -2 + 5 -1 + 5 0 + 6 S(λ 1 , λ 2 ) = -4 -3 -2 1 3 4 7 8 -4 1 3 4 6
We get {y 1 , ..., y 5 } = {-4, 1, 3, 4, -2}. This gives

S( λ 2 , λ 1 ) = -4 -2 1 3 4 -4 -3 1 3 4 6 7 8
and finally ( λ 2 , λ 1 ) = (5.5.5.4.4.3, 4.4.3.1). Observe that both (λ 1 , λ 2 ) and ( λ 2 , λ 1 ) have rank equal to 38.

Let F be the flip involution on the set of bipartitions defined by F (µ 1 , µ 2 ) = (µ 2 , µ 1 ). We introduce the map Φ

∞ (s 1 ,s 2 ) = F • R ∞ (s 1 ,s 2 )
. With the previous notation, we have

Φ ∞ (s 1 ,s 2 ) : Π 2 (n) → Π 2 (n) (λ 1 , λ 2 ) → ( λ 1 , λ 2 ).
By using the crystal isomorphism introduced in (4) and the properties of the R-matrix R ∞ (s 1 ,s 2 ) we have just recalled, we get the following proposition.

Proposition 3.4. Assume that l = 2 and e = ∞. Consider m + and m - separated by the wall m

(s 1 ,s 2 ),∞ 1,2 such that m + = (m + 1 , m + 2 ) with m + 1 -s 1 - (m + 2 -s 2 ) > 0 and m -= (m - 1 , m - 2 ) with m - 1 -s 1 -(m - 2 -s 2 ) < 0. (1) The map Φ ∞ (s 1 ,s 2 ) defines a g ∞ -crystal isomorphism from G ∞,m + ,s to G ∞,m -,s . (2) The map (Φ ∞ (s 1 ,s 2 ) ) -1 = R ∞ (s 2 ,s 1 ) •F is a g ∞ -crystal isomorphism from G ∞,m -,s to G ∞,m + ,s . (3) The map Φ ∞ (s 1 ,s 2 )
is the unique crystal isomorphism between G ∞,m + ,s and G ∞,m -,s which preserves the rank of the bipartitions. 1 1 More generally, this also shows there is only one graph isomorphism from G ∞,m + ,s to G ∞,m -,s preserving both the labelling of the arrows and the rank of the bipartitions.

Assume that l ∈ N. To construct a crystal isomorphism between two Fock spaces G ∞,m,s and G ∞,m ′ ,s where m and m ′ are separated by the wall m := m s,∞ i,j (and only by this wall), we define the map:

Φ s,∞ m→m ′ : Π l (n) → Π l (n) (λ 1 , . . . , λ l ) → (µ 1 , . . . , µ l ), such that µ k = λ k if k = i, j and (µ i , µ j ) = Φ (s i ,s j ),∞ (λ i , λ j ) if m i -s i -(m j -s j ) > 0, (Φ (s i ,s j ),∞ ) -1 (λ i , λ j ) otherwise. Proposition 3.5. Φ s,∞ m→m ′ defines a g ∞ -isomorphism of crystal between G ∞,m,s and G ∞,m ′ ,s .
Proof. Set δ = ms and δ ′ = m ′s. By Proposition 3.1, since we can move in each chamber without changing the crystal structure we can assume that δ and δ ′ have distinct coordinates. We can also assume that m and m ′ are very close to each other but belong to distinct half-spaces defined by the wall m. So the coordinates of δδ ′ are small and we have δ iδ j > 0 and δ ′ iδ ′ j < 0 (or δ iδ j < 0 and δ ′ iδ ′ j > 0). We will assume δ iδ j > 0 and δ ′ iδ ′ j < 0, the arguments being analogous when δ iδ j < 0 and δ ′ iδ ′ j > 0. Let σ be the permutation of {1, . . . , l} corresponding to the decreasing reordering of δ. One can then choose the coordinates of δδ ′ sufficiently small so that the permutation of {1, . . . , l} yielding the decreasing reordering of

δ ′ becomes σ ′ = σ • (i, j). We then have Φ s,∞ m→m ′ = (σ ′ ) -1 • R ∞ (s i ,s j ) ⊗ Id l-2
•σ where we also denote by σ and σ ′ the crystal isomorphims of type (4) mapping G ∞,m,s and G ∞,m ′ ,s on the JMMO structures G ∞,σ(s) and G ∞,σ ′ (s) , respectively. Therefore, Φ s,∞ m→m ′ is also a crystal isomorphism between the Fock spaces G ∞,m,s and G ∞,m ′ ,s .

We now describe the canonical g e -isomorphisms when e ∈ N is finite. Consider a wall: m s,e i,j,N := {(m 1 , . . . , m l ) |s im i -(s jm j ) = N.e}. Assume that m and m ′ are separated by this wall (and only it) and that:

(9) s i -m i -(s j -m j ) > N.e and s i -m ′ i -(s j -m ′ j ) < N.e.
To define a g e -crystal isomorphism between the crystal G e,m,s and G e,m ′ ,s , let us set s = (s 1 , . . . , s i -N.e, . . . , s l ). Then an element m is in the wall m s,e i,j,N if and only if m is in the set

m ∞ := m ∞, s i,j := {(m 1 , . . . , m l ) |(s i -N.e) -m i -(s j -m j ) = 0}
, where s = (s 1 , s 2 , . . . , s i -N e, . . . , s l ). The set m ∞, s i,j is a wall for s in the case where e = ∞ and we known that the map Φ s,∞ m→m ′ of Proposition 3.5 is a g ∞ -crystal isomorphism between G ∞,m, s and G ∞,m ′ , s . Theorem 3.6. Φ s,∞ m→m ′ is a g e -crystal isomorphism between G e,m,s and G e,m ′ ,s .

Proof. Let define σ and σ ′ from Set δ = ms and δ ′ = m ′s as we did in the previous proof. By Proposition 5.2.

2 of [6] R ∞ ( s i , s j ) ⊗ I l-2 is a g e -crystal isomorphism between the JMMO structures G ∞,σ( s) and G ∞,σ ′ ( s) . Since we have Φ s,∞ m→m ′ = (σ ′ ) -1 • R ∞ ( s i , s j ) ⊗ Id l-2
• σ, we get that that Φ s,∞ m→m ′ is a g e -crystal isomorphism between G e,m, s and G e,m ′ , s (see ( 4)). Now note that s and s are two multicharges that are equivalent modulo e. So we have in fact the equalities G e,m, s = G e,m,s and G e,m ′ , s = G e,m ′ ,s (see Remark 2.4). The result follows.

To simplify, we will denote by Ψ s m→m ′ the above bijection obtained by crossing a single wall when m and m ′ satisfy [START_REF] Losev | Supports of simple modules in cyclotomic Cherednik categories O[END_REF]. The above result allows to compute certain g e -crystal isomorphisms between two arbitrary Fock spaces F e,m,s and F e,m ′ ,s by composing the g ∞ -crystal isomorphisms of Theorem 3.6.

Remark 3.7. In [START_REF] Jacon | A combinatorial decomposition of higher level Fock spaces[END_REF], we gave a simple criterion deciding whether a multipartition λ = (λ 1 , . . . , λ l ) is a highest weight vertex in a Uglov crystal structure. Using the discussion in §2.4, this criterion can be easy adapted to the crystal structure G e,m,s .

Example 3.8. We will here consider the same example as [9, ex 5.6], We assume that s = (0, 0) and e = 2. We take n = 3. Then we have 3 hyperplanes to consider :

m (0,0),2 1,2,-1 := {(m 1 , m 2 ) ∈ Q 2 | m 2 -m 1 = -2}, m (0,0),2 1,2,0 := {(m 1 , m 2 ) ∈ Q 2 | m 2 -m 1 = 0}, m (0,0),2 1,2,1 := {(m 1 , m 2 ) ∈ Q 2 | m 2 -m 1 = 2}. The 2-partitions of 3 are : (∅, 1.1.1), (∅, 2.1), (∅, 3), (1, 1.1), (1, 2), (1.1, 1), (1.1.1, ∅), (2, 1), (2.1, ∅), (3, ∅)
Now we pick one parameter in each of the four associated chambers :

(1) Take m[1] = (m 1 , m 2 ) ∈ Q 2 such that m 2 -m 1 < -2. (2) Take m[2] = (m 1 , m 2 ) ∈ Q 2 such that 0 > m 2 -m 1 > -2. (3) Take m[3] = (m 1 , m 2 ) ∈ Q 2 such that 2 > m 2 -m 1 > 0. (4) Take m[4] = (m 1 , m 2 ) ∈ Q 2 such that m 2 -m 1 > 2.
The following table gives the isomorphisms computed using our procedure, the notation (⋆) indicates that the associated bipartitions are highest weight vertices for the parameters considered.

m[1] m[2] m[3] m[4] (⋆) (∅, 1.1.1) (∅, 1.1.1) (1.1.1, ∅) (1.1.1, ∅) (∅, 2.1) (∅, 2.1) (2.1, ∅) (2.1, ∅) (⋆) (∅, 3) (1.1.1, ∅) (∅, 1.1.1) (3, ∅) (1, 1.1) (1, 1.1) (1.1, 1) (1.1, 1) (1, 2) (∅, 3) (3, ∅) (2, 1) (⋆) (1.1, 1) (1, 2) (2, 1) (1, 1.1)
(1.1.1, ∅) (1.1, 1) (1, 1.1) (∅, 1.1.1) (2, 1) (2, 1) (1, 2) (1, 2)
(2.1, ∅) (2.1, ∅) (∅, 2.1) (∅, 2.1) (3, ∅) (3, ∅) (∅, 3) (∅, 3)

Wall crossing bijections for Cherednik algebras

We now give an interpretation of the above isomorphisms in the context of rational Cherednik algebras. In [START_REF] Losev | Supports of simple modules in cyclotomic Cherednik categories O[END_REF], Losev has introduced certain combinatorial maps between the sets of simple modules in categories O for rational Cherednik algebras corresponding to different parameter values. We are going to explain how these maps (called "wall-crossing bijections") are connected with our crystal isomorphisms. We refer to [START_REF] Losev | Rational Cherednik algebras and categorification[END_REF] for more details on the representation theory of Cherednik algebras and for problems we are interested in in this section. 4.1. Rational Cherednik algebras. Let H κ,s (n) be the rational Cherednik algebra associated with the complex reflection group of type W := G(l, 1, n) acting on h := C n . As a vector space, this algebra is S(h * ) ⊗ CW ⊗ S(h) (where S(V ) denotes the symmetric algebra of the vector space V ). It admits a presentation by generators and relations for which we refer to [5, §2.3].

Importantly, this presentation depends on a parameter s := (κ, s) ∈ C × C l . This parameter is the one used in [START_REF] Losev | Supports of simple modules in cyclotomic Cherednik categories O[END_REF][START_REF] Losev | Rational Cherednik algebras and categorification[END_REF] as well as in [START_REF] Gordon | On category O for cyclotomic rational Cherednik algebras[END_REF] (the reader may look at the relations between the different parametrizations given in [5,

§2.3.2])

We will consider the category O n,s for this algebra whose simple objects are parametrized by the set Π l (n), which also indexes the set of irreducible representations of the complex reflection group W in characteristic 0. Remark 4.1. An important problem in this theory is to compute the support of a simple module in the category O n,s parametrized by an l-partition λ. This in particular leads to a classification of the finite-dimensional simple modules in this category.

As explained in [9, §4] and [10, §3. 1.3], in most of the questions relative to the study of the category O we can assume (and we will do in the sequel) the following condition is satisfied. Condition 4.2. In the rest of the paper we assume that:

(1) κ = r e is a positive rational number where r and e are relatively prime, (2) r.s j ∈ Z for any j = 1, . . . , l.

In particular, we have now s := (κ, s) ∈ Q × Q l . Let us denote by S the subset of Q × Q l satisfying (1) and [START_REF] Geck | Representations of Hecke algebras at roots of unity[END_REF].

In [START_REF] Shan | Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras[END_REF], Shan has introduced an action of the quantum group g e on a Fock space defined from a categorical action on the direct sum over n of the categories O n,s . This action heavily depends on the choice of the parameter s. It also induces a structure of g e -crystal on the set of l-partitions. This crystal structure can be defined by using a relevant total order ≤ on z-nodes as we did in Section 2.2. Consider γ = (a, b, c) and γ ′ = (a ′ , b ′ , c ′ ) two such z-nodes. Definition 4.3. We set γ ≤ γ ′ if and only if

κl(b -a + s c ) -c ≤ κl(b ′ -a ′ + s c ′ ) -c ′ .
The associated oriented graph is denoted by G s with indexing set I s .

Remark 4.4. The order ≤ is the reverse of the one used in [START_REF] Losev | Supports of simple modules in cyclotomic Cherednik categories O[END_REF] where the crystal operators are also slightly defined differently. Nevertheless, the graph structures obtained are the same in both papers.

Observe this is indeed a total order since the equality

κl(b -a + s c ) -c = κl(b ′ -a ′ + s c ′ ) -c ′ ,
implies that l divides cc ′ . But 0 ≤ |cc ′ | < l so we have in fact c = c ′ and γ = γ ′ . We are going to see that the graph structure G s coincides with a graph structure already defined which is a crystal structure up to reparametrization of the colors. Since r and e are relatively prime, we have eZ + rZ = Z and eZ ∩ rZ = erZ. Then, for any integer a there exists a unique pair (c, d) such that a = ed + rc and c ∈ {0, . . . , e -1}. Since we have rs j ∈ Z for any j = 1, . . . , l, we can set: [START_REF] Losev | Rational Cherednik algebras and categorification[END_REF] rs j = ed j + rc j , where c j ∈ {0, . . . , e-1}. Note then that c j is equivalent to s j modulo κ -1 Z. Recall Definition 2.2. We have the following elementary lemma.

Lemma 4.5. The map ψ :

Z/eZ → I s i(mod e) → i(mod κ -1 Z)
is well defined and is a bijection Proof. Since e = rκ -1 , we have eZ ⊂ κ -1 Z. This implies that we have a natural map from Z/eZ to Q/κ -1 Z. By the comment preceding the lemma, this map is surjective onto I s . As Z ∩ κ -1 Z = eZ, it is injective. This concludes the proof.

For j = 1, . . . , l write also

m j = s j - j κl = s j - je rl ∈ Q,
and set m = (m 1 , . . . , m l ) ∈ Q l .

4.2. Relation with extended JMMO Fock space structure (2). In the following proposition, we will consider the oriented graph G s with indexing set I s and the crystal graph G e,m,c which indexing set Z/eZ where c = (c 1 , . . . , c l ) (see [START_REF] Losev | Rational Cherednik algebras and categorification[END_REF]).

Proposition 4.6. The colored oriented graph G s is equivalent to the crystal G e,m,c . More precisely, we have an isomorphism of oriented graphs θ between G e,m,c and G s and according to the notation of §2.2, the following maps :

Ψ Π l (n) → Π l (n) λ → λ , ψ Z/eZ → I s i(mod e) → i(mod κ -1 Z).
Proof. First, we show that two boxes have the same residue for (κ, s) if and only they have the same residue for (1/e, c). This follows from the fact that, for two nodes (x, y, i) and (x ′ , y ′ , j) of an l-partition, we have

x -y + s i = x ′ -y ′ + s j + κ -1 Z if and only if r(x -y + s i ) = r(x ′ + y ′ + s j ) + eZ,
because κ = r e . By using [START_REF] Losev | Rational Cherednik algebras and categorification[END_REF] we get r(xy + c i ) = r(x ′ + y ′ + c j ) + eZ.

Now both (xy + c i ) and (x ′ + y ′ + c j ) are integers. As (e, r) = 1, we have

x -y + c i = x ′ + y ′ + c j + eZ,
which is what we wanted to show. Observe also that the residue of (x, y, i) for (1/e, c) and (κ, s) are respectively equal to (xy + c i )(mode) and (xy + s i )(modκ -1 Z), and we have (

x -y + c i ) ≡ (x -y + s i )(modκ -1 Z).
It just remains to show that the order ≤ on z-nodes corresponds to the order m .

Let γ = (x, y, i) and γ ′ = (x ′ , y ′ , j) be two nodes of the l-partition. Then we have the equivalences

γ ≤ γ ′ ⇐⇒ κ(x -y + s c ) - c l = κ(x ′ -y ′ + s c ′ ) - c ′ l ⇐⇒ x -y + m i ≤ x ′ -y ′ + m j .
which is exactly what we wanted. Remark 4.7. Combining this proposition with Remark 3.7, we obtain a criterion to decide whether an l-partition is a highest weight in G s . The finitedimensional simple modules in the associated category O of the Cherednik algebra are in particular parametrized by l-partitions satisfying this criterion. However it is not a sufficient condition for being finite-dimensional. To obtain a necessary and sufficient condition, one has to take into account the action of the Heisenberg algebra. We refer to [START_REF] Gerber | Triple crystal action in Fock spaces[END_REF] for other results in this direction and a detailed investigation of the crystal action of the Heisenberg algebra on G s . Remark 4.8. One can also combine the results of [START_REF] Losev | Supports of simple modules in cyclotomic Cherednik categories O[END_REF] together with the approach in [START_REF] Jacon | Crystal isomorphisms for irreducible highest weight Uv( sle)-modules of higher level Algebras and Representation[END_REF]. Given a parameter s, the crystal structure G s is equivalent to a crystal G e,m,c (by the previous proposition) which is isomorphic to a JMMO crystal G e,s (by §2.4). Now we described in [START_REF] Jacon | Crystal isomorphisms for irreducible highest weight Uv( sle)-modules of higher level Algebras and Representation[END_REF] the crystal isomorphisms on the JMMO crystals G e,s , s ∈ Z l corresponding to the action of the extended affine symmetric group on Z l . This eventually leads to a rank preserving bijection between G s and a crystal structure G s ′ associated to an asymptotic parameter where the support can be calculated thanks to the result in [START_REF] Losev | Supports of simple modules in cyclotomic Cherednik categories O[END_REF]. As all the rank-preserving bijections involved here leave the support invariant, this gives another way to compute the support of a simple module by reduction to the asymptotic case.

Let us now review the wall crossing bijections. Fix n ∈ N. In the set of parameters S, one can define certain hyperplanes called essential walls. Set h j = κm j = κs jj/l, j = 1, . . . , l. Definition 4.9. Recall n is fixed. Given i, j distinct in {1, . . . , l}, an essential wall parametrized by (i, j) is a set of parameters s = (κ, s) ∈ S such that there exists an integer a satisfying:

(1) |a| < n, (2) m i -m j = a, (3) s i -s j -a ∈ κ -1 Z.
Consider two parameters s = (κ, s) and s ′ = (κ ′ , s ′ ) in S satisfying : [START_REF] Shan | Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras[END_REF] κκ ′ ∈ Z and ∀j ∈ {1, . . . , l} κs jκ ′ s ′ j ∈ Z and such that s and s ′ are separated by one essential wall.

Remark 4.10. Assume s and s ′ satisfy [START_REF] Shan | Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras[END_REF]. This means that there exist k ∈ Z and t ∈ Z such that r ′ s ′ j = rs j + ke and r ′ = r + t.e (here κ ′ = r ′ e ). By [START_REF] Losev | Rational Cherednik algebras and categorification[END_REF], for each j = 1, . . . , l, there exists a unique (c j , c ′ j ) ∈ {0, . . . , e -1} 2 and (d j , d ′ j ) ∈ Z 2 such that

rs j = ed j + rc j , r ′ s ′ j = ed ′ j + r ′ c ′ j .
We obtain: r ′ s ′ j = rs j + ke = ed j + rc j + ke = ed j + r ′ c jtec j + ke = e(d jte + k) + r ′ c j .

We thus have c ′ j = c j and d ′ j = d jte + k. By Proposition 4.6 G s and G s ′ are respectively isomorphic to G e,m,c and G e,m ′ ,c for good choices of m and m ′ . But G e,m ′ ,c and G e,m,c are isomorphic as crystal graphs because the associated Fock spaces have the same multicharge. So G s and G s ′ are isomorphic as soon as s and s ′ satisfy [START_REF] Shan | Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras[END_REF].

In [9, Prop. 5.9], Losev defines a bijection wc s→s ′ between l-partitions which is an isomorphism of graphs in the sense of §2.2. Proposition 4.11. Given s and s ′ in S satisfying [START_REF] Shan | Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras[END_REF] and separated by the essential wall parametrized by (i, j), there is a bijection wc s→s ′ on the set of l-partitions such that if µ = wc s→s ′ (λ) we have

• λ k = µ k if k = i and k = j,
• (µ i , µ j ) = wc (κ,s i ,s j )→(κ ′ ,s ′ i ,s ′ j ) (λ i , λ j ) where wc (κ,s i ,s j )→(κ ′ ,s ′ i ,s ′ j ) is the unique graph isomorphism between G s and G s ′ preserving both the labelling of the arrows and the rank of the bipartitions.

Let us look at the essential hyperplanes and check that they are the same as the ones define in §2.3. Assume that we are in an essential hyperplane. This implies that there exists a ∈ Z such that s is ja ∈ κ -1 Z and m im j = a.

Thus there exists x ∈ Z such that s is j = a + κ -1 x. We obtain:

e(d id jx) = r(c jc i + a).

This implies that c jc i = -a + eZ because e and r are relatively prime. Thus, we have :

c j -m j -(c i -m i ) ∈ eZ.
Conversely, assume we have c jm j -(c im i ) = eN with N ∈ Z with the above relation between the parameters. We obtain that m im j = c ic j + eN.

Let us then set a = c ic j + eN which is in Z. Now by [START_REF] Losev | Rational Cherednik algebras and categorification[END_REF] we have s i = κ -1 d i + c i and s j = κ -1 d j + c j . We thus get s is ja = κ -1 (d id j ) + (c ic j )a = κ -1 (d id j + rN ) ∈ κ -1 Z.

We now want to check that the wall crossing bijections correspond to our bijections defined in §3.2.

Theorem 4.12. The wall crossing bijection wc s→s ′ corresponds to Ψ c m→m ′ .

Proof. Since both wc s→s ′ and Ψ c m→m ′ modify the component of indices i and j in the l-partition it suffices to show that for any bipartition (λ 1 , λ 2 ) we have wc (κ,s i ,s j )→(κ ′ ,s ′ i ,s ′ j ) (λ i , λ j ) = Ψ (c i ,c j ) (m i ,m j )→(m ′ i ,m ′ j ) (λ i , λ j ). But wc (κ,s i ,s j )→(κ ′ ,s ′ i ,s ′ j )

and Ψ

(c i ,c j ) (m i ,m j )→(m ′ i ,m ′ j ) are graph isomorphisms which preserve the labelling of the arrows and the rank of the bipartitions. Since there is only one such isomorphism, they coincide. Remark 4.13. Assume now s := (κ, s) where κ is a rational negative number. To each l-partition λ = (λ 1 , . . . , λ l ) we associate its conjugate λ # = ((λ l ) # , . . . , (λ 1 ) # ) where (λ c ) # is the conjugate of the partition λ c for any k = 1, . . . , l. There is a natural bijection between the nodes of λ and λ # which maps γ = (a, b, c) ∈ λ on γ # = (b, a, lc). Set s # := (-κ, s # ) where s # = (-s l , . . . , -s 1 ). The map # then defines a graph isomorphism twisted-equivariant for the labelling from the graphs G s to G s # which sends each l-partition λ on λ # and each arrow λ z → µ on λ # -z → µ # . This implies that the wall crossing maps for κ and -κ coincide up to conjugation by #.


Zakariya Boufaida

Julien Boisse
email: julien.boisse@univ-lorraine.fr

Stéphane André
email: stephane.andre@univ-lorraine.fr

Laurent Farge

A Mechanical Study of a Glass Fabric-Thermoplastic Resin Composite: 3D-DIC and X-ray tomographic observations explained by numerical simulations based on a spectral solver

Keywords: fabric composite materials, spectral solver, CraFT, DAMASK, Digital Image Correlation, damage

Introduction

Industry's widespread use of composite materials by the industry is still of paramount importance especially in the field of transport for evident reasons linked to structure alleviation, fuel economy and environmental concerns. Nowadays composite materials are not just used for the fabrication of accessories (appearance parts) for vehicles and enter into the design and production of structural parts (car industry) and high performance parts (aeronautics). This shift brings additional constraints and thus more research is continually required into how to produce better predictive calculations. One major research theme driven by a permanent search for innovation derives from the multiple possibilities of selecting the basic materials for the building blocks of a composite material:

-Different reinforcement materials -either technical or bio-sourced, -Different armor architectures (e.g. types of fabrics, laminated or 3D textile structural composites...), -Wide range of possible materials for the matrix, especially in the present context of replacing thermoset materials by thermoplastics for recycling purposes, -Different techniques to modify the reinforcement material-matrix adhesion which is a key point for the global mechanical performance. To analyze the influence of the internal structure of the material on the overall mechanical properties, it is essential to take into consideration both averaged fields and full local fields. For instance, damage and fracturing are heavily dependent on the local details of stress or strain fields. It is thus essential to possess reliable and efficient tools for this which are also able to produce calculations systematically, moving easily from one type of architecture to another and correctly taking the structure at different scales into account. All of this means that modelling issues in this field are increasingly important.

Initially, analytical homogenization approaches were favoured to predict macroscopic equivalent material properties. Then, the joint development of powerful computers and opensource or commercial software for simulation engineering has led to a majority of studies on composites taking up the challenge through discretization methods of the Finite Elements type [Sierles et al., 2001[START_REF] Yu | Non-orthogonal constitutive equation for woven fabric reinforced thermoplastic composites[END_REF][START_REF] Barbero | Finite element modeling of plain weave fabrics from photomicrograph measurements[END_REF][START_REF] Komeili | The effect of meso-level uncertainties on the mechanical response of woven fabric composites under axial loading[END_REF], Piezel et al., 2012[START_REF] Liu | Lightweight design of carbon twill weave fabric composite body structure for electric vehicle[END_REF][START_REF] Jia | Nonlinear viscoelastic multi-scale repetitive unit cell model of 3D woven composites with damage evolution[END_REF]. These tools have one important limitationobtaining highly resolved calculations requires very thin meshing in multiple areas which results in very high computational costs. An alternative has been developed over the past two decades. The CraFT code (Composite response and Fourier Transforms [http://craft.lma.cnrsmrs.fr/, Moulinec andSuquet, 1994, 1998]) and the DAMASK code (Düsseldorf Advanced MAterial Simulation Kit [http://damask.mpie.de/, [START_REF] Roters | DAMASK: the Düsseldorf Advanced Material Simulation Kit for studying crystal plasticity using an FE based or spectral numerical solver[END_REF][START_REF] Eisenlohr | A spectral method solution to crystal elasto-viscoplasticity at finite strains[END_REF]) were developed to avoid this difficulty by using a spectral approach. These two codes rely on the use of 3D Fourier Transforms and are very well-suited to modelling the behavior of complex materials at the bulk scale where a periodic pattern (Representative Elementary Volume -REV) can be found. These types of spectral methods have also proved very efficient in other fields of engineering sciences like heat transfer [START_REF] Maillet | Solving the Heat Equation through Integral Transforms[END_REF]. Regarding the study presented here of plain weave reinforced composites, these spectral solvers give very precise and fast simulations of the 3D behavior of a clearly defined macroscopic REV. Strictly speaking, they do not require any meshing and produce highly resolved simulations of materials presenting very strong heterogeneities of properties and scales. Our research led us to prefer the CraFT code developed by Moulinec andSuquet (1994,1998). The authors initial goal was to develop an efficient method for numerically solving the local equilibrium equations in the case of complex heterogeneous materials whose microstructure could be obtained directly from experimental imaging (microscopy, tomography…). The method was first applied in the framework of elasticity and extended to the case of materials with arbitrary phase contrast and elasto-plastic and linear hardening behavior [Michel et al., 2011]. It remains nevertheless restricted to small transformations (kinematic framework of infinitesimal strains).

The aim of this work is to present a specific study of a glass fiber plain weave fabric / thermoplastic resin composite which provides mechanical modelling of the whole material in a quasi-static test and compares the results in terms of strain field measurements obtained with 3D-Digital Image Correlation (DIC). Most of the experimental features and results on this material (including fatigue analysis and mechanical spectroscopy) have been explained and reported elsewhere [START_REF] Boufaida | Influence of the fiber/matrix strength on the mechanical properties of a glass fiber/thermoplastic matrix plain weave fabric composite[END_REF] and are only used here to validate computational results. As the results are particularly convincing, simulations of the associated stress fields will also be shown to enhance knowledge and analysis of damage phenomena which is proved consistent with post-mortem observations yielded by X-ray tomography. One important point to stress regarding the study is that it concerns in-plane shear loading as this allows the intensification of solicitations at the fiber/matrix interfaces.

The paper is structured as follows. In section 2, the modelling tool will be presented. Results of the simulation in terms of strain fields will be given in Section 3 and discussed in relation with their experimental counterparts. Section 4 will focus on predicted stress fields to investigate the damaging aspect of this material, especially regarding the role of the matrixstrand debonding. We summarize our findings and give ideas regarding future prospects in section 5.

Spectral solver for composite simulations

For the well-information of the reader about the spectral approach and its conceptual simplicity, we shall first recall the basic mathematical principles applied to the local equilibrium problem. More details can be found in [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF]. Numerical simulations provided by CraFT and DAMASK for the same test-case will be given to show how well the algorithm works. Lastly, the model microstructure based on the reproduction of real specimen will be detailed along with explanations about how to reproduce it numerically and the associated material properties.

Numerical method

On a REV V , the local problem consists of equilibrium equations (1-a), behavior laws for the different phases (1-b) and boundary and interface conditions (1-c). For the whole study, perfectly bonded interfaces between all phases are considered (continuous evolution of the displacement and stress variables). The boundary conditions were assumed to be periodic which is in phase with the modelling of a REV as a repeated pattern of a given microstructure. All of this gives

  0 div   on V (1-a)               *    x c x u x c x u x Ex    :: (1-b) * #, #  un  (1-c)
where   cx is the rigidity tensor. As a function of space, it was indeed specified through the values of the mechanical properties associated to each "pixel" of the numerical image of the microstructure (having multiple images in the third direction to create the 3D structure). Additionally, eq.(1-b) shows that the strain field is decomposed into its average

  () x  u x E 
corresponding to the prescribed strain onto V and a fluctuation term

  * () ux 
. Eq.(1-c) points out that the fluctuating term * u is assumed to be periodic (symbol # ) while the local stress at the boundaries is anti-periodic (symbol #  ). To solve this problem, an auxiliary problem is defined, based on the assumed knowledge of a homogeneous, isotropic and linear elastic reference material whose rigidity tensor is known as 0 c . The equation below can then be substituted for eq. (1-b)

        0* ()  x c u x E x     : (1-b) with     * ( ) ( )  x c x u E    :
a priori unknown, and 0 ( ) ( )  c x c x c  .

In practical terms, the reference tensor 0 c is determined by the user and affects the convergence rate of the algorithm. For a two-phase composite material for example, it can be initialized with the average of the Lamé coefficients on both phases.

In a second step, the solution of (1) is expressed in real and Fourier spaces. In the real space, considering the Green operator 0 ()  x verifying the null divergence of the stress tensor along with periodic boundary conditions and homogeneous elastic tensor 0 c , the complete strain solution of eq.( 1) can be obtained by linearity property, as

    0 ( ) ( )    u x x x E  * (2)
Its mirror counterpart in Fourier space (notation and transformed variable ξ ) is

    0 ()        : , ( )     0 ε 0 E (3)
The way the Green operator is computed can be found in many textbooks [T. [START_REF] Mura | Micromechanics of Defects in Solids[END_REF]. This is done directly in the Fourier domain (see Appendix A of [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF] for further details). The solution to problem (1) then needs to be recast into the solution of equation ( 4) known as the Lippman-Schwinger periodic equation,

    ( ) : ( ) ( ) ( ) : : ( ) , ( )                0 0 ε u E - c ε u ε - c ε 0 ε 0 E    (4)
which can be solved numerically within a fixed-point algorithm and by performing the jumps between real and Fourier spaces with FFT algorithms available in mathematical libraries. The algorithm follows the path presented below (with 1  F and F denoting inverse and direct Fourier Transform).

The convergence test is based on the decrease and transition below a prescribed value of a relative error defined as to ensure that equilibrium is reached. In the computing code, this algorithm applies to the discretized coordinates of pixels in real space and to the corresponding discretized frequencies in Fourier space.

      : ( ) , : ) ( ) : ; ? ) : ĉ) ˆˆ) ( ) ( ) : , ( )
)) 0 0 1 Initialization V Iterate i 1 known a Convergence test exit b d e f i i 1                              0 i i i i i i i i i i i+1 i i+1 i+1 i+1 ε x E x ε σ = c x ε σ σ τ = σ c ε τ τ ε ξ - ξ τ ξ 0 ε 0 E ε ε  F F F
The CraFT code has been improved over the past ten years and validated for many different test cases. This study provides new validation of the code by comparing CraFT results with the results obtained with the DAMASK code. The test case consists of the simulation of a periodic unit cell containing 64 randomly-placed circular fibers (Figure below), subjected to a constant (average) deformation rate corresponding to the simulation of the woven tow behavior of the fabrics used for composites. Calculations were performed in the pure elastic case (Figs 1-a, 1-b) and in the case of a matrix assumed to be plastic with linear hardening (Figs 1-c, 1-d). Exactly the same results were obtained. Very small differences in the equivalent strain distribution can be seen by a close look at Figs. 1a-1b and 1c-1d. However at the meso-scale, in terms of the resulting stress-strain curves, the agreement is perfect as is shown by Figure 2 which corresponds to the case of Figs. 1cd, where the matrix is considered to have elastic-plastic behavior with isotropic linear hardening. In terms of computational efficiency, the above simulations required a CPU time of 2000s to 4700 s (depending respectively on whether the pure elastic or elastoplastic case was processed) on 2 over 8 cores of a standard computer (Intel® Xeon(R) CPU W3520 @ 2.67GHz × 8 ).

Input microstructure model

The composite material under study is modelled at the mesoscale, i.e. the scale corresponding to the internal structure of the fabric which structures the mechanical stress/strain fields. In all following figures, the (XY) or (12) planes correspond to the plane of the plain weave lamina. Thus the through-thickness (or stacking) directions Z or 3 are the same in both frameworks. The (12) directions refer to the Cartesian framework defined by the averaged warp (1) and weft (2) tow directions. The (XY) directions refer to the testing machines system of axis (X=tensile direction).

Microstructure generation

The plain weave microstructure was ideally reproduced with the general characteristics of the real material (Fig. 3) as provided by the weaving company or as measured. The 2D plane view of the REV is a square of about 7.94mm in size and consists of four intertwined identical tows surrounded by the matrix. Optical microscopy was used to characterize tows by a cross section with an elliptic shape of Overall Strain on the REV

The 3D modelling of the REV was performed using Solidworks® and is shown in Figure 3 which clearly shows the distribution of materials within sections at different depths of the fabric lamina following the tow undulations. The tows are shown here with the same 'color' but in the material description under CraFT, two sets of properties are given with same mechanical properties but in different orders, so as to take into account their different orientations. The real laminate composite is simulated as a superposition of nine plies. For this, two additional layers (one pixel size) on the bottom and up surfaces were considered as having null mechanical properties. Each ply was divided into 9 pixels along direction 3 which gives 83 pixels for the whole discretization along this direction. In the (12) plane, a unit cell was made up of 443 443  pixels which conserves the pattern's real aspect ratio and gives a resolution of about 7.94 / 443 18 / m px  .

Figure 3: 3D numerical REV corresponding to the real woven and material distribution for different slices.

2.2.2. The mechanical properties at the meso-scale.

Once this geometrical description of the REV has been made, mechanical properties must be attributed to each "phase" of the material. Only the elastic behavior is considered for each phase and Table 1 lists the values of all relevant moduli. The resin was a newly referenced thermoplastic (Elium® -Arkema) considered to be homogeneous and isotropic and whose properties were characterized experimentally (using Dynamic Mechanical Analysis-DMA for the elastic modulus determination) or from a well-established data base (Poisson coefficient for acrylic material). E-Glass fibers with well known mechanical properties and diameters in the 15 20 m  range were used to perform the tows. Each tow was modelled as a unidirectional composite whose mechanical behavior can therefore be described with 5 independent elastic constants, for example 11

E , 22 E 12  , 12
G and 23 G (1 is the average fiber direction in the tow). Many micromechanical models are available to calculate these homogenized elastic constants, for instance the Chamis model [START_REF] Chamis | Simplified composite micromechanics equations for hygral, thermal and mechanical properties[END_REF], the Hopkins and Chamis model (HC) [START_REF] Hopkins | A unique set of micromechanics equations for high-temperature metal matrix composites[END_REF] and the Composite Cylinder Assemblage model (CCA) [START_REF] Rosen | Thermomechanical properties of fibrous composites[END_REF]. These models give more or less the same results but it should be noted that only the CCA model provides the 23 G value (see table 1). For the simulations shown in this paper we used the Chamis model for 11

E , 22 E 12 
and 12 G , and the CCA model for 23 G . However, we checked that the conclusions obtained during our study remain qualitatively and quantitatively true even if other micromechanical models were used to evaluate the elastic constants in the tows. The variations of elastic constants expressed in the (1, 2, 3) coordinate system due to the fiber undulation effect were not taken into account. To check the validity of this approximation, we modelled the tow undulation using a sinusoidal function and found the variations of 11 E to be less than 5%. The elastic constants in the tows are indicated in table 1 for a tow along the 1 direction.

In-plane shear loading.

As mentioned in the introduction, in-plane shear loading conditions lead to the harshest conditions being applied to the fiber/matrix interface. In the (XY) plane, a tensile load was applied in the X-direction, at 45  from the warp (1) and weft (2) fabric directions. The following plane stress tensor was applied:

( , ) ( , ) / / / / XY 12 2 2 0 2 2 00                         (5)
where  was considered to be the average loading stress imposed on the whole REV in CraFT software. Our goal in section 4 will be to analyse the stress field at the mesoscale which is linked to this uniform constant stress at the macroscale. It is evident from the tensor defined in (5) that in the direction of a tow (1 for example), the longitudinal stress 11  derives only from the fiber response whereas both the shear stress 12  and transverse stress 22  "work" in order to introduce fiber/matrix debonding.

In the following, we simulated the behavior of the above described composite in the case of tensile tests performed with warp and weft tows oriented at 45  with respect to the tensile direction (ASTM D3518 or NF ISO 14129) and compared the simulated results to experimental ones. For both sections 3 and 4, all quantities with upperscripts M and m will denote values corresponding respectively to the Macroscale and the mesoscale.

Analysis of strain fields

Qualitatively, it is possible to compare the mesoscopic strain field m XX  obtained by

CraFT simulation (Fig. 4a) and by experimental DIC measurements (Fig. 4b) using ARAMIS software. The X-axis denotes the tensile direction, while the 1 and 2 directions are correspond to the mutually orthogonal axis of the tows. It is clear that the exact same pattern was obtained for strain distribution which displays a strong correlation with the fabric geometry. The small squared areas of high strains was found to be located in the rich resin zones, those of small strains, in the rich fiber zone (interlaced tows corresponding to regions 1 and 2 in Fig. 4a). The areas with intermediate strains (bands surrounding the interlaced regions and forming a cross pattern) correspond to the undulation nodes of the tows where the resin forms strong junctions between the laminates. On this qualitative comparison, we can also highlight the important difference in spatial strain resolution between the simulation (highly resolved spatial resolution set as 1/Np where Np=443 is the number of pixels chosen for the REV size) and the DIC measurements. The best resolution achieved for the latter was about one fifteenth the VER size ( [START_REF] Boufaida | Influence of the fiber/matrix strength on the mechanical properties of a glass fiber/thermoplastic matrix plain weave fabric composite[END_REF], see also Fig. 5a). The ratio of simulation versus experimental resolution was found to be within a factor of 30. 12) axis systems (eq. ( 5)). We naturally obtained

REV XX M XX M XX E   
. Table 2 gives these results (1 st row) along with those obtained using the above-mentioned micromechanical models (2 nd row) applied to an equivalent laminate made up of three layers: i)

Two UD (UniDirectional) composite layers respectively oriented at   45 and   45 with fiber volume fraction corresponding to that of the tows ( 0.7

tows f V  ) and
ii) a layer made of pure resin, which is added to account for the resin pockets.

The relative thickness of these layers was chosen to comply with the overall composite fiber volume fraction ( 0.47

f V  ).
For the CraFT simulation, the in-plane elastic constants in the two composite layers were calculated using the Chamis model (see table 1). The results were found to be very close for XX E  (within 3%) and for the M G 12 values corresponding to the CraFT simulation and those calculated with the equivalent UD laminate approximation. G  obtained from the tensile curve was found to be 30% lower than the models. The discrepancy obviously came from an experimental under-estimation of our measurement. The stress-strain curve obtained experimentally during   45 tensile testing with strains measured using 3D-DIC was not linear [START_REF] Boufaida | Influence of the fiber/matrix strength on the mechanical properties of a glass fiber/thermoplastic matrix plain weave fabric composite[END_REF], even in the strain range where the M G 12 shear modulus needed to be evaluated according to the ASTM D3518 standard i.e. for . This suggests that the resin had a noticeable viscoelastic effect in this kind of stress state. As a result, the elastic behavior needs to be looked for at very early times of the tensile test which is unfortunately when the machine response can be smoothed by the feedback loop resulting in a lower estimation of the Young Modulus. This fact was identified when measuring the instantaneous modulus of semicrystalline polymers -a visco-elastic physical model needs to be applied to the whole tensile curve to obtain a good estimation of the modulus [START_REF] Blaise | Advantages of a 3-parameter Reduced Constitutive Model for the Measurement of Polymers Elastic Modulus using Tensile Tests[END_REF]. Therefore, to avoid this drawback, the experimental measurement of XX E  provided in Table 2 was obtained through DMA performed on a Bose 3300N fatigue machine. DMA enables the obtention of the "true" elastic modulus of a visco-elastic material (real part of the complex modulus). The agreement between the XX E  values obtained by the CraFT simulation and with the DMA experiment was found to be very good (within 0.5%). Proof of the discrepancy in the 12 G  values attributed to visco-elasticity can be seen in the following calculations -for the   45 fiber orientation, the relation between XX E  and the composite elastic constants related to the    in eq.6 is very small which can be verified. The M G 12 value that can be calculated with eq.8 was therefore obtained with realistic elastic constant measurements, in other words without any bias to visco-elasticity. Eq.6 gives the new experimental value of

GPa G M 67 . 4 12 
. The difference with the value obtained with the CraFT simulation (

GPa G M 82 . 4 12 
) is now only about 3%. This confirms that the discrepancy between the M G 12 value measured during   45 tensile testing and the value obtained with the CraFT solver can indeed be attributed to visco-elastic effects which are not accounted for in the numerical approach.

Figure 5b shows the three meso XX  strain profiles obtained theoretically (solid lines) and experimentally (dashed line). There was clear evidence of the difference in resolution in the 2D patterns (Fig. 5a) and also in the curves of Fig. 5b. The DIC measurements therefore correspond to a kind of averaging of the strain distribution over non negligible zones. It was therefore impossible to get a good resolution of strong strain gradients and to capture strong strain variations which were still present because of the transitions from rigid tows to the softer resin material. As a result, the highly deformed zones associated with the resin pockets appeared to be magnified to some extent but a strong quantitative validation of both results can be obtained by looking at the profiles of Fig. 5b. The experimental and theoretical strain profiles were found to be very similar. Of course the experimental curve appears very noisy with an enlarged base-peak as a result of the low measurement resolution. The theoretical profiles were once again nearly symmetrical but there were two peaks at the tow/resin interface which could not be present on the experimental curve for the same reason. This suggests that the maximum strains are located exactly at the interface in the resin phase. The ratio between the strains in the fiber rich and resin rich regions (Fig. 5a) was found to be approximately 3 both experimentally and by simulation.

In an attempt to reproduce the experimental results more closely, we considered the highly resolved strain patterns to be real objective measurements and tried to reproduce the spatial filtering operated when using our 3D-DIC system. The whole procedure is detailed in the appendix. In shortly, the highly resolved strain field was integrated to obtain the displacement field (in the small displacement-gradient theory framework). For this, the heterogeneous component of the strain (considered as periodic over the REV) was transformed in the Fourier domain to produce the Fourier transform of the displacement which was back-transformed in the real domain. From this highly-resolved displacement field, we used an averaging procedure to simulate the behavior of the Aramis software and obtain the transformation gradient tensor at a smaller resolution. Figure 6 (or A-2 in the Appendix) shows a new strain pattern obtained numerically after lowering the resolution and the corresponding numerical and experimental profiles along directions 1 or 2. The peaks found at the resin-tow interfaces can be seen to have disappeared while the strain profiles now match the experimental ones much better (apparent enlargement of the peak bottom). strain along the 45°lines after applying a smoothing procedure to CRAFT results.

Stress fields analysis

Following on from the validation provided by numerical and experimental comparisons of strain fields, the stress field obtained numerically was then investigated. Within the scope of damage initiation through fiber-matrix debonding, experimental investigations based on X-ray tomography imaging were found to confirm the results.

For a woven fabric composite material, the stress field is totally governed by the mesoscale resulting from the textile geometry. The average stress applied to a micrometric REV (inside the tows) does not correspond to the average macroscopic stress but to the mesoscopic stress in the region where the micrometric REV is taken. The detailed stress field structure at the mesoscale is required to accurately analyze the damage phenomenon related to the fiber-matrix debonding in a fabric. In the following, the stress field is analyzed for the tows with their fibers oriented in the 2-direction (central square of Fig. 4a or Fig. 7 below). We focused on the 12 m  and 11 m  stress components which were the most likely be able to test the fiber-matrix resistance because the 22 m  component of the mesostress tensor would be mainly supported by the fibers (as in a UD composite with applied load along the fibers). Figure 7 shows the 12 m  stress map obtained from CRAFT simulations which is given here in the fiber framework 1 ( ,1, 2,3) RO and for a macroscopic applied stress 50 XX MPa   . Highly-sheared regions appeared to be located in the tows at the tow-resin interface. The contrast in shear stress was of about factor 5 when crossing this interface. This plane shear test effectively produces a high shear stress level at the mesoscale in order to solicit the matrix-tow and fiber-matrix adhesion. The plot of Figure 8 corresponds to the profile of the 12 m  stress along the tensile direction. The tow can be seen to be highly stressed on its edges (stress drop again of a factor 5-6) and the resin in the pocket zones was subjected to strong stress gradients. Indeed the shear stress was about 5 MPa at the interface and 15 MPa in the center as compared with a macroscopic stress of 12 2 25 XX MPa      which was obviously mainly supported by the tows (Figures 7 and8). In the fiber region (at the centre of the REV of Fig. 8a), and in view of the fiber/matrix bond resistance estimation, the ratio of the max-min shear stress max min 12 12

 was approximately 1.3 which is not particularly strong. The 11 m  stress map (Figure 9-a) provides additional information which shows that the regions of highest stress were very small and located in the corners of the tows. There, the mechanical solicitation was oriented perpendicularly to the direction of the fibers in the tow and could (i) introduce fiber-matrix debonding with further cracks propagation and (ii) create important damage in the form of cavities in the pocket resin. This is precisely what can be observed in Figure 9-b which presents an X-ray tomography snapshot of the real specimen tested in such plane shear test. Moreover, heat build-up experiments reported in Boufaida et al. (2015 -Figs 5a and 5b) and consisting of the detection of a change in thermomechanical regime under fatigue excitations showed that the typical stress initiating the so-obtained damage yield is of the order of 25 MPa . In fact, this value drives the XX   value of 50MPa taken for the simulations presented here which give a Von Mises stress in the resin pocket in the 28 Mpa -33 MPa range (depending on whether the center or the edges of the zone are considered). It is of the order of 68 MPa in the corner of the crossing tows. This value is much greater than the stress at rupture of acrylic resin which is of the order of 44 r MPa 

. All these numerical and experimental results therefore match very reasonably and show that the CRAFT tool and associate spectral solver are very efficient and have interesting prospects for engineering applications on composite materials.

Lastly it should be noted that in the fiber region, the ratio of max/min values ,max ,min 11 11 mm  was of about 2. This is quite a large value and can be seen as a strong stress concentration at the mesostructure scale although it occurred in the same microstructural object, namely the tow. Following on from this observation, future study of the behavior of the heterogeneous tow with respect to damaging phenomena could be an interesting research direction.

Conclusion

This study presented numerical simulation results of the mechanical behavior of a woven fabric composite in the pure elastic case. Despite the complex geometry and strong heterogeneous properties of the Representative Elementary Volume (acrylic resin, Glass fibers arranged in warp and weft tows at 90°), the results serve as an explanation for in-depth experimental observation in terms of strains (direct comparison with full-field DIC measurements) and stresses (indirect comparison through the characterization of damage initiation). All data for the simulations was provided by independent characterizations performed by the authors and introduced, in some cases, into micromechanical models to determine properties at the mesoscale. It would be a good idea for future research based on this spectral solver to deal with modelling of the tows themselves in the stress field configuration obtained at the mesoscale to improve the overall numerical predictions.

Figure 1 :

 1 Figure 1 : Simulations of the equivalent deformation field of a woven tow with randomly distributed circular fibers using CRAFT code (a -c) and DAMASK code (b -d).

Figure 2 :

 2 Figure 2: Comparison of the results (i) published in Moulinec and Suquet (1998, Fig.6), (ii) obtained by CRAFT for the random arrangement shown in Fig.1 (solid line), (iii) obtained by DAMASK for the same arrangement (square dots).

Figure 4 :

 4 Figure 4: a) Numerical strain field XX  (the 1 and 2 squares indicate the warp and weft

  Figure 5 : a) Numerical and experimental strain patterns of meso XX  around a resin pocket; b)

Figure 6 :

 6 Figure 6 : Numerical and experimental profiles of the meso XX strain along the 45°lines after applying a smoothing procedure to CRAFT results.

Figure

  Figure 7: 12 m  stress map (CRAFT Simulation for 50 XX

Figure

  Figure 8: 12 m  stress profile along the tensile direction of Figure 9.

Figure

  Figure 9: a) 11 m  stress map (CRAFT Simulation for 50 XX

Table 1 :

 1 Mechanical properties of the different phases of the studied composite (1 = average tow direction).

				E (GPa)			
	Elium® resin	3.6				0.37
	E-Glass fiber	73				0.22
	Tow		Model	11 E (GPa)	22 E (GPa)		12	12 G (GPa)	23 G (GPa)
	tows V 	0.7	Chamis	52.2	17.6	0.265	6.57	Not provided
	f
			HC	52.2	15.3	0.265	5.71	Not provided
			CCA	52.2	19,1	0.259	6.01	7.17

Table 2 :

 2 Comparison of acroscopic elastic moduli

						XX E  and 12 G 
	E	XX 			XX  XX 	12 G 		2	12 12   
					REV				REV

Acknowledgments:

The authors wish to thank the Région Lorraine for financial support and PhD grant, Alain Gérard, technician at LEMTA who provided technical assistance for the experiments, Hervé Moulinec (LMA) for a fruitful introduction to CraFT and guidance in using it.