Silvio Capobianco
email: silvio@cs.ioc.ee

Pierre Guillon
email: pguillon@math.cnrs.fr

Camille Noûs
email: camille.nous@noussommesluniversite.fr

A characterization of amenable groups by Besicovitch pseudodistances

Keywords: Besicovitch distance, Følner sequences, submeasures, amenability, non-compact space, symbolic dynamics

The Besicovitch pseudodistance defined in [BFK99] for onedimensional configurations is invariant by translations. We generalize the definition to arbitrary countable groups and study how properties of the pseudodistance, including invariance by translations, are determined by those of the sequence of finite sets used to define it. In particular, we recover that if the Besicovitch pseudodistance comes from a nondecreasing exhaustive Følner sequence, then every shift is an isometry. For non-Følner sequences we prove that some shifts are not isometries, and the Besicovitch pseudodistance with respect to some subsequence even makes them non-continuous.

Introduction

The Besicovitch pseudodistance was proposed by Blanchard,Formenti and Kůrka in [BFK99] as an "antidote" to sensitivity of the shift map in the prodiscrete (Cantor) topology of the space of 1D configurations over a finite alphabet. The idea is to take a window on the integer line, which gets larger and larger, and compute the probability that in a point under the window, chosen uniformly at random, two configurations will take different values. The upper limit of this sequence of probabilities behaves like a distance, except for taking value zero only on pairs of equal configurations: this defines an equivalence relation, and the resulting quotient space is a metric space on which the shift is an isometry, or equivalently, the distance is shift-invariant.

The original choice of windows is X n = [-n : n], the set of integers from -n to n included. This notion can be easily extended to arbitrary dimension d ≥ 1, taking a sequence of hypercubic windows. If we allow arbitrary shapes, the notion of Besicovitch space can be extended to configurations over arbitrary groups; in this case, however, the properties of the group and the choice of the windows can affect the the distance being or not being shift-invariant. An example of a Besicovitch pseudodistance which is not shift-invariant is given in [Cap09], where it is also proved that, if a countable group is amenable (cf. [CGK13] and [CSC10, Chapter 4]), then the Besicovitch distance with respect to any nondecreasing exhaustive Følner sequence is shift-invariant. The class of amenable groups is of great interest and importance in group theory, symbolic dynamics, and cellular automata theory.

In this paper, we explore the relation between the properties of Besicovitch pseudodistances over configuration spaces with countable base group and those of the sequence of finite sets used to define it. We introduce a notion of synchronous Følner equivalence between sequences, and a related order relation where one sequence comes before another sequence if it is synchronously Følner-equivalent to a subsequence of the latter. This notion, on the one hand, generalizes that of Følner sequences, and on the other hand, allows us to compare the Besicovitch distances and submeasures associated to different sequences. In particular, we prove that an increasing sequence of finite sets is Følner if and only if every shift is an isometry for the corresponding Besicovitch distance: this provides the converse of [Cap09,Theorem 3.5]. Finally, we give conditions for absolute continuity and Lipschitz continuity of Besicovitch submeasures with respect to each other.

Background

We use the notation X Y to mean that X is a finite subset of Y . We denote the symmetric difference of two sets X and Y as X∆Y . We write

a n ∼ n→∞ b n if lim n→∞ a n /b n = 1 and a n = o n→∞ b n if lim n→∞ a n /b n = 0. For α ∈ R we put α = max{m ∈ Z | m ≤ α}.

Submeasures

The following definition is classical (see for instance [Sab06]).

Definition 1. A submeasure over a set G is a map µ : 2 G → R {+∞} such that: 1. µ(∅) = 0; 2. µ(W ) < ∞ if W is finite; 3. µ(V ∪ W ) ≤ µ(V ) + µ(W ) for every V, W ⊂ G.
If G and A are two sets, the difference set of two functions x, y :

G → A is the set ∆(x, y) = { i ∈ G| x(i) = y(i)}.
Any submeasure over G gives rise to an associated pseudodistance over A G :

d µ (x, y) = µ(∆(x, y)) ∀x, y ∈ A G .
Remark 1. The topological space corresponding to such a pseudodistance is homogeneous in the following sense: the balls around every two points y and z are isometric. Indeed, identify A with the additive group Z/ |A| Z. Then for every y, z ∈ A G the map ψ y,z : A G → A G defined by ψ y,z (x)(i) = x(i) -y(i) + z(i) for every x ∈ A G and i ∈ G is an isometry between any ball around y and the corresponding one around z.

We say that submeasure µ is absolutely continuous (resp. α-Lipschitz, for some α > 0) with respect to submeasure

ν if ν(W ) = 0 =⇒ µ(W ) = 0 (resp. µ(W ) ≤ αν(W )) for any W ⊂ G.
Remark 2. Let ε, δ > 0, µ, ν two submeasures on G, and z ∈ A G . The following are equivalent.

1. For every set W ⊂ G, µ(W ) ≥ ε =⇒ ν(W ) ≥ δ. 2. For every x, y ∈ A G , d µ (x, y) ≥ ε =⇒ d ν (x, y) ≥ δ. 3. For every x ∈ A G , d µ (x, z) ≥ ε =⇒ d ν (x, z) ≥ δ.
Consequently, the identity map, from space A G endowed with d ν onto space A G endowed with d µ , is continuous (resp. α-Lipschitz) if and only if µ is absolutely continuous (resp. α-Lipschitz) with respect to ν. In that case the identity is even absolutely continuous.

Shifts and translations

If A is an alphabet, G is a group, and g ∈ G, the shift by g is the function σ g : A G → A G defined by σ g (x)(i) = x(g -1 i), for every x ∈ A G and i ∈ G. A map ψ from A G to itself is shift-invariant if ψσ g = σ g ψ for every g ∈ G. Note that ∆(σ g (x), σ g (y)) = g∆(x, y) for every x, y ∈ A G and g ∈ G.

Since the maps ψ y,z from Remark 1 are shift-invariant, one can see that the shift is continuous, Lipschitz, etc in every x if and only if it is in one x.

Given g ∈ G, let gµ(X) = µ(g -1 X) for every X ⊂ G. Then d µ (σ g (x), σ g (y)) = d g -1 µ (x, y), that is, the shift by g, within space A G endowed with d µ , is topologically the same as the identity map, from A G endowed with d µ onto space A G endowed with d g -1 µ . Remark 1 can then be rephrased into the following.

Remark 3. If G is a group, g ∈ G, and A G is endowed with d µ , then σ g is continuous (resp. α-Lipschitz) if and only if g -1 µ is absolutely continuous (resp. α-Lipschitz) with respect to µ. In that case, the shift by g is even absolutely continuous.

Besicovitch submeasure and pseudodistance

Among classical examples of submeasures are the ones that induce the Cantor topology, the shift-invariant Besicovitch pseudodistance, the Weyl pseudodistance (see [HM17,Def 4.1.1]. . . We will focus on the Besicovitch topology. Let X and Y be nonempty sets and let (X n ) be a nondecreasing sequence of finite subsets of X. We may or may not require that (X n ) be exhaustive, that is,

n X n = X. Let us denote P ( W | V ) = |W ∩V | |V | (by convention, this is +∞ if V = ∅). The Besicovitch submeasure µ (Xn) : 2 X → [0, 1] is defined by: µ (Xn) (W ) = lim sup n P ( W | X n ) . The Besicovitch pseudodistance is d (Xn) = d µ (Xn ) .
For example, if X = N, Y = {0, 1}, and

X n = [0 : n -1],
x(i) = 0 for every i ∈ N and y ∈ {0, 1} N is the characteristic function of the prime numbers, then d (Xn) (x, y) = 0. The topology of the quotient space is very different from the prodiscrete topology.

We will now concentrate on the case of nondecreasing sequences (X n ).

3 Følner equivalence and Besicovitch submeasures

Følner equivalence

Let (X n ) and (Y n ) be nondecreasing sequences of finite subsets of G. We say that they are synchronously Følner-equivalent if

lim n→∞ |X n ∆Y n | |X n | = 0 . Proposition 1. Consider nondecreasing sequences (X n ) and (Y n ). The follow- ing are equivalent. 1. (X n ) and (Y n ) are synchronously Følner-equivalent. 2. |X n ∩ Y n | ∼ n→∞ |X n | ∼ n→∞ |Y | n . 3. |X n | ∼ n→∞ |Y n | and |X n \ Y n | = o n→∞ (|X n |).
Corollary 1. The synchronous Følner equivalence is an equivalence relation.

The proofs are left to the reader (see [CGN] for details).

We also denote (

X n ) (Y n ) if (X n ) is synchronously Følner-equivalent to a subsequence (Y mn ). Equivalently, lim n→∞ min m∈N |X n ∆Y m | |X n | = 0 .
To be convinced of the equivalence, note that the minimum is reached by some m n for each n ∈ N, because (Y m ) is nondecreasing and X n is finite. Thanks to symmetry of synchronous equivalence, we also have that (

X n ) (Y n ) if and only if lim n→∞ min m∈N |Xn∆Ym| |Ym|
= 0. We say that they are Følner-equivalent, and write (

X n ) ∼ (Y n ), if both (X n ) (Y n ) and (Y n ) (X n )
. This is the case if they are synchronously Følner equivalent, but the converse is false. As counterexamples, one can consider twice the same sequence, but with repetitions on both sides that are longer and longer, and not synchronized. If one wants to obtain strictly increasing sequences, repetitions can be replaced by very slowly increasing sequences (point by point).

Remark 4. It is easy to see that is a preorder relation. In turn, Følnerequivalence, being defined as the equivalence corresponding to the preorder , is an equivalence relation.

Proposition 2. Assume that |X n | ∼ n→∞ |Y n |. Then (X n ) and (Y n ) are syn- chronously Følner-equivalent if and only if (X n ) (Y n ). Proof. Assume (X n ) (Y n ) (the converse implication is trivial). Let n, m ∈ N. If m ≤ n, then |X n \ Y n | ≤ |X n \ Y m | and |Y n \ X n | ≤ |Y n \ Y m | + |Y m \ X n | since (Y n ) is nondecreasing. Summing up, |X n ∆Y n | ≤ |X n ∆Y m | + |Y n \ Y m |. Symmetrically, if n ≤ m, |X n ∆Y n | ≤ |X n ∆Y m | + |Y m \ Y n |. Overall for every m ∈ N, we get |X n ∆Y n | ≤ |X n ∆Y m | + ||Y m | -|Y n ||. If we apply this with (m n ) the subsequence from the definition of , which is such that (X n ) ∼ (Y mn ), we have |X n ∆Y mn | = o n→∞ (|X n |), and by Proposition 1 (applied to (X n ) and (Y mn )), |Y mn | ∼ n→∞ |X n | ∼ n→∞ |Y n |. Summing up, we deduce that |X n ∆Y n | = o n→∞ (|X n |).

Comparing Besicovitch submeasures

A basic tool in our set constructions will be the following elementary remark.

Remark 5. If (X n ) is nondecreasing and exhaustive, then for every finite set W and every ε > 0, there exists

n (Xn) (W, ε) such that ∀n ≥ n (Xn) (W, ε), P ( W | X n ) < ε and W ⊂ X n .
We deduce the following, which will be useful in our constructions. Lemma 1. Let (X n ) be a nondecreasing exhaustive sequence of an infinite group G. Let W = i∈N W i where ∅ = W i G for each i ∈ N, such that, for every n ∈ N, there are at most finitely many i's such that W i ∩X n = ∅ (this is the case, for example, if the W i 's are pairwise disjoint); in that case j n = max Wj ∩Xn =∅ j is well-defined for every n. Then:

1. µ (Xn) (W ) ≥ lim sup i→∞ max m∈N P ( W i | X m ) . 2. If there is a sequence (ε n ) converging to 0 such that n (Xn) ( i<jn W i , ε n ) ≤ n for every n ∈ N, then: µ (Xn) (W ) = lim sup i→∞ max m∈N P ( W i | X m ) .
3. In general, there exists a nondecreasing integer sequence l such that, denoting

W l = i∈N W li : µ (Xn) (W l ) = lim i→∞ max m∈N P ( W li | X m ) .
Proof.

1. Let (m i ) ∈ N N be such that P ( W i | X mi ) = max m∈N P ( W i | X m ).
We know that this sequence goes to infinity (even though it may not be nondecreasing), because only finitely many W i 's intersect each X m , but they all intersect at least one. Hence, µ (Xn) (W ) ≥ lim sup i→∞ P ( W | X mi ). We get the desired inequality by noting that W i ⊂ W . 2. Point 1 already gives one inequality. For the converse:

µ (Xn) (W ) = lim sup n→∞ P   i<jn W i ∪ W jn ∪ i>jn W i X n   ≤ lim sup n→∞   P   i<jn W i X n   + P ( W jn | X n ) + P   i>jn W i X n     ≤ lim sup n→∞ ε n + max m∈N P ( W jn | X m ) + 0 ≤ lim sup n→∞ ε n + lim sup n→∞ max m∈N P ( W jn | X m ) ≤ 0 + lim sup i→∞ max m∈N P ( W i | X m ) .
The last inequality comes from the fact that the sequence (j n ) is nondecreasing (because (X n ) is nondecreasing), and not upper-bounded (because the W i 's are nonempty), so it goes to infinity. 3. Let us define some sequence l by recurrence, from any seed l 0 ∈ N. Assume that l n is defined, and write k n = n (Xn) ( j≤n W lj ). Choose any l n+1 such that for every m ≥ l n+1 , W m does not intersect X kn-1 (this is possible by assumption). If j n = max W l j ∩Xn =∅ j, then n (Xn) ( j<jn W lj ) = k jn-1 . By definition, W lj n does not intersect X kj n -1-1 . Since W lj n intersects X n , we can deduce that n > k jn-1 -1. This means that (W li ) satisfies the hypothesis of Point 2.

Replacing the lim sup by a lim can be achieved by taking a subsequence.

Lemma 2. Let ε, δ > 0, and (X n ), (Y n ) be nondecreasing and exhaustive. The following are equivalent.

For every

W ⊂ G, if µ (Yn) (W ) ≥ ε, then µ (Xn) (W ) ≥ δ. 2. lim inf n∈N max m∈N ε |Y n | -|Y n \ X m | |X m | ≥ δ.
If m n realizes the maximum for each n ∈ N, and if ε < 1, then these properties imply that

δ ε ≤ lim inf n∈N |Y n | |X mn | ≤ lim sup n∈N |Y n | |X mn | ≤ 1 -δ 1 -ε .
In particular, the properties imply that δ ≤ ε.

Proof.

Let

n | ≤ 1-δ 1-ε , provided that ε < 1.
2⇒1 If property 2 is satisfied and µ (Yn) (W ) ≥ ε, then:

µ (Xn) (W ) ≥ lim sup n→∞ P ( W ∩ Y n | X mn ) ≥ lim sup n→∞ |W ∩ Y n | -|Y n \ X mn | |X mn | = lim sup n→∞ ε |Y n | -|Y n \ X mn | |X mn | + |W ∩ Y n | -ε |Y n | |Y n | |Y n | |X mn | ≥ lim inf n→∞ ε |Y n | -|Y n \ X mn | |X mn | + lim sup n→∞ |W ∩ Y n | |Y n | -ε lim inf n∈N |Y n | |X mn | ≥ δ + 0 δ ε
= min { k| |Y n \ X k | ≤ ε |Y n |}, because for large k, Y n \ X k = ∅ (because (X k ) is exhaustive and Y n is finite). By noting that (Y n ∩ X kn ) \ X kn-1 = (Y n \ X kn-1 ) \ (Y n \ X kn ) (by convention X -1 is empty), we can write that |(Y n ∩ X kn ) \ X kn-1 | = |Y n \ X kn-1 |-|Y n \ X kn |, which is bigger than ε |Y n | -|Y n \ X kn |, by minimality of k n . Hence (Y n ∩ X kn ) \ X kn-1 admits a subset Z n of cardinality |Z n | = ε |Y n | -|Y n \ X kn |. Define W n = (Y n \ X kn ) Z n . Note that W n ⊂ Y n , and that ε -1 |Yn| < P ( W n | Y n ) ≤ ε.
The W i satisfy the hypotheses of Lemma 1, so that Point 3 gives l ∈ N N , with µ (Xn) (W l ) = lim i→∞ max m∈N P ( W li | X m ). By construction, we have:

P ( W i | X m ) = P ( Y i \ X ki | X m ) + P ( Z i | X m ) = |Y i ∩ X m \ X ki | + |Z i ∩ X m | |X m | . If m < k i , then X m ⊆ X ki , and Z i ∩ X m ⊆ Z i ∩ X ki-1 = ∅, so that this quantity is 0. On the contrary, if m ≥ k i , then Z i ⊆ X ki ⊆ X m , and
Y i ∩ X m \ X ki = (Y i \ X ki ) \ (Y i \ X m )
, so that:

P ( W i | X m ) = |Y i ∩ X m \ X ki | + |Z i ∩ X m | |X m | = |Y i ∩ X m \ X ki | + |Z i | |X m | = |Y i \ X ki | -|Y i \ X m | + ε |Y i | -|Y i \ X ki | |X m | ≤ max m∈N | ε |Y i | | -|Y i \ X m | |X m | < δ by hypothesis.
Taking the limit, we get that µ (Xn) (W l ) < δ. On the other hand, applying now Point 1 of Lemma 1 to sequence (Y n ):

µ (Yn) (W l ) ≥ lim i∈N max m∈N P ( W li | Y m ) ≥ P ( W li | Y li ) = ε .
The previous lemma now allows to characterize the main properties of interest for comparing two Besicovitch submeasures.

Theorem 1. Let (X n ) and (Y n ) be nondecreasing and exhaustive.

1. µ (Yn) is λ-Lipschitz with respect to µ (Xn) , where λ > 0, if and only if

∀ε > 0, lim inf n→∞ max m∈N |Y n | -1 ε |Y n \ X m | |X m | ≥ 1 λ .
2. µ (Yn) is absolutely continuous with respect to µ (Xn) if and only if it is Lipschitz.

µ

(Yn) ≤ µ (Xn) if and only if (Y n ) (X n ). 4. µ (Yn) = µ (Xn) if and only if (Y n ) ∼ (X n ).
One can even see from the proof that (Y n ) (X n ) if and only if there exists

ε ∈]0, 1[ such that ∀W ⊂ G, µ (Xn) (W ) < ε =⇒ µ (Yn) (W ) < ε.
Proof.

1. Just note that the λ-Lipschitz property of µ (Yn) is equivalent to the properties in Lemma 2, for every δ and ε = λδ, and hence to:

lim inf n∈N max m∈N |Y n | -1 ε |Y n \ X m | |X m | ≥ 1 λ .
2. From Lemma 2, µ (Yn) is absolutely continuous with respect to µ (Xn) if and only if

∀ε > 0, lim inf n→∞ max m∈N |Y n | -1 ε |Y n \ X m | |X m | > 0 .
From Point 1, this is equivalent to the existence of some λ such that µ (Yn) is λ-Lipschitz with respect to µ (Xn) .

Let (m

n ) ∈ N N satisfy lim n→∞ |Yn∆Xm n | |Yn| = 0. Then lim n∈N |Y n | -1 ε |Y n \ X mn | |X mn | = lim n∈N |Y n | |X mn | 1 - 1 ε lim n∈N |Y n \ X mn | |Y n | = 1 .
We can conclude by Point 1.

Conversely, suppose that

lim inf n∈N |Y n | -1 ε |Y n \ X mn | |X mn | ≥ 1 .
By the last inequalities in Lemma 2, we know that lim n∈N

|Yn| |Xm n | = 1. More- over, lim n→∞ |Y n \ X mn | |X mn | ≤ lim n→∞ ε |Y n | |X mn | -ε lim inf n∈N |Y n | -1 ε |Y n \ X mn | |X mn | = ε -ε = 0 .
By Point 3 of Proposition 1, we obtain that (Y n ) (X n ). 4. This is direct from the definitions and the Point 3.

The following is direct from Theorem 1 and Remark 2.

Corollary 2. If (X n ) and (Y n ) are nondecreasing and exhaustive, then (Y n ) (X n ) (resp. (Y n ) ∼ (X n ))
if and only if the identity map from A G endowed with d (Xn) onto A G endowed with d (Yn) is 1-Lipschitz (resp. an isometry).

Here are particular classes of sequences, where the proposition can be applied.

Corollary 3. Let (X n ) and (Y n ) be nondecreasing and exhaustive.

1. If there exist λ > 0 and a sequence (m n ) such that lim inf n→∞ P (

X n | Y mn ) ≥ 1 λ and X n ⊂ Y mn , then µ (Xn) is λ-Lipschitz with respect to µ (Yn) . 2. If for cofinitely many n ∈ N, Y n ⊂ X n+1 and lim inf n→∞ P ( X n | X n+1 ) ≥ λ, then µ (Xn) is λ-Lipschitz with respect to µ (Yn) . 3. On the other hand, if |X n | ∼ n→∞ |Y n | but (X n ) and (Y n ) are not (syn- chronously) Følner-equivalent , and n (Ym) (X n , ε n ) = n + 1 for some real sequence (ε n ) converging to 0, then µ (Xn)
is not absolutely continuous with respect to µ (Yn) .

Proof.

1. For every ε > 0, lim inf n→∞ max m∈N |X n | -1 ε |X n \ Y m | |Y m | ≥ lim inf n→∞ |X n | -1 ε |X n \ Y mn | |Y mn | = lim inf n→∞ |X n | |Y mn | ≥ 1 λ .

Apply Point 1 with m

n = min { m ∈ N| X n ⊂ Y m }; the hypothesis is that m n is ultimately n + 1. 3. Suppose |X n | ∼ n→∞ |Y n | and (X n ) and (Y n ) are not synchronously Følner- equivalent. By Proposition 2, (X n ) (Y n ), that is, ε = lim sup n→∞ |Xn\Yn| |Yn| > 0. We can write lim inf n→∞ |Xn|-1 ε |Xn\Yn| |Yn| = 0.
By the second assumption, for every m > n, X n \ Y m = ∅ and |Xn| |Ym| ≤ ε n . We get:

max m∈N |X n | -1 ε |X n \ Y m | |Y m | ≤ max |X n | -1 ε |X n \ Y n | |Y n | , ε n .
Putting things together, lim inf n→∞ max m∈N |Xn|- 2. µ (Y ln ) = µ (X ln ) , for every increasing sequence (l n ) ∈ N N . 3. µ (Y ln ) is absolutely continuous with respect to µ (X ln ) , for every increasing sequence (l n ).

Proof.

1 =⇒ 2 If (X n ) and (Y n ) are synchronously Følner equivalent, then so are (X ln ) and (Y ln ) for every increasing (l n ) ∈ N N . We conclude thanks to Theorem 1. 2 =⇒ 3 This is obvious. 1 =⇒ 3 If (X n ) and (Y n ) are not synchronously Følner-equivalent, then there exists an infinite set I ⊂ N and a real number α > 0 such that ∀n ∈ I, |Xn∆Yn| |Xn| ≥ α. This implies that for every increasing sequence (l n ) ∈ I N , (X ln ) and (Y ln ) are not synchronously Følner-equivalent. We can take an increasing sequence (l n ) ∈ I N such that n (Ym) (X ln , ε ln ) = l n+1 , for some real sequence (ε n ) converging to 0. Then (X ln ) and (Y ln ) satisfy the assumptions for Point 3 of Corollary 3.

Shift

If G is a group and (X n ) ∼ (gX n ), then we say that (X n ) is (left) g-Følner ; (X n ) is right g-Følner if (X -1 n ) is left g -1 -Følner. Since |X n | = |gX n |,
Proposition 2 says that it is enough to require (X n ) (gX n ), and in this case, (X n ) and (gX n ) are even synchronously Følner-equivalent.

A (left) Følner sequence for a countable group G is a g-Følner sequence for every g ∈ G. A countable group is amenable if and only if it admits a Følner sequence: see [CSC10, Chapter 4] also for many equivalent definitions.

The following is a rephrasing of Corollary 2.

Corollary 5. Let G be a countable group and let (X n ) be a nondecreasing exhaustive sequence. There are nondecreasing non-Følner sequences for which the shift is Lipschitz (but not an isometry) in Z d . Here's an example: X n = ( -n, n ∪ 2 -n, n ) d . Indeed, for every n, 1 + X n ⊂ X 2n and |X2n| |Xn| = (8n-1) d (4n-1) d , which converges to 2 d when n goes to infinity. We conclude by Point 1 of Corollary 3, with m n = 2n and α = 2 d . But the shift is not an isometry because the sequence is not Følner: µ((2Z) d ) = 2 d /3 d > µ((2Z + 1) d ) = 1/3 d .

"Dually" to shifts, we can define the propagation π g : A G → A G by π g (x)(i) = x(ig). A block map (see [LM95] for G = Z) is, in essence, a composition of a radius-0 function with a product of propagations. The same characterizations are true for propagations as for shift maps, to which we can derive the following: Corollary 7. A nondecreasing exhaustive sequence (X n ) of finite subsets of a f.g. group G is right Følner if and only if for every increasing sequence (l n ) ∈ N N , every block map with neighborhood size k is k-Lipschitz for d (X ln ) .

Conclusions

We have presented a way to compare Besicovitch submeasures (in terms of absolute continuity, Lipschitz continuity, equality) thanks to the sequences of finite sets which describe them. In a shift space (with respect to a finitely generated group) endowed with the Besicovitch topology, we have derived conditions on the defining sequence for the shift maps to be continuous, Lipschitz or isometries. As part of this, we gave another characterization of f.g. amenable groups.

Future work will involve the study of other topological and dynamical properties (cf. [CGN]) or extension to configuration spaces on possibly uncountable groups. The latter would require the use of the more general notions of directed set and of net, and although the definition of Besicovitch pseudodistance and submeasure would be immediate to extend, the techniques used to prove the main lemmas could need a major revision.

  us start by proving the final inequalities. Suppose lim inf n∈N Then on the one hand, it is clear that lim inf n∈N ε|Yn| |Xm n | is even bigger, which gives the first inequality. On the other hand, since |Y n \ X mn | ≥ |Y n | -|X mn |, we can see that lim inf n∈N (ε -1) Yn

				ε|Yn|-|Yn\Xm n | |Xm n |	≥
	δ. Xm n	+ 1 ≥ lim inf n∈N	ε|Yn|-|Yn\Xm n | |Xm n |	≥ δ, which
	gives that lim sup n∈N	|Yn| |Xm

  Corollary 4. Let (X n ) and (Y n ) be nondecreasing and exhaustive. Assume that |X n | ∼ n→∞ |Y n |. Then the following are equivalent. 1. (X n ) and (Y n ) are synchronously Følner-equivalent.

	1 ε |Xn\Ym| |Ym|	is 0. We con-
	clude by Point 2 of Theorem 1.

  1. (X n ) is g-Følner if and only if µ (Xn) = µ (g -1 Xn) if and only if the shift by g is an isometry. 2. (X n ) is Følner if and only if every shift is an isometry. 3. If G is finitely generated (see below) then it is amenable if and only if there exists a nondecreasing exhaustive sequence (X n ) of finite subsets of G such that every shift is an isometry. Note that one implication of Point 3 was already stated in [Cap09, Theorem 3.5], but the proof contains a confusion between left and right Følner. A group G is finitely generated (briefly, f.g.) if E G exists such that for every g ∈ G there exists e 1 , . . . , e n ∈ E ∪ E -1 such that e 1 • • • e n = g. Remarkably (cf. [Pet, Lemma 5.3]) if a f.g. group is amenable, then it has a nondecreasing exhaustive Følner sequence. In addition, if the size of the balls grows polynomially with the radius, then they form a Følner sequence, so Point 3 of Corollary 5 generalizes [HM17, Cor 4.1.4]. Corollary 6. Let G be a finitely generated group. 1. If (X n ) is the sequence of balls with respect to some generating set of cardinality α, then every shift is α-Lipschitz. 2. If g ∈ G, a nondecreasing exhaustive sequence is g-Følner if and only if all of its subsequences yield a Besicovitch pseudodistance for which the shift by g is continuous. 3. G is amenable if and only if it admits a nondecreasing exhaustive sequence of finite subsets of which all subsequences yield a Besicovitch distance for which every shift is continuous. |E| + 1) n . We can apply Point 2 of Corollary 3. 2. This comes from Corollary 4. 3. This comes from Point 2.

	The first point generalizes [HM17, Prop 4.1.3]. Note that it still applies in non-
	amenable groups, but the shifts are no longer isometries, and there is a sub-
	sequence of balls with respect to which the Besicovitch pseudodistance makes
	them non-continuous.
	Proof.

1. If E is the generating set and E n the corresponding radius-n ball, then E 0 = {e} where e is the identity of G and

E n+1 = (E ∪ E -1 ) • E n , so |E n | ≤ (2

Corresponding author. This research was supported by the Estonian Ministry of Education and Research institutional research grant no. IUT33-13.


O De Almeida

E Bessard

G Bernhart

INFLUENCE OF PROCESSING PARAMETERS AND SEMI-FINISHED PRODUCT ON CONSOLIDATION OF CARBON/PEEK LAMINATES

Keywords: long-fibre thermoplastic composite, fibre impregnation, PEEK

Five millimetres thick laminates of carbon fibre PEEK matrix composite are processed with a fast thermo-compression moulding process using different parameter sets, namely temperature, pressure and consolidation time. Laminates consolidation is evaluated according to the architecture and quality of semi-finished product and process conditions.Microstructural observations show that impregnation level largely depends on the initial degradation of the matrix. Indeed, semi-finished product manufacturing can induce PEEK degradation that reduces the fluidity of the matrix and therefore impregnation.

Introduction

Semi-crystalline thermoplastic composites exhibit high toughness and excellent solvent resistance that make then suitable for high performance application. Among the thermoplastics, poly-ether-ether-ketone (PEEK) is a reference material for the aeronautic industry since the 80's: the unidirectional prepreg tape AS4/APC2 of Cytec Engineering Materials Inc. represents a first choice material for structural applications. However, this thermoplastic semi-finished product exhibits poor flexibility due to the high melting point of PEEK (Tm ≈ 343°C) and complex shape composite parts are difficult to manufacture with UD prepreg tapes [START_REF] Meyer | Hot forming behavior of non-crimp fabric peek/c thermoplastic composites[END_REF]. Carbon fabrics exhibit a convenient drapability that allows composite parts manufacturing with undevelopable geometries. But above melting temperature, thermoplastics viscosity is much higher than that of thermosets and "dry impregnation" of fibre yarns with thermoplastic resins is still not conclusive [START_REF] Mayer | Macro-and micro-impregnation phenomena in continuous manufacturing of fabric reinforced thermoplastic composites[END_REF] [START_REF] Bourban | Material phenomena controlling rapid processing of thermoplastic composites[END_REF]. For these reasons, various semi-finished products have been developed during the last decade: commingled yarns used in crimp and non-crimp fabrics [START_REF] Ye | Consolidation of unidirectional CF/PEEK composites from commingled yarn prepreg[END_REF], powdered fabrics, film stacking [START_REF] Jespersen | Film stacking impregnation model for a novel net shape thermoplastic composite preforming process[END_REF]. Their specificity is to get around the difficulty of dry impregnation by reducing the distance the thermoplastic matrix has to cover to obtain a consolidated laminate. On the other hand, such products are interesting for out of autoclave processing routes that still need to be optimized. In this study, processing behaviour of three different carbon fibre PEEK semi-finished products is evaluated using a fast thermo-compression moulding process based on induction heated dies. Thermal sensitivity of PEEK is first analysed in order to propose a process window and the quality of the semi-finished products is discussed. Laminates microstructure induced by the process conditions is then discussed for the different products. Out of autoclave consolidation of PEEK laminates is a delicate issue to the extend that thermoplastics can be sensitive to oxidation. In particular, in air environment at high temperature, above its melt temperature, PEEK can exhibit a reactive behaviour leading to the formation cross-links [6][7]. In order to identify a processing window, measurements of the melt behaviour of a neat PEEK have been performed with a plate-plate shear viscometer. The samples were rapidly heated to an isothermal temperature condition and the viscosity was measured through time (Figure 1). The results show that in a first stage, PEEK viscosity remains constant and a classical thermal dependence is observed. But degradation of PEEK progressively occurs resulting to an increase of the PEEK viscosity that is all the more rapid that temperature is high. The characterisation of the thermal degradation of PEEK then reveals that the use of high processing temperature for PEEK laminates consolidation is not necessarily beneficial. Low viscosity can favour impregnation but if degradation occurs before total impregnation, the advantage of fluidity can be considered as totally lost. Considering that the advantage of low viscosity is lost from the moment where viscosity increase is of 20%, an acceptable residence time can be defined for each isothermal processing temperature. The resulting processing window is depicted in Figure 2.

Semi-finished products

Three different carbon / PEEK semi-finished products containing 50vol% carbon fibre were investigated, all of them based on PEEK from Victrex (151G grade). One of the products is a Pi-Preg fabric provided by PORCHER Composites as a 2-faces powdered carbon fabric (5H Satin, 3k, 285g.m -2 ). Powdered fabrics are manufactured by heating to melt a fabric previously sprinkled with PEEK powder. As show in Figure 3, the microstructure of this product appears as a dry fabric covered with PEEK droplets. The second studied material is a commingled quadriaxial NCF consisting of a blended combination of carbon reinforcing yarns and PEEK yarns (585g.m -2 ). PEEK fibres were obtained by melt spinning and the average diameter of PEEK filaments is 20μm. Figure 4 shows that the distribution of the filaments within the blankets is not very homogeneous. The microstructure exhibits aeras with high concentration of PEEK or carbon. A composite obtained by stacking of PEEK films and carbon fabrics (Serge 2x2, 285g.m -2 , 6k) was also evaluated. PEEK 38μm thick films were supplied by Victrex and the stack sequence was made laying alternatively PEEK films and carbon fabrics so as to satisfy a carbon content of 50vol%. The three laminate configurations were compared to the reference prepreg AS4/APC2 from Cytec which fibre volume ratio is 0.6. The conventional autoclave processing cycle for AS4/APC2 consists in applying a constant pressure of 5 bars at a temperature of 400°C for 100 minutes. Nevertheless, as displayed in Figure 5, AS4/APC2 tape is already highly impregnated before processing and shorter thermal cycle can be used to manufacture laminates in an out of autoclave configuration, using simply a vacuum bag in an oven. 4 Product quality Since PEEK exhibits a high sensitivity to high processing temperatures, the initial quality of the studied semi-finished products had to be characterized. Calorimetry (DSC) measurements were performed on the different PEEK matrices and the melting temperature was used to quantify the possible initial degradation level of PEEK.

As a reference, a sealed calorimeter cap has been prepared with about 3mg of neat virgin PEEK and degradation cycles were performed on the sample. The cycles consisted in applying cooling (20°C.min-1) and heating (10°C.min-1) treatments to the sample between room temperature and 400°C (20 minutes isothermal step). The melt temperature was measured during the heating stage for 15 cycles and the evolution is plotted in Figure 6. Figure 6 shows that PEEK degradation process lead to a fast decrease of the melt temperature of PEEK with cycles. After 15 cycles, the melt temperature decreases from 343°C to about 340°C, i.e. with an average rate of 0,2°C per cycle.

The same procedure was applied to the PEEK matrix of the semi-finished products and 3 cycles were performed on the samples in such a way to reduce measurements uncertainty. The melt temperature measured for the 3 different products is reported in Table 1. The values allow ranking the different matrices used in the semi-products as a function of their initial degradation level. PEEK matrix of the powdered fabric exhibits the highest melt temperature of 343,2°C that is consistent with the value of non degraded polymer. On the other hand, the melting temperature of PEEK filaments from the commingled NCF and the melting temperature of PEEK from the 38μm film are respectively 341,1°C and 340°C. This indicates that initial commingled NCF and film stacking are processed with degraded PEEK that will be damaging for fibre impregnation.

Semi

Compression moulding process

Laminates were made of different number of plies so as to obtain a final fully impregnated 5mm thick reinforced composite. Thermo-compression was performed with an inductive heating mould (Cage system® technology developed by Roctool) mounted on an instrumented press. This technology allows high heating and cooling rates and therefore offers new optimization possibilities [START_REF] Bessard | Manufacturing of structural thermoplastic composites with fast compression moulding process in SEICO[END_REF]. In this study, plies of 200x200mm 2 were heated to the processing temperature (resp. cooled) at a constant heating rate of 50°C.min-1 (resp. 20°C.min-1). Pressure was applied at the beginning of the isothermal temperature step and maintained until the end of cooling to avoid deconsolidation. By using this fast thermo-compression moulding process, the exposure time to high temperature can be controlled and according to the preliminary evaluation of the processing window, different exposure time / processing temperature combinations were thus chosen to manufacture C/PEEK laminates:

-5 minutes at 420°C, -10 minutes at 400°C, -10 minutes at 380°C, -20 minutes at 380°C.

These different configurations were used to evaluate the influence of processing parameters on impregnation of the powdered fabric product. The different semi-finished products are then compared so as to rank impregnability of the products.

Influence of processing parameters on powdered fabrics impregnation

Laminates manufactured with powdered fabrics were processed with a consolidation pressure of 8 bars. For each condition, a sample was machined in centre of the laminate and the polished specimens were observed with an optical microscope.

The microstructure resulting from the different consolidation time and temperature conditions are displayed in Figure 7. Consolidation in configuration 1 and 2 leads to a fully impregnated laminate, with a porosity level of about 2-2,5% as determined by chemical dissolution. However, consolidation quality, when performed at 380°C for only 10 minutes, turns out to be lower and numerous micro-porosities are visible. As a consequence, processing window of powdered fabrics can be refined by adding a insufficient processing consolidation zone (Figure 8). This area illustrates the processing temperature / consolidation time couples that lead to a poor impregnation level (porosity rate of about 3%). The shape of this area is approximated in considering that high fluidity with short processing time as well as high viscosity with long processing time lead to the same impregnation level.

Influence of semi-finished products on impregnation

The same processing procedure has been performed to manufacture laminates from commingled NCF and from film stacking products. Figure 9 shows the microstructure resulting from consolidation of commingled NCF laminates with a pressure of 8 bars for two extreme processing conditions. In contrary to powdered fabrics, in both cases, impregnation is incomplete and macro-porosities are observable. The microstructure consecutive to a processing condition of 5 minutes at 420°C exhibits macroporosities in both carbon yarns and rich polymer zones while 20 minutes at 380°C lead to even worth consolidated microstructure with millimetric porosities in polymer areas. To improve consolidation quality of commingled NCF, a higher pressure is required. As shown in Figure 10, porosity can be reduced to a micrometric size by applying a pressure of 15 bars. The microstructure still exhibits a disperse micro-porosity which rate is close to 3,5%. Finally, laminates manufactured from film stacking configuration were processed at 400°C for 10 minutes with an applied pressure of 7 bars. As displayed in Figure 11, the microstructure pictures exhibit enlarged areas of unimpregnated fibres depicted as large dark ellipses in the centre of the transverse yarns. To the extend that carbon fibre bundles are 6k carbon yarns, lower impregnation level could have been expected in comparison with the powdered product and to enhance the degree of consolidation, residence time and/or the molten resin fluidity and/or the pressure must be increased. Nevertheless, the advanced degradation of the PEEK films may also be responsible for some extend for this poor impregnation.

Conclusions

The analysis of PEEK thermal stability shows a rapid degradation process. It induces an increase of the melt behaviour that must be taken into consideration when processing reinforced PEEK laminates. High temperature processing may favour impregnation at first but degradation may stop the PEEK percolation process within the carbon bundles for longer consolidation time.

Consolidation of different semi-finished products was evaluated as function of processing conditions. The characterization of semi-finished product matrices shows that degradation was already initiated before composite processing (i.e. during semi-product manufacturing) and as a consequence laminates exhibited distinct degrees of impregnation when manufactured with an equivalent processing parameter set. Moreover, each semi-finished product led to different porosity type depending on the initial PEEK carbon architecture.

Figure 1 .

 1 Figure 1. Time evolution of shear viscosity of PEEK for different isothermal conditions

Figure 2 .

 2 Figure 2. PEEK Processing window

Figure 3 .

 3 Figure 3. Powdered fabric from Porcher Composites (a) surface aspect (b) thickness microstructure

Figure 4 .

 4 Figure 4. Commingled quadriaxial NCF (a) surface aspect (b) microstructure showing the commingling quality

Figure 5 .

 5 Figure 5. Semi-finished product APC2-AS4

Figure 6 .

 6 Figure 6. Influence of heating and cooling cycles on the melt temperature of PEEK

Figure 7 .Figure 8 .

 78 Figure 7. Microstructure of powdered fabric laminates (a) 5 minutes at 420°C (b) 20 minutes at 380°C (c) 10 minutes at 380°C

  behaviours of the commingled product cannot only be attributed to the commingling organisation since polymer distribution is supposed to favour impregnation. The initial PEEK quality may probably also have influenced the impregnability of this semifinished product.

Figure 9 .

 9 Figure 9. Microstructure of commingled NCF (a) 5 minutes at 420°C (b) 20 minutes at 380°C

Figure 10 .

 10 Figure 10. Microstructure of a commingled NCF laminate processed 10 minutes at 400°C with a consolidation pressure of 15 bars.

Figure 11 .

 11 Figure 11. Film stacking microstructure after 10 minutes at 400°Cwith a consolidation pressure of 8 bars

Table 1 .

 1 Melt temperature of the semi-finished products PEEK matrix (average of 3 cycles)

	-finished product	Melt temperature
	Powdered fabric	342,2°C
	Commingled NCF	341,1°C
	Film	340,0°C


L Escale

O De Almeida

G Bernhart

J F Ferrero

COMPARISON OF THE IMPACT RESISTANCE OF CARBON/EPOXY AND CARBON/PEEK COMPOSITE LAMINATES

Keywords: polymer matrix composites, carbon fibres, impact behaviour, energy absorption

This study deals with the impact resistance of usual composite materials. Numerous results available in the open literature concern damage tolerance studies of thermoset laminates reinforced with unidirectional carbon fibre plies, as they are today mainly used in aeronautical structures. However, thermoplastic composites represent a growing interest with the aim of future structural applications. The influence of the composite matrix and the reinforcing carbon armour on the impact behaviour of composite laminates was then analysed over a wide range of impact energy: drop weight test, Charpy test and gas gun test. Dissipative mechanisms were analysed (delamination, elastic energy storage and fibre breakage) and related to the armour architecture and polymer matrix. The results show that UD plies favour delamination and therefore energy dissipation. However, the effect of composite matrix was not observed.

Introduction

Next generation aircraft may introduce contra-rotative propulsion systems for energy saving purposes. This introduces new requirements on fuselage related to the open rotor configuration, which pale rupture may generate high velocity small fragments. This problematic is particularly delicate since organic composite materials are envisaged for the next generation fuselages. Nevertheless, the current design of composite aircraft structures, as regards to impact resistance, only consider compression resistance after low speed / low energy impact for fatigue life prediction of damaged laminates; and laminates impact energy absorption is still relatively unknown. The aim of this study was therefore to evaluate the impact behaviour of various carbon fibre composite laminates as regards to the reinforcing armour, the polymer matrix and the impact energy.

Materials and testing methods

In order to analyse the influence of the matrix nature on the impact behaviour, i.e. thermoset vs. thermoplastic, three different resins were tested: M21 (epoxy thermoset), PEEK and PPS (semi-crystalline thermoplastics). The influence of the carbon armour was also evaluated by comparing unidirectional reinforcements to carbon 3k satin 5H fabrics.

Quasi-isotropic 5 mm thick laminates of 200 mm x 200 mm dimensions were manufactured from different semi-finished carbon fibre products (table 1). A conventional processing cycle in autoclave, as recommended by Hexcel, was used to process T700/M21 UD prepregs. Cytec AS4/APC-2 prepregs were consolidated in an electrically heated press at 395°C during 35 minutes with a constant pressure of 7 bars. Finally, PEEK and PPS carbon fabrics, provided by Porcher Composites were processed with the Cage System® inductive moulding technology developed by Roctool using a short consolidation cycle [START_REF] Paris | Étude et modélisation de la polymérisation dynamique de composites à matrice thermodurcissable[END_REF]. Drop weight testing was performed on 100 mm x 150 mm laminates supported by a 75 mm x 125 mm frame according to the AITM 1-10010 testing method [START_REF]Airbus Industrie Test Method: AITM 1-0010: Determination of compression strength after impact[END_REF]. An impact energy of 55 J was obtained with a spherical impactor of 25 mm diameter and the impact load was recorded during the impact thanks to a load cell mounted on the impactor. Charpy tests were performed on unnotched samples of 10 mm x 70 mm dimensions with an impact energy of about 300 J. According to the ISO 179 standard method, the samples were struck through laminate thickness by the hammer and the impact force was measured. Gas gun tests consisted in propelling a bearing ball of 25 mm diameter on 200 mm x 200 mm composite plates with a gas gun. The plates were only supported by a 170 mm x 170 mm frame and the maximum ball velocity was 124 m.s -1 as recorded before impact by a highspeed camera. Residual ball speed after perforation was also recorded with a second highspeed camera.

Results and discussion

Drop weight tests

The variation of the measured load during drop weight tests is displayed in figure 1a for the different laminates. The curves first exhibit a linear elastic domain but rapidly followed by a slight decrease of rigidity depicted by oscillations. Then, because of the low impact energy imposed in drop weight tests the curves exhibit a loop shape characteristic of elastic energy restitution (bounce of the impactor) [START_REF] Rahme | Experimental investigation of impact on composite laminates with protective layers[END_REF]. Thanks to the load measurement during impact, the projectile velocity and therefore the energy variation can be determined by a double integration (figure 1b) [START_REF] Petit | Impact and compression after impact experimental study of a composite laminate with a cork thermal shield[END_REF]. The residual energy corresponds to the absorbed energy by damaging. The graphs of figure 1 clearly show the influence of the reinforcing armour on the impact behaviour of composite laminates. Indeed, the force-displacement curve well as the energy curve is similar for T700/M21 and AS4/APC2 but visibly different from the PEEK and PPS fabric laminates, which exhibit a similar behaviour. This distinct behaviour of both populations is noticeable on the force-displacement curves (figure 1a). T700/M21 and AS4/APC2 curves exhibit oscillations with large amplitude that are characteristic of delamination, while woven laminates oscillations show evidence of fibre breakage. Moreover, as displayed in figure 1b, woven laminates mainly absorbed the 55J energy of the drop weight test by damaging, while unidirectional prepreg laminates can store about 50% of the impact energy by elasticity. This result is confirmed by the resulting permanent indentation that was measured with a dial comparator 48 hours after the impact (figure 2b). Woven laminates exhibit a large permanent indentation of about 1.5mm in comparison to the 0.5mm of the UD prepreg laminates. PPS Porcher, which confirms the influence of the reinforcement structure on composite impact behaviour. The difference between T700/M21 and AS4/APC-2 may be explained by the different number of plies of these laminates (table 1). According to the literature, the delaminated area increases at each interface from the impacted side to the opposite side and lead to a damage cone [START_REF] Guinard | A 3D damage analysis of low-velocity impacts on laminated composites[END_REF]. In the present study, dividing the total delaminated area by the number of interface lead to the same value for both laminates.

Charpy tests

The variation of the measured load during Charpy tests is drawn in figure 3 for each material.

The area under the force curve is proportional to the energy absorbed by damaging. In this way, the advantages of this test form are immediately apparent. Where the traditional test results only in a measurement of total absorbed energy and a fractured specimen, the instrumented test provides a measure of the specimen response which can be interpreted in terms of events. As illustrated in figure 3a, the response of a typical composite Charpy impact specimen exhibits three distinct regions: pre-initial fracture (1), initial fracture (2), and post-initial fracture (3) [START_REF] Toland | Impact testing of carbon-epoxy composite materials in "Instrumented impact testing[END_REF]. The region (1) corresponds to the elastic response of the composite beam and precedes the fracture (2) that occurs subsequently by fibre breaking (small oscillations in the load curve) or interply shear delamination (large subsequent oscillations). Integration of these curves then allows to quantify the distribution of the impact energy through these three mechanisms. The result is displayed in figure 4 as a bar diagram. As displayed in figure 4, the influence of the reinforcing armour is marked: unidirectional laminates absorb about 8 to 9.3J while woven composites only dissipate 5J. Similar to drop weight tests, a large part of the impact energy is stored by elasticity. It represents more than 50% of the dissipated energy for both prepreg laminates. Nevertheless, woven laminates then only dissipate energy by fibre fracture while UD laminates exhibit in addition delamination process which contribution can account for more than fibre fracture dissipation. The below observations of fractured specimens confirm the previous conclusions (figure 5). Delamination is clearly visible on T700/M21 and AS4/APC2 specimens and in particular on the picture of AS4/APC2 specimen: delamination occurred on the entire length of the Charpy specimen. It leads to a diffuse fracture for these materials. On the contrary, PEEK and PPS woven laminates exhibit a straight fracture and fibre breakage is concentrated in the centre of the specimen.

Gas gun impact tests

For all different materials, gas gun tests led systematically to perforation. Thanks to the highspeed cameras, impact and residual velocities after perforation were measured and the absorbed energy was calculated with equation 1. Eimpact corresponds to the incident impact energy and Eresidual is the residual energy after perforation, both determined from the steel ball velocity with equation 2. mball corresponds to the mass of the steel ball and v(t) is the velocity of the ball at the instant t.

(1)

(2)

residual impact absorbed E E E - = ( ) ( ) t v m t E ball ball 2 2 1 × = (b) (a) (c) (d)
The specific energy of each material is depicted in figure 6. The results show similar dissipated energy for materials with similar armours. Indeed, T700/M21 and AS4/APC-2 absorb 30 % more energy than PEEK and PPS Porcher. Pictures of the damaged laminates confirm again the relation between the armour and the damage mechanism (figure 7). T700/M21 and AS4/APC2 exhibit a rear face with a wider damaged area than PEEK and PPS woven composites, and the main damage mechanism of UD laminates is inter-plies delamination as already observed for drop weight tests and Charpy tests. On woven laminates, the perforation is mainly due to out-of-plane shear that induced local fibre facture around the impactor.

Conclusions

The influence of the composite matrix and the reinforcing carbon armour on the impact behaviour of composite laminates was analysed through three different tests ranging from 55J to 500J. The results obtained for prepreg laminates are systematically better than the woven composites. The fractured samples as well as the measurements show that unidirectional plies favour delamination and the activation of this mechanism may be beneficial to dissipate impact energy. Elasticity also appeared as an important mechanism of energy storage and a high strength may then be advantageous to enhance impact properties of laminates. However, PEEK and PPS woven laminates exhibit similar impact resistance. The different tests were then not conclusive to evaluate the influence of matrix nature on the impact performances.

Figure 1 .

 1 Figure 1. Curves of force versus displacement (a) and energy versus time (b) obtained experimentally

Figure 2 .

 2 Figure 2. Delaminated areas (a) and permanent indentations (b) after impact

Figure 3 .

 3 Figure 3. Curves of force and energy absorbed versus time for T700/M21 (a), AS4/APC-2 (b), PEEK Porcher (c) and PPS Porcher (d)

Figure 4 .

 4 Figure 4. Comparison between energy absorbed by the various mechanisms

Figure 5 .

 5 Figure 5. Fractured Charpy specimens for T700/M21 (a), AS4/APC-2 (b), PEEK Porcher (c) and PPS Porcher (d)

Figure 6 .Figure 7 .

 67 Figure 6. Comparison between specific energies absorbed

Table 1 .

 1 Material characteristicsThree different impact tests were performed on laminates so as to characterize the influence of impact energy on the composite resistance and energy absorption: drop weight test, Charpy test and gas gun test. Sample dimensions and test specifications are listed in table 2 and table3.

	Material designation	T700/M21	AS4/APC-2	PEEK Porcher PPS Porcher
	Resin	Epoxy	PEEK		PPS
	Reinforcing armour	UD		Satin 5
	Number of plies	24	40	16
	Stacking sequence	[90°,+45°,0°,-45°]3S [90°,+45°,0°,-45°]5S	[90°,+45°,0°,-45°]2S
	Thickness (mm)	5,8	5,4	4,9	4,6
	Fibre content in vol. (%)	62	60	51	55
	Test	Length (mm)	Width (mm)	Thickness
	Drop weight	150	100
	Charpy	70	10	Laminate thickness
	Gas gun	200	200

Table 2 .

 2 Sample characteristics

	Test	Impactor mass (g)	Impact velocity (m.s -1 )	Impact energy (J)
	Drop weight	2079	7,27	55
	Charpy	21900	5,23	299,5
	Gas gun	64	121-124	468-491

Table 3 .

 3 Test specifications


Emeline Bessard

Olivier De Almeida

Gérard Bernhart

Melt state behaviour of PEEK and processing window interpretation for fast compression moulding process

Keywords: thermoplastic, thermo-compression, PEEK PACS: 81.05.Qk, 47.57.Qk, 61.05.C-, 61.30.-v, 07.20.Fw

published or not. The documents may come

Melt state behaviour of PEEK and processing window interpretation for fast compression moulding process

INTRODUCTION

PEEK/Carbon composites are usually processed in a hot press or by autoclave at a temperature set between 380 and 400°C [1,2]. The autoclave process is usually preferred (APC2) but the processing time and the use of vacuum bag still limit the development of these thermoplastic composites. Compression moulding is more convenient but the air environment induces fast degradation of the polymer matrix [START_REF] Day | [END_REF]6]. Nowadays, induction technology (Cage System ® from Roctool) allows mould heating and cooling kinetics of about 100°C/min. Fast compression moulding is thus possible, but processing conditions (applied stress, compression temperature, …) still need to be optimized.

This study focuses on the PEEK matrix behaviour in the fast compression moulding process. The melt rheology has been characterized as well as its evolution with time. Then, the influence of the cooling conditions on the degree of crystallinity is analysed by calorimetry, in order to propose ways of optimization.

MATERIAL AND EXPERIMENTAL

The investigated PEEK has been provided by Victrex as powder, granules and injected plates. This is a neat polymer that is commercialized with the reference 151G. Its melt temperature is 343°C as measured by DSC.

All the rheological properties of the samples were measured in the melt state with a Haake Mars apparatus equipped with a plane-plane geometry. Dynamic mechanical behaviour have been measured between 0.01 and 100Hz with applying a shear strain of 1% for different temperatures between 380 and 420°C. This strain level has been chosen in the linear viscoelastic domain where the viscosity is described by a linear relation of the elastic and viscous moduli.

The Differential Scanning Calorimetry investigation has been performed with a TA-Instrument Q20 previously calibrated with indium and zinc. The DSC sealed caps have been prepared from 5 to 8mg of PEEK powder and in order to impose the cooling conditions from a nuclei-free melt, the samples have been systematically heated from room temperature to 390°C and held 10 minutes in the molten state. The degree of crystallinity has been determined from the melting peak obtained with a heating rate of 10°C/min in considering an melt enthalpy of 130J/g [3]. The cooling conditions have been carried out with the DSC up to 20°C/min but higher cooling rates have also been achieved by quenching samples in nitrogen or cold water.

WAXS has been used to identify the crystalline structure of the samples. All X-Ray diffraction tests have been performed with a Phillips X-Ray diffractometer and the specimens have been prepared from PEEK powder or plates heated to the melting point and then cooled at different cooling rate. Quenched specimens have also been analysed by WAXS.

EVOLUTION OF PEEK VISCOSITY WITH TIME IN AIR ENVIRONMENT FIGURE 1. Melt rheology of PEEK

Figure 1 shows the viscous behaviour of PEEK in the molten state between 380 and 420°C. PEEK exhibits a conventional plastic behaviour for high strain rates which can be fitted with a power law with an Arrhenius parameter of 35000 J/mol [START_REF] Agassant | La mise en forme des matières plastiques[END_REF].

The low viscosity of PEEK at high temperature is a way to optimize the compression moulding process of PEEK composites. It may facilitate fibre impregnation and so on composite consolidation. Nevertheless, the air environment as a large influence on the stability of PEEK in the melt state and in order to characterize the influence of thermal degradation in air, the evolution of the PEEK viscosity with time has thus been measured for different temperatures (figure 2). The results show that PEEK viscosity remains mostly constant during a few minutes but quickly increase for longer time of exposure due to crosslinking [START_REF] Day | [END_REF]6]. At a temperature of 400°C, this thermal degradation occurs after 15 minutes and the viscosity, initially of about 350 Pa.s, reaches 800 Pa.s after 35 minutes. Therefore, thermal degradation limits the processing conditions in conventional compression moulding process and the use of processing temperature over 400°C is impossible. With induction technology, the high heating and cooling kinetics allow to control the time of exposure of PEEK to high temperature. Processing temperatures of about 420°C may thus be used but the processing time must be limited to a few minutes.

INFLUENCE OF COOLING CONDITIONS ON PEEK CRYSTALLINITY Influence of cooling conditions on the degree of crystallinity

High cooling kinetics may reduce considerably the processing time of PEEK composites. Nevertheless, it induces low crystallinity and so it may have an influence on the composite properties. The influence of cooling conditions on the degree of crystallinity of PEEK have thus been investigated and the cooling conditions conciliating fast cooling and high crystallinity have been prospected.

The influence of constant cooling rates on crystallinity has been firstly studied by DSC (cooling rates up to 20°C/min). Quenched specimen have also been prepared but the degree of crystallinity could not be determined by DSC due to cold crystallisation for high cooling rates. The volume mass fraction of these samples have then been estimated from WAXS scans. Figure 3 shows the evolution of the crystallization and melting heat flux of PEEK for different low cooling rates. The measured degree of crystallinity is reported in table 1. With increasing the cooling rate, PEEK crystallinity decreases drastically. For a cooling rate of 20°C/min, the degree of crystallinity is only 31%. In order to improve the degree of crystallinity without increasing the cycle time, an isothermal cristallization can be performed. While cooling the PEEK samples from the melt state to room temperature with a constant rate 20°C/min, the cooling has been stopped during 5 minutes at different temperatures from 300°C to 317°C and the degree of crystallinity has been measured. As presented in table 1, isothermal crystallisation performed between 300°C and 313°C favours PEEK crystallinity. For higher isotherm temperatures, the crystallisation is not initiated and only occurs during the cooling stage.

Origin of double melting peak

However, the heat curve exhibits two melting peaks which have been taken into account when measuring the of crystallinity (figure 4). Indeed, the major melting peak at 343°C is accompanied with a second peak which appears for a temperature of 7°C over the isotherm temperature. It must be noticed that a second peak was already distinguishable for slow cooling rates (1°C/min, figure 3).

FIGURE 5. X-ray diffraction scans of PEEK cooled with extreme rates

In order to analyse the origin of this second melting peak, X-ray diffraction has been performed on a sample cooled at 1°C/min and on a sample quenched in nitrogen from the melt state. The first sample exhibits a second melting peak around 310°C (low cooling rate) while it is supposed that high rates inhibit the apparition of this second peak (quenched specimen). The figure 5 shows the diffraction scans obtained for the two specimens. One can see that both scans exhibit the same diffraction peaks which positions correspond to the orthorhombic structure of PEEK crystal. The ratio of amorphous and crystalline structure is the only difference between the specimens and this ratio is in agreement with the influence of cooling rate on the crystallinity. The two meting peaks may thus be characteristic of the same crystalline structure.

The apparition of a second melting peak has also been analysed by DSC. A sample previously cooled at 20°C/min with an isotherm at 300°C during 5 minutes has been up to 315°C during 1 minute and then cooled again at 20°C/min. By heating the sample up to 315°C, the crystalline structure of the lower peak is supposed to be melt. On a second sample, the same procedure has been applied but during the second cooling stage, a 5 minutes isotherm at 300°C has been performed.

The figure 6 shows the heating thermogrammes of both samples. The first specimen still exhibits two melting peaks : the major peak temperature is still 343°C but the second one appears at T=320°C i.e. 5°C over the isotherm temperature. On the other hand, the second thermogramme presents three distinct peaks : the same two peaks at T=343°C and T=320°C as observed for the first sample and a new one at a temperature of 306°C. Actually, this last peak can be considered as the consequence of the isotherm at 300°C during the cooling stage while the heating temperature of 315°C induces a new endothermic peak to a higher temperature.

Thus, theses results suggest that the presence of a double melting peak is the consequence of a physical ageing, due to a reorganisation of the polymer chains. This phenomenon is in agreement with the conclusions of Wei et al. and contradict the hypothesis of a second crystalline phase [7].

CONCLUSIONS

High heating and cooling kinetics are new ways of optimisation of fast compression moulding process of PEEK composites. On one hand, higher moulding temperatures may be favourable but the processing time must be carefully controlled due to the fast thermal degradation of PEEK above its melt temperature. On the other hand, fast cooling induces poor matrix crystallinity and it may have negative influence on the composite properties. Moreover, the analysis of the origin of the second melting peak suggests that physical ageing occurs in PEEK and that this peak could be considered as a crystalline phase when measuring the matrix crystallinity. An isotherm crystallisation at 300°C is thus a way to keep a high degree of crystallinity. In future work, the influence of carbon on the melt behaviour should be investigated because it would induce a different nucleation process and thus a different crystallisation [8,9].

FIGURE 2 .

 2 FIGURE 2. Viscosity as a function of time for different temperatures (f=1Hz, ε 0 =1%)

FIGURE 3 .

 3 FIGURE 3. DSC thermogrammes of PEEK for different cooling rates

FIGURE 4 .

 4 FIGURE 4. DSC thermogrammes of PEEK for different isothermal crystallisation

FIGURE 6 .

 6 FIGURE 6. Influence of heat temperature on DSC thermogrammes of PEEK

TABLE 1 .

 1 degree of crystallinity of PEEK for different cooling conditions (X m % +/-0.5)

	Cooling rate	Anisothermal		Isothermal crystallisation
	(°C/min)	crystallisation	300°C	310°C	313°C	317°C
	1	46.5
	10	41.3
	20	31,5	35,6	35,9	34,5	31,9
	Quenched in nitrogen	<20