Samir Adly
email: samir.adly@unilim.fr

Abderrahim Hantoute
email: ahantoute@dim.uchile.cl

Bao Tran Nguyen

Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operators $

Keywords: Lyapunov stability, lsc Lyapunov pairs and functions, invariant sets, differential inclusions, maximal monotone operators, normal cones, subdifferentials Mathematics Subject Classification (2010): 37B25, 47J35, 93B05

, while the invariance criteria are still written by using only the data of the system. So, no need to the explicit knowledge of neither the solution of this differential inclusion, nor the semi-group generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for a-Lyapunov pairs of lower semi-continuous (not necessarily weakly-lsc) functions associated to these differential inclusions. The lack of differentiability of the candidate Lyapunov functions and the consideration of general invariant sets (possibly not convex or smooth) are carried out by using techniques from nonsmooth analysis.

Introduction

We provide sufficient and, in many different interesting situations, necessary criteria for the invariance property of closed subsets with respect to the following differential inclusion, given in a Hilbert space H, ẋ(t) ∈ f (x(t)) -Ax(t), x(0) = x 0 ∈ dom A, a.e. t ≥ 0 [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF] where A is a maximal monotone operator which is subject to a Lipschitzian perturbation f . Equivalently, we establish many primal and dual explicit criteria for a-Lyapunov pairs and functions associated to the differential inclusion above. The current work extends and improves some of the results given in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF] on weakly closed invariant sets and weakly lower semi-continuous a-Lyapunov pairs. The domain of A does not need to be closed, nor the values of A are supposed to be bounded or even nonempty. Thus, the scope of the equation above goes beyond the differential inclusions treated in [START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Aubin | Viability Theory[END_REF][START_REF] Clarke | Nonsmooth Analysis and Control Theory Nonlinear Analysis[END_REF][START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF][START_REF] Frankowska | A measurable upper semicontinuous viability theorem for tubes[END_REF], where the right-hand side is generally represented by a cusco set-valued mapping (in particular, with nonempty and weak*-compact multi-valued operator). It is the monotonicity of A which compensates the lack of compacity in our differential inclusion, while the maximality of this operator guaranties, among other properties, the existence and the regularity of solutions. These two facts are also essential when checking the invariance of closed sets.

In front of the lack to a direct access to the explicit calculus of either the solution of the inclusion above or to the semi-group generated by A, the current work aims at finding weaker conditions for the invariance of closed sets, which only appeal to the fresh input data, namely the maximal monotone operator and the Lipschitz mapping. These conditions are applicable to a large variety of closed sets which do not need to be convex or smooth. Our approach fits the general scope and the main ideas behind Lyapunov's stability, which consists of looking for an adjacent function to the system described by the inclusion above; namely, an energy-like function which decreases along the trajectories and, so, under some extra usual conditions, forces the system to converge towards its equilibrium state and to remain there. Since our analysis allows to deal with extended-real valued functions, the invariance of a set occurs as long as the associated indicator function is a Lyapunov's function. However, our approach is more geometric since we first establish criteria for the invariance property and next deduce the adequate conditions for Lyapunov pairs and functions.

Invariant sets associated to general differential inclusions/equations have been the subject of extensive research during the last decades; namely, in relation with differential inclusions involving cusco mappings in their right-hand side (see, e.g., [START_REF] Aubin | Viability Theory[END_REF]). First results dealing with Lyapunov pairs and functions associated to the differential inclusions above have been first established in [START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF][START_REF] Pazy | On the Lyapunov method for evolution equations governed by accretive operators[END_REF] in the case of homogeneous systems; that is, f ≡ 0. Pazy's criteria for a-Lyapunov pairs are given by means of directional-like derivative using the Moreau-Yoshida approximation of the operator A. This result has been extended to the general inclusion above in [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF][START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF], with the use of implicit criteria depending heavily on the semi-group generated by the maximal monotone operator A. Recently, different criteria for weakly lower semi-continuous a-Lyapunov pairs have been investigated in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF].

The need of more explicit conditions, not depending on the semi-group generated by A, is of utmost importance for many reasons, one of which is that the inclusion above is sometimes evoked as a companion tool to analyze other differential inclusions. In that case, the operator A may not be known explicitly, and this facts makes the access to its semi-group more complicated. For instance, in our work [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets[END_REF] we have investigated the existence of solutions to a differential inclusion governed by the normal cone to a prox-regular set ( [START_REF] Poliquin | Local differentiability of distance functions[END_REF]), by rewriting it in the form of (1) with A being some intrinsic maximal monotone operator to this prox-regular set. Such an operator A is not known explicitly but it processes enough information in order to check the invariance of the involved prox-regular set with respect to [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF]. This was sufficient to get the desired existence results; for more details, we refer the reader to [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets[END_REF].

Invariant sets are also referred to in the wide literature as viable sets [START_REF] Aubin | Viability Theory[END_REF][START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Aubin | Set-valued analysis[END_REF], and are of crucial use in many domains, as in economic, renewable resources, biology, diseases propagation, control processes of species and so on. It is manifest, in recent papers [START_REF] Grosshans | Invariant sets forced by symmetry[END_REF][START_REF] Ruggieri | Exact solutions for a nonlinear model of dissipative media[END_REF], that the investigation of certain algebraic varieties is sufficient to characterize invariant sets forced by symmetries. Lyapunov pairs and functions are used extensively in dynamic systems and control theory, among many other applications; see, e.g., [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF][START_REF] Brogliato | Nonsmooth mechanics. Models, dynamics and control[END_REF].

In this work, we provide different criteria to characterize those sets which are invariant with respect to the differential inclusion [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF]. Only the data, A and f, will be appealed to and no need to solve explicitly the equation. These invariant results are then rewritten as criteria for a-Lyapunov pairs, which are crucial for Lyapunov stability of [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF]. Because the sets we consider are not necessary convex or smooth, and the candidate Lyapunov functions are not necessarily sufficiently regular, we use techniques of nonsmooth analysis (e.g.. [START_REF] Clarke | Nonsmooth Analysis and Control Theory Nonlinear Analysis[END_REF][START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory[END_REF][START_REF] Rockafellar | Variational Analysis[END_REF]).

The organization of the paper is as follows. After an introductory section to present the main notations and tools which are used through this work, we give in Section 3 the main invariance criterion in Theorem 4, using the normal cone to the nominal set. Other corollaries follow in order to simplify this invariance criterion and provide equivalent primal and dual conditions. In Section 4, we apply the previous invariance result to investigate a-Lyapunov pairs associated to differential inclusion (1).

Notation and preliminary results

Let (H, •, • , • ) be a Hilbert space, with origin θ. Given a set S ⊂ H, by S and S * we denote the closure of S and the polar of S, respectively, where

S * := {x * ∈ H | x * , x ≤ 0 for all x ∈ S}.
The indicator and the distance functions are respectively given by I S (x) := 0 if x ∈ S; +∞ if x ∈ S, and d S (x) := inf{ xy : y ∈ S} (in the sequel we shall adopt the convention inf ∅ = +∞). For δ ≥ 0, we denote P δ S the (orthogonal) δ-projection mapping onto S defined as

P δ S (x) := {y ∈ S : x -y 2 ≤ d 2 S (x) + δ 2 };
for δ = 0, we simply write P S (x) := P 0 S (x). It is known that P S is nonemptyvalued on a dense subset of H \ S ( [START_REF] Clarke | Nonsmooth Analysis and Control Theory Nonlinear Analysis[END_REF]). For an extended-real valued function ϕ :

H → R := (-∞, +∞] , we denote dom ϕ := {x ∈ H | ϕ(x) < +∞} and epi ϕ := {(x, α) ∈ H × R | ϕ(x) ≤ α}. Function ϕ is lower semi-continuous (lsc, for short) if epi ϕ is closed. The contingent directional derivative of ϕ at x ∈ dom ϕ in the direction v ∈ H is ϕ (x; v) := lim inf t→0 + ,w→v ϕ(x + tw) -ϕ(x) t .
A vector ξ ∈ H is called a proximal subgradient of ϕ at x ∈ H, written ξ ∈ ∂ P ϕ(x), if there are ρ > 0 and σ ≥ 0 such that

ϕ(y) ≥ ϕ(x) + ξ, y -x -σ y -x 2 , ∀ y ∈ B ρ (x),
where

B ρ (x) ( =: B(x, ρ)) is the closed ball centered at x ∈ H of radius ρ > 0. The vector ξ is called a Fréchet subgradient of ϕ at x, written ξ ∈ ∂ F ϕ(x), if ϕ(y) ≥ ϕ(x) + ξ, y -x + o( y -x ), ∀ y ∈ H;
and a basic (or Limiting) subgradient of ϕ at x, written ξ ∈ ∂ L ϕ(x), if there exist sequences (x k ) k and (ξ k ) k such that

x k ϕ → x, ξ k ∈ ∂ P ϕ(x k ), ξ k ξ,
where refers to the weak convergence in H, and

x k ϕ → x means that x k → x together with ϕ(x k ) → ϕ(x).
If x / ∈ dom ϕ, we write

∂ P ϕ(x) = ∂ F ϕ(x) = ∂ L ϕ(x) = ∅.
If S is a closed set and s ∈ S, we define the proximal normal cone to S at s as N P S (s) = ∂ P I S (s), the Fréchet normal cone to S at s as N F S (s) = ∂ F I S (s), the limiting normal cone to S at s as N L S (s) = ∂ L I S (s), and the Clarke normal cone to S at s as N C S (s) = coN L S (s). Equivalently, we have that N P S (s) = cone(P -1 S (s)s), where P -1 S (s) := {x ∈ H | s ∈ P S (x)}. The Bouligand tangent cone to S at x is defined as

T S (x) := v ∈ H | ∃ x k ∈ S, ∃ t k → 0, st. t -1 k (x k -x) → v as k → +∞ .
We also define the Clarke subgradients of ϕ at x as the vectors ξ ∈ H such that (ξ, -1) ∈ N C epi ϕ (x, ϕ(x)), and denote ∂ C ϕ(x) the Clarke subdifferential of ϕ at x. The singular subdifferential of ϕ at x, written ∂ ∞ ϕ(x), is the set of vectors ξ ∈ H for which there are sequences [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory[END_REF]Theorem 2.38]). It is known that every ξ ∈ H such that (ξ, 0) ∈ N P epi ϕ (x, ϕ(x)) belongs to ∂ ∞ ϕ(x) and, moreover, there exist sequences as in the definition before but with

x k ϕ → x, ξ k ∈ ∂ P ϕ(x k ) and λ k → 0 + such that λ k ξ k ξ; equivalently, ξ ∈ ∂ ∞ ϕ(x) iff (ξ, 0) ∈ N L epi ϕ (x, ϕ(x)) (see
λ k ξ k → ξ instead of λ k ξ k ξ (see [18, Lemma 2.37]). Observe that ∂ P ϕ(x) ⊂ ∂ F ϕ(x) ⊂ ∂ L ϕ(x) ⊂ ∂ C ϕ(x).
For all these concepts and properties we refer to [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory[END_REF][START_REF] Rockafellar | Variational Analysis[END_REF].

We shall use the following version of Gronwall's Lemma:

Lemma 1. (Gronwall's Lemma [3]) Let T > 0 and a, b ∈ L 1 (t 0 , t 0 + T ; R) such that b(t) ≥ 0 a.e. t ∈ [t 0 , t 0 + T ]. If an absolutely continuous function w : [t 0 , t 0 + T ] → R + satisfies, for 0 ≤ α < 1, (1 -α)w (t) ≤ a(t)w(t) + b(t)w α (t) a.e. t ∈ [t 0 , t 0 + T ], then w 1-α (t) ≤ w 1-α (t 0 )e t t 0 a(τ )dτ + t t 0 e t s a(τ )dτ b(s)ds, ∀t ∈ [t 0 , t 0 + T ].
Next, we review some facts about monotone and maximal monotone operators. Given a set-valued operator A : H ⇒ H, which we identify with its graph, we denote its domain by dom

A := {x ∈ H | Ax = ∅}. Operator A is monotone if x 1 -x 2 , y 1 -y 2 ≥ 0 for all (x 1 , y 1 ), (x 2 , y 2 ) ∈ A.
We say that A is maximal monotone if A is monotone and coincides with every monotone operator containing its graph. In such a case, it is known that Ax is convex and closed for every x ∈ H; moreover, for every λ > 0 there exists a unique vector J λ x ∈ (id +λA) -1 (x), which is the resolvent of the (maximal monotone) operator A, while

A λ x := x-J λ x λ is the Moreau-Yoshida approxima- tion of A. If S ⊂ H is a closed convex set, we denote S 0 := {y ∈ S | y = min z∈S z }; in particular, we write A 0 x := (Ax) • , x ∈ dom A.
Associated with a maximal monotone operator A : H ⇒ H we consider the differential inclusion given in (1) :

ẋ(t) ∈ f (x(t)) -A(x(t)), a.e. t ≥ 0, x(0) = x 0 ∈ dom A,
where f : H → H is a given (L-)Lipschitz continuous mapping. Every solution of differential inclusion (1) will be denoted by x(•; x 0 ). We introduce the concept of invariant sets (see, e.g., [START_REF] Aubin | Viability Theory[END_REF][START_REF] Clarke | Nonsmooth Analysis and Control Theory Nonlinear Analysis[END_REF][START_REF] Clarke | Approximate invariant and Differential Inclusion in Hilbert Spaces[END_REF]): Definition 1. A set S ⊂ dom A is said to be invariant for [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF] provided that x(t; x 0 ) ∈ S for every x 0 ∈ S and every t ≥ 0.

We also recall the following result on the existence of solutions of (1); for more details, we refer to [START_REF] Br | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF].

Proposition 2. For any x 0 ∈ dom A and T > 0, system (1) has a unique continuous solution, which is the uniform limit on [0, T ] of x λ (•; x 0 ) (as λ ↓ 0), where x λ (•; x 0 ) is the solution of the differential equation

ẋλ (t) = f (x λ (t)) -A λ (x λ (t)), x λ (0) = x 0 .
Moreover, the following holds:

(i) For all s, t ≥ 0 and all y 0 ∈ dom A we have that

x(s; x(t; x 0 )) = x(t + s; x 0 ), x(t; x 0 ) -x(t; y 0 ) ≤ e Lt x 0 -y 0 . (ii) If x(t 0 , x 0 ) ∈ dom A for some t 0 ≥ 0, then d + x(t 0 ; x 0 ) dt = (f (x(t 0 ; x 0 )) -Ax(t 0 ; x 0 )) 0 . (iii) The function t → d + x(t;x 0 )
dt is right-continuous at every t ≥ t 0 , where t 0 ≥ 0 is such that x(t 0 ; x 0 ) ∈ dom A, and we have

d + x(t; x 0 ) dt ≤ e L(t-t 0 ) d + x(t 0 ; x 0 ) dt .

Invariant sets

In this section, we achieve our first goal to characterize those closed sets in the Hilbert space H, which are invariant with respect to differential inclusion (1):

ẋ(t) ∈ f (x(t)) -A(x(t)), t ∈ [0, ∞), x(0) = x 0 ∈ dom A;
the unique solution of this inclusion is written x(•; x 0 ).

It is worth observing that whenever differential inclusion (1) possesses a strong solution starting from S (x 0 ∈ S), which is an absolutely continuous function such that x(t; x 0 ) ∈ dom A for all t > 0, each invariant closed set S ⊂ dom A satisfies the condition

S = dom A ∩ S.
(2)

However, this condition may not be true when only weak solutions exist. This is why we shall assume in what follows that our invariance candidate sets satisfy this "almost necessary" condition.

Remark 1. Theorem 4 below gives the main invariance criterion, given in (3), for closed sets with respect to differential inclusion (1), using only the data in (1) which are the operator A and the mapping f. Hence, explicit calculus of either the solution or the semigroup generated by A are not required. Criterion (3) extends and adapts some of the results given in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF] on weakly closed invariant sets. Its geometric meaning is very similar to the classical ones established in [START_REF] Clarke | Nonsmooth Analysis and Control Theory Nonlinear Analysis[END_REF][START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF] for differential inclusions of the form

ẋ(t) ∈ F (x(t)),
with a w * -compact, nonempty and convex multifunction F . In our case, condition (3) takes into account that the right-hand side in [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF] , which is governed by a general maximal monotone operator, may have empty or unbounded values. As well, another crucial difference between (1) and the last inclusion above is that our analysis also allows the initial condition in (1) to start from the larger set dom A. Thus, the scope of our analysis goes beyond the differential inclusions treated in [START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Aubin | Viability Theory[END_REF][START_REF] Clarke | Nonsmooth Analysis and Control Theory Nonlinear Analysis[END_REF][START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF][START_REF] Frankowska | A measurable upper semicontinuous viability theorem for tubes[END_REF]. First invariance criteria for differential inclusions involving maximal monotone operators have been given in [START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF] (see, also, [START_REF] Br | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF]) without considering the Lipschitzian perturbation. Such results have been extended in [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF][START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF] to maximal monotone operators which are subject to Lipschitz perturbations, using criteria which depend on the semi-group of contractions generated by -A. Compared to [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF][START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF] (see, also, references therein), condition (3) relies exclusively on the geometry of C as in [START_REF] Clarke | Nonsmooth Analysis and Control Theory Nonlinear Analysis[END_REF][START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF].

Before we state the main theorem of this section, Theorem 4 below, we give the following lemma.

Lemma 3. Given a closed set S ⊂ H and an m ≥ 0, we denote

S m := x ∈ S ∩ dom A | (f (x) -Ax) 0 ≤ m . Then the set S m is closed. Proof. Take a sequence (x k ) k ⊂ S m such that x k → x (∈ S)
. Without loss of generality, and taking into account the norm-weak upper semi-continuity of the maximal monotone operator A, we conclude that the sequence (

P Ax k (f (x k ))) k weakly converges to some z ∈ Ax. Then (f (x) -Ax) 0 ≤ f (x) -z ≤ lim inf k→∞ f (x k ) -P Ax k (f (x k )) = lim inf k→∞ (f (x k ) -Ax k ) 0 ≤ m, so that x ∈ S m .
Theorem 4. Given a closed set S ⊂ dom A ∩ S, we assume that for every x ∈ S ∩ dom A there exist m, r > 0 such that P Ax (f (x)) ≤ m and

sup ξ∈N P S m (y) min y * ∈Ay∩B(θ,m) ξ, f (y) -y * ≤ 0 for all y ∈ B(x, r). ( 3
)
Then S is invariant for (1).

Proof. We fix x 0 ∈ S ∩ dom A and ε > 0. Let m, r > 0 be as in the current assumption (with x = x 0 ), and choose an M > 0 such that

f (y) -Ay ∩ B(θ, m) ⊂ B(θ, M ) for all y ∈ K := S m ∩ B(x 0 , r). ( 4
)
We also choose sufficiently small numbers t, δ > 0 and a sufficiently large integer

N such that max{6M 2 t2 , 8δ 2 } < r 2 2 , δ < t N , (5) max (
M 2 + 4M + 1) t2 N , M 2 t2 N 2 + 2δ 2 < ε 2 4 . ( 6
)
We denote by π := {t 0 , t 1 , ..., t N } the uniform partition of the interval [0, t]. We put d(π) := max 0≤i≤N -1 (t i+1t i ) = t N and, by (4), we choose an element

s * 0 ∈ f (x 0 ) -A(x 0 ) such that s * 0 ≤ M . We consider the the function z 0 (t), t ∈ [t 0 , t 1 ] such that ż0 (t) = s * 0 , t ∈ [t 0 , t 1 ], z 0 (0) = x 0 ,
and denote z 1 := x 0 + s * 0 t 1 . We pick ŝ1 ∈ P δ K (z 1 ). Then there exists a pair (y 1 , s 1 ) such that s 1 ∈ K, y 1s 1 ∈ N P K (s 1 ) and (see, e.g., [START_REF] Clarke | Approximate invariant and Differential Inclusion in Hilbert Spaces[END_REF][START_REF] Radulescu | Geometric Approximation of Proximal Normals[END_REF])

max{ y 1 -z 1 , s 1 -ŝ1 } ≤ δ, (y 1 -s 1 ) -(z 1 -ŝ1 ) ≤ 2δ,
as well as (see [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets[END_REF]Lemma 4] )

s 1 -x 0 2 ≤ 6 z 1 -x 0 2 + 8δ 2 = 6t 2 1 s * 0 2 + 8δ 2 < 6 t2 M 2 + 8δ 2 < r 2 ;
hence, s 1 ∈ int(B(x 0 , r)) and, so, N P K (s 1 ) = N P S m (s 1 ). Consequently, by the current assumption of the theorem, we find

s * 1 ∈ (f (s 1 ) -A(s 1 )) ∩ B(θ, M ) such that y 1 -s 1 , s * 1 ≤ 0. With this vector s * 1 in hand, we consider the function z 1 (t), t ∈ [t 1 , t 2 ], such that ż1 (t) = s * 1 , t ∈ [t 1 , t 2 ] z 1 (t 1 ) = z 1 .
By repeating the arguments used above, for each i ∈ 2, N -1, we consider the function

z i (t), t ∈ [t i , t i+1 ], such that żi (t) = s * i , t ∈ [t i , t i+1 ] z i (t i ) = z i-1 (t i ) =: z i ,
and the corresponding elements

(ŝ i , y i , s i , s * i ) such that ŝi ∈ P δ K (z i ), y i -s i ∈ N P K (s i ) = N P S m (s i ), s * i ∈ [f (s i ) -A(s i )] ∩ B(θ, M ), y i -s i , s * i ≤ 0, max{ y i -z i , s i -ŝi } ≤ δ, (y i -s i ) -(z i -ŝi ) ≤ 2δ.
Now, we are going to prove that the absolute continuous trajectory z(•), defined on [0, t] as z(t

) := z i (t) = z i + (t -t i )s * i for t ∈ [t i , t i+1 ], satisfies d S (z(t)) ≤ ε, ∀t ∈ [0, t], (7)
s i -z(t) ≤ 2ε, ∀t ∈ [t i , t i+1 ]. (8)
Indeed, for any 1 ≤ i ≤ N -1, one has

d 2 K (z i+1 ) ≤ z i+1 -ŝi 2 = z i+1 -z i 2 + z i -ŝi 2 + 2 z i+1 -z i , z i -ŝi = (t i+1 -t i )s * i 2 + d 2 K (z i ) + δ 2 + 2d(π) s * i , z i -ŝi ≤ M 2 d 2 (π) + d 2 K (z i ) + δ 2 + 2d(π) s * i , y i -s i + 2d(π) s * i , (z i -ŝi ) -(y i -s i ) ≤ d 2 K (z i ) + (M 2 + 4M + 1)d(π)(t i+1 -t i ),
which gives us

d 2 K (z i+1 ) ≤ d 2 K (z 1 ) + (M 2 + 4M + 1)d(π)(t i+1 -t 1 ) ≤ z 1 -x 0 2 + (M 2 + 4M + 1)d(π)(t i+1 -t 1 ) ≤ (M 2 + 4M + 1)d(π) t ≤ (M 2 + 4M + 1) t2 N < ε 2 4 . ( 9
)
This shows that, for every t ∈ [t i , t i+1 ],

d 2 S (z(t)) ≤ d 2 K (z(t)) = d 2 K (z i (t))) = d 2 K (z i (t i ) + (t -t i )s * i ) ≤ 2d 2 K (z i ) + 2(t -t i ) 2 M 2 ≤ ε 2 2 + 2d 2 (π)M 2 ≤ ε 2
and (7) follows. Inequality (8) also follows since that for every t ∈ [t i , t i+1 ]

s i -z(t) 2 ≤ 2 z(t) -z i 2 + 2 s i -z i 2 ≤ 2(t -t i ) 2 M 2 + 4 s i -ŝi 2 + 4 z i -ŝi 2 ≤ 2(t -t i ) 2 M 2 + 4d 2 K (z i ) + 8δ 2 ≤ 2d 2 (π)M 2 + ε 2 + 8δ 2 ≤ 2ε 2 ,
where in the last inequality we used [START_REF] Aubin | Set-valued analysis[END_REF]. Now, let x(t) be the (strong) solution of (1) starting at x 0 , and denote

l i (t) := s i -z(t), t ∈ [t i , t i+1 ], so that ż(t) = s * i ∈ f (s i ) -A(s i ) = f (z(t) + l i (t)) -A(z(t) + l i (t))
. Hence, by using the monotonicity of A we get

f (z(t) + l i (t)) -ż(t) -f (x(t)) + ẋ(t), z(t) + l i (t) -x(t) ≥ 0,
which leads us, using [START_REF] Aubin | Viability Theory[END_REF] and ( 8) together with the L-Lipschitzianity of f , to

ż(t) -ẋ(t), z(t) -x(t) ≤ 2ε f (z(t) + l i (t)) -ż(t) -f (x(t)) + ẋ(t) + z(t) -x(t) f (z(t) + l i (t)) -f (x(t)) ≤ 2ε ż(t) -ẋ(t) + 2εL z(t) + l i (t) -x(t) + L z(t) -x(t) z(t) + l i (t) -x(t) . So, if C is any constant such that ż(t) -ẋ(t) ≤ C for all t ∈ [0, t] (as ż(t) ≤ M, and x(•) is Lipschitz on [0, t]), we get ż(t) -ẋ(t), z(t) -x(t) ≤ 2εC + 4εL z(t) -x(t) + L z(t) -x(t) 2 + 4ε 2 L.
Next, by applying Lemma 1 to the function z(

•) -x(•) 2 + 2εC+4ε 2 L L we get, for all t ∈ [0, t] z(t) -x(t) ≤ 4ε 2 L + 2εC L 2 e Lt + 4ε(e Lt -1),
implying that, in view of ( 7) and ( 8),

d S (x(t)) ≤ d S (z(t)) + z(t) -x(t) ≤ 4ε 2 L + 2εC L 2 e L t + 4εe L t.
Consequently, by the arbitrariness of ε we conclude that x(t) ∈ S for every t ∈ [0, t]. Moreover, as x( t; x 0 ) ∈ S ∩ dom A, by the same argument as above we find t > 0 such that for every t ∈ [0, t] (recall Proposition 2)

x(t + t; x 0 ) = x(t; x( t; x 0 )) ∈ S ∩ dom A; that is, x(t) ∈ S for every t ∈ [0, t + t]. This proves that x(t) ∈ S for every t ≥ 0. Finally, if x 0 ∈ S ∩ dom A, we take a sequence (x k ) ⊂ S ∩ dom A such that x k → x 0 .
As we have just shown, for every k ≥ 1 we have that x(t; x k ) ∈ S for every t ≥ 0. Thus, since S is closed, as k → +∞ we deduce that x(t; x 0 ) ∈ S for every t ≥ 0.

The proof of Theorem 4 shows actually the following: Corollary 5. Given a closed set S ⊂ dom A ∩ S and x 0 ∈ S ∩ dom A, we assume that for some m, r > 0 such that P Ax (f (x 0 )) ≤ m it holds

sup ξ∈N P S m (y) min y * ∈Ay∩B(θ,m)
ξ, f (y)y * ≤ 0 for all y ∈ B(x 0 , r).

Then there exists t > 0 such that x(t; x 0 ) ∈ S for all t ∈ [0, t].

As we show in the corollary below the criterion of Theorem 4 becomes necessary if the maximal monotone operator A has a minimal norm section, which is locally bounded relative to its domain. As typical examples of such operators there are normal cones to closed convex sets, and the subdifferential mapping of lsc convex functions, which are Lipschitz relative to their domains. To fix this concept we say that the operator A is locally minimally bounded on S, if for every x ∈ S ∩ dom A there exist m, r > 0 such that

A 0 y ≤ m for all y ∈ S ∩ dom A ∩ B(x, r). ( 10
)
This condition is less restrictive compared with the local boundedness of A relative to S, which means that for every x ∈ S ∩ dom A there exist m, r > 0 such that y * ≤ m, ∀y * ∈ Ay, y ∈ S ∩ dom A ∩ B(x, r). [START_REF] Brogliato | Nonsmooth mechanics. Models, dynamics and control[END_REF] Obviously every locally bounded operator is locally minimally bounded.

Then the following result gives necessary and sufficient simpler criteria for the invariance of closed sets with respect to differential inclusion (1), using the normal cone mapping to S, N S , which stands for either the proximal normal cone N P S or the Fréchet normal cone N F S .

Corollary 6. Let S ⊂ H be a closed set satisfying [START_REF] Grosshans | Invariant sets forced by symmetry[END_REF]. Then the following statements are equivalent, provided that A is locally minimally bounded on S, (i) S is an invariant set for (1);

(ii) for every

x ∈ S ∩ dom A f (x) -P Ax (f (x)) ∈ T S (x); (iii) for every x ∈ S ∩ dom A sup ξ∈N S (x) ξ, f (x) -P Ax (f (x)) ≤ 0; (iv) for every x ∈ S ∩ dom A and every m ≥ f (x) -P Ax (f (x)) sup ξ∈N S (x) inf x * ∈(f (x)-Ax)∩B(θ,m) ξ, x * ≤ 0;
and the following assertion, when A is locally bounded relative to S, (v) for every x ∈ S ∩ dom A

sup ξ∈N S (x) inf x * ∈f (x)-Ax ξ, x * ≤ 0.
Proof. We fix x ∈ S ∩ dom A. The implication (iii) =⇒ (iv) is immediate, while the implication (ii) =⇒ (iii) follows because T S (x) ⊂ (N S (x)) * . In the same line, implication (i) ⇒ (ii) follows easily by observing that

(f (x) -A(x)) 0 = d + x(•; x) dt (0) = lim t↓0 x(t; x) -x t ∈ T S (x).
Thus, we only need to prove that (iv) ⇒ (i). If (iv) holds, by the current local boundedness assumption of A 0 on S ∩ dom A we pick m, r > 0 such that (f (y) -Ay) 0 ≤ m for all y ∈ B(x, 2r) ∩ S ∩ dom A. Hence, ξ, x * ≤ 0, and (i) follows, according to Theorem 4. Suppose now that A is locally bounded on S ∩ dom A, and consider the intermediate assertion (iv) for every x ∈ S ∩ dom A and every large enough m ≥ (f (x) -Ax) 0 we have that sup

B(x, 2r) ∩ S ∩ dom A = S m ∩ B(x,
ξ∈N S (x) inf x * ∈(f (x)-Ax)∩B(θ,m) ξ, x * ≤ 0.
As we see from the proof above (namely, the implication (iv) ⇒ (i)), we have that (iv) ⇒ (i), so that (v) ⇒ (iv) ⇒ (i). The proof of the corollary is finished because the implication (iv) =⇒ (v) is immediate.

In the following corollary we deduce another sufficient condition for the invariance of closed sets, using the Moreau-Yoshida approximations of A. Observe that we do not require here that set S satisfies condition (2).

Corollary 7. Given a closed set S ⊂ H, we suppose that for every bounded subsets

B of S lim inf λ↓0 sup y∈B sup ξ∈N P S (y) ξ, f (y) -A λ y ≤ 0.
Then S is invariant set for (1).

Proof. Fix an x ∈ S and let x(•; x) be the corresponding solution of (1). Given an r > 0 we let λ k , k ≥ 1, be such that λ k ↓ 0 and sup

ξ∈N P S (y) ξ, f (y) -A λ k y ≤ 0 for all k ≥ 1 and y ∈ B(x, r) ∩ S. ( 12
)
If ε < r 4 and t > 0 are such that x(t; x) ∈ B(x, r 4 ) for all t ∈ [0, t], then for large enough k ≥ 1 the solution

x λ k (•; x) of the differential equation ẋ(t) = f (x(t)) -A λ k (x(t)), x(0) = x, satisfies (see Proposition 2) x(t; x) -x λ k (t; x) ≤ ε < r 4 ; (13) hence, x λ k (t; x) ∈ B(x, r
2 ) for all t ∈ [0, t]. On the other hand, since A λ k is Lipschitz continuous, for large enough m > 0 we have B(x, r) ∩ S = {z ∈ B(x, r) ∩ S | A λ k z ≤ m}. So, according to Corollary 5, [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF] ensures that for some t > 0, say t ∈ (0, t), it holds

x λ k (t; x) ∈ S for all t ∈ [0, t]. Since x λ k (t; x) ∈ B(x, r
2 ) for all t ∈ [0, t], we infer that x λ k (t; x) ∈ B(x, r 2 ) ∩ S for all t ∈ [0, t]. Consequently, by [START_REF] Clarke | Nonsmooth Analysis and Control Theory Nonlinear Analysis[END_REF] we get d S (x(t; x)) ≤ ε for all t ∈ [0, t]. Then, as ε → 0, we deduce that x(t; x) ∈ S for all t ∈ [0, t]. Finally, the invariance of S follows by using the semi-group property of the solution x(•; x) (see again Proposition 2).

We consider now the special case where f ≡ θ, so that our differential inclusion (1) takes the simpler form ẋ(t) ∈ -Ax(t), x(0) = x 0 ∈ dom A. [START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF] In this case, the criterion of Theorem 4 becomes also necessary as the following corollary shows. Here too N S m stands for either N P S m or N F S m .

Corollary 8. Let S ⊂ H be a closed set satisfying [START_REF] Grosshans | Invariant sets forced by symmetry[END_REF]. Then the following statements are equivalent: (i) S is an invariant set of ( 14);

(ii) for every x ∈ S ∩ dom A

-A 0 x ∈ T S m (x) for all m ≥ A 0 x ;
(iii) for every x ∈ S ∩ dom A and for every m

≥ A 0 x sup ξ∈N S m (x)
ξ, -A 0 x ≤ 0;

(iv) for any x ∈ S ∩ dom A and every m

≥ A 0 x sup ξ∈N S m (x) inf x * ∈(-Ax)∩B(θ,m) ξ, x * ≤ 0.
Proof. As in the proof of Corollary 6, the implications (ii) =⇒ (iii) and (iii) =⇒ (iv with N S m = N F S m ) =⇒ (iv with N S m = N P S m ) are immediate. For the implication (i) =⇒ (ii), we assume that S is an invariant set of [START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF]. If x ∈ S ∩ dom A, then for a given m ≥ A 0 x we have

A 0 x(t; x) = d + x(t; x) dt ≤ d + x(0; x) dt = A 0 x ≤ m, for all t ≥ 0.
Hence, x(t; x) ∈ S m for all t ≥ 0 and we deduce that -A 0 x = d + x(0;x) dt ∈ T S m (y), yielding (ii). Finally, the implication (iv with N S m = N P S m ) =⇒ (i) is direct from Theorem 4.

To show how can our Theorem 4 be applied we consider the following example, which is treated in details in [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets[END_REF] in order to study the existence and the stability of solutions of differential inclusions involving the normal cone to a prox-regular set.

Recall that a closed set C ⊂ H is said to be uniformly r-prox-regular (r > 0) if for every x ∈ C and ξ ∈ N P C (x) ∩ B(θ, 1) we have ( [START_REF] Poliquin | Local differentiability of distance functions[END_REF])

ξ, y -x ≤ 1 2r y -x 2 for all y ∈ C.
Example 1. Let C ⊂ H be a uniformly r-prox-regular set and consider the associated differential inclusion

ẋ(t) ∈ g(x(t)) -N C (x(t)), a.e. t ∈ [0, T ], x(0) = x 0 ∈ C, ( 15
)
where g is a Lipschitz mapping on H. According to [4, Lemma 6(c)], let T : H ⇒ H be a maximal monotone operator such that for some m ≥ 0 it holds, for all y ∈ C,

N C (y) ∩ B(0, m) + m r y ⊂ T (y) ⊂ N C (y) + m r y,
and consider the associated differential inclusion

ẋ(t) ∈ g(x(t))+ m r x(t)-T x(t), a.e. t ∈ [0, T ], x(0) = x 0 ∈ C (⊂ dom T ). ( 16
)
This inclusion perfectly fits the form of differential inclusion [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF]. Then we make appeal to Theorem 4 to prove that the set C is invariant for (1), so that

ẋ(t) ∈ g(x(t)) + m r x(t) -T x(t) ⊂ g(x(t)) -N C (x(t)),
providing us with a solution for [START_REF] Clarke | Approximate invariant and Differential Inclusion in Hilbert Spaces[END_REF]. We refer to [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets[END_REF] for more details.

Lyapunov pairs and functions

In this section, we apply the results of the previous section to derive different criteria for a-Lyapunov pairs with respect to differential inclusion (1) :

ẋ(t) ∈ f (x(t)) -A(x(t)), t ∈ [0, ∞), x(0) = x 0 ∈ dom A,
whose unique solution is written x(•; x 0 ). Similar criteria to ours have been established recently in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF] in the case of weakly lsc Lyapunov pairs. Definition 2. We say that a pair (V, W ) of proper lsc functions V, W : H → R with W ≥ 0, is (or forms) an a-Lyapunov pair (a ≥ 0) with respect to system (1) if, for every x 0 ∈ dom A,

e at V (x(t; x 0 )) + t s W (x(τ ; x 0 ))dτ ≤ e as V (x(s; x 0 )), for all t ≥ s ≥ 0.
Observe that (V, W ) is an a-Lyapunov pair with respect to system (1) iff for every x 0 ∈ dom A there exists a t > 0 such that (see, e.g., [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF]Proposition 3.2])

e as V (x(s; x 0 )) + s 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ), for all s ∈ [0, t] .
We may assume without loss of generality that W is Lipschitz continuous on every bounded set (see, e.g., [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF]Lemma 3.1] or [START_REF] Clarke | Nonsmooth Analysis and Control Theory Nonlinear Analysis[END_REF]Theorem 1.5.1]). While, concerning function V, one need to suppose the following condition

V (x) = lim inf y dom A → x V (y) for every x ∈ dom V, ( 17
)
which is in fact necessary for V to be a Lypunov function in many important cases (for instance, when differential inclusion (1) possesses a strong solution).

Theorem 9. Given two proper lsc functions V : H → R satisfying [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF], W : H → R + , and a real number a ≥ 0, we assume that for every x ∈ dom V ∩dom A there are m, r > 0 such that P Ax (f (x)) ≤ m and, for all y ∈ B(x, r),

sup ξ∈∂ P (V +I A m )(y) inf y * ∈Ay∩B(θ,m) ξ, f (y) -y * + aV (x) + W (x) ≤ 0.
Then (V, W ) forms an a-Lyapunov pair with respect to system (1).

Proof. We fix T > 0 and x 0 ∈ dom V ∩ dom A. Following the discussion made before the current theorem we may suppose without loss of generality that W is Lipschitz continuous on every bounded set containing the trajectory {x(t; x 0 ), t ∈ [0, T ]}.

Let us define the maximal monotone operator  :

H × R 4 ⇒ H × R 4 and the Lipschitz function f : H × R 4 → H × R 4 as Â(x, μ) := (Ax, θ R 4 ), f (x, μ) := (f (x), 1, 0, 1, 0), and, given a
fixed μ 0 ∈ R 4 , consider the associated differential inclusion given in H × R 4 by ẏ(t) ∈ f (y(t)) -Â(y(t)), a.e. t ∈ [0, T ]; y(0) = (x 0 , μ 0 ), (18)
whose unique solution is y(t) := (x(t), t, 0, t, 0) + (θ, μ 0 ), t ∈ [0, T ] (with x(t) := x(t; x 0 )).

For each n ≥ 1, we consider the lsc function

V n : H × R 3 → R defined as V n (x, α, β, γ) := e aγ V (x) + (α -β)g n (α) + l 2 (α -β) 2 , ( 19
)
where g n is an l-Lipschitz extension of the function

W (x(•; x 0 )) -1 n from [0, T ] to [-1, T + 1]; hence, ∂ C g n (α) ⊂ B(0, l) for all α ∈ [0, T + 1]. (20)
We denote S := epi V n , so that S = S ∩ dom Â, by [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF], and

epi(V n + I A m ×R 3 ) = S ∩ Âm =: S m . ( 21
)
We also denote y 0 := (x 0 , θ R 3 , V (x 0 )) ∈ S ∩ dom Â. Let m, r > 0 be as in the current assumption, corresponding to x 0 , and choose r < r small enough such that for all (x, α, β, γ) ∈ B((x 0 , θ R 3 ), r)

g n (α) -e aγ W (x) + 2l |α -β| ≤ -1 2n . ( 22
)
Take y := (y 1 , μ 1 ) ∈ B(y 0 , r)∩S m , with y 1 := (x 1 , α 1 , β 1 , γ 1 ), and pick (ξ, -κ) ∈ N P S m (y). Due to [START_REF] Poliquin | Local differentiability of distance functions[END_REF] and [13, Exercise 1.

2.1], (ξ, -κ) ∈ N P S m (y) = N P epi(V n +I A m ×R 3 ) (y) ⊂ N P epi(V n +I A m ×R 3 ) (y 1 , V n (y 1 )) ; hence, κ ≥ 0. If κ > 0, say κ = 1 for simplicity, then ξ ∈ ∂ P (V n + I A m ×R 3 )(y 1 )
and, thanks to [START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF], we find

ξ 1 ∈ ∂ P (V +I A m )(x 1 ) and ς ∈ ∂ P g n (α 1 ) ⊂ ∂ C g n (α 1 ) such that ξ ∈ (e aγ 1 ξ 1 , g n (α 1 ) + (α 1 -β 1 )(ς + l), -g n (α 1 ) + l(β 1 -α 1 ), ae aγ 1 V (x 1 )) .
Since y ∈ B(y 0 , r) ∩ S m we have that x 1 ∈ B(x 0 , r) ∩ A m ∩ dom V and, so, by the current assumption, there exists an

x * 1 ∈ Ax 1 ∩ B(θ, m) (this last set being weak*-compact) such that ξ 1 , f(x 1 ) -x * 1 + aV (x 1 ) + W (x 1 ) ≤ 0.
Then we obtain (recall ( 20) and ( 22)

) (ξ, -1), (f (x 1 ) -x * 1 , 1, 0, 1, 0) = e aγ 1 ξ 1 , f(x 1 ) -x * 1 + g n (α 1 ) + (α 1 -β 1 )(ς + l) + ae aγ 1 V (x 1 ) = e aγ 1 ( ξ 1 , f(x 1 ) -x * 1 + aV (x 1 ) + W (x 1 )) + g n (α 1 ) -e aγ 1 W (x 1 ) + (α 1 -β 1 )(ς + l) ≤ g n (α 1 ) -e aγ 1 W (x 1 ) + 2l |α 1 -β 1 | ≤ -1 2n . ( 23
)
If κ = 0, then thanks to [START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF] we find ξ 2 ∈ H such that ξ = (ξ 2 , θ R 3 ), with the property that there are sequences

λ k ↓ 0, z k V +I A m -→ x 1 , ζ k ∈ ∂ P (V + I A m )(z k ) such that λ k ζ k → ξ 2 as k → ∞. By the current assumption, for each large enough k so that z k ∈ B(x 0 , r) there exists z * k ∈ Az k ∩ B(θ, m) such that ζ k , f(z k ) -z * k + aV (z k ) + W (z k ) ≤ 0.
Because A is maximal monotone and (z * k ) k is bounded, we can find an

x * 2 ∈ Ax 1 ∩ B(θ, m) such that ξ 2 , f(x 1 ) -x *
2 ≤ 0; hence, by multiplying the last inequality above by λ k and taking the limit as k → ∞,

(ξ, 0), (f (x 1 ) -x * 2 , 1, 0, 1, 0) = ξ, f (x 1 ) -x * 2 ≤ 0. (24)
According to Corollary 5, ( 23) and ( 24) imply the existence of some t := t(n) ∈ (0, T ] such that for every t ∈ [0, t],

(x(t), t, 0, t, V (x 0 )) ∈ S; in other words, e at V (x(t)) + tg n (t) + l 2 t 2 ≤ V (x 0 ) and, so, for every t ∈ [0, t]

e at V (x(t)) + t 0 W (x(τ ))dτ ≤ e at V (x(t)) + t 0 (g(t) + l(t -τ ))dτ + t n ≤ V (x 0 ) + t n . ( 25
)
Now, we claim that for all t ∈ [0, T ]

e at V (x(t)) + t 0 W (x(τ ))dτ ≤ V (x 0 ) + e (1+a)t n . ( 26
)
To prove this claim we define

t * := sup{t ∈ [0, T ] | inequality (26) holds on [0, t]}.
Indeed, from (25) and the lsc of V, it follows that (26) holds at t * . If t * < T, we denote y * := (x(t * ), θ R 3 , V (x(t * ))) and we easily check that y * ∈ S ∩ dom Â. Then, arguing as with y 0 above, we arrive at a relation which is similar to (25); that is, there is some t > 0 such that for all t ∈ [0, t]

e at V (x(t; x(t * ))) + t 0 W (x(τ ; x(t * )))dτ ≤ V (x(t * )) + t n . ( 27
)
Hence,

e a(t+t * ) V (x(t + t * )) + t+t * 0 W (x(τ ))dτ ≤ e a(t+t * ) V (x(t + t * )) + t+t * 0 W (x(τ ))dτ + (e at * -1) t 0 W (x(τ + t * ))dτ = e at * (e at V (x(t + t * )) + t 0 W (x(τ + t * ))dτ -t n ) + t * 0 W (x(τ ))dτ + e at * t n ≤ e at * V (x(t * )) + t * 0 W (x(τ ))dτ + e at * t n ≤ V (x 0 ) + e (1+a)t * n + e at * t n .
Consequently, due to the inequality e γ ≥ 1 + γ, we obtain that for all t ∈ [0, t]

e a(t+t * ) V (x(t + t * )) + t+t * 0 W (x(τ ))dτ ≤ V (x 0 ) + e (1+a)(t+t * ) n ,
leading us to a contradiction with the definition of t * . Now, the claim being true, we take the limit in (26) as n goes to +∞ to obtain that

e at V (x(t)) + t 0 W (x(τ ))dτ ≤ V (x 0 ) for all t ∈ [0, T ].
Finally, if x 0 ∈ dom V, then by the current assumption [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF], there exists a sequence (x k ) k≥1 ⊂ dom V ∩ dom A such that x k V → x 0 . Thus, from the last inequality above we conclude that e at V (x(t; x k )) + t 0 W (x(τ ; x k ))dτ ≤ V (x k ) for all t ∈ [0, T ] and all k ≥ 1.

Hence, as k goes to +∞, the lsc of V and Proposition 2 ensure that

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) for all t ∈ [0, T ],
showing that (V, W ) is an a-Lyapunov pair.

As in the case of the invariance of closed sets, the criterion of Theorem 9 takes a more simpler form when the maximal monotone operator A, or its minimal norm section, A 0 , is locally bounded (see [START_REF] Br | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF]). Here, ∂V stands for either ∂ P V or ∂ F V.

Corollary 10. Given two proper lsc functions V, W : H → R, such that W ≥ 0 and (17) holds, and a number a ≥ 0, we assume that A is minimally locally bounded relative to dom V. Then the following statements are equivalent.

(i) (V, W ) is an a-Lyapunov pair for (1); (ii) for any

x ∈ dom V ∩ dom A sup ξ∈∂V (x) ξ, (f (x) -Ax) 0 + aV (x) + W (x) ≤ 0; (iii) for any x ∈ dom V ∩ dom A V (x; (f (x) -Ax) 0 ) + aV (x) + W (x) ≤ 0;
Moreover, if in addition, [START_REF] Brogliato | Nonsmooth mechanics. Models, dynamics and control[END_REF] holds, then the above statements are also equivalent to (iv) for any

x ∈ dom V ∩ dom A sup ξ∈∂V (x) inf x * ∈Ax ξ, f (x) -x * + aV (x) + W (x) ≤ 0; (v) for any x ∈ dom V ∩ dom A inf v∈Ax V (x; , f(x) -v) + aV (x) + W (x) ≤ 0.
Proof. First, the implications (iii)(with

∂ = ∂ F )⇒ (iii)(with ∂ = ∂ P ) ⇒ (ii) follow since that ∂ P ⊂ ∂ F and σ ∂ F V (x) ≤ V (x; •). (i) ⇒ (iii). Fix x 0 ∈ dom V ∩ dom A. Since (V, W
) is an a-Lyapunov for (1), we have that for all t > 0

V (x(t; x 0 )) -V (x 0 ) t + e at -1 t V (x(t; x 0 )) + 1 t t 0 W (x(τ ; x 0 ))dτ ≤ 0, while Proposition 3 ensures that lim t↓0 x(t; x 0 ) -x 0 t = d + x(0; x 0 ) dt = (f (x 0 ) -Ax 0 ) 0 .
Hence, using the lsc of V together with the continuity of x(•, x 0 ),

V (x 0 ; (f (x 0 ) -Ax 0 ) 0 ) ≤ lim inf t↓0 V (x(t; x 0 )) -V (x 0 ) t ≤ -aV (x 0 ) -W (x 0 ), (28)
leading us to (ii).

(ii)(with ∂ = ∂ P ) ⇒ (i). We fix x 0 ∈ dom V ∩ dom A. From the one hand, by the boundedness assumption of A 0 , for a large m ≥ 0 there exists an r > 0 such that

B(x 0 , r) ∩ dom V ∩ dom A ⊂ A m . ( 29
)
On the other hand, we have that

∂ P (V + I A m )(x) ⊂ ∂ P V (x) for all x ∈ B(x 0 , r 2 ). (30)
Indeed, if ξ ∈ ∂ P (V + I A m )(x) for x ∈ B(x 0 , r 2 ), there exist δ > 0 and ρ ∈ (0, r 2 ) such that

(V + I A m )(z) ≥ V (x) + ξ, z -x -δ z -x 2 ∀ z ∈ B(x, ρ).
Take z ∈ B(x, ρ 4 ) ∩ dom V (⊂ B(x 0 , r)). By [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF] together with (29), there exists a sequence (z

n ) n ⊂ B(x, ρ) ∩ dom V ∩ A m such that z n → z and V (z n ) → V (z).
Since each z n satisfies the last inequality above, by taking the limit as n → ∞

we arrive at V (z) ≥ V (x) + ξ, z -x -δ z -x
2 and the inclusion (30) follows. At this stage, from (29) and the Lipschitzianity of f there exists some M ≥ m such that, for all x ∈ B(x 0 , r),

P Ax (f (x)) ≤ f (x) + A • x ≤ f (x) + m ≤ M, which shows that (f (x) -Ax) 0 ∈ f (x) -Ax ∩ B(θ, M ). Since ∂ P (V + I A M ) ⊂ ∂ P (V + I A m ), in view of (30), assumption (ii)(with ∂ = ∂ P ) implies that, for every x ∈ B(x 0 , r 2 ) sup ξ∈∂ P (V +I A M )(x) inf x * ∈Ax∩B(θ,M ) ξ, f (x) -x * + aV (x) + W (x) ≤ sup ξ∈∂ P V (x) ξ, (f (x) -Ax) 0 + aV (x) + W (x) ≤ 0.
Thus, (i) follows from Theorem 9. Finally, if A is locally bounded on dom V, then from the first part of the proof one only needs to verify the implication (iv) =⇒ (i), the proof of which is similar to the one of "(ii) ⇒ (i)" that we did above.

In the following corollary we provide criteria for a-Lyapunov pairs, which use the Moreau-Yoshida approximation of A.

Corollary 11. Let V, W and a be as in Corollary 10, and let ∂ be such that

∂ P ⊂ ∂ ⊂ ∂ C . If there exist λ 0 > 0 such that for all λ ∈ (0, λ 0 ] sup ξ∈∂V (x) ξ, f (x) -A λ x + aV (x) + W (x) ≤ 0, ∀ x ∈ dom V, then (V, W ) is an a-Lyapunov pair for (1). Proof. Fix x 0 ∈ dom V and t ≥ 0. If x λ (•; x 0 ) is the solution of the differential equation ẋλ (t) = f (x λ (t)) -A λ (x λ (t)), x λ (0) = x 0 (λ ∈ (0, λ 0 ]), (31)
then, according to Corollary 10(ii), the pair (V, W ) is an a-Lyapunov pair of (31); that is,

e at V (x λ (t)) + t 0 W (x λ (τ ))dτ ≤ V (x 0 ) for all t ≥ 0.
Hence, the conclusion follows as λ ↓ 0.

We consider now the case when f ≡ 0 so that differential inclusion (1) reads

ẋ(t) ∈ -A(x(t)), x(0) = x 0 ∈ dom A. ( 32
)
In the following theorem ∂ stands for either ∂ P or ∂ F .

Corollary 12. Let V, W : H → R be two proper lsc functions, such that W ≥ 0 and (17) holds, and let a ≥ 0. Then the following statements are equivalent: (i) (V, W ) is an a-Lyapunov pair for (32);

(ii) for every x ∈ dom V ∩ dom A and every m

≥ A 0 x sup ξ∈∂(V +I A m )(x)
ξ, -A 0 x + aV (x) + W (x) ≤ 0;

(iii) for every x and m as in (ii)

sup ξ∈∂(V +I A m )(x) inf x * ∈-Ax∩B(θ,m) ξ, x * + aV (x) + W (x) ≤ 0;
(iv) for every x and m as in (ii)

(V + I A m ) (x; -A 0 x) + aV (x) + W (x) ≤ 0;
(v) for every x and m as in (ii) inf v∈-Ax∩B(θ,m)

(V + I A m ) (x; v) + aV (x) + W (x) ≤ 0.
Proof. The implications (ii) ⇒ (iii), (iv) ⇒ (v), (iv) ⇒ (ii), and (v) ⇒ (iii) are immediate. To prove that (i) ⇒ (iv), we fix x 0 ∈ dom V ∩ dom A and m ≥ A 0 x 0 . According to Proposition 2, for any t ≥ 0 we have that

-A 0 (x(t; x 0 )) = d + x(t; x 0 ) dt ≤ d + x(0; x 0 ) dt = -A 0 x 0 ≤ m;
that is, x(t, x 0 ) ∈ A m for all t ≥ 0. Hence, since x(t,x 0 )-x 0 t → -A 0 x 0 as t ↓ 0, provided that (V, W ) is an a-Lyapunov pair for (32) we obtain, by arguing as in the proof of (28),

(V + I A m ) (x 0 ; -A 0 x 0 ) ≤ lim inf t↓0 (V + I A m )(x(t; x 0 )) -(V + I A m )(x 0 ) t = lim inf t↓0 V (x(t; x 0 )) -V (x 0 ) t ≤ -aV (x) -W (x),
giving rise to (iv). Finally, the conclusion of the corollary follows because the implication (iii) ⇒ (i) holds according to Theorem 9.

We obtain the following corollary, which can be find in [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF]; the original version of this result was established in [START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF] Corollary 13. Let V, W : H → R be two proper lsc functions, such that W ≥ 0, and let a ≥ 0. If condition [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF] and, for every x ∈ dom V, lim inf λ↓0 V (J λ (x)) -V (x) λ + aV (x) + W (x) ≤ 0, then (V, W ) is an a-Lyapunov pair for (32).

Proof. We fix x ∈ dom V ∩ A m for some large m ≥ 1. Since A λ x ∈ A(J λ x) and A λ x ≤ A • x ≤ m, we infer that J λ x ∈ A m and, so, using the current assumption, (V + I A m ) (x 0 ; -A 0 x 0 ) ≤ lim inf t↓0 V (J λ (x)) -V (x) t ≤ -aV (x) -W (x).

The conclusion follows then from Corollary 12(iv).

Corollary 10 obviously covers the case when A is the null operator, where (1) becomes a usual differential equation stated in the Hilbert space H as ẋ(t) = f (x(t)), a.e. t ≥ 0, x(0) = x 0 ∈ H.

(33)

The following characterization is known when ∂ is the viscosity subdifferential as defined in [17, Definition 2.7], while the case of weakly lsc a-Lyapunov pairs can be found in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF]. We close this section by analyzing a typical example of Lyapunov pairs. Example 2. Assume that a function V : H → R is a proper, convex and lsc, and consider the differential inclusion ẋ(t) ∈ -∂V (x(t)).

Then the pair (V, (∂V ) 0 2 ) is a Lyapunov pair, so that for every x 0 ∈ dom V V (x(t, x 0 )) + t 0 ẋ(τ , x 0 )

2 dτ ≤ V (x 0 ) for all t > 0.

To see this fact we fix x ∈ dom V ∩ dom ∂V. Since A λ (x) ∈ A(J λ (x)) for every λ > 0 (A = ∂V ), condition [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF] holds and one has that

V (J λ (x)) -V (x) ≤ -A λ (x), x -J λ (x) = - 1 λ x -J λ (x) 2 .
Hence, lim inf λ↓0 V (J λ (x))-V (x) λ

+ A 0 x 2 ≤ lim inf λ↓0 V (J λ (x))-V (x) λ + 1 λ 2 x -J λ (x) 2 ≤ 0 ,
and Corollary 13 (together with Proposition 2) applies.

Conclusion and further research

We gave different conditions for the invariance of closed sets, which only involve the input data, represented by the maximal monotone operator and the Lipschitz mapping. These conditions are applicable to a large variety of closed sets which do not need to be convex or smooth. The current work extends and improves some of the results given in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF] and dealing with weakly closed invariant sets and weakly lower semi-continuous a-Lypunov pairs. It will be our aim in a forthcoming work to apply the current results to specific differential equations/inclusions where the underlying maximal monotone operator is not known explicitly. This will make the access to the corresponding semi-group more easier, namely regarding the behavior at infinity of trajectories.

  2r), and, since S = S ∩ dom A, for every y ∈ B(x, r) ∩ S m , N S m (y) = N S m ∩B(x,2r) (y) = N S∩dom A∩B(x,2r) (y) = N S∩dom A (y) = N S (y). So (iv) gives us, for every y ∈ B(x, r) ∩ S m , sup ξ∈N S m (y) inf x * ∈(f (y)-Ay)∩B(θ,m)

Corollary 14 .

 14 Let V, W and a be as inCorollary 10, and let ∂ be such that∂ P ⊂ ∂ ⊂ ∂ C .Then the following statements are equivalent:(i) (V, W ) is an a-Lyapunov pair for differential equation (33), (ii) for every x ∈ dom V sup ξ∈∂V (x) ξ, f (x) + aV (x) + W (x) ≤ 0, (34) (iii) for every x ∈ dom V V (x; f (x)) + aV (x) + W (x) ≤ 0.Proof. In view of Corollary 10, we only need to check that (i) =⇒ (ii) (with ∂ = ∂ C ), and this easily follows from the relation∂ C V = co{∂ L V + ∂ ∞ V }.Indeed, assume that (i) holds and takeξ ∈ ∂ L V (x) and ζ ∈ ∂ L,∞ V (x). By the definition of ∂ L V (x) we choose sequences ξ k ∈ ∂ P V (x k ) such that x k V → x and ξ k ξ. Then, by (i), ξ k , f(x k ) + aV (x k ) + W (x k ) ≤ 0 for all k ≥ 1,and, so, as k → ∞, we deduce that ξ, f (x) + aV (x) + W (x) ≤ 0. Similarly, we choose sequencesx k V → x and λ k ↓ 0 such that ζ k ∈ ∂ P V (x k ) and λ k ζ k ζ.Then, by arguing as above we deduce that ζ, f (x) ≤ 0, which in turn yields ξ + ζ, f (x) + aV (x) + W (x) ≤ 0, and this gives us (ii) (with ∂ = ∂ C ) by convexification.

$ Research partially supported by Conicyt-Fondecyt grant no.

1151003, Conicyt-Pcha/Doctorado Nacional/2014-63140104, and Conicyt-Redes no. 150040.


Jamal Fajoui

David Gloaguen
email: david.gloaguen@univ-nantes.fr

Emmanuel Girard

Ronald Guillén

Elastoplastic behaviour of zirconium alloy cladding tubes after cold rolling

Complementary methods have been used to analyse residual stresses in zirconium alloy tubes which were manufactured by cold rolling: X-ray diffraction and scale transition model. A modified elasto-plastic self-consistent model (EPSC) has been used to simulate the experiments and exhibits agreement with experimental data. X-ray diffraction analysis in rolling direction shows different stress values for the analysed planes respectively. The measured strains were generated by an anisotropic plastic deformation. Plastic incompatibility stress on X-ray measurements should be taken into account so as to make a correct interpretation of the experimental data.

I. Introduction

Pressurized heavy water reactors use zirconium alloys due to their low neutron absorption cross-section, low irradiation creep and high corrosion resistance in reactor atmosphere. Due to their hcp structure, Zr single crystals are usually highly anisotropic. Consequently, strong residual stresses are generated in fabricated components of Zr alloy during mechanical treatments. These stresses play an important role in subsequent deformation and modify the mechanical properties of the reactor core components. This work concerns the manufacturing process of Zr alloy cladding tubes. These tubes are manufactured by cold rolling. To reach its final size, the material undergoes three cold-rolling passes in a particular mode called cold pilgering. To understand the influence of plastic anisotropy on the mechanical properties, the last rolling pass has been completely studied. X-ray diffraction techniques are used to characterize the mechanical state at the macro-and mesoscopic levels of the specimens. The experimental data show the appearance of important second order stresses which depend on crystallographic planes. A modified EPSC approach is adopted to describe the plasticity evolution and determine the active deformation systems. A formulation of the crystal behaviour is proposed. Experimental results have been compared with those given by the modified model for Zr alloy tubes. The EPSC formulation provides an accurate description of the intergranular plastic residual stresses induced by the anisotropic plastic behaviour of single crystals. An estimation of the first-order stresses is also proposed.

II. Modified self-consistent approach

The plastic flow can take place when the Schmid criterion is verified, i.e. slip occurs if the resolved shear stress g τ on a system g is equal to the critical value g c τ depending on the hardening state of the slip system. A complementary condition which states that the increment of the resolved shear stress must be equal to the incremental rate of the critical resolved shear stress (CRSS) has to be verified simultaneously. In small strain formulation, one has:

g c g τ τ = = ..σ g R & and g c g τ τ & & & = = σ R g .. ( 1
)
where g R is the Schmid tensor on a system g is the double scalar product. If g γ & denotes the slip rate on a system g, the Schmid criterion is thus given by:

0 g g c = γ ⇒ τ < & g τ (2.a) 0 and g g c g g c g = γ ⇒ τ < τ τ = τ & & & (2.b) 0 and g g c g g c g > γ ⇒ τ = τ τ = τ & & & (2.c)
The main problem is to determine which combination of slip systems will be really activated at each step of the plastic deformation path. In this case, all possible combinations of potentially active systems must to be scanned to find one that satisfies the two preceding conditions simultaneously. Running time considerations become the main task of the model. Base on the work of Ben Zineb et al. [START_REF] Ben Zineb | fifteenth French conference of mechanic[END_REF], the relations (2a), (2b) and (2c) can be expressed with the following equation:

g g g M τ = γ & & (3)
The slip rate is linked to the resolved shear stress through a function g M . This formulation is based on the crystalline rate-dependent flow rule. On the other hand, the temporal variable does not play any role in this approach. The hardening parameter g M is given by:

( ) ( ) ( ) ( )       τ +       τ +                                   - τ τ + β = g 0 g 0 g c g 0 g k th 1 2 1 k th 1 2 1 1 k th 1 2 1 M & (4)
With equation ( 4) and after some algebric calculations, the constitutive relation which links the overall stress rate and strain rate in the grain is then given by: ( )

σ σ R s ε .. R 1 g - =       ∑ + = .. .. .. g g M (5)
l is the elastoplastic consistent tangent moduli tensor. The other mechanical variables are determined by the usual relations given by the EPSC model [START_REF] Lipinski | [END_REF], taking into account the equations ( 3) and (5).

III. Experimental procedure III.1 Crystallographic texture analysis

X-ray diffraction analysis has been performed with a four circles XRD3003PTS SEIFERT goniometer. The K α copper radiation has been used as the X-ray The study of samples having a curved geometry not only requires an acurrate positioning of the samples but also take into account the geometric effects induced by XRD directional feature. A theoretical model based on a ray-tracing method was used to take into account the geometric effects which modify the collected intensities [3]. Correction coefficients accounting for the "geometrical texture" of the samples as well as absorption corrections are calculated for each measurement point. Figure 1 presents the initial and final textures. In the final state, the {0002} PF shows a bimodal ditribution of the intensity maxima in the NT-TD plane at about 30° of the ND axis. This is typical for this kind of sample.

III.2 Evaluation of residual stresses

The average strain for diffracting grains can be written as [4]: incompatibilities between the diffracting and the non diffraction volumes. σ I is the macroscopic stress tensor. The second term in the above equation can be approximated by :

d V IIpi I ij ij d V ) hkil , , ( ) hkil , , (
d V IIth d V IIpi ) hkil , , ( q ) hkil , , ( ψ φ ε = ψ φ ε (7) Where d V IIth ) hkil , , ( ψ φ ε
is calculated from the EPSC model and averaged for crystallites having the (hkil) normal to the orientation characterized by the ψ and φ angles. q is a constant factor (a fitting parameter) introduced in order to find the real magnitude of plastic strains. In this case, it is possible to evaluate the magnitude of the first as well as the second order residual stresses using eqs. (6,7) with a nonlinear fitting procedure associated with the scale transition model. The relation (6) shows clearly that the measured strain cannot be identified to the macroscopic strain if the material presents anisotropic properties. The presence of intergranular strain after a mechanical solicitation influences the measured strain. Generally, the interpretation of experimental data is based on the unjustified assumption

that d V IIpi ) hkil , , ( ψ φ ε = 0.
We have determinated the evolution of internal stresses due to plastic anisotropy in deformed samples along the rolling direction for the last rolling step at different total strain. These experiments were carried out on a XRD3003PTS SEIFERT goniometer with Cr K α and Cu K α radiations. An Ω goniometric assembly with a position Sensitive Detector were used. Three plane families were studied with Cr radiation: {

IV. Simulation data

The texture was introduced in the model by a set of 1000 grains characterized by Euler angles and by weights which represent their volume fraction. The main active slip systems are assumed to be: the prismatic systems, the firstorder pyramidal slip (pyr<c+a> and pyr) and { } 2 1 10 twinning. The critical resolved shear stresses (CRSS) used here, τ = 95 MPa for prismatic mode, τ = 175 MPa for pyrmode, τ = 310 MPa for pyr<c+a> slip system and τ = 180 MPa for tensile twinning, give a correct description of the experimental data. We considered a linear hardening law [5], the coefficient H gr is equal to H gg for any deformation mode r:

∑ = r r γ g H g τ & &
. The hardening coefficients were found to be 80 MPa for prismatic slip, 170 MPa and 300 MPa for pyramidal slip (pyrand pyr<c+a> respectively) and 180 MPa for twin mode.

V. Results

Applying eq. ( 6) and ( 7) and fitting the results from the model to the experimental data, the longitudinal stress I 11 σ and the q factor have been found for each step of the rolling process. plane. X-ray measurements show the effective existence of plastic anisotropy. As can be seen from eq. ( 6), the measured stress depends on a function of the analysed plane family. Strain incompatibilities are present at the mesoscopic level in the material and consequently, the stresses obtained by XRD depend on the plane. The diffracting crystals are not the same for each case, which allows us to deduce that different second order stresses exist, linked to a strong anisotropic plastic deformation for these two plane families. The term The method employed in this study enables the magnitude of the macro-and meso-scopique residual stresses to be evaluated using information from an EPSC approach.

VI. Conclusions

A modified algorithm has been used for computing the mechanical response of a single crystal. Numerical results, obtained at the different scales, show the relevance of this approach. This approach has been developed in order to simulate the mechanical response of Zr alloy tube after cold pilgering. Stresses observed by X-ray diffraction are significantly different from one plane family to another. These results can be explained by the presence of mesoscopic stresses of plastic in the material. Self-consistent model had been used to quantify the stress differences between these plane families. The choice of prismatic slip as principal deformation mode can explain the opposite pseudomacrostresses values for the two studied planes. These predicted results can only be obtained for a single set of slip modes and hardening parameters. This analysis shows that X-ray diffraction stress analysis could constitute an effective validation or identification of a model. Nevertheless, the influence of second order strain must be taken into account to get a correct interpretation of X-ray diffraction results for hexagonal material.

Figure 1 :

 1 Figure 1: Initial (a) and final (b) { } 0002 and { } 0 1 10 PF.

  is the average over diffracting grains for the hkil reflection. F ij (φ,ψ,hkil) are the X-ray elastic constants and

  2θ = 137.2° and { 0004 } at 2θ = 125.6°. With Cu radiation, the three plane analysed were: 117.8°. The direction φ = 0°c orresponds to the rolling direction (parallel to the tube axis).

  Figure2shows the evolution of residual stress for each plane as a function of the axial position z along the rolling direction (z varies between 0 and 300 mm).

  role in a correct interpretation of different measurements of residual stresses.

Table 1

 1

						gives the result for the final tube (195%
	total deformation). I 11 σ value reaches 60 MPa.
							q	I 11 σ (MPa)
	ε	IIpi	, ( ψ φ	,	) hkil	d V	0.081 0.202 ±	60 ± 30
	ε	IIpi	≠ 0 hkil ) , ( ψ , φ	d V	-	{ } 4 1 10 plane	{ } 2 2 20 plane
			= 0			189± 16	-226±24

Table 1 :

 1 results of evaluation of σ I and q parameter with two different assumptions, i.e.

		ε	IIpi	, ( ψ φ	,	hkil	)	d V	= 0 or
	≠ 0
	In order to express the influence of the second order plastic
	strain on the correct interpretation of experimental data an
	additional test has been performed. The procedure
	presented above was applied for the X-ray measurements
	but with the assumption that	ε	IIpi	, ( ψ φ	,	) hkil	d V	= 0. The
	results are presented in table 1. For σ 11 , the two planes exhibit an opposite behaviour: traction for { } 4 1 10 plane and compression for { } 2 2 20

& l & &


F.-X Keller
email: keller@inrs.fr

F Chata

Characterization of wood dust emission from hand-held woodworking machines

Keywords: dust emission, wood dust, hand-held machines, concentration measurement, air cleaning, aerosols

This paper focuses on the prevention of exposure to wood dust when operating electrical hand-held sawing and sanding machines. A laboratory methodology was developed to measure the dust concentration around machines during operating processes. The main objective was to characterize circular saws and sanders, with the aim of classifying the different power tools tested in terms of dust emission (high dust emitter versus low dust emitter). A test set-up was developed and is described and a measurement methodology was determined for each of the two operations studied. The robustness of the experimental results is discussed and shows good tendencies. The impact of air-flow extraction rate was assessed and the pressure loss of the system for each machine established. For the circular saws, 3 machines over the 9 tested could be classified in the low dust emitter group. Their mean concentration values measured are between 0.64 and 0.98 mg/m 3 for the low dust emitter group and from 2.55 and 4.37 mg/m 3 for the high dust emitter group. From concentration measurements, a machine classification is possibleone for sanding machines and one for sawing machines -and a ratio from 1 to 7 is obtained when comparing the results. This p. 4 classification will be helpful when a choice of high performance power tools, in terms of dust emission, must be made by professionals.

p. 5

INTRODUCTION

Wood dust is classified at the French national level on the list of carcinogen materials (INRS ED974) and the decree of December 23 rd 2003 fixed a value for the mandatory occupational exposure limit to wood dust of 1 mg.m -3 effective from July 1 st 2005 in France.

In the short and medium terms, occupational exposure to wood dust can cause skin and respiratory diseases: eczema, conjunctivitis, rhinitis, asthma and pulmonary fibrosis. In the long-term, it can be at the origin of primary cancers of the nasal cavities and of the ethmoïd bone. In France, the population exposed to wood dust during their occupational activities is greater than 300,000. Professional diseases associated with this activity are recognized according to table N°47 of the Social Security general regime and to table N°36 of the agricultural regime. On average, more than 120 cases of occupational illness linked to wood dust, including 70-80 cancers, are listed each year for the general regime (DRP, 2016).

Reinforced prevention methods were imposed under the Carcinogen, Mutagen, Reprotoxic decree of February the 1 st 2001: risk evaluation, substitution by a less dangerous agent (rarely applicable to wood), working in a closed system and methods of collective protection p. [START_REF] Chata | Estimation by inverse methods of the emission profile of handheld woodworking machines[END_REF] (capture at the source), training of staff, regular assessment of exposure (annual control of Occupational Exposure Limit values), following exposure, medical monitoring, etc. Today, industrial managers in this profession have to choose machines without having any information concerning their characteristics in terms of dust emission. Effectively, the suppliers and manufacturers of hand-held machines are not currently obliged to notify the users of the dust emission levels of their apparatuses. Furthermore, the machine directive is vague concerning the subject of dust emission and the European norms are not necessarily reapplicable to hand-held machines.

Woodworking requires the use of hand-held machines to carry out representative tasks by the worker. For example, amongst the most commonly encountered we can count sanding and sawing. Hand-held machines are amongst those that have the highest emission levels and in most cases dust capture systems are not of effective design (BIA, 2003). Following the "wood dust" campaign from March to June 2008 [START_REF] Lamy | Occupational exposure to wood dust -Results from the French national campaign 2008[END_REF] initiated by the French Ministry of Labour, 3105 companies (in the building sector, wood and sawmills, furniture building) were visited. The hand-held machines were found to be connected to an aspiration system in only 20% of cases, and in more than 60% of cases the OEL value was exceeded.

p. 7

A previous study, entitled "Evaluation of the capture performances for 3 types of handheld woodworking machines -INRS study" that concerned hand-held electric machines from the workshop or construction site [START_REF] Muller | Evaluation des performances de captage de trois types de machines à bois portatives[END_REF] concluded that many machine hood collection devices were not effective. The operations studied were sawing, ripping and sanding. The aim was to draw up an overview of the situation in terms of dust capture of existing material but not to develop a standardizable method.

Trials in companies were carried out to evaluate the occupational exposure of the workers using the most efficient materials identified during laboratory tests [START_REF] Muller | Evaluation des performances de captage de trois types de machines à bois portatives[END_REF]. Individual samples were taken from 22 workers from 13 carpentries (manufacture of beams, doors, staircases, etc.). The results showed that in situations where good occupational practices were respectedfrequent cleaning of the workshops using a centralised vacuum system, the exclusive use of vacuum toolsthe occupational exposure measured varied from 0.4 to 1.1 times the OEL value for sanding operations and from 0.6 to 1.3 times the OEL value for sawing (OELv = 1mg.m -3 ) [START_REF] Lamy | Occupational exposure to wood dust -Results from the French national campaign 2008[END_REF][START_REF] Muller | Evaluation des performances de captage de trois types de machines à bois portatives[END_REF]. All the study results depend on respect of the suction flow rates recommended for each type of machine. These flow rates can only be ensured during the whole period of work if industrial vacuums are used and are regularly cleaned.

p. 8

Following on from these different observations, the aim of this study was to develop a methodology to measure wood dust emissions around hand-held machines. Measures of the suction flow and pressure loss were also integrated into this approach. In parallel to the progress of this study, normalization tasks in the context of CEN/CENELEC (European Committee for Electrotechnical Standardization) were undertaken to attempt to impose at least a labelling of the machines, with the aim of informing the user of an indication of the dust emission level generated, the recommended suction flow rate and the associated pressure loss, and of unifying the diameters of the attachment ducts linked to the vacuum network. This latter information is needed to allow the fitter to size the suction centre so that it is adapted to the machines connected to the network. In the long-term, the benefits of the study should lead to completing the labelling of the machines according to the dust emissions that they generate (classes A, B, C and D for example).

In the frame of this study, two types of hand-held machines were tested. We have selected 4 sanders and 9 circular saws. Sanding is one of the most widespread machining procedures for woodworkingmainly of raw wood. It generates rather fine wood dust. Sawing is also an extremely widespread operation, used to cut both raw wood and chipboard. The dust created p. 9

during sawing is of a different particle type from that issued from sanding and circular saws are very emissive machines (Muller and Fontaine 2010). The dust shape also differs. These two characteristics have an influence on the response of the sensors used during tests [START_REF] Harper | Determining particle size distributions in the inhalable size range for wood dust collected by air samplers[END_REF]. Wood dust morphologies induce a bias during the measurement of concentration with optical counters. The determination of the shape factors and densities of these dusts is not straightforward (Chata, 2015 and[START_REF] Gorner | Laboratory study of selected personal inhalable aerosol samplers[END_REF].

Choice of the test method

Several standard methods exist to determine the emission of a polluting source. In this study we chose the EN 1093-3 norm as the reference to establish the test procedure.

We placed the pollutant source in a ventilated cabin (Figures 1 and2), in which we controlled the rate of air flowing through the test chamber. The dimensions of the main cabin are length (L) = 4.5 m; width (l) = 3 m; height (H) = 3 m. Downstream, a converging portion leads to an extraction conduit of 1 m in diameter and 15 m in length. An extraction ventilator linked to this conduit allows the airflow rate in the cabin to be regulated over a range from 8,500 to 50,000 m 3 .h -1 . During our tests, a fixed flow rate of approximately 13,000 m 3 .h -1 is maintained, so as to have a 0.4 m.s -1 mean horizontal air velocity. This speed guarantees a controlled flow in the cabin and particle transport that predominates over sedimentation. We are focusing on particle with aerodynamic diameter in the order of 10 µm to 100 µm (inhalable dust) and with a density of ~1000 kg.m -3 . For this type of particle the sedimentation velocity is in the range of 5.10 -3 m.s -1 which is 80 times lower than the air velocity fixed in the cabin. The ventilated cabin was adapted to standardize the airflow, from upstream to downstream, using a diffusing canvas at the entrance, (Chata, 2015 and[START_REF] Welling | Wood dust particle and mass concentrations and filtration efficiency in sanding of wood materials[END_REF]. A workbench on which the machining operations were carried out was installed in the cabin. The different sampling points were placed downstream in the ventilated cabin and a detailed experimental and CFD study was performed in order to characterized the sensor p. 11 positions and the concentration field at these locations (Chata, 2015). All the tests carried out during this study for sanding and sawing operations were performed with a robot (KR60 KUKA Roboter GmbH, Augsburg Germany). This allowed the reproducibility of machining cycles during testing to be ensured as fully as possible. A standard document EN 50632-1 applied to woodworking machines is developed and aims at measuring the amount of dust in a closed room around the machine and its extraction system. This standard is aiming at measuring the dust generated in a workshop placing different sensors in the room plus one on the operator but it is not directly linked to a pollutant emission rate measurement.

MATERIALS AND METHODS

A ventilated test cabin was developed to measure dust emission from hand-held wood saws and sanding equipment. The characterization of emissions from electrical hand-held machines was carried out by the intermediary of dust measurements generated during machining operations. Trial cycles of the machining processes were developed and are described in this article. From these measurements it is possible to achieve a classification of p. 12 the different machines and confirm that some materials are more efficient than others in terms of dust collection hoods.

The orbital sanders used during these tests are all connected to the electrical sector (no battery-driven machines were used) and have the plate diameters and references indicated in Table I below. The papers used during the measurements were ordered from the corresponding machine manufacturers and the grit type is specified:

The circular saws tested are all connected to the electrical sector (no battery-driven machines were used) and show the characteristics defined in Table II below (blade diameter, blade thickness, number of teeth, the electric power of the motor and the max. rotational speed):

During the tests, all the machines used were connected to a Local Exhaust Ventilation (LEV) system (Spanesi 4BD 11kW -Spanesi France s.a.r.l) by means of plug connectors present on the machines (each protection hood (provided by the tool manufacturer), diameter and nozzle shape differs depending on the constructors' choice). The Local Exhaust p. [START_REF]Electric motor-operated tools -Dust measurement Procedure -Part 1: General requirements[END_REF] Ventilation system is located outside of the ventilated cabin in order to not affect the pollutant emission of the tested machines. Description of the sanding set-up (Figure 1):

We developed a test procedure for sanding that consists of a linear, reproducible and controlled machining cycle. The cycle duration was fixed at 10 minutes, corresponding to 10 "there and backs" on the working table. The forward speed was kept constant to 1 cm.s -1 and the machining rotational speed was set to the maximum for each sanders. Before and after each operating cycle, the wooden plank was weighed to determine the amount of wood removed during the test (Sartorius balance LA8200Saccuracy: hundredth of gram). The dimensions of the planks are: length 800 x width 790 x thickness 16 mm. For sanding operations we have selected beech rectified raw wood as it is a common type of wood widely used in the industry. All the machines were connected to the LEV system via a high vacuum central unit. Two vacuum flow rates were tested for each machine -80 m 3 .h -1 , which is recommended by the French prevention institution to obtain satisfactory collection performances (INRS ED6052, 2009), and 40 m 3 .h -1 which is representing a degraded airflow p. 14 in order to evaluate in the laboratory the loss of collection efficiency for a diminished flow rate. This second flow rate allows the impact of a reduction in suction on the emission from machines to be estimated. A 0 m 3 .h -1 flow rate was tested once in order to confirm the impact of the suction flow rate on the concentration values (Table III). Each of these tests was repeated at least three times to assess the dispersion and variability of the measurement of dust concentration. The sanding paper was replaced at the beginning of the test for each machine. The working methodology is composed of the following phases: 1) start of the measurement devices and the machine, 2) waiting time (2 min), 3) machining operation (10 min), 4) end of the machining operation (machine stopped), 5) waiting time (2min), 6) stop of the measurement devices.

Description of the sawing set-up:

A method was setup to mimic a sawing cycle. A first step in the methodology was done with 25 linear cuts during a 7 minutes cycle time. Then a second step was carried out with 4 linear cuts during a 3 minutes cycle. Planks of chipboard were used for the test. The planks had dimensions of length 800mm x width 790 mm x thickness 16 mm. The volume and p. 15 weight of the cut wood are known for each circular saw tested. For each machine, new blades were used for these trials to eliminate any possible impact of blade blunting. All the machines were connected to a high vacuum suction system and the maximum suction flow rate was measured for each machine. These measurements were carried out with a 28 mm diameter connection tube. Each test was repeated at least three times. A schematic view of the test cabin used for sawing trials is shown in Figure 2. The working methodology is composed of the following phases: 1) start of the measurement devices and the machine, 2) waiting time (2 min), 3) machining operation (7 min/3 min), 4) end of the machining operation (machine stopped), 5) waiting time (2min), 6) stop of the measurement devices.

Experimental method developed

The measurements of dust concentration were carried out using three types of sensor. An

Measurement of the pressure losses and the suction flow rate

Pressure loss was measured for each machine tested in order to evaluate the pressure restriction level from the machines and show that some machines require more energy to move the extraction air at 80 m 3 /h. The measurement of the flow rate was made by the intermediary of a calibrated Venturi tube, a manometer (MP200, Kimo SA) and a flow meter (Preso, Badger Meter Inc.) placed in the vacuum duct. This measurement was carried out for each of the machines before the start of measurements of dust concentration. It was done on sanders that were turned off, for which the rotating plate was not in contact with the workstation. For each circular saw test, the nominal vacuum suction was fixed as the maximum suction that could be attained with the high vacuum installation. The pressure loss measurement was established for each machine tested. This test was carried out for each circular saw before beginning measurements of dust concentration. It was performed on machines that were turned off.

p. 18

Measurement of dust concentration

The wood dust emission measured during a sanding or sawing cycle is characterized by signals recorded with the different sensors. These signals have similar appearances irrespective of the sensor used. However, their heights are different due to instrument sensitivity, measurement type and measurement frequency. When the concentration signals recorded for some 10 minutes of sanding were analysed (Figure 5), a measurement plateau was observed. This plateau represents the establishment of the emission of wood dust in the air, generated by the sander. From the experimental values measured with the different sensors during the time cycle we calculated a moving average of concentration which is then used and plotted.

p. 19

RESULTS AND DISCUSSION

In the next two graphs (Figures 3 and4) we show an example of the measurement of the particle distribution size of wood dust generated by either sanding or sawing. Particle size distributions upon sanding and sawing are centred at around ~3µm: the dust issued from the operations is relatively fine and it could be confirmed that the particles do not perturb the airflow in the cabin. However, a difference can be noted between sanding and sawing dusts at the level of 5 -15 µm. The particle volume distribution shows a clearer difference between the measurements carried out with sanding dust and with sawing dust. The sawing curve gives an optimum centred at 10 µm that is approximately two-fold higher than that of sanding. This difference is a factor that explains the higher concentration generated by sawing operations.

Machines pressure losses

The pressure losses are generated by changes in the section of the air passage in the vacuum duct, as well as by the different machine hoods. Some restrictions of passage section invoked pressure losses that can be extensive. These pressure losses are directly linked to the p. 20 energy consumption of the high vacuum extraction system. As an example for sanders, at a flow rate of 80 m 3 .h -1 , the pressure loss varied between 10 kPa and 17 kPa. As an example, for circular saws, at a 130 m 3 .h -1 flow rate (order of magnitude of flow rate measured during our tests), the pressure loss values vary between 20 kPa and 35 kPa.

Sanders

Preliminary trials showed an increase in the concentration measured in the ventilated cabin when the suction flow was decreased from 80 m 3 .h -1 to 0 m 3 .h -1 (Table III). These measurements were made using a single sander. Later, only two suction flow rates were taken into account (80 and 40 m 3 .h -1 ).

After a first series of tests carried out to setup the measurement methodology, we performed systematic assays following the sanding cycle that uses two detection apparatuses (TEOM and APS). The total of the results are grouped in the following two graphs (Figures 6 and7). It can be seen that machine n° 1 (with an 80 m 3 .h -1 extraction flow rate) releases more dust than the three others (by a factor of 7 approximately). In Figure 7, it can be seen once again that reducing the vacuum strength (from 80 to 40 m 3 .h -1 ) diminishes the collection efficiency. Only machine n° 3 retained relatively high collection efficiency.

p. 21

From these measurements, we calculated the average and standard deviation for each machine tested, for the two flow rates. The results obtained using the two measurement apparatuses (TEOM and APS) were found to be homothetic.

 the relative positions of the different machines were identical, irrespective of the instrument used.

 the use of one measurement system or the other led to the same classification of the machines (low dust emitter versus high dust emitter).

By carrying out a variance analysis on the experimental values, the inter-machine variability could be calculated for 80 and 40 m 3 /h and this was found to be significant with respect to test reproducibility [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]. For that we used the Fisher test (F), which allowed two variances to be compared according to their ratio, provided that this ratio does not exceed a theoretical value (F CRITICAL in the Fisher table). For example, starting from measured values (80 m 3 .h -1 flow rate) with TEOM we calculated F TEOM ≃ 342, APS gave F APS ≃ 287, compared to an F CRITICAL of ≃ 3.4. For reference, the values with a 40 m 3 .h -1 flow rate are F TEOM ≃ 44 and F APS ≃ 223, compared to F CRITICAL ≃ 3.6. This confirmed the inter-machine variability and allowed an inter-comparison of the different machines.

p. 22

Circular saws

The sum of the results for a first set of five machines is regrouped in Figure 8 (first step).

Both the TEOM concentration values (black cross) and the APS concentration values (blue circle) are plotted for machines 1 to 5. As for the sanders, we carried out an uncertainty calculation to determine whether or not it was statistically possible to differentiate the machines from each other on the basis of dust emission measurement. Starting from this variance analysis, we calculated the inter-machine variability. For the values measured with the TEOM equipment the results were: F TEOM ≃ 90 and for APS: F APS ≃ 50, with an F CRITICAL value of ≃ 3.0. This allowed us to conclude that inter-machine variability is significant with respect to test reproducibility. For sawing tests, we observed a standard deviation of the plateau stability of about 10 to 50 % and regarding the test repetitions, we obtained a standard deviation ranging from 5 to 40 % (Figure 8). Despite these high standard deviations, the machines can be clearly discriminated from each other p. 23

• The tests were of a long enough duration (reduction of the impact of plate stability)

• The test protocol allowed a "satisfactory" reproducibility for the establishment of a classification (of the 5 machines, two "families" could be distinguished)

• Starting from the TEOM measurements, machines 1,3 and 5 generated much less dust than machines 2 and 4 did (a ratio of 8:1)

• For four of the 5 machines, the conclusion concerning the classification is identical irrespective of the experimental device used (TEOM or APS) Following the first series of tests with five circular saws, which allowed us to setup the measurement methodology, and to verify its robustness, we widened our trials by adding four additional machines to the five circular saws previously used (second step). The tests for the 9 circular saws were carried out and the results are shown in the following figures. To simplify the procedure, only the concentration values measured with TEOM are shown (for similarities in tendencies between TEOM and APS). The main results are shown in Figure 9 andTable V.

We have plotted the TEOM concentration values (blue circle) for several measurements done for machines 1 to 9. The extraction flow rates applied to the machines during the tests are also plotted (black triangle). This figure demonstrates that machines 1, 5 and 7 give lower p. 24 concentration values than the other machines. The ratio between the lowest average and the highest is approximately 7. Two groups of machines can be distinguished: one (low dust emitter) where the machines show an average below 1.5 mg.m -3 (between 0.6 and 1 mg.m -3 ) and the other (high dust emitter) where the machines have an average above 2 mg.m -3 (ranging from 2.4 to 4.4 mg.m -3 ). These values could be used for relative comparisons of tools. We carried out a variance analysis test for these 9 machines. From this, we calculated the inter-machine variability. The TEOM value is F TEOM ≃ 51 with an F CRITICAL value of ≃ 3.2. This allowed us to conclude, as above, that the inter-machine variability is significant compared to the reproducibility of the tests.

The average concentration values are regrouped for each machine, with the corresponding vacuum flow rates (Figure 9). The flow values were between 109 and 134 m 3 .h -1 , whereas the emission of the machines varied by a factor of 7. In Table IV the values of dust concentration and the wood mass removed by the machines during the test cycle are summarized. The mass removed is in a range from 68 g to 106 g. There is no clear correlation between the dust concentration measured and the wood mass removed. As an example for machine n°1, concentration = 1 mg.m -3 and wood mass = 76 g, for machine n°4, concentration = 3.7 mg.m -3 p. 25 and wood mas = 76 g. This is due to the fact that some machine hoods are designed in order to capture more efficiently wood dust from the cutting process.

In order to perform more experimental testing, two circular saws were selected from the previous results: machine n°5, which is part of the lowest emissive group and machine n°2, which is part of the highest emissive group. We performed series of tests at different date and plotted the results of dust concentration measured with TEOM and APS equipment (Figure 10 and Table VI). The last points show the influence of the blade exchange from machine n°2 to machine n°5. No significant impact can be noted. The difference in terms of dust concentration measured with the two real-time instruments is clearly stated between the two machines.

p. 26

CONCLUSION

During this study, a methodology for the measurement of dust emission by hand-held electric wood-working machines was set-up. The results obtained show that it is possible to develop a classification of the machines for a given type of machining (sawing or sanding).

An analysis of the robustness of the methodology was carried out and this allowed the discrimination between different machines to be confirmed (circular saws and sanders). The measurement of dust concentration is according to the EN 1093 -3 standard and is simple and accessible for the manufacturers of hand-held machines, at the level of its implementation. We established a correlation between concentration measurements in conduits with a particle counter (APS) and gravimetric measurements in the cabin (TEOM).

The studies using circular saws initially concerned tests on 5 machines, to confirm the stability of the measurement method. Afterwards, 9 circular saws were tested, allowing a classification of these machines to be developed. The studies concerning 4 orbital sanders validated the methodology for a second type of machining and a classification is presented.



  EN 1093 -1: Choice of test methods  EN 1093 -3: Method on the test bench to measure the emission rate of a pollutant  EN 1093 -8: Parameter of pollutant concentration, method on the test bench  EN 1093 -11: Decontamination index p. 10

  APS 3321 particle counter (Aerosol Particle Sizer -TSI Incorporated, Shoreview, MN 55126 USA) allows the real-time particles size distribution and emitted wood dust concentration to be obtained by the measurement of the time of flight of the particles [from 0.5 µm to 20 µm logging time 1s]. A microbalance (TEOM Series 1400 -Tapered Element Oscillating p. 16Microbalance -Thermo Electron Corporation, Greenbush, NY 12061, USA) measures the real-time (logging time 10s) total mass of dust deposited on a filter. The measurement flow rate is set to 2.5 l.min -1 and the concentration measured is the max. concentration analysable without a size-selective sampler. Three real-time particle counters (Lighthouse HH3016 -Worldwide solutions, Fremont CA 94538 USA) class the particles [from 0.3 µm to 25 µmlogging time 1s] into 6 channels and give a concentration value for each of these channels, as well as the overall concentration of the particles. The 5 sensors were placed in the ventilated cabin downstream of the dust source for measurements carried out on the sanders. For measurements pertaining to circular saws, the APS is positioned in the ventilation cabin extraction duct. Indeed, the emissions from sanders are much reduced compared to those from circular saws. The concentration figures measured by APS in the conduit for sanding tests would entail major uncertainties in measurements and would not necessarily allow different machines to be discriminated from each other. For sanders the quantity of dust that has to fly to the APS in the conduit coupled with the equipment size-selective property (max. 20 µm) will led to smaller values compared to what will be measured in the ventilated cabin downstream of the dust source. For the machining operations a moving average was used in order to establish the concentration values measured by the different real-time sensors. The p. 17

  average was done for a 600 sec. measuring time for sanding and a 180 sec. measuring time for sawing.

  Starting from measurements of dust concentration and classification of the machines, it is possible to propose recommendations to the profession. The importance of the vacuum flow imposed on the dust extraction nozzles present on the different machines was confirmed.

FIGURE 1 :

 1 FIGURE 1: Ventilated cabin -View from above -Sanding.

FIGURE 2 :

 2 FIGURE 2: Ventilated cabin -View from above -Sawing.

FIGURE 3 :

 3 FIGURE 3: Granulometric distribution in terms of number of wood dust entities.Measurements were carried out with APS.

FIGURE 4 :

 4 FIGURE 4: Granulometric distribution in terms of wood dust volume. Measurements were carried out with APS.

FIGURE 5 :

 5 FIGURE 5: Concentration signals -Machine n°4 -40 m 3 .h -1

FIGURE 6 :

 6 FIGURE 6: Results for sanders (80 m 3 /h, paper 120).

FIGURE 7 :FIGURE 8 :

 78 FIGURE 7: Results for sanders (40 m 3 /h, paper 120).

FIGURE 9 :

 9 FIGURE 9: Concentration measurements and vacuum flow rate for 9 circular saws.

FIGURE 10 :

 10 FIGURE 10: Average concentrations for 2 circular saws.

TABLE I :

 I Sanders tested

	Brand	Reference	Diameter (mm)	Grit type
	Festool	WTS150/7	150	P120
	Bosch	GEX150	150	P120
	Black and Decker	KA191E	125	P120
	Makita	BO5041	125	P120
				p. 41

TABLE II :

 II Circular saws tested

	Brand	Reference	Blade	Blade	Number	Power	Max. rotational
			diameter	thickness	of teeth	(W)	speed
			(mm)	(mm)			(tr/min)
	Festool TS75EBQ	210	2.4	Z36	1600	3550
	Hitachi	C7BUY	190	2.0	Z18	1300	5500
	Mafell	KSP55F	210	2.4	Z36	1600	4400
	Metabo	K555	160	2.2	Z18	1200	5600
	Bosch	GKS65	190	2.0	Z18	1300	5000
	Makita	SP6000	165	2.2	Z48	1300	5800
	Ryobi	RWS	190	2.2	Z24	1600	5000
	Virutex SRI174T	160	2.8	Z28	1150	5500
	Dewalt	DW5520	165	2.0	Z48	1300	4200
							p. 42

TABLE III :

 III Dust concentration emitted as a function of the suction flow rate -Machine n°2

	Suction flow	Min	Max	Mean
	rate (m 3 /h)	Concentration	Concentration	Concentration
		(µg/m 3 )	(µg/m 3 )	(µg/m 3 )
	80	10	30	15
	40	100	300	200
	0	1000	10000	5000
				p. 43

TABLE IV :

 IV Average concentrations and wood mass removed for 9 circular saws

	Machine	Mean Concentration	Wood removed
		(mg/m 3 )	(g)
	1	1.0	76
	2	4.4	83
	3	2.4	91
	4	3.7	76
	5	0.8	68
	6	2.6	83
	7	0.6	76
	8	3.8	106
	9	3.1	83
			p. 44

TABLE V :

 V Measurement values for 9 circular saws

	Machine	Nomber of test	Concentration Mean concentration mg/m 3 mg/m 3	Standard deviation	LEV Q (m 3 /h)
		1	1,04
	1	2 3	0,97 0,96	0,98	0,046	115
		4	0,94
		1	4,47
	2	2 3	4,27 4,17	4,37	0,176	111
		4	4,55
		1	3,15
	3	2 3	2,23 2,48	2,43	0,537	134
		4	1,88
		1	3,65
	4	2 3	4,07 3,90	3,71	0,369	113
		4	3,22
		1	0,42
		2	0,61
	5	3 4	0,67 1,17	0,76	0,255	115
		5	0,84
		6	0,82
		6	1,57
	6	6 6	2,44 3,39	2,55	0,765	101
		6	2,82
		1	0,96
	7	2 3	0,77 0,55	0,64	0,286	132
		4	0,30
		1	4,60
	8	2 3	3,80 3,71	3,82	0,588	121
		4	3,17
		1	3,33
	9	2 3	3,10 2,85	3,06	0,206	129
		4	2,97

p. 45

TABLE VI :

 VI Additional measurement values for 2 circular saws

	Machine	Number of test	Concentration TEOM	Mean concentration TEOM	Standard deviation	Concentration APS	Mean concentration APS	Standard deviation	LEV
			µg/m 3	µg/m 3		µg/m 3	µg/m 3		m 3 /h
		1	4144			602
		2	4136			635
		3	3774			582
		4	3973			679
	2	5 6	4744 4247	4353	340	775 652	694	83	110
		7	4647			854
		8	4603			747
		9	4605			691
		10	4658			721
		1	769			166
		2	1126			221
		3	896			188
		4	567			130
		5	923			190
	5	6 7	1019 813	948	207	202 182	193	30	110
		8	858			174
		9	851			184
		10	1102			222
		11	1078			218
		12	1368			239

p. 27

Several flow rates were tested on the sanders and the corresponding concentration levels were measured. Possible optimization of machine protection hoods was observed and measurements of pressure loss established the progress points. This methodology allowed the development of systematic measurements of dust levels around hand-held machines. The information provided about the equipment concerning dust emission from hand-held machines will be useful for prevention and will make the choice of a machine showing good dust collection performances possible.

p. 28

ACKNOWLEDGEMENTS

The author designed and executed the study and has sole responsibility for the writing and content of the manuscript.