Sholom Schechtman

Stochastic Proximal Subgradient Descent Oscillates in the Vicinity of its Accumulation Set

We analyze the stochastic proximal subgradient descent in the case where the objective functions are path differentiable and verify a Sard-type condition. While the accumulation set may not be reduced to unique point, we show that the time spent by the iterates to move from one accumulation point to another goes to infinity. An oscillation-type behavior of the drift is established. These results show a strong stability property of the proximal subgradient descent. Using the theory of closed measures, Bolte, Pauwels and Ríos-Zertuche [6] established this type of behavior for the deterministic subgradient descent. Our technique of proof relies on the classical works on stochastic approximation of differential inclusions, which allows us to extend results in the deterministic case to a stochastic and proximal setting, as well as to treat these different cases in a unified manner.

Introduction

Let d be a positive integer, let X be a nonempty, closed and convex set and let f, g : R d Ñ R be two locally Lipschitz functions. In this note, we study the behavior of the stochastic proximal subgradient descent (SPGD):

x n`1 P prox γn g,X px n ´γn v n `γn η n`1 q ,

where prox γn g,X is the proximal operator for the function g on X (see Eq. [START_REF] Davis | Stochastic subgradient method converges on tame functions[END_REF] for a definition), pγ n q is a sequence of stepsizes, pη n q is a noise sequence and for each n P N, v n is in the set Bf px n q of Clarke's subgradients of f at x n .

Let N X pxq be the normal cone of X at x. It is known (see [START_REF] Davis | Stochastic subgradient method converges on tame functions[END_REF], [START_REF] Majewski | Analysis of nonsmooth stochastic approximation: the differential inclusion approach[END_REF]) that, under mild conditions on f , g and pη n q, every limit point of px n q is included in the set Z :" tx : 0 P Bf pxq `Bgpxq `NX pxqu. The proof leans on the seminal paper of Benaïm, Hofbauer and Sorin [START_REF] Benaïm | Stochastic approximations and differential inclusions[END_REF] (see also [START_REF] Benaïm | Dynamics of stochastic approximation algorithms[END_REF], [START_REF] Duchi | Stochastic methods for composite and weakly convex optimization problems[END_REF]), which analyzes Eq. ( 1) as an Euler-like discretization of the differential inclusion (DI): 9 xptq P ´Bf pxptqq ´Bgpxptqq ´NX pxptqq .

(

) 2
While the sequence px n q is known to converge to Z, recent work [START_REF] Rios-Zertuche | Examples of pathological dynamics of the subgradient method for lipschitz path-differentiable functions[END_REF] shows that in principle, it might not converge to a unique point. In [14, Section 2] Ríos-Zertuche considers the deterministic subgradient descent (that is to say g " 0, X " R d , η n " 0) and constructs f , which verifies main assumptions of nonsmooth optimization (such as Whitney stratifiability or Kurdyka-Łojasiewicz inequality) but the limit set of px n q is equal to Z " tx : x " 1u. This encourages a more precise study of Eq. [START_REF] Aubin | of Grundlehren der Mathematischen Wissenschaften[END_REF].

In [START_REF] Bolte | Long term dynamics of the subgradient method for lipschitz path differentiable functions[END_REF] the authors, using the theory of closed measures, show that in the case of the deterministic subgradient descent the convergence to Z arises in a structured manner. First, they prove that if x, y are two distinct accumulation points of px n q, then the time that the iterates spend to get from a neighborhood of x to a neighborhood of y goes to infinity. Second, in a first approximation their results imply that if x is an accumulation point of px n q, then

ř n i"1 γ i v i 1 x i PBpx,δq ř n i"1 γ i 1 x i PBpx,δq ÝÝÝÝÑ nÑ`8 0 ,
(see [START_REF] Bolte | Long term dynamics of the subgradient method for lipschitz path differentiable functions[END_REF]Th. 7] or Section 3 for a precise statement). Intuitively speaking, this means that even if x n ´x0 " ř n i"0 γ i v i does not converge, on average, the drift coming from the subgradients compensate itself and vanishes at infinity. This behavior captures an oscillation phenomenon of the iterates around the critical set. Results of this type show a strong stability property of the deterministic subgradient descent.

In practical settings, when the function f is either unknown or computation of its gradient is expensive, the deterministic gradient descent is often replaced by its stochastic version, in many cases, this may lead to a faster convergence (see e.g. [START_REF] Bottou | Optimization methods for large-scale machine learning[END_REF]). Proximal methods, on the other hand, along with the regularizer function g, are widely used to regularize the initial problem of minimizing f . Depending on the choice of g, we can, for instance, preserve the boundedness of the iterates [START_REF] Duchi | Stochastic methods for composite and weakly convex optimization problems[END_REF] or promote the sparsity of solutions [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]. It is therefore interesting to establish stability results of the type [START_REF] Bolte | Long term dynamics of the subgradient method for lipschitz path differentiable functions[END_REF] for the SPGD.

In this work we investigate further the questions of oscillations of the SPGD. Our contributions are threefold. First, we show that the time spent by the SPGD to move from one accumulation point to another goes to infinity. Second, we establish an oscillation-type behavior of the drift. These two results extend [START_REF] Bolte | Long term dynamics of the subgradient method for lipschitz path differentiable functions[END_REF]Th. 7.] to a stochastic and a proximal setting. Finally, our technique of proof doesn't rely on the theory of closed measures used in [START_REF] Bolte | Long term dynamics of the subgradient method for lipschitz path differentiable functions[END_REF] but is build upon the classical works on stochatic approximation of differential inclusions ( [START_REF] Davis | Stochastic subgradient method converges on tame functions[END_REF], [START_REF] Duchi | Stochastic methods for composite and weakly convex optimization problems[END_REF], [START_REF] Benaïm | Stochastic approximations and differential inclusions[END_REF]). We feel that this approach gives a simpler proof and allows us to treat the deterministic, the stochastic and the proximal cases in a unified manner.

Paper organisation. In Section 2, we recall some known facts about Clarke subgradient, path differentiable functions and differential inclusions. Our main results are given in Section 3. Section 4 is devoted to proofs.

Preliminaries

Notations

For S Ă R d , we denote cl S its closure and conv S its closed convex hull. For a function F : R d Ñ R, we denote ∇F its gradient. Constants will usually be denoted as C, C 1 , C 2 . . . , they can change from line to line. For a sequence px n q, we denote acctx n u its set of accumulation points. The space of continuous functions from R `to R d will be denoted as CpR `, R d q, we endow this set with d the metric of uniform convergence on compact intervals, that is to say dpx n , yq ÝÝÝÝÑ nÑ`8 0 ðñ @T ą 0, sup hPr0,T s

x n phq ´yphq ÝÝÝÝÑ nÑ`8 0 .

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Perturbed solutions of differential inclusions

We say that H : R d Ñ R d is a set valued map if for each x P R d we have that Hpxq is a subset of R d . Consider the DI:

9 xptq P Hpxptqq . (4)
We say that an absolutely continuous curve (a.c.) x : R `Ñ R d is a solution to (4) starting at x P R d , if xp0q " x and Eq. ( 4) holds for almost every t ě 0, we denote S Hpxq the set of these solutions and S H " Y xPR d S H pxq. We say that H is upper semi continuous at a point x P R d if for every U a neighborhood of Hpxq, there is δ ą 0 such that y ´x ď δ ùñ Hpyq Ă U . We say that H is upper semi continuous (usc) if it is upper semicontinuous at every point. We have the following existence result.

Theorem 1 ([1]

). Assume that, for each x in R d , Hpxq is nonempty, convex and compact, and there is a constant C ě 0 s.t. supt v : v P Hpxqu ď Cp1 ` x q. Assume that H is usc, then for every x P R d , the set S Hpxq is nonempty.

For a set-valued map H and δ ą 0, we denote H δ pxq " tv P Hpyq : y ´x ď δu. In this work we will be interested in perturbed solutions of the DI associated to H.

Definition 1 ([3]). Assume that H " ř l i"1 H i ,
where each H i is a set valued map from R d to R d . We say that an a.c. curve X P CpR `, R d q is a perturbed solution of the DI associated to H if the following holds. i) There is a function b : R `Ñ R `and a locally integrable function ρ : R `Ñ R d s.t. for almost every t ě 0, we have:

9 Xptq ´ρptq P l ÿ i"1 H bptq i pXptqq .
ii) lim tÑ`8 bptq " 0 .

iii) For every T ą 0, we have:

lim tÑ`8 sup 0ďhďT ż t`h t ρpuq du " 0 .
The following theorem states that, in some sense, when t goes to infinity a perturbed solution shadows a solution of the corresponding DI. Its proof can be found in [3, Proof of Th.

4.2].

Theorem 2. Let X be a perturbed solution associated with H " ř l i"1 H i and assume that each of the H i satisfies the assumptions of Th. 1. Then the family tXpt `¨q : t P R `u is relatively compact (in CpR `, R d q) and we have:

lim tÑ`8
dpXpt `¨q, S H q " 0 Remark 1. Strictly speaking, in [START_REF] Benaïm | Stochastic approximations and differential inclusions[END_REF], a perturbed solution to the DI 9

xptq P Hpxptqq was required to satisfy 9

Xptq ´ρptq P H bptq pXptqq, where H " ř l i"1 H i . Nevertheless, the proof of [3, Th. 4.2] goes through with our definition.

Given a set A, we say that a continuous function ϕ : R d Ñ R is a strict Lyapunov function for the DI (4), if for every t ą 0 and x P S Hpxq , we have that ϕpxptqq ă ϕpxq if x R A and ϕpxptqq ď ϕpxq otherwise. If such a ϕ exists, then more can be said about the behavior of a perturbed solution.

Theorem 3 ([3, Th. 3.6 and Prop. 3.27]). Assume that we are in the context of Th. 2 and let ϕ be a strict Lyapunov function for a set A Ă R d . Assume, moreover, that ϕpAq is of empty interior, we have:

L X :" č tě0 cl tXpuq : u ě tu Ă A and ϕ is constant on L X .
In this note, we will be primarily interested in two particular set valued maps. Consider f : R d Ñ R a locally Lipschitz function. By Rademacher's theorem, f is differentiable almost everywhere. The set Bf pxq of Clarke subgradients of f at x is defined as follows:

Bf pxq " convtv : there is a sequence x i Ñ x s.t. f is differentiable at x i and ∇f px i q Ñ vu .

(5) The set tx : 0 P Bf pxqu of Clarke-critical points contains local extrema (see [START_REF] Clarke | Nonsmooth analysis and control theory[END_REF]). The map Bf : R d Ñ R d is usc and for every x in R d , Bf pxq is nonempty, compact and convex. Given a convex set X Ă R d , the normal cone of X is a set valued map N X : R d Ñ R d , defined as:

N X pxq " tv : xv, y ´xy ď 0, @y P X u .

For each x P X , N X pxq is a closed convex subset of R d .

Semialgebraic and definable functions

An important case to which our results apply, is when f, g and X are semialgebraic, or more generally definable. We say that a set A Ă R N is semialgebraic if it can be written as a finite union and intersection of sets of the form tx : P pxq ď 0u, where P : R N Ñ R is some polynomial. A function is semialgebraic if its graph is a semialgebraic set. While they may be nonsmooth, semialgebraic functions present strong regularity properties. Among other things, they are C k differentiable on a dense open set (for any k ě 0) and stable under many elementary operations such as composition, sum, multiplication. A generalization of this notion, which preserve the aformentioned structural properties, is the one of definableness in an o-minimal structure. While we will not mathematically define this notion here, let us mention that any semialgebraic function, as well as the exponential and the logarithm are definable (hence, also their composition). This explains their ubiquity in the optimization literature. Up to our knowledge, the first work to exploit the link between optimization and definableness was [START_REF] Bolte | Clarke subgradients of stratifiable functions[END_REF]. An interested reader can find more on definability and its usefulness in optimization in [START_REF] Ioffe | An invitation to tame optimization[END_REF], and more details in [START_REF] Van Den Dries | Geometric categories and o-minimal structures[END_REF], [START_REF] Coste | An Introduction to O-minimal Geometry[END_REF], [START_REF] Davis | Stochastic subgradient method converges on tame functions[END_REF].

Path differentiable functions

We say that a locally Lipschitz function f : R d Ñ R is path differentiable if for any a.c. curve x : r0, 1s Ñ R d , for almost every t P r0, 1s: pf ˝xq 1 ptq " xv, 9

xptqy @v P Bf pxptqq.

By [4, Proposition 2], every convex, concave, semialgebraic or definable function is path differentiable. Moreover, if another function g : R d Ñ R is path differentiable, then f `g is also path differentiable [4, Corollary 4]. From a similar point of view, if X is a convex set, then for any a.c. curve x : r0, 1s Ñ R d , for almost every t P r0, 1s:

xv, 9 xptqy " 0 @v P N X pxptqq .

Consider now f, g : R d Ñ R path differentiable, X Ă R d a convex set and x a solution to the DI (2). Using Eq. ( 7) and ( 8) and the fact that Bpf `gq Ă Bf `Bg, we obtain pf `gqpxptqq ´pf `gqpxp0qq " ´ż t

0 9 xpuq 2 du . (9)
This implies that pf `gqpxptqq ă pf `gqpxp0qq if xp0q R Z. In other words, f `g is a strict Lyapunov function for the DI (2).

Main results

Consider pΩ, Ξ, Pq a probability space and pη n q a sequence of random variables with values in R d . Define prox γ g,X : R d Ñ R d , the proximal operator for g on X with a step γ:

prox γ g,X pxq " arg min yPX tgpyq `1 2γ y ´x 2 u . (10)
We study Eq. ( 1) under the following assumptions.

Assumption 1.

i) The set X is a closed convex subset of R d .

ii) The functions f, g : R d Ñ R are locally Lipschitz continuous.

iii) There is a filtration pF n q nPN , such that pη n q is a martingale difference sequence adapted to it, and x n is F n measurable for every n P N.

iv) The sequence of stepsizes pγ n q is nonnegative and such that ř `8 i"0 γ i " `8.

Note that if g is nonconvex, prox γ g,X pxq is a set in R d . However, as soon as x n`1 is chosen in a measurable manner (relatively to η n`1 and x n ), px n q will be adapted to pF n q. Such a choice is always possible (see e.g. [START_REF] Davis | Stochastic subgradient method converges on tame functions[END_REF]).

By [15, 10.2 and 10.10], we can rewrite Eq. (1) as:

x n`1 " x n ´γn pv n `vg n `vX n q `γn η n`1 , (11)
where v g n P Bgpx n`1 q and v X n P N X px n`1 q.

Assumption 2.

i) Almost surely, sup n x n ă `8.

ii) There is q ě 2 such that

`8 ÿ i"0 γ 1`q{2 i ă `8 , (12)
and, for any compact set

K Ă R d , sup nPN Er η n`1 q 1 xnPK |F n s ă `8 . (13)
Assumptions of this type are standard in the field of stochastic approximation. Assumption 2-(i) prevent the algorithm to diverge. Note that it is superfluous if X is compact. Otherwise it can be obtained by a proper choice of the regularizer g (see [START_REF] Duchi | Stochastic methods for composite and weakly convex optimization problems[END_REF]).

Let τ n " ř n i"1 γ i be the discrete time of the algorithm. Define the linearly interpolated process X P CpR `, R d q by: Xptq " x n `t ´τn γ n`1 px n`1 ´xn q for τ n ď t ă τ n`1 .

Following [START_REF] Benaïm | Stochastic approximations and differential inclusions[END_REF] we will show that X is a perturbed solution of the DI (2). The next two assumptions ensure us that f `g will be a Lyapunov function for the DI (2).

Assumption 3. The functions f and g are path differentiable. Assumption 4 is a classical Sard-type condition. It ensures the fact that if x is a solution to the DI (2), with xp0q P Z, then x is constant. As established in [START_REF] Bolte | Clarke subgradients of stratifiable functions[END_REF], it is satisfied as soon as f, g and X are definable.

The next two propositions are not new and can be found in one way or another in e.g. [START_REF] Davis | Stochastic subgradient method converges on tame functions[END_REF], [START_REF] Bolte | Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning[END_REF], [START_REF] Majewski | Analysis of nonsmooth stochastic approximation: the differential inclusion approach[END_REF], [START_REF] Bolte | Long term dynamics of the subgradient method for lipschitz path differentiable functions[END_REF]. Nevertheless, since our set of assumptions is slightly different and their proof is a simple application of Section 2.2, for completeness, we include it in Section 4.1. Proposition 4. Let Assumptions 1 and 2 hold, then the family pXpt `¨qq tě0 is relatively compact. Moreover, if a sequence t n Ñ `8 and x P CpR `, R d q is such that dpXpt n `¨q, xq Ñ 0, then x is a solution to the DI (2). Proposition 5. Under Assumptions 1-4, the set acctx n u is included in Z and f `g is constant on acctx n u.

The next theorem tells us that even if acctx n u is not a single point, the time that it takes to px n q to go from one accumulation point to another goes to infinity. This is an extension of [6, Th. 6.i), Th. 7.i)], to the best of our knowledge this result is new in a stochastic and proximal setting. Theorem 6. Let Assumptions 1-4 hold. Let x, y be two distinct points in acctx n u. Consider two sequences n i , n j , with n i ď n j , such that x n i Ñ x and x n j Ñ y. Then τ n j ´τn i Ñ `8.

Under Assumptions 1-3, the same result is true if pf `gqpxq ď pf `gqpyq.

As it is shown in [START_REF] Rios-Zertuche | Examples of pathological dynamics of the subgradient method for lipschitz path-differentiable functions[END_REF], it is possible that acctx n u is not reduced to a unique point. Nevertheless, Th. 6 implies that the "nonconvergence" happens in a very slow manner. Asymptotically, the time spent by the algorithm to move from one accumulation point to another goes to infinity.

We now investigate the question of oscillations. Given U, V two open sets, such that cl U Ă V , we will call I " rn 1 , n 2 s a maximal interval related to U, V if the set X n 2 n 1 :"

tx n 1 , x n 1 `1, . . . , x n 2 u is such that X n 2 n 1 Ă V , X n 2 n 1 X U ‰ H and either x n 1 ´1 or x n 2 `1 is not in V .
The next two results are an extension of [START_REF] Bolte | Long term dynamics of the subgradient method for lipschitz path differentiable functions[END_REF]Th. 7] to a stochastic setting.

Theorem 7 (Long intervals). Let Assumptions 1-4 hold. Consider x P acctx n u and U, V two neighborhoods of x such that cl U Ă V . For i P N, denote I i " rn i 1 , n i 2 s a sequence of distinct maximal intervals related to U, V . Then, either one of I i is unbounded or τ n i 2 ´τn i 1 Ñ `8.

Theorem 8 (Oscillation compensation). Let Assumptions 1-4 hold, and fix U , V and I i as in Th. 7. Denote A " Ť I i , then

ř n i"1 γ i pv i `vg i `vX i q1 A px i q ř n i"1 γ i 1 A px i q ÝÝÝÝÑ nÑ`8 0 . ( 14
)
Th. 8 gives an intuitive explanation of why Th. 6 holds. Indeed, while the drift coming from one iteration v i `vg i `vX i might not go to zero (as it happens for such a simple example as f pxq " x , g " 0 and X " R d ), on average, it compensates itself. Th. 6 and 8 suggest that the algorithm oscillates around its accumulation set, while the center of these oscillations moves in acctx n u with a vanishing speed.

Let us finish with a remark on the Eq. ( 14). At first sight, maximal intervals in Th. 8 and Th. 7 may seem artificial. A more satisfactory result would be

ř n i"1 γ i pv i `vg i `vX i q1 U px i q ř n i"1 γ i 1 U px i q ÝÝÝÝÑ nÑ`8 0 , ( 15
)
where U is an open neighborhood of an accumulation point x. Looking at the proof of Th. 8, to obtain Eq. ( 15), we could think of defining maximal intervals as

I i " rn i 1 , n i 2 s such that tx n i 1 , . . . , x n i 2 u Ă U and x n i 1 ´1, x n i 2 `1 R U .
Unfortunately, for this type of intervals we dont have an equivalent of Th. 7, i.e. it may very well be that the quantity τ n i 2 ´τn i 1 is bounded. Actually, it is not very hard to show, that for the function from [14, Section 2], there are x, U such that Eq. ( 15) is false. Nevertheless, as explained in [START_REF] Bolte | Long term dynamics of the subgradient method for lipschitz path differentiable functions[END_REF], Eq. ( 14) is a good approximation of Eq. ( 15). Indeed, apply Th. 8 with U and V " U δ , where U δ " ty P R d : Dz P U, z ´y ă δu, then, as an approximation, we have

lim δÑ0 lim nÑ`8 ř n i"1 γ i`1 pv i `vg i `vX i q1 A px i q ř n i"1 γ i`1 1 A px i q « lim nÑ`8 ř n i"1 γ i`1 pv i `vg i `vX i q1 U px i q ř n i"1 γ i`1 1 U px i q .

Proofs

In the following we will denote x n`1{2 " x n ´γn v n `γn η n`1 and N pT, nq " inftj ě n s.t. τ j ´τn ě T u .

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Proof of Prop. 4 and 5

To put ourselves in the context of Section 2.2 we need to alter the map ´Bf ´Bg ´NX in a way that it verifies assumptions of Th. 2 and 3. While this section is slightly technical, conceptually, we just find a set-valued map G verifying assumptions of Th. 3 and s.t.

x n`1 P Gpx n q. A convinced reader may want to skip to Section 4.2.

We start with two technical lemmas.

Lemma 1. Under Assumptions 1 and 2, almost surely, for every T ą 0, we have:

lim nÑ`8 sup nďjďN pT,nq j ÿ i"n γ i η i`1 " 0 . ( 17
)
As a consequence, the sequence p x n`1{2 q is almost surely bounded.

Proof. Indeed, since almost surely sup x n ă `8, for each δ ą 0, there is C ą 0 s.

9 xptq P ´B f pxptqq ´Bgpxptqq ´r N X pxptqq , (18)
and that x remains in Bp0, Cq X X , then x is a solution to the DI (2). Finally, denoting r Z " tx : 0 P B f pxq `Bgpxq `r N X pxqu, we have the equality r Z X Bp0, Cq " Z X Bp0, Cq.

Proof. Let Π C`1 : R d Ñ R d be the projection on Bp0, C `1q. Define f pxq " f pΠ C`1 pxqq, gpxq " gpΠ C`1 pxqq. By construction, we have that v n P B f px n q and v g n P Bgpx n`1 q and that v n , v g n are bounded by L f , L g the Lipschitz constants of f and g. Hence, since x n`1{2 is bounded, there is

C 2 s.t. supt v X n : n P Nu ă C 2 . Defining r N X pxq " tv : v ď maxpC 2 , L f , L g q, v P Π X pxqu
, where Π X is a projection on X , proves the first claim. The two other statements immediately follow from our construction.

To prove Prop. 4 it remains to show that X is a perturbed solution to the DI (18). For t P rτ n , τ n`1 q, we define ρptq " η n`1 and bptq " x n`1 ´xn . The condition on ρ immediately follows from Lemma 1. The condition on b follows from the following lemma. Proof. By Lemma 1, we have that x n`1{2 ´xn ÝÝÝÝÑ nÑ`8 0, moreover, we have:

gpx n`1 q `1 2γ n x n`1 ´xn`1{2 2 ď gpx n q `1 2γ n x n ´xn`1{2 2 .
Therefore,

1 2γ n x n`1 ´xn 2 ď gpx n q ´gpx n`1 q ´1 γ n xx n`1 ´xn , x n ´xn`1{2 y ď x n`1 ´xn ˜Lg ` x n ´xn`1{2 γ n ¸, and
x n`1 ´xn ď γ n L g ` x n ´xn`1{2 , which finishes the proof.

To finish the proof of Prop. 4 consider t n Ñ `8 and x s.t. dpXpt n `¨q, xq Ñ 0. Then, by Th. 2, x is a solution to the DI (18), moreover, it remains in Bp0, Cq X X , therefore, it is also a solution to the DI (2).

For the proof of Prop. 5, notice that f `g is path differentiable (as a composition of path differentiable functions). Then, in the same way as in Section 2.4, we have that f `g is a strict Lyapunov function for the DI (18) and for the set Z. Since acctx n u " L X Ă cl Bp0, Cq, by Th. 3 we have that L X Ă r Z X cl Bp0, Cq Ă Z, and that f `g is constant on acctx n u.

Moreover, if for some h ě 0, pf `gqpxphqq " pf `gqpxp0qq, then xph 1 q " xp0q for every h 1 P r0, hs. If additionally Assumption 4 holds, then:

xphq " xp0q, @h P R `. (20)
Proof. By Prop. 4, x is a solution to the DI (2), and the first result follows by Eq. ( 9). Under Assumption 4, we have that xpR `q Ă acctx n u Ă Z, hence, by Prop. 5, we have that pf `gq ˝x is constant. Using Assumption 3, we have for all h P R `, 0 " pf `gqpxphqq ´pf `gqpxp0qq " ´ż h

0 9 xpuq 2 du . (21)
This implies that ş h 0 9 xpuq 2 du " 0. Hence, 9 xphq " 0 for almost every h P R `and we obtain Eq. (20).

Suppose that there is T ą 0 such that τ n j ´τn i ď T . The sequence Xpτ n i `¨q is relatively compact, and after extraction it converges to x a solution to [START_REF] Benaïm | Dynamics of stochastic approximation algorithms[END_REF]. Extract once again to have τ n j ´τn i Ñ h. Then Xpτ n j q ´Xpτ n i q Ñ xphq ´xp0q " y ´x , and we obtain a contradiction with Lemma 4.

Proof of Th. 7

The next lemma is the key ingredient for the proofs of Th. 7 and Th. 8. Proof. Suppose that we have ε ą 0 and two sequences n k and n k ď j k ď N pT, n k q, such that for n k large enough:

j k ÿ i"n k γ i pv i `vg i `vX i q ą ε .
This implies:

x j k ´xn k `jk ÿ i"n k γ i η i`1 ą ε .
Extract a sequence such that Xpτ n k `¨q converges to x and τ j k ´τn k Ñ h, with h ď T . Then x j k Ñ xphq and x n k Ñ xp0q, but xphq ´xp0q ě ε which is impossible by Lemma 4.

Suppose that no I i is unbounded, then we can choose n i P I i " rn i 1 , n i 2 s such that x n i P U . Since x n i 2 `1 is in V c , after extraction x n i Ñ y 1 and x n i 2 `1 Ñ y 2 , with y 2 ‰ y 1 , moreover:

τ n i 2 `1 ´τn i ´γn i 2 `1 ď τ n i 2 ´τn i 1 . (22)
By Th. 6, the first term of this inequality tends to infinity.

Proof of Th. 8

Take I i as in Th. 7, and A n " Ť iďn I i . Define u n " a n b n " ř γ i pv i `vg i `vX i q1 An px i q ř γ i 1 An px i q . Then,

u n`1 " a n `ř γ i pv i `vg i `vX i q1 I n`1 px i q b n `ř γ i 1 I n`1 px i q . ( 23
)
Fix ε ą 0, by Lemma 5, there is n 0 such that, for n ě n 0 , ř j k i"n k γ i pv i `vg i `vX i q ď ε. Decompose I i " rn i 1 , n i 2 s " Ť 1ďkďK i ra i,k , a i,k`1 s, with a i,1 " n i 1 and a i,k`1 " mintN pT, a i,k q, n i 2 u. We obtain:

u n`1 "
a n `řkďKn ř a n,k`1 i"a n,k γ i pv i `vg i `vX i q b n `řkďKn ř a n,k`1 i"a n,k γ i ď a n `pK n qε b n `pK n ´1qT .

By Th. 7, we have that K n Ñ `8 and, therefore, for n large enough:

u n`1 ď a n `2pK n ´1qε b n `pK n ´1qT .

Hence, by induction:

u n`j ď a n `2ε ř n`j´1 k"n pK i ´1q b n `T ř n`j´1 k"n pK i ´1q

.

Therefore, lim u n ď 2ε T . Since ε is arbitrary, this finishes the proof.

Assumption 4 .

 4 The set of Clarke critical values tf pxq `gpxq : x P Zu has an empty interior.

Lemma 3 .

 3 Under Assumptions 1 and 2, almost surely, we have that x n`1 ´xn ÝÝÝÝÑ nÑ`8 0.

Lemma 5 .

 5 Under Assumptions 1-4, we have sup

  t. if we denote A " t@n P N x n ď Cu, then PpAq ą 1 ´δ. Define ηn`1 " η n`1 1 xn ďC , then Erη n`1 |F n s " 0 and sup nPN Er ηn`1 q s ă `8. Hence, by [2, Prop. 4.2], we have Lemma 2. Let Assumptions 1 and 2 hold. Let A P Ξ be a probability one set on which px n q and px n`1{2 q are bounded, and let C be a random variable s.t. x n ă C and C is finite valued on A. Then, for each ω P A, there are two globally Lipschitz functions g, f : R d Ñ R and a bounded set-valued map r N X : R d Ñ R d s.t. in Eq. (11) we have v n pwq P B f px n pwqq, v g n pwq P Bgpx n`1 pwqq and v X n pwq P r N X px n`1 pwqq. Moreover, if x is a solution to the DI:

	sup nďjďN pT,nq	ř j i"n γ i ηi`1 ÝÝÝÝÑ nÑ`8	0. Since δ is arbitrary, Eq. (17) follows.

Fundings

This work was supported by Région Île-de-France.


MD Margaux Petitjean

MD, PhD Émilie Béquignon

MD Françoise Zerah

MD Maxime Fieux

PhD Bruno Louis

MD, PhD André Coste

MD Sophie Bartier
email: sophie.bartier@aphp.fr

Covid-19 pandemic: do surgical masks impact respiratory nasal functions?

Keywords: Covid-19, face mask, nasal respiratory functions, rhinomanometry, nasal mucosa. Running head: surgical masks and nasal respiratory functions

Introduction

Because of Coronavirus disease 19 (COVID-19), the use of surgical masks has become a worldwide recommendation [START_REF] Kohanski | Review of indoor aerosol generation, transport, and control in the context of COVID-19[END_REF] , which has however encountered some resistance in the general population, justified by complaints of nasal discomfort 2 . To date, the functional and architectural impacts of wearing a surgical mask on nasal respiratory functions have not been investigated.

The objective of this prospective study was to evaluate any consequences of surgical mask use on nasal respiratory functions and geometry, measured by rhinomanometry, acoustic rhinometry and nasal compliance.

Methods

We conducted a prospective monocentric study including adult volunteers without septal perforations or complete nasal obstruction (medical and paramedical staff of a French teaching hospital, and patients scheduled for rhinomanometry) between January and April 2021.

Questionnaires and measurements were performed three times: i) T1, after 30 minutes of wearing a mask and with the mask still on, each volunteer was asked to complete a visual analogue scale (VAS) questionnaire of nasal discomfort symptoms (nasal obstruction, rhinorrhea, pruritus, shortness of breath, sweating and mouth breathing), followed by nasal entrance temperature anterior rhinomanometry, acoustic rhinometry and nasal compliance measurements immediately after the mask was removed; ii) T2, the volunteer then spent 30 minutes without a mask on, and, thereafter, completed the same questionnaires and measurements; and iii) T3, the volunteer put the mask back on and immediately underwent anterior rhinomanometry to assess the barrier resistance generated by the surgical mask itself (supplementary Figure 1). As recommended in the Riga conference consensus 3 , the logarithmic effective resistances (LReff) were chosen to evaluate the pressure flow recorded by anterior rhinomanometry (Low and very low LReff < 0.89 Pa.mL -1 .s ; High and very high LReff > 1.09 Pa.mL -1 .s). Methods are detailed in Supporting Information.

Results

For the 50 adult volunteers included, (characteristics in supplementary Table 1), the nasal obstruction VAS (/10) for the right nasal fossa was significantly higher at T1, compared to the VAS at T2: 1.8 ± 2.1 and 1.3 ± 1.9 respectively (p = 0.012). The VAS of other symptoms (headaches, mouth breathing, sweating, respiratory oppression, pruritus and rhinorrhea) (/10) were significantly higher at T1 than at T2 (Table 1). LReff at T1 were significantly reduced (right = 0.89 ± 0.33 Pa.mL -1 .s; left = 0.84 ± 0.37 Pa.mL -1 .s) compared to LReff at T2 (right = 1.12 ± 0.36 Pa.mL -1 .s; left = 1.09 ± 0.33 Pa.mL -1 .s) [Right and left nasal fossae: p < 0.001].

LReff with the masks re-donned (T3) were significantly increased (right = 1.64 ± 0.48 Pa.mL - 1 .s; left = 1.56 ± 0.46 Pa.mL -1 .s) compared to LReff at T2 [Right and left nasal fossae: p < 0.0001] (Figure 1).

The MCA-1 at T1 was significantly larger (right = 0.62 ± 0.21 cm²; left = 0.67 ± 0.21 cm²) than that at T2 (right = 0.48 ± 0.18 cm²; left = 0.52 ± 0.26 cm²) [Right and left nasal fossae: p < 0.0001] (supplementary Figure 2). Nasal compliance at C1 was significantly higher at T1 than T2 (0.02 ± 0.019 and 0.017 ± 0.013 cm²/cmH 2 0 respectively; p = 0.036) (supplementary Figure 3).

The temperature at the nasal entrance was significantly higher at T1 (31.2 ± 1.00 °C) than it was at T2 (27.9 ± 1.60 °C), p < 0.0001.

Discussion

Wearing a surgical mask provides positive effects on functional nasal respiratory parameters. Indeed, LReff were significantly reduced after wearing a mask for 30 minutes, while MCA-1 and compliance at the nasal valve were significantly higher. Rarely (if ever) the crosssectional nasal areas and rhinomanometry scores are off kilter when people are complaining about mask usage. People complained about subjective matters as expected [START_REF] Isaac | Correlations between acoustic rhinometry, subjective symptoms, and endoscopic findings in symptomatic children with nasal obstruction[END_REF] .

Scarano et al. reported increased humidity behind masks [START_REF] Scarano | Facial Skin Temperature and Discomfort When Wearing Protective Face Masks: Thermal Infrared Imaging Evaluation and Hands Moving the Mask[END_REF] and we reported significantly warmer temperatures (mean +3.4 °C). When the nose is exposed to cool, dry air, the nasal mucosa releases vasoactive amines and leukotrienes 6,7 that increase both superficial mucosal blood flow, which, in turn, leads to increased nasal resistances [START_REF] Hayes | Continuous nasal positive airway pressure with a mouth leak: effect on nasal mucosal blood flux and nasal geometry[END_REF] . When wearing a mask, the exhaled air, instead of being cooled away from the nasal vestibule, warms newly inhaled air. As a result, LReff and areas of sections are decreased when wearing a mask.

The VAS-assessed worsening of rhinologic symptoms when wearing a mask in our study is in line with the meta-analysis performed by Kisielinski et al. [START_REF] Kisielinski | Is a Mask That Covers the Mouth and Nose Free from Undesirable Side Effects in Everyday Use and Free of Potential Hazards?[END_REF] . These observations point to a discordance between the discomfort expressed by a subject when wearing a mask for 30 minutes (lower VAS score) and the aforementioned positive effects of doing so on the subject's nasal mucosa (significantly reduced LReff in between T2 and T1). This paradoxical result may be explained by the fact that the first LReff measurements (T1) were performed with no tissue interposing between the nasal fossa and the rhinomanometer. At T3, the rhinomanometer applied on the mask reduced the effective surface used to breath which explain the increase of LReff.

There are some limitations to our study. Firstly, we chose healthy volunteers as subjects, among which a majority of healthcare workers, more used to wear surgical masks for hours per day. Secondly, the measurements' sequence (T1, T2, T3) was not randomised. Indeed, regarding the worldwide sanitary crisis (mask required to enter the hospital), it was impossible to make measurements before any mask use. Prospective studies in patients with chronic rhinologic symptoms or other types of masks (cloth, surgical masks, N95, KN95, …) and about the longer-term impacts of wearing face masks and the related physiological changes on nasal mucosa are required.

Tables and figures

Table 1: VAS (visual analogic scale) evaluation of subjective symptoms with and without a mask for 30 minutes.

Legend: VAS scored for a maximum of 10. Values presented as mean ± standard deviation (SD).

  www.hal.inserm.fr/inserm-03676709Submitted on 24 May 2022

Figure 1 :

 1 Figure 1: Logarithmic values of the effective resistance of the entire breath (LReff) in

Table 1 :

 1 VAS (visual analogic scale) evaluation of subjective symptoms with and without a mask for 30 minutes.

	Evaluation item,	With a mask	Without a mask	p-value
	VAS	for 30 minutes	for 30 minutes
	Headache	0.72 ± 1.70	0.18 ± 0.77	0.009
	Mouth breathing	2.70 ± 3.18	0.46 ± 1.03	< 0.0001
	predominant
	Sweating	2.38 ± 2.91	0.22 ± 0.91	< 0.0001
	Respiratory	1.60 ± 2.24	0.10 ± 0.36	< 0.0001
	oppression
	Pruritus	1.28 ± 1.94	0.06 ± 0.31	< 0.0001
	Rhinorrhea	1.68 ± 2.29	0.48 ± 1.22	0.001

Legend: VAS scored for a maximum of 10. Values presented as mean ± standard deviation (SD).

This study was not sponsored by any external financial support.


Catalin I Pruncu
email: c.i.pruncu@bham.ac.uk.

T T Pham

A Dubois

M Dubar

L Dubar

Morphology of surface integrity as effect of cold forging of aluminum alloy

Keywords: aluminum alloy, cold forging, wear characteristics, FEM

Aluminum and its alloys represent a common raw material for components released through a cold machining routine (i.e. forging, cold heading and rolling process). They offer easy manufacturing and high performance of plastic strength, together with light weight, long lifespan and easy recycling; and are heavily used in the transport industry [1] [2]. However, during processing the sample/tool interfaces can generate sticking mechanisms that produce growth in the friction values and high wear rate, leading to irreversible damage of the workpiece surface.

The surface morphology provides characteristics that allow detection of the damage amplitude.

.

Optical non-destructive observations permit the detection of the quantity of wear elements as per material transfer from one side (specimen surface) to the other side (contactor surface). This paper presents the mechanisms of the adhesive layer, of material transfer, that are directly proportional to the new roughness stature. A robust Finite Element Method (FEM) analysis was embedded to establish a relationship between time, the plastic deformations, friction coefficients and surface defects.

Introduction

Lightweight and robust structures have proven useful for innovative applications within the automotive industry [START_REF] Farrahi | Stress-strain time-dependent behavior of A356.0 aluminum alloy subjected to cyclic thermal and mechanical loadings[END_REF]. A good surface finish during the manufacturing routine represents an important parameter that sustains the emergent trend of lightweight structures. Aluminum and its alloys may fit within these requirements, being an efficient solution in terms of processing and cost. Most studies reveal the great achievement of aluminum alloys, by using cold manufacturing (cold forging, cold heading and cold rolling process), as a principal raw material.

However, the contact conditions between tool and workpiece are vital in optimization of such processes. Tribological tests have been developed to better understand this complex phenomenon of friction and wear mechanisms during metal forming processes: for example, the ring compression tests, alternative ring-compression tests, pin-on-disc configuration, a modified cylinder-on-cylinder test with a spring loading system, and/or a new load-scanning test method, among others [START_REF] Noh | Deformation characteristics at contact interface in ring compression[END_REF], [START_REF] Robinson | Study on ring compression test using physical modelling[END_REF], [START_REF] Tan | Friction studies at different normal pressures with alternative ring-compression tests[END_REF], [START_REF] Podgornik | Comparison between different test methods for evaluation of galling properties of surface engineered tool surfaces[END_REF], [START_REF] Wang | A new forming method of solid bosses on a cup made by deep drawing[END_REF]. The contact pressure is a common parameter that affects the friction coefficients while substantial interface contact behavior occurs [START_REF] Shih | Effects of Contact Pressure and Strain on Friction in Sheet-Metal Forming[END_REF]. Friction stress at tool/workpiece interfaces depends on the local peaks of temperature that gradually increase as the sliding velocity rises and may be further accentuated by the increase of billet radius causing higher dissipation of friction energy [START_REF] Bay | A friction model for cold forging of aluminum, steel and stainless steel provided with conversion coating and solid film lubricant[END_REF] and of course of tool radii [START_REF] Sanchez | Experimental Investigation of Friction Effects Enhanced by Tool Geometry and Forming Method on Plane Strain Sheet Metal Forming[END_REF].The positioning of the surface and the reduction in sample height, at the contact interface, during the upsetting procedure also lead to an increase in the values of friction coefficients [START_REF] Azushimaa | Coefficient of friction at interface of lubricated upsetting process[END_REF]. Boric acid films which can be firmly adhered to clean aluminum surfaces by spraying their methanol solutions may provide extremely low friction coefficients (~0.04); thus, preventing direct contact and preserving the surface qualities [START_REF] Wei | Dry Lubricant Films for Aluminum Forming[END_REF].

Heinrichs et al. [START_REF] Heinrichs | Influence of tool steel microstructure on initial material transfer in metal forming -In situ studies in the SEM[END_REF], [START_REF] Heinrichs | Mechanisms of material transfer studied in situ in the SEM : Explanations to the success of DLC coated tools in aluminium forming[END_REF] applying multi-stage sliding tests, detected a gradual increase in the value of friction forces due to progressive sticking. It was found that the friction coefficient is directly related to the amount of workpiece material transferred to the tool which in turn depends on the microstructure of the tool steel. It appears that the coefficient of friction (COF) can increase with each pass, until it reaches saturation level. This progression is accompanied by a gradual increase in the number of transferred layers of material, which lead to roughening of the tool surface [START_REF] Heinrichs | New understanding of the initiation of material transfer and transfer layer build-up inmetal forming-In situ studies in the SEM[END_REF].

3 Kirkhorn et al. [18] reproduced the forming routine by a tribo-tester based on a flat-die strip drawing. Considering the contact area, normal pressure, sliding speed, lubricant and surface characteristics, they found that tribological behavior depends mainly on tool microstructure and the contact pressure is governed by lubrication conditions. [START_REF] Wang | A new forming method of solid bosses on a cup made by deep drawing[END_REF] confirmed this concept by performing compression drawing tests in order to obtain the formation of bosses. They found that it is possible to acquire the prominent form of "bosses" with moderate punch pressure using only one side of the blank lubricated. This report presents details concerning the adhesive layer of material transfer and its mechanism produced during the forming process. It is observed that the amount of surface covered by the material transfer depends on plastic deformation. The embedding of the numerical simulations allows further improvement of sticking mechanisms. A relationship between plastic deformations, friction coefficients and surface defect deposition was recognized.

Material characteristics

Samples made of aluminum alloy Al6082 with 2.0 mm thickness were considered in this research. Aluminum alloy 6082 is a medium strength alloy with good weldability, brazeability, corrosion resistance, formability and machinability. This grade is used in substitution to the conventional 6061 alloys in many structural applications in which improved mechanical properties are required [START_REF] Mocko | Dynamic Properties of Aluminium Alloys Used in Automotive Industry[END_REF]. Rods, bars and machining stocks, seamless aluminum tubing, structural profiles and custom profiles are widely used in transport and structural applications in which high stress resistance is essential [START_REF] Davies | Materials for Automobile Bodies[END_REF]. Mechanical properties of the Al6082 T6 studied are as follow: the Yield stress, Re ~280 (MPa); the Ultimate strength, Rm ~333 (MPa); the Young's modulus, E ~69 (GPa) and the Elongation at fracture, A ~12%. The chemical compositions of this material include Mg~0.60, Si~0.7, Fe~0.24, Cu~0.06, Mn~0.9, Cr~0.02, Zn~0.06, Ti~0.02.

The mechanical behavior of the Al6082 material was assessed via a quasi-static tension test, with a crosshead speed displacement of 0.5 mm/s. During the testing process, tensile strain developed in specimens was measured with an extensometer. The average stress-strain (-) curves derived from six experimental tests are plotted in Figure 1.

Parameters and set up

The nominal dimensions of the 46 specimens tested in this study are as follows: L=40 mm, l=10 mm, and h=2 mm, surrounded in Fig. 2. The geometry of the contactors was reproduced from [START_REF] Lazzarotto | Identification of Coulomb's friction coefficient in real contact conditions applied to a wire drawing process[END_REF]; except for the contour radius that is 20 mm. Upsetting sliding tests were performed with respect to the following conditions:

 penetration depth (contactor within the specimen), h: 0.04, 0.07, 0.095, 0.12, 0.18 mm;  sliding velocity, V S : 10mm/s (for 22 samples tested), 60mm/s (for 22 samples tested); The initial roughness of the contactor and of the specimen is shown in Figure 3. Both surfaces are orthotropic. USTs are performed so that the sliding direction corresponds to the longitudinal direction "L" of the specimens. It can be noticed that all surfaces have a uniform level of roughness asperities in any region of the inspected surfaces.


The surface of the specimen may become scratched during the UST. This involves damage accumulation on both specimen and contactor surfaces. The rate of surface degradation may be expressed as a function of penetration depth due to possible transfer of material from specimen to the contactor; more localised and severe degradation of the contactor surface will occur as a result of increasing penetration depth.

Very early damage of the sample surface may be avoided and the steady state friction phase can be detected performing tests when the specimens are lubricated (i.e. by molybdenum disulphide (MoS2) solid lubricant). This medium may reduce the direct contact between contactor and workpiece sample, and finally lead to a significant decrease of wear damage.

Numerical modelling

To obtain a better understanding of the plastic deformation activity produced at the tool/workpiece interface while forming was replicated, a finite element model analysis was created for each experimental test. Experimental results were used as boundary conditions for numerical simulations.

2D solid body models were modelled for the samples subjected to plane strain with nominal dimensions, as detailed in Section 3. The fixed stand and the contactor were modelled as rigid elements. The "model" was discretized into implicit CPE4R elements, having the element size of 0.1 mm determined via convergence analysis. In the numerical computation, the coefficient of friction was inserted from the experimental UST results according to [START_REF] Lazzarotto | Identification of Coulomb's friction coefficient in real contact conditions applied to a wire drawing process[END_REF].

A mathematical power curve =K×n MPa as indicated in the E646 ASTM International Standard [START_REF]Standard Test Method for Tensile Strain-Hardening Exponents[END_REF] was embedded to detect the parameters of the stress-strain curve, and later inserted in ABAQUS 6.12. Thus, the material strength coefficient K=498 and hardening exponent n=0.101 were used.

Statistical setup of plastic deformation

The statistical determination was verified by taking into account 22 (i.e. forming tests, due to the same range of plastic strain shown for both simulations; 10 and 60mm/s) trial experiments performed under the USTs. Plastic strain achieved in the vicinity of the contact zone ranged from 0.006 to 0.112. Sturges' rule was used to define "classes" of plastic strain [START_REF] Scott | Sturges' and Scott's Rules[END_REF] delimiting the large range of values. The classes were delimitated for a limited number as: k=1+3.332*Log (N), where N=22 (tests)

The range of plastic deformation Ep for different values was associated to an imposed penetration depth.

Ep= max - min [START_REF] Yoshimura | Precision forging of aluminum and steel[END_REF] Where:  max of 0.112 (i.e. a maximum of plastic deformation generated at the interface between contactor/specimen using a penetration depth h=180m);  min of 0.006 (i.e. a maximum plastic deformation generated at the interface between contactor/specimen using a penetration depth of h=40m).

The length of each grade is obtained from the relationship:

L=Ep / k (3)
The allocated classes, corresponding to upper and lower limits defined on the basis of Eqns. [START_REF] Moreira | Fatigue crack growth in friction stir welds of 6082-T6 and 6061-T6 aluminium alloys: A comparison[END_REF][START_REF] Yoshimura | Precision forging of aluminum and steel[END_REF][START_REF] Hase | Wear elements generated in the elementary process of wear[END_REF], that make possible the determination of the mean values and corresponding standard deviations are summarized in Table 1.

Assessment of surface integrity

Quantification of material transfer onto the contactor

The thickness of a local layer of material transferred on the tool was evaluated by inspecting the tool surface before and after the forming sequence. For this purpose, the maximum ten-point height (Rz) was determined using Eq. ( 4). In this measurement, the difference between the averages of the 5 highest and 5 lowest points on the surface was computed. Figure 4 shows a standard measure chain for a typical specimen investigated in this research.

5 ) 5 ... 2 1 ( ) 5 ... 2 1 ( V V V P P P Rz       (4)
The difference between the final and initial form of the profile thickness corresponds to the thickness of the layer, of aluminum material, transferred from the specimen to the contactor.

Figure 5 provides details of the amount of material transferred onto the contactor surface as a function of the range of plastic strain. It appears that the thickness of the transfer layer increases with plastic deformation. The amount of material deposited onto the tool surface may not be significant at low plastic strain, as in the deformation grades Ep1 to Ep4 (see Figure 5). Also, it seems that the transferred material spreads homogenously on the tool surface and does not depend on sliding direction.

Correlation between optical profilometry and SEM observations is of fundamental importance because it may provide fundamental insight regarding the UST procedures and allow the detection of when breaking of the lubricant film occurs. In particular, 3D-Optical profilomety may help in detecting the real thickness of the surface defect deposited on the tool surface. On the other hand, the SEM observations provide topographical and morphological details on the surface defect deposition. Details of major observations are introduced in Figure 6.

Figure 6 shows the 3D-profilomety and SEM patterns recorded for the contactor surface at different levels of plastic strain. The SEM topography of sliding tracks reveals some typical rolling marks achieved on the rolled aluminum surfaces, appearing as narrow and deep wear scars extended along the sliding direction. The aluminum rolled surfaces may generate as a strong presence of aluminum-iron oxides that could be a consequence of the mechanical mix of the removed and transferred material during the mechanical routine [START_REF] Ruiz-Andrés | Wear behaviour of aluminium alloys at slow sliding speeds[END_REF]. It can be seen that the amount of material transferred from the workpiece to the contactor increases with plastic deformation and almost covers the whole surface of the tool for the very large plastic deformation of grade Ep5 (see Figure 5 (g). The same behavior (i.e. massive materials transfer) was observed in case of dry forging (i.e. without lubricant), as shown in Figure 6 (h). Obviously, even for a small plastic deformation (Ep1) the SEM patterns starts to emphasize the wear adhesion as the primary type of defect encountered on the tool surface, in good agreement with [START_REF] Heinrichs | Influence of tool steel microstructure on initial material transfer in metal forming -In situ studies in the SEM[END_REF] studies that have indicated the aluminum adherence on the tool surface starts on the first contact. The transfer mechanism may develop through adhesion of aluminum to steel and delamination of aluminum particles which are subsequently smeared on the aluminum surface.

The initial interactions between aluminum and iron at asperity level and, with progressive transfer, it changes to that of aluminum and aluminum. This was very well reflected, in Figure 6, in the iron content present on the worn surface as found by SEM analysis. The edges of the plastically deformed material is cracking up leading of the detachment of a debris [START_REF] Pramila Bai | Effect of Load on Dry Sliding Wear of Aluminum-Silicon Alloys[END_REF]. In this process, the wear particles are transferred away from the center of the contact area leaving the underlying metal exposed. In case of smooth wear, the surface is enriched in Si but when galling occurs the silicon is strapped aside and the hard silicon surface is probably responsible to some degree for the damage, as well as iron transfer to the aluminum surface [START_REF] Spyros | Wear and Galling of 356-T6 Aluminum-on-Steel in Low Amplitude Reciprocating Sliding in the Presence of Synthetic Lubricants in HFC-134a Atmosphere[END_REF]. Figure 7 plots the values of coefficient of friction with respect to the level of plastic strain developed in zone II. An average friction coefficient value corresponding to a standard deviation evaluated for each level of plastic strain was computed. The MoS 2 lubricant used in the UST routine helps to reduce the value of the coefficient of friction from 0.5-0.7 (values determined from the two non-lubricated USTs.) to 0.06-0.15.

Evolution of friction

It is obvious, that the Coulomb friction coefficient has a low value as long as the plastic strain remains in the Ep1 to Ep4 grade range for an initial speed of 10mm/s, and is related to the thickness value of material transfer (max. 5 m) [START_REF] Dubar | Effects of Contact Pressure, Plastic Strain and Sliding Velocity on Sticking in Cold Forging of Aluminium Billet[END_REF]. This process may occur due to material transfer that is fully immersed in the MoS 2 "layer" (since the thickness of the lubricant is 10m) applied before initiating the UST process. The results are consistent with the findings of Masters Moreover, insignificant fluctuations in coefficient of friction values may derive from having imposed different process parameters, small variations of temperature, roughness discrepancies, or solid lubricant inhomogeneity. However, differences in coefficient of friction values were not found to be statistically significant.

When the processing speed is changed to 60mm/s, the plastic strain within the range Ep1 to Ep3 produces a gradual exponential growth of COFs followed by stabilization. The Ep5 plastic strain section reveals a new slight intensification of COF rate. A major statistical variation was achieved described by the large area of material transfer. Galling damage occurs when the lubricant fails to separate the two surfaces and adherence followed by the transfer of aluminum onto the steel surface takes place. In this case, aluminum wear and galling occurs via a process involving crack formation and propagation and delamination of particles from the metal surface.

The presence of a lubricant or anti-wear films prevent the hard contact metal-to-metal contact and damage of the aluminum surface [START_REF] Spyros | Wear and Galling of 356-T6 Aluminum-on-Steel in Low Amplitude Reciprocating Sliding in the Presence of Synthetic Lubricants in HFC-134a Atmosphere[END_REF].

The workpiece surface characteristics

The process of damage occurring on the tool surface may strongly affect the surface integrity of the workpiece surface. As long as the degradation of tool surface occurs, and the thickness of material deposition grows (as a proportion of plastic strain) on the top of the tool surface, the machined surface appears to become more and more degraded, showing visible craters and cracks. A consistent relationship between tool damage and workpiece integrity has been established, as emphasized in Figure 8.

The results acquired by 3D-optical profilometry, in Figure 8 a), show a map with the evolution of workpiece surface while the plastic strain increases. It seems that at low plastic strain (Ep1-Ep2) the macroscopic modification on the workpiece surface produces an isotropic structure that presents the same stature in all directions, especially for low speed. However, when the stain increases to Ep5 the orthotropic modification on the workpiece surface appears, as tool degradation evolves while increasing the material transfer, as per Ep3 to Ep5. The change in the machining speed may accelerate further the progress of surface degradation, as can be observed in Figure 8b) in respect to Figure 8c). The SEM observations permit a better interpretation of the structure morphology achieved by profilometry measurements. When the imposed penetration develops a range of plastic deformation that covers the Ep1-Ep2 interval, the workpiece surface processed presents homogenous morphology with a low value of roughness. However, the rough pitch rises in the Ep3 interval (see Figure 8a, b, andc for Ep3). The workpiece surface behaves almost orthotropically with a significant number of peaks and shallow valleys in the Ep4-Ep5 interval. The numerical simulation (in Figure 8d) provides insight as to how plastic deformation acts inside of the workpiece structure that is of vital importance to understand the evolution of surface integrity. The numerical model reveals that at low plastic strain, from Ep1 to Ep2 interval, a homogeneous material behavior occurs over the entire workpiece depth that means this loading produces only superficial modification of the workpiece surface. However, when the loading increases, the stresses tend to accentuate near the interface when the value of Ep reaches the grade Ep5; the contact generates large deformation at the interface between tool and workpiece and the sticking mechanism is very unpredictable, creating an orthotropic surface.

Corroborating the numerical simulation with the profilometry measures and SEM observations permits the validation of the numerical tool as a design solution to predict the surface integrity of the workpiece and thus predict with accuracy when the tool needs replacement.

Conclusion

This research highlights the relationships between equivalent plastic deformations developed in a workpiece during cold forming, friction coefficients, and material transfer caused by the hard surface interactions. A formal-numerical method was validated as a design solution to predict the surface integrity of the workpiece and thus predict with accuracy when the tool needs replacement. The experimental and numerical approaches used in this work prove that in the presence of lubrication (MoS2), the thickness of the aluminum transfer onto the tool surface may have very low effects on the workpiece surface integrity for a plastic strain in the range Ep1-Ep4; the average thickness of the aluminum layer transferred onto the tool surface is between 1.5 and 19.5 m; the coefficients of friction measured during the cold forming tests present a low value while the size of aluminum deposited onto the tool surface belongs to grade Ep4, especially for low speeds up to 10mm/s. In that case, the thickness of material deposition on the tool surface may be approximately 5m; friction coefficients show a critical value when the plastic deformation reaches the Ep5 interval and the tool surface presents a critical surface defect. This process may lead to significant wear surface events for both counterparts (tool/workpiece). The progress of wear on the tool surface can be attributed to the adhesion mechanism.

To further improve the understanding of the cold forming processes and to optimize the results may require the introduction of an extended approach focused on different geometrical and physical parameters (for reference; changing the tool shape/material, by using different types of lubricants (solid or liquid), processing temperature (from 23° to possibly 100°), multi-pass strategy, etc).

  True sliding distance: 40 mm;  Specimen and contactor temperatures: room temperature (≈ 20°C)  Mean arithmetic surface roughness of the contactors: Ra ≈ 0.3 µm;  Lubricant: dry test and lubricated MoS 2 (thickness of solid lubricant layer about 10 µm); Each test was performed with a new contactor.

  et al. [28] who measured a uniform value of coefficient of friction in strip draw tests on aluminum alloys for strain levels up to 0.10. Although solid wax lubricants were utilised in the studies by Masters et al. [28], the results gathered in the present study prove good correspondence. As expected, the coefficient of friction strongly increases as deformation grade evolves from Ep4 to Ep5, as a consequence of the major level of thickness of material transfer measured for domain Ep5 (practically 20 m).

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Stress-strain curve of Al6082 aluminum alloy: a) Engineering stress-strain curve; b) True stress-strain curve.

Figure 4 .

 4 Figure 4. Map of measurement profile a) theoretic curve, b) before test, and c) after test (in zone I, Ep1)

Figure 5 .Figure 6 .

 56 Figure 5. Variation of thickness of the material layer deposited on the tool surface in the UST: a)direction transverse to sliding and b) longitudinal (i.e. sliding) direction; b).

Figure 7 .Figure 8 .

 78 Figure 6. Comparison between 3D-optical profilometry (Figures 6a through 6d) and SEM (Figures 10e through 10h (i.e. red-aluminum transfer, and green-contactor surface), magnification 50X) patterns recorded for the contactor surface (zone I) at different grades of equivalent plastic strain with or without lubrication: Ep1 (a)-(e), Ep4 (b)-(f), Ep5 (c)-(g), No Lub. (d)-(h).

Table 1 .

 1 Class of plastic strains corresponding to different penetration depths

	Grade	Ep1	Ep2	Ep3	Ep4	Ep5	No Lubr.
	Strain	0.006-	0.027-	0.048-	0.069-0.09 0.09-0.112
	Range	0.027	0.048	0.069
	Strain mean	0.014	0.035	0.052	0.073	0.107	0.518
	value

Masters, I.G., Williams, D.K., Roy, R.,. 2013, Friction behaviour in strip draw test of prestretched alluminium alloys. International Journal of Machine Tools & Manufacture 73, 17-24.

Acknowledgements

The present research work has been supported by International Campus on Safety and Intermodality in Transportation, the Nord-Pas-de-Calais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, and the National Center for Scientific Research. The authors gratefully acknowledge the support of these institutions