IN WHAT SENSE IS KNOWLEDGE THE NORM OF ASSERTION?

Pascal Engel Université de Geneve Summary .

The knowledge account of assertion (KAA) is the view that assertion is governed by the norm that the speaker should know what s/he asserts. It is not the purpose of this article to examine all these criticisms nor to try to give a full defence of KAA, but only to defend it against the charge of being normatively incorrect It has been objected that assertion is governed by other norms than knowledge, or by no norm at all. It seems to me, however, that a number of these criticisms are based on a number of misunderstandings of the notion of a norm and of the way it can regulated a given practice. Once we spell out in what sense knowledge can play a normative role in this context, the KAA appears much more plausible.

1.

Preliminary formulations of the knowledge account

The knowledge account of assertion (KAA), although it has probably been present within philosophy at least since the Pyrrhonian sceptics1 , was introduced within contemporary analytic philosophy by G.E. [START_REF] Moore | Common place book[END_REF], revived by Michael Slote (1972) and Peter [START_REF] Unger | Ignorance[END_REF] and Timothy [START_REF] Williamson | Knowledge and its Limits[END_REF] 2 . It has since received as much approval as criticism. It consists in at least the following three claims:

(i) There is a norm for the speech act of assertion (ii) This norm is unique and constitutive of assertion (iii) This norm is that one must assert that P only if one knows that P Each of these claims can be contested. One can reject the idea that there is a norm for the practice of assertion: maybe there are no rules or norms for this practice, and just things that people say in various circumstances, which can all equally count as assertions. One can also reject the idea that if there is a norm it is unique and constitutive. There might be several kinds of norms, including ones which do not involve any epistemic position on the part of the asserter. And one can reject the knowledge norm itself, and propose a weaker one, while accepting that the norm of assertion is some epistemic status. The other obvious candidates are the weaker norms of truth and a norm of belief:

(KT) One must assert that P only if P is true (KB) One must assert that P only if one believes that P or perhaps some higher condition than mere belief, alternatively :

(KRB) One must assert that P only if one rationally believes that P (KWB) One must assert that P only if P is warranted (KPB) One must assert that P only if P is highly probable

In this article, I will not really discuss the issue of whether one should defend (KT) or one of the norms (KB), which all presuppose that the norm of assertion involve some epistemic status, low or high, on the part of the speaker. I shall be mainly concerned with the sense in which either of these can be said to a norm of assertion. For that, I shall also suppose that the norm for assertion is at least as strong as (KN). Even if we assume that (i)-(iii) are correct, there are, nevertheless, several formulations of (iii), which are not equivalent. The most basic one says that an assertion that P is correct if and only the speaker knows that P, hence obeys the prescription:

(KN) One must: assert that P only if one knows that P This is Williamson's (2000: 243) formulation. Moore's own version is distinct: "By asserting that P, you imply, though you do not assert, that you know that P (Moore 1962: 277).

( MKN) To assert that P is to imply that one knows that P Peter Unger's formulation differs too: "Asserting that something is so entails not just representing the thing as being so, but representing oneself as knowing that it is." (Unger 1975: 256) (SPKN) To assert that P is to represent oneself as knowing that P Let us call this, with [START_REF] Pagin | Is assertion social?[END_REF], the "self-presentation account". The self-presentation account is clearly the counterpart of similar claims which have been proposed for the belief account of assertion (KB). Thus Dummett has suggested:

"A man makes an assertion if he says something in such a manner as deliberately to convey the impression of saying it with the overriding intention of saying something true" (Dummett 1973: 300) and Davidson similarly :

"What is understood [by the hearer of an assertion by a speaker] is that the speaker, if he has asserted something, has represented himself as believing it -as uttering a sentence he believes true, then. But it is not a convention, it merely part of the analysis of what assertion is. To assert is, among other things, to represent oneself as believing what one asserts". [START_REF] Davidson | Communication and Convention[END_REF] The self-presentation analysis transposes this claim from belief to knowledge.

As one sees, these formulations are not equivalent and their differences can generate a lot of ambiguities. In the first place it is not clear whether they are supposed to give a defining feature of assertion, or at least to give a necessary condition of it, as (KN) does, or whether they are giving the condition of a successful assertion. On the first reading, a statement which would not conform to the (KN) norm would fail to be an assertion. On the second reading, it might be an assertion but a defective one. In the second place (KN) and the other two formulations (MKN) and (SPKN) differ in that the latter two do not require that the speaker actually knows that P, but only that he conveys to his audience the belief that he does, whereas the former does require that the speaker actually knows that P, and is compatible with the following:

(KNB) One must: assert that P only if one believes that one knows that P There is at least one version of the self-presentation account (SPKN) which is clearly defective, for reasons which have been clearly brought up by Peter [START_REF] Pagin | Is assertion social?[END_REF].

Suppose that a speaker represents herself as knowing that P by simply saying (1) I hereby represent myself as knowing that P But an utterance of (1) is obviously not an assertion that P, for conveying to an audience, successfully or not, the belief that one knows that P cannot amount to asserting that P.

When one asserts that P, what one says implies that P, and is incompatible with its falsity.

(1), however, is compatible with the falsity of P. This is not simply because one may lie:

one may commit oneself, through some sort of act -for instance a statement before a tribunal, which legally ties one to the truth of P, without it being the case that p is actually true. Asserting that P is more than committing oneself to some audience, through some sort of social commitment. In other words, it is something different from assuring someone that P. It is saying that P with the aim of saying something true, and not simply saying something which one represents to others as true, and that one believes as true. An actor on the stage might, in this sense, represent himself as knowing that P without asserting that P. For the same reason, the self-presentation account does not work when the speaker is supposed to commit herself to the belief that P:

(2) I hereby represent myself as believing that P is just as inadequate as (1) for the same reasons. The same goes for

(3) I hereby represent myself as committed to the truth that P which is no more an assertion that P than (1). 3This may have to do with the reason why Moore says that to assert that P is not to assert that one knows that P, but only to imply it. But it remains to be seen in what sense there is such an implication in assertion. The inadequacy of the formulations (1), ( 2) and (3) does not rule out the self-presentation account in itself, since there may be more adequate formulations of it (I shall come back to that below). I shall however, assume first that the correct account is (KN).

Familiar objections to KAA

Several kinds of arguments have been advanced in favour of (KN). The most familiar ones appeal to linguistic evidence. Asserting that P invites the question "How to do you know?" when the assertion is improper, or "You don't know that", when it is false.

Similarly asking the question whether P seems to imply that one ignores whether P. The point is that when a speaker asserts that P, and does not know that P, he can be criticised, or has to withdraw his assertion. Similarly if you ask whether P while you already know that P, you have in some sense failed to ask a question, or have violated Grice's maxim of quality. This linguistic evidence is not very compelling, though. In a number of contexts, utterances that P can elicit questions like: "What are your reasons for believing P?" or "Do you believe that?" which suggest that reasonable grounds for belief, or simply truth, are the proper presuppositions of assertion. And some questions are just ironical. For instance the following dialogues make perfectly good sense:

(4) "LakeLeman is polluted." -What are your reasons to believe this? (5) " Hillary Clinton will be the next president" -Do you believe this ? (6) "Have you not received the Nobel prize yet ?" -You are kidding! An argument based on Moore's paradox has also been advanced in favour of KN. It consists in remarking that sentences like (7) It is raining, but I do not know it are just as absurd as the familiar Moorean sentences such as "It is raining but I don't believe it" or " I believe it is raining but it's not". Indeed sentences like (7) are much more clearly absurd than these. But one may remark that (7') It is raining, but I do not have very good reasons for believing it.

does not make sense either. So why should the presence of "know" in ( 7) make the difference?

The lottery argument is perhaps the most discussed (Williamson 2000: 246-47). The point is that an assertion made only on the basis of a very high degree of belief can be straightforwardly rejected. In a fair lottery of 10 000 tickets where A has bought one ticket, if B says: "Your ticket did not win", A is entitled to assert : "But you do not know that !" Here too there is a defence in terms of rational belief. Someone could as well answer: "But you do not have enough grounds to believe this!"

Another related argument appeals to the role of knowledge in practical reasoning. [START_REF] Hawthorne | Knowledge and Lotteries[END_REF] presents the following piece of reasoning coming from an individual participating in a fair lottery:

(1) My ticket will not win (2) If I keep my ticket, I shall win nothing (3)

If I sell my ticket, I'll get 10 pence (4)

Hence I must sell my ticket

The reasoning is wrong, because the individual is not entitled to (1) in the first place, for his premise is not known. The point is that knowledge plays an indispensable role in practical reasoning. But here too it is not clear that rational belief could not be the appropriate norm: we seem entitled to affirm things of which we are not absolutely certain in the sense of knowledge [START_REF] Douven | Assertion, Knowledge and Rational credibility[END_REF].

Other objections to (KN) concentrate on the many circumstances where assertion seems to be proper even when one does not know that P [START_REF] Weiner | Must we Know what we Say ?[END_REF]. They include cases of lying (a liar does not even believe what he says, hence does not know it) , ironical assertions, gossips where one merely "says" that P in the coffee house without asserting it seriously, cases of assertions about future events ("The next American president will be Barack Obama") which we feel perfectly entitled to assert even though we do not know that they are true, cases of agents who are in a Gettier situation, where they are justified to believe something true merely by accident ("Nogot owns a Ford", " This is a barn façade"), or cases where the circumstances allow us to assert something for reasons which have nothing to do with any epistemic status. For instance, I may assert "That's your train!" in a circumstance where I am not sure that it is your train, although it is very important for you to take a train (if it were not your train, you would not be entitled to criticize me for not knowing) (Williamson 2000: 256). Or I may say to you, when you are worried by a very serious problem: "That's not a big deal", when I actually know that it's a big deal.

Another similar kind of objection involves situations where assertions are proper in the presence of other norms than purely epistemic ones. Jennifer [START_REF] Lackey | Norms of Assertion[END_REF] has presented cases of what she calls "selfless assertions", where a subject, for purely non epistemic reasons does not believe that P, but is aware that P is well supported by epistemic reasons, hence is led to assert that P, although he still does not believe that P, hence does not know that P. She gives the case of a racist juror who because of his prejudices is led to believe that a black man is guilty although he recognises the racist origin of his prejudice and asserts that the man is innocent, or the case of a creationist teacher who nevertheless is able to recognise the scientific evidence against her religious beliefs and is led to assert evolutionist claims. According to Lackey, in those cases, the agent, qua asserter is not subject to criticism -she is actually subject to praise for having surmounted her previous beliefs -and offers an assertion which is both true and epistemically flawless, although she can be criticised for her beliefs, and because of these, she does not have knowledge. The upshot of these examples is that it is a mistake to require proper assertion to pass through the doxastic states of the asserter. Notice also that they are meant to be counterexamples both to KN and to KB (or one of its version such as KRB), hence to any account of assertion according to which the asserter has to occupy a certain epistemic status.

It is not clear, however, that such cases are genuine counterexamples to KAA and to KN in particular. In the first place it is not clear that although the subjects have religious or racist beliefs which run counter their assertions, they do not make these assertions on the basis of their evidence and of their knowledge (of evolutionary theory, of the innocence of the black man). In the second place, it is not clear that these cases are not cases where someone accepts that P although he or she does not believe that P. One can accept, for pragmatic or for non epistemic reasons of some kind (profession, role), and hence assert that P, without believing that P. At this point both the defender of KAA and its critic can have an answer4 . On the one hand, the defender of KAA will want to say that to assert that P on the basis of an act of acceptance is merely to simulate assertion, or giving a fake assertion (not necessarily lying). On the other hand, the critic of the KAA will insist that these cases count as genuine assertions, although the subject is not in the appropriate epistemic status required by KAA or KB (or one of its versions). We meet again here the ambiguity already noticed above, between the claim that KAA states a norm for the correctness of assertions and the claim that it defines what an assertion is.

The critic of KAA can claim that the case of selfless assertions, although they do not conform to the norm of correctness of assertion, count as assertions nevertheless.

At this point is becomes important to turn to the issue which is implicit from the beginning: in what sense is assertion subject to a norm?

Which norm do we follow in assertion?

The objections above show that there is a strong disagreement on this point. Two problems have to be distinguished. The first one is whether knowledge (or for that matter some sort of high epistemic status) is the only norm of assertion. In other terms, are our reasons for asserting a given sentence only epistemic and are the reasons for our being criticized in our assertions? One of the main reasons adduced in favour of the idea that knowledge is the norm of assertion by its defenders is that assertions which are not made in presence of proper knowledge can be "criticised". But are our reasons to criticize an assertion only epistemic reasons? The second problem is the problem of the relationship between the norm and its application.

Timothy Williamson addresses the first problem by replying to the kind of counterexamples mentioned in the previous section involving cases of assertion made in the absence of knowledge or belief in the asserted sentence, such as lying, speaking non seriously or making "selfless"assertions:

"Such cases do not show that the knowledge rule is not the rule of assertion. They merely show that it can be overridden by other norms not specific to assertion. The other norms do not give me warrant to assert p, for to have such warrant is to satisfy the rule of assertion" (Williamson 2000: 256) The problem is whether an assertion can remain an assertion when it is uttered in the absence, or in the relative unimportance, of the epistemic status of the sentence uttered. Indeed the person who says "That's you train" intends to say this because there is an obvious cost in not asserting it, and because it is better in such cases to err on the side of prudence. There is no doubt that we make assertions for reasons which can be practical.

In many of these cases we make, as I noticed above, acts of pragmatic acceptance without belief or without knowledge. It is clear that there can be assertions made for such pragmatic reasons, just as there can be can be (pragmatic) reasons for wanting to believe distinct from ( epistemic) reasons for belief . There is a clear sense in which a belief which is held for reasons which fall short of being epistemic -for instance a self deceptive belief or one which we aim to have to secure a form of comfort-still counts as a belief, so why could not assertions which are made for reasons which fall short of being epistemic, or which happen to be epistemically weak fail to count as assertions?

Commenting upon the example of "That's your train" and of assertions out of Gettierised beliefs, Jennifer Lackey says: "There is a clear sense in which speakers who assert reasonably believed falsehoods and Gettierized beliefs are not subject to criticism. For the faultiness of the assertions in such cases result from infelicitous circumstances, not from any sort of blameworthy behaviour on the part of the asserters." ( Lackey 2007: 597) But the issue is precisely this: what kind of blameworthiness is at stake here? In the case of selfless assertions, the speakers are not blameworthy because they make assertions which are, although epistemically defective, perfectly proper. In the same manner, the liar, the joker, the ironist or the man who wants to reassure you by saying "That's no big deal" can be praised or blamed for their assertions for reasons why are distinct from their being epistemically valuable or not valuable. There is no doubt that they make assertions. This is not the point at issue, and we can here remove the first ambiguity noticed in § 1 above: the KAA does not say that it is a defining feature of assertion (a necessary and sufficient condition) that it is governed by the norm of knowledge. It is perfectly compatible with KAA that some assertions violate the norm, and that other norms or dimensions of evaluation of assertions come into play. But from the fact that such norms can come into play in a number of assertions it does not follow that the other norm, the KN (or for that matter the KB or KRB norms) ceases to operate.

Compare again with the contrast between reasons to believe and reasons for wanting to believe: the two can coexist or can conflict within a subject (for instance in selfdeception as it usually characterised), but the fact that it can be in some cases rational in a pragmatic sense to hold a belief for which one has only very weak epistemic reasons or evidence does not show that the main standard of evaluation of a belief is not truth or evidence. Similarly the fact that one can praise or blame an assertion in some cases for pragmatic reasons does not mean that the dimension of evaluation remains its epistemic status.

One might object that this answer relies too much on a parallel between belief and assertion, and that it is not a norm of assertion that it aims at truth or at knowledge contrary to the parallel claim about belief. 5 Indeed it seems that in the case of assertion, we are entitled to adopt whichever norm we intend to adopt, depending on our specific aims, hence that the norms here are hypothetical imperatives. In the case of belief, they may be categorical. But is it the case? Certainly the liar, the ironist or the friendly asserter do make assertions, and they make them in the absence of knowledge (and of belief) of what they assert. But does it means that the point of their assertions was not truth or knowledge? If the objective of asserting that P were not to aim at one's audience recognising the truth of P (or believing or knowing that P -the kind of epistemic norm is not here in question -) assertions would be pointless, even for the liar. Certainly the ties between belief and truth or knowledge are stronger than those between assertion and truth or knowledge, but it does not imply that they stop to operate in the case of assertion. The KN norm on assertion seems sometimes to be the view that our assertions should be so serious that only a Victorian clergyman -or an overly scrupulous scientist -could actually make assertions. But even the joker and the ironist have to recognise, if only implicitly the norm of truth and knowledge on assertion.

4..What it takes to obey the norm of assertion

But it is precisely at this point -and it our second problem with the very idea of a norm of assertion -that we may ask: is not the norm KN too strong? Who will be ever able to obey it? Isn't it actually so strong that no one will ever be able to apply it? This was in part what was involved in the lottery case: there are actually many propositions (perhaps most of our beliefs) upon which we can have only a very high degree of belief, and no certainty or knowledge: if it were required that we know them, we would be entitled to assert nothing, or most of our assertions would fail6 . Hence the KN norm would be trivial. After all, ought is supposed to imply can. Bernard Williams says:

It has been claimed that the norm attached to assertion is knowledge, in the sense that in asserting that p one represents oneself as knowing that p. The force of the claim might be thought to lie in the idea that one can be criticized for asserting what one does not know ("you should not have said this if you did not know"). This represents the criticism as a criticism of an assertor, but the speaker, after all, may not be in position himself to apply the norm effectively, because, at the point of asserting that p, he may reasonably think that he knows that p when he does not. In such a case, it is not reasonable to criticize him, in the sense, at least of blaming him for not having remained silent rather than speaking, or for not having said that p only in some qualified or doubtful way. It may be said, as it can in the case of many other rules, that he has broken the norm, but that in these circumstances he is not to blame for doing so. But then we still need to know the supposed norm applies, and what the consequences of its obtaining are supposed to be." (Williams 2002: 76-77) Here some defenders of the KAA account often use a distinction put forward by Keith DeRose between the primary and the secondary propriety or impropriety of a norm: "As it happens with other rules, a kind of secondary propriety / impropriety will arise with respect to KN. While those who assert appropriately (with respect to this rule) in a primary sense will be those who actually obey it, a speaker who broke this rule in a blameless fashion ( one who asserted something she didn't know, but reasonably thought she did know, would in some secondary sense be asserting properly." (De Rose 2002: 180) In other words, one could violate a norm of assertion in the primary sense, while not violate it in the secondary sense, hence not being open to criticism in the primary sense; alternatively one can violate the norm in the secondary sense and not be criticisable in the primary sense. Some writers, like Matt [START_REF] Weiner | Must we Know what we Say ?[END_REF], use this distinction to defend the norm of truth (KT) for assertion against the KN norm: the speaker may not be able to respect the stronger norm (KN), but he has to respect in any case the weaker one to make a proper assertion: the only case where one is liable to criticism, on the view in question, is the case where the asserter has no grounds for believing the uttered sentence to be true (as in the paradoxical sounding (7') above). Now I agree with Lackey (2007: 603-607) that this distinction is spurious, because there is only one way of violating a norm: simply by not doing or believing what it prescribes. The fact that I am unaware that I crossed the yellow line does not make me less liable to receive the fine from the policeman who notices my bad driving behaviour. That may excuse me, but that does not change the fact that I have violated the rule. The fact that one is unaware of violating the norm does not change one's status with respect ot the norm.

Williams' objection and DeRose's distinction, however, although they do no signal different ways of respecting or violating a norm, point to another distinction which it is important to make, between the statement of a norm and its regulation. On the one hand a norm can be formulated in an objective sense, as a correctness condition of the form Practice P is correct only if it has feature C or as an imperative One must: perform practice P only if one satisfies feature C KT, KB and its variants, as well as KN as formulated above are formulated in terms of such correctness and imperative conditions.

But on the other hand there are the conditions under which a norm is regulated, or its regulation conditions. These are the subjective conditions under which the correctness conditions are accessed by a given individual and are implemented in his psychology. 7 The former are independent of what the subject believes to be the norm, whereas the latter are so dependent. Of course the two do not have to be completely independent, for otherwise one could not see how the norm is regulated, but they need not necessarily coincide.

In the case of the knowledge norm for assertion, the relevant distinction is between (i) the question whether a subject's assertion are conform to the norm, and (ii) whether the subject actually believes that his or her assertion are conform to the norm. Many criticism of the KN rule actually assume that conformity to the norm implies (ii). This is why, as we saw above in § 1, a number of formulations consider that the norm KN has to include the correctness conditions that the subject "represents himself" as knowing the content of the assertion, or some other kind of "self-presentation". As we saw, the self presentation account in the form (1) above, is clearly inadequate. But the KNB form , which Williamson (2000: 260) formulates as:

(the BK rule) : One must: assert P only if one believes that one knows P Is not obviously so inadequate. As Williamson notes, saying that the asserter has only to believe that he knows, or take himself to know, can explain many conservational phenomena which the KN account explain, without asking a correctness condition as strong as knowing the 7 Although there are many ways of doing this distinction, I was inspired by the way in which [START_REF] Shah | How truth Governs Belief[END_REF] formulates a similar distinction with respect to the norm of truth for belief. See Engel (2008). content of the assertion. But Williamson rightly objects to it that even if we add to BK the condition that the belief has to be rational (for one might irrationally believe that one knows), one can rationally believe oneself to know that P while not knowing that P, hence violating the norm KN (Williamson 2000: 262). I agree with him that the right move, for the defender of KN is not to relativise it to a belief or to some mental attitude whereby the subject represents to himself the norm.

It is certainly difficult to say that an agent does not need in any sense to have access to a norm in order to follow it, and familiar attempts to reduce rules and norms to dispositions or rules to tacit knowledge of rules fail in this respect. But it is not clear that one needs to actually believe anything about the norm in order to apply it and to perform assertion. Someone who asserts a sentence in the future tense -"there will be sea fight"-obviously does not know, and does not believe that he knows, the existence of the future state of affairs described by the sentence. Nevertheless he makes an assertion, and he conforms himself to the KN norm. A liar obviously does not believe what he asserts, hence he does not know it. He nevertheless makes an assertion. It does not matter here whether we say that the KN norm is a conventional feature of assertion in a language in general, or whether it is a commitment of individual speakers. The point is that the norm is presupposed even when other norms, such as truthfulness, social norms, or other determinants of what we may want to say, are in place. The fact that the norm can be in many cases violated, overridden by other norms, or be applied in a very loose and relaxed way in many conversational circumstances does not show that the norm is not in place. 8 Similarly the fact that it is most often (in most of the cases) very hard to implement does not show that it is not in place. We certainly sometimes do not have the slightest idea of what it would take for us to know the things that we assert. But is does not imply that we do not have any idea of what it would take to know anything. In so far as the standard of knowledge is with us in quite many cases (I would argue for many so called "Moorean" propositions, such as that there are trees in the garden across my window), it is enough for us to have it for most assertions. It is not because we cannot attain certain kinds of knowledge that the norm is not in place. In this sense, ought does not necessarily imply can.

Conclusion

8 I here concur completely with Williamson 2000 : 258-59 I have not, in this paper, tried to examine all the objections that have or can be raised against the KAA. Neither I have dealt with the issue whether another epistemic norm than knowledge ( warrant or rational credibility) should not be preferred to KN. In fact most of the arguments that I have presented are compatible with the view that some high epistemic status, be it knowledge or not, is a norm of assertion, although there are some good reasons, which I have not given here, to prefer the knowledge account. My main point has consisted in objecting that the knowledge norm is confused or inapplicable does not threaten the claims of KAA, once we understand what it takes to be a norm of assertion. 9 9 versions of this paper have been read in 2007 in York, Nottingham, Geneva and Nancy. For their comments and encouragements I thank Tom Baldwin, Steven Barker, Berit Brogard, Greg Currie, Igor Douven Julien Dutant, Frank Lihoreau, Kevin Mulligan, Paul Noordhof, Ortto Bruun, Tom Stoneham. I am also indebted to the participants of a workshop that I organised at the University of Paris IV in january 2006, Manuel Garcia Carpintero, David Owens, Maria van der Schaard, and Peter Pagin, whose 2006 article inspired me much.

Although it has, to my knowledge, not been often remarked, the Pyrrhonian sceptic's refusal to assent fully to propositions is based on the implicit recognition that knowldedge is the norm of assertion. seeFrede's (1984) classical article.

The knowledge account surfaces also in some writings byRuth Marcus, such as her (1983), where she holds that to believe that P implies that P is possible, and one's realisation that P is impossible leads to a withdrawall of one's claim to believe, and in appropriate conditions of one's having asserted that P. In other words, realisation that one does not know that P, since P cannot but be false when P is impossible, amounts to the realisatin that one is not entitled to assert that P.

[START_REF] Pagin | Is assertion social?[END_REF] . Pagin develops in his 2005 a general strategy for assessing claims about speech acts.

For the distinction between believing and accepting, seeCohen 192, Engel 2000. Jennifer Lackey considers the case of professional assertions, of lies and other cases, but considers that they are clear violations of the norm knowledge of assertion.

See e.g.Boghossian 2003 p.39 : "The linguistic version of the normativity thesis, in contrast with its mentalistic version, has no plausibility whatsoever; and the reason for this is that it is not a norm of assertion that we should aim at the truth, in the way in which it is a norm of belief to do so." I have defend the normativity claim for belief in[START_REF] Engel | Laws and Models in science, London: King's college Publications 2008 "Belief and Normativity[END_REF]

It is interesting here to note that this kind of objection has also been raised against normative account of belief in terms of truth: it is said that the norm for belief cannot be truth, since many truth cannot be believed and many truth are too trivial to be believable (seeBoghossian op cit, Engel 2005). But since the norm of belief for assertion is arguably weaker than the knowledge norm KN, the charge of being too strong would apply even for the weaker norm (KB)


Wei Liu

Dominique Guines
email: dominique.guines@insa-rennes.fr

Lionel Leotoing

Eric Ragneau

Identification of sheet metal hardening for large strains with an in-plane biaxial tensile test and a dedicated cross specimen

Keywords: biaxial tensile test, cruciform specimen, hardening law, sheet metal forming, anisotropic yield criterion

In this work, an in-plane biaxial tensile test of cruciform specimen is performed to identify the hardening behaviour of metallic sheets under large strains. Firstly, an optimal shape of the specimen is suggested. Then, a biaxial tensile test is carried out for an aluminium alloy AA5086. Experimental forces on the two axes of the specimen are measured during the test and strains in the central area of the specimen are post-treated by means of Digital Image Correlation (DIC) technique. Finally, by considering different yield criteria, the associated hardening laws are identified thanks to an inverse procedure based on a Finite Element (FE) modeling of the biaxial tensile test and on the experimental data mentioned above. The identified biaxial flow curves are then compared with the ones from the classical uniaxial tensile test.

Introduction

Sheet metal forming processes are widely adopted in industries to produce thin-wall parts.

Nowadays, with increasing demands for safety, lower weight or reduced fabrication costs, new materials and innovative forming processes emerge. In order to reduce the amount of the time-consuming and expensive trial-and-error processes, Finite Element (FE) simulations are intensively used to analyze the capabilities of sheet metal forming processes. To improve the accuracy of FE models, the material must be characterized for conditions close to the ones encountered in practice [START_REF] Meinders | Numerical modeling of advanced materials[END_REF]. In sheet metal forming processes, the material is deformed under 2/30 multi-axial states and is subjected to large strains. Hence, the identification of material constants, corresponding to the different behaviour models (yield criterion, plastic hardening …), under biaxial tensile state has become a basic issue for the characterization of phenomenological models.

Characterization of material behaviour under biaxial tensile states can be performed by different tests such as: (i) hydraulic bulge tests of circular specimen [START_REF] Gutscher | Determination of flow stress for sheet metal forming using the viscous pressure bulge (VPB) test[END_REF], (ii) tension-internal pressure tests of tubular specimen [START_REF] Kuwabara | Advances in experiments on metal sheets and tubes in support of constitutive modeling and forming simulations[END_REF], through-thickness compression tests using stacked specimens [START_REF] An | Analytical and experimental study of frictional behavior in throughthickness compression test[END_REF][START_REF] Steglich | Mechanical Testing of Thin Sheet Magnesium Alloys in Biaxial Tension and Uniaxial Compression[END_REF] or (iii) biaxial tensile tests of flat cruciform specimen. Considering the inplane biaxial tensile test, a cruciform specimen is directly loaded along two perpendicular directions. This test presents several interests for the characterisation of material behaviours: first, it is a frictionless test without any contact between the tested specimen and tools; moreover, many strain paths ranging from uniaxial tension to biaxial tension can be encountered in the same test in different regions of the specimen; finally, by changing the displacement ratio between the two perpendicular axes, different linear or non linear biaxial strain and stress states can be obtained in the central region of the specimen. The main drawback of this test is related to the design of the cross specimen. Very recently, the international standard ISO16842 [START_REF]International Standard ISO16842, Metallic materials -Sheet and strip -Biaxial tensile testing method using a cruciform test piece[END_REF] has been proposed. This standard specifies the testing method for measuring the biaxial stress-strain curves of sheet metals subject to biaxial tension at an arbitrary stress ratio. The test piece is made of a flat sheet metal and has a uniform thickness. The measured biaxial stress-strain curves are used to determine contours of plastic work of the sheet samples. This standard is based on the research work of Kuwabara et al. [START_REF] Kuwabara | Measurement and analysis of differential work hardening in cold-rolled steel sheet under biaxial tension[END_REF] and considers that stresses are calculated analytically by assuming an equivalent cross section. This shape cannot be used for hardening identification since the strain level measured in the central zone is very low when necking appears in the arms. For low strains, many studies have been performed to determine the initial yield surface and estimate yield criterion parameters.

Muller et al. [START_REF] Muller | New experiments for determining yield loci of sheet metal[END_REF] have presented an optimized geometry of cross tensile specimen with notches at corners to obtain a large zone of homogeneous deformation in the central area of the specimen. The beginning of plastic deformation is determined by the heat dissipation due to the plastic deformation. Naka et al. [START_REF] Naka | Effects of strain rate, temperature and sheet thickness on yield locus of AZ31 magnesium alloy sheet[END_REF] have proposed a specimen with notches at the corners and two slits in the arms to make stresses and strains as homogeneous as possible in the central gage section. Effects of strain rate, temperature and sheet thickness on the yield locus of AZ31 magnesium alloy sheet were investigated. Merklein et al. [START_REF] Merklein | Characterization of yielding behavior of sheet metal under biaxial stress condition at elevated temperatures[END_REF], always on magnesium alloy AZ31, have designed a cruciform shape with six slits of various lengths in More recently, the development of optical full-field measurement techniques for analysing heterogeneous strain fields, such as DIC technique, has led to the development of new methodologies for characterising the plastic behaviour of materials [START_REF] Avril | Stress Reconstruction and Constitutive Parameter Identification in Plane-Stress Elasto-plastic Problems Using Surface Measurements of Deformation Fields[END_REF]. The most common approach is based on inverse analysis methodologies, which consist in the calibration of the material model by the best set of material parameters that minimise the gap between predicted and experimental results. In this context, several works in literature have been proposed.

Teaca et al. [START_REF] Teaca | Identification of sheet metal plastic anisotropy using heterogeneous biaxial tensile tests[END_REF] have suggested two types of specimen shapes which can generate heterogeneous strain fields. The first shape with a hole in the central zone covered the stress range from uniaxial tension to that corresponding to plane strain and the second shape with four slots in each arm covered the stress range from uniaxial tension to equibiaxial tension.

The strain was calculated by DIC technique and the inverse strategy of parameter identification, including the FE model of the specimen, is performed to determine some of the constants of the Ferron, Makkouk and Morreale (FMM) yield function. The other parameters were determined by means of the classical uniaxial tensile test. Zhang et al. [START_REF] Zhang | Potential of the cross biaxial test for anisotropy characterization based on heterogeneous strain field[END_REF] proposed a parameter identification process with a single biaxial tensile test for complex Bron and Besson anisotropic yield model for two metallic sheets (AA5086 and DP980). An identification stage based on the minimization of experimental and numerical principal strains along a specified path in the gauge area of the cruciform specimen was performed. Prates et al.

[14] used a single equibiaxial tensile test of cruciform specimen to simultaneously identify the parameters of Hill 48 yield criterion and Swift isotropic hardening law by mixed numericalexperimental method. A cruciform shape, with tapered arms, was proposed. Nevertheless, this work was not experimentally validated. As seen in this literature review, the biaxial tensile test associated with cruciform specimen has been extensively used to determine the yield locus of sheet metal. Unfortunately, the equivalent strain level reached in these applications is usually very small, but sufficient for initial or subsequent yield locus identification.

Several attempts of cruciform shape design to reach large strains have also been presented.

Green et al. [START_REF] Green | Experimental investigation of the biaxial behaviour of an aluminum sheet[END_REF] have proposed specimen geometry with a central section thinner than the rest of the specimen and seven parallel slots in each arm. The cruciform specimens were deformed up to effective strains of approximately 15% in biaxial stretching, along seven different linear 4/30 strain paths. An iterative procedure coupled with finite element analysis was performed to determine the various yield functions and biaxial flow curves. Tasan et al. [START_REF] Tasan | In-Plane Biaxial Loading of Sheet Metal until Fracture[END_REF] have demonstrated that a significant thickness reduction in the centre is necessary to reach fracture under a biaxial strain path. Abu-Farha et al. [START_REF] Abu-Farha | On the Development of Viable Cruciform-Shaped Specimens: Towards Accurate Elevated Temperature Biaxial Testing of Lightweight Materials[END_REF] have optimised two shapes of cruciform specimen to study micro-structural deformation mechanisms under biaxial strain state for lightweight materials at elevated temperature. The arms of the first geometry were tapered and a circular profile is used to define the thickness evolution. The other geometry has notches between arms and a circular flat-bottomed recess in the gauge area. However, in works presented above, only relatively low values of equivalent plastic strain (close to those obtained in uniaxial tension) are reached before plastic instabilities occur.

With the aim to determine forming limit strains of sheet metal, some studies have been led to define an appropriated cruciform shape with an onset of necking in the center of the specimen.

Different strain paths must be tested to cover the whole forming limit curve (FLC), and in all cases, the rupture of the specimen must occur at the central point. Considering these requirements, Yu et al. [START_REF] Yu | Design of a cruciform biaxial tensile specimen for limit strain analysis by FEM[END_REF] have suggested a cruciform specimen with two thickness reductions in the central region to obtain strain limits and study the influence of complex loading paths by finite element simulations. But this work was not validated by experiments.

Zidane et al. [START_REF] Zidane | Development of an in-plane biaxial test for forming limit curve (FLC) characterization of metallic sheets[END_REF] have adopted a two-steps thickness reduction in the central zone. The first zone is a square one with a constant thickness and the second one is a circular zone with a progressive thickness reduction defined by a circular profil. With this specimen, the rupture is always observed at the central point where various strain paths can be obtained by different speed ratios on the two arms. The whole FLC of aluminium alloy AA5086 has been determined. For the same specimen, Leotoing et al. [START_REF] Leotoing | Cruciform shape benefits for experimental and numerical evaluation of sheet metal formability[END_REF] have numerically investigated the influence of both the yield criteria and hardening laws on the prediction of FLC. The main drawbacks of the proposed specimen shape are the complex geometry of the central area and the initial thickness of 4mm required for the two thickness reductions.

In this work, in order to show the potential of the in-plane biaxial tensile test to capture the hardening behaviour of sheet metal for large strains, different shapes of cruciform specimen are numerically investigated. Based on FE results, a shape with thickness reduction at the central zone, slits along the arms and notches between two arms is proposed from a parametric study. The designed shape of cruciform specimen is validated by quasi-static equibiaxial tensile tests performed on an aluminium alloy AA5086. The strain in the central 5/30 area of the specimen is calculated by DIC method. Then, experimental data, corresponding to forces along the two arms and strains in the central part are used to determine the biaxial flow stress curve of the material for large equivalent strain (up to 30%). The parameter identification is based on the FE model of the in-plane biaxial tensile test and on the minimization of the difference between experimental and calculated principal strains. In this identification stage, three yield criteria are evaluated (Mises, Hill48 and Bron and Besson).

Results obtained from the biaxial tensile test are finally compared with those from uniaxial tensile test.

Shape design of cruciform specimen

As shown in bibliography overview, there is no standardized in-plane cross specimen which exhibits large strains in equibiaxial strain state. Very few researches about characterization of plastic behaviour of metallic sheets by biaxial tests on cruciform specimen under large strains have been reported. In order to obtain rupture (and consequently large strains) under different stress states and not only under the classical uniaxial tension, some cruciform specimen shapes, more or less complex, have been investigated. Very few of them have been validated by experiments, except the shape proposed by Zidane et al. [START_REF] Zidane | Development of an in-plane biaxial test for forming limit curve (FLC) characterization of metallic sheets[END_REF], which is based on a two-step thickness reduction (from as received thickness 4mm to final thickness 0.75mm).

Nevertheless, the use of this shape for any sheet thickness seems difficult, particularly for sheets with small initial thicknesses. Another limitation of the cruciform shape with a progressive central thickness reduction defined by a curved profile is that large deformations are localized in the centre of the specimen leading to strongly heterogeneous strain fields on the specimen surface. For the inverse procedure of identification of material parameters, one possibility is to consider whole strain fields on the specimen surface during the test and to minimize the difference between numerical and experimental strain fields as previously performed by other authors [21]. In this study, the identification procedure aims to be simple and for this purpose just the strain evolution in the specimen central point during the test is analysed. Consequently, it has been chosen to define a constant central thickness zone where an almost homogeneous strain field is expected. To obtain robust values, the principal strains, used in the identification loop, are obtained by the average of strains calculated over some points located around the central point of the specimen, as shown hereafter.

6/30

To define a dedicated shape design which covers a large strain range, the in-plane equibiaxial tensile test is investigated by means of FE simulations carried out with the software ABAQUS. In order to investigate the different shapes, an isotropic Mises yield criterion is first chosen and an isotropic hardening behaviour is assumed. This hardening behavior is introduced through the true stress-plastic strain law (Fig. 1) from uniaxial tensile test.

Fig. 1 True stress-plastic strain curve of AA5086

To evaluate the potential of different shapes, simulations are analysed for a value of 20% of the major principal strain 1 ε . In a first step, the equibiaxial tensile test is simulated on the basic shape given Fig. 2a. The S3R shell elements are used in the FE model (Fig. 2b). Due to symmetries, only one quarter of the geometry is considered. The specimen can be divided into three zones: the central zone, the arms and the intersection of the arms. Fig. 3 shows the major principal strain 1 ε , the equivalent plastic strain p ε and the strain state (ratio of in-plane principal strains 21 ε /ε ) for the basic specimen shape. At the same time, the value of 20% is reached for both the major principal strain and the equivalent plastic strain at the intersection between two consecutive arms (Fig. 3a and3b), which corresponds approximately to a uniaxial tensile state in this area ( 21 ε /ε 0.5  ). The equivalent plastic strain in the central zone is very low (below 2%, as shown in Fig. 3b) and strain state corresponds to an equibiaxial one ( 21 ε /ε 1  , Fig. 3c). Strain localisation at corners and onset of necking in arms limit the strains in the specimen central zone. than radii, so notches have been preferred in this study (Fig. 4a). The thickness reduction is realized only on one side of the sheet and not in a symmetrical way to simplify the specimen manufacturing. Linear tetrahedral elements of type C3D4 are used to mesh the specimen (Fig. 4b). Potential of this shape is illustrated by simulation results given in Fig. 5 where the thickness e and the radius r of the thickness-reduced zone are respectively equal to 0.5mm and 5mm. Compared to the basic shape, this second shape exhibits larger major principal strain ( 1 8% ε 12%  ) in the central thickness-reduced zone (Fig. 5a). In this zone, an equibiaxial strain state is observed (Fig. 5c) and an equivalent plastic strain of 20% is reached.

It must be noticed that the maximum value of 1 ε is again first reached at the junction between two consecutive arms (i.e. in notches) despite of the proposed shape modifications. As shown in Fig. 5c, in this zone a ratio 21 ε /ε ranging from -0.2 to -0.5 is observed, typical of a stress state close to the uniaxial tension. The comparison between Fig. 3a and 5a shows that shape modifications have transferred the deformation of the specimen from arms, where the major principal strain is now less than 4%, to the central zone. So, on the basis of results presented above, a parametric study has been performed. Five parameters have been introduced: radius r of the thickness-reduced zone, position D and radius R of the notches, and finally, positions 1 S and 2 S of the slots, as shown in the Fig. 6.

Concerning, the thickness of the reduced central zone (parameter e in Fig. 6), the smaller the thickness is, the larger the strains will be. Nevertheless, to avoid early rupture in this zone due to defects induced by the machined process (milling or lathing), the thickness is limited to a value of 0.6mm. In previous works [START_REF] Zidane | cruciform specimen shape for large strains has been proposed -Considering three different yield criteria, the Voce's hardening law has been identified[END_REF] on the same alloy, authors have shown that a The mesh of the FE model, discretized with linear tetrahedral C3D4 solid elements, is presented Fig. 7. The equivalent plastic strain at the central zone reaches 30% when the major principal strain is about 20% near the ends of slots and in the fillet radius between the two different thickness zones (Fig. 8). The interest of slots is clearly shown since the maximum strain localization appears now in the thickness-reduced zone. The so-defined specimen shape presented in Fig. 6 with optimal dimensions (Tab. 1) will be used in the following to characterize the hardening behavior of an AA5086 aluminium alloy up to large equivalent plastic strain under equibiaxial tensile state. An area around the central thickness reduction zone is discretised by subsets and treated by DIC (Fig. 10). The size and distance of square subsets are respectively 32pixels×32pixels and 16pixels×16pixels which leads to 14 strain calculation points along each direction (X and Y).

Considering the image resolution and the dimensions of the filmed zone, a resolution of 0.037 mm/pixel is obtained. The displacement vectors of the gauge zone are shown in Fig. 11, it can be seen that the position of the central point is nearly kept fixed. Details on strain calculations are given in [START_REF] Zhang | Potential of the cross biaxial test for anisotropy characterization based on heterogeneous strain field[END_REF].

The equivalent strain, major and minor principal strain fields in the central zone of specimen at the time t=1.54s are shown in Fig. 12, Fig. 13 and Fig. 14 respectively. As expected, the equivalent strain reaches 30% just before the crack of specimen (t=1.612s), whereas the principal strains are about 16% for the major strain and 14% for the minor strain. The 13/30 maximum deformation appears in the fillet between the flat central zone and the non-reduced thickness zone on the central axis of the specimen (transverse direction). where m and n are the numbers of points to be averaged in each direction and t is the time.

Here, the average strains are calculated for m=n=3, corresponding to a square area 15/30

. Inverse analysis procedure

The cross section of the central zone of the specimen is not constant and stress and strain fields are not homogeneous. Then, stress or strain cannot be calculated in an analytical way.

Hence, inverse analysis is adopted to identify the material constants of the hardening law, as shown in Fig. 16. The simulated principal strain curves at the centre of the FE model are compared with the experimental ones. To carry out the optimization process, the software platform ModeFRONTIER [24] is used and the Simplex algorithm is chosen. The optimization algorithm Simplex was preferred in the identification process. Like hill climbing algorithms, the Simplex method may not converge to the global minimum and can stop at local optima. To be sure that the global minimum was found, several optimizations were launched with various initial sets of parameters. The Simplex method presents the advantage of using p+1 vectors (with p the number of parameters) and then facilitates the search of the global minimum in the p-dimensional space. The error function Q between simulated and experimental strains is defined as follows:

        2 2 2 2 1 2 1 1 exp 1 exp 2 2 exp 2 exp ( , ) n n n n i i i i i i sim sim sim sim i i i i Q                     [2]
Here, n is the total number of the time points of simulation; 1sim

Finite element model of the in-plane biaxial tensile test

Previously to the calibration stage, a FE model must be defined and the element size should be chosen small enough to capture the deformation localisation. Due to the symmetry, a quarter of the FE model of cruciform specimen is built. For the material model, anisotropic

Hill48 yield criterion and a Voce hardening law both identified in a previous work by Zidane [START_REF] Zidane | cruciform specimen shape for large strains has been proposed -Considering three different yield criteria, the Voce's hardening law has been identified[END_REF] have been used. The experimental forces Fx exp and Fy exp are applied to FE models, which include the slight out of synchronization observed between the two axes. Based on these assumptions, a parametric study with successive refined meshes has been led and appropriated element sizes have been chosen (linear tetrahedral C3D4 element). A final element size of 0.1mm in the central zone, notches and slit ends is adopted (Fig. 17a) and will be considered as a reference. Unfortunately, with this mesh, FE simulations are very time-consuming. So, to keep simulation times compatible with the inverse procedure which requires many simulations, a FE model with 2D linear shell element mesh is proposed. The shell model has been divided into three zones with different thicknesses: (i) the central flat thickness-reduced zone with t=0.625mm, (ii) a transition zone corresponding to the fillet radius with a thickness of 1.31mm and (iii) the last zone with the 2mm initial thickness. The thickness of the transition zone has been chosen as the average value of thicknesses of the central zone and the initial sheet. From successive simulations, adequate element sizes have been defined in these three zones of the specimen.

The equivalent plastic strain fields simulated by FE models with solid and shell elements are presented in Fig. 17. Distributions of the equivalent plastic strain are almost the same for the two models, although the localisation at the edge of central thickness-reduced zone in the solid model is more pronounced than in the shell model. The evolution of major and minor principal strain at the central zone has been also compared between the two models, as shown in Fig. 18. It confirms that the major and minor principal strain curves are very close for the two models. It can be concluded that the proposed shell mesh refinement is well adapted to precisely describe the behaviour of the cruciform specimen shape proposed in Fig. 6. The 3D element mesh with the appropriated element size in the central zone requires 814708 linear tetrahedral elements of type C3D4, leading to a computational time of 31096s (processor Intel 3.2 GHz with 16 Go RAM). The same specimen, meshed with shell elements, requires 4660 S4R elements, leading to a computational time of 222s, which corresponds to a ratio of 140 between the two computational times on the same computer.

Material models

To describe the material behaviour, an elasto-plastic model is adopted. The elasticity is described by Hooke's law with Young's modulus

Yield functions

For the identification of hardening laws, three yield functions   ij   , written in function of stress tensor components, are defined as follows for plane stress conditions:

Von Mises:   2 2 2 3 2 xy yy xx yy xx ij            [3]
Hill48:

      2 2 2 2 2 yy xx yy xx ij N H H F H G              [4] The constants 632 . 0  F , 671 . 0  G , 329 . 0  H and 460 . 1  N
, with G+H=1 for plane stress conditions, are calculated from the three plastic anisotropic coefficients 0 0.49 r  , 45 0.62 r  and 90 0.52 r  , determined in uniaxial tension by the ratio of width plastic strain rate to normal plastic strain rate for the three directions (0°, 45° and 90°) from the rolling one [START_REF] Zhang | Calibration of anisotropic yield criterion with conventional tests or biaxial test[END_REF].

The parameters of the advanced phenomenological Bron and Besson yield function for AA5086 have been identified from a single biaxial test in a previous work [START_REF] Zhang | Calibration of anisotropic yield criterion with conventional tests or biaxial test[END_REF]. This yield function is based on:

    1 1 a K a kk ij k           [5] with   k b k k 1    [6]            

Hardening law

A modified form of Voce's hardening has been adopted as follows [START_REF] Diot | Forming process of a 5083 aluminium alloy: constitutive model covering a large range of temperature[END_REF] to limit the saturating behaviour of classical Voce's formulation: The three identified hardening laws from the biaxial test are with the experimental flow true stress-strain curve determined from an uniaxial test (Fig. 20). It can be concluded that the flow stress curves identified with Mises and Hill48 yield criteria are not in good agreement with the analytical flow stress curve from uniaxial tensile experiments. The curve identified with Mises criterion strongly underestimates the hardening of the material, whereas Hill48 criterion overestimates hardening for low strains (below 7%) and underestimates for larger strains. As expected, due to the advanced formulation of Bron and Besson yield criterion, the flow stress curve identified with this criterion is very close to the uniaxial results below 20%. Nevertheless, the biaxial test permits an accurate evaluation of hardening until an equivalent plastic strain of 30%. As it can be seen in Fig. 20, very different hardening behaviours can be identified depending on the choice of the yield criterion. Considering the type of loading (biaxial), this result is rather logical and permits to validate the whole modeling of the plastic behaviour of the material thanks to an adapted choice of both yield criterion and hardening. In Fig. 19, three points corresponding to the stress-state at the central point of the cross specimen have been added for the three yield functions. As it can be seen on this figure, due to the slight out of synchronization in the two tensile forces on each axis of the cross specimen, the stress state obtained in the specimen center is not exactly biaxial but is almost the same for the three tested yield criteria. This is confirmed by the temporal evolution of the stress path presented in Fig. 21 through the 21 /

 ratio. The 21 /  stress ratio evolution is not presented for the elastic part, since the same elastic behaviour is consider the yield criteria. As it can be observed in Fig. 21, the material follows a quasiconstant path in the stress space. The strain path, at the cross specimen center, is presented Fig. 25, mainly for the plastic strain range. Experimental 21 /

 ratio be evaluated precisely at the beginning of the test for small levels of strains. As it can be observed in the Fig. 25, due to the slight out of synchronization and to the anisotropy of the material, this experimental ratio is always inferior to one and quite constant (between 0.6 to 0.8) for time ranging from 0.6 to 1.5s.  ratio decreases. This effect is due to the divergent evolution of experimental and calculated minor strains as it can be observed in Fig. 24. The other two simulated strain paths from Mises and Hill48 criteria overestimate the 21 /  ratio, more especially for the Hill48 criterion.

Conclusion

In this work, the potential of the in-plane biaxial tensile test on cross specimen to characterize the hardening behaviour of metallic sheets under large strains has been investigated. Due to the complex specimen shape, exhibiting non-homogeneous strain and stress fields, a numerical FE model of the biaxial tensile test used in an inverse calibration procedure is needed to identify constants of the hardening behaviour. To propose a simple 27/30 calibration stage of these hardening material constants, the strain evolution of the central point of the specimen is considered in the identification procedure.

The proposed cruciform specimen shape, with constant thickness central zone, permits to reach an experimental biaxial flow stress curve for AA5086 up to 30% of equivalent plastic strain. The identification strategy is based on both experimental local principal strains at the central point of the specimen and global force measurements along the two arms of the specimen. The hardening behaviour has been identified for both isotropic (Mises criterion) and anisotropic plastic criteria, a simple one (Hill48) and an advanced one (Bron and Besson).

Comparison between the identified biaxial flow stress curves and the experimental one from uniaxial tensile test (below 20%), shows a good agreement if an appropriate yield criterion is considered. This is confirmed by the strain path in the center of the specimen which is better predicted by Bron and Besson yield function. Consequently, due to the biaxial loading, both the yield criterion and the hardening are validated for the modeling of the sheet material behaviour.

Like the bulge test, in-plane biaxial tensile test permits also the identification of hardening behaviour up to large strains. Nevertheless, contrary to the bulge test, strain path is controlled by actuator displacements in the two main directions of the cross specimen and effects of strain path changes can be investigated without any restriction about the type of non-linear path. This feature provides interesting perspectives for the identification of advanced hardening behaviours like isotropic-kinematical combined models. Strain rate dependent hardening sensitivity can be also studied by this procedure since the cross shape and the servo-hydraulic testing machine permit to reach intermediate strain rates (below 100s -1 ) encountered in sheet metal forming processes.

  determine initial and subsequent yield loci. Once again, the yield stresses are calculated by the measured forces and corresponding cross-sections. Strain fields are calculated by Digital Image Correlation (DIC) technique.

A

  2mm thickness sheet metal of aluminium alloy AA5086 is considered. The elastic parameters of this alloy are Young's modulus MPa

Fig. 3

 3 Fig. 2 Basic specimen shape

Fig. 5

 5 Fig. 4 Specimen shape with thickness reduction and notches

30 Fig. 6

 306 Fig. 6 Dimensions of optimized cruciform specimen (in mm)

Fig. 8 Fig. 9

 89 Fig. 7 3D mesh of the optimized cruciform specimen shape

Fig. 10 Fig. 13 Fig. 15 .

 101315 Fig. 10 Gauge zone Fig. 11 Displacement vectors

( 1 .

 1 Fig. 15, have been filtered in order to be applied as boundary conditions in the FE model. Due to small mechanical gaps introduced in connections of the different links used to clamp the four arms of the specimen at each actuator, a slight out of synchronization is observed between the two X and Y axes.

Fig. 15

 15 Fig. 15 Experimental principal strains and forces



  Fig.16 Flowchart of inverse procedure for hardening identification

Fig. 17

 17 Fig. 17 Comparison of simulated equivalent plastic strain (PEEQ) fields from 3D solid element mesh (a) and 2D shell element mesh (b).

Fig. 18

 18 Fig. 18 Comparison of principal strain at central point for 3D and 2D mesh

  behaviour, the associated flow rule is assumed and an isotropic hardening is introduced. Three yield criteria are considered, one isotropic and two anisotropic models. The first anisotropic one is the well-known Hill48 criterion and the second one is the advanced Bron and Besson (B&B) criterion[START_REF] Bron | A yield function for anisotropic materials -Application to aluminum alloys[END_REF]. The first two plastic criteria are already introduced in 0environment. For the Bron and Besson criterion, the user subroutine UMAT was used.

  The shapes of the three yield functions are shown in Fig.19.

Fig. 19

 19 Fig. 19 Mises, Hill48 and Bron and Besson yield functions

  has been identified (Tab. 2) with the three yield criteria (Mises, Hill48 and Bron and Besson), respectively. Concerning the identification stage, a uniformly distributed design of experiment (DOE) with 150 samples, based on a Sobol sequence in modeFRONTIER, have been performed for each yield criteria and the cost function (Eq. 2) has been evaluated for each sample. Based on these results, optimizations with Simplex algorithm have been launched from the best samples obtained in the DOE in order to converge to the global minimum and to avoid the problem of local optima.

Fig. 20 Fig. 21 30 Fig. 22

 20213022 Fig. 20 Comparison of flow stress curves

Fig. 25

 25 Fig. 25 Experimental and simulated strain path ratios


Martin Tanguy
email: martin.tanguy@mines-paristech.fr

Aldo Napoli
email: aldo.napoli@mines-paristech.fr

A methodology to improve the assessment of vulnerability on the maritime supply chain of energy

Keywords: maritime network, maritime territories, network modelling, maritime safety and security. I

the globalization of trade is due to the transportation possibilities and the standardization (containerization of freight). The dependency of the economy to the sea and to the merchant navy has increase this last decade. This process forms a worldwide maritime network between the different locations of production and consumption. This network, representing between 80 % and 90% of world traffic is a major economic concern, including freight distribution, raw materials or energy. Rodrigue demonstrates[1] the economic dependency of energy is increasing in the industrialized countries (North America, Europe, East Asia). The inter-regional trade of oil was 31 million bbl/day in 2002 and is expected to grow up to 57 bbl/day in 2030 [2]. Most of the international traffic use a maritime way, where may occur disruptions. For example, the Suez crisis (1956-1957) caused a closure of the canal, reducing the throughput capacity of transportation. This disruption cost a 2 millions of barrels lost per day. This article focuses on vulnerability of the energy supply, and proposes a methodology to formalize and assess the vulnerability of the network by taking into account the spatial structure of maritime territories.

INTRODUCTION

Today, the supply chain of energy has a global dimension, connecting major consumers (East-Asia, North America) to major producers (Middle-East, Russia). Most of this trade is done on the maritime territories, which represent 40% of the supply of petroleum and 25% of gaz (Institut Supérieur d'Economie Maritime). The energy security is a major issue for global economy. The International Energy Agency (IEA) defined energy security as "the uninterrupted availability of energy sources at an affordable price" [START_REF]Energy eupply security report[END_REF]. To ensure the security of the supply chain, the understanding of possible disruptions is necessary. The aim of IEA is to "maintain and improve systems for coping with oil supply disruptions". The severity of a disruption can be measured by energy lost, level of commercial inventories, the duration of the disruption and the available spare production capacity. Disruptions on the supply chain can be due to environmental causes (extreme weather conditions), maritime activities (collisions, infrastructures failures) or deliberate acts like piracy and terrorist attack. To ensure theses disruptions, two emergency responses are used by IEA, the increase of supply, by stockdraw and production surge, and demand reducing, by restraint and switching energy type. This network can be model by the use of graph. Indeed, geography and spatial reasoning can highlight new knowledge about supply chain security. The aim of this work is to propose a methodology to formalize and asses the vulnerability of the network by the access of spatial possibilities offered by ocean spatial structure. To achieve it, a state of the art of different modelling of network is detailed in Section II. Then, Section III describes the spatial structure of the ocean and effects of "territorialization" on maritime territories. Section IV focus on vulnerability measurement of the network. Section V presents the methodology and the different steps to assess the vulnerability. Finally, the paper ends with a discussion on the further works.

II. GRAPH THEORY AND MARITIME NETWORK

MODELLING

Modelling the supply chain by a network offers several advantages. Graphs can describe the topological relation existing between consumers and production through space and quantifies the flow of energy with a ponderation of the graph. This section first describes the properties of graphs then the existing modelling of maritime network and these limitations to access vulnerability.

A. Properties of graphs

Graph theory is mainly used for network modeling. Most of the studies on maritime networks use graph theory. Indeed, network analysis is used to studyy transport network forms and structures [START_REF] Rodrigue | The geography of transport systems[END_REF]. Networks are modelled by graphs for several reasons:

• The main strength of graph theory is to allow the description of the topological relationship between several nodes, describing links between locations. This allows studying the connectivity and the degree distribution of every location in a topological space. Those concepts are essential for the study of networks.

• In the case of spatial network the vector and geometric proprieties are very useful. Vectors properties are used to add a direction to a link. For transportation modelling, this property is used to model flows between locations. Geometrics properties can be used to insert distance into the model. This allows spatializing the network in a Euclidian space.

It is essential for the modelling of transport.

• Graph theory allows also the description of relationships through the graph. These properties based on the concept of path (a course between components into the graph) and cycle (a path have the same origin and destination) allow the study of relationships between various parts of the graphs.

Theses mathematical properties of graphs are primarily used for the static and dynamic modelling of a transport network.

B. Maritime network modelling

Few works have been done on modeling of maritime networks. These studies focus on several research questions on maritime network modelling:

• The topological properties of the network and formalization of these transportation networks. Most of them use nodes as ports and links as a maritime flow between ports.

• The configuration of the maritime network and its relationship with the land areas.

• The identification of groups (global regions, clusters…) into the maritime network.

Joly [START_REF] Joly | Structuration des lignes maritimes régulières de navires porte-conteneurs[END_REF] was the first to model the maritime networks using graph theory, with data from the Naval Llyod's insurance company. Each point corresponds to a port, each relation to a link. Veenstra et al. [START_REF] Veenstra | Analysing container flows in the Caribbean[END_REF] model the maritime networks in three levels: the first level corresponds to infrastructure (ports), the second level corresponds to the ports used by shipping lines and the third one to quantification of the previous levels (e.g. transported volumes, frequencies of transport). The goal of their research is to measure the attractiveness of a port within a network. The purpose of the study of Hu and Zhu [START_REF] Hu | Empirical analysis of the worldwide maritime transportation network[END_REF] is to demonstrate that the shape of global maritime network is a "network of small world" which is characterized by the presence of cliques (a sub-graph where all the nodes are connected) and "scale free", following a power law, with high degrees of correlation nodes (hubs) and lower ones (spokes).

Ducruet [START_REF] Ducruet | Network diversity and maritime flows[END_REF] models the network with a multigraph. The result is a hierarchy of ports and shipping lines depending of nodal regions. Zavitsas [START_REF] Zavitsas | The Vulnerability of The Petroleum Supply Chain[END_REF] proposes a model to determine the vulnerability of the maritime supply chain. His methodology consists to create a grid on the maritime space and simulate paths by a shortest path algorithm to obtain a network. Xu et al. [START_REF] Xu | Evolution of regional inequality in the global shipping network[END_REF] propose a network based by grouping seaboards to analyze the evolution of the maritime network between different parts of the world. Kaluza et al. [START_REF] Kaluza | The complex network of global cargo ship movements[END_REF] propose a model to compare the different network forms by ships type (container ships, bulk dry carriers, oil tankers…).

All of these modelling are illustrated in (Fig. 1). The figure compares the different structure of maritime network proposed in scientific papers. Circles represent maritime space, nodes the different locations linked (ports for most of them) and links the structuration of network. Links differentiation correspond to the 3 levels of network in Veenstra et al [START_REF] Veenstra | Analysing container flows in the Caribbean[END_REF] modelling and different type of ships into Kaluza et al [START_REF] Kaluza | The complex network of global cargo ship movements[END_REF] modelling.

Figure 1. Maritime network modelling using graph theory

C. Limitation of these works in our case study

However, these works are limited to provide useful information for safety and security on the sea. These models provide any information on the maritime territory crossed by ships. Indeed, the proposed models use ports as nodes and links as roads. The only exceptions are some checkpoints such as channels (Suez, Panama) used in the model of Ducruet [START_REF] Ducruet | Network diversity and maritime flows[END_REF] In his modelling, Zavitsas [START_REF] Zavitsas | The Vulnerability of The Petroleum Supply Chain[END_REF] uses the shortest route between the port of departure (supply) and arrival (demand) through a cost-distance matrix and justifies this choice by the lack of available information on maritime territory. Veenstra et al [START_REF] Veenstra | Analysing container flows in the Caribbean[END_REF] don't model the sea routes because they do not consider the maritime space as a physical space "There are no physical links between airports and between seaports that have the characteristic of a line". The work of Hu & Zhu [START_REF] Hu | Empirical analysis of the worldwide maritime transportation network[END_REF] only focuses on the shape of the maritime network and do abstraction of space.

A spatial reasoning about the space crossed by ships is missing or not covered in the existing works on maritime flows modelling. Due to the specific research question behind these modelling, mainly focuses on relationship between ports, maritime territories are not present into the conceptual framework. Indeed, there are no modelling approach who tries to understand the relationship between transportation and the spatial structure of the ocean. But ocean isn't an empty and homogeneous place. Moreover, to measure the vulnerability by topological and geometric features, a specific model of maritime space is needed. Indeed, measure the influence of ports between them and their topological properties is not sufficient under this research.

To obtain this model, it will be necessary to integrate the structuring elements of maritime territory in the previous section to formalize a network capable of representing the topology of the maritime area. The problematic of this study is to find a new methodology to build this network, and understand relationships between the economical use of ocean (the maritime network) and the structuration of maritime space (maritime territories). Our hypothesis is that dynamic structure of maritime space under the effect of risky situations may induce disruptions on the network, and affects the performance of the supply chain of energy.

TERRITORIALIZATION AND SPATIAL STRUCTURATION OF MARITIME SPACE

Our goal in this work is to formalize a network based on the spatial structure of ocean. To achieve it, we have to understand how the maritime territories are forms. Indeed, through these maritime territories, the maritime network is forms by the used of space into the maritime territories. With this methodology, we can understand the structuration of the maritime network.

A. Typology of network by structuration

Three types of network are distinguish depending to their spatial structuration [START_REF] Rodrigue | The geography of transport systems[END_REF]: strictly defined and delimited, vaguely defined and delimited, or immaterial. In a strictly defined and delimited network, the space occupied by the transportation network is strictly reserved for the use of transport and can be identified on a map. An appropriation of this space for transportation purposes can be established, the network structure is dependent on infrastructure. This is the case of road and rail networks. In the vaguely defined and delineated networks, as maritime and aerial networks, the space can be shared by multiple activities, there is no appropriation of space, only rights of way. Infrastructures are exclusively areas of departure and arrival. Finally, some networks, such as wireless communication networks are immaterial and have no spatiality

In the case of maritime transportation, the possibility of the maritime space isn't constraint by an infrastructure and can use all the opportunities available at sea.

B. Spatial propriety of maritime space

Between ports, the network crosses maritime space, which corresponds to the entire ocean. This space may be considered by its topological properties, connexity and connectivity:

• Connexity means that the maritime space does not have any discontinuity within its space and maritime space therefore is totally connected. This means that from a port, it is theoretically possible to reach all the other ports.

• Connectivity is measured by the degree of freedom (k) is theoretically infinite as the ocean is not limited by a network-support (as is the case for road or rail) and therefore, there is an infinite number of ways to connect the ports (Fig. 2)

Figure 2. Connectivity of maritime space However, this degree of freedom is only theoretical, because in practice other structural logics (e.g economical, legal) are sea routes and spatial constraints for the transportation within the maritime space.

C. Territorialization and spatial constaints

This process of maritimization has an impact on maritime spaces. This increasing use of space as for transport and exploitation has resulted in a territorialization of maritime spaces. Di Meo [START_REF] Méo | La géographie culturelle : quelle approche sociale ?[END_REF] defines the concept of territory, distinguishing its two components: an infrastructure and a superstructure. Infrastructure refers to concrete aspects of territories: it consists of a geographical instance through the construction of concrete space (ports, off-shore structures ...) and economic proceeding involves a spatial organization of transportation and exploitation in the territory. These instances are managed by the superstructure, consisting in a representation of the territory and a political and legal case involving power over the space considered, at the same time international (Montego Bay convention) and national (e.g. EEZ, territorial seas).

This territorialization process forms geographical, legal and economic constraints, related to maritime territories.

Constraints are formalized as barriers on the maritime network and explain the overall structure of the maritime network.

Rodrigue [START_REF] Rodrigue | The geography of transport systems[END_REF] distinguishes two types of barriers for transport networks: the absolute and relative barriers.

The absolutes barriers prevent all movements within the network. In a maritime network, this is the land/sea interface which corresponds to the limit of the maritime space. Indeed, the land/sea interface is a discontinuity and therefore is characterized by an impassable limit.

Relative barriers reduce and restrict the degree of freedom and the space connectivity. Thus, they affect the routes used to connect two parts of the network. They correspond to the capes, straits, channels, iceberg, restricted area, weather, and any element affecting the freedom of use in maritime space.

These structural elements (physical constraints and political boundaries) explain the configuration of maritime lines between the ports. Indeed, they have an influence by degree of constraint on the maritime network structure.

IV. NETWORK VULNERABILITY MEASUREMENT

The structure of the maritime network has an impact on the maritime network vulnerability. The main problematic of transport network is the accessibility of locations. This accessibility is measured by the degree of connectivity into the network. To assess the vulnerability of a network, several indicators are used.

A. Typology of vulnerability measurements

Connective and access vulnerability, robustness and reliability are the three mains indicators to measure the impact of disruptions. These three concepts are complementary and rely on different conceptualizations of network security:

1) Connective and access vulnerability Zavitsas [START_REF] Zavitsas | The Vulnerability of The Petroleum Supply Chain[END_REF] defined network vulnerability both as a probability of occurrence of a disruption on a network and as a degree of degradation resulting in a performance drop once the disruption performed.

The two fundamentals notions to measure the vulnerability are accessibility and connectivity. D'Este and Taylor [START_REF] D'este Et | Modeling network vulnerability at the level of the national strategic transport network[END_REF] differentiate the "connective vulnerability" and "access vulnerability". The connective vulnerability involves the configuration of the links. It is related to the extension of the shortest paths, a link between two nodes is vulnerable when the degradation following a disruption increases the cost of travel between two nodes. The access vulnerability focus on nodes in the graph: a node is vulnerable if a disruption greatly reduces its accessibility.

2) Robustness

Robustness refers to the network's ability to normally function despite a malfunction of a component following a disruption [START_REF] Snelder | The importance of alternative routes for the robustness of a road network[END_REF]. Static robustness refers to the ability to maintain connectivity after random removal of part of the network. Dynamic robustness is interested in redistribution of flows after a disruption.

3) Reliability

Reliability is a user-oriented view of vulnerability. The most common definition is from Wakabayshi and Iida [START_REF] Wakabayashi | Upper and lower bounds of terminal reliability of road networks: an efficient method with boolean algebra[END_REF] which is the probability that the system works properly for a period when the user tries to use the system.

B. A structural vulnerability of transport networks

The work of Gleyze [START_REF] Gleyze | La vulnérabilité structurelle des réseaux de transport dans un contexte de risques[END_REF] distinguishes the different aspects of the vulnerability of a network. This work is interesting for our research because the conceptual framework used in this work isolates structural measurement for the network.

Gleyze [START_REF] Gleyze | La vulnérabilité structurelle des réseaux de transport dans un contexte de risques[END_REF] identifies three parts in the vulnerability of networks: the material part, the structural part and the functional part. The material part refers to the potential of the network infrastructure damage (e.g. roads, railways, stations, ports), the structural aspect determines the network relational (ability to connect each part of the network) and the functional aspect indicates the vulnerability for the transportation services offered by the network. These three vulnerabilities are linked, for example, if a road is vulnerable to a flood (material vulnerability), it will directly affect the potential of the relational network (structural vulnerability) and therefore the use of the network (functional vulnerability). This distinction isolates the spatial structure through the conceptualization of the notion of structural vulnerability. The network is described as a mediator, a relational object linking different parts of the ocean (in this case of study). Through this conceptual framework, we can use spatial properties on the network (topology, geometric) to measure the network vulnerability.

C. Measurement of network efficacity and solicitation

Gleyze uses two main measures to determine the structural vulnerability: network efficiency, which is the degree of degradation of the disruption, and the network solicitation, which is the structural response of the network use after a disruption.

The network efficiency measures vulnerability facing network disruption by the degradation of effectiveness at local and global levels. The average distance is used to measure for each given component its accessibility. In case of disruption, overall efficiency is calculated by the additional length (distance, time or cost) to connect all points of the network.

The second measure used is the network solicitation. It allows the understanding of the impact of a disruption in terms of redistribution of relationships on the network. It measures the total weight of the shortest paths on each component. After a disruption, it can distinguish which component will be solicited and therefore how the network uses the structure to respond to the disruption. This conceptual framework and associated measurements are useful in our work. Nevertheless, to use the concept and measurement of structural vulnerability, we have to adapt them to our specific study case. Gleyze work focuses on urban networks but maritime transportation has is own specificity: the transportation possibilities are not a set of way but a surface with infinite deviational possibilities. To obtain an assessment of maritime network, we have to propose a model taking into account theses specificities.

V. MODELLING THE MARITIME NETWORK VULNERABILITY

The territorialization of maritime space results in a spatial differentiation. This process creates heterogeneity into this space, and maritime territories have different features depending of their location and spatial content. For transportation, this can have effect on the safety and security at sea. Some areas within the maritime space facilitate or hinder transportation. According to the spatial features of territories, maritime space can ease or reduce transportation possibilities. For example, a high traffic density area or a stormy weather area in the ocean can reduce the accessibility and affects the efficacy of transport.

A. Measuring vulnerability at sea

We aim to measure efficacy to improve knowledge on the relationship between the ocean as spatial structure and the network as spatial use of this structure. As seen before we will use the framework of structural vulnerability defined by Gleyze and these measurements: efficacy and solicitation. Vulnerability can be access by the difference of efficacy and solicitation between a regular configuration of the network and a network configuration after disruption.

A disruption causes a deviation of regular use of the spatial structure (e.g weather condition). To avoid these impacted spaces, ships have to crossed additional distance. According to Gleyze [START_REF] Gleyze | La vulnérabilité structurelle des réseaux de transport dans un contexte de risques[END_REF] efficacy is the total length to reach all nodes into the graph. In our study case, we focus on the total length to reach ports. Solicitation is measured by the number of shortest paths transiting thought a corposant of the network. Basically, this corresponds to the maritime territories crossed by ships. After a disruption, territories used for the deviation became shortest paths for several destinations, causing a change into the maritime territories. This means they are variations into each maritime territory.

In maritime transport, additional distance becomes a lengthening of the total time of travel. This additional time is an extension of the cost of each travel. Time lost has a global impact on the economy, by a decrease of transport efficacy. To model it, we have to add locations reached during a disruption.

B. Creating location into the maritime network

The first step in the process is to create locations within the maritime space. These locations are elements that guide and influence ships direction and speed during travel. These locations do not necessarily have the same ability to influence the trip. These are capes, channels, straits, TSS as checkpoints for the trip, or barriers for navigation like stormy area or piracy zone. Each location within the space can produce knowledge on the territory and spatial differentiation.

These constraints are the structural elements of the maritime territory. The primary advantage of this approach is to create geographic data for a better understanding of maritime territories. In this figure(fig), 3 ports are link by the maritime space and 3 constraints exist into the space. C1 and C2 are checkpoints (TSS, good weather) and C3 a zone to avoid (piracy zone).

C. Network formalization

The second step in the process is to link these constraints (checkpoints or barriers). A sea route can be defined as a directed path between ports of origin and destination with a succession of passages and intermediate ports, Paths can be defined like a sequence of location to reach or to avoid (in case if the barriers is on the shortest between two nodes): AB= (A,c1,c2,B) AC= (A,c1,C) BC= (B,c2,-c3,C) For example, a trip form port B to port C will start to B, go thought node c2, avoid node c3 and reach port C. This step formalizes topological structure to the maritime network by maritime territory constraints. This formalization allows the topological representation of the use of space for transport (fig. 4). ). This results in a graph, and represents the maritime space and the constraints of territory. Ports, as entry and exit door of maritime areas are located outside, and maritime constraint are inside the maritime space.

D. Measering maritime network vulnerability

From this formalization, the maritime network structure is able to measure the vulnerability of the network into the maritime space. A disruption corresponds to a change of state of attractive constraints into repulsive constraints. Zavitsas [START_REF] Zavitsas | The Vulnerability of The Petroleum Supply Chain[END_REF] demonstrate that the strongest disturbances are related to the closure of ports and channels (attractive to a repulsive place for navigation).

By using structural vulnerability measurement established by Gleyze [START_REF] Gleyze | La vulnérabilité structurelle des réseaux de transport dans un contexte de risques[END_REF] we can assess the maritime network.

VI. FURTHER WORKS AND CONCLUSION

Further works will focus on the modelling of the territory by a spatial agent based model, as propose by Langlois [START_REF] Langlois | Simulation des systèmes complexes en géographie[END_REF] to highlight effects of constraints in the maritime space. This methodology will allow to define territorial behaviors (attractive or repulsive for navigation) and to measure the influence and territorial dynamics on the navigation. Indeed, these attractive or repulsive constraints can affect a entire territory for navigation and change the constraint degree and attraction of a place into the ocean it is possible to know the influence of a place on the maritime space. Depending on the behavior of the place (checkpoint or absolute barrier), the territory can be split into several locator agent. These spatial agent can be attractive, repulsive or neutral (no significant effect on navigation) depending of the constraints into the territories.

Ultimately, the goal of this study is to demonstrate the spatial structure effects on spatial behaviors and conversely, spatial behaviors change the configuration of the spatial structure. To achieve it, multi-agent model will be used to simulate the dynamic of territories (attractive and repulsive zones) and its possible effects on spatiotemporal behaviors of ships. This can contribute to a better comprehension of the worldwide maritime network, its formalization and evolution, its performance as a spatial mediator, and its impact on the global economy.

Figure 3 .

 3 Figure 3. Ports and constraints into the maritime spaceThis methodology is used to provide information on the maritime territories to formalize the network.

Figure 4 .

 4 Figure 4. Proposed topological structure of maritime network