Isidora Stojanovic
email: isidora.stojanovic@ens.fr

De Se

Ed S Assertion

Goldberg

Ed Sanford Goldberg

Assertion

Keywords: de se attitudes, centered contents, the essential indexical, the first-person pronoun, subjectivity, disagreement, same-saying

De se attitudes, that is, attitudes that we have about ourselves in a first-personal way, have long been recognized as interestingly different from other attitudes. However, speech acts and, in particular, assertions that we make about ourselves have barely begun to draw philosophers' attention. This chapter discusses some recent proposals that aim to bridge the gap between the significance of the de se phenomena in thought and the way that we express those attitudes in language. Section 1 provides some background on the de se and the essential indexical. Section 2 surveys proposals that make use of centered contents in modeling assertion and communication. Section 3 discusses the main motivations for the idea that centered contents are not only the contents of de se attitudes, but also of the corresponding assertions.

Introduction: thinking of oneself as oneself

While the concept of the self has always been of interest to philosophers, the tight connection between first-personal attitudes and action has been brought to attention through the work of [START_REF] Castañeda | On the Logic of Attributions of Self-Knowledge to Others[END_REF], [START_REF] Anscombe | The First Person[END_REF], Perry (1977, 79) and [START_REF] Lewis | Attitudes De Dicto and De Se[END_REF], inter alia. The core issue that first-personal attitudes raise, also known as the problem of the essential indexical [START_REF] Perry | The Problem of the Essential Indexical[END_REF] or of de se attitudes [START_REF] Lewis | Attitudes De Dicto and De Se[END_REF], amounts to the observation that we can have beliefs and desires that happen to be de facto about ourselves, yet such beliefs and desires will not motivate the right sort of action, unless they are also beliefs and desires that we have about ourselves as ourselves, in a first-personal mode. The easiest way to see the problem is by way of examples: 1

The messy shopper John Perry once followed a trail of sugar on the supermarket floor, pushing his cart down the aisle on one side of a tall counter and back the aisle on the other, seeking the shopper with the torn sack to tell him he was making a mess. With each trip around the counter, the trail became thicker. But Perry seemed unable to catch up. Finally it dawned on him. He was the shopper he was trying to catch. Perry believed at the outset that the shopper with a torn sack was making a mess. But he did not believe that he himself was making a mess. It is only when he came to believe the latter that he stopped following the trail around the counter and rearranged the torn sack in his cart.

The grizzly bear attack

Fenrong and Deeti are hiking in the wilderness. They encounter a grizzly bear, which sets out to attack Fenrong. Both Fenrong and Deeti believe that a bear is about to attack Fenrong, and both of them want Fenrong to be safe. But they act differently. Fenrong drops to the ground in a fetal position and covers her neck with her hands, while Deeti reaches for a bear pepper spray in her bag and sprays it towards the bear.

These scenarios pose a challenge for theories which rely on these two tenets:

Tenet 1: a content-driven account of action It is the content and only the content of the agent's attitudes that predicts and explains how a rational agent will behave or act.

Tenet 2: the propositionality of attitude contents

The content of any given attitude is best modeled by propositions.

The challenge posed by the messy shopper case is that John Perry had all along the relevant propositional attitudes. He believed all along the proposition that the guy with a torn sack of sugar, namely, John Perry, was making a mess, and he desired all along that this guy would rearrange his torn sack and stop making a mess. But since Perry had those attitudes without realizing that he was the person about whom he had them, he didn't look into his own cart. The puzzle is, what other beliefs, desires or intentions did Perry come to have when it finally dawned on him that he was the messy shopper, and that led to a change in his behavior? 1 Example 1 is almost literally taken from Perry (1979: 22), while example 2 is slightly modified from [START_REF] Perry | Frege on Demonstratives[END_REF]. There are many other examples in the literature: the crazy Heimson who believes to be Hume, the amnesiac Lingens lost in the Stanford library [START_REF] Perry | Frege on Demonstratives[END_REF], the author of the Hiker's Guide to the Desolation Wilderness who can't figure out which way to go [START_REF] Perry | The Problem of the Essential Indexical[END_REF], the two gods in [START_REF] Lewis | Attitudes De Dicto and De Se[END_REF] who have all the propositional knowledge available, yet don't know who they are, Kaplan who sees himself in a mirror with his pants on fire [START_REF] Kaplan | Demonstratives[END_REF], etc. Some of these examples, such as Perry's Lingens or Lewis's two gods, are primarily targeted against the idea that general, descriptive knowledge can provide de se knowledge; others, such as the Heimson-Hume case, involve subtle issues about personal identity.

The challenge posed by the grizzly bear case is that both Deeti and Fenrong have beliefs, desires and intentions that, when reduced to their propositional content, are the same: the belief that Fenrong is about to be attacked by a bear, the desire that she should avoid the attack, the intention that the bear should be prevented from attacking Fenrong. What is it, then, that explains that they behave so differently?

The problem of the essential indexical and of the de se, like almost any other important philosophical problem, defies unanimity. Not only is there no consensus on the solution to the problem; there is not even agreement that there is a single, welldelineated, distinctive problem that must be traced to the first-personal character of the attitudes at stake. There is controversy regarding the scope of the phenomena to be accounted for: are they distinctively about the first-personal way in which we think about ourselves, or do they generalize to all sorts of ways in which we think about all kinds of individuals, objects, events, and what not? Similar puzzles arise when we have a belief about a moment of time without realizing that the time at stake is now, or about a place without realizing that the place at stake is here, or about somebody without realizing that the person at stake is the person standing right in front of us. Over the past forty years, the literature on the problem of the essential indexical and de se attitudes has moved forward tremendously, both in terms of understanding the underlying phenomena and clarifying what the problems are and which theories they target. 2 What matters for the purposes of this chapter is that first-personal attitudes pose a certain challenge, whether or not it is distinctive and unique. The chapter examines how this gets reflected at the level of assertion and communication. With this in mind, the remainder of this section recalls the proposals from John Perry and David Lewis.

Perry's way out of the problem was to give up Tenet 1. The gist of Perry's view is that the content of a person's attitudes is not the only thing that explains the person's actions. What is equally crucial to action explanation is how these attitudes are connected to one another and how they contribute to the person's overall cognitive architecture. While Perry's views on the matter haven't ceased to develop over the past four decades (see e.g. Perry (ms.) for his most recent ideas on the topic), it will suffice here to schematize the proposal, as it was developed in [START_REF] Perry | Frege on Demonstratives[END_REF](1979). The idea is to distinguish, for each attitude, between its content and its "role". In the messy shopper example, the belief that Perry had before realizing that he was causing a mess has the same content as the belief that he had afterwards. The change in his beliefs occurred not at the level of content but at the level of role. Only the posterior belief has a motivating role and is such that whoever has it will look for a torn sack in their own cart.

2

A lot of the critical discussion of the problem of the essential indexical and the de se, including [START_REF] Millikan | The Myth of the Essential Indexical[END_REF], [START_REF] Brinck | The Indexical 'I': the First Person in Thought and Language[END_REF], [START_REF] Stojanovic | For Whom Is the Problem of the Essential Indexical a Problem?[END_REF], [START_REF] Spencer | Is There a Problem of the Essential Indexical?[END_REF] or Magidor (2017), ultimately serves to clarify what the different problems are, rather than call into question the idea that first-personal attitudes pose a real challenge. For a recent defense of the idea that the challenge posed by the de se is distinctive indeed, see e.g. [START_REF] Ninan | What is the Problem of De Se Attitudes?[END_REF]. In the grizzly bear example, the beliefs that Deeti and Fenrong have are beliefs with the same content, beliefs that are true under the same circumstances, but nevertheless, beliefs that differ in their roles. Only Fenrong's belief is first-personal and, although it has the same content as Deeti's belief, Fenrong and Deeti are in different belief-states. Had Deeti been in the same belief-state as Fenrong, she, too, would drop to the ground and would act just like Deeti.

Lewis's way out of the problem was to give up Tenet 2. Given Lewis's view about propositions, modeled as sets of possible worlds, to believe a propositional content is to be able to distinguish between worlds that, for all you believe, may turn out to be the actual world, from those that may not. Lewis's diagnostics of the problem of the de se is that we need contents that allow us not only to distinguish between worlds, but also, given one and the same world, to distinguish between possible locations that one may turn out to occupy within that world. In other words, not all beliefs are propositional; some are "self-locating" beliefs. [START_REF] Lewis | Attitudes De Dicto and De Se[END_REF] proposes that just as the content of a propositional attitude is modeled by a set of possible worlds, the content of a self-locating attitude is modeled by a set, only not merely of possible worlds, but rather of centered worlds, that is, agent-time-world triples. For example, the content of a belief that one would express by saying "I am making a mess" is that content true at an agent, a time, and a world, if and only if the agent is making a mess at that time and in that world. Put in a more traditional jargon, centered contents are properties; the content of a person's de se belief that they are making a mess is the property of making a mess. Lewis's solution to the messy shopper is that there is also a change in the content of Perry's attitudes. For it is only when Perry realizes that he is the one with a torn sack of sugar that he acquires a belief whose content is the property of making a mess. In the grizzly bear example, Lewis suggests that Deeti and Fenrong have beliefs with different contents: Fenrong's belief has for its content the property of being about to be attacked by a bear, while the content of Deeti's belief is the proposition that Fenrong is about to be attacked by a bear.

One important further element is needed to complete Lewis's picture. It is the idea that to have a self-locating belief is to self-ascribe the content of that belief. When Perry realizes that he is making a mess, he comes to self-ascribe the property of making a mess. Now, [START_REF] Lewis | Attitudes De Dicto and De Se[END_REF] generalizes all attitudes, including propositional ones, to the idea of attitude self-ascription. The general schema is that to believe a proposition p is to self-ascribe the property of inhabiting a world in which p is true. This being said, for many third-personal beliefs, there can be other alternatives. For example, in the grizzly bear attack case, Deeti could be self-ascribing the property of visually attending to a scene in which a bear is about to attack Fenrong. 3 3 As has been pointed out by many, the mechanism of self-ascription plays a crucial role in Lewis's account so it may be said that he, too, gives up Tenet 1. Similarly, if we take into account Perry's complete writings, especially [START_REF] Perry | Thought without Representation[END_REF], we see that he is not committed to Tenet 2 either.

2

Centered content in assertion and communication

An immediate advantage of Perry's approach over Lewis's is that it sits well with the idea that one and the same content serves equally well as the content of an affirmative sentence that a person utters in context, the content of the belief that she expresses with such an utterance, the content of the assertion that she thereby makes, and, last but not least, the content that she communicates to her interlocutors. This conception of content as the appropriate level that fulfills all these functions is taken even further in the work of David [START_REF] Kaplan | Demonstratives[END_REF], who identifies it with 'what is said'. Consider the grizzly situation transposed to the level of discourse:

1. Fenrong: "I am about to be attacked by a bear." 2. Deeti (looking at Fenrong): "She is about to be attacked by a bear." 3. "Fenrong and Deeti both said/asserted/claimed that Fenrong was about to be attacked by a bear."

The fact that Fenrong is referring to herself using the first-person pronoun and that Deeti is referring to her using the third-person pronoun does not seem to have any impact on the contents that they assert, or on the information that they communicate in (1) and ( 2). This intuition is further supported by the fact that a report such as (3) strikes us as true. For Perry and Kaplan, these intuitions are easy to accommodate: the content of Fenrong's and Deeti's beliefs as well as of their utterances is the proposition that (at a certain time t) Fenrong is about to be attacked by a grizzly bear. The relevant cognitive differences between Fenrong and Deeti, which account for the differences in their behavior, are captured at the level of belief-states (or at the level of character, to use Kaplan's terminology).

In Lewis's view, on the other hand, the content of Fenrong's belief is the property of being about to be attacked by a bear, while the content of Deeti's belief is a different property, namely, depending on how one decides to incorporate third-person reference into the view, the property of standing in a relevant relation to an individual who is about to be attacked by a bear, or the property of inhabiting a possible world in which Fenrong is about to be attacked by a bear. For Lewis, then, the question of how belief content relates to the content that a person asserts and communicates is far from trivial.

While Lewis himself, to my knowledge, never claimed that centered contents play a role in assertion and communication, over the past decade, surprisingly many such proposals have seen light. The aim of this section is to give a brief and non-exhaustive survey of those. 4 Some of the details need to be postponed to section 3, in which I will 4 I am grouping the views in two families. The term 'recentering' is from Weber (2013), while 'uncentering' occurs both in [START_REF] Kindermann | Varieties of Centering and De Se Communication[END_REF] and [START_REF] Pagin | De Se Communication: Centered or Uncentered?[END_REF]; I am not aware that it is used by any of the proponents of such views. [START_REF] Kindermann | Varieties of Centering and De Se Communication[END_REF] uses the term 'multicentering', from [START_REF] Torre | Centered Assertion[END_REF], for the views that I will discuss in section 2.2. under the label 'sequence-relativizing'. For survey, comparison, and critical discussion of several among the views discussed here, see e.g. [START_REF] Kindermann | Varieties of Centering and De Se Communication[END_REF], [START_REF] Pagin | De Se Communication: Centered or Uncentered?[END_REF], or look at the motivation for the idea that certain kinds of assertionde se assertionsare best modeled by centered contents.

Uncentering and recentering

The most straightforward way of extending Lewis's proposal regarding de se attitudes to assertion and communication is to say that although the content of Fenrong's belief in ( 1) is the property of being about to be attacked by a bear, the content of her assertion is a different one, namely, an "uncentered" one, such as the proposition that (at a certain time t) Fenrong is about to be attacked by a bear. The combination of a Lewisian theory of attitude content and a Kaplanian theory of discourse content, at least for sentences such as (1), has been the preferred option for many theorists, including [START_REF] Egan | Epistemic Modals, Relativism and Assertion[END_REF][START_REF] Egan | Relativist Dispositional Theories of Value[END_REF], [START_REF] Moss | Updating as Communication[END_REF] and [START_REF] Kölbel | The Conversational Role of Centered Contents[END_REF]. As Egan (2012: 576) puts it: "Given [a certain] acceptance conditions-based story about the theoretical role of content in an account of assertion and communication, we definitely do not want to go for a semantic theory that assigns de se content to indexical sentences. That combination is a big disaster. […] So we ought not to believe that indexical sentences have self-locating content. We ought instead to believe the usual sort of Kaplanian theory."

The idea that the content of a person's de se belief need not be the same as the content that this person can reasonably hope to communicate to others is reminiscent of Frege's view, at least as it is presented in this often-quoted passage (1918/1967: 25-26):

Everyone is presented to himself in a particular and primitive way, in which he is presented to no one else. So, when Dr. Lauben thinks that the has been wounded, he will probably take as a basis this primitive way in which he is presented to himself. And only Dr. Lauben himself can grasp thoughts determined in this way. But now he may want to communicate with others, perhaps in the sense of "he who is speaking to you at this moment", by doing which he makes the associated conditions of his utterance serve for the expression of his thought.

While Frege has been often criticized for the idea of people having thoughts that cannot be grasped by others, the combination of a Lewisian account of attitude content and a Kaplanian account of speech content is not committed to such ungraspable thoughts. For anyone who thinks that they are about to be attacked by a bear grasps, in some relevant sense, the belief that Fenrong has when she thinks that she is about to be attacked by a bear. What the view does share with Frege's view is a demarcation between attitude content and communicated content. 5 And that may or the Appendix in [START_REF] Recanati | Mental Files in Flux[END_REF].

5

As can be seen from the passage quoted from [START_REF] Egan | Relativist Dispositional Theories of Value[END_REF], proponents of this sort of view Isidora Stojanovic, De Se Assertion (chapter for the Handbook of Assertion, ed. S. Goldberg) 7 may not be a problem, depending on how one conceives of the thought-language interface more generally.

Let me also stress that some among the uncentering views, Egan's and Kölbel's in particular, hold that in certain cases, it is the centered content that gets asserted and communicated. The cases at stake are those in which subjective matters, such as matters of taste, are under discussion; I will return to them in section 3.1.

Let's turn to the recentering view proposed in Weber ( 2013). Weber's aim is to see whether, and how, we can maintain the idea that de se attitudes have centered content, as well as the idea that the utterances based on such attitudes express centered contents, without running into what Egan calls "a big disaster". Weber proposes that the speaker literally expresses a belief with a centered content, but the belief that the hearer acquires is not the one that the speaker expresses. Rather, they acquire a belief with a content suitably related to one expressed by the speaker. The acquired content is determined by the content expressed by the speaker together with the hearer's beliefs about how she is related to the speaker's context. To see how this works, consider the following variant of the bear example: 4. Deeti, speaking to Fenrong: "You are about to be attacked by a bear."

Weber's desideratum is that if Fenrong assents to the truth of Deeti's utterance, then based on this, she ought to be able to self-ascribe the property of being about to be attacked by a bear. But the belief that Deeti expresses does not have that property for its content; rather, its content is the property of addressing someone who is about to be attacked by a bear, and that is also the content of Deeti's assertion in (4). On Weber's model, what Fenrong needs to do is put together her belief that Deeti expresses the above content with her belief that she is the person whom Deeti is addressing in (4). Based on this, Fenrong acquires the belief that she is the one who is about to be attacked by a bear; that is, she comes to self-ascribe the property that she herself would express were she to say "I am about to be attacked by the bear." 6

Sequence-relativizing

For Lewis, de se attitudes, that is, attitudes that we have about and hold towards typically take the asserted content to align with the communicated content, rather than with the attitude content. Theoretically, there is also room for the view in which the asserted content is the content of the speaker's belief, rather that the communicaFted content. Given that, as argued by Paul Grice and widely endorsed in contemporary pragmatics, there are independent reasons to keep apart what is asserted and what is communicated, such a view is worth exploring. (The view from [START_REF] Weber | Centered Communication[END_REF], discussed below, could be seen as such a view.) 6 While Weber's model is cast squarely within a Lewisian centered content account, there are other models that share the spirit of the proposal. One such is put forward in [START_REF] Maier | Why My I Is Your You. On the Communication of De Se Attitudes[END_REF]; it is cast within Discourse Representation Theory [START_REF] Kamp | From Discourse to Logic[END_REF] but improves on the latter by clearly distinguishing the speaker's production perspective from the hearer's interpretation perspective. ourselves, are central, and all the other attitudes, de re and de dicto, derive from them. When Deeti and Fenrong look at the grizzly bear, call him Grizz, and when they both think that he is ferocious, their attitudes are only derivatively about Grizz. According to Lewis, the content of Deeti's and Fenrong's beliefs is the property of inhabiting a possible world in which Grizz is ferocious, and it is a property that both of them selfascribe. 7 Now, one may want to make room for the idea that Deeti's belief that Grizz is ferocious is a genuinely de re belief that Deeti has about Grizz, rather than a de se belief that Deeti has about herself. Here is a way to account for this idea without giving up the spirit of a Lewisian account. For Deeti to have the de re belief that Grizz is ferocious is for her to ascribe ferocity to Grizz directly. In other words, just as we may self-ascribe properties, we may ascribe properties to other individuals or objects. Similarly, we may ascribe relations to pairs of individuals or objects. Imagine that you are trying to choose between two melons, Mel, on your right, and Mil, on your left, and you believe that Mel is bigger than Mil. On a standard Perry-Kaplan account, the content of your belief is the singular proposition that Mel is bigger than Mil, and Mel and Mil come under suitable representations (or Kaplanian "characters"), such as "melon on my right" and "melon on my left". On a standard Lewisian account, the content of your belief is the property of having on your right a melon that is bigger than a melon that you have on your left, and you self-ascribe that property. On the proposed variant of Lewis's account, the content of your belief is, simply, the relation of being bigger than, and it is this relation that you ascribe to the ordered pair (Mel, Mil).

Recall from section 1 that an important set of motivations for a theory of belief is to account how having certain beliefs, desires and intentions may lead to action. We want an account that predicts that if you prefer a bigger melon, and you think that Mel is bigger than Mil, you will pick out Mel rather than Mil. In the Perry-Kaplan account, the connection between you and Mel, which secures that your action bear on Mel rather than some other melon, is mediated by your representation of Mel as the melon on your right. In Lewis's account, Mel's ending up being the melon that you pick out is secured by the fact that it uniquely satisfies the descriptive content of your thought. In the present account, the connection derives from your wordly relation to Mel, that is to say, from the fact that it lies right there on your right, that you are looking at it and touching it. Your external relation to it allows you to ascribe properties and relations to it directly, and to consequently act upon it. Of course, more would need to 7 This is the standard move for Lewis to convert de re attitudes into de se attitudes, but there are other options. Deeti may be self-ascribing the conjunctive property of inhabiting a world in which Grizz is ferocious and being related in such-and-such way to Grizz, while Fenrong may be self-ascribing a slightly different conjunctive property, being related to Grizz in such-and-such other way. Alternatively, the selfascribed property may be that of being related in such-and-such way to some ferocious grizzly bear, so that their beliefs would remain the same if it were some other bear instead of Grizz, or even no bear at all. In any case, it is essential to Lewis's account that all those contents are ultimately properties that the attitude holder self-ascribes.

be said about the details of such an account. But the main idea is to allow for contents that are not propositional, but that, pace Lewis, are not always self-ascribed either, but can be ascribed to other individuals and objects and sequences thereof.

The idea to relativize contents to sequences and to use the resulting notion of content in an account of assertion and communication was proposed independently in [START_REF] Stojanovic | What is said: an inquiry into reference, meaning and truth. Isidora Stojanovic, De Se Assertion[END_REF][START_REF] Stojanovic | Speaking about Oneself[END_REF], [START_REF] Ninan | De Se Attitudes: Ascription and Communication[END_REF] and Torre (2012). For Ninan and Torre, the main objective is to extend Lewis's account of de se attitudes to discourse. Both Ninan and Torre aim to maintain the idea that communication succeeds when the common ground gets updated with the asserted content, but to avoid the consequence that when Fenrong asserts "I am about to be attacked by a bear" her interlocutors acquire a de se belief that they are about to be attacked by a bear. For me, the driving motivations were to allow for de re attitudes that do not boil down to de se attitudes, and make room for direct reference and for the idea that property-like and relation-like contents can be believed of and asserted about objects and individuals directly. 8 What the three sequence-relativizing accounts have in common is the idea that contents are no longer to be modeled as functions that map centered worldsthat is, individual-time-world triplesto truth values, but rather, they map sequences (a1, a2,... an, t, w) to truth values. 9 Since there are significant differences between, on the one hand, the accounts put forward by Ninan and Torre, and on the other, the account that I proposed, I will present them separately.

Sequence-relativizing (1): Ninan's and Torre's proposal

The idea common to Ninan and Torre is that the notion of a centered world should allow for there being more than one center. This is why their approach is also known by the name of "multi-centering". For simplicity, let us focus on the version given in [START_REF] Ninan | De Se Attitudes: Ascription and Communication[END_REF], in which there are only two "centers", occupied by the speaker and the hearer. The proposal combines two ideas. First, the content of an assertion is a paircentered content; that is to say, it may be modeled as a function that maps quadruples (x, y, t, w) to truth values. Second, to each conversation there corresponds a conversational sequence, represented as an ordered pair (a, b). The role of a conversational sequence is to "stabilize" the center, in the following sense. The first member of the quadruple, viz. x, is interpreted by the first member of the conversational sequence, viz. a; and analogously, y is interpreted by b. Furthermore, if 8

The relevant notion of direct reference and aboutness is in the spirit of Donnellan's view (e.g. [START_REF] Donnellan | Reference and Definite Descriptions[END_REF]), which contrasts with the more popular Kripke-Kaplanian view according to which a mere use of a proper name already secures direct reference. A more recent source of inspiration is [START_REF] Dickie | Fixing Reference[END_REF], whose focus is on a cognitive rather than linguistic notion of reference. I discuss the motivations related to the notion of direct reference in [START_REF] Stojanovic | What Is Said, Linguistic Meaning, and Directly Referential Expressions[END_REF] and [START_REF] Stojanovic | Prepragmatics: Widening the Semantics-Pragmatics Boundary[END_REF]. On the topic of de re thought, see also the essays in Jeshion (ed., 2010). 9

For simplicity, I present here a generalization. However, in [START_REF] Ninan | De Se Attitudes: Ascription and Communication[END_REF], the sequence only contains two individuals, namely, the speaker and the addressee.

a utters a first-person sentence, using the pronoun 'I', the content that she asserts is the property λx: P(x, t, w); if she utters a second-person sentence, using the pronoun 'you', then it is the the property λy: P(y, t, w). Mutatis mutandis, if b utters a first-person sentence, the content will be the property λy: P(y, t, w); if she utters a second-person sentence, it will be the the property λx: P(x, t, w).

To see how this works, suppose that Deeti is talking to Fenrong. Deeti has a de te attitude, that is to say, an attitude about her interlocutor, that she is in danger. Let the context be such that its conversational sequence is (Fenrong, Deeti). Fenrong tells Deeti:

"You are in danger."

As Deeti is the second member of the conversational sequence, she assumes the role of y in the quadruple (x, y, t, w). Hence the content that she expresses is λx: x is in danger at t in w. Since Fenrong is the first member of the conversational sequence, she assumes the role of x, and thereby comes to ascribe to herself the above content and to believe what she herself could express by saying "I am in danger".

For both Ninan's and Torre's proposals, it is crucial that, within a conversation, the speaker and her interlocuters converge on what the relevant conversational sequence is. This is a crucial step that makes the transitions from 'I' to 'you' and from 'you' to 'I' possible. It is how a hearer can acquire a de te attitude on the basis of the speaker's de se assertion, and conversely, a de se attitude on the basis of a de te assertion whose addressee they are. How precisely a conversational sequence gets settled upon in a given context is a separate and non-trivial issue. [START_REF] Pagin | De Se Communication: Centered or Uncentered?[END_REF] raises the following objection to multi-centering proposals. (He targets Torre, but the same issue arises for Ninan.) For a conversation to be successful, the speaker and the hearer must agree on a conversational sequence, yet to do so, they must resort to some communication mechanism that cannot be captured in terms of a Lewisian theory of de se attitudes. Pagin further points out that if the way in which the speaker and the hearer establish a conversational sequence is mediated by some propositional content, then the way in which the speaker communicates de se information to the hearer must also rely on propositional content, thereby considerably weakening the idea that de se information ought to be captured non-propositionally.

Although Pagin has a point here, I submit that he is wrong to assume that settling on a conversational sequence must proceed by antecedent communication. To make such an assumption would be to assume that before somebody makes an utterance, they must first communicate to their interlocutors that they are the speaker. But this would be absurd. By the mere fact of making an utterance, the person makes available to their interlocutors the information that they are the speaker. Conversational sequences are established by a similar kind processor so I suggest. The speaker and the hearer coordinate on a conversational sequence by means that are not properly semantic but involve joint attention and similar processes that are at play when we make various things contextually salient and when we refer to things. 10

Sequence-relativizing (2): Stojanovic's proposal

I would now like to present the main gists of the proposal that I have been defending over the last decade. 11 Like Lewis, I reject the idea that attitude content must be propositional. Unlike Lewis but like several other authors discussed here, I also reject the idea that asserted content must be propositional. Unlike the uncentering and recentering proposals but like Ninan and Torre, I endorse the idea that indexical sentences also express such non-propositional contents. Another feature that my proposal shares with Ninan's and Torre's is that it models those contents not merely as mappings from centered worlds to truth values, but rather, as mappings from larger sequences of the form (a1, a2,... an, t, w) to truth values. 12 What is peculiar to my proposal is the idea that, for any parameter in the sequence, the speaker may, in principle, self-ascribe it, or they may ascribe it to something or somebody else. To see how this works, let us look at some examples. Consider again: 6. Fenrong: "I am in danger." 7. Deeti (looking at Fenrong): "She is in danger."

My proposal is that the content associated with both ( 6) and ( 7) is simply the property of being in danger. However, in ( 6), Fenrong self-ascribes this property, while in ( 7), Deeti ascribes this property to Fenrong. Since the terms 'self-ascribe' and 'selfascription' have a strong cognitive connotation, and since our focus is assertion, let me coin the terms 'self-assert' and 'self-assertion'. Then in (6), Fenrong self-asserts the property that in (7), Deeti asserts of, or about, Fenrong.

(6) is taken to be a case of self-assertion because Fenrong is using the first-person pronoun. More generally, we expect there to be a correlation between the use of the first person pronoun and self-assertion, the use of the second person pronoun and de te assertion, the use of the demonstrative pronoun 'this' and assertions about proximal things, the use of 'that' and assertions about distal things, and so on. Nevertheless, 10In [START_REF] Stojanovic | Prepragmatics: Widening the Semantics-Pragmatics Boundary[END_REF], I discuss the properties of the processes that underlie demonstrative and indexical reference, and I argue that they are neither semantic nor pragmatic but belong in what I label prepragmatics. Alternatively, if we work with a model according to which conversation always takes place against a common ground à la Stalnaker (2002), the reply to Pagin would be that common ground need not be initially established by communication. For yet another very insightful picture of the processes at stake, see [START_REF] Gilbert | On Social Facts[END_REF][START_REF] Gilbert | Mutual Recognition, Common Knowledge and Joint Attention[END_REF].

11

An early version of the proposal was given in [START_REF] Stojanovic | What is said: an inquiry into reference, meaning and truth. Isidora Stojanovic, De Se Assertion[END_REF], though one that does not explicitly appeal to a Lewisian account of de se attitudes. Subsequent versions were circulated over several years under Stojanovic (ms.), leading to [START_REF] Stojanovic | The Problem of De Se Assertion[END_REF] and [START_REF] Stojanovic | Speaking about Oneself[END_REF].

12

The sequences should actually contain (at least) two time parameters and two world parameters, but since nothing relevant to the present discussion hinges on this, I will stick with the simplified framework; see [START_REF] Stojanovic | Speaking about Oneself[END_REF] for the more general version. such correlations are not imposed by or encoded in the contents associated with the sentences that contain those pronouns. Rather, it is the pronouns' lexical function to indicate what kind of linguistic action the speaker is performing: whether they are referring to themselves, and relatedly, asserting something about themselves, or whether they are addressing their interlocutors and asserting (or, as the case may be, asking or commanding) something about them, and so on.

Before I wrap up this presentation, there are two more things to note. First, in the proposed account of ( 6) and ( 7), sequences are not needed; centered worlds would do. In this respect, the proposal differs from Ninan's and Torre's, whose accounts appeal to multicentering already for such simple sentences. The reason why I need sequences is that one and the same assertion can often be about more than one individual or object. Consider: 8. Deeti to Fenrong: "I will help you."

My proposal is that the content that Deeti expresses is a relation, namely, the one that holds, at time t, between two people whenever there is some later time t' at which the first helps the second. This content is a combination of self-assertion and assertion de te: Deeti self-asserts it as regards the first relatum, and asserts it about Fenrong as regards the second relatum.

The second thing to note is that there can be cases where it may remain unclear whether the content is self-asserted or asserted about somebody else. Consider: 9. Bator: "There is a shelter nearby."

The content of ( 9) is, once again, a property, namely, the one that is satisfied by an individual or object whenever there is a shelter nearby this individual or object. But, in contrast with ( 6) and ( 7), the sentence in (9) does not linguistically indicate whether Bator self-asserts this content or asserts it about something or somebody else. In many contexts, by uttering ( 9), Bator will be saying that there is a shelter nearby himself. But suppose that he is a ranger who is talking to Fenrong over a loudspeaker. Then he will likely be asserting this content not about himself, but about her, telling her that there is a shelter nearby her. In such a case, (9) becomes an instance of de te assertion. And yet in another context, Bator may not be addressing Fenrong but may still be asserting (9) about her; for instance, if he tells (9) to his fellow ranger as the two of them are watching the scene from distance.

3

De se phenomena in language

As must have become clear from the discussion so far, the driving motivation for many among the authors who use centered contents in their accounts of assertion and communication comes from the desire to put together a Lewisian account of de se attitudes with the view that the content that a speaker asserts and communicates is the content of the speaker's underlying belief. However, this is not the only motivation, and for some not even the most important one. As we will see shortly, some authors, such as [START_REF] Stephenson | Judge Dependence, Epistemic Modals, and Predicates of Personal Taste[END_REF], do not commit themselves to a Lewisian theory of de se attitudes, but use centered contents in modeling assertion-related phenomena.

There are two main sets of motivations for using centered contents in accounts of assertion and communication. The first are related to subjectivity. The most discussed cases involve judgments of personal taste, although the scope may extend to aesthetic and moral judgments, epistemic judgments, and even vagueness. The second set of motivations relies on the observation that de se assertions, that is, assertions that we make about ourselves using the first-person pronoun, are interestingly different from other assertions. Although the two sets of phenomena may be related, they have been mostly kept apart. Accordingly, I will discuss them separately.

Asserting one's taste

One of the most debated issues in philosophy of language in the past decade or two has been the question of how judgments of personal taste can at the same time be deeply subjective and give rise to disagreements. Here is an illustration: 10. Deeti: "Monopoly is boring." 11. Fenrong: "No, Monopoly isn't boring."

The issue is complex and I don't hope to address it in this chapter. (For an early discussion, see [START_REF] Kölbel | Truth Without Objectivity[END_REF]; for an overview of the debate, see e.g. [START_REF] Marques | Disagreeing in Context[END_REF], [START_REF] Stojanovic | Context and Disagreement[END_REF] or [START_REF] Zeman | Contextualist Answers to the Challenge from Disagreement[END_REF]). My more modest goal is to show how centered contents have been used in different attempts to account for such disagreements. [START_REF] Stephenson | Judge Dependence, Epistemic Modals, and Predicates of Personal Taste[END_REF] proposes that the truth value of sentences involving predicates of personal taste (such as 'boring' and 'delicious') and epistemic modals (such as 'might') be relativized to a so-called judge parameter, along with the world and the time parameter. 13 The resulting picture yields a notion of centered contenta content that must be evaluated for truth at a world, a time, and an individual. Importantly, however, the notion of a "judge" is to be distinguished from that of the speaker. The judge, that is, the person at whom the content is evaluated for truth, may be the speaker of the utterance that expresses the relevant content, but need not. Thus when Deeti utters (10), she assumes the role of the judge in taking the content that she asserts to be true, since Monopoly is indeed boring as evaluated from her perspective.

13

For predicates of personal taste, [START_REF] Stephenson | Judge Dependence, Epistemic Modals, and Predicates of Personal Taste[END_REF] follows [START_REF] Lasersohn | Context Dependence, Disagreement, and Predicates of Personal Taste[END_REF]. In this chapter, for reasons of space, I leave aside the discussion of epistemic modality; but see Cariani (forthcoming). Note that Stephenson's proposal departs from Lasersohn's in that she holds that the judge-dependence is also reflected in the syntax through a silent pronoun PRO that takes its value from the judge parameter.

But consider what it would take for Bator to accept Deeti's claim: Monopoly would need to be boring to him. 14 (10) stands in contrast with judgments of taste that are explicitly first-personal, such as: 12. Deeti: "I find Monopoly boring."

To accept Deeti's claim in ( 12), Bator only needs to trust it that Deeti is accurately reporting her taste; his own appreciation of Monopoly becomes irrelevant. [START_REF] Stephenson | Judge Dependence, Epistemic Modals, and Predicates of Personal Taste[END_REF], like [START_REF] Egan | Epistemic Modals, Relativism and Assertion[END_REF][START_REF] Egan | Relativist Dispositional Theories of Value[END_REF][START_REF] Egan | There's Something Funny About Comedy: A Case Study in Faultless Disagreement[END_REF], [START_REF] Kölbel | The Conversational Role of Centered Contents[END_REF] or Moss (2013), works with a Stalnakerian model of assertion and communication [START_REF] Stalnaker | Assertion[END_REF](Stalnaker , 2002)), according to which the goal of assertion is to lead to an update of the common ground. For Stalnaker, the common ground is modeled as a set of possible worlds; namely, those that, given what has been accepted in the conversation, could turn out to be the actual world. But since contents are now to be evaluated for truth not only at worlds and times but also at judges, there is a non-trivial question of how to think of the common ground. Stephenson's proposal is that it be modeled as a set of judgetime-world triples (which is tantamount to centered worlds). She further proposes that, given a conversation, the judge element in all these triples represents the plurality of the group of participants in the conversation. However, she opts for a norm of assertion that only requires the speaker to believe the asserted content to be true as evaluated at triples containing him or her as the judge: "The norm of assertion is crucially weak in a certain sense. In order for A to assert that S, A only needs to believe that S is true as judged by A, [but does not need to believe that S is true as judged by the whole group of conversational participants.] But if A's assertion is accepted by the other speakers and added to the common ground, it has the same effect as adding the proposition that S is true as judged by the group of conversational participants" (Stephenson 2007: 701). 15 As we have seen, [START_REF] Stephenson | Judge Dependence, Epistemic Modals, and Predicates of Personal Taste[END_REF] appeals to centered contents to account for predicates of personal taste and epistemic modals, without touching on the topic of de se attitudes. The proposals in [START_REF] Egan | Epistemic Modals, Relativism and Assertion[END_REF][START_REF] Egan | Relativist Dispositional Theories of Value[END_REF] and [START_REF] Kölbel | The Conversational Role of Centered Contents[END_REF] are fairly similar to Stephenson's, except that both of them appeal explicitly to Lewis's account of de se attitudes.

Speaking about oneself

In this final section, I'd like to discuss the phenomena that have motivated my own account, presented in section 2.2.2. In a nutshell, the phenomena highlight interesting properties of assertions that people make about themselves, that is, de se assertions, that are revealed in how people perceive and report what has been said or asserted. The phenomena at stake also pose a challenge for the mainstream accounts of assertion and what is said, especially regarding the contribution of the first person pronoun, a challenge that I call "the problem of de se assertion" in [START_REF] Stojanovic | The Problem of De Se Assertion[END_REF]. I am not aware that the problem has been discussed or even properly acknowledged elsewhere, except in [START_REF] Pearson | The Sense of Self: Topics in the Semantics of De Se Expressions[END_REF][START_REF] Pearson | A Judge-Free Semantics for Predicates of Personal Taste[END_REF]. 16 In what follows, I primarily wish to present the problem and then show it can be handled with the help of sequencerelativized contents. I will only sketch en passant why it is a problem for the mainstream view; for in-depth argumentation, see [START_REF] Stojanovic | The Problem of De Se Assertion[END_REF][START_REF] Stojanovic | Speaking about Oneself[END_REF].

Recall, from the beginning of section 2, that the Perry-Kaplan account appears to fare better than Lewis's when it comes to accounting for our intuitions regarding what people say and assert. Intuitively, what Fenrong said in (1), repeated below as ( 13), and what Deeti said in (2), repeated as ( 14), seem to be the same content, viz. the proposition that Fenrong is about to be attacked by a bear: 13. Fenrong: "I am about to be attacked by a bear." 14. Deeti (looking at Fenrong): "She is about to be attacked by a bear." But now consider: 15. Bator: "I am about to be attacked by a bear."

There is also a strong intuition that Fenrong and Bator are saying the same thing; for each of them is saying that they are about to be attacked by a bear. The fact that whenever two people both say/assert something about themselves in a first-personal way, we may perceive them as saying/asserting the same thing, is further supported by a systematic ambiguity that we see in speech reports. Consider: 16. Deeti said that she was about to be attacked by a bear, and so did Bator. 17. Deeti: "I am about to be attacked by a bear. Bator said that, too." 18. Deeti and Bator both said that they were about to be attacked by a bear. 16 [START_REF] Pearson | The Sense of Self: Topics in the Semantics of De Se Expressions[END_REF] summarizes her work as follows: "We adopt Lewis ' (1979) proposal that attitudes de se involve a self-ascription of a property and investigate how this view of mental content is reflected in natural language. The implementation favored is a strong version of Lewis' position: root and embedded clauses are uniformly treated as being of a property type." Regrettably, I am not able here to do justice to the complex set of motivations that Pearson discusses, many of which stem from the syntax and semantics of attitude-ascriptions.

In ( 16) and ( 17) there remains an ambiguity as to whether Bator said that Deeti was about to be attacked by a bear, or that he was about to be attacked by a bear; an ambiguity that, in line with a similar well-known syntactic ambiguity, may be called the strict-sloppy ambiguity in speech-reports. 17 The fact that the sloppy reading, disambiguated in (18), is semantically (albeit not necessarily pragmatically) available whenever the target sentence contains the first-person pronoun supports the idea that when two people assert the same thing, each about himself or herself, they are easily taken to assert the same thing tout court. Now, it has been often pointed out that the notion of what is said is versatile and that we report what other people said in ways that often do not reflect the content of what they said. [START_REF] Lewis | Index, Context, and Content[END_REF] famously objected to Kaplan's identification of 'content' with 'what is said', based precisely on the intuition that in cases such as ( 13) and ( 15), we easily hear Fenrong and Bator as "saying the same thing". However, Lewis did not hypothesize that there was anything peculiar about first-personal discourse. Rather, he thought that the notion of what is said was, as it were, up for grabs, and that it was a poor guide to any semantic insights about language. My aim in [START_REF] Stojanovic | The Problem of De Se Assertion[END_REF][START_REF] Stojanovic | Speaking about Oneself[END_REF] was to demonstrate that there is something special about de se assertion, and that it is this, rather than the versatility of what is said, that accounts for the availability of sloppy readings in same-saying reports.

A first piece of evidence that the first-personal assertions behave differently is that if we take a pair of sentences that differ minimally from ( 13) or ( 15), in which we replace the first person pronoun by a third person pronoun, then sloppy readings are no longer immediately available: 19. Bator (pointing at Fenrong): "She is about to be attacked by a bear." 20. Tarek (pointing at Deeti): "She is about to be attacked by a bear." 21. ?? Bator and Tarek said the same thing. 22. Bator (pointing at Fenrong): ?? "She is about to be attacked by a bear. Tarek said that, too."

While in the case of the de se assertions in ( 13) and ( 15) the difference in the firstpronoun pronouns' reference did not impede us from hearing the two utterances as same-saying, in the case of de re assertions, it does. To forestall a misunderstanding, I am not claiming that a sloppy report is never available for third-personal sentences. For instance, the same-saying report (22) becomes correct if the reporter makes it explicit that the two speakers were referring to different individuals: 23. Bator (pointing at Fenrong): "She is about to be attacked by a bear. Tarek said that, too, but about somebody else."

17

The ambiguity is typically discussed with respect to verb-phrase ellipsis. Thus "Deeti greeted her boss, and so did Bator" is ambiguous between the strict reading, viz. that Bator greeted Deeti's boss, and the sloppy reading, viz. that he greeted his own boss.

Even if (23) suggests that sloppy reports for de re assertion may be unavailable for pragmatic reasons, there remains an asymmetry between the de se and the de re that calls for an explanation.

A second piece of evidence that speaks in favor of a privileged status of de se assertion is that we can easily perceive different speakers who assert something about themselves as asserting the same thing, even if the sentences that they are using differ considerably:

24. Bator (talking to Tarek on Sunday 17 December): "I saw your sister last night at a concert in my neighborhood." 25. Fenrong (on Thursday 21 December): "I saw Deeti on Sunday night at a concert in Poblenou."

In a situation in which it is common knowledge that Deeti is Tarek's sister and that Bator leaves in Poblenou, Bator and Fenrong are easily perceived as saying and even asserting the same thing:

26. Both Bator and Fenrong said that they saw Deeti at a concert in Poblenou on Sunday night. 27. Bator asserted that he saw Deeti at a concert in his neighborhood on the night of Sunday 17 December, and Fenrong asserted it, too.

To be sure, as it stands, (27) remains ambiguous between reporting Fenrong as asserting that Bator saw Deeti at a concert that night vs. that she herself saw her at that concert. 18 We have again a strict-sloppy ambiguity for de se assertion, yet one that, unlike the case of ( 13)-( 15), cannot be explained by appealing to a sameness of the sentences that the speakers utter.

For reasons of space, I will not delve here into the problems that these phenomena pose for the mainstream view. 19 Instead, I would like to sketch how the account from section 2.2.2. handles these cases. The proposal combines three ideas:

(I) the content of a sentence is modeled as a mapping from sequences of the form (a1, a2,... an, t, w) to truth values; (II) for any parameter in the sequence, the speaker may, in principle, self-assert the content, or they may assert it about something or somebody else; (III) intuitions about what a speaker has said or asserted are derivative upon the semantics and pragmatics of discourse reports and same-saying. I have already discussed (I) and (II), so let me turn to the general idea behind (III).

18

In fact, ( 27) is three-ways ambiguous, since there is also a reading on which Fenrong is reported to have seen Deeti at a concert in her own neighborhood.

19

See [START_REF] Stojanovic | The Problem of De Se Assertion[END_REF][START_REF] Stojanovic | Speaking about Oneself[END_REF]. In particular, I show that the proposal to explain the same-saying in the case of ( 24)-( 25) by allowing for combinations of content and character to play the role of 'what is said' does not succeed, for there are many such combinations that do not give rise to sloppy reports.

Our practices of reporting what has been said, asserted, claimed, expressed, etc. and of reporting different people to have said, asserted, claimed, expressed, etc. the same thing are governed by a complex set of syntactic, semantic and pragmatic rules. From these, we can extrapolate two central cases. For any given a parameter in the contentrelativizing sequence:

(i) when two speakers self-assert the same content with respect to that parameter, they may be reported as same-saying (that is, as saying/asserting/claiming/expressing etc. the same thing);

(ii) when two speakers assert the same content about the same object or individual with respect to that parameter, and this is known to the reporter, they may be reported as same-saying (that is, as saying/asserting/claiming/expressing etc. the same thing). 20 Let me now show how we can the predictions for the cases previously discussed. The same-saying in ( 13)-( 14) and ( 13)-( 15) falls out straightforwardly. In ( 13) and ( 14), Fenrong and Deeti assert the same content, namely, the property of being attacked by a bear, and they both assert it about the same person, Fenrong. In ( 13) and ( 15), Fenrong and Bator assert again the same content, namely, the property of being attacked by a bear, and they both self-assert it.

Turning to ( 19) and ( 20), Bator and Tarek assert the same content, which is, again, the property of being attacked by a bear, but they assert it about different people, namely Fenrong vs. Deeti. They thus fail to fulfill the conditions under which it would be correct to report them as same-asserting. In other words, the reason why (21) does not strike us as a correct report is not that different contents have been asserted; rather, given the semantics of same-saying reports, the default interpretation of ( 21) is that if the contents asserted by Tarek and Bator are sensitive in truth value to an individual, they must have been either self-asserted or asserted about the same person. Nevertheless, this default interpretation can be cancelled, as in (23).

In ( 24)-( 25), Bator and Fenrong assert the same content, which corresponds to a three-place relation that obtains of a sequence (x, y, z, t, w) iff x sees y at z, at time t in world w. With respect to x, both Bator and Fenrong self-assert this content; with respect to y, z and t, they both assert it about the same things, events or times, namely, about Deeti, the concert in Poblenou, and Sunday 17 December. The conditions in (I) and (II) are thus met, predicting that reports such as (26) or ( 27) are correct, which is a welcome prediction. * * * My main aim in this chapter has been to discuss various accounts that try to bridge the gap between de se attitudes and assertion. The focus has been on accounts that appeal to centered contents, inspired by [START_REF] Lewis | Attitudes De Dicto and De Se[END_REF]. There are alternative proposals regarding the first-personal character of assertions, tightly connected with view about propositions and speech-acts, such as [START_REF] Moltmann | Two Kinds of First-Person Oriented Content[END_REF] or Hanks (2015), that I have left out for reasons of space. De se assertion constitutes a fertile area of topics where 20 Let me stress that (i) and (ii) are extrapolations from a more complex set of correctness conditions for same-saying reports; see Stojanovic (2016: 212-214) for details.

This is debatable, and my own stance regarding sentences such as (10) is that they are interpretable in more one way, the idea being that, depending on the context, (10) may be understood as reporting Deeti's own experience of Monopoly (viz. that it is boring to her) or it may be understood as a kind of generic claim (viz. that it is boring to people in general); see[START_REF] Stojanovic | Talking about taste: Disagreement, implicit arguments, and relative truth[END_REF]Stojanovic ( , 2017, forthcoming), forthcoming). If this is so, then there won't be any straightforward rule on how accepting (10) would lead to an update of the common ground, for what it takes to accept (10) would also depend on the context. Fortunately, this complication may be ignored for the purposes of the present discussion.15Stephenson thus endorses what may be seen as a hybrid view of assertion: she is committed to Stalnaker's model, but at the same time that holds that assertion has a norm. I thank Sanford Goldberg for pointing out this peculiar feature of Stephenson's account.

Acknowledgments

I would like to thanks Sanford Goldberg for the invitation to contribute this chapter and for providing helpful comments on the first draft. At an institutional level, I'd like to acknowledge ANR-10-LABX-0087 IEC and ANR-10-IDEX-0001-02 PSL.


R El-Hachem

E Roziere

F Grondin

A Loukili

INFLUENCE OF SULPHATE SOLUTION CONCENTRATION ON THE PERFORMANCE OF CEMENTITIOUS MATERIALS DURING EXTERNAL SULPHATE ATTACK

Despite the various studies on concrete submitted to an external sulphate attack in aqueous environments, this aggressive factor threatening the durability of concrete is not completely understood. A study has been made in our laboratory in order to explain the effect of the sulphate solution concentration during external sulphate attack. Length and mass measurements are registered continuously; leached calcium and hydroxide ions are also calculated. In parallel, a scanning electron microscopy observation is realized. The results show the strong influence of the sodium sulphate concentration on the kinetics of degradation: higher is the concentration, faster is the attack. It should be noted that the degradation mechanism is not really affected by the sulphate concentration which leads us to reconsider the fact that the relative proportions of expansive products depend on the sulphate concentration.

1.

INTRODUCTION

While sustainable development is one of our major aims, designing a durable concrete becomes a necessity. External sulphate attack, which could be defined as a reaction between sulphate ions dissolved in solution and hydration cement products causing damages, is a real threat for concrete durability in an aggressive environment. Sulphate ions are present in ground water, sea water, soils, waste waters [START_REF] Biczók | Concrete corrosion and concrete protection[END_REF], [START_REF] Figg | Field studies on sulfate attack on concrete[END_REF]. The reaction mentioned above leads to formation of gypsum, ettringite, and or thaumasite and causes cracking, spalling, softening, expansion, loss of strength and other forms of damage can occur [START_REF] Monteiro | Time to failure for concrete exposed to severe sulfate attack[END_REF].

Physical and chemical factors can influence this attack such as the pH of the solution and its concentration, the associated cation, the binder type, the water/binder ratio and the curing conditions.

Many studies were made to explain the mechanism of sulphate attacks and the influencing factors. However the complicated nature of this attack has made these studies incomplete or conflicting in several times especially that most of them do not link between physical degradation and formation of expansive products.

The aim of this study is to highlight and explain the effect of sulphate concentration on the performance of the cementitious materials in order to draw a clear schematic diagram joining causes to effects. First we will present our experimental procedures used to evaluate this attack than we will expose our results (mass and length measurements, kinetics of leached Ca 2+ and OH -ions and visual and SEM observations) in order to try to explain this phenomenon.

EXPERIMENTAL PROGRAM

Mortar specimens, cast as 2x2x16 cm prisms, have been mixed with cement of type CEM I 52.5 N. Siliceous sand (Palvadeau) with a grain diameter size inferior to 2mm was chosen to avoid reactions due to calcium carbonate present in limestone aggregates [START_REF] Ferraris | Sulfate Resistance of concrete: A New Approach and Test[END_REF], [START_REF] Ferraris | Developing a more rapid test to assess sulfate resistance of hydraulic cements[END_REF]. The waterto-cement ratio (w/c) has been chosen equal to 0.6. Mortar prisms were cured in a 100% relative humidity (RH) for 24 hours until they were demolded. Then they were stored in lime water for 14 additional days. Mortars prisms were submerged at day 15 into three different solutions of sodium sulphate (SO 4 2-

) with three different concentrations (3g/l; 10g/l and 30g/l of SO 4

2-

). In parallel, two specimens were submerged in tap water. The prisms had gauge studs at both ends in order to measure expansion with length comparators. In addition, a resin layer was applied at these ends and also coated at 5 mm on the side. This type of protection gives us more representative and reproductive results for the length change of the entire specimen and not a part of it [START_REF] Ferraris | Sulfate Resistance of concrete: A New Approach and Test[END_REF], [START_REF] Ferraris | Developing a more rapid test to assess sulfate resistance of hydraulic cements[END_REF]. The device used for the sulphate test is illustrated in figure 1. Tests have been developed to study the effect of the concentration in controlled conditions at constant pH. The pH was regulated at 7.5 by adding a solution of nitric acid 0.5mol/l. In addition, the solutions were renewed every 15ml of added acid to maintain the concentration gradients as a constant and to avoid the interference between nitrate and cement hydrates. In fact, a neutral pH is meant to be representative of field conditions [START_REF] Ground | The American Well Owner[END_REF].

(b)

Mass and expansion were measured periodically. In parallel, a Scanning Electron Microscopy (SEM) observation and a Thermogravimetry Analysis (TGA) were made to provide us information about the products that have been formed during the attack in order to explain the obtained experimental results. Mass and length of specimens have been measured periodically each time the solution is renewed (Figures 2 and3). In parallel, the quantity of leached hydroxide ions was assessed through the added volumes of nitric acid HNO 3 solution, and that of leached calcium by the titration of the solutions (Figures 4 and5). In figures 2 and 3, we can see that for any sulphate concentrations, mortar's specimens have approximately the same response to the attack but with a delay in time with low concentrations. We can identify two stages in the evolution of the mass and the length:

RESULTS AND DISCUSSION

Chemio-mechanical analysis

-a first stage corresponding to the period of latency, when no significant expansion can be measured with a relative mass decrease caused by the leaching of portlandite, -a second step corresponding to a mass and length increase due to swelling of damaged mortar specimens.

These results are quite similar to those obtained in other studies where these 2 phases are also distinguished on mortars immersed in sodium sulphate solutions at different concentrations varying from 2.1 g/l [START_REF] Planel | Les effets couplés de la précipitation d'espèces secondaires sur le comportement mécanique et la dégradation chimique des bétons[END_REF] to 30 or 34 g/l [START_REF] Santhanam | Effects of gypsum formation on the performance of cement mortars during external sulphate attack[END_REF], [START_REF] Shanahan | Cement composition and sulfate attack Part I[END_REF]. If we correlate these results to the mass and length variations, we can say that at the first stage, leaching is the dominant phenomenon which explains this relative decrease in mass. While in the second stage, the formation of expansive products will dominate inducing expansion and mass augmentation. We have also observed these phenomena with the nucleation of macro-cracks and by using the SEM method.

Visual observation

All the specimens at different concentrations were suffering from the same symptoms (figure 6).

The first visual signals of sulphate attack were longitudinal cracks on the borders of the samples. These cracks appeared at the same time when the increase of mass and length was detected. When these cracks are more developed, transversal cracks are detected and lead to a two pieces broken specimen. E. Rozière [START_REF] Rozière | Etude de la durabilité des bétons par une approche performantielle[END_REF] and D. Planel [START_REF] Planel | Les effets couplés de la précipitation d'espèces secondaires sur le comportement mécanique et la dégradation chimique des bétons[END_REF] explained in their theses these surface cracks. The formation of expansive products induce tensile strength on the central part of the specimen relatively unattacked which explains these longitudinal cracks followed by the transversal cracks leading to the disintegration of the specimen.

This length variation is accompanied with a bending phenomenon represented clearly in figure 6-d. The upper face of the specimen is affected by bleeding that's why the sulfate ions may penetrate more easily from this face than the others so a more rapid expansion can occur. One of the solutions is to reduce the w/c ratio to 0.55 or 0.5.

SEM observation

After several observations on a SEM for different specimens exposed at three different levels of sodium sulphate concentrations at two different times, at the beginning and at the end of expansion, we can illustrate the microstructure evolution of mortars exposed to sulphate attack in figure 7. At the intermediate stage, gypsum is present in zones 1 and 2. A large crack is dividing the sample in 2 parts (figure 8): one totally disintegrated and another much more resistant.

At the end of expansion, zone 1 is totally leached and there is no more gypsum in this region. Also, C-S-H is decalcified leading to a non cohesive paste. In fact, the Thermogravimetry Analysis confirms these observations. The peak of portlandite is almost absent in region 1. In contrary, in region 4, the quantity of portlandite is most equal to that observed in the control specimen. One can observe that large voids are present in the interfacial transition zone (ITZ) around aggregate and the principal crack is much more developed. Gypsum is still present in zone 2. In fact, gypsum crystals would develop in the ITZ pores which lead to cracking and spalling and cohesion decrease. Gypsum could be found in different forms: plates, clusters, etc., as shown in figures 9 and 10. Mehta [START_REF] Mehta | Concrete, Structure, Properties and Materials[END_REF] explained the presence of gypsum in the ITZ: microcracks existed from the beginning in this zone. These microcracks, larger than capillary pores, are a preferential path for the creation of interconnections leading to a more permeable system. So the diffusion of sulphate ions is preponderant in this zone. Ettringite is not very stable at low pH (below 11.5~12) [START_REF] Santhanam | Sulfate attack research -Whiter now?[END_REF]. That's why it couldn't be found at the external leached zone. In fact, a study made by Mehta [START_REF] Mehta | Evaluation of sulfate-resisting cements by an new test method[END_REF] shows that at low pH, the principal mechanism of degradation is due to gypsum formation and the quantity of C 3 A (11.3% in our case) is not an influencing factor. At a constant 7.5 pH, ettringite may be transformed to gypsum and its presence is not quite certain. ) have shown that gypsum formation is the main cause of degradation. Also, based on the study of Mehta [START_REF] Mehta | Evaluation of sulfate-resisting cements by an new test method[END_REF], we have shown that gypsum formation has an expansive and destructive character and it is not only the formation of ettringite that leads to expansion even with a cement rich in C 3 A.

We should note that many factors are involved in the sulphate attack phenomenon. To study the effect of the sulphate concentration, other factors had been taken as a constant which may affect the obtained results. Our study is still limited to these conditions and can not be taken for a general conclusion for sulphate attack.

This study will be completed by other tests that will take into consideration the effect of the curing conditions (hydration degree and saturation rate) in order to explain and understand this complicated attack.

Figure 1 :

 1 Figure 1: Schematic diagram (a) and device (b) used for sulphate test

Figure 2 :Figure 3 :

 23 Figure 2: Mass change data for 3, 10 and 30g/l of Na 2 SO 4 solutions

Figure 4 :Figure 5 :

 45 Figure 4: Kinetics of OH -leached ions in 3, 10 and 30g/l of Na 2 SO 4 solutions

Figure 6 :

 6 Figure 6: a) Longitudinal cracks: first visual symptoms of sulphate attack. b) Expansion of the attacked specimen compared to control specimen. c) Transversal cracks leading to the fracture of the specimen. d) Buckling phenomenon of the attacked specimen compared to control specimen.

Figure 7 :

 7 Figure 7: Schematic illustration (left) of the attacked specimen (right).

Figure 8 :

 8 Figure 8: Large crack dividing the specimen in two parts

Figure 9 :

 9 Figure 9: Energy dispersive spectroscopy analysis on an attacked mortar sample.

Figure 10 :

 10 Figure 10: Different forms of gypsum crystals observed on each of the three specimens immersed in 3, 10 and 30 g/l of Na 2 SO 4 solution

Figure 11 :

 11 Figure 11: Ettringite spherulite existing in zone 2 cross section


Marwen Bouasker
email: marwen.bouasker@univ-nantes.fr

Frédéric Grondin
email: frederic.grondin@ec-nantes.fr

Pierre Mounanga
email: pierre.mounanga@univ-nantes.fr

Abdelhafid Khelidj
email: abdelhafid.khelidj@univ-nantes.fr

-F Grondin

- A Khelidj

Early Age Autogenous Deformations of Cement-Based Materials Multi-scale Micromechanics Modelling

Keywords: cement paste, autogenous shrinkage, early stage, modelling, experiments

come

Early age autogenous deformations of cement-based materials

Introduction

Cement-based materials experience, at an early stage, dimensional variations caused by cement hydration heat release, physico-chemical evolution of the material and moisture transfers with the surrounding environment. In sealed conditions and without significant thermal variations, the free deformation observed is defined as autogenous shrinkage: it is caused firstly by the volume difference between the reaction products and the corresponding reactants (cement and water). This chemical shrinkage creates partially-saturated pores, within the material structure, which selfdesiccate in the course of the hydration process. The self-desiccation of the porous network induces internal capillary tension and an external contraction of the material [START_REF] Justnes | Total and external chemical shrinkage of low W/C-ratio cement pastes[END_REF]. Typical autogenous shrinkage curves usually present two main phases, separated by the setting period: an accelerated phase, corresponding to chemical shrinkage, also called Le Chatelier's contraction, which is followed by a second phase characterized by a flattening of the strain curve (self-desiccation shrinkage).

Models based on empirical relations [START_REF] Tazawa | Chemical shrinkage and autogenous shrinkage of hydrating cement paste[END_REF] have been developed with the aim of calculating the strains due to the autogenous shrinkage in cement pastes and mortars. However, these macroscopic models do not take into account the microstructure heterogeneities. The homogenization methods have been developed for the determination of the equivalent mechanical properties of heterogeneous materials [START_REF] Bornert | Homogénéisation en mécanique des matériaux[END_REF]. Recently, these methods have been applied to the calculation of the mechanical properties [START_REF] Bernard | A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials[END_REF][START_REF] Sanahuja | Modelling elasticity of a hydrating cement paste[END_REF] and the autogenous shrinkage of cement pastes at an early stages [START_REF] Pichler | A multiscale micromechanics model for the autogenousshrinkage deformation of early-age cement-based materials[END_REF]. However, the model developed by Pichler et al. does not consider the chemical phase of autogenous shrinkage observed before setting.

In this work, we propose a multi-scales model to follow the evolution of cement paste autogenous strains from the end of mixing up to 48 h of hydration. From the mass fractions and mechanical properties of each main cement phase (C 3 S, C 2 S, C 3 A, C 4 AF and gypsum), the cement Blaine fineness and the water-to-cement ratio of the mixture, this model enables to calculate the chemical shrinkage and the selfdesiccation shrinkage of cement paste. The paper firstly presents the experimental part of the work. The multiscales micromechanics model is then described. Finally, the numerical and experimental results are compared.

Experimental study

The experimental validation tests have been carried out on a cement paste with a W/C ratio of 0.40. Ordinary Portland cement (CEM I 42.5 from Saint Pierre La Cour plant) was used. Its Blaine fineness is 339 m²/kg and its Bogue's composition is 62% C 3 S, 11% C 2 S, 8% C 3 A and 8% C 4 AF. All testing -Vicat setting times, chemical shrinkage, hydration degree and autogenous shrinkage -have been performed at a constant temperature of 20 ± 0.5°C.

Vicat needle tests have been made in accordance with European standards NF EN 196-3. Chemical shrinkage was measured by hydrostatic weighing, ensuring permanent water saturation of the pore volume of the material, according to the protocol described by [START_REF] Mounanga | Predicting Ca(OH) 2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF]. The hydration degree is obtained by loss on ignition method [START_REF] Bouasker | Influence of limestone filler on chemical shrinkage and hydration of cement pastes at an early stage[END_REF]: it consists of measuring the nonevaporable (chemically-bound) water content of the sample heated between 105 and 950°C. The hydration degree at time t is calculated as the ratio between the amount of chemically-bound water content at time t and the amount of water necessary for the complete hydration of cement.

Concerning autogenous shrinkage measurement, the hydrostatic weighing method has been used. An original testing device equipped with a rotary system has been developed in order to prevent the specimen from bleeding and sedimentation inside the latex membrane. Paraffin oil is used as the immersion liquid in order to limit liquid absorption by the sample.

Micromechanics modelling

Evolution of the phases during hydration

The modeling approach requires the determination of the evolutions of the volume and mass fractions of each phase of the material (unhydrated cement, water and hydrates) at an early stage. They are calculated on the basis of the chemical equations defined below:

CH CSH H S C 3 . 1 3 . 5 3 (1) CH CSH H S C 3 . 0 3 . 4 2 (2) 26 3 32 3 6 2 3 H S A C H H S C A C (3) 30 3 3 32 3 6 2 4 FH CH H S A C H H S C AF C (4) 12 4 32 3 6 3 3 4 2 H S A C H H S A C A C (5) 3 12 4 32 3 6 4 2 2 3 12 2 FH CH H S A C H H S A C AF C (6) 6 3 3 ) , ( 6 H F A C H A C (7) 3 6 3 4 ) , ( 10
FH CH H F A C H AF C (8)
Each cement component has a different hydration rate, which depends on its normalized affinity ) ( ~X A and its characteristic time X , according to Atkins [START_REF] Atkins | Physical Chemistry[END_REF]:

) ( ~ A dt d X X (9)
From the knowledge of the hydration degree ) ( X , it is then possible to calculate the volumes of clinker phases ) ( X V remaining in the mix [START_REF] Bernard | A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials[END_REF]:

)) ( 1 ( ) ( 0 t f V t V X X C X (10) where 0 C
V is the initial cement volume and X f the initial volume fraction of cement. In the same way, the amount of water remaining is equal to:

) ( ) ( 0 t V V t V X X E E E avec E E X X X C E C X E M n M f n V V / / 0 ( 11
)
where 0 E V is the initial volume of water, X E V is the volume of water used to hydrate the clinker X and the ratio E X n n / refers to the number of water moles E n consumed to hydrate one mole of clinker X n with an apparent density of

X C f .
The volume of the hydrated elements is computed from the volume of the reactants according to this chemical reaction:

P m P m P P P P R n R n R R R R V n V n V n V n V n V n 2 2 1 1 2 2 1 1 (12) with: n j j j i P i t C t V 1 ) ( ) ( avec i i P j j j C R i C j i M n M f n V C / / 0 (i=1,m) (13)
where i and j are respectively the indices associated with products and reactants whose mole number is R i n and P j n respectively. i is the product density and M i its molar mass.

The volume fractions computed from Equation ( 12) and the mechanical properties of each constituent of the cement paste microstructure provided in literature [START_REF] Bernard | A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials[END_REF] enable us to simulate the homogenized macroscopic behavior of the cement paste.

Equivalent homogeneous properties and deformation

Two major stages in the microstructural evolution of cement paste are considered: before the setting, the cement paste is "fluid" and the solid phases are in suspension in water. The capillary porosity (water and gas) of the matrix and the solid phases are chosen as inclusions. After the setting, the cement paste becomes "solid" and we consider a matrix formed by the CSH-phase in which are included the other solid elements (anhydrous cement and hydrates), as well as the pores. The elastic coefficients of each phase have been identified experimentally [START_REF] Bernard | A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials[END_REF]. By including in an iterative way the different inclusions in the matrix according to their volume fraction calculated above, and by using the self-consistent scheme method, the evolution of the homogenized cement paste Young's modulus can be simulated.

For the calculation of the equivalent strain, the self-consistent scheme method is used, distinguishing pores at different pressures [START_REF] Lemarchand | Contribution de la micromécanique à l'étude des phénomènes de transport et de couplage poromécanique dans les milieux poreux: application aux phénomènes de gonflement dans les géomatériaux[END_REF]. We consider pores containing water and air (whose volume fraction is equal to the sum of volume fractions of water and air) at the capillary pressure and pores containing ettringite at the pressure generated by its formation [START_REF] Ping | Mechanism of sulphate expansion II: validation of thermodynamic theory[END_REF]. The capillary pressure is computed using the following empirical law proposed by [START_REF] Coussy | The equivalent pore pressure and the swelling and shrinkage of cement-based materials[END_REF]:

(1 1/ ) ( ) ( 1)

Analysis and discussion of the experimental and numerical results

Application on a cement paste

Figure 1 shows a good correspondence between the results of hydration degree modelling and the experimental values obtained by loss on ignition. Besides, we observe that the evolution of chemical shrinkage is closely related to the progress of cement hydration. Figure 2 provides the simulation results of self-desiccation shrinkage and capillary tension plotted as a function of time. The curve shape indicates that self-desiccation shrinkage directly depends on the capillary tension rate.

Figure 3 shows that the multi-scales modelling enables us to correctly predict the evolution of Young's modulus after 24 h of hydration. The differences observed between numerical and experimental results between 0 and 24 h can be attributed to the strong viscous component which characterizes the behaviour of early-age cement paste and which is not taken into account in the present version of the model.

In Figure 4, the comparison between chemical and autogenous shrinkage results as a function of time of hydration highlights the good agreement between simulation and measurement results from the first contact between cement and water up to 24 h. After 24 h, the numerical results underestimate the experimental autogenous shrinkage values. This underestimation could be explained by a « real » capillary tension higher than that calculated with the model. It should also be noted that the numerical values of chemical shrinkage are slightly higher than the experimental values.

Cement properties effect

The cement fineness is defined by the cement particle size distribution. Previous experimental research works have shown that an increase in the cement fineness causes an acceleration of the cement hydration and autogenous shrinkage of cement pastes [START_REF] Bentz | Early-age properties of cement-based materials: I. Influence of cement fineness[END_REF]. In the model presented in this paper, the cement fineness is an input parameter for the calculation of hydration. Calculations have been performed to quantify the influence of this parameter on a cement paste with W/C = 0.4. The cement composition is given in section 2. Calculated results on the autogenous shrinkage and Young's modulus are plotted in Figure 5. It is observed that the lowest cement fineness gives the highest autogenous shrinkage and Young's modulus values. Indeed, higher cement Blaine fineness induces faster water consumption and therefore more intense capillary tension at 24 h. Besides, cement pastes reach more rapidly their final mechanical properties.

Conclusions

The numerical results presented in this paper demonstrate the capacity of the multi-scales modelling approach to simulate with a certain accuracy the cement hydration rate and the chemical shrinkage evolution. The Young's modulus is correctly estimated after 24 h of hydration. Between 0 and 24 h, the viscoelastic part of the cement paste's behaviour, which is not taken into account in the present version of the model, probably explains the differences between experimental and numerical results. Concerning autogenous shrinkage, simulated and measured values are closed at very early age and diverge after 24 h. Sensitivity tests have shown the influence of the cement fineness on the evolutions of autogenous strain and Young's modulus. Improvements of the model are in progress in order to better simulate the evolution of internal capillary tension and refine the simulation of cement paste autogenous shrinkage at an early stage ( 24 h).

  37.55 MPa and m = 2.1648 are the material parameters and S represents the water saturation rate of the cement paste.

Figure 1 .Figure 2 .Figure 3 .Figure 4 .

 1234 Figure 1. Evolution of hydration degree and chemical shrinkage (IS: initial setting time, FS: final setting time)

Figure 5 .

 5 Figure 5. Effect of the Blaine fineness on the autogenous shrinkage and Young's modulus at 24 h of hydration


Icíar Alfaro

François Gratecap

Arnaud Poitou

Elías Cueto

Francisco Chinesta

Numerical Simulation of Friction Stir Welding Processes

  L'archive ouverte pluridisciplinaire

INTRODUCTION

Friction stir welding (FSW) is a process that, although in its development stage, has been successfully used to join pieces of materials with poor weldability. Briefly speaking, it achieves welding of the pieces by employing a rotating ping that provokes both extremely high plastic deformation and also a high heat generation. It is represented in Fig. 1. In FSW, two pieces of sheet or thin plate are joined by inserting a specially designed rotating pin into the adjoining edges of the sheets to be welded and then moving it all along the seam. At first, the sheets or plates are abutted along edge to be welded and the rotating pin is sunken into the sheets/plates until the tool shoulder is in full contact with the sheets or plates surface. Once the pin is completely inserted, it is moved with a small nuting angle in the welding direction. Due to the advancing and rotating effect of the pin and shoulder of the tool along the seam, an advancing side and a retreating side are formed and the softened and heated material flows around the pin to its backside where the material is consolidated to create a high-quality, solid-state weld.

From the numerical point of view, such a process presents several challenging difficulties. First of all, the extremely large deformation present during the process always introduces numerical problems. Second, there is a strong coupling between these large deformation and heat generation, that in turn affects the behavior of the material. Meshless methods allow for a Lagrangian description of the motion, while avoiding the need for remeshing. Thus, the nodes, that in our implementation transport all the variables linked to material's history, remain the same throughout the simulation. Mappings between old and new meshes are not necessary and hence the avoidance of numerical diffusion. Among the many meshless methods available nowadays, we have chosen the Natural Element Method (NEM) [START_REF] Cueto | Overview and recent advances in Natural Neighbour Galerkin methods[END_REF]. It possesses some noteworthy advantages over other meshless methods that will be described shortly.

THE NATURAL ELEMENT METHOD

The Natural Element Method is a meshless method that employs natural neighbour interpolation in a Galerkin framework to solve the weak form of the problem. Consider a model composed by a cloud of points

N = {n 1 , n 2 , . . . , n m } ⊂ R d ,
for which there is a unique decomposition of the space into regions (the Voronoi diagram) such that each point within these regions is closer to the node to which the region is associated than to any other in the cloud (see figure 2):

T I = {x ∈ R d : d(x, x I ) < d(x, x J ) ∀ J = I}, where d(•, •) is the Euclidean distance function.
The dual structure of the Voronoi diagram is the Delaunay triangulation. Two nodes sharing a facet of their Voronoi cell are called natural neighbours and hence the name of the technique. Equivalently, the second-order Voronoi diagram of the cloud is defined as

T IJ = {x ∈ R d : d(x, x I ) < d(x, x J ) < d(x, x K ) ∀ J = I = K}.
The most extended natural neighbour interpolation method is the Sibson interpolant [START_REF] Sibson | A Vector Identity for the Dirichlet Tesselation[END_REF]. Consider the introduction of the point x in the cloud of nodes. Due to this introduction, the Voronoi diagram will be altered. Sibson [START_REF] Sibson | A Vector Identity for the Dirichlet Tesselation[END_REF] defined the natural neighbour coordinates of a point x with respect to one of its neighbours I as the ratio of the cell T I that is transferred to T x when adding x to the initial cloud of points to the total volume of T x . In other words, if κ(x) and κ I (x) are the Lebesgue measures of T x and T xI respectively, the natural neighbour coordinates of x with respect to the node I is defined as

φ I (x) = κ I (x) κ(x) .
FSW processes involve large deformation and high velocities of the rotating pin that make the elastic strains to be negligible. Although an elastic recovery exists, it is obvious negligible as a first approximation. The obvious advantage of this assumption is that the material can then be modelled as a non-newtonian (rigid-visco-plastic) fluid. This assumption is known as the flow formulation in the forming processes community. Thus, the essential variables of the problem will be velocities and pressures. From the numerical point of view, such a process presents several challenging difficulties. First of all, the extremely large deformation present during the process always introduces numerical problems. Second, there is a strong coupling between these large deformation and heat generation, that in turn affects the behavior of the material. A fully coupled thermo-mechanical formulation has been considered by means of a Picard algorithm in order to take into account this great amount of heat generated in the process. Meshless methods allow for a Lagrangian description of the motion, while avoiding the need for remeshing. Thus, the nodes, that in our implementation transport all the variables linked to material's history, remain the same throughout the simulation. Mappings between old and new meshes are not necessary and hence the avoidance of numerical diffusion. Among the many meshless methods available nowadays, we have chosen the Natural Element Method (NEM) [START_REF] Cueto | Overview and recent advances in Natural Neighbour Galerkin methods[END_REF]. It possesses some noteworthy advantages over other meshless methods that can be found in the before mentioned references.

NUMERICAL RESULTS

We consider here a very simplified two-dimensional, plane stress, model of the union of two plates. The model is composed by 240 nodes whose Delaunay triangulation is re-computed at each time-step. This is very fast process, which actually consumes only a few time, if compared with the equilibrium iterations. The pin is assumed to rotate at 1000 rev/min and to advance at 1mm/s. Despite that the usual friction coefficient between the pin and the sheet has been established to be around 0.5 we have considered here perfect adhesion between them. This will overestimate the strain produced by the pin rotation. The time increment chosen was 10 -4 seconds. In Fig. 4 (top) the equivalent strain rate is depicted. But the most interesting feature of the method, if compared to Eulerian or ALE formulations is, in our opinion, the possibility of tracking the flow of the material. This is shown in Fig. 4 (bottom). The formation of layers, also experimentally seen, is also found in this simulation. These layers suggest the presence of chaotic flow during the welding process.

CONCLUSIONS

A novel model has been constructed for the simulation of FSW processes based on the use of the Natural Element Method in an updated Lagrangian framework. This allowed us to track the layers of material during the process. The obtained flow patterns, also confirmed by experimental evidence, suggested the presence of chaotic flow during the process.

Figure 1 .

 1 Figure 1. Schematic representation of the FSW process.

Figure 2 .

 2 Figure 2. Delaunay triangulation and Voronoi diagram of a cloud of points. On the right, an example of degenrate (non-unique) triangulation.

Figure 3 .

 3 Figure 3. Definition of the Natural Neighbour coordinates of a point x.

Figure 4 .

 4 Figure 4. Top: Evolution of the equivalent strain rate during the beginning of the rotation. 10th and 90th time steps. Bottom: Evolution of the phases through the welding process.