Loïc-Michel Perrin

Kanuri Saharan

Polysemous Qualities and Universal Networks, Invariance and Diversity

The topic of this paper is the conceptual organization of polysemous prototypical qualities. This study, based on data collected in 24 languages, makes use of a single notional space composed of 110 notions. This space enables us to separately represent the polysemies observed in each language as well as polysemous patterns observable in several languages in order to contrast the variability specific to each language with the linguistic invariance. The results show that what is common in the language sample is based on recurring polysemies organized in networks. This method will also be useful in explaining how the linguistic variability is built up. Indeed, some of the qualities involved in these networks always take part in polysemous associations specific to only one language. Such qualities, called federative notions, are characterized by the fact that they are regularly involved in polysemous patterns, and across numerous languages.

Polysemous Qualities

Linguistic Discovery 8.1:259-280 5). Section 6 will conclude with a comparison between the semantic organization of the polysemous qualities and Lazard's (1992) approach to spatial organization and grammaticalization phenomena.

Language Sample and Data

Composition of the corpus

.

Introduction

The purpose of this article is to provide an analysis of the semantic organization of qualities involved in polysemous patterns. Following a joint study1 on the typology of adjectives and qualification in twenty-two African languages, to which I added French and English, this paper is an attempt to apply the semantic map method to represent the polysemous patterns of quality expressions.

It will be shown that what is common between the semantic maps of the language sample is not exactly a high number of recurring polysemous patterns observable in other languages but rather the existence of underlying cognitive frameworks. These frameworks called universal networks (see Section 4.1.) form the level of invariance on which the recurring polysemies are built up.

It is also an attempt to apply this method to language specific polysemous patterns and to explain how these unique polysemous patterns are made up. These patterns are characterized by two properties: (i) they are found in only one language of the sample and (ii) they almost always involve at least one quality, called federative notion, which is characterized by a particular semantic behavior. Indeed, these federative notions are defined by the fact that they are regularly involved in polysemous patterns, across numerous languages (see Section 4.2.); e.g., the federative notion [A] occurs in various unique polysemous patterns (e.g. [A, B], [A, C]) to which may be added recurring patterns cross-linguistically (e.g. [A, D] ,[A,E]).

After a short presentation of the language data and the theoretical framework (Section 2), this article will then examine and discuss a sample of the semantic maps, highlighting both recurring polysemous patterns and unique polysemous patterns (Section 3). The paper will aim at explaining the linguistic variability, which hinges on the semantic and cognitive invariance (Section 4), and the major principles involved in the elaboration of each semantic map (Section The data itself contains 110 qualitative concepts (see Appendix 1). In order to avoid ambiguities due to the polysemy of some English lexical items, French or German have been used instead. For example, the English word sharp can characterize something that has a pointed end as well as something that cuts easily. So, in this case, the German word SPITZ was used to designate the fact that something has a pointed end, and the word SHARP was retained to designate something that cuts easily.

At the beginning, we drew up a corpus composed of 113 qualitative notions. The selection of the qualitative notions was based on the list of prototypical qualities given by Dixon (2004:3-5), with the exception of color which constitute a particular system2 in some African languages. It should be remembered that Dixon distinguishes two kinds of semantic types typically associated with the adjective class (see Table 2). The first one is relative to four core semantic types, which are typically associated with both large and small adjective classes, and the second one is relative It does not seem that this opposition is significant in the present study. Each observation presented in this paper concerns notions relative to core semantic types as well as peripheral ones, without distinctions. The polysemous connections in the above-mentioned qualitative notions were observed in all 24 languages of the sample. A further nine notions had to be added because they occurred at least twice in polysemous patterns in the language sample. For example, we added the notion ACID because sixteen languages contain the pattern SOUR/ACID. Similarly, we added the notion CALM since it is involved in two different polysemous patterns: CALM/COLD in Bijogo and CALM/SOLID in Yulu. These additions concerned the following notions: ACID, CALM, CONSTANT, COWARDLY, FREQUENT, MAIGRE, NASTY, SALT, and SUPERFICIAL. Moreover, we also removed from the initial data the few qualitative notions which were never involved in a polysemy (HARD-WORKER, MULTICOLOURED, WHOLE, SQUARE) as well as those which were part of a marginal polysemous pattern (that is a pattern occurring in only one language and containing a notion which does not appear in the initial corpus). This concerns the notions UGLY, HUNGRY, DRUNK, JEALOUS, ACTIVE, DARK, RUSTY and THIRSTY. Thus, some fifteen polysemous patterns were not taken into consideration in the database. 3In the language sample, 256 polysemous patterns were found (see Appendix 2). 148 of them are particular to one language, 7 are shared by a minimum of 10 (up to 16) languages, and 16 are shared by 5 to 9 languages. Thus, polysemies which are language specific are more numerous than those attested cross-linguistically (roughly, 58% vs. 42%).

Theoretical framework concerning the notion of "polysemy"

By -polysemy‖ one usually refers to the fact that a same form is used to refer to two (or more than two) different notions. From a synchronic viewpoint, one can distinguish two kinds of polysemous phenomena: synonymy and strict polysemy [START_REF] Jacquet | Polysémie lexical. Sémantique et traitement automatique des langues[END_REF]. Synonymy is used Linguistic Discovery 8.1:259-280 whenever the different meanings of a polysemous word can be express by another word, as illustrated in Figure 1: (2) ACID/SOUR (two synonyms in French, English…) a. These wines taste sour  these wines taste acidic b. Ces vins ont un goût aigre  Ces vins ont un goût acide One talks about strict polysemy whenever there is no other word (no synonym) able to express the different meanings of a polysemous word (see Figure 2). (3) nooy (Wolof), soft (English) a. pleasant to touch (DOUX/French) b. Not hard (MOU/French)

The fact that a language uses the same word to refer to different notions, while other languages use different words, seems to prove that this language resorted to a polysemous process. Conversely, the fact that no language uses different words for two distinct notions seems to prove that an analysis in terms of separate representations is difficult (Haspelmath 2003:239). Among the one hundred and ten qualitative notions, it is always possible to observe two different notions unconnected in one language but which may be involved in a polysemous pattern in at least one other language. For example, even if the polysemous pattern ACID/SOUR occurs in sixteen languages, there are eight languages in which there is no specific word for the expression of these two notions simultaneously. It is thus possible to assume that each qualitative notion used in the corpus is vindicated. All the polysemies referenced in the database were collected from dictionaries as well as from responses elicited from native speakers by linguists who are specialists in the languages investigated. But, it is clear that the number of polysemies observed in each language is debatable. For instance, the study which I carried out on Wolof and French (my native language) shows three times more polysemies in French (see Figure 5) than in Wolof (see Appendix 3d). One could also compare the Trésor de la langue Française4 , which gives more than thirty synonyms for the notion DRY in French, whereas the French semantic map given in the present paper contains only two patterns. This variation is related to the degree of specificity of the meanings developed by a polysemous word. Indeed, following [START_REF] Jacquet | Polysémie lexical. Sémantique et traitement automatique des langues[END_REF], the different meanings of a polysemous word are only valid in a limited set of contexts. And the more numerous and varied the contexts, the less the meaning is specific. For example, the word sec (DRY) in French is synonymous with STINGY only when describing a person. But sec can also describe a low-fat (MAIGRE) entity, for instance a person, a diet, etc. Therefore, the pattern DRY/MAIGRE is more salient than the pattern DRY/STINGY in French. Nevertheless, one can consider the data valid insofar as the observed regularities only concern the salient polysemous patterns.

It is necessary to specify what is understood by -universal‖ in this paper. In order to account for the organization of the polysemous qualities, a pattern is considered as universally polysemous if it tends to be recurring across languages and cultures in more than two different languages. -Universal‖ is not defined by a systematic rule (remember that the so-called -universals‖ in typology always have exceptions), but by a tendency, or at best a potentiality, based on the observation of actual recurring polysemous patterns attested in the data.

A pattern recurring only twice in the language sample could be considered the result of a sheer coincidence or the result of a single polysemous process. In the latter case, one can posit that what is significant from a cognitive viewpoint is the fact that two different peoples with two different cultures, having had no contact whatsoever, have developed the same cognitive and linguistic process (metaphor, metonymy, generalization of a signified, specialization of a signified, cohyponymic transfer-Blank 2000; [START_REF] Koch | Pour une approche cognitive du changement sémantique lexical: Aspect onomasiologique[END_REF]Koch & 2004)). Whether such a capacity was developed in a few or many languages is not what is at stake: some universal phenomenon may be more recurring than others.

In order to reduce the possibility of the sheer coincidence of polysemous patterns but not to leave aside less recurring polysemous patterns, it seemed reasonable to limit the present study to patterns recurring at least three times cross-linguistically. Note that this is a working hypothesis to be further tested with in-depth studies on the basis of areal and genetic distributions. The consequence of such a methodological approach is that the more recurring a phenomenon, the higher its potential universality.

About the semantic maps

The polysemous connection between concepts is represented by means of a line linking them together. The schema (the diagram) thus obtained symbolizes the semantic network, the semantic map, of the polysemous connections as observed in a particular language. The notions are organized on the map so as to bring close together the most frequent polysemous patterns attested in the data cross-linguistically and make visible the network that they build. This Linguistic Discovery 8.1:259-280 organisation does not imply that the physical closeness between notions on the semantic maps is significant. Only the links matter-for practical reasons, it was impossible to represent the semantic proximity according to closeness on the semantic map when some qualitative notions are involved in up to thirteen different polysemous patterns. Furthermore, it must be kept in mind that the qualitative notions related to each other on the map are not necessarily expressed by the same word in each language; they may also represent two polysemous words having the same notion in common.

Presentation of the Semantic Maps

To start with, a sample of six semantic maps will be presented and discussed from two points of view: (i) the polysemous networks of each language and the quantitative specificities relative to the contrasting opposition -unique‖ vs. -recurring‖; (ii) the possible genetic or areal (borrowing) bias of some polysemous patterns.

Diversity and invariance from a qualitative viewpoint

The semantic maps presented in this paper represent the semantic associations5 observed in six languages: Bambara (Figure 4), French (Figure 5), Cerma, Gbaya, Jola, and Wolof (see Appendix 3a-d). It is not necessary to present all the semantic maps because there is no semantic map which looks like another, even partly. It simply seems that there are a number of invariants, i.e., polysemous patterns that can be observed in several languages. This also holds true for genetically related languages. If it is not debatable that genetically related languages may share polysemies, we think that this phenomenon is very rare. Let us take the case of the Wolof and Jola semantic maps (see Appendix 3c-d). Figure 3 represents the polysemous associations observed simultaneously in both languages (the red lines represent the patterns shared by both languages, the black ones the patterns observed only in Wolof, and the blue ones the patterns observed only in Jola). These two African languages both belong to the Atlantic North sub-group and are geographically close (both are spoken in Senegal); moreover, Wolof is the major vehicular language spoken by over 80% of the Senegalese population including some of the Jola people. Nevertheless, even though some semantic associations shared by both languages may imply genetic and/or areal features, the semantic maps for each language are really very different. In fact, there are only four patterns which are shared simultaneously by these two languages: ACID/SOUR, GOOD/GENEROUS, LITTLE/SMALL, and FULL/NUMEROUS. Linguistic Discovery 8.1:259-280 And patterns specific to one language are in black. Following the Bambara semantic map's example, it can be observed that polysemies particular to Bambara only (FAST/HOT, THIN/YOUNG, THIN/SMALL and HEAVY/SOLID) are scarce as compared with the nineteen polysemies6 also observable in other languages in the database. This remark applies to the five other languages discussed in this section and holds true for all the languages of the corpus (see Appendix 3a-d) including the two Indo-European ones (see Figure 5)-each semantic map shows more recurring patterns than specific patterns.

Diversity and invariance from a quantitative viewpoint

Whereas most of the semantic patterns of a language can be observed in other languages, still 58% of all conceptual pairs in the database are specific to only one language (148 specific polysemies vs. 108 recurring polysemies, see Appendix 2). In other words, the majority of the polysemous patterns observed for each language are recurring associations cross-linguisticallyin fact, 3/4 on average-while there is a majority of particular semantic associations in the database. These are only outward discrepancies. They are due to the fact that the recurrence of some polysemous patterns is so important that all languages are concerned: each language has an average of 6 specific polysemies (148 polysemies for 24 languages), i.e., one fourth of the polysemies observed on each semantic map, but, although only 18 polysemous patterns are shared by 4 languages (7 % of the database), each language contains an average of 3 polysemies Linguistic Discovery 8.1:259-280 observable in 4 languages (72 polysemies for 24 languages).

A quantitative comparison between two genetically and areally close languages highlights the problems linked to polysemous patterns, which may be due to genetic and/or borrowing reasons. For instance, the four polysemous connections7 common to Wolof and Jola can also be observed in other languages of the corpus, but only FULL/NUMEROUS is attested in another Atlantic language, namely Bijogo.

Among the 256 polysemies attested in the database, only two patterns may be explained by a genetic reason: FULL/NUMEROUS in the three Atlantic languages (Bijogo, Jola, and Wolof), EXPENSIVE/HARD in the two Gur languages (Cerma and Nateni).

As for polysemies due to borrowings in the case of geographical proximity, African languages only count six polysemous patterns that may be linked to areal factors (in some cases in addition to genetic factors): COLD/WET and EXPENSIVE/HARD (Cerma, Nateni), COLD/SLOW (Cerma, Nateni, Bambara, Tigemaxo), WHITE/CLEAN (Jola, Balante, Bambara), DICK/SOLID (Bambara, Balante), and BIG/ROUND (Yulu, Gbaya). Just as we assumed that a polysemous pattern shared by only two languages has one chance out of two of being the result of the same linguistic and cognitive process, there is no evidence to help one decide whether these semantic connections are due to universal, genetic, or areal factors. In consequence, these eight polysemous patterns represent the error margin.

Of course the number of possible borrowed polysemies increases a bit if one takes French and English into consideration since both languages are also spoken in almost all the African countries (but to various extents according to particular socio-linguistic situations and to speakers); the quantitative data might not be significant any more. For example, the pattern BIG/ROUND in Gbaya (see Appendix 3b) and Yulu8 also occurs in French9 (see Figure 5) and could be attributed to a borrowing from French. Still, we think that the sample of African languages is representative enough of this kind of genetic phenomenon.

One could add that lexical borrowing as well as polysemy are generally the consequence of a lexical gap-a language has no word to designate an entity or a concept [START_REF] Choi | Borrowing as a semantic fact[END_REF]. Moreover, even if a polysemous pattern is borrowed from another language, this pattern is initially triggered by a particular linguistic and cognitive process; it is thus highly probable that speakers reconceptualized the borrowed polysemous connection because of its lexical salience. Still, even if we do not question the hypothesis of genetically shared or borrowed polysemies, we think that they are marginal and cannot invalidate our working hypothesis.

Universal Polysemous Networks & Federative Notions

We will now deal with the representation of notional space for conceptual facts observable in several languages. We call conceptual map the spatial representation which enables us to represent polysemous patterns shared by several languages. The purpose is to highlight (i) the existence of networks made of recurring polysemous patterns, and (ii) the existence of qualitative notions which are regularly involved in polysemous patterns and across many languages. Furthermore, we will also take up a position on the psychological interpretation of the presented networks.

Linguistic Discovery 8.1:259-280

Recurring polysemous patterns and universal networks

The conceptual map presented in Figure 6 represents the recurring associations observed in at least three languages. The patterns shared by only three languages are represented by dotted lines, and the patterns shared by a minimum of four languages are represented by full lines. It can be observed, with very few exceptions, that the notions involved in these recurring patterns do not form a set of separate pairs but are organized in networks in the sense that these notions are related to each other. The polysemous patterns observed in three languages only are also significant since they do not change anything in the principle of networks-most of the threelanguages patterns are related to a pattern shared by a minimum of four languages. So, despite the fact that these semantic patterns imply a margin of error, they show that patterns with a low frequency of occurrence seem also to be built on notions involved in universal networks. The universal polysemous networks may explain what is common between each semantic map. Nevertheless, it is impossible to decide on the mental reality of such cognitive networks, or on the mental reality of the semantic maps. In order to prove such assumptions, we would first have to define what a mental reality is-is it related to the cultural representations shared by native speakers or is it related to metalinguistic representations built by linguists, or both? Whatever the answer, the fact that some qualitative notions are organized in networks does not mean that these networks reflect a pre-conceptual organization. -Network‖ here refers to the fact that some concepts are related to each other within the framework of polysemous connections, nothing else. Still, we can at least suppose that these recurring polysemous connections hinge on common conceptual reasoning based on basic cognitive experiences, potentially shared by each individual. In our opinion, these universal networks can indeed be assimilated to a potential stock of polysemies. But this does not mean that all speakers share all these connections in their minds. Actually, these patterns are based on universal cognitive abilities which can be developed very Linguistic Discovery 8.1:259-280 easily but which may also be hidden. The present study cannot explain why each language chooses certain recurring patterns rather than others.

Polysemous valence and federative notions

It is now necessary to introduce a new concept called the polysemous valence of a notion. In other words, the question is to determine the number of polysemous connections that a quality may involve whatever the number of languages concerned. For example, the notion GAY is connected to only one notion: it occurs in the pattern GAY/PROUD only observed in Cerma; so the number of polysemous valence of GAY is 1. The notion ACID has a polysemous valence of 4 since it can occur in four different patterns (see Figure 7). One of them, ACID/SALT, occurs in only one language-Bijogo; but the others can be observed in a minimum of two languages: ACID/SOUR (16 languages), ACID/BITTER (4 languages), and ACID/NASTY (2 languages). Similarly, the polysemous valence of HARD is 13 since this notion is involved in 9 patterns shared by up to 10 languages, as well as in 4 patterns specific to only one language. 10 With such a method, we can bring out the notions that are often involved in semantic patterns (even if it is not a recurring polysemous pair) and those which are involved in only one semantic pattern, as Table 3 shows. 10 See Figure 8 hot, constant, boiled, lose, proud, long, open, rotten, wise, sharp, near, round, frequent, ill, salt, far, maigre, rough, dry 3 droite, ripe, generous, stinky, fast, silent, old, spitz, wrong, handicapped, polite, lazy, painful, stubborn, shy, fearful 2 flat, cool, bent, blunt, dressed, lié, full, limping, numerous, new, gay, loud, stubborn, tight 1 A second important principle is called federative notions. These are notions defined by two properties: the number of polysemous valences as well as the number of languages concerned by these various connections. We can thus eliminate notions for which the polysemous valence is not really meaningful since the semantic connections are only limited to a few languages. For example, the notion EXPENSIVE is involved in seven patterns which only occur in four languages; CLEVER though only occurs in five different patterns but across eleven languages. From a quantitative viewpoint, we limited the set of federative qualities which are involved in a minimum of five polysemous patterns and across a minimum of six languages. These are SOUR, NASTY, CLEAN, GOOD, DOUX, BITTER, BAD, PURE, CLAIR, CHEAP, MOU, NARROW, WARM, SMALL, LITTLE, HEAVY, YOUNG, DIFFICULT, HARD, GESUND, FAT, DICK, DICKFLUSSIG, SOLID, STRONG, SMOOTH, COLD, DENSE, WIDE, WEAK, LÉGER, LARGE, BIG, EASY, THIN, CLEVER, and SLOW (italicized in Table 3).

Finally, in Figure 6, we contrasted the federative notions (written in bold red) with the universal cognitive networks. Most of the federative notions are involved in networks shared by a minimum of four languages. Only five federative notions occur in a network shared by three languages: NASTY, BAD, CHEAP, FAT, and DENSE.

From the Invariance to the Diversity

The spatial representations of polysemous patterns for each language (by means of semantic maps) as well as the patterns shared by several languages (by means of a conceptual map) proved to be very useful in bringing out a certain number of regularities concerning the organization of the semantic maps.

Considering such regularities, we will try and understand how the linguistic variability hinges on the invariance in order to model the organization of polysemous patterns represented Linguistic Discovery 8.1:259-280 in each semantic map. In other words, we will account for the correlation between the cognitive networks and the federative notions on the one hand, and the varied semantic maps built by languages on the other hand.

Organisation and elaboration of the semantic maps

In order to prove this assumption, we will contrast the federative notions on each semantic map (in bold red type above) by means of maps. The federative notions are of consequence in the organization of these semantic networks in the sense that whenever a particular pattern occurs (represented by a black line), it almost always involves a federative notion (82% of the polysemies specific to only one language). So, from a dynamic viewpoint, we can assume that whenever a quality is involved in recurring polysemies, and if this quality is a federative notion, then this quality should occur in other polysemous patterns specific to a particular language.

To illustrate the fact that the federative notions have a major impact on the creation of each semantic map, let us take the case of the federative notion HARD which is involved in thirteen polysemous patterns, as illustrated in Figure 8. Since this notion occurs in a universal network (see Figure 6), we should observe numerous languages which contain some of the nine recurring patterns involving the notion HARD (the connections in full lines in the above schema). But since the polysemous valence of HARD is 13, this notion also occurs in four other patterns which are particular to one language (the connections in dotted lines).

So, we can conclude that a quality which is a -federative notion‖ takes part in a universal network and may occur in one or several polysemous patterns observable in only one language.

The federative notions enable us to understand how the interaction between these levels of invariance (the universal networks and the federative notions) and the linguistic variability is built. In fact, the federative notions form the hard core of the universal networks, and all the other patterns, especially those which are unique, are built from these federative notions. Besides, we previously observed a similar semantic phenomenon when we noticed that most of the patterns shared by three languages were built on notions involved in the universal networks.

Modeling

In order to sum up all these observations and to model the system related to the elaboration of semantic maps (i.e. to explain how both kinds of invariants organize the semantic maps of each language), we will use schemas symbolizing the interactions between the cognitive level (represented by the conceptual map) and the linguistic level (represented by the different semantic maps). So, at the cognitive level, between these miscellaneous qualities (symbolized by the letter ‗X' in Figure 5), there are several universal networks based on common cognitive experiences and potentially shared by all speakers. And within these universal frameworks, we observed that some qualitative notions are more regularly involved in polysemous patterns than others-these are the federative notions (the letters ‗X' in bold circles in Figure 9). At the linguistic level, as a first stage, we can first notice on each semantic map some polysemous connections, which come out of the universal networks. But each language does not systematically resort to the same patterns, as illustrated in Figure 10a. As a second stage, some federative notions build particular polysemous connections specific to only one language (these are indicated by dotted lines on each semantic map in Figure 10b). So, we obtain two semantic maps different from each other.

Linguistic Discovery 8.1:259-280

Conclusions

To conclude this study, we would like to set some limits to our observations. Indeed, it is always possible to provide some patterns that go against our assumptions, e.g., the polysemies specific to only one language that do not involve a federative notion.11 But counterexamples are relatively rare. Moreover, as we said previously, we do not claim that the presented modeling reflects some systematic rules, but rather it reflects some regularities. We also do not forget that these regularities, based on a small sample of qualitative concepts, could be refined in a larger sample. This sample is made of a list of prototypical adjectives as given by [START_REF] Dixon | Adjective classes in typological perspective. Adjective classes: A crosslinguistic typology[END_REF]. So it is possible that the semantic behaviors that we modeled could be only specific to these notions. Other qualities, or even other kinds of concepts could give a different result.

Nevertheless, the presented regularities seem to prove the existence of several conceptual sets organized into networks and which are independent of the linguistic variability. These networks are based on linguistic and cognitive processes that can be easily developed (i.e. potentially shared by numerous languages). They seem to form the universal framework of the recurring polysemies. Furthermore, the fact that the polysemous patterns observable in only one language always involve a particular set of notions characterized by a particular polysemous behavior (i.e. the federative notions) reveals that the linguistic diversity is related to the invariance since the federative notions are all involved in universal networks. Besides, the federative notions seem to make up the semantic hardcore of the universal networks.

Furthermore, it seems possible to correlate the semantic regularities reflecting the organization of polysemous qualities with Lazard's (1992:427-434) grammaticalization cognitive model. If we conceive of the set of possible grammaticalized notions as located in a multidimensional space, we can observe that: 1) some areas of this conceptual space are such that most of the languages construct grammatical tools in these particular areas […]; and 2) some parts of the areas which act as -fields of grammaticalization‖ have preferential status. So, as Lazard did, we observed two levels of invariants. Namely, at the first level, there are some universal invariants shared by almost all the languages-what we called universal polysemous networks. And at the second level, some of the notions, which participate in these universal networks, have a preferential status because they are more frequently involved in polysemous connections than others in the same semantic field-what we called federative notions.

Linguistic Discovery 8.1:259-280

With regard to the semantic organization of grammaticalized notions or with regard to the semantic organization of polysemous qualitative notions, the underlying framework is always the same; it consists of two levels of invariance on which the linguistic variability hinges. But all in all, Lazard's idea about the organization of semantic spaces is really essential because his approach is based on a scale principle and not on a quantitative threshold level. First of all, it is difficult to define the quantitative limits which enable a statement about the universal character of a linguistic feature. The typical features of recurring semantic phenomena are fixed in accordance with a continuum; and what is observable for a highly recurring feature is also potentially valid for a low recurring feature. Second, the semantic features characteristic of one or a few languages seem to be related to the recurring ones since they are built on the basis of those recurring facts. In this sense, a typological method which consists of a contrastive study between what is highly recurring and what is unique neglects an important part of the corpus: all the features shared by few languages.

Figure 1 :

 1 Figure 1: Schematic structure and examples of synonymy

Figure 2 :

 2 Figure 2: Schematic structure and examples of strict polysemy

Figure 3 :

 3 Figure 3: Wolof and Jola semantic map

Figure 4 :

 4 Figure 4: Bambara semantic map

Figure 5 :

 5 Figure 5: French semantic map

Figure 6 :

 6 Figure 6: Conceptual map

Figure 7 :

 7 Figure 7: Polysemous patterns shared by the notion ACID

Figure 8 :

 8 Figure 8: Polysemous patterns shared by the notion HARD

Figure 9 :

 9 Figure 9: At the cognitive level

Figure 10a :

 10a Figure 10a: At the linguistic level (Stage 1)

Figure 10b :

 10b Figure 10b: At the linguistic level (stage 2)

Table 1 :

 1 Language sample

  Linguistic Discovery 8.1:259-280 to three peripheral semantic types, which are typically associated with medium-sized and large adjective classes:

		Core semantic types		Peripheral semantic types
	Dimension	big, small, long, short, wide, deep...	Physical property	hard, soft, heavy, wet, rough, strong, clean, hot, sour; well, sick, tired, dead, absent...
	Age	new, young, old...	Human propensity	jealous, generous, ashamed, eager... happy, cruel,	clever, proud,
		good, bad, lovely, atrocious,	Speed	fast, quick, slow...
	Value	perfect, odd, strange, curious, necessary, crucial, important,
		lucky...
	Color	black, white, red...

Table 2 :

 2 Dixon's semantic types associated with the adjective class

  in Section 5.1.

	Polysemous notions			Valence
	hard, dick			13
	Weak			12
	solid, cold, good			11
	strong, big, doux, narrow			10
	dickflüssig, gesund, slow, bad			9
	wide, easy, heavy, cheap, difficult, léger			8
	clean, bitter, fat, pure, mou, expensive, thin			7
	warm, clair, dense, large, small, sour, nasty, smooth, wet, rude, little	6
	young, empty, delicious, clever, lichter, sweet, calm, poor, straight	5
	short, acid, beautiful, white, shallow, stingy, fragile, deep, dirty, true, raw, idiot	4
	brave,
	ACID	SOUR	NASTY
		BITTER	SALTED
	Linguistic Discovery 8.1:259-280

Table 3 :

 3 Valence and polysemous notions

Appendix 1. List of the 110 notions studied

	healthy	en bonne santé	GESUND	SUPERFICIAL superficiel	oberflächlich
	GOOD	bon	gut	SWEET	sucré	süß
	HANDICAPPED infirme	behindert	TIGHT	serré	eng
	ENGLISH HARD ACID HEAVY BAD HOT BEAUTIFUL foolish BENT ILL BIG LARGE BITTER LAZY BLUNT light BOILED not dense	FRANCAIS dur acide lourd mauvais chaud (brûlant) beau IDIOT courbé malade gros grand, vaste amer paresseux émoussé LÉGER cuit espacé	DEUTSCH hart sauer schwer schlecht heiß schön dumm krumm krank groß groß bitter faul stumpf leicht gekocht LICHT(ER).	ENGLISH THIN LIMPING TRUE LITTLE WARM LONG WEAK untied WET LOUD WHITE thin WIDE soft WISE NASTY WRONG	FRANCAIS mince boiteux vrai petit, jeune chaud long faible humide détaché bruyant blanc MAIGRE large MOU sage méchant faux	dünn DEUTSCH wahr hinkend warm klein schwach lang LOSE feucht weiß laut weit mager weise weich boshaft falsch
	BRAVE tied up	courageux LIÉ	mutig festgebunden	NARROW YOUNG	étroit jeune	eng jung
	CALM	calme	ruhig	NEAR	proche	nah
	CHEAP	bon-marché	billig	NEW	nouveau	neu
	light	CLAIR.	hell	NUMEROUS	nombreux	zahlreich
	CLEAN	propre	sauber	OLD	vieux	alt
	CLEVER	malin	schlau	OPEN	ouvert	offen
	COLD	froid	kalt	PAINFUL	douloureux	schmerzhaft
	CONSTANT	constant	beständig	POOR	pauvre	arm
	COOL	frais	frisch	POLITE	poli	höflich
	COWARDLY	lâche	feige	PROUD	fier	stolz
	DEEP	profond	tief	PURE	pur	rein
	DIFFICULT	difficile	schwierig	RAW	cru	roh
	DELICIOUS	délicieux	schmackhaft	RIPE	mûr	reif
	DENSE	dense	dicht	ROTTEN	pourri	verdorben
	thick	épais	DICK	ROUGH	rugueux	rauh
	thick	épais (non-liquide) DICKFLÜSSIG	ROUND	rond	rund
	DIRTY	sale	schmutzig	RUDE	impoli	unhöflich
	soft	DOUX	sanft	SHALLOW	peu profond	seicht
	DRESSED	habillé	angezogen	SALT	salé	salzig
	right	DROITE	rechts	SHARP	tranchant	scharf
	DRY	sec	trocken	SHORT	court	kurz
	EASY	facile	einfach	SHY	timide	schüchtern
	EMPTY	vide	leer	SILENT	silencieux	still
	EXPENSIVE	cher	teuer	SLOW	lent	langsam
	FAR	loin	fern	SMALL	petit (de taille) klein
	FAST	rapide	schnell	SMOOTH	lisse	glatt
	FAT	gras	fett	SOLID	solide	fest
	FEARFUL	peureux	ängstlich	SOUR	aigre	sauer
	FLAT	plat	flach	pointed	pointu	SPITZ
	FRAGILE	fragile	zerbrechlich	STINGY	avare	geizig
	FREQUENT	fréquent	häufig	STINKY	malodorant	stinkend
	FULL	plein	voll	STRAIGHT	droit	gerade
	GAY	joyeux	fröhlich	STRONG	fort	stark
	GENEROUS	généreux	freigiebig	STUBBORN	têtu	stur
	Linguistic Discovery 8.1:259-280			Linguistic Discovery 8.1:259-280

PICS n°

(2004)(2005)(2006): -Typologie des adjectifs et de la qualification dans les langues africaines‖ Llacan (Langage, Langues et Cultures d'Afrique Noire -CNRS) / Universität Bayreuth (Afrikanistik I & II). The collaboration regarding the semantic study of polysemous qualities involved Dymitr Ibriszimow, Eva Rothmaler and Holger Tröbs (University of Bayreuth), Loïc-M. Perrin and Paulette Roulon (Llacan-CNRS).

Actually, such systems are defined by the fact that they mix the concept of color with the concept of brightness.

Especially for constraints relating to the organization of the semantic maps space.

The Trésor de la Langue Française (TLF) is a large 16 volume dictionary of the French language, published by the Institut National de la Langue Française (INaLF, former laboratory of the C.N.R.S.)

At this point, the links' colors are not important.

ACID/SOUR, STINGY/BITTER, WHITE/CLEAN, DOUX/MOU, NEAR/SHORT, LITTLE/SMALL, SLOW/COLD, SMALL /NARROW, WARM/HOT, WARM/FAST, SMALL/YOUNG, HEAVY/DICK, DICK/SOLID, SOLID/HARD, WIDE/LARGE, WIDE/BIG, BIG/LARGE, LÉGER/FRAGILE, and FAR/LONG.

ACID/SOUR, GOOD/GENEROUS, LITTLE/SMALL, and FULL/NUMEROUS.

These two languages are spoken in the Central African Republic.

-une femme ronde‖: a big lady (lit. a round lady).

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X -Semantic Map 1. -Semantic Map 2. X X X X X X X X X X X X X X X X X X X X X Conceptual Map

In fact, 18 % of the specific polysemies.X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X -Semantic Map 1.-Semantic Map 2.

Linguistic

 Discovery 8.1:259-280

Appendix 2. List of the polysemous patterns observed in 24 languages

No. of languages concerned

Polysemous patterns observed in the corpus

No. of patterns


Gilles Besnard
email: gilles.besnard@lmt.ens-cachan.fr

François Hild
email: francois.hild@lmt.ens-cachan.fr

Jean-Michel Lagrange
email: jean-michel.lagrange@cea.fr

Philippe Martinuzzi
email: philippe.martinuzzi@cea.fr

Stéphane Roux
email: stephane.roux@lmt.ens-cachan.fr

Michel Lagrange

Analysis of necking in high speed experiments by stereocorrelation

Keywords: Digital Image Correlation, Full-eld measurements, Necking, Spatial registration, Stereocorrelation, Temporal registration. 1 Introduction

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

the behavior of materials under dynamic loading conditions. In most cases, the load and mean strain experienced by samples are extracted from the analysis of the signals delivered by strain gauges glued on the input and output bars (i.e., usually quite far from the sample itself ). More recently, with the development of (stereo)correlation techniques [START_REF] Sutton | Advances in Two-Dimensional and Three-Dimensional Computer Vision[END_REF][START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF], it is possible to measure 3D displacement and 2D strain elds. When applied to pictures acquired by high speed cameras, the previous kinematic data are measured directly on the sample surface [START_REF] Tiwari | Assessment of High Speed Imaging Systems for 2D and 3D Deformation Measurements: Methodology Development and Validation[END_REF]. These techniques then allow for various studies to be performed to monitor high speed experiments. For instance, Barthelat et al. [START_REF] Barthelat | Dynamic torsion testing of nanocrystalline coatings using high-speed photography and digital image correlation[END_REF] analyze torsion tests on nanocrystalline coatings of tungsten carbide-cobalt cermets, Elnasri et al. [START_REF] Elnasri | Shock enhancement of cellular structures under impact loading: Part I Experiments[END_REF] compression tests on aluminum and nickel foams, Gilat et al. [START_REF] Gilat | Full eld measurement during a tensile split Hopkinson bar experiment[END_REF] tensile tests on copper samples, and Besnard et al. [START_REF] Besnard | A space-time approach in digital image correlation: Movie-DIC[END_REF] necking in tantalum. Kurigulige et al. [START_REF] Kurigulige | Measurement of transient deformation using digital image correlation method and high-speed photography: application to dynamic fracture[END_REF] follow crack propagation of an impacted sample. Tiwari et al. [START_REF] Tiwari | Application of 3D image correlation for full-eld transient plate deformation measurements during blast loading[END_REF] and Besnard et al. [START_REF] Besnard | Characterization of necking phenomena in high speed experiments by using a single camera[END_REF] analyze experiments during during blast loadings.

In most of the above mentioned cases, the measured strain levels do not exceed 10%. The experiments reported herein aim at monitoring the inception and development of necking. This analysis therefore requires to measure strain levels well above 100%. The aim of the present study is to show that such measurements are feasible, even at quite high strain rates, which imply dealing with low denition pictures (typically of the order of a few tens of kpixels).

Once these results are available, the next step will consist in comparing them with nite element simulations, which are not presented herein.

The paper is organized as follows. First, the experimental conguration is introduced. Second, the stereocorrelation technique implemented herein uses space and time registrations based upon a global approach to digital image cor-relation [START_REF] Besnard | Characterization of necking phenomena in high speed experiments by using a single camera[END_REF]. New initialization procedures need to be implemented to measure large displacements and large strains. Third, the results of the stereocorrelation analysis are presented to analyze necking in pure aluminum.

High speed experiments

The tested samples are either cylinders 5 mm in diameter, or parallelepipeds with a height of 8 mm and a width of 2.5 mm. Both geometries have gauge lengths varying from 10 to 40 mm (Figure 1). Their ends are threaded so that they can be xed on the input and output bars end. Dierent gauge lengths are tested to investigate size eects on the occurrence of multiple necking. Pure aluminum was chosen in the tests reported herein because of its very high ductility. Its Young's modulus is equal to 69 GPa, Poisson's ratio is equal to 0.33, and its density is equal to 2,700 kg.m -3 . In the present case, the tensile loading is created by the impactor that hits the shoulder of the input bar (Figure 2). The input bar then applies the load to the sample as in any classical conguration. The relative motion between the two bars allows the load to be applied to the sample.

A stereovision setup consisting of two identical PHOTRON APX RS fast cameras is used to determine the 3D shape of the external surface and the corresponding 3D displacements. All optical devices (e.g., lenses, supports) are identical for both cameras. Figure 1(a) shows the stereovision setup for which special attention was paid to lighting conditions. The two cameras are connected together and electronically synchronized so that pictures are acquired with a time delay less than a few 100 ns [16]. To protect the objective lenses from heat, a screen is put below them. For the eight tests presented herein, the frequency of acquisition varies between 45,000 and 52,500 fps with a (low) denition ranging from 37 to 57 kpixels (Tables 1 and2). The denition / acquisition rate is the result of a compromise so that enough pictures are available to perform a successful registration to analyze necking in samples with very small diameters.

Because of the chosen picture denition and acquisition rate, the displacement amplitudes between two consecutive pictures will be large and special pre-correction procedures are called for to allow the spatial and temporal registrations to be successful when using a global approach to DIC [START_REF] Besnard | Characterization of necking phenomena in high speed experiments by using a single camera[END_REF] applied to cylinders.

Stereocorrelation applied to high speed experiments

Extraction of a 3D shape is performed through stereovision exploiting at least two view points [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF]. When applied at dierent instants of time, it also possible to measure 3D surface displacements by stereocorrelation (or 3D-DIC), and thus the components of the displacement gradient in the tangent plane of any point of the observed surface. In the following analyses, a global approach to stereocorrelation will be used. It consists in using nite element meshes with 4-noded quadrilateral elements (i.e., Q4-DIC [START_REF] Besnard | Finite-element displacement elds analysis from digital images: Application to Portevin-Le Chatelier bands[END_REF]). Consequently, the spatial and temporal registrations will be performed with Q4-DIC. The interested reader will nd details on the practical implementation of the 3D-DIC code in

Ref. [START_REF] Besnard | Characterization of necking phenomena in high speed experiments by using a single camera[END_REF]. The Q4-DIC procedure rests on multi-resolution analyses in which successive linearizations and corrections are performed via a modied Newton algorithm [START_REF] Besnard | Finite-element displacement elds analysis from digital images: Application to Portevin-Le Chatelier bands[END_REF]. In the present case, the image denition is small so that very few scales can be used. Consequently, another route has to be followed when complex shapes (e.g., cylinders) are studied. It is based on the initialization of the correlation procedure.

For the two (spatial and temporal) registrations, the problem to be solved is of a dierent nature. For the spatial registration, the main challenge is related to the fact that the diameter of the sample is small so that large deformations occur when the image pairs are to be registered to reconstruct the 3D shape. For the temporal registration, large displacements and strains occur between two consecutive instants of time so that the coarse-graining technique cannot be applied as usual [START_REF] Besnard | Finite-element displacement elds analysis from digital images: Application to Portevin-Le Chatelier bands[END_REF] since the denition of the pictures is too small. Consequently, two dierent procedures are developed.

Initialization of the spatial registration

The aim of this rst pre-correction is to increase the surface of cylinders that can be monitored by stereovision, and the quality of the initial reconstruction. It consists in the evaluation of the low frequency components of the displacement elds. The latter ones have a large amplitude. When evaluated, the pictures are pre-corrected so that the classical registration techniques will be successful since the subsequent corrections are small in amplitude. It is also based on the a priori knowledge of the sample geometry that has, as a rst approximation, a symmetry of revolution.

The very rst reconstruction is performed without any information. It is restricted to a small part of the picture where the deformations are not too large.

From the reconstructed shape, a cylinder is sought that interpolates best, in the least squares sense, the cloud of reconstructed points. From this information, it is possible to compute for any point of the surface, its location in the pictures acquired by both cameras as shown in Figure 3. This projection gives the point locations and the displacement to apply to predict where any point in the picture of one camera has moved in the picture of the other one.

This rst estimate is used as initialization of the global approach. Because of the curvature of the object, the apparent displacement amplitudes are very important on the edges of the picture and very small in the central part.

When the pre-correction procedure is used, this eect is accounted for and the corrections are almost uniform in amplitude over the whole region of interest.

This makes the registration much easier and more robust. The eect of this pre-correction is to increase the surface that can be reconstructed as shown in Figure 4. A 50% increase of the reconstructed surface is obtained and the root mean square dierence between the reconstructed surface and a perfect cylinder is equal to 41 µm prior to pre-correction, and 19 µm afterward. This quantity is comparable in magnitude to the roughness of the reconstructed surface.

For parallelepipedic samples, the present procedure was not used since the shape of their external surfaces is easier to register. As such, the present precorrection does not apply to the temporal registration for both sample series. Another procedure will be followed.

Initialization of the temporal registration

The main objective is to decrease the reconstruction uncertainty by precorrecting the large displacements that occur between consecutive pictures. Two routes can be followed. The rst one consists in updating the reference picture so that large displacements can be measured by considering a series of pictures [START_REF] Chevalier | Digital Image Correlation used to Analyze the Multiaxial Behavior of Rubber-Like Materials[END_REF][START_REF] Hild | Multi-Scale Displacement Field Measurements of Compressed Mineral Wool Samples by Digital Image Correlation[END_REF], namely, the reference picture of the current analysis is the deformed one of the previous step. However, the larger the number of pictures, the larger the measurement uncertainty for the pictures as their number increases (i.e., the measurement uncertainties are gradually cumulated since the reference picture is updated in the analysis). The second is to use a rst estimate as an initialization of the temporal registration in the longitudinal direction. This initial estimate, which corresponds to the result of the analysis corresponding to route no. 1, no longer requires to update the reference picture. Otherwise, the number of measurement points decreases dramatically because of the low denition of the pictures.

In the transverse direction, a contour-detection technique (i.e., based upon thresholded gradients) is used and coupled with the evaluation of the axis of the cylinder, which was determined for the rst 3D shape. From these two quantities, a rst estimate of the radial displacement is obtained. When the initial (Figure 5 For parallelepipedic samples the transverse direction is treated in the same way as the longitudinal direction. With these initial guesses, the correlation algorithm is run and only consists in correcting for small deviations induced by the various hypotheses made during the initialization step. In particular, no image updating is needed when a series of temporal analyses has to be carried out.

The second advantage of the proposed strategy is related to the elements used in the Q4-DIC approach. Their sizes could be decreased thereby allowing for a richer analysis, in particular within the necks. An example of picture stabilization is shown for cylinder A in The transverse displacements are more modest in amplitude, but the main difculty is related to the shape of the surface. Had this initialization step not been considered, the analysis would not have converged.

In the following section, necking under dynamic loading condition will be analyzed by using the procedure developed herein.

Experimental results

In the following study, the size of the elements used in Q4-DIC to register spatially and temporally the dierent pictures is chosen to be equal to 8 pixels.

This size is a compromise between the measurement resolution and the spatial resolution [START_REF] Besnard | Finite-element displacement elds analysis from digital images: Application to Portevin-Le Chatelier bands[END_REF]. The latter is critical here since low denition pictures are used (Tables 1 and2).

Analysis of the 3D shape of samples with necks

The 3D shape of cylinders is obtained by using the following steps:

• Reconstruction of the surface in its reference conguration. The spatial registration makes use of the above discussed pre-correction procedure.

• Extraction of the cylinder axis by tting, in the least squares sense, the reconstructed shape with a cylinder.

• Evaluation of the distance of each 3D point with respect to the cylinder axis, and the corresponding radius.

• Estimation of the average radius in each plane perpendicular to the cylinder axis.

For the parallelepipedic samples, the shape of the external surface does not require any special treatment when the temporal pre-correction procedure described above is utilized.

The mean reconstruction uncertainty is of the order of 30 µm for 8-pixel elements and all the experiments reported hereafter. It was evaluated by propagating the registration uncertainty for the reconstruction process [START_REF] Besnard | Caractérisation et quantication de surfaces par stéréocorrélation pour des essais mécaniques du quasi statique à la dynamique ultra-rapide[END_REF]. The reason for this high value is due to the small denition of the images so that the physical size of one pixel varies between 50 and 83 µm (Table 1 and2), and the small element size (i.e., the smaller the element size, the larger the uncertainty associated with the correlation procedure [START_REF] Besnard | Finite-element displacement elds analysis from digital images: Application to Portevin-Le Chatelier bands[END_REF]). It is worth noting that thanks to the pre-correction techniques introduced previously, the uncertainty level is independent of the number of analyzed pictures since there is no need to update the reference picture.

Figure 7 shows the reconstructed outer surface for the four tested cylinders at the end of each image sequence. In the same gure, the pictures recorded by the left camera are also shown for comparison purposes. Even though the strains of the cylinders are very high in the neck, it is possible to follow their development with the correlation procedure adapted to low denition pictures. This is particularly the case for samples C and D for which only 64 pixels were available in the height of the picture (i.e., a very small angular part was visible by both cameras). For the parallelepipedic samples, the picture recorded by the right camera is shown for times close to fracture in Figure 8. The longitudinal strain elds are also given. All necks are visible.

To follow the development of necking in cylinders, Figure 9 shows the change of the radius prole with the picture number. As the denition of the pictures decreases (i.e., from samples A to D), the uncertainty level in the reconstruction increases (i.e., the apparent roughness increases from samples A to D).

This eect is due to the decrease of the angular part that is monitored. From these data, necking is clearly revealed, and leads to radial variations of more than 1.3 mm to be compared with an initial radius on the order of 2.5 mm.

It is possible to monitor the formation of necks by analyzing the longitudinal prole of the mean radius. At the beginning of the experiment, the radius decreases uniformly along the cylinder axis. Then, the prole becomes less uniform and one (samples A and D) or possibly two necks (samples B and C) are formed.

Loading condition

Apart from the change in the reconstructed 3D shape, it is possible to use the 3D displacement elds. Their mean component at both ends of the analyzed region of interest allows us to evaluate the mean longitudinal strain, which was checked to coincide to within measurement uncertainties with the mean major principal strain ⟨ϵ 2 ⟩. Figure 10 shows the change of the latter with time for the four tested cylinders. For each of the four tests, a virtually constant strain rate is observed, even after the rst plateau. From these data it can be concluded that the mean strain rates vary from 1600 s -1 (sample A) to 11 -1 (sample D). It is worth noting that even though the strains localize within the neck, the mean strain rate remains virtually constant as a result of the impactor traveling at a constant velocity (Table 3). In the Appendix, it is shown that these trends can be understood from the Lagrange diagram of the setup and the dierent states in the bars and the sample. The same trend is observed for the four parallelepipeds (Figure 11). The mean strain rates vary from 1400 s -1 (sample E) to 600 s -1 (sample H), see Table 4. These results

indicate that the experiments reported herein were strain rate controlled on a macroscopic level.

Strain elds

The strains are computed in the tangent plane of each considered surface point.

Because of the extreme conditions encountered herein (especially with cylinders), for instance the radius variations along the longitudinal axis (Figure 9), the strain elds that are derived from the projection of the displacement onto the local tangent plane will be uctuating more than usual. Further, in the case of large transformations, special care has to be exercised. Experimentally, only in-plane components of the deformation gradient F are accessible.

Consequently, some additional hypotheses are to be made to evaluate large strains. In the present case, nominal (or Cauchy-Biot) strains are computed.

It is assumed that the normal vector to the surface is an eigen direction. In each element Ω e , the mean in-plane deformation gradient F p reads

F p = 1 p + ∫ ∂Ωe u p ⊗ p ds (1)
where 1 p is the in-plane unit tensor, u p the in-plane projection of the 3D displacement vector u, p the outward normal to the element edges ∂Ω e lying on the surface tangent plane, and s the curvilinear abscissa. The mean inplane nominal strain tensor ϵ ϵ ϵ p is computed by considering the in-plane right stretch tensor S p associated with the polar decomposition of F p

ϵ ϵ ϵ p = S p -1 p (2)
In the present case, the strain resolution is of the order of 1 %. This value was determined by analyzing the very rst pictures for which no loading was applied. This value degrades as the shape of the reconstructed surface becomes noisier.

With the reported strain rates (Tables 3 and4) and the integration time of each experiment (Tables 1 and2), it is concluded that the mean strain increment during picture acquisition is less than 1 % (i.e., the estimated strain resolution). Consequently, even though strain rates larger than the mean value occur in the necks, it is believed that the integration time remains small enough not to bias the strain measurements.

As for the reconstructed surface, only the mean major strain ϵ 2 for each longitudinal position is analyzed. Figures 12 and13 show the major strain proles for the same pictures as those shown in Figures 9 for the cylinders. From these results it can be seen that the strains are mostly concentrated in necks after their inception and their levels signicantly lower elsewhere. For sample A, the mean major strain is equal to 52 %, and the strains in the neck exceed 180 %. For sample B, the strain levels in the neck reach values greater 300 %.

For samples C and D, the strains become even higher to reach 600 %. All these values are achieved by using at most 2 × 55 pictures in the correlation analysis. For the parallelepipedic samples, the strain elds at the end of the load history are shown in Figure 8. Very high strain levels are observed and the presence of one or two necks is observed.

In Figures 12 and13, the estimate of the neck length is depicted. Its physical size is given in Tables 3 and4. It varies between 1.3 mm (sample B) and 2.7 mm (sample F). No clear dependence with the strain rate is observed.

The comparison between the mean major strain and the average value in the neck(s) is shown in Figures 14 and15. From these data, the onset of localized necking is determined by looking for the instant when both curves depart from each other. It is also worth noting that in all cylinder-shaped specimens where two necks appear, the rst to initiate is never the one that will become dominant. This conclusion is dierent for parallelepipeds. In all cases where two necks appear, only one eventually develops. The strain levels corresponding to the inception of the dominant neck are reported in Tables 3 and4. No clear conclusion can be drawn from the analysis of the results in terms of strain rate dependence. It is worth remembering that the underlying elasto-plastic behavior of aluminum itself is, as a rst approximation, strain-rate independent [START_REF] Lindholm | Eect of strain rate on yield strength tensile strength and elongation of three aluminium alloy[END_REF].

Conversely, it can be concluded that the inception strain is a random variable with a rather large scatter. Such an observation could support the hypothesis that neck inception is controlled by initial geometric imperfection that cannot be resolved, rather than being dictated by the constitutive law.

Analysis of multiple necking

On the picture of the deformed conguration (Figure 7 At the beginning of the experiment, the radius reduction is uniform along the cylinder axis. From picture no. 21 and on, the proles become less uniform.

For picture no. 31, two necks are clearly formed and have about the same size.

After image no. 31, one neck continues to develop while the other one has stopped growing. The former will eventually lead to the failure of the sample.

With Figure 14(b), it is possible to conclude that neck inception did not occur at the same time, and that one of the necks concentrated the strains very early on after their joint formation. For sample B, the two necks could grow during about 300 µs, and the distance between the two necks is equal to 9.5 mm.

Similarly, from Figure 15(b,d), the time duration of both neck growths is equal to 250 µs for sample F, and 500 µs for sample H. The neck separation is approximately equal to 8.5 mm for sample F and 24 mm for sample H.

The previous analysis can be discussed in terms of Mott's model [START_REF] Mott | Fragmentation of Shell Cases[END_REF]. When multiple necks form, the regions connecting them do not yield but gradually unload. Each neck behaves independently as long as the information about unloading has not reached other necks. This information is carried by an unloading wave travelling at a speed of v y . The unloading wave caused by the rst formed neck will reach its neighbors and stop their growth before the inverse interaction occurs. In the present case, v y ≈ 40 m/s. This value is obtained from the above analysis. If the unloading wave is assumed to propagate at a speed of √ σ y /ρ [START_REF] Von Karman | The propagation of plastic deformation in solids[END_REF][START_REF] Taylor | The Scientic Papers of G. I. Taylor[END_REF], and when σ y ≈ 5 MPa for pure aluminum, a value v y ≈ 43 m/s is found. This result therefore validates the applicability of Mott's model. However, the fact that the rst neck does not become domi-nant is related to the particular loading conditions of split Hopkinson bars for which constant strain rate regimes are separated by a constant strain regime (Figures 10 and11).

Summary

A stereocorrelation technique was applied to the analysis of necking in high speed tensile tests. The spatial and temporal registration is based on a global approach to digital image correlation using Q4 elements. The low denition of the pictures due to the high acquisition frequency required pre-corrections of the temporal registration to capture large displacements and strains in cylinders. An additional pre-correction procedure of the spatial registration was needed since the radius of the sample was small. Both pre-correction techniques allowed for the reconstruction of 3D shapes in a large part of the external surface of the sample, and the measured strain levels was shown to exceed 600 % in the dominant neck prior to nal failure by using 8-pixel Q4 elements and 2 × 55 pictures.

Some additional gain to the previous results can be obtained by still improving the temporal registration. To decrease the measurement uncertainties, it is possible to resort to a spatiotemporal registration in which the displacement eld is not only decomposed over a spatial basis, but also a temporal one. The early results indicate that such approaches are able to capture localized phenomena [START_REF] Besnard | A space-time approach in digital image correlation: Movie-DIC[END_REF][START_REF] Besnard | Analysis of Image Series through Digital Image Correlation[END_REF]. Their extension to stereocorrelation is currently implemented and will be tested.

By following the change of the surface and longitudinal strain proles, the inception strain of necks is determined. In some cases two necks are formed and one of the necks is shielded by the other one, thereby interrupting its further growth. Only one neck is fully developed in all the experiments on cylinders reported herein. For long parallelepipeds two necks could grow simultaneously for a longer time. This result shows that neck formation is the result of a competition between multiple neck inceptions and the growth of dominant necks that obscure others as proposed by Mott [START_REF] Mott | Fragmentation of Shell Cases[END_REF]. The analysis of strain elds shows that the local levels are signicantly larger than the global ones. This is to be expected from any localized phenomenon such as necking. In the present case, a ratio of up to 15 is found between local and global strain levels.

When analyzing the dominant neck of the eight experiments reported herein, it is not possible to conclude about a strain rate sensitivity of their inception strain. The same conclusion could be drawn when the neck width was analyzed. Conversely, the inception strain itself is very scattered, and calls for a probabilistic treatment.

Last, these results are very useful to validate constitutive models describing necking. Moreover, the measured end displacements of the region of interest can be used as boundary conditions to the numerical simulations so that there is no need to model the complete split Hopkinson bar setup, but only a very small region of the sample. Further, the strain heterogeneities during the whole test were monitored and constitute a discriminating benchmark test for numerical modeling.

• State VII is characterized by V V II ≈ -V /2 and σ V II ≈ ZV /2. The incident bar is in tension again, but with stress and velocity levels less than those observed in state I.

• State VIII, which is similar to state III, corresponds to a new tensile loading of the sample.

• Similarly, state IX is very close to states II and V. 3 for the four tested samples. 39

11 Mean major principal strain as a function of time for the four parallelepipedic samples. The analysis of these data yields the strain rate reported in Table 4 for the four tested samples. 40 samples. The analysis of these data yields the strain rate reported in Table 3 for the four tested samples.

  (a)) and current (Figure 5(b)) contours are compared, transverse motions are estimated. The transverse displacement is assumed to keep the geometry axisymmetric so that from the analysis of a single contour it is possible to evaluate the displacement of any point of the external surface.

Figure 6 (

 6 e) when compared with the raw information (Figure 6(c)) and with the reference picture (Figure 6(a)). The corresponding longitudinal and transverse displacements are shown in Figure 6(b,d). The range of longitudinal displacements is of the order of 100 pixels, which is very large when considering only 15 pictures with a low denition.

  (b)) and on the longitudinal strain elds (Figure 8(b,d)) two necks are visible at the end of the experiments. The aim of the following analysis is to understand the history of the formation of these two necks. With these data, it is possible to follow the change of the mean longitudinal strain as a function of time and space, and therefore to monitor the formation of necks by analyzing the corresponding proles. The time evolution of the measured proles is shown in Figure 9(b).

  tested sample. Aluminum samples to be tested in tension with split Hopkinson pressure bars (b). 30 2 Principle of the tensile setup of split Hopkinson pressure bars used herein. 31 3 Projection of points of interest on an image pair of sample A. 32 4 Visualization of the maximum angle of the reconstructed shape prior to (sector delimited by continuous lines) and after (sector delimited by dotted lines) correction for sample A. 33 5 Contour detection (red lines) in the reference (a) and deformed (b) congurations of sample A. 34 6 Eect of pre-correction for temporal registration (sample A). 35 7 Pictures of the left camera when the necks are formed, and corresponding reconstructed surface for the four cylindrical samples. The color map corresponds to the radius. 36 8 Pictures of the right camera when the necks are formed, and corresponding longitudinal strain eld for the four parallelepipedic samples. 37 27 Radius as a function of longitudinal coordinate along the gauge length for the four cylindrical samples. The picture number is indicated. 38 10 Mean major principal strain as a function of time for the four cylindrical samples. The analysis of these data yields the strain rate reported in Table

12Fig. 1 .

 1 Fig. 1. Experimental setup used to monitor necking by using two high speed cameras

Fig. 2 .

 2 Fig. 2. Principle of the tensile setup of split Hopkinson pressure bars used herein.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Projection of points of interest on an image pair of sample A.

Fig. 5 .

 5 Fig. 5. Contour detection (red lines) in the reference (a) and deformed (b) congu-

Fig. 6 .

 6 Fig. 6. Eect of pre-correction for temporal registration (sample A).

Fig. 8 .

 8 Fig. 8. Pictures of the right camera when the necks are formed, and corresponding

Fig. 9 .

 9 Fig. 9. Radius as a function of longitudinal coordinate along the gauge length for

Fig. 10 .

 10 Fig. 10. Mean major principal strain as a function of time for the four cylindrical

Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .

 11121314 Fig. 11. Mean major principal strain as a function of time for the four parallelepipedic

Fig. 15 .

 15 Fig. 15. Mean major principal strain as a function of time for the four parallelepipedic

Table 1

 1 Experimental information of the four tests on cylinders.

	Test	A	B	C	D
	Gauge length (mm)	10	20	30	40
	Striker velocity (m/s)	18.7	24.2	26.6	26.6
	Picture denition (pixels)	384 × 96 640 × 80 768 × 64 896 × 64
	Physical size of 1 pixel (µm)	52	63	78	78
	Acquisition rate (fps)	52,500	50,000	52,500	45,000
	Integration time (µs)	4.0	4.1	3.9	4.0

Table 2

 2 Experimental information of the four tests on parallelepipeds.

	Test	E	F	G	H
	Gauge length (mm)	10	20	30	40
	Striker velocity (m/s)	19.1	24.2	26.7	26.7
	Picture denition (pixels)	384 × 112 512 × 96 570 × 80 768 × 80
	Physical size of 1 pixel (µm)	71	83	80	80
	Acquisition rate (fps)	50,000	50,000	50,000	45,000
	Integration time (µs)	4.0	4.0	4.1	3.9

Acknowledgments

The authors wish to thank Damien Delaplanche, Nicolas Granier, and Gilles Roy of CEA Valduc, Stéphane Pattofatto and Han Zhao of LMT-Cachan

Appendix

Lagrange's diagram of the Hopkinson bar setup (Figure 2) is shown in Fig- ure 16. For all the tested geometries, a rst loading ramp is followed by a plateau and a subsequent ramp before failure is observed. For the larger samples, a third ramp was needed. However, the results reported herein stop at the end of the second ramp (see shaded area). From Figure 16, it can be concluded that the reected signal coming from the transmitter bar has not reached the sample.

The following analysis is carried out by assuming that the impedance of the sample Z s is very small compared to that of the bars Z. The reections on the free surface of the transmitter bar are not accounted for since they do not alter signicantly the various states in the incident bar. The stress and velocity states are denoted by roman numbers in Figure 2. Let -V denote the velocity of the striker.

• State I is characterized by V I = -V /2 and σ I = ZV /2. In that case, the incident bar is in tension.

• For state II, V II = 0 and σ II = 0. The striker is now motionless.

• State III is characterized by V III ≈ -V and σ III ≈ -Z s V . The velocity of the interface between the incident bar and the sample is approximately equal to the initial velocity of the striker. The sample is loaded in tension.

• For state IV, V IV ≈ -V /2 and σ IV ≈ -ZV /2. The incident bar is in compression.

• State V is characterized by V V ≈ 0 and σ V ≈ 0. The sample is no longer loaded in tension.

• For state VI, V V I ≈ -V and σ V I ≈ 0.