Research on the relationship between emotion and (natural) language generally considers them from two complementary perspectives:

 an emotional state, a passion, an affect; 2. the semantic universe of signs of a language (either in the linguistic sense of la langue 1 or in the semiotic sense of le langage) that conceptualize and clarify the cultural vision, the meaning shared by a community of speakers about emotions, passions and affects.

Thus, in linguistics, we meet the distinction between expressive and descriptive words (Kövecses 1995): expressive words are, for example: "shit!" (expressing anger) or "wow!" (expressing a surprise); descriptive words are, on the other hand, lexemes of the "anger" genre which manifest, express a certain (cultural) vision of the emotional experience of anger (Greimas 1983).

In this article we will look at some of the themes dealt with in this dual perspective. After providing a quick overview of linguistic research on emotion (chapter 1), we propose to quickly explore these two perspectives by focusing more on:

1. the specific semantic space that characterizes our everyday understanding, the linguistically, historically and culturally situated common sense conceptions of the emotional experience (chapter 2 and 3); 2. the expression or staging of an emotional experience in the form of signifying textual environments called emotional display or emotionscape using signs and sign systems of the natural language and any other (non-verbal) languages (chapter 4).

1) Language based conceptions of emotion

As we can imagine, there are many and varied types of research on the question of (semantic) conception and expression of emotion or emotional experience through signs and systems of verbal (and non-verbal) signs. They are based on philosophical and theoretical assumptions and conceptions of language that are sometimes highly divergent and even contradictory.

Here, for example, we find research that, from a cognitive or neurobiological perspective, looks for traces of basic emotions in human language. P. Johnson-Laird and K. Oatley state for example: "According to our theory, there is a set of basic emotion modes that correspond to internal signals that can impinge on consciousness. These modes -happiness, sadness, anger, fear, disgust -should be universally accepted as discriminable categories of direct experience" (Johnson-Laird et Oatley 1989:85). In a very characteristic way these two researchers summarize the work of a cognitive (psycho-)linguist as a work that is essentially based on three entities: emotion (i.e. the emotional experience), the concept of emotion (i.e. the type of emotional experience) and a description of the emotional experience (i.e. a kind of account of what one feels, of an experienced emotion).

"An emotion such as embarrassment is what you feel; a concept is a mental construct that enables you to categorise your experience as one of embarrassment; and a description is a way of putting your experience, presumably by way of its categorisation, into words. The meanings of words are concepts -those concepts that have been dignified by a word for the purposes of communication" (Johnson-Laird et Oatley 1989:84).

Following P.N. Johnson-Laird and K. Oatley, "emotions serve a communicative function both within the brain and within the social group" (Johnson-Laird et Oatley 1989:84). Considering as established the existence of five "basic emotion modes" (i.e. happiness, sadness, anger, fear and disgust) which, according to P.N. Johnson-Laird and K. Oately, form "subjective primitive experiences" (Johnson-Laird et Oatley 1989:89), the work of the cognitive linguist and the psychologist consists essentially in classifying and categorizing the emotional vocabulary (lexicon) of a language according to these five basic emotion modes (Johnson-Laird and Oatley 1989, p. 89).

As well known, one of the major research issues is whether or not there exists a small number of basic emotions. The question is of whether there are specific emotional states which are irreducible to each other and which, together, form a canonical basis of man's emotional experience (or even more generally, of other species such as primates) -regardless of the cultural and linguistic constraints that undeniably affect man's emotional experience, his visions of them, and the verbal and non-verbal expressions he uses to communicate them.

We owe to the American psychologist Robert [START_REF] Plutchik | Emotion: Theory, research, and experience[END_REF] the famous wheel of emotions in which we distinguish eight basic emotions: anger, fear, disgust, sadness, trust, joy, anticipation, surprise. Each of these basic emotions experiences variations in intensity. For example, the basic emotion trust is somehow in the middle of the basic emotion of simple acceptance (of someone or something) and the basic emotion of boundless admiration). Basic emotions can form (dyadic) constellations with other emotions to give rise to more nuanced emotional themes or topoi. For example, the two basic emotions anticipation and joy can form the emotional topos of optimism. This approach to basic emotions has become extremely popular under the name of the circumplex model of emotion (J. [START_REF] Russell | A cirumplex model of affect[END_REF]. Another known and widely recognized theory is, of course, that of the American psychologist Paul Ekman (Ekman 1972;[START_REF] Ekman | What is Meant by Calling Emotions Basic[END_REF] who, through comparative studies of facial expressions, first specifies a list of six basic emotions (anger, disgust, fear, happiness, sadness, surprise) before completing it with a list other basic emotions such as embarrassment, excitement, guilt or again relief and satisfaction. These basic emotions are driven by a specific type of behavioral schemas called "affect programs" (Ekman 1972) that constitute stereotypical responses to the basic needs of an organism (a human, an animal) -needs such as defending oneself against predators or, more generally, protecting oneself against adversity, fighting to feed oneself and one's offspring, reproducing oneself, strengthening oneself by grouping together with other similar agents, etc.

The question of the existence of a small set of basic emotions is one of the most debated issues among experts in the field. It is complicated by the fact that the term basic emotion has different meanings depending on whether it is used in language sciences, psychology or biology [START_REF] Celeghin | Basic Emotions in Human Neuroscience: Neuroimaging and Beyond[END_REF]. Thus, according to [START_REF] Celeghin | Basic Emotions in Human Neuroscience: Neuroimaging and Beyond[END_REF], basic in (conceptual) semantics refers rather to the basic concept in a branch (taxonomy) of concepts, the basic concept being the father of all the other concepts of a branch (cf. below [START_REF] Wierzbicka | La quête des primitives sémantiques : 1965 -1992[END_REF]. In psychology, a basic emotion is an emotion that cannot be reduced to other emotions. In biology, finally, a basic emotion refers rather to a type of innate mechanism of regulation between an organism and its close environment.

A particular way of dealing with the question of basic emotions is the one proposed in cognitive linguistics (Lakoff and Johnson 1980;Lakoff 1987), which brings them sometimes closer to the notion of stereotype [START_REF] Hilary | The meaning of 'meaning[END_REF][START_REF] Putnam | Representation and Reality[END_REF] and sometimes to that of prototype [START_REF] Eleanor | Natural categories[END_REF]. The notion of stereotype is considered as a kind of belief shared by a community, as a form of doxa [START_REF] Bourdieu | Le sens pratique[END_REF] or common sense [START_REF] Geertz | Local Knowledge. Further Essays in Interpretive Anthropology[END_REF]. The representations and models that social actors (individual persons, groups, …) possess of an emotion are part of the doxa or common sense, which can vary from one community (of gender, belief, lifestyle, etc.) to another and from one socio-cultural and historical context to another [START_REF] Kelly | Gender-Emotion Stereotypes are Context Specific[END_REF][START_REF] Caprariello | Social Structure Shapes Cultural Stereotypes and Emotions: A Causal test of the Stereotype Content Model[END_REF].

The concept of prototype, on the other hand, is used rather to identify and categorize objects, people or -in our case -feelings, affects, etc. in a gradual way as more or less representative examples of a given category. Thus, in cognitive psychology and psycholinguistics, there exists a whole series of works that offer, through different tests and surveys, lists of lexical expressions of feelings that seem to be more prototypic (more, so to speak, typical) than others for a given population. Thus, for example, the lexical expressions "happiness", "anger", "hate", "love", "anxiety", etc. represent more typical emotional experiences (more representative for a given population) than, for example, experiences expressed by terms such as "respect", "calm", "boredom", etc. (cf., for example, [START_REF] Fehr | Prototype categorization of emotion[END_REF][START_REF] Fehr | Concept of emotion viewed from a prototype perspective[END_REF].

One -very interesting -approach to the prototypicality of emotions is that of psychologist James Russell2 [START_REF] Russell | In defense of a prototype approach to emotion concepts[END_REF]Barrett & Russell 2015) who considers that the different basic emotions identified in Paul Ekman's theory are deployed according to typical scripts or scenarios.

A basic emotion such as sadness possesses, like all the other basic emotions, a script, a scenario (for more explanations, see chapter 3 below). Thus, the concrete emotional experiences that an individual can make in his life are compared and evaluated in relation to the typical script of the basic emotion sadness and categorized (by the individual) as authentically sad or more or less sad or as somehow sad experiences mixed with other emotional values, etc.

In order to return to linguistic (and semiotic) research on emotion, however, we should not believe that it is all exclusively dependent on the cognitive turn that has profoundly affected the disciplinary field of language sciences. On the contrary, there is a great variety and richness of research projects and programs (very different from those we just have mentioned) that are more particularly interested in the conception and expression of emotional experience in a historical and comparative perspective. The analysis, for example, of linguistic expressions of emotion in a natural language is used here to reconstruct the historically, socially and culturally situated meaning of emotionality, to reconstruct the meaning of emotional experience embodied by a specific social actor, ethnic community, social group, gender community, etc. (cf. [START_REF] Lutz | Ethnopsychology compared to what?[END_REF][START_REF] Lutz | Unnatural Emotions: Everyday Sentiments on a Micronesian Atoll and their Challenge to Western Theory[END_REF][START_REF] Lutz | Engendered emotion: Gender, power, and the rhetoric of emotional control in American discourse[END_REF]Athanasiadou and Tabakowska 1998).

One challenge of this type of research consists in the scientific denunciation of ethnocentric visions of emotion, of emotional experience -ethnocentric visions which, according to researchers adopting a historical and comparative approach to the conception and expression of emotion, threaten the different variants of basic emotion theory and, more generally, cognitive and neuroscientific approaches to emotion. J. Harkins and A. Wierzbicka (2001) cite a whole series of comparative examples of the notion of anger. The meaning of this notion obviously varies from one language to another, which makes problematic the postulate of a basic emotion anger which would possess a universal, culturally invariant meaning. Such a basic emotion seems rather to be a particular (scientific) mental construct belonging to a specialized language (of cognitive psychology or neuroscience) used to designate, for example, recurrent patterns of neurological activities, physio-psychological sensations, etc. On another note, according to this comparative linguistic and sociocultural reasoning, the assumption that there is a strong and significant correlation between a type of mimic expression (facial, for example) and an emotional category (for example: anger) must be considered and problematized in the most varied contexts and observation frameworks and not be taken as evidence. Anna Wierzbicka's interesting position should be mentioned here. While recognizing the linguistic and cultural specificity of an emotional category such as sadness or anger, she nevertheless considers the existence of a kind of transcultural metalanguage composed, as she says, of a set of simple and universal concepts, making it possible to speak, to account for an emotion in a given language or culture. In particular, this metalanguage allows the linguist to analyze the core of the meaning of emotional lexemes in different languages and to compare them. Status and composition of the metalanguage in question resembles the conceptual structure in cognitive semantics (Jackendoff 1983). For A. Wierzbicka, the metalanguage used by the linguistlexicologist to analyze lexical expressions in a given language is composed of conceptual primitives that she believes can be found in all human languages [START_REF] Wierzbicka | La quête des primitives sémantiques : 1965 -1992[END_REF](Wierzbicka , 1996)). These conceptual primitives form a small meta-lexicon of about thirty expressions [START_REF] Wierzbicka | La quête des primitives sémantiques : 1965 -1992[END_REF] and can be considered either -from an almost cognitivist or mentalist perspective -as part of an alphabet of human mind and reasoning [START_REF] Wierzbicka | La quête des primitives sémantiques : 1965 -1992[END_REF] or, on the contrary, as a tool, as an analytical tertium comparationis used by the linguist to describe and compare verbal entities of two different languages.

2) Lexicon and emotional common sense

There exists a whole tradition of research -in linguistics but also in philosophy and anthropology -which is interested in what are sometimes called popular beliefs, sometimes folk theory, sometimes still doxastic beliefs. Generally speaking, this research deals with what anthropologist C. [START_REF] Geertz | Local Knowledge. Further Essays in Interpretive Anthropology[END_REF] called common sense. Common sense constitutes an epistemic frame of reference used ("naturally", "spontaneously", "obviously"...) by people who form a social actor (a group, a community, a class...) to act, interact and communicate. Common sense can be broadly understood as a topical field (a field of topoi or of commonplaces) that: 1) ... is shared by all persons forming a social actor (for example, a community of speakers of a language)3 ; 2) .... qualifies what is obvious, natural, what constitutes, for a social actor (a community of speakers) a kind of fundamental semantics in the sense of A.J. [START_REF] Greimas | Sémiotique. Dictionnaire raisonné de la théorie du langage[END_REF]; 3) .... forms constructions of meaning presupposed by secondary modeling systems in the sense of Y. Lotman (2009) including, for example, myths and mythologies, narrative literature, etc.

A topos is a framework, a model or a view that thematizes (interprets) a given (intentional) object (here: the intentional objet emotion). More technically speaking, the topos is the semantic configuration that characterizes the meaning of any (intentional) object for a given actor or community of actors (such as, for instance, a speech community). By thematizing an object or a domain, a topos affirms something about this reference object by qualifying it or by charging it with a set of characteristics that possess a value -i.e. a relevance -for the given actor. In discussing the concept of the topos in the work of Ernst R. Curtius (1956) and in referring to Quintilian, the German philosopher Otto Pöggeler (1973) insists that a topos is not a mere concept but represents rather a vision or a model, a scheme that serves to recognize, to deal with situations and data, to solve problems, to discuss and argue, etc. 4 In this sense, a topos represents a vision on which an actor holds because he trusts this vision which constitutes for him an evidence, an evidential basis for his doing.

The hypothesis here is that it is more particularly the lexical part -the vocabulary -of a natural language that enables the researcher to elicit the semantic structure of a supposed common sense, in our case: of the common sense of what emotions are, of qualifying and evaluating emotional experiences and of dealing with emotional experiences. Of course, the lexicon of a language constitutes only one kind of data among many others that can be used to explain the semantic structure of the meaning of a popular theory of the intentional objet emotion. For example, research in discursive psychology [START_REF] Edwards | Emotion discourse[END_REF][START_REF] Scheff | Microsociology: Discourse, emotion, and social structure[END_REF] or in psycholinguistics and psycho-semiotics on emotional discourse [START_REF] Katriel | Exploring Emotion Discourse[END_REF] regularly uses the comments (verbatims) of populations of interviewees speaking about, expressing emotional experiences and witnessing on emotionally charged events. (Oral or written) discourse is indeed the main input of a wide variety of research devoted to the reconstruction of emotional topoi that are at work in the different social fields of communication -in media communication, in institutional communication, in political communication, etc. [START_REF] Mangham | Emotional discourse in organizations[END_REF][START_REF] Loseke | Examining emotion as discourse: Emotion codes and presidential speeches justifying war[END_REF][START_REF] Steunenberg | Between Reason and Emotion: Popular Discourses on Turkey's Membership of the EU[END_REF].

Let us develop a little bit more this idea of common sense topoï of emotion in considering quickly a French lexical corpus of emotion. This corpus is provided by the TLFi (Trésor de la Langue Française informatisé)5 , the computerized version of the TLF (Trésor de la Langue Française)6 , the exceptional dictionary of the nineteenth and twentieth centuries French in 16 volumes published between 1971 and 1994. A second resource is the Dictionnaire électronique des synonymes7 of the CRISCO research laboratory of the university of Caen. Let us note two particularly interesting methods offered by these two resources and that we were able to test in a work on the French lexeme "politesse" and its associated lexical field (Stockinger 2018).

First method: The method of lexical cliques. A clique is a subgraph whose vertices are all adjacent two-by-two. A lexical clique is therefore a group of lexemes positioned around a reference lexeme which are adjacent to each other. Clique lexemes therefore have a greater semantic proximity between them than with other lexemes that are part of the same lexical field. The French lexeme "émotion" knows 85 cliques8 -the lexical field, on the other hand, is composed of 65 synonyms (completed by about ten antonyms).

Second method: The method of visualization of semantic space. Figure 1 shows a visualization of the space of the semantic universe of the (lexical) topos of emotional behavior. This semantic space has a triangular shape on which are distributed, on the one hand, the semantic regions specific to the 65 lexical expressions which maintain with the reference expression "émotion" a relationship of partial synonymy and, on the other hand, the already mentioned 85 lexical cliques.

(figure 1) Among the 65 lexical expressions grouped around the reference lexeme "émotion", the closest, according to TLFi and CRISCO, are in descending order "agitation" (agitation, excitement), "trouble" (confusion), "passion", "émoi" (agitation, turmoil), "exaltation" "inquietude" (concern, worry, anxiety), "transport", "saisissement" (sudden emotion), "délire" "commotion" (shock), "sensation" (sensation, feeling), "affres" (torments, "in the throes of") "angoisse" (anxiety), "choc", "fureur" (fury), "transe", "épouvante" (dismay, terror, horror), "vertige" (vertigo), "sensibilité" (sensitivity), "douleur" (pain), "crainte" (fear), "choc", "trauma". Each of these expressions (as suggested in figure 1) offers a vision, a particular view of the intentional object emotion, of the emotional experience. They designate topoi that qualify (thematize, describe, explain, ...) emotion, emotional experience sometimes as a puzzled mental state, sometimes as a passionate mental state, sometimes as an embarrassed mental state, sometimes as an exalted mental state etc. Of course, as figure 1 shows, while offering a great diversity of visions of emotional experience, the topoi in question are often very close to each other, overlap, some are included in the semantic space occupied by others, less precise, offering more diffuse understandings but also more easily employable (in daily life communication, for instance) to account for an emotional experience. This is the case, for example, of the topos of emotion is an embarrassed mental state that offers a very circumscribed, very precise vision of the emotional experience compared to that, much more general, much more diffuse proposed by the topos of emotion is a restless mental state. The former only covers a particular kind of emotional experience, the latter takes into account a much broader kind, may be appropriate to characterize emotional experiences that have little to do with emotional embarrassment. Some lexical expressions that are part of the list of 65 partial synonyms belong to a large number of lexical cliques. This is the case, for example, of the French lexemes "agitation" (found in about thirty cliques), "trouble" (belonging to about twenty cliques) and "passion" (integrating about fifteen cliques). Other lexical expressions are only part of very few cliques. For example, the French lexemes "trauma", "plaisir" or "embarras" can only be found in one single clique. The lexical expressions belonging to the greatest number of cliques are semantically the closest to the French reference expression "émotion". In other words, the semantics of the expressions "agitation", "trouble" and "passion" are closer to the semantics of "émotion" than the semantics of expressions such as "plaisir", "trauma" or "embarras". This variation in semantic proximity between expressions belonging to a field of partial synonyms can be interpreted as a process of semantic specialization from a common semantic background which is provided, in our case and as figure 1 shows, by the semantic space expressed by the French lexeme "émotion" 9 .

For example, the French lexeme "embarras" participates only in one single clique formed by the following expressions: {"embarras", "trouble", "malaise", "émotion"}. Like any clique, also this one is composed by, first, the reference lexeme "émotion" which designates the triangular semantic reference space (figure 1) and, second, by the lexical head from which a clique is constructed. In our case, the lexical head is "embarras". In its role as the lexical head of the considered lexical clique, "embarras" refers to its core semantic space which remains invariant through its own synonymic lexical field. It constitutes, in other words, the semantic background common to all the lexical expressions and cliques that maintain with it (i.e. the French lexeme "embarras") a relationship of partial synonymy -lexical expressions of which the lexeme "émotion", among others, is a part 10 .

The two remaining expressions of the lexical clique of which the lexeme "embarras" is the lexical head -the lexemes "trouble" (disorder, confusion) and "malaise" (faintness, malaise) - 9 The semantic background should not be interpreted in the sense of a more elementary or more general semantic category (for example in the sense of a semantic type) but rather in the sense of a semantic magma from which emerge and are constituted by specialization the different semantic regions which delimit the meaning of a lexical expression belonging (in our case) to the synonymic field of the lexeme "émotion". Orientation and degree of specialization of a semantic region (in the semantic reference space of the lexeme "émotion") are provided by the composition and number of cliques in which a particular lexeme participates. 10 According to the Dictionnaire des Synonymes du CRISCO, the field of partial synonyms of the French lexeme "embarras" brings together a hundred lexemes and some 133 lexical cliques (http://www.crisco.unicaen.fr/des/synonymes/embarras) finally make it possible to interpret the orientation and degree of specialization of the semantic region specific to the lexeme "embarras" as a partial synonym of the lexeme "émotion". This semantic region, intuitively speaking, is in an area that is also occupied by the semantic regions of the French lexeme "trouble" and some other lexemes such as the French lexeme "agitation" and which is, on the other hand, closer to the area where one finds the semantic regions of French lexemes such as "douleur" (pain), "inquiétude" (concern, worry, anxiety) or "affolement" (panic).

Let us return to the lexemes and lexical cliques composing the synonymic field of the French lexeme "émotion". Together, they thus contribute to the formation by specialization of particular semantic regions that are more or less variable in terms of surface area and density (i.e. of the settlement of a region by lexical expressions and lexical cliques) from a common semantic background that is represented by the lexeme "émotion". Figure 1 shows some remarkable semantic regions. One of the most important regions is that which delimits the radius of the semantic space of the French lexeme "agitation" (agitation, excitement) within the semantic universe of "émotion". This region occupies a central position in the semantic reference space. Its surface area is one of the most important in the semantic reference space -it is larger compared with those of many other regions. It is, finally, densely populated by a large number of lexical cliques (represented in figure 1 by small squares) and partially synonymic lexemes. Here, among many other lexical expressions, we find the French expressions such as "trouble" (disorder), "movement", "affres" (torments, "in the throes of"), "angoisse" (anguish), "inquietude" (concern, worry, anxiety), "malaise" (faintness, malaise), "excitation" (excitement), "transport", "passion", "transe" or even "delire" (delirium). In other words, it is here, in this region, where the main thing happens in terms of common conception and understanding of the intentional object emotion or emotional experience.

Figure 1 identifies further semantic regions: a second region at the left end of the reference space (region designated by the French lexeme "choc"); a third region adjacent to the second region on the right and designated by the French lexeme "sensation" (sensation, feeling); a fourth and a fifth region (designated respectively by the French lexemes "angoisse" (anguish) and "exaltation" (exaltation, elation, ecstasy)) which are at the two opposite poles to the right of the reference space; finally a sixth region (designated by the French lexeme "passion") which largely overlaps the first region designated by the French lexeme "agitation". Each region is characterized by the co-presence of a set of lexical expressions and cliques and offers a vision, a particular "view" of the intentional object of emotion.

Taking a closer look at the lexicographical definitions offered by the TLFi of the lexemes that are part of the synonymic field of the reference lexeme "émotion", as well as at the organization of some 80 lexical cliques, the particular meaning of these different regions (i.e. the folk theories, the doxastic beliefs that these regions offer of the intentional object "émotion") can be explained with the help of a small set of criteria:

1. Emotion is a mental state which is distinguished by a quantitative modulation (which can lead to a qualitative modification) of a preceding mental state, notably by the quantitative modulation of the relationship of well-being between the actor-subject and the world (or the other). This criterion is manifest, among others, quite clearly in the following French lexemes: "bouleversement" (shake up, upset), "ébranlement" (shaking, undermining), "affection" (affection), "agitation" (agitation, excitement), "trouble" (confusion, turmoil), "traumatisme" (trauma), "choque".

2. The mental state of a quantitatively modulated (and eventually qualitatively modified) well-being is characterized by an internal instability. This instability can be exemplified, among other things, by oppositions like alm vs troubled, agitated state or quick-tempered state vs atonic, motionless state, etc. One detects the presence of this type of opposition in lexemes such as "sensation" (sensation, feeling), "saisissement" (amazement, shock), "excitation" (excitement), "exaltation" (exaltation, ecstasy) or again "alarme" (alarm).

3. The instability of the mental state of modulated well-being is in a way the result of the tension between two opposing forces: an agonistic force and an antagonistic force. The dynamic between these two forces (cf. L. Talmy 2000) is present throughout the concerned lexicon and is quite evident in lexemes such as "branle-bas" (commotion, disturbance), "secousse" (concussion), "choque", "épouvante" (terror, dismay) or "passion".

4. The instability of the mental state of modulated well-being may present a variable degree of intensity. The variable degree of intensity seems to be a consequence, among others, of a greater or lesser asymmetry between the agonistic force and the antagonistic force. This asymmetry translating into varying degrees of intensity can be observed in certain series of lexemes such as: a) "angoisse" (anguish), "terreur" (terror), "affres" (torments, "in the throes of") and "épouvante" (dismay, terror, horror); b) "agitation", "transport", "enthousiasme", "exaltation", "fureur" (fury) and "délire (delirium).

5. The specific quality of the mental state of well-being, which is more or less unstable and intensely modulated, is thematized with reference to different modes of perception, including, for example: a. proprioception (perception relating to the sensations of balance, movement, acceleration, etc.), traces of which can be found in lexemes such as "vertige" (vertigo), "transport" "exaltation" or "enthusiasme"; b. thermoception (perception relating to the sensation of heat), traces of which can be found in lexemes such as "ardeur" (ardour), "bouillonnement" (boiling, ferment) or "fièvre" (fever) 11 ; c. nociception (perception of the sensation of pain) present in lexemes such as "passion", "douleur" (pain), "traumatisme" or again "malaise" (faintness, malaise).

6. Finally, the more or less intensely modulated mental state of well-being is necessarily evaluated in reference to the axiological dimension of phoria. The dimension of phoria evaluates and prioritizes any emotional experience in relation to continua such as pleasure vs displeasure, joy vs suffering, attraction vs repulsion, etc. This evaluation and prioritization can be seen quite clearly in the semantics of lexemes such as "crainte" (fear), "embarrass" (embarrassment), "angoisse" (anguish) or "passion", "exaltation", "enthousiasme", "plaisir" (pleasure) and "transport" (transport of delight).

Together, the six criteria contribute to the structuring and organization of the semantic universe of the French lexeme "émotion". This semantic universe is differentiated into a whole series of semantic regions. Semantic regions correspond, in the context of the semantic universe of the lexeme "émotion", to the extension of the meaning of lexemes that are part of the list of (partial) synonyms of our he reference lexeme. Each region is internally structured and has a more or less rich and nuanced intensional semantic landscape as witnessed by the lexical cliques that populate a region. The semantic landscape specific to a region can be explained, interpreted by the criteria introduced above. It forms the semantic configuration characteristic of the topos or topical field of a semantic region -topos or topical field which expresses, let us repeat it, a view, a particular vision of the intentional object emotion.

Thus, for example, the topos of emotional experience in the sense of an agitated mental state (topos which is delimited by the extension of the semantic region of the French lexeme 11 Let us refer here to Z. Kövecses' stimulating analyses of the metaphorical system by the means of which the emotional experience is conceptualized in different languages and literatures and in which thermoception and, more generally, the (popular) physics of heat and pressure seem to play a decisive role [START_REF] Kövecses | Emotion Concepts[END_REF][START_REF] Kövecses | Metaphor and Emotion. Language, Culture, and Body in Human Feeling[END_REF].

"agitation" within the reference space of the lexeme "émotion") has a rich internal structure allowing it to take into charge of, to classify, to evaluate, to narrate, ... very varied emotional experiences which can come under one or more modes of perception, be more or less intense, show various phoric (euphoric or dysphoric) potentials, be the result of variable configurations between agonistic and antagonistic forces, etc.

But the (lexical) topos of the agitated mental state is only one particular topos among about sixty other topoi each of which offers a specific view, a specific understanding of the emotional experience. They are part of the meaning, of the sense of emotional experience that is (more or less) common, which is (more or less) shared by the language community of (competent) French speakers.

Without being able to continue here the discussion of the semantic universe of the French lexeme "émotion", let us underline nevertheless that it would be interesting to explore further the following three directions of research:

1) the specificity or, on the contrary, the cultural and historical generality of the semantic universe represented by figure 1 in comparing it on the one hand with those of other languages and language families and, on the other hand, with those underlying the sociolectal conceptions manifesting themselves in different cultural ecosystems such as those of literature, cinema, advertising, politics, etc.

2) the convergences (or, on the contrary, the divergences) of the results between the various researches devoted to the semantic and, more widely, rhetorical and discursive universes of the emotional object (here, we think more particularly of the research works in lexical semantics (Greimas 1983;[START_REF] Greimas | Sémiotique des passions: Des états de choses aux états d'âme[END_REF][START_REF] Wierzbicka | L'amour, la colère, la joie, l'ennui : la sémantique des émotions dans une perspective transculturelle[END_REF][START_REF] Wierzbicka | Emotions across Languages and Cultures: Diversity and Universals[END_REF] and in cognitive semantics [START_REF] Kövecses | Emotion Concepts[END_REF][START_REF] Kövecses | Language and Emotion Concepts[END_REF][START_REF] Kövecses | Metaphor and Emotion. Language, Culture, and Body in Human Feeling[END_REF]Lakoff and Johnson 1980 ;Lakoff 1987);

3) the epistemological relationships that may exist between this common sense vision of the intentional object emotion and the common sense(s) underlying the different theories or specialized elaborations of this notion (we think, for example, of the various classifications of "basic" or more specialized emotions, of the circumplex model of emotions, of the model of the prototypic emotional episode, etc.).

3) From common sense to the scientific elaboration of emotion

Let us consider quickly the third perspective mentioned above, i.e. that of considering the thematic and epistemological relationships between conceptions of common sense that natural languages provide us with regarding emotion and those of specialized constructions, theories coming from research on emotion in the different scientific disciplines.

It goes without saying that to scientifically study emotion -the emotional dimension -in language or any other system of signs, we first need a global understanding, a "vision" of this object called "emotion" -understanding or vision which, at least in its general outlines, should be considered as being shared by researchers in this field.

In consulting the specialized literature -in (social, clinical, cognitive, genetic…) psychology, neuroscience, language studies and linguistics -we encounter a whole series of (not always congruent) attempts at definition. These attempts at definition are based, for example, on the conceptual history of the notion emotion in philosophy (which goes back, as far as European history is concerned, at least to Aristotle and his rhetoric), on the etymology of the lexical expression "emotion" (coming from the Latin noun "motio"), on the comparison of this notion with related notions such as passion, feeling, affect, mood, temperament and, of course, on a critical analysis of the use of this notion in scientific research since at least the 19th century (cf., for example, Rimé 2005).

A general vision, which seems to be quite commonly shared, is that emotion is a thymic state, i.e. a specific mental state which is characterized, among other things, by a phoric value (i.e. falling within the continuum between pleasure vs displeasure or joy vs pain), by a relatively short duration as well as by a variable intensity and internal instability. Emotion -the emotional state -also forms a particular type of thymic state that can be distinguished, for example, from mood, temperament or emotional disorder (Rimé 2005 ;[START_REF] Annie | Emotions et affects. Contribution de la psychologie cognitive[END_REF]Robbins & Judge 2014) 12 .

A particular emotional state of a social actor (a person or a group of persons in the case of a possible collective emotional experience) is caused or triggered by an event -by an external stimulus -that affects (or is likely to affect) the allostasis, i.e. the state of mental balance of an actor who is the subject (of this mental state). From a cognitive perspective, special attention is given to the act of emotional (environmental) assessment. By observing the element that can potentially disturb his allostasis, the actor-subject compares it to a standard (which can be a personal experience, a collective tradition, an explicit norm...). Comparing the observed element's behavior to a standard allows the actor concerned to assess the incident (or "incident rate") that the element hypothetically possesses on his given mental state. The evaluation of the value -the meaning -of the event or the external stimulus puts the actor-subject in any case in a particular mental state of alert. This mental state of alerting, generally of short duration, comprises more particularly two central continuous dimensions: the dimension of tension (or activation) and the dimension of phoria (of pleasure or displeasure). It characterizes, in other words, mental imbalance, i.e. mental instability caused by the disruptive element -instability which is more or less intense and more or less euphoric or, on the contrary, dysphoric. The state of mental imbalance is the (more or less probable) cause of a response on the part of the concerned actor-subject in the form of a concrete behavior, i.e. a (somatic, verbal or non-verbal) doing or acting in order to find a new mental balance.

The behavioral response of the concerned actor is expressed in the form of different signs or complexes of signs constituting texts lato sensu (emotional displays or emotionscape, cf. chapter 4). Thus, Nico Frijda and his collaborators distinguish several categories of behavioral patterns that are at the actor's disposal to express, to stage his emotional state, i.e. his joy, his fear, his sadness, his anger, his trouble, his excitement etc. [START_REF] Frijda | The Emotions[END_REF] thus identifies some eight behavioral patterns, among which he counts, for example, the behavioral pattern of the positive approach; the behavioral pattern of aggression, the behavioral pattern of panic, the behavioral pattern of submission or the behavioral pattern of self-affirmation. As the most diverse studies in (social) psychology show, the 12 Temperament, according to the specialized literature (Rimé 2005; Piolat & Bannour 2008; Robbins & Judge 2014), is a mental state composed of very stable affective traits that accompany the individual throughout his existence and contribute to the organization of his character. Thus, we speak of neurotic, neurasthenic, extroverted, etc. temperament. Mood is understood as a state, an internal disposition specific to the individual and whose existence does not depend on a "contextual stimulus" (Robbins & Judge 2014). The mental state designated by the term "mood" has a very variable life span between "a few minutes to a few weeks" (Rimé 2005) that can be transformed into a temperament, a character trait. Thus one can be in a "good" or "bad mood", but also a "childish mood", "gay", "sad", etc. Finally, emotional disorders are "pathological syndromes" (Rimé 2005) such as anxiety disorders, depression, mania or phobia. They are, in other words, mental states that characterize inappropriate responses of the subject to his environment or mental states characteristic of a subject whose horizon of meaning is inadequate in relation to the specificities of his environment. On the other hand, what all these types of thymic state have in common is the presence of positive or negative affects, i.e. a phoric axis establishing itself between an euphoric pole (of "pleasure") and a dysphoric pole (of displeasure or pain), as well as an axis of excitation (of excitability) between a calm state and an excited state. different behavioral patterns identified by Frijda can be expressed through a wide variety of (kinesic, verbal, somatic, etc.) signs and sign complexes which testify that they are always part of a specific personal or collective history and culture. This informal presentation of the main stages and activities that make up the typical path -or thread -of an emotional experience is problematized, according to different points of view and theoretical references, in the specialized literature, particularly in that which favors a cognitive approach to the emotional experience (Stanley Schachter, James A. Russell, Lisa F. Barrett, Klaus Scherer, Nico Frijda, ..) 13 . The history (the life cycles) of an emotional experience is thus very often described by a set of recurring structural characteristics organized around a typical scenario or script that psychologists or cognitivists call the prototypical emotional episode (Russell and Barrett 1999). A prototypical emotional episode is: « (a) complex process that unfolds over time, involves causally connected events (antecedent, appraisal, physiological, affective and cognitive changes; behavioral response; self-categorization), has one perceived cause, and is rare. Its structure involves categories (anger, fear, shame, jealousy, etc.) organized as a fuzzy hierarchy and horizontally organized as a part of a circumplex" (Russell et Barrett 1999: 805).

Considering this scientific view at the intentional object emotion, one is struck by the profound similarities that seem to exist between it and the visions and views of emotion that we were able to meet during our examination of the semantic universe of the French lexeme "emotion" and its list of partial synonyms.

The possible -cognitive, historical or epistemological -reasons for these similarities and their consequences for scientific reasoning and discourse would, in our opinion, be amply worthy of further questioning. In any case, considering these similarities, we can ask ourselves the question of the relationship between scientific constructions of an object of knowledge such as that of emotion and common-sense constructions of it -and thus question research in this domain on the meaning and value of its doing.

4) Emotionscape or the expression and interpretation of emotion

Constitutive for any sign is the distinction between on the one hand the perceptible elements which express, which stage an emotional experience and on the other hand the meaning of these elements for all concerned actors, both for the actor who is the subject of the emotional experience (who "lives it") and for the direct or indirect participants of this experience (who observe it, "read" it, interpret it, interact with the actor-subject of the experience, etc.). An emotional experience, as everyone knows, can be expressed and staged using a wide variety of signs: acoustic signs (example: shouting) and verbal signs (example: interjections), kinesic signs (example: facial expressions), physiological signs (example: heartbeat), object-signs (example: mourning clothes), behavioral signs (example: panic reactions) ritual signs (example: love rites), and so on.

Using terms such as emotional display [START_REF] Robinson | Emotion Display as a Strategy for Identity Negotiation[END_REF][START_REF] Wege | Emotion displays in media: a comparison between American, Romanian, and Turkish children's storybooks[END_REF] or emotionscape [START_REF] Gartner | The self-help revolution[END_REF][START_REF] Haverkamp | Synesthetic Design. Handbook for a Multisensory Approach[END_REF]Garde-Hansen and Gorton 2013), we try to consider all the obvious signs that contribute either the expression of an emotional state and, more particularly, a specific affective value of an emotional state, or its interpretative reconstruction.

Like any textscape (Stockinger 2017), an emotionscape is a signifying environment [START_REF] Greimas | Sémantique Structurale[END_REF][START_REF] Greimas | Sémiotique. Dictionnaire raisonné de la théorie du langage[END_REF] composed of signs by means of which an actor who is the subject of an emotional experience, signifies his particular affective state. As an environment at least partially perceptible to an observer, the emotionscape provides the data by means of which every actor in the role of the observer "reads" and interprets these signs produced by the subject of an emotional experience. The interpretation may or may not be in accordance with the actorproducer's intentions, may be based on an ambient common sense culture or, on the contrary, may come from specialized knowledge, an allegedly scientific theory, etc.

Among the multiplicity of signs or sign systems that can compose an emotionscape are, for example, facial, postural, gestural and kinesic expressions (Ekman 1982, Philippot 2007). We also find the different forms of neurophysiological activity, the physiological and bodily sensations specific to the actor who is the subject of an experience. These activities, which form a whole "symbolique corporelle" (i.e. in English "bodily symbolism", [START_REF] Breton | Anthropologie des émotions[END_REF], are not always directly discernible by an observer. This is the case, for example, with accelerated heart rate, dry mouth, chills, characteristic neurological signs, etc. Identifying and interpreting such signs and sign patterns requires sometimes specific skills and technologies such as functional neuroimaging [START_REF] Pichon | Neuro-imagerie et neuroscience des émotions[END_REF][START_REF] Harrison | The Embodiment of Emotional Feelings in the Brain[END_REF].

Emotionscape is also, of course, partly linguistic in nature (for a more detailed discussion, see [START_REF] Ortner | Text und Emotion[END_REF]. Verbal and paraverbal signs play a dual role in the organization and manifestation of emotionscape. They are primarily used for the verbal (and paraverbal) expression of an emotional experience and, more particularly, the (more or less intense, more or less pleasant or unpleasant) affective value of an emotional state. This is called the emotive mode in the use of linguistic signs.

But verbal signs are also used for reflexive purposes, i.e. to express a metadiscourse on an emotional experience in the form of comments, testimonies, descriptions, explanations, etc. This metadiscourse, which can again be provided with a specific emotional dimension, serves primarily to "objectify" an emotional experience -that which the speaker is experiencing or has experienced or that which one assumes someone is experiencing, has experienced. This distinction corresponds, roughly speaking, to that made by John Lyons between the "subjective experience self" and the "objective observing self" (Lyons 1982) and also to that made by Charles Bailly between the "mode pur" ("objective", detached, reflexive) and the "mode vécu" ("subjective", engaged, spontaneous) in the verbal behavior of a speaker (quoted in Hübler 1997).

The verbal and para-verbal sign that contributes to the expression or staging of an emotional state (in a conversation, for example, or in a work of fiction) is part of a whole emotional language, sometimes also called emotive language. This emotional mode or use of verbal or para-verbal signs concerns all the constitutive layers of language and speech:

phonetic and prosodic layer: particular intonation, characteristic voice vibration, acceleration of speech flow, repetition of certain words or parts of speech; lexico-semantic layer: use of de-personalization lexemes (in French, this is the case, for example, of lexemes like "bastard", "dog", "snotty», etc.), phrases expressing an affective value; discursive and rhetorical layer 14 : intensive repetition (Blanche-Benveniste 1990) of certain words or parts of speech, emotive interjections (Kleiber 2006), emotive exclamations, emotive topicalization (de [START_REF] Beaugrande | Topicality and emotion in the economy and agenda of discourse[END_REF]Hübler 1997 ;[START_REF] Egidi | Emotional language processing: How mood affects integration processes during discourse comprehension[END_REF] ; particular kinds of genres and speech acts such as insult or intimidation ; discursive construction modes simulating an emotional state [START_REF] Auger | De la violence verbale pour une sociolinguistique des discours et des interactions[END_REF], etc.

The verbal (and paraverbal) expression of emotion is a particularly important object of research in psycholinguistics and second language acquisition (cf. H. Sarter 2012). In psycholinguistics, we are thus interested in the commonly attested use of a "motherese" (or baby talk) to establish, on the part of the adult, the first exchanges the infant. In all cultures, there exists a form of motherese or baby talk which is characterized, among other things, by the existence of specific and repetitive rhythmic, intonational and melodious patterns used for the "good" management of subjective and intersubjective emotions: "... emotional content conveyed via melody represents the first step of a hierarchical coding of meaning into words, leading from self-evident, "body-near" messages to abstract symbols and notions " [START_REF] Wermke | From Emotion to Notion. The Importance of Melody[END_REF]. Starting from such a background of language and communication, the little child gradually develops, thanks to his successive personal experiences and in interaction with his social environment15 , his representations, his theories of emotions and emotional states, i.e. a form of what is called in specialized literature an internal-state language (ISL; cf. for example [START_REF] Meins | Social relationships and children's understanding of mind: Attachment, internal states and mind-mindedness[END_REF].

Finally, to return to the different types and systems of signs composing an emotionscape, it is not limited to the systems of verbal, para-verbal, somatic and physiological signs. It is "populated" with all kinds of practices, activities, objects, signifying places that play the role of signs through which the actor, who is the subject of an emotional experience, stages his state of mind. Thus, we know well that material objects (artifacts, natural objects...), works (photos, poems, prayers, incantations...), ritual activities, symbolic places (places of meditation...), periods (moments of relaxation...), people of reference, etc. serve or can serve as sign carriers, form real media systems to stage an emotional experience (see, for example, the excellent and very complete study by Tilman Habermas on "beloved objects" [START_REF] Habermas | Geliebte Objekte. Symbole und Instrumente der Identitätsbildung[END_REF]).

Anthropological and semiotic research comparing different forms of expression and staging of an emotional experience make us aware that an actor's emotional state is necessarily rooted in an affective culture [START_REF] Breton | Anthropologie des émotions[END_REF][START_REF] Reddy William | The Navigation of Feeling. A Framework for the History of Emotions[END_REF], i.e. a culture of emotions and affective values. The signs used and the multimodal textual organization of an emotionscape make us more attentive to the fact that emotions must be studied in their socio-cultural and historical context. It is a perspective that is often neglected or treated as a simple epiphenomenon in the more naturalizing approaches adopted by researchers in cognitive psychology, cognitive sciences and neurosciences (cf. [START_REF] Crapanzano | Réflexions sur une anthropologie des émotions[END_REF]Bernard 2015). Thus, anthropology and the history of emotions [START_REF] Reddy William | The Navigation of Feeling. A Framework for the History of Emotions[END_REF]Santangelo and Lee 2018 ;[START_REF] Schuler | Introduction: Historicizing Asian Community-Based Emotion Practices[END_REF][START_REF] Glushkova | From Constant Yearning and casual Bliss to Hurt Sentiments: An Emotional Shift in Varkari Tradition (India)[END_REF]) make us sensitive to the fact that there exist very significant variations in the expression, staging and recognition of relevant emotional themes (such as fear, anger or surprise) between different (ethnic, linguistic, religious, social,...) cultures. There are, in other words, emotional regimes (W.M. Reddy 2001), universes of meaning (i.e. topoi, topical fields) and (bodily, verbal, visual, acoustic, etc.) symbols of expression that characterize the affective culture of a person, a social group or a social organization (cf. S. Fineman 2000; A. Roux 2013) monumental work Institutio oratoria (1979). Heinrich Lausberg (1949[START_REF] Lausberg | Handbuch der Literarischen Rhetorik[END_REF] offers a methodical presentation of the use of discursive pathos (i.e. the affective burden of speech) in classical rhetoric to solve a stasis (i.e. a crisis, a conflict) through discussion, through debate.

Taking into account the inevitable cultural contextualisation of the emotional experience raises with acuity the problem of the ("controlled", "objective") interpretation of an emotional experience based on its signs of manifestation, including neurological signs that today enjoy an aura of objectivity that is not self-evident. In a similar way to an interpreter of written or audiovisual texts, the emotionscape produced by an (collective or individual) actor who is the subject of an emotional experience offers the interpreter (expert or not) the opportunity to induce the meaning of its constitutive expressions and -depending on the interpreter's culture (i.e. according to the knowledge and beliefs to which the interpreter refers; see also Barrett 2017) -to explain the emotional state underlying the production of emotionscape. As Bliss-Moreau (2017) rightly points out, by considering the activities, postures, facial expressions, etc. of an animal, we hypothetically reconstruct a meaning "behind" these signs, i.e. what the animals or people feel, their emotional states, etc., that is, the meaning of the signs. It simply means that we interpret according to our point of view, according to our knowledge and beliefs (according to our language game, to use Ludwig Wittgenstein's well-known expression) the expressions of the other -animal or human being.

Here arises the as well epistemological as practical problem if there are objective signatures of emotions whose (however controversial) status is to form (in the biological sense) a basic, in other words, an innate emotion. These objective signatures should make it possible to understand objectively the state of mind of an actor who is the subject of an emotional experience. For example, the simple presence of an objective signature (such as a facial expression, a gesture or a characteristic neurological pattern) in a primate's emotionscape should somehow "deliver" his state of mind (cf. Bliss-Moreau 2017 for a critical discussion of this classic vision).

Conclusion

This article proposes to quickly explore a certain number of researches devoted to the relationship between language and emotion. Of course, this exploration remains partial. Its objective was not, however, to provide a fair picture of the abundant research in this field. Instead, it has targeted a few -in our opinion -central themes for a transdisciplinary approach of the (intentional) object emotion, of emotional experience that proposes to describe emotion, emotional experience under the following three major angles:

1. as an experience that takes place according to a script, a typical scenario that can vary from one social actor to another (i.e. from one sociocultural and historical context to another); 2. as an experience that is both part of man's common "natural baggage" and is understood and conceptualized by means of visions -of beliefs -that are part of the common sense, of the semantic foundation of every situated social actor (such as a sociolinguistic community); 3. as an experience that is expressed, staged, communicated and shared using a huge diversity of signs and sign systems forming emotional displays, emotionscapes.

These three angles constitute many major themes in research on emotion which cross the various disciplines concerned (from anthropology to (cognitive) neurosciences including linguistics, semiotics, the human sciences and philosophy) without one in particular being able to claim to be able to explore them alone or to hold a complete vision of them.

The French « la langue » corresponds to the « natural language » whereas « le langage » in French recovers all sign systems (and not only only the verbal sign system of natural languages)

See James Russell's website: http://emotiondevelopmentlab.weebly.com/principal-investigator.html

In a style close to that of Michel Foucault's archaeology of knowledge (1969), Hans R. Curtius writes: « Die Toposforschung gleicht der 'Kunstgeschichte ohne Namen' im Gegensatz zur Geschichte der einzelnen Meister »(Curtius, 1973(Curtius, [1938]:14) ]:14)

Cf. also Walter Veit who affirms that « Begriffsgeschichte (…) ist schon keinesfallsTopik. » (1973, p. 205)

http://atilf.atilf.fr/

For more information (in French), cf. http://www.atilf.fr/spip.php?rubrique77

See : http://www.crisco.unicaen.fr/des/

See : http://www.crisco.unicaen.fr/des/synonymes/%C3%A9motion

We want to quote here the excellent article of[START_REF] Scherer | Emotions are emergent processes: they require a dynamic computational architecture[END_REF] where we find a very sophisticated model of the emotional script based on the mathematical catastrophe theory (René Thom). This comprehensive modelling takes into account the contributions of four central approaches to the phenomenon of emotion: basic emotion theory (defended mainly by Paul Ekman and Carroll Izard); early constructivist theory (Stanley Schachter); current constructivist theory (James A. Russell, Lisa F. Barrett) and appraisal theory (Klaus Scherer).

As well known, it is rhetoric that has long been interested in the use of emotion -of pathos -in public discourse as a resource, as a means of getting the public to adhere to the cause, to the point of view defended by the speaker. A classic reference here is the Latin rhetorician Quintilian (1st century AD) with his

As stressed by Ulrike Lüdke: « Bedeutungen von Zeichen werden vom Kind nicht statisch durch eine rein kognitive Übernahme eines feststehenden, eindeutigen Inhalts in sein individuelles, isoliertes mentales Konzept und Lexikon erworben, sondern sie werden wahrnehmungsbasiert und damit narrativ und "embodied" über komplexe sozial-intersubjektive Verhandlungs-und Validierungsprozesse von polyphonen Ausgangsmodellen sukzessive in wechselseitige, affektiv bewertete Repräsentationen ("mutual representations") überführt. »(Lüdke, 2012, 108).


Frédéric Lebon

Sylvie Ronel-Idrissi

F Lebont

S Ronel-Ldrissi:

Asymptotic studies of Mohr-Coulomb and Drucker-Prager soft thin layers

Keywords: asymptotic studies, elasto-plasticity, non-associated laws, interface laws, finite elements

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Introduction

The objective of this paper is to analyse soft thin layers in order to replace them by interface laws.

The aim of our work is to study non-linear soft materials; especially, we focus on the case of non- associated elastic-plastic materials of Mohr-Coulomb and Drucker-Prager kind. These materials are described in the implicit standard materials framework based on the bipotential theory [START_REF] Hjiaj | A variational inegality-base formulation of the frictional law with non-associated sliding rule[END_REF].

During the last two decades several authors have developed asymptotic theories applied to thin layers [START_REF] Suquet | Discontinuities and plasticity[END_REF], [START_REF] Ait-Moussa | Modelisation et etude des singularites d'unjoint colle[END_REF], (Klarbring 199l), (Licht, Michaille 1996). [START_REF] Lebon | Numerical study of soft adhesively bonded joints in finite elasticity[END_REF], [START_REF] Bayada | Asymptotic and numerical analysis for unilateral contact problem with Coulomb's friction between an elastic body and a thin elastic soft layer[END_REF]. The idea of this work is to replace a thin layer bonded with two substrata by an interface law which keeps in memory the mechanical and geometrical characteristics of the layer. The motivation of this analyse is the complexity of the numerical approach:

• the thickness of the layer is small regarding the substrata dimensions • the stiffness of the layer is weak compared with the substrata rigidities. The theory consists in considering that the geometrical and mechanical parameters of the layer tend to zero and to analyse the limit problem. In this limit problem, the layer vanishes geometrically and an interface law replaces it. Using this theory, simplified models of interface are obtained which are easier to compute by finite element software. On the one hand, this theory permits to justify empirical interface laws that one can find in the literature and on the other hand, to find new interface laws. In previous papers, different kinds of behaviour (elasticity, visco-elasticity, plasticity, . . . ) and kinematics tProfessor +Assistant Professor (small perturbations, plates, finite deformations, ... ) have been dealt with. Our contribution in this work is concerned with non-associated elastic-plastic behaviours.

The paper is . organised as follows: in the second section, the mechanical problem is presented, notations are specified and the constitutive law of the thin layer is formulated in the implicit standard materials framework. The third section is devoted to the theoretical results. The mathematical background is given and the matched asymptotic expansion method is applied to the problem with a non-linear behaviour. The mathematical results are commented. In section 4, numerical examples are presented in order to validate the theory and to quantify the terms obtained in section 3. In the last section, we close by giving conclusions and perspectives.

The mechanical problem

1. Problem definition

We consider two elastic bodies perfectly bonded with a third one which is very thin. For simplicity, we work only in two dimensions. The structure is denoted Q with boundary an and is referred to the local frame ( 0, x~o x 2 ). On a part of the structure I] a surface load is applied. The structure is embedded in part T 0 .

We denote (Fig. 1):

Q' = { x = (x 1 , x 2 ) E !2/lx 2 1 > ~} £2! = {X= (XI, x,) E Q/±x2 > ~} Q± = {x = (xl> x 2 ) E Q/±x 2 > 0} !lo=!l+ua_ B' = { x = ((x 1 , x 2 ) E !2/lx,l) < ~} S! = { x = (x 1 , x 2 ) E !l/±x 2 > ~} S = {x = (xi>x 2 ) E Q/x 2 >0}
e: thin layer thickness

!(au' au:)

Small strains, eij(u') = 2 ax; + if;, S! is the interface between the adhesive and the adherent lJ' is the thin layer Additive decomposition of strains: e(u"J=e'(u")+e"(u"), respectively elastic strain and plastic strain. S is the surface to which the adhesive tends geometrically The two bodies {4' (adherent) are supposed to be elastic and the joint If' (adhesive) to be elastic- plastic. In the next section, deiaiis of the behaviour of the thin layer are given.

Denoting by aijki the elasticity parameters, we have to solve the following problem:

Find (u', cr') such as:

cr: .. = 0 in!2 0 !],) e ( ') . n' crij = aijkl ekl u lllJo.& + behaviour law in B' u'=0 on ro cr'n =F on T 1 lu'l,. = 0 and lcr' nl,• = 0 ' ' [ ],
• is the jump on the boundary S~ and n is the external normal unit.

' 2.2. Behaviour in the thin layer

(I)
In this section, we describe the behaviour of the material in the thin layer. We consider that it obeys a non-associated elastic-plastic law. This kind of material is not in the family of generalized standard materials [START_REF] Nguyen | Materiaux elasto-visco-plastique et elastoplastique a potentiel generalise[END_REF] but in the class of implicit standard materials described in terms of the hipotential theory [START_REF] Hjiaj | A variational inegality-base formulation of the frictional law with non-associated sliding rule[END_REF]. In the case of generalized standard materials, we need to define a potential and a pseudo-potential of dissipation. In the framework of implicit standard materials, the behaviour is described introducing a hi-potential b and a hi-potential bp which depend on two tensorial fields, the stress tensor cr and the elastic strain tensor e' (resp. the plastic-strain rate tensor eP).

The behaviour law is built from the derivation of these two hi-potentials: In the case of a Drucker-Prager material, the hi-potential of dissipation is written:

bp( u, e") = ce~ + (tg8-tg<p)(sm-cJIIePII + Xx,(eP) + Xxa( u)
where XA is the indicator function of set A:

XA(x) = 0, if x E A, x,.(x) = + oo otherwise
Sets Ka and K, are defined by:

(2) c is the cohesion, cp is the friction angle and e is the dilatance angle.

(3) (4
In the case of a Mohr-Coulomb material, the hi-potential of dissipation is written:

Sets Ka and K, are defined by:

Ka = { U, lltJI<;; C-tgcpna} K, = {.r',e~<':O} (5) (6) (7) (8) (9)
(10)

(11) (12) (13)
where ta and na are respectively the shear and normal stresses associated to the stress tensor u. In the following, the material will be supposed isotropic.

Theoretical results

1. Mathematical background

The idea of matched asymptotic expansions [START_REF] Eckhaus | Asymptotic Analysis of Singular Perturbations[END_REF]) is to find two expansions of the displacement u' and the stress rr in the powers of e, that is, an external one in the bodies and an internal in the joint, and to connect these two expansions in order to obtain the same limit. We will obtain relations in the internal expansions that we will express from values that intervene in the external expansions.

1. 1. External expansions

The external expansion is a classical expansion in powers of e:

u' (X~o Xz) = u 0 (X~o Xz) + e u 1 (x,, Xz) + ... , ( 14
) m _ _ 1 (du~ du7) e .. - + ' 1 2 dXj Jx, ( 15
) (16) (17)

1.2. Internal expansions

In the internal expansion, we proceed to a blow-up of the second variable. Let y 2 = x 2 /e. The internal expansion gives:

u' (x,, xz) = v 0 (x 1 , yz) + e v 1 (x" Yz) + ... , m avr m Jv~+! m 1(dv~ dv~+l) e" = Jx,' e, = -ay;• e,, = 2 Jx, + Jy, ' e ') ( -I -I 0 I eii(u x 1 , y 2 ) = e eij + eu + ceu+ ... , We use the convention v"' = 0, m < 0, "m = 0, m < -1
where m is the expansion order.

Continuity conditions

(18) (19)
(20)

(21) (22)
The third step of the method consists of the connection of the two expansions. We choose some intermediate points defined by Xz = ± S E 1 , 0 < t < 1, S E ]0,+ oc(. When E tends to zero, Xz tends to 0' and y 2 =x 2 /e tends to ±oc. The principle of the method consists in assuming that the two expansions give the same asymptotic limits, that is:

(i) v 0 (x,, ±oc) = u 0 (x 1 , 0'),
(ii) -r-1 (x 1 , ±oc) = 0, (iii) -r 0 (x" ±oc) = a 0 (x 1 , 0').

Mathematical results

1. Equilibrium equations at order 2

(23) (24) (25)
We develop the equation div a= 0. In this paragraph, we consider the equilibrium equations at order -2; we obtain:

Thus, we get 0 -1 a'<;z = 0 ay, ( 26
) (27)
The elasticity law in the bodies gives: Thus, we have Jv" -a :..:..;_ -I2j2 (}yz' () vo a:, = 0 andv 0 (x 1 , y,) = v 0 (xi), IYzl > 112 We have r,i = 0 in the bodies and, due to the matched conditions v 0 (x1. ±ly 2 1)=u 0 (xJ. o•).

Equilibrium equations at order 1

We have ar",

-' =0 dyz
The connection conditions give with y 2 = x 2 /e tends to ±<x when e tends to zero.

Elasticity of the thin layer

The asymptotic expansions give:

where (e:j )'is the elastic part of e~.

(28)

(29) (30) (31)
Note that before the beginning of plastification, the plastic strain is equal to zero, so the strain is reduced to the elastic strain.

We have supposed that the layer is thin and soft, that is, the thickness is small (e--70) and the stiffness coefficients are small (A--70, !1--70). As we can see in the expansions, the identification of the different orders depends on the relative behaviour of the Lame coefficients A and 11 with respect to the thickness e. In fact the limit contact law depends on the two ratios A !e and 11 !e. We have nine possible relative variations of these two ratios corresponding to the behaviour of the coefficients (zero, a positive value, infinity). In the following, we note j = lim,. 0 fie. In particular, we denote by I= lim,..., 0 Ale and 'ji = lim...., 0 we the limits associated to Lame coefficients. In the following, only one case is presented, that is the more representative case where the limits are positive values. All the other cases can be deduced easily.

Before the beginning of plastification, the identification of the terms of order zero gives:

o -dv~ rdxi) = !1- dYz 0 ( ) ('+ 2 ;;)dv~ 'r 22 X 1 = IL r-;:}., vn ( 33
) (34)
By integration of Eq. ( 33), Eq. ( 34) and using the connection conditions Eq. ( 23), Eq. ( 25), we find the elasto-static case which is now classical (Ait-Moussa 1989):

CJn = KL [u] (35)
Matrix KL is diagonal and its diagonal terms are equal to j1 and X + 2j1 respectively. [] is the jump on boundary S.

Elasto-plasticity

The most interesting case is when the plastic threshold is reached. For a Mohr-Coulomb material this threshold

I -1 -1 o . I < -1 -1 o l £ tr tr+ ... <0, c-tg rp £ n, + n,+ ... is replaced by
where t} is the shear stress part of -r 1 and n} is the normal stress associated to '1" 1 .

For a Drucker-Prager material this threshold

II -1 -1 o II < -1 -1 o )
£ S-r: +s,+ ... '5:.c-tgqJ E Srm+Srm+ ...

is replaced by

where s} is the deviatoric part of -r 1 and s}, is the pressure associated to '1" 1 •

We obtain from Eq. ( 32)

where (ei/f is the plastic part of ei/. Using a notation similar to Eq. ( 35), we denote by 112

[uj] = 2 J (e~i)P dy 2

-1/2 and 112 p 1 f --1 p ---1 p [u 2 ] = =--= (A(e 11 ) +(A+ 2,U)(e 22 ) )dy, A+ 2.U -uz (36) (37) (38)
(39)

(40) (41) (42)
(43)

Considering that the elastic part (e~il' is equal to zero, it seems convenient to suppose that (e~ilP is equal to zero too (that is the deformation e]~ is small), and thus 1/2

[ui] = f (e;:i)" dy, -112

Note that this hypothesis is confirmed numerically.

So, we obtain from Eq. ( 32)

and integrating Eq. ( 3)

a 12 = jl([u~]-[t/;]) a,, = eX+ 2jl)([u;J-[~])
• Drucker-Prager condition:

[uP] = -k 0 psn, k 0 p > 0 • Mohr-Coulomb condition: [uP] = -kMct, kMc> 0
Vector t corresponds to the direction of the shear stress.

(

) (45) 44
(47)

(48)
The limit problem is quite different in this non-linear case. The plastic yield and the plastic law depend on a local problem. Local problem means that they do not depend on the stress vector but on all the components of the stress tensor in the thin layer (Eq. ( 47), Eq. ( 48)). On the one hand the thin layer vanishes from a geometrical point of view(£--> 0) and on the other hand, in the limit problem, we have a strong coupling between the interface law (Eq. ( 45), Eq. ( 46)) and a problem in the thin layer because the local stress tensor is a priori unknown in Eq. ( 47), Eq. ( 48).

Local problem

In order to determine the threshold and the direction of sliding, we have to solve the following elasticplastic problem (here Drucker-Prager model, results obtained for Mohr-Coulomb model are of the same kind) in the rectangle domain S x [ -1/2, 112]:

a,i = A.(e,,)' {jii + 2f.J.(e,i)' lis II$ c-tgrpsm If llsll < c-tgrpsm then ep = 0 If lis I I = c-tgrpsm then ep = -ks (49)
Due to the fact that the solution does not depend on the thickness, we have to solve only a "onedimensional problem". We observe that we obtain a "pseudo-penalized-Coulomb" law of friction. Note that if the direction of flow is equal to (or approximated by) x 1 , we find the classical Coulomb law of friction.

Numerical results

Geometry of the examples

In this section, we present three numerical tests. The first one is a long square bar bonded with a rigid obstacle. The width of the bar is equal to 100 mm and the thickness of the thin layer is equal to 1 mm. A load is applied on the left part of the structure. The second example is the same bar with a load applied both on the left part and on the top of the structure. The load on the top is twice smaller than the load on the left part. These examples are treated considering that the thin layer obeys Drucker-Prager behaviour. Details on the mechanical characteristics are given in Fig. 2.

The third example is a dovetail assembly. Due to the symmetry of the problem, only a half of the structUre is considered. A thin layer is bonded between the two elastic parts of the assembly. The layer is oblique with regards to the loads. The loads are applied at the bottom of the structure. The dimensions and the mechanical characteristics are given in Fig. 3. As for the previous examples, the material in the thin layer obeys Drucker-Prager elasto-plasticity. The set of these three examples is quite general and representative of the mechanical phenomena induced by non-linearities and the small characteristics of the thin layer. In the following section, numerical results obtained using these three examples are presented and commented.

Numerical synthesis

1. Elastic domain

The computations are done using ANSYS software (ANSYS 2002). In the first part of the n\]merical synthesis, we observe the results in the elastic domain (Fig. 4). Fig. 4(a) correspbnds to example 1 at step 13, Fig. 4(b) corresponds to example 2 at step 15 and Fig. 4(c) to example 3 at step 30. Ih these figures, we represent the tangential stress/tangential displacement ratio. As expected from the theory, this ratio is close to Jl IE, where Jl is the second Lame coefficient, Jl = E/2(1 + v) (Eq. ( 35)). We observe that examples 1 and 2 give the same value because the thin layers have the same mechanical characteristics. A similar result is obtained for the normal components. In this case the limit is equal to (I+ 2Jl)IE, with A= Ev /(1-v 2 ). These results establish the validity of our theory in the elastic case; note that due to boundary effects, this theory is not valid at the extremities.

a)

Step 13 ~ Ratio tangential stress (Mpa)ltangential 1 Step 24 • Rallo rormalstresa (Mpa) lnonnal dlsplaeement (mm) along the fin layer

. -H --------------1 -8.001!103 t f ----------------1 -~Hr------------------~ = 9 , 0 ' l £ . Q . ) + + ---------------I ~ ~~~---------------~ ~•~~H--------------------------1 t5 7,00E+03 tl-------------------1 8.001!+03 -1 -----------------i 5~~----------------_ , MOE+03 ~~-------------.,l b ,~~H--------------------------; ~u~~------------------------~ um~r-------------------------~ •~~~------------------------~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Nodes c) Slep 30 -Ratio tangential stress (Mpa)ltangentlal displacement (mm) along lhe thin layer 3,50E+04 .,....----------------------. 3,00E..04 -1 ---------------t t <S 2,50E..04 +-------------------H "i'l b 2,00E+04 . y -------------------1 1,50E+04 ~----------------------1 1.00E+04 -+--....-..,.....--,--.....----.--.-----.-"T""'
3,50E+04 <"~ 3.00E+04 ~2,5oe+04 ~ 2,00E+04 1,50E+04 1,00E+04 r I ' I " "' '

Elastic-plastic domain

We consider now the elastic-plastic domain (Fig. 5). In the elastic part, we find the previous results again. The first two examples give a stiffness density of 33000 N/mm 3 and the last example gives 64800 N/ mm 3 , these values correspond to (A+2J1)/e (Eq. ( 35)). In the plastic zones (Eq. ( 39)), we do not find a constant value. In the next sections, we analyse more precisely these zones. Note in the third example that the points of discontinuity correspond to a normal displacement and a normal stress close to zero together: the thin layer is not crushing or pulling out on these nodes.

Plastic yield

We have shown in the theoretical part, that in the limit problem it is necessary to solve a local problem coupled with the global one. In the local problem (Eq. ( 49)), there are two significant quantities, which intervene: the plastic yield and the plastic flow. The aim of this section is to quantify the level of each term in the plastic yield. In fact, we want to analyse if it is possible to replace the "real" plastic yield in which all the terms of the stress tensor are considered, by a "simplified" one in which only the terms of the stress vector on the surface are taken into account.

The real plastic yield is defined by:

llsll ~ c-tgcp s m (50)
The simplified plastic yield is defined by: a)

Step We compare the real plastic yield with the simplified one which use only the terms corresponding to the stress vector. In the last formula (Eq. ( 51)), the stress tensor is replaced by a n®,n, where ®s is the symmetrical tensorial product (ssimp is the deviatoric part of ((j n): s(a n) and Sm(simp)=tr(ssimp)). Fig. 6 shows the differences between the two plastic yields for the three examples (only the right hand side of Eq. ( 50) and Eq. ( 51) are represented). We observe a very low difference for the three examples and that this difference does not modify the initiation of the plastiftcation. The gap is maximum in the elastic zone but generally remains lower than 5%. In the plastic zones, this gap decreases to 4% for the first two examples and to 2% for the last one. Fig. 7 shows the relative difference of the two yields for example 3 along the surface of the thin layer at step 50. As a conclusion of this study, we have shown that our simplification is valid and that it is possible to work only with the stress vector for the computation of the plastic yield.

Plastic strain

The aim of this section is to quantify the level of each component of the plastic strain (Eq. (3), Eq. ( 47), Eq. ( 48)). Fig. 8 shows for the three examples that the components of the plastic strain vector that is e~ and e{z are preponderant. eft is very small compared to the two other components of the tensor (Eq. (43), Eq. (44)). As an example, in the case of the dovetail assembly, for the node 16 (at the beginning of the interface) at step 50, e~ = -3.64x10-4, e-b_ = 8.09xlo-5 and e 11 = -2.92x10-{j. eft is less than 1% of the value of e 1 ~ and close to 4% of eb_ and thus can be neglected. We show in Fig. 9 differences of the Von Mises norm of the "real" plastic strain and the "simplified" one. In the simplified version the plastic strain tensor e P is replaced by e P n ®sn. The gap for the third example is close to 1%.

Conclusion of the numerical synthesis

As a conclusion, the numerical results obtained in this paper show that the local problem introduced in the theoretical study can be neglected, that is to say that the interface law can be written only in terms of stress vector. We obtain a compliance law (regularized Coulomb law), well known in the literature (Fig. 10).

Step 50: Equivalent plastic strain along the thin layer l: 2+0E-04 ~ 1,56-04

... -••••••--•--•-•••••• --•••••••• ... 3 1 0&04 ••-•-- ••------••---•-•-••---•--•----••--•••••-----
.:r s:os.os • 1--_.,.,--_ _ _ _ _ _ _ ,.......,-----j O,OE+OO .J....---~o:=--.---...... =-....::loo.,-J 0 100 200 300 400 Nodos Fig. 9 Equivalent plastic strain along the substratum/thin layer interface: example 3 at step 50

(u] Fig. 10 The limit law: compliance law of contact

Conclusions

In this paper, we have analysed theoretically and numerically the asymptotic behaviour of a soft thin layer obeying to a non-associated elastic-plastic law (Drucker-Prager or Mohr-Coulomb) bonded with elastic solids; the results obtained by these two models are of the same kind. Theoretically, we have shown that when the mechanical and geometrical characteristics of the layer tend to zero, an interface law is obtained. This law couples a local problem with a global one. The numerical study has permitted to quantify the influence of the local problem and has shown that this local problem can be neglected. A classical law of compliance has been obtained. Numerical results obtained are representative of all nonassociated elastic-plastic models.

A perspective of this work is to introduce damage and cracks in the thin layer in order to obtain laws which take into account the rupture between the layer and the substrata. These laws can be used to model the gluing of mechanical structures, the fibber/matrix interface in composites, the mortar/brick interface for masonries or the composite/structure interface in reinforcement techniques. Another important point is the implementation of these laws in a finite element computational software.

Fig

  Fig. I Geometry of the problem

)

  Ka = { U, lis I I <; ; c-tgcp sm} K, = {ep, e~ <' : rgelle:;ll} sm = ~tr(U), (hydrostatic pressure) s = U-s,ld, (stress deviator tensor) 1 ~ = 3tr(ei') e:; = eP-~Jd llall = J~a,jaij for any tensor a (Von Mises norm)

Fig. 2

 2 Fig. 2 The first two examples: a bar bonded with a plane (the dimensions are in mm)

  dleplacament (mm) along the 1t1n layer b) Step 15 • Ratio tangential atreas (MpaYtangential dla~ (mm) tlong the thin layer '~It::=:=:=:=:=:=:=:=:=:==: ~ 1, 106+00

  Fig. 4 Tangential stress (MPa)/tangential displacement (mm) ratio in the elastic domain along the interface substratum/thin layer a) example 1 at step 13 b) example 2 at step 15 c) example 3 at step 30

Fig. 5

 5 Fig. 5 Normal stress (MPa) /nonnal displacement (mm) ratio in the elastic-plastic domain along the interface substratum/thin layer a) example 1 at step 22 b) example 2 at step 24 c) example 3 at step 50

Fig. 6

 6 Fig. 6 Plastic yields along the substratum/thin layer interface a) example 1 at step 22 b) example 2 at step 24 c) example 3 at step 50

Fig. 7 Fig. 8

 78 Fig.7Relative difference (referred to the real plastic yield) between the plastic yields (real and simplified) along the substratum/thin layer interface (example 3 at step 50)