Taous Tatiana
email: tatianataous@gmail.com

The Dico form: an autonymous marker

Keywords: token-reflexivity, metadiscourse, enunciative disconnection, aspectual value, pragmatics

In this paper we review a specific use of the dico form: its use as autonymous marker. From a corpus of Latin texts (from the archaic period to the Flavian Age), we try to determine if this use is correlated to specific linguistic configurations. After reminding that the dico form could be used in the self-quotation with different values (over-enunciative and autonymous) often connected with the morphosyntactic form of the quoted speech, we study more specifically the linguistic conditions of emergence of the dico form as autonymous marker and review different criteria: contextual, syntactic, semantic and pragmatic. In the last part of this paper, we propose to focus our attention on two phenomena, particularly relevant: the autodiaphonic repetitions and the case particularities of the word used in autonymy with dico.

Introduction

The notion of autonymy is relatively difficult to grasp. 1 Indeed, if the use of autonymous units is a frequent phenomenon in languages, 2 the conceptualization of this phenomenon is late and subject to discussion, evolution and precision. 3 Since the end of the 19 th century, 4 linguists and philosophers of language agree with the idea that the words of natural languages can be used in two ways in discourse: 5 a "common" use, i. e. that the words are used in usage, with actualization of their meaning in language, and a "less common" use ("moins ordinaire", Brabanter 2005: 13) where, used in mention, the words play with the relation between signified and signifiant and so they belong to the metalanguage, and no longer to the language. 6 In linguistics, the identification of autonymy represents a real difficulty and constitutes one of the major problems. The first theoreticians of autonymy (see Brabanter 2005: 14 and 18), and especially the followers of the demonstrative theory (see Brabanter 2005: 26-29), thought that the inverted commas were a relevant and decisive criterion in order to recognize a word used in autonymy. However the utilization of inverted commas is never systematic and reveals, in discourse, an "overt form of the shown heterogeneity" 7 (Authier-Revuz 1984), but this discursive heterogeneity may also be hidden: so the interlocutor needs some sufficient knowledge in order to grasp the heterogeneity.

But even with overt heterogeneity, the inverted commas are not the only sign: to indicate an autonymous use, the speaker may use not only typographic signs, only visible with the written forms, like italics, inverted commas or red slashes of mediaeval manuscripts (see Rosier-Catach 2003: 2), but also morphological and syntactic signs, visible with written and spoken language. These signs belong to rules which are parallel (and sometimes conflicting) with grammatical rules of the common and usual language: 8 in French, the utilization or the non-utilization of the determiner (see, in (1a) and (1b), the underlined units) disagrees with the syntax of usual clauses. 9 In (1c), the inflectional form of the verb (fr. s'entend) depends neither on the number of the noun nor on the article with which it occurs. Also are selected the no-marked gender and number in French, i. e. masculine and singular. A similar phenomenon can be noticed about (1d): the speaker selects the masculine of the definite article, whereas the grammatical use requires the feminine with the French noun faucille. The utilization of comments in brackets or parenthetical and disjunct sentences is also a clear indication of the autonymous use, whereas the body language and the suprasegmental indications can play a very important role in this identification. 10 (1) a. fr. × Sieur est aujourd'hui hors d'usage. b. fr. CES tu vois m'excèdent au plus haut point.

metalinguistic words ("Verb" is a noun). See Rosier-Catach (2003: 6-7) and Nicolas (2005b: 47-50). The mediaeval copyists have also been sensitive to these uses where the word seems to be understood in its materiality. For a complete approach, see Rosier-Catach 2003. 5 That is Quine's opposition between "use" and "mention" (Quine 1981 [1940]: 23-27; see also Lyons 1977: 5-10). 6 See, for example, Rey-Debove 1971: 90, Fleck 2015: 1. 7 "Une forme marquée de l'hétérogénéité montrée". 8 Nicolas 2005a8 Nicolas : 96, 2005b: 45. 9 : 45. 9 It is possible that the utilization of this determiner with the parts of speech, which commonly do not need it, is due to an "amazing" invention of mediaeval Latin of scholars ("une invention remarquable du latin savant médiéval"), indicated by Rosier-Catach (2003: 4-6): the French article li and its allomorphs are used in order to indicate an autonymous use; the French article is positioned before Latin words in hybrid phrases, like le homo / li homo (with a noun), del albus (with an adjective) or li legit (with an inflected verb). 10 For example, we think about the "mimed" inverted commas which go with the speech of some speakers and indicate visually the autonymous use or, more generally, the discursive heterogeneity in the speech. According to Rosier-Catach (2003: 2), "there are no univocal typographic or linguistic markers which allow recognizing autonymous uses". 11 However, on the occasion of a colloquium dealing with the (in)direct speech act in Latin, we have worked on the selfquotation, indicated by the use of the forms of lexeme dico / aio and we have proposed a comparative and contrastive study of these verbal forms. But, unlike aio, the specificity of dico is that this verbal form can be considered a real autonymous marker. 12 We have tried to explain this phenomenon, but it was impossible to develop the complete analysis. 13 That is why, in this paper, we would like to work on the uses of dico as autonymous marker and add some indications about this verbal form, especially concerning the morphosyntactic and pragmatic behavior of this "overt form of the shown heterogeneity" (Authier-Revuz 1984).

The verbal form dico in self-quotation

1. Self-quotation: definition

A speaker may integrate alterity into his own utterance; this phenomenon is current in discourse. In the mind of the locutor, this integration may be deliberate and explicitly indicated: for example, by an explicit reference to the popular parole (the way of "comme on dit"), or by a reference to the individualized alterity (the way of "comme il/elle dit"). 14 Discursive alterity may also be deliberate, but hidden; then, it may be involuntary: the locutor is so full of another speech that he does not know that he is quoting a speech which does not belong to him. 15 In these three configurations, alterity is more or less consciously integrated by the speaker. Nevertheless there is another way of speaking: the way of "comme je dis" (Rey-Debove 1971: 92). Quotation becomes "self-quotation". Rosier (2008: 126) defines selfquotation as a "quotation by oneself of an utterance which seems like an anterior or a socalled anterior utterance in the speech". 16 When the Latin locutor is using the verbal formdico, he repeats his first utterance and stresses his own verbal act: he becomes his own repeater (see Taous 2016: 13-14). In this analysis of verbal form-dico, we have decided to consider a corpus from ancient Latin texts to neo-classical Latin texts (namely from the archaic period to the Flavian period), 17 in order to propose a large view of configurations where dico can be used. 18 11 "Il n'existe pas de marques typographiques ou linguistiques univoques des usages autonymiques". 12 See also Rosier-Catach (2003: 5) and Nicolas (2005b: 49 and note 10). 13 For this comparative work, see Taous 2016. 14 See Rey-Debove (1971: 92) and the three ways of speaking: the way of "it is said", the way of "he/she says/is saying" and the way of "I say/am saying". 15 Concerning this idea that "necessarily, the words always are 'the words of everyone'" (fr. "les mots sont toujours, inévitablement, 'les mots des autres'"), see Authier-Revuz 1984: 99-100. 16 "Citation par soi d'un dire qui se présente dans le discours comme un dire antérieur ou supposé tel". 17 For a periodization of Latin literature, see Flobert 1975 Nevertheless, in these three examples, the dico form must not be considered as an autonymous marker: its utilization indicates that the speaker reaffirms his posture of locutor in his utterance act. So the morphosyntactic class of the quoted unit, introduced by dico, is very important because it can be a criterion of the pragmatic value to attribute to the utterance. Morphosyntactically the quoted speech can be an infinitive clause, a juxtaposed clause (with imperative mood or indicative mood (present and future)), a dependent clause with subjunctive, 21 an accusative, an adverb or another case. Our corpus proves that there is a correlation between the pragmatic and enunciative interpretation of the quoted speech, introduced by dico, and the morphosyntactic class of this same speech.

Finally, besides the discursive context and the morphosyntactic class of the quoted speech, a last parameter should be considered, namely the position of dico on the syntagmatic axis in comparison with the quoted speech.

A triple relation is established between the position of dico, the morphosyntactic class of the quoted speech and the pragmatic-enunciative interpretation to attribute to this reported speech. Consequently the autonymous interpretation of the quoted speech often coincides with juxtaposition / accusative / other cases, on the one hand, and inserted position of dico, 22 dico are distributed in this way: 100 occ. (archaic period I), 028 occ. (archaic period II), 158 occ. (archaic period III, limited to Cic. Catil., Dom., Har. resp., Leg. agr., Prov., Verr. and all the letters), 016 occ. (Augustan Age (= period IV)), 120 occ. (Julio-Claudian dynasty (= period V)), 022 occ. (Flavian Age (= period VI)). 19 See (2a). The notion of "diaphonic repetition" (= fr. "reprise diaphonique") has been introduced by Roulet (1985: 71), quoted by Rosier (2008: 31). In verbal interaction, a locutor Y reacts to the utterance of his allocutor (= locutor X): so he repeats an unit of this speechconsidered as pragmatic connectiveand reintegrates it in his own utterance in order to indicate his point of view (pragmatic effect). 20 For more examples, see Taous 2016: 13-19. 21 For the opposition between paratactic clause and hypotactic clause, see Sznajder 1996. For the detail of numbers of the morphosyntactic classes of the quoted speech, see Taous 2016: 19. 22 Quoting speech (i. e. the dico form) can be prepositional (i. e., before the quoted speech (= fr. "positionnement on the other hand. The "over"-enunciative interpretation (or emphatic and contrastive interpretation) coincides with the infinitive clause / dependent clause with subjunctive / adverb, 23 on the one hand, and the prepositional position of dico, on the other hand.

In the following table, we have selected some examples from Plautus, Cicero and Seneca, 24 in order to illustrate the linguistic relations indicated supra. We have chosen the three morphosyntactic classes which more often coincide with the autonymous interpretation, that is to say the juxtaposed clause, the accusative 25 and the non-accusative case forms. The gathered data allow evaluating the degree of predictibility of the semantico-pragmatic interpretations of the quoted speech (over-enunciative; autonymous) in comparison with its own morphosyntactic class (juxtaposed clause; accusative; non-accusative case): The data collected for the three writers confirm that the dico form governing an accusative is to interpret as an autonymous marker in 2/3 of the examples. Then, the examples for the "over"-enunciative interpretation are due to cataphoric phenomena: the accusative, cataphoric, announces the message content which often is an infinitive clause from a morphosyntactic point of view.

So we can also explain the frequency of juxtaposed clauses with the "over"-enunciative interpretation, especially in Plautus' and Seneca's examples (see 14 occ. and 22 occ.). The juxtaposition with cataphoric reference constitutes a borderline case, because it is used in the two semantico-pragmatic contexts. In fact the speaker highlights his utterance (emphatic value) and also stresses this cataphoric "production" introducing his speech (autonymous value). Two tendencies co-exist in the corpus, when the dico form is inserted between the cataphoric adverb and the juxtaposed clause. So the semantic value of the adverb is essential to determine the semantico-pragmatic orientation to give to this self-quotation. In the first en prolepse")), postposed (i. e., after the quoted speech (= fr. "positionnement en analepse")) or inserted (= fr. "positionnement par insertion"). With respect to this French terminology, see Cerquiglini 1981: 25 andTaous 2016: 10-11. 23 Concerning the fact that juxtaposition with cataphoric reference constitutes a borderline case in this configuration, see infra. For the methodological detail, see Taous 2016: 20-23. 24 For the reason why these Latin writers have been selected, see 25 When it is not considered as a complement like uera or uerbum.

tendency, the adverb has a discourse role: as cataphoric, it announces the message content and retakes the utterance into the discourse coherence; in the second tendency, cataphora is implicit, because it is satisfied by the context. That is why the adverb (or the adverb phrase) does not announce any message content, but gives some indications about the modus (see Cic. Dom. XCVI: maxuma uoce dico), speaking time (see Cic. Verr. II V 140: breuissime… dico) and iterative aspect of the speech (see Cic. Att. XIII 25, 3: etiam atque etiam dico) or value judgment of the speaker concerning his utterance (see Cic. Catil. I 3: dico aperte, Cic. Att. VI 1, 3: plane dico…).

Finally, we can see that Cicero and Seneca often use dico with a non-accusative case form. This syntactic rule, which seems very different from the syntactic convention of usage with dicere, is an overt sign indicating that dico is an autonymous marker in these configurations. Some occurrences have been difficult to interpret, especially when there was a prepositional phrase with de + ablative or with a spatial meaning. We have considered three criteria as decisive: in the first we have decided to consider all the context, especially in the (non-)utilization of auto-diaphonic repetitions or words referring to the spatiotemporal setting; in the second one (syntactic), we have considered that the prepositional phrase was to interpret either as a complement governed by the verb (autonymous interpretation), or as an adverbial stressing the hic (et nunc) of the speaker and thus taking the utterance in his sphere (in (3a), see hodie, adsunt). So dico should be interpreted as over-enunciative. The last criterion, semantic, results from the syntactic interpretation: dico does not mean "I am saying", but "I am talking" / "I am saying this". In the table 2, we have tried to summarize the three linguistic criteria from the examples quoted in (3):

(3) a. Cic

Semantic criterion

"say 'X'…" YES NO "talk" NO YES So the three Latin writers approximately confirm that there is an interrelation between morphosyntactic class of the quoted speech, position of dico and semantico-pragmatic interpretation. However some occurrences constitute borderline cases, so that we agree with Reix (1981: 279), when he says that "the natural metalanguage is our current and ordinary conversation about the language" and that, "in the verbal interaction, we are compelled to comment on our speech, especially in the frame of reference and signification" (we have stressed some words). 26 So "over"-utterance act and autonymy do not constitute dichotomous notions. That is why it is finally difficult to evaluate with precision the place of autonymy in the self-quotation. We will only say that autonymy plays a major role, so that we have decided to describe precisely the autonymous dico from examples drawn from the archaic period to the Flavian period.

Dico and the autonymous interpretation of the quoted speech

1. Autonymous uses and pragmatics

Dico as an autonymous marker often appears in epistolary and philosophic genres, although Livius and, above all, Quintilianus use autonymous dico here and there. In these uses, locutors want to explain or confirm a formulation, considered opaque by them if this is not emphasized by a metadiscourse comment, or finally, to stress a witty remark (see (7) for an example in comedy). In Latin (conforming to what is observed in other languages of the world), the word quoted in autonymy is various, not only as semantic category ((in)animate), but also as syntactic class (noun, inflected verb, adverb, phrase, clause…). The syntactic function of dico is different, because it depends on the context of the occurrences: either dico constitutes the verbal nucleus of the clause and should be considered as verbal predicate (see (4a) and (4b)), or it is inserted into a clause which possesses its own inflected verb; in these conditions, dico indicates a break in syntagmatic chain, a change of enunciative level. Indeed dico introduces a comment clause, a parenthetical clause which tries to justify the relevance of selected words or clauses, used in usage, and proposes a metalinguistic analysis of these units (see (5a) and (5b)): 27 the speaker slides from a linguistic plane, where the meaning of words is understood in usage, to a metalinguistic plane.

(4) a. Sen After reading various examples, we have noticed that dico as an autonymous marker generally coincides with auto-diaphonic repetitions (see (4) and (5b), for example) or is used immediately after the word, utilized in usage (see (5a)). According to Reix (1981: 279), a speaker chooses an autonymous procedural step, because he wants to clarify a reference, regarded as ambiguous, or to develop a signification, in order to be sure that the allocutor really grasps all the relevance of the selected words. In a comment unit concerning the reference (see (5a)), the autonymy is utilized to disambiguate and so possesses a pedagogical scope where the receiver determines the metadiscourse of the locutor. By contrast, in a comment unit concerning the signification, the explicature allows the speaker to reveal to the allocutor the relevance of the selected words. So the speaker draws the attention of the receiver to the subtlety of his utterance, with this idea that, in discourse, each chosen word signifies and is efficient. By theses different uses the speaker works on ethical proof.

On this matter, the example (4b) is interesting to analyze. Thus Seneca seems to stress no more the selected lexeme (the verb posse) than the chosen mood (subjunctive), while the indicative is expected. Indeed, with the causal structure non quod/ut (+ subjunctive), sed quia/ut (+ indicative), the verb used in the second part of the causal clause is inflected in the indicative. Seneca explicitly justifies the use of the subjunctive possit and guides the reader towards an interpretation as potential or counterfactual, as it is suggested at the end of the extract (Sine amico quidem numquam erit): the subjunctive contains a semantically full value; thus we must not consider it as a syntactic use, determined by the hypotaxis. In fact this metadiscursive comment clause allows conceptualizing the philosophic idea according to which the sage (sapiens), alone, is still with his friends and so is connected with the discourse coherence and makes the Stoic paradox evident. By contrast, Cicero's autonymous comment units (nouns, phrases or clauses) are very different: they are not comments, but excursus, that means metadiscursive digressions where the speaker breaks the chain of his utterance and adds an occasional and (so-called) insignificant remark.

2. Autonymous uses and repetitions: anaphoric or diaphonic function?

We have considered that the criterion of repetition was very important to recognize dico as autonymous marker. So the procedural step of analysis, generally used to explain anaphoric reference, will be utilized here: we must identify anaphoric substitutes and clarify the semantic relationship between these substitutes and their antecedents. Nevertheless, it is a little inexact to claim that anaphoric repetition is the same thing as repetition in the autonymous contexts, because this inaccuracy hides one of specificities of the autonymous discourse: the importance to the metalinguistic plane. Indeed, anaphora (in the strict sense) is connected with the topical and textual coherence of a speech and is at the linguistic level. By contrast, auto-diaphonic repetitions are not only correlated to anaphora, but above all they add the idea that there is an enunciative disconnection: if there is repetition, this is connected with the metadiscourse.

Auto-diaphonic repetitions have a special status which can explain the morphosyntactic diversity of words quoted in autonymy and the differences of syntactic rection, in comparison with the grammatical usage. Indeed, with dico infinitive clauses are expected, dependent clauses with subjunctive (with ut or interrogative word), accusative whose semantic category refers to the discourse28 and prepositional phrases with de + ablative. But when it is an autonymous marker, dico can govern not only a noun inflected in the accusative, even if it pertains to the animate semantic category (see (5a) and (6a)), or in another case form (see (6b)), but also an adverb (see (5b)) or an inflected verb, juxtaposed (see ( 4 Concerning the repetitions, the Latin writers of the corpus offer two tendencies: either the speaker produces a first utterance which is literally repeated, so that he justifies and explains his terminological choices (see (4a), (4b), (5b), (6a) and (6c)), or the speaker develops one of the words of the previous utterance differently, so that he clarifies his idea or disambiguates an homonymic unit (see (5a) and (6b)). The way of repeatingliterally or differentlyis very important to grasp the intentions of the speaker. So in a literal repetition (i. e. the prototype … X, dico X, quia...), dico contains a progressive aspect (« X, I am saying here and now -X, because… »). The utterance introduced by dico constitutes a break, a temporal stop where the speech spreads a reflexive parenthesis about the language.29 Thus the intent of the locutor is to stress the relevance of his lexical selection (indicate a syllepsis, a witticism…): thanks to the explanation and explicature of the speaker, the receiver reads behind the word and understands that the speech is pre-built and so is not insignificant. In some examples, dico again introduces a correction (see (5b), (6a) and (6c)): at the sentence level, the epanorthosis always coincides with an interrogative sentence (like quid dico… X?); at the pragmatic level, these corrections confirm the intellectual honesty of the locutor.

But in a non-literal repetition (i. e. the prototype … X, dico Y), dico contains a desiderative aspect and indicates a pedagogic precision ('X, I mean Y'). In these configurations, dico can change its syntactic category and after a derivation by conversion be grammaticalized as an adverb. The French translations 'c'est-à-dire', 'à savoir' would confirm this analysis. In the following table, we propose to condense our different remarks: We can see that the two occurrences of dico in (8a) are different: in the first one, dico suggests a metadiscourse referring to the relevance of the signification (from there, the literal autodiaphonic repetition), whereas, in the second one, dico introduces a metadiscourse on the reference: dico opens a pedagogic parenthesis. The speaker repeats the prepositional phrase in portum and reveals all the stakes and dangers correlated to this invasion (in portum = in urbem). The first utterance, non-autonymous (in ipsum portum), announces some constituents of the last reformulation (in urbis intumam partem). So dico explicitly introduces the metadiscursive digression which must be understood as a simple epanorthosis, in the beginning, but as a political remark, at the end. Ernout and Thomas (2002 [1953]: 135) notice that dico can be used in a sentence containing its own syntactic organization. Dico often repeats the unit with the case form in which it is inflected in the sentence (see (6a), (6b), ( 7), (8a) and (9a)), 30 except with the nominative case (see (10a)): in this configuration, dico repeats the unit with accusative, the usual case expected with this verb in its non-autonymous uses.

The case particularities of the word used in autonymy with dico

The examples of the corpus approximately confirm the Ernout and Thomas' remarks (see (10)), while some nominative / accusative case forms are ambiguous and do not allow any sound conclusion. 31 The example (10c) is interesting to consider, because it indicates the differences of syntactic behavior between dico and inquam concerning the syntactic rules: so, dico used as an autonymous marker has not reached the degree of grammaticalization of inquam, since it keeps its verbal prerogatives (rection with accusative), whereas inquam, higher on the scale of grammaticalization, 32 seems to become an adverb of evidentiality:

(10) a. Cic. Att. 4, 16, 8 However some examples reveal that it is possible to find the nominative case governed by dico used as an autonymous marker. To the two examples of the corpus ((12a) and (12b)), we can add the text of a tabella defixionis where the locutor indicates the grammaticality of the two syntactic rules (dico with accusative; dico with nominative) and this, on the same syntagmatic chain. In this tablet, the elements in square brackets indicate the philologists' reconstructions from the comparison of five tablets dated from the end of Republic; and concerning the inflectional ending the text is categorical: nominative (or vocative?) and accusative co-occur in the same utterance:

(12)
31 See (11). In (11a), is the second occurrence of the accusative me governed by dico or does dico literally repeat the accusative me, governed by inuitarunt in the non-autonymous utterance? In the same way, in (11b), breues can be interpreted either as accusative governed by dico or as nominative repeated by dico with the case form of the non-autonymous utterance. The nominative nullae would confirm this last interpretation and would reveal that the nominative with dico is possible, despite (10b), of similar structure (Rem dico? Immo uero existimatio). 32 The grammaticalization of inquam, detectable from syntactic indications (nominative vs accusative), has been facilitated by the defective conjugation of this verb and by its very conditioned uses, unlike dico, very frequent and usual (as shows its future in Romance languages). We will venture the hypothesis that the text of the tablet certainly offered a vocative (Proserpina) and an accusative (Acheruosiam). The accusative form, because it suggests considerable syntactic dependency, would be comparable to the role of the subjunctive in comparison to the indicative and so would be less spontaneous than a vocative. Indeed the change of the vocative into an accusative indicates the importance of the syntactic dependency on the verbal form dicere. It is possible that the nominative-vocative with dico has been used for notifying in writing the intonation of the appellative. Also, the nominativevocative would be used whenever the intent would be to imitate the intonation contour of the utterance. In these conditions, Latin speakers would have attributed to these units in nominative-vocative a rising intonation, non-existent in the frame of the accusative.

The use of nominative and vocative cases, indicated here, is not an isolated phenomenon. So Rébeillé-Borgella (2012: 86) has noticed some similar facts with φωνεῖν and uocāre, in the New Testament (Johan. XIII 13): the quoted examples seem syntactically surprising, since from a referential point of view the element in the accusative (με and me in the examples) is connected with an object complement in the vocative (magister et Domine) or in the nominative (Ὁ διδάσκαλος καὶ Ὁ κύριος). This syntactic deviance contributes not only to indicate the autonymous utilization, but also to confer on the utterance a particular value: the utilization of nominative / vocative cases in texts which are embedded in the oral form and which consider the utterance act as magic (see the Tabellae defixionis) and efficient (see, in the Apocrypha, the Λόγια Ἰησοῦ) weapon could be due to diaphasic reasons. In these specific and particular texts, it was necessary to imitate in writing the uttered word, in order to give back all its performativity. The phenomenon could have been generalized in the less symbolically charged texts.

Then a confusion between diverse cases could have been made (see, perhaps, Quintilianus in (12b)) and the vocative-nominative could no longer have been saved to imitate the rising intonation of the locutor but only for syntactic reasons: in some contexts, the autonymous unit was coreferential with a nominative (here, pignora) and so it has been inflected in the case form of the word with which it was connected (namely, in the nominative), with dico as adverb. The suprasegmental value, originally attributed to the vocative, has progressively disappeared because of the morphological neutralization between vocative and nominative; vocative may have been reinterpreted as nominative (no-marked case form) which has been saved, when the word with which it was coreferential was subject of the sentence (see pignora in our example). In our opinion Quintilianus deliberately selects the nominatives liberi and propinqui, whereas he could have used the accusative, since the neuter pignora, noun expanded by liberi, parentes and propinqui, is morphologically ambiguous: so Quintilianus wittingly selects a nominative, because liberi, propinqui and parentes are subjects of adfert (agreement with the latter) too. Dico is thus completely grammaticalized, since it has lost the verbal prerogatives which it still kept in these configurations, namely with an expanded nominative.

Conclusion

While this paper is not exhaustive, we have tried to prove the original behavior of dico as autonymous marker. The diverse examples have indicated that dico was used for introducing a digression or a metadiscursive comment. Two pragmatic attitudes have been shown and correlated to precise morphosyntactic features: when the point of incidence of the metadiscourse is connected with a comment on the reference, dico contains a desiderative aspect ("I mean" / "I want to say") and coincides with a no-literal autodiaphonic repetition. The speaker thus tries to clarify his idea in order to be clearly understood. But when the point of incidence of the metadiscourse is connected with a comment on the signification, dico so contains a progressive aspect and the speaker comments on the relevance of the word used in autonymy or, by contrast, in an intellectual honesty, he tries to correct his utterance. This second configuration is to connect with a literal autodiaphonic repetition where the speaking time seems to stop and where the linguistic construction is made at the same time as the speech is uttered.

For this reason, the verb aio, symbolically very charged, could not be used in these autonymous configurations: indeed, the trend towards the grammaticalization, which ultimately concerns dico in some occurrences, reveals that this linguistic evolution was perhaps less compatible with aio.

Bibliographical references

Authier-Revuz, Jacqueline. 1984. Hétérogénéité(s) énonciative(s). Langages 73. Les Plans d 'énonciation. 98-111. Bearth, Thomas & Fennig, Charles. 2015. French-English Glossary of Linguistic Terms, see https://feglossary.sil.org/french-linguistic-terms (accessed 13 May 2017).

Brabanter (de), Philippe. 2005. Philosophes du langage et autonymie: une déjà longue histoire. Histoire Épistémologie Langage 27(1). La linguistique baltique. 11-43. Carnap, Rudolph. 2002 [1937]. The Logical Syntax of Language, translated by A. Smeaton. Chicago and La Salle: Open Court Classic.

2. 2. Self-quotation with dico: parameters of analysis

  and the following note.

		Locutors use the dico form in three configurations: (auto)diaphonic repetitions, 19
	emphatic contexts (in (2), see At ego, tu, nunc) and contrastive contexts (in (2c), see percipe
	vs dico): 20
		(2)
		a. Plt. Poen. 472-474:
		AN. … sexaginta milia HOMINVM uno die
		VOLATICORVM manibus occidi meis.
		LY. VOLATICORVM HOMINVM? AN. Ita deico quidem.
		b. Plt. Amph. 612-613:
		SO. AT EGO NVNC, Amphitruo, dico: Sosiam seruom tuom
		praeter me alterum, inquam, adueniens faciam ut offendas domi
		c. Lucr. III 135-136:
		... TV cetera percipe dicta.
		NVNC animum atque animam dico coniuncta teneri.
	18	We	have	listed	the	Latin	examples	from	Brepols'	web	site	(see	on
	http://clt.brepolis.net.janus.biu.sorbonne.fr/llta/Default.aspx, accessed 27 August 2013). The 444 examples of

TABLE 1

 1 Semantico-pragmatic interpretation and morphosyntactic form of the quoted speech (number of occ.)

	Writer	Plaut.		Cic.		Sen.
		"over"-	autonymous "over"-	autonymous	"over"-	autonymous
	Interpretation	enunciative		enunciative		enunciative
	Juxtaposed	14	01	04	09	22	06
	clause
	Accusative	03	06	02	12	10	21
	Non-accusative	01	////	03	06	////	20
	case forms

  /clt.brepolis.net.janus.biu.sorbonne.fr/llta/Default.aspx (accessed 27 August 2013).

								Taous 2016: 8-9. The examples come from
	Brepols'	web	site,	which	integrates	several	data	banks	like	LASLA.	See	on
	http:/

  . Dom. XXXIV: Videsne me non radicitus euellere omnis actiones tuas neque illud agere, quod apertum est, te omnino nihil gessisse iure, non fuisse tribunum plebis, hodie esse patricium? Dico apud pontifices, augures adsunt. b. Sen. Epist. XIV 11 : Ad philosophiam ergo confugiendum est; hae litterae, non dico apud bonos sed apud mediocriter malos infularum loco sunt.

	Spatiotemporal setting	NO	See (3a) : hodie, adsunt
		2. Syntactic criterion
	Complement governed by dico	YES	NO
	Adverbial	NO	YES
		TABLE 2
	Some difficult examples: Synthesis of linguistic criteria.
		About dico apud…
		Autonymous	"Over"-enunciative
		interpretation	interpretation
		1. Contextual criterion
		YES
	Auto-diaphonic repetition	See (3b) : APVD bonos sed	SOMETIMES
		APVD mediocriter malos
			YES

  . rhet. Suas. V 4: Si uenerit, uincemus. Hoc non est diu colligendum: de eo dico VINCEMVS quem uicimus. Sed ne ueniet quidem. b. Sen. Epist. IX 5: Ita sapiens se contentus est, non ut uelit esse sine amico, sed ut possit. Et hoc, quod dico POSSIT, tale est: amissum aequo animo fert. Sine amico quidem numquam erit.

	(5)
	a. Cic. Prov. XVIII:
	An Ti. Gracchus -PATREM dico, CVIVS utinam filii ne degenerassent a grauitate
	patria! -tantam laudem est adeptus, quod tribunus plebis solus…?
	b. Cic. Verr. II 4, 6:
	Nuper homines nobiles eiusmodi, iudices -sed quid dico NVPER? Immo uero MODO AC
	PLANE PAVLO ANTE -uidimus qui…

  Scrib. Larg. Compos. XXII 23: His fere utor primis diebus collyriis non sine CETERIS AVXILIIS, prout res postulat, ABSTINENTIA dico, SANGVINIS DETRACTIONE. c. Cic. Epist. XII 3, 1: Primum in statua quam posuit in rostris inscripsit "Parenti optime merito", ut non modo sicarii sed iam etiam parricidae IVDICEMINI. Quid dico IVDICEMINI? IVDICEMVR potius.

	) and
	(6c)):
	(6)
	a. Liv. XXXII 21, 13:
	Cur igitur nostrum ille auxilium absens petit potius quam praesens NOS, socios ueteres,
	simul ab Nabide ac Romanis tueatur? NOS dico? Quid ita passus est Eretriam
	Carystumque capi?
	b.

TABLE 3

 3 Semantico-pragmatic specificities of dico as autonymous marker Cic. Verr. II 5, 96: … statim sine ullo metu IN IPSVM PORTVM penetrare coeperunt. Cum IN PORTVM dico, iudices,explanandum est enim diligentius eorum causa qui locum ignorant -IN VRBEM dico ATQVE IN VRBIS INTVMAM PARTEM uenisse piratas. b. Sen. Benef. V 4, 2: Nam si turpe est beneficiis uinci, non oportet A PRAEPOTENTIBVS VIRIS accipere beneficium, quibus gratiam referre non possis, A PRINCIPIBVS dico, A REGIBVS, quos eo loco fortuna posuit, ex quo largiri multa possent pauca admodum et inparia datis recepturi. LIBERTVM ego habeo SANE NEQVAM HOMINEM, HILARVM dico, ratiocinatorem et clientem tuum. b. Cic. Leg. agr. II 95: HAEC QVI PROSPEXERVNT, MAIORES NOSTROS dico, Quirites, non eos in deorum immortalium numero uenerandos a nobis et colendos putatis? c. Sen. Dial. VII 15, 6: Quae autem dementia est potius trahi quam sequi? Tam me hercules quam stultitia et ignoratio condicionis est suae dolere, quod deest aliquid tibi aut incidit durius, aeque mirari aut indigne ferre EA, QVAE TAM BONIS ACCIDVNT QVAM MALIS, MORBOS dico,

	Point of incidence of the metadiscourse	Comment on the reference	Comment on the signification
	(Proto)type of the	Different / no-literal	Literal
	autodiaphonic repetition	(… X, dico Y…)	(… X, dico X…)
	Aspectual value of dico	Desiderative	Progressive
	Intents of the locutor	Pedagogic intent (clarify)	Ethic intent (read under the word)
	We can again specify the semantico-referential behavior of the non-literal autodiaphonic
	repetitions. Indeed, the quoted speech can use a characterization which clarifies the
	antecedent (see ferreo in (7)) or paraphrase words referring to a semantic relationship
	between…
	… meronym/holonym (see, in (8a), in portum (part) paraphrased in in urbem (all)),
	… hyperonym/hyponym (see (8b): the ablative praepotentibus uiris is repeated by
	principibus, regibus),
	… generic/specific (see (9)). In this last configuration, a nominal relative clause, uttered
	at first, can be specified (see (9b) and (9c)):
	(7) Plt. Curc. 690-691:
	Delicatum te hodie faciam CVM CATELLO ut accubes,
	FERREO ego dico…

:

  Itaque CAESARIS AMICI (ME dico ET OPPIVM, disrumparis licet) in monumentum illud quod tu tollere laudibus solebas […] contempsimus sexcenties sestertium. b. Cic. Att. XVI 15, 5: Me RES FAMILIARIS mouet. REM dico? Immo uero EXISTIMATIO. c. Cic. Att. VI 2, 4: His ego duobus generibus facultatem ad se aere alieno liberandas aut leuandas dedi, uno quod omnino NVLLVS in imperio meo SVMPTVS factus est (NVLLVM cum dico non loquor ὑπερβολικῶς: NVLLVS inquam, ne terruncius quidem)…

	(11)
	a. Cic. Verr. II 4, 25:
	Mamertini ME publice non inuitarunt. ME cum dico, leue est.
	b. Cic. Epist. I 14, 1:
	BREVES litterae tuae -BREVES dico? Immo NVLLAE.

  Cic. Fam. IX 22, 3: Cum loquimur "terni", nihil flagiti dicimus; at cum "bini", obscenum est? Graecis quidem, inquies. Nihil est ergo in uerbo, quoniam et ego Graece scio et tamen tibi dico "BINI", idque tu facis quasi ego Graece, non Latine, dixerim. b. Quint. Inst. VI 1, 24: Adfert in his momentum et aetas et sexus et pignora, LIBERI, dico, ET PARENTES ET

	PROPINQVI.
	c. Tabellae deuotionis (A. Ernout 1916: 100 A, 1-13 = W. Sherwood Fox 1912: A, 1-13,
	pp. 19-21):
	[B]ona pul[chra P]roserpina, Plutoni[s
	u]xsor, seiue [me SALVIAM] deicere oportet,
	eripias salu[tem], corp[us], colorem, uires,
	uirtutes Au[on]ia[es] […]
	[…] [Qu]are hanc uictimam
	[tibi] trado, [Proserpin]a, seiue me
	[PROS]ERPINA, se[iue me ACH]ERVOSIAM dicere
	[opo]rtet…

a.

See Reix (1981:

279): "le métalangage naturel est donc notre discours familier quotidien sur le langage"; "dans la communication, nous sommes obligés de commenter nos dires, notamment dans le cas de désignation et de signification".27 Concerning dico and its role in parenthetical clauses, seeCharnavel (2007: 100-108; 172).

So contrast me dico / Capuam dico with uera dico: we can identify the non-autonymous uses of dico by the semantic proximity between the accusative (uerbum, uera…) and dicere. Nevertheless, an animate accusative can be governed by dicere, but the meaning is "honour somebody".

Concerning the notion of opacification defined as the interposition of metalinguistic units, seeCharnavel (2007: 286) quoting Authier-Revuz.

Other examples: Cic. Att. XVI 15, 5 (with the dative); Liv. V 3, 9 (with the adnominal genitive); Liv. XL 27, 11 (with a dative governed by occurret); Sen. rhet. Contr. I, praef., 10 (with a dative governed by placeat); Sen. rhet. Contr. III, praef., 12 (prepositional phrase); Quint. Inst. IV 2, 79 (with the ablative of the comparative).


Raffaele Russo
email: raffaele.russo@ehu.eus

Samuel Forest

Franck Andrés

Girot Mata

Thermomechanics of Cosserat medium: Modeling Adiabatic Shear Bands in Metals

Keywords: Cosserat medium, Thermodynamics, Elastoviscoplasticity, Characteristic length, Adiabatic Shear Band

During most metal manufacturing processes, the medium deforms by generating large quantities of plastic strain at relatively high strain rates, inevitably inducing rises in temperature. Metals characterized by low thermal conductivity properties might locally retain high temperatures, consequently undergoing thermal softening. The classical balance laws governing the continuum equilibrium show severe mesh sensitivity if they were numerically discretized through Finite Element Methods. Furthermore, the plastic deformation tends to localize in narrow areas whose characteristic length is comparable to grain size, thereby requiring the adoption of theories able to predict sizeeffects. In this manuscript we demonstrate that the Cosserat medium is able to overcome these issues related to manufacturing processes simulation. We first provide a thermodynamically-consistent description of the Cosserat medium, and then we propose a method to calibrate the two additional characteristic lengths introduced by the Cosserat medium description by enriching the model with the TANH stress flow rule under adiabatic conditions.

Introduction

Manufacturing processes of metallic materials can be found in almost every engineering field, e.g. aerospace, naval and civil. Performing the numerical simulations of the processes, however, still represents a big challenge in many aspects. The material, in fact, during the process, experiences prohibitive deformation fields, consequently reaching very high temperatures, close to the material melting point. From the physical standpoint, a properly-adapted model is necessary to obtain results from the simulation that would describe the real medium development.

During machining, for example, the metal is shaped into its final form through removal of small quantities of material, which is shredded from the final piece through shear deformation mainly. The material locally experiences high plastic deformation, which induce, if the adiabatic condition is met (the thermal diffusion is characterized by a lower velocity than heat production rate), to locally retain high temperatures, thus leading to material thermalsoftening. Upon material softening, a severe mesh dependence is experienced whether these equations were to be solved through Finite Element Modeling (FEM) [START_REF] De Borst | Gradient-dependent plasticity: Formulation and algorithmic aspects[END_REF][START_REF] Chambon | One-dimensional localisation studied with a second grade model[END_REF][START_REF] Wang | Viscoplasticity for instabilities due to strain softening and strain-rate softening[END_REF]. On top of that, an accurate, thermodynamically-consistent material description must be employed to correctly capture the thermal softening induced by high plastic deformation [START_REF] Aldakheel | Coupled thermomechanical response of gradient plasticity[END_REF][START_REF] Lele | A large-deformation strain-gradient theory for isotropic viscoplastic materials[END_REF][START_REF] Cheng | A multi-mechanism model for cutting simulations based on the concept of generalized stresses[END_REF][START_REF] Wcis Lo | Local and non-local thermomechanical modeling of elastic-plastic materials undergoing large strains[END_REF][START_REF] Pamin | Gradient-enhanced large strain thermoplasticity with automatic linearization and localization simulations[END_REF]. Furthermore, the fact that during these processes the deformation is prone to localize in narrow areas makes it necessary to adopt numerical models which are able to predict size-effects [START_REF] Fleck | Strain Gradient Plasticity[END_REF][START_REF] González | Numerical analysis of the indentation size effect using a strain gradient crystal plasticity model[END_REF][START_REF] Yuan | Identification of the intrinsic material length in gradient plasticity theory from micro-indentation tests[END_REF][START_REF] Joshi | An Explanation for the Size-Effect in Machining Using Strain Gradient Plasticity[END_REF][START_REF] Wu | Modeling of flow stress in orthogonal micro-cutting process based on strain gradient plasticity theory[END_REF][START_REF] Marchand | An experimental study of the formation process of adiabatic shear bands in a structural steel[END_REF][START_REF] Nguyen | A nonlocal coupled damage-plasticity model for the analysis of ductile failure[END_REF][START_REF] Poole | Micro-hardness of annealed and work-hardened copper polycrystals[END_REF][START_REF] Stölken | A microbend test method for measuring the plasticity length scale[END_REF].

Since the processes of interest are highly characterized by shear deformation, the medium undergoes large material rotations at the locations where the deformation localizes. This line of reasoning, coupled with the fact that the deformations are localized in areas whose size is comparable with the grain size, led us think that the rotations experienced by the micro-structure, whose effects are normally neglected during homogeneous deformation, would assume a major role in characterizing the material response during these processes. The medium description which fits with this description is the Cosserat theory [START_REF] Cosserat | Theorie des corps deformables[END_REF].

The Cosserat theory finds many applications in geomechanics, because the mechanical behavior of the soil, being a granular material at macroscopic level, is characterized by a size effects already at macroscopic level [START_REF] Mühlhaus | The thickness of shear bands in granular[END_REF][START_REF] Tordesillas | Shear band evolution and accumulated microstructural development in Cosserat media[END_REF][START_REF] Rattez | Numerical Analysis of Strain Localization in Rocks with Thermo-hydro-mechanical Couplings Using Cosserat Continuum[END_REF]. However, besides geomechanics, the Cosserat model has also been extensively used both for homogeneous metal materials [START_REF] Grammenoudis | Hardening rules for finite deformation micropolar plasticity: Restrictions imposed by the second law of thermodynamics and the postulate of Il'iushin[END_REF][START_REF] Bauer | Micropolar hyper-elastoplasticity: constitutive model, consistent linearization, and simulation of 3D scale effects[END_REF] and polycrystal materials [START_REF] Forest | Cosserat modelling of size effects in the mechanical behaviour of polycrystals and multi-phase materials[END_REF][START_REF] Neff | A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations[END_REF][START_REF] Ask | A Cosserat-phase-field theory of crystal plasticity and grain boundary migration at finite deformation[END_REF]27], and many authors focused on different aspects of the Cosserat medium description. Steinmann and Willam investigated on the localization properties of the Cosserat model in elasto-plastic materials under infinitesimal deformations in case the loss of ellipticity was caused by negative material tangent operator [START_REF] Steinmann | Localization within the Framework of Micropolar Elasto-Plasticity[END_REF]. De Borst and Sluys demonstrated that the presence of an internal length in a von Mises plastic model formulated for the Cosserat medium can be used to avoid mesh dependency when localization phenomena were simulated [START_REF] De Borst | Simulation of strain localization: a reappraisal of the cosserat continuum[END_REF]. Most recently, Sabet and de Borst demonstrated that the Cosserat medium description retains ellipticity also in case the material model is characterized by a non-associated rate-independent plastic flow [START_REF] Sabet | Structural softening, mesh dependence, and regularisation in non-associated plastic flow[END_REF]. Kratochvíl et al. investigated over the characteristic lengths introduced by the Cosserat medium description from a physical point of view, by relating the Cosserat wryness to the Nye's dislocation tensor, thus interpreting the wryness tensor as a measure of geometrically necessary dislocations [START_REF] Kratochvíl | On physically motivated mesoscale Cosserat model of shear band formation[END_REF]. Khoei et al. drew a distinction between torsional and bending characteristic lengths, and they investigated the effect of these lengths on the shear bandwidth during different localization processes [START_REF] Khoei | 3D finite element modeling of shear band localization via the micro-polar Cosserat continuum theory[END_REF]. Many lines of work can also be found in literature proposing a thermodynamically-consistent model of the Cosserat kinematics [START_REF] Grammenoudis | Hardening rules for finite deformation micropolar plasticity: Restrictions imposed by the second law of thermodynamics and the postulate of Il'iushin[END_REF][START_REF] Bauer | Micropolar hyper-elastoplasticity: constitutive model, consistent linearization, and simulation of 3D scale effects[END_REF][START_REF] Forest | Elastoviscoplastic constitutive frameworks for generalized continua[END_REF][START_REF] Forest | Nonlinear microstrain theories[END_REF][START_REF] Rattez | Numerical Analysis of Strain Localization in Rocks with Thermo-hydro-mechanical Couplings Using Cosserat Continuum[END_REF].

The problems related with metal manufacturing simulations are already well known issues, and previous researches adopted a strain gradient theory approach to overcome them [START_REF] Wang | Adiabatic Shear Localization for Steels Based on Johnson-Cook Model and Second-and Fourth-Order Gradient Plasticity Models[END_REF][START_REF] Huang | Gradient Plasticity : Implications to Chip Formation in Machining[END_REF][START_REF] Joshi | An Explanation for the Size-Effect in Machining Using Strain Gradient Plasticity[END_REF][START_REF] Royer | Finite strain gradient plasticity theory for high speed machining[END_REF], however, to the best of of the author's knowledge, no research has been conducted so far on the application of the Cosserat medium for metal manufacturing simulations. Therefore, in this manuscript, the authors aim to cover this gap by providing a thermodynamically-consistent Cosserat medium description, which would be able to predict size effects even for relatively high temperatures, while retaining a compatible physical description and without loss of ellipticity. Furthermore, via the provided framework, it would be possible to unambiguously and uniquely correlate plastic deformations with temperature variations, allowing the calibration procedure to be performed through comparison of thermal fields.

In order to reproduce the same phenomena occurring in manufacturing processes, the Hat-Shaped Specimen [START_REF] Peirs | The use of hat-shaped specimens to study the high strain rate shear behaviour of Ti-6Al-4V[END_REF] was chosen as benchmark test. The combination of specific load case and specimen geometry drives the shear deformation to localize in a narrow band that takes the name of Adiabatic Shear Band (ASB). The same ASB develops during machining operations [START_REF] Ye | Critical cutting speed for onset of serrated chip flow in high speed machining[END_REF][START_REF] Calamaz | A new material model for 2D numerical simulation of serrated chip formation when machining titanium alloy Ti-6Al-4V[END_REF][START_REF] List | Strain, strain rate and velocity fields determination at very high cutting speed[END_REF][START_REF] Molinari | Adiabatic shear banding and scaling laws in chip formation with application to cutting of Ti-6Al-4V[END_REF][START_REF] Wang | Adiabatic Shear Localization for Steels Based on Johnson-Cook Model and Second-and Fourth-Order Gradient Plasticity Models[END_REF]. Regarding the material model to be plugged in the Cosserat framework, the Johnson-Cook (JC) model is widely used in machining simulations [START_REF] Peirs | The use of hat-shaped specimens to study the high strain rate shear behaviour of Ti-6Al-4V[END_REF][START_REF] Li | Numerical simulations on adiabatic shear behaviour of 921A steel pure shear hat-shaped specimens[END_REF][START_REF] Chen | Finite element simulation of highspeed machining of titanium alloy (Ti-6Al-4V) based on ductile failure model[END_REF][START_REF] Cahuc | Mechanical and Thermal Experiments in Cutting Process for New Behaviour Law[END_REF][START_REF] Guo | Dynamic material behavior modeling using internal state variable plasticity and its application in hard machining simulations[END_REF], however, it underestimates the softening effect on the flow stress [START_REF] Guo | Dynamic material behavior modeling using internal state variable plasticity and its application in hard machining simulations[END_REF]. Many models have been derived from the JC to overcome this issue, and we decided to use the TANH model among the others [START_REF] Calamaz | A new material model for 2D numerical simulation of serrated chip formation when machining titanium alloy Ti-6Al-4V[END_REF].

The next section provides the description of the kinematics, energetic and thermodynamic aspects of the Cosserat model. The choices of the specific form of the Helmholtz free energy and plastic models are given in Sections 2.2 and 3.2 respectively. The description of the selected benchmark test geometry and load conditions are given in Section 3. Finally, in Sections 3.3 and 3.4 the methodology used to calibrate the additional characteristic lengths introduced by the Cosserat medium are proposed.

Notation and Recalls of Tensorial Algebra

Einstein's convention is employed in the whole manuscript. Tensors are indicated through compact notation or index notation, regardless of capital or not-capital letters:

a =⇒ a i ; a : =⇒ a ij ; a : =⇒ a ijk ; a : : =⇒ a ijkl ;
The outer product is indicated with ⊗ and it operates in the following manner:

A : ⊗ B : = C : : =⇒ A ij B kl = C ijkl ;
Single and double contraction operators are respectively indicated as follows:

A : • B : = C : : =⇒ A ijk B klm = C ijlm ; A : : B : = C : =⇒ A ijk B jkl = C il ;
Divergence and gradient operators are represented as:

A : ⊗ ∇ = B : =⇒ ∂A ij ∂x k = B ijk ; A : • ∇ = B =⇒ ∂A ij ∂x j = B i ;
where x i is the i-th Cartesian coordinate of an Eulerian space. When handling with skew-symmetric second order tensors, it might be easier to define the first order tensor associated to it as:

A = - 1
2 : : B : ;

(1

)
where : is the Levi-Civita permutation symbol.

Cosserat Model

Kinematics and Balance Laws

The Cosserat model [START_REF] Cosserat | Theorie des corps deformables[END_REF] enhances the Classical Continuum Mechanics (CM) model by introducing three additional degrees of freedom, being the microstucture infinitesimal rotation around the three axis. The degrees of freedom of the continuum can then be written as:

{u i , θ i }, i = 1, 2, 3 (2)
where u i is the spacial displacement along the i-th direction, and θ i is the micro-structural rotation around the i-th axis. The superscript × indicates the fact that the quantity is a pseudo-vector. As function of these degrees of freedom, the deformation measures in the Cosserat medium are defined as:

e : = u ⊗ ∇ -Θ : ;

(3)

k : = θ ⊗ ∇; (4)
where the nabla operator ∇ is referring to the initial frame of reference and Θ :

is the skew-symmetric second order tensor associated with the pseudo-vector θ:

Θ : = -: • θ; (5)
The first deformation measure, e : , is commonly referred to as Cosserat deformation, while the second, k : , is addressed as Cosserat wryness. It must be noted here that the former incorporates the difference between material infinitesimal rotation and micro-structure rotation in addition to the strain measure usually used in linearized (small deformation) CM as can be seen in the following:

e : = sym (u ⊗ ∇) + skew (u ⊗ ∇) -Θ : ; (6)
where sym and skew indicate the symmetric and the skew-symmetric parts of a tensor respectively. The balance laws governing the Cosserat medium equilibrium can be derived by explicitly expressing the internal and external power related to the deformation measures defined in Eq. ( 3) and (4):

P (i) = Ω σ : : ė : + µ : : k : dΩ; (7)
where σ : is the power conjugate to the Cosserat strain rate and µ : is the power conjugate to the Cosserat wryness rate; in literature they are commonly referred to as stress and couple stress tensors. It is important to highlight here that due to the not-symmetric nature of the Cosserat strain and Cosserat wryness, both the stress and couple stress tensors are not symmetric as well. Equivalently, the external power can be defined as:

P (e) = Ω f • u + c • θ dΩ + ∂Ω t • u + m • θ dS; ( 8
)
where f is the body force per unit volume conjugate with displacement, c is the body couple per unit volume conjugate with micro-structure rotation, t is the surface traction conjugate with displacement, and finally m is the surface couple per unit surface conjugate with micro-structure rotation. The internal power can be equivalently written as:

Ω σ : : ė : + µ : : k : dΩ = Ω 2σ • θ -u • σ : • ∇ -θ • µ : • ∇ dΩ+ ∂Ω u • σ : • n + θ • µ : • n dS; (9)
where the vector σ is the axial vector associated with the skew-symmetric part of the stress tensor σ : as in Eq. 1, and n is the unit vector normal to the outer surface of the domain Ω. The equality between internal and external power can be expressed in its weak form as:

Ω u • σ : • ∇ + f dΩ + Ω θ • 2σ + c + µ : • ∇ dΩ+ + ∂Ω u • t -σ : • n dS + ∂Ω θ • m -µ :
• n dS = 0; (10) from which, given the arbitrariness of u and θ, the strong forms of the linear and angular momentum balances can be derived as:

σ : • ∇ + f = 0; (11)
µ

: • ∇ + 2σ + c = 0; (12)
which are vectorial partial differential equations in the variables u and θ, bounded by:

σ : • n = t; (13)
µ : • n = m; (14)

Clausius-Duhem inequality

The material behavior can be explicitly stated in terms of stress-strain relationship or by expressing the Helmholtz free energy and resorting to the Clausius-Duhem inequality. Given that our objective is to characterize the energetic behavior of the Cosserat medium, in this work we employ the second method. The following decompositions of strain and wryness are adopted:

e : = e : e + e : p + e :

th ;

k : = k : e + k : p + k :

th ; [START_REF] Dong | In-situ measurement of Ti-6Al-4V grain size distribution using laserultrasonic technique[END_REF] in which the relation between elastic, plastic and thermal deformation has an additive character, due to the linearization performed under small displacements and small rotations assumptions. It is then necessary to define the state variables of the medium, and in this work they are the following:

{e : e , k :

e , T, α};

namely, elastic strain, elastic wryness, temperature and a hardening variable, which can be a tensor of any rank. Based on the choice of the state variables, the Helmholtz free energy density per unit volume can be written as:

ψ = ψ e : e , k :
e , T, α ;

Here a more explicit definition of the Helmholtz free energy, linearized with respect to the initial conditions {e : e 0 = 0, k : e 0 = 0, T 0 , α 0 = 0}, is provided in a still quite generic form. It assumes quadratic potentials for the elastic deformation measures, typical energy definition related to thermal deformations, standard energy definition related to heat exchange and an additional term concerning the stored energy relative to plastic deformations:

ψ e :
e , k :

e , T, α = 1 2

e : e : Λ : :

: e : e + 1 2
k : e : C : :

: k : e -(T -T 0 ) P : : e : e -(T -T 0 ) P : k : k : e - 1 2 ρ C ε T 0 (T -T 0 ) 2 + ψ α ; ( 19
)
where Λ : : and C : :

are the fourth-order elastic stiffness tensors, relating stressstrain and couple stress-wryness respectively, P : and P : k are second order ten- sors relating thermal variations with elastic expansions or wrynesses respectively, C ε is the specific heat capacity, ρ is the density and ψ α is the part of the Helmholtz free energy that accounts for variation in the recoverable energy of the continuum due to variation in the material yield limit, e.g. hardening or softening. Subsequently, we employ the first and second thermodynamic principles to write the Clausius-Duhem inequality and to explicitly express the dissipation of the continuum. From the first thermodynamic principle:

u = σ : : ė : + µ : : k : -q • ∇ + r; ( 20
)
where u is the internal energy rate density per unit volume, not to be mistaken with the displacement rate, q is the heat flux and r is the heat production.

The second thermodynamic principle reads:

η - r T + q T • ∇ ≥ 0; ( 21
)
where η is the entropy rate density per unit volume. Considering that the Helmholtz free energy is defined as:

ψ e : e , k : e , T, α = u -T η; (22)
and by expressing its differential as:

ψ = ∂ψ ∂e : e ė : e + ∂ψ ∂k : e k : e + ∂ψ ∂T Ṫ + ∂ψ ∂α α = u -T η -Ṫ η; (23)
the second thermodynamic principle assumes the following form:

σ : -

: th ∂T Ṫ - ∂ψ ∂α α - q • (∇T ) T ≥ 0; (24)
from which, assuming that the elastic strain, elastic wryness and thermal variation are fully recoverable, the followings can be derived:

σ : = ∂ψ ∂e : e ; ( 25
) µ : = ∂ψ ∂k : e ; (26)
η = - ∂ψ ∂T + σ : : ∂e : th ∂T + µ : : ∂k : th ∂T ; (27)
and, by expanding Eq. ( 25) and ( 26), the following relation between thermal deformations, elastic deformations and stress/couple stress can be found: : :

-1 : P : k (T -T 0 ) = C : : : k : e -k : th ; (29)
where:

e : th = Λ : :

-1 : P

: (T -T 0 ) = Γ : (T -T 0 ) ; (30)
k : th = C : :

-1 : P : k (T -T 0 ) = Γ : k (T -T 0 ) ; (31)
and the tensors Γ : and Γ : k are operators used to compute the thermal expan- sion/wryness. The remaining terms in Eq. ( 24) define the medium dissipation, here referred to as Π:

Π = σ : : ė : p + µ : : k : p - ∂ψ ∂α α - q • (∇T ) T ≥ 0; (32)
Furthermore, we can write the heat equation from Eq. ( 20) and ( 23) as:

σ : : ė : p + µ : : k : p -q • ∇ + r = ∂ψ ∂α α + T η; (33)

Specific Choices of the Material Model

Elastic part

From Eq. ( 25) and ( 26) it can be observed that the form of the constitutive equations of stress and couple-stress directly derives from the chosen form of the Helmholtz free energy, it is then essential to give a proper shape to this potential. The Helmholtz free energy form adopted in Eq. ( 19) is still quite general, and many mechanical behaviors can be modeled using this form. The most simple model used for isotropic materials can be expressed as [START_REF] De Borst | Localisation in a Cosserat continuum under static and dynamic loading conditions[END_REF]:

σ : = λ
µ : = ξ tr k : e I : + 2 β sym k : e + 2 γ skw k : e ; ( 35
)
where I : is the Identity second-order tensor, and the following conditions on the material parameters must be valid:

     3λ + 2G ≥ 0 G ≥ 0 G c ≥ 0 &      3ξ + 2β ≥ 0 β ≥ 0 γ ≥ 0 (36)
In this constitutive model, λ and G are the usual Lamé parameters and G c , ξ, β and γ are additional material coefficients. In the 2D case, it is customary to assume β = γ for simplicity [START_REF] De Borst | Simulation of strain localization: a reappraisal of the cosserat continuum[END_REF]. For our computations, we are prone to use relatively high values of G c when compared to G, such that the mismatch would be characterized by a stiffer response, and the value of skw{u ⊗ ∇} would be negligible if compared to sym{u ⊗ ∇}. In this case, the Cosserat theory would recover the characteristics of a couple-stress theory, also known as constrained Cosserat theory [START_REF] Koiter | Couple Stresses in the Theory of Elasticity, I & II[END_REF]. There is, however, a conceptual difference between the hydrostatic term of the wryness and its deviatoric counterpart: while, in fact, the former is related to variation of the micro-structure rotation along the axis in which it is defined (torsion), the latter expresses the variation of the micro-structure rotation along the other two directions (bending), and in general these two deformations might be characterized by different responses. After a simple dimensional analysis, the following elastic characteristic length of the Cosserat medium can be defined:

l el = β G (37)
To be noted that in case no simplifications were done on the constitutive behavior of the Cosserat model, other characteristic lengths were identifiable, as done in other researches [START_REF] Khoei | 3D finite element modeling of shear band localization via the micro-polar Cosserat continuum theory[END_REF].

Viscoplastic part

In the same way we defined the material response by defining the Helmholtz free energy instead of the stress-strain relation, it is possible to explicitly state the shape of the viscoplastic potential associated to the plastic deformation. A general shape of the multi-criterion Cosserat viscoplastic potential is retrieved [START_REF] Forest | Elastoviscoplastic constitutive frameworks for generalized continua[END_REF] where two potentials are defined, one related to e :

p and one related to k :

p . They can be modeled using the general viscoplastic potential form as the one proposed by Chaboche in case of standard continuum mechanics [START_REF] Chaboche | Constitutive equations for cyclic plasticity and cyclic viscoplasticity[END_REF] :

Ω = D k + 1   f σ : , p, T D   k+1 + L ξ + 1   g µ : , p k , T L   ξ+1 ; ( 38
)
where Ω is the viscoplastic potential, D and L are the drag stress and couple stress, k and ξ are parameters to be calibrated and f and g are over-stress and over-couple-stress between the Macaulay brackets. If we assume associated plasticity and explicitly give the form of Ω, the plastic behavior would be completely defined. This is because the plastic strain rate and plastic wryness rate can be written as:

ė : p = ∂Ω ∂f ∂f ∂σ : , k : p = ∂Ω ∂g ∂g ∂µ : ; ( 39
)
from which, the first ratios are equal to the rates of the viscoplastic multipliers:

ṗ = ∂Ω ∂f , ṗk = ∂Ω ∂g (40)
and the second ones define the direction of the plastic flows, being equal to the one of the associated stress/couple stress:

∂f ∂σ : = n : σ , ∂g ∂µ : = n : µ ; (41)
The choice of expressing the viscoplastic potential as in Eq. ( 38) would impose the calibration of additional parameters related with both the contributions coming from plastic deformation and plastic wryness. However, the expression of the viscoplastic potential usually used with the Cosserat medium is simplified by assuming a single criterion, thus a single viscoplastic multiplier, which is common to both plastic deformation and plastic wryness:

Ω = D k + 1   f σ : , µ : , p, T D   k+1 ; (42)
and:

ṗ = ∂Ω ∂f , ė : p = ṗ ∂f ∂σ : , k : p = ṗ ∂f ∂µ : ; ( 43
)
with f depending both on stresses and couple stresses:

f = f (σ : , µ : , p, T ) = σ eq (σ : , µ : ) -R(p, T ); (44)
where R represents the radius of the yield surface above which plasticity is activated. In the Cosserat framework, several equivalent stress measures can be adopted :

σ eq = 3 2
a 1 σ : : σ : + a 2 σ : : σ :

T + bµ

: : µ : ; (45)
σ eq = 3 2
a 1 σ : : σ : + a 2 σ : : σ :

T + b 1 µ : : µ : + b 2 µ : : µ : T ; ( 46
)
σ eq = 3 2 a 3 σ
: sym : σ : sym + a 4 σ : skw : σ :

T skw + b 3 µ
: sym : µ : sym + b 4 µ : skw : µ :

T skw ; ( 47
)
where the first one was proposed by de Borst [START_REF] De Borst | Simulation of strain localization: a reappraisal of the cosserat continuum[END_REF][START_REF] De Borst | A generalisation of theory for polar continua[END_REF], the second one by Forest [START_REF] Forest | Elastoviscoplastic constitutive frameworks for generalized continua[END_REF], and the third one is based on the invariant decomposition of a second order tensor [START_REF] Zheng | Theory of Representations for Tensor Functions-A Unified Invariant Approach to Constitutive Equations[END_REF]. The operator σ : returns the deviatoric part of the tensor σ : . Based on the dimensional analysis of the coefficients appearing in the Equations above, and assuming that the terms a i are equal to each other, as well as the b i terms, the plastic characteristic length arises from the model as:

l p = a b ; (48)
The last remark regarding the completeness of the plastic model concerns the definition of the plastic part of the Helmholtz free energy, that is, ψ α in Eq. [START_REF] Forest | Elastoviscoplastic constitutive frameworks for generalized continua[END_REF]. This is, in general, equal to [START_REF] Śloderbach | Thermodynamic potential of free energy for thermo-elasticplastic body[END_REF]:

ψ α = α 0 π(ξ)dξ; ( 49
)
where π is a general stress thermodynamically associated to the internal variable α. In most plastic evolution laws the equivalent cumulative plastic strain is used as hardening parameters, such that ṗ = α, and π is essentially the stress thermodynamically conjugated to the equivalent cumulative plastic strain.

Given the integral formulation of this term, its closed form does not necessarily exist, and it could be eventually found once the plastic flow rules are defined.

Adiabatic Temperature Evolution

The time variation of the state variables is then to be defined. Given total strain and wryness increment, the elastic stain/wryness, viscoplastic multiplier and thermal rates must be evaluated. The derivation of the adiabatic temperature rate can be found in the appendix, and it is here reported:

Ṫ = σ : : ė : p + µ : : k : p - ∂ψ α ∂α α + T ∂ 2 ψ α ∂T ∂α α ρT C ε T 0 + ∂C ε ∂T (T -T 0 ) T 0 - ∂ 2 ψ α ∂T 2 T ; ( 50
)
which in case of further simplification becomes:

Ṫ = σ : : ė : p + µ : : k : p - ∂ψ α ∂p ṗ ρT C ε T 0 ; ( 51
)
Note that it is possible, with this formulation, to evaluate the part of plastic work converted into adiabatic temperature increment without any additional calibration to be performed. Many researches can be found in literature where the effects of hardening/softening on the temperature increment is incorporated in a unique parameter, namely the Taylor-Quinney coefficient Ξ [START_REF] Taylor | The latent energy remaining in a metal after cold working[END_REF], that requires calibration in different condition:

Ṫ = Ξ ρC ε σ : : ė : p + µ : : k : p ; (52)
By using this temperature evolution law and the evolution of the plastic deformations defined in Eq. ( 43), the elastic predictor can then be written as:

ė : e = ė : -ė : th -ė : p = ė : -Ṫ Γ : - ṗ ∂f ∂σ : ; (53)
k

: e = k : -k : th -k : p = k : -Ṫ Γ : k - ṗ ∂f ∂µ : ; ( 54
)

Model Implementation and Characteristic Lengths Calibration

The model described in the previous section is here proposed to be a suitable solution for manufacturing processes simulations. The main reason is the embodied property of the model to enlarge the domain of the vectorial solution space {u} for which the ellipticity of the PDEs is retained. In most of the plastic models employed to simulate manufacturing processes, the strain hardening effect overcompensates the thermal softening, thus the PDEs rarely experience a loss of ellipticity. However, when high temperatures are expected to develop, material softening could most likely exceed the strain hardening, but not many material models are able to properly predict the flow stress at high temperatures (≈ 0.8 T m -melting temperature), therefore thermal softening might not be correctly captured. The Johnson-Cook model, widely used in manufacturing processes, is known to underestimate the softening effect [START_REF] Guo | Dynamic material behavior modeling using internal state variable plasticity and its application in hard machining simulations[END_REF], and its implementation in a strain-gradient theory was already documented [START_REF] Wang | Adiabatic Shear Localization for Steels Based on Johnson-Cook Model and Second-and Fourth-Order Gradient Plasticity Models[END_REF]. Due to the limitation of the JC model in capturing material softening, many modified versions model can be found in literature. Among the others, we decided to use the TANH model [START_REF] Calamaz | A new material model for 2D numerical simulation of serrated chip formation when machining titanium alloy Ti-6Al-4V[END_REF]. Both the Classical CM model and the Cosserat model have been fetched with this material description. Therefore, acknowledging the lacking of a proper material plastic model able to predict flow stress models at high operational temperature, in this section we limit ourself to demonstrate the procedure by which the two additional Cosserat characteristic lengths might be calibrated. Given our final object of employing the Cosserat medium to simulate manufacturing processes, the calibration procedure will be performed by modeling a Hat-Shaped Specimen under compression [START_REF] Peirs | The use of hat-shaped specimens to study the high strain rate shear behaviour of Ti-6Al-4V[END_REF]. The Hat-Shaped test is characterized by the development of an ASB inside the specimen, followed by an increase in temperature due to the high levels of localized plastic strain. Since most of the manufacturing processes that export material from the final piece are designed to do so through the development of ASBs and subsequent material removal, this example could represent a simplified condition of the one occurring during manufacturing. In fact, the calibration procedure has been performed precisely while looking at the fields developed at the ASB location.

The geometry of the specimen was taken from the work of Peirs et al. [START_REF] Peirs | The use of hat-shaped specimens to study the high strain rate shear behaviour of Ti-6Al-4V[END_REF]. Although the same geometry and material properties were used, the plastic behavior chosen for this test is different than the one adopted by Peirs et al. The specification of the plastic model and the material properties will be assessed in Section 3.2, and aim to reproduce the behavior of the Titanium alloy Ti6Al4V. Given the axial-symmetry of the specimen, only half of the cross section has been modeled in 2D under axial-symmetric conditions.

In Figure 1 the geometry and the boundary conditions of the benchmark tests are reported. A vertical downward velocity of 5.5 m/s was imposed at the top surface of the specimen, and the simulations lasted until a vertical negative displacement of -0.275 mm was reached. The simulations have been produced using the FEM solver Z-set [START_REF] Mata | non-linear material and structure analysis suite[END_REF], in which the Cosserat element description and material behavior are implemented. Quadratic interpolation has been used to discretize displacement and rotation fields between nodal values, thereby quadratic elements have been used.

Mesh-independence Simulations & Mesh Size Choice

One of the reasons for us to employ a Cosserat medium description is to avoid the mesh dependency developed when the material softens at high temperatures. Therefore, a preliminary verification was performed in order to be sure that the Cosserat medium description would be able to produce meshinsensitive results. Simulations of the Hat-Shaped specimen under compression were carried out using the Classical CM and the Cosserat descriptions both fetched with a plastic behavior that reproduces pure softening. For these simulations, the mechanical and thermal properties reported in Table 1 were used. In addition to the ones indicated in the table, the couple stress moduli β and γ were both equal to 0.114 MPa•mm 2 . Although this model might not be descriptive of the behavior of any material, this numerical exercise allow us to verify the mesh independence in the most sever condition of pure softening. The flow stress adopted for this exercise reads:

R(p) = R 0 + H p; ( 55
)
where H is the softening modulus, which was set equal to -30 MPa. As it can be seen from Eq. ( 55), the equivalent cumulated plastic strain in this case is causing the same effect that is usually induced by the temperature on the flow stress. According to Eq. ( 55), the material softening is induced whenever plastic strain cumulates, while normally the softening is a thermally-activated phenomena. The same simulation was performed for two different mesh size, a finer mesh of 20 µm x 15 µm and a coarser mesh of double the size for both the Classical CM and the Cosserat medium, and the plastic distributions across the ASB were compared. In Figure 2 the comparison is appreciable.

The equivalent plastic strain distribution has been plotted across the ASB. While the Classical CM description exhibits a strong mesh-dependency, the Cosserat medium does not show such dependency. More discussion related to the differences in cumulative plastic distribution between Classical CM and Cosserat medium will be provided in the Section 3.3, but this results clearly shows that the Cosserat medium can retain mesh independence results under conditions for which the Classical CM cannot.

Mesh Size

In order to ensure the mesh size to be smaller than the elastic characteristic length of the model, a preliminary investigation was made to find the characteristic length for which the simulations would converge. In this analysis, the assumptions that both a i and b i are equal to 0 in Eq. ( 46) were made. Consequently, both the deformation and the wryness are fully elastic. Under this assumptions, the Hat-Specimen under compression was simulated using an initial mesh size of 20 µm x 15 µm, and using the material properties as Table 1. The results in terms of load-displacement of the Hat-Shaped Specimen are reported in Figure 3. The values of the elastic characteristic length range from 10 1 mm to 10 -4 mm. The curves achieve convergence already for values of the elastic characteristic length of 10 -2 mm, therefore suggesting the adoption of a mesh size, along the direction orthogonal to the ASB, smaller than 100 µm. Subsequently, using a mesh size of 40 µm x 30 µm can be considered a safe choice. Furthermore, for all the simulations, quadratic elements are used, so that the distance between the nodes spanning the element edges is ulteriorly halved.

Specifications of the Plastic Model

Although the model previously described defines the plastic behavior of the medium by providing the explicit function of the plastic potential Ω, most of the models used in literature are simply provided through their flow stress definition.

The flow stress used in this manuscript derives from the one developed by Calamaz et al. [START_REF] Calamaz | A new material model for 2D numerical simulation of serrated chip formation when machining titanium alloy Ti-6Al-4V[END_REF], but the dependence of the flow stress on the rate of the equivalent plastic strain has been omitted in order to simplify the model

N]

l el = 10 1 l el = 10 0 l el = 10 -1 l el = 10 -2 l el = 10 -3 l el = 10 -4 Fig. 3: Convergence analysis of the elastic characteristic length performed on the load-displacement graph of the Hat-Shaped Specimen using b i = a i = 0 as assumption in Eq. [START_REF] Śloderbach | Thermodynamic potential of free energy for thermo-elasticplastic body[END_REF]. Values of the characteristic lengths are given in millimeters.

and focus only on strain hardening and thermal softening behaviors. This flow stress model has been used several times in literature [START_REF] Hor | Modelling, identification and application of phenomenological constitutive laws over a large strain rate and temperature range[END_REF][START_REF] He | Simulation analysis of the influence of dynamic flow stress behavior on chip formation[END_REF], but it has never been coupled with the Cosserat medium description. The radius of the yield surface is defined as:

R = A + Bp n exp (p a ) 1 - T -T 0 T m -T 0 m D + (1 -D) tanh 1 (p + S) c ; (56)
where:

D = 1 - T T m d ; (57)
S = T T m b ; (58)
and the flow rule of the viscoplastic multiplier p follows:

ṗ = exp f R C -1 = exp 1 C σ eq R -1 -1; (59)
It must be remembered that since we are using a single viscoplastic multiplier, ṗ depends on both plastic strain and wryness contributions. The material properties are listed in Table 1, and they belong to the Titanium alloy Ti6Al4V, and the TANH flow stress model parameters have been summarized in Table 2. Table 2: Visco-plastic coefficients used in the TANH model [START_REF] Peirs | The use of hat-shaped specimens to study the high strain rate shear behaviour of Ti-6Al-4V[END_REF].

E [MPa] ν Gc [MPa] ρ [kg/m 3 ] Cε [J/kg•K] T 0 [K] Tm [K] Γ [1/K] Γ k [1/K•

Elastic characteristic length calibration

In our case, since we are modeling in 2D, there is no torsional component in the wryness (k e ii = 0, ii = 1, 2, 3), therefore the parameter α does not have any influence and it must not be calibrated. Throughout the calibration process the value of G c was set constant and imposed to be higher than the second Lame parameter (shear modulus), such that the Cosserat model would assume similar nature of a strain gradient theory. The value of G c was set to be equal to 114000.0 MPa. The calibration of the elastic characteristic length can be performed by simply varying the coefficient β in Eq. [START_REF] Mühlhaus | The thickness of shear bands in granular[END_REF]. The Classical CM framework can be retrieved by imposing the elastic characteristic length to vanish, and in this case the deformation localizes as if it was unaffected by the "diffusing" effect induced by not-vanishing k : . The calibration procedure shall start from large values of β and then its values should progressively reduce. Differently from the analysis done in the previous section, for the current calibration procedure only the condition b 1 = b 2 = 0 was used in Eq. [START_REF] Śloderbach | Thermodynamic potential of free energy for thermo-elasticplastic body[END_REF], that is, the plastic characteristic length was set equal to infinity. Consequently, the developed curvature is fully elastic, but the strain is also plastic, and ṗ depends on the stress only in Eq. [START_REF] Śloderbach | Thermodynamic potential of free energy for thermo-elasticplastic body[END_REF].

The calibration procedure could be performed by comparing data of different nature between experiments and simulations. Following the complete thermodynamic description outlined in Section 2 we could compare the temperature distribution in the ASB, especially because this can be easily measured through thermal camera or, in case the plastic deformation localizes inside the specimen, by positioning thermocouples at the location where the ASB is expected to develop. Since the magnitude of the characteristic length influences the area over which the developed deformation spreads (thickness of ASB), many thermocouples might be necessary to capture both the thermal distribution and its peak value. To be noted here that in adiabatic conditions the thermal variation is only driven by local development of plastic deformation, since there is no time for the temperature to diffuse, thus the measured temperature is a direct indication of the developed plastic deformation. Six values of the elastic characteristic length were used, from 10 1 mm to 10 -4 mm.

The effects of the elastic characteristic length have been investigated by looking both at the load-displacement curve and at the temperature distribution at the ASB.

As already observed in case of a hyper-elastic material model, the adoption of an elastic characteristic length, whose order of magnitude is comparable with the geometry size, extends the stiffening effect to a macroscopic level, inducing the whole system to behave stiffer. In case a plastic component is introduced in the Cosserat strain, the same behavior is observed, as inferred by Figure 4a. It can be observed from this Figure that the over-prediction of the initial elastic stiffness is negligible for values of the characteristic length smaller or equal than ≈ 10 -1 mm. Furthermore, it can be appreciated from the same Figure the trend of the load-displ curve of converging toward the results produced for a Classical CM simulation for values of the characteristic length that approach zero. The curve, however, should not overlap with the one obtained from the adoption of the TANH model in a Classical CM framework because, although the curvature is providing a null energetic contribution, the part of the Cosserat strain accounting for the mismatch between macro and micro rotation is still playing a role.

In Figure 5 the thermal fields across the ASB are reported. From the Figure, it is possible to identify the effect of the characteristic length on the thermal distribution. The elastic characteristic lengths indicate the range over which the thermal field, in this case, is spreading: smaller elastic characteristic length will induce the fields to spread over smaller domains. It can be noted that the temperatures reached in the Classical CM case are lower than the ones in the Cosserat model with a characteristic length of 10 -4 mm, however, it should be reminded that the results of this simulation strongly depend on the mesh size.

Plastic characteristic length calibration

The calibration of the plastic characteristic length has been done under the hypothesis that a 1 = a 2 and b 1 = b 2 in Eq. [START_REF] Śloderbach | Thermodynamic potential of free energy for thermo-elasticplastic body[END_REF]. The characteristic plastic length was chosen in the range between 10 0 mm and 10 -4 mm. From the previous calibration procedure it was evident that the elastic characteristic length for this problem had to be smaller or equal than 10 -2 mm for the deformation to localize. However, excessively low values of β induce the development of a relatively low values of couple stress if compared to the stress, therefore obliging the adoption of high values of the coefficient b in Eq. ( 46) if we wanted the couple stresses to have an effect on σ eq . Thus for the plastic characteristic length calibration, an elastic characteristic length of 10 -2 mm has been chosen. Once again, we look at the effects on the global load-displacement graph and to the temperature distribution inside the ASB, which are reported in Figures 6a and6b respectively.

In Figure 6a it can be observed that the load-displacement curves are bounded between the curves obtained for l pl = 10 0 mm and l pl = 10 -4 mm, which represent the asymptotic results for larger or smaller characteristic plastic lengths respectively. This behavior suggests that the plastic development is simultaneously driven by both strain and wryness in case the characteristic plastic length belongs to the range ≈ [10 -1 mm÷10 -3 mm], and that it is purely dominated by strain or wryness only in case the characteristic plastic length falls respectively before or after this domain.

If we look at the temperature distribution across the ASB in Figure 6b, it shows an increasing peak of predicted temperature at the center of the shear band, and a narrower localization of the thermal field with a decreasing plas- tic characteristic length. The maximum difference between the peaks of the temperatures belonging to the different plastic characteristic lengths measures 65 K. This difference lies well beyond the resolution of instruments, like thermocouples or thermal camera, thus the effect of different plastic characteristic lengths could be probed. Nonetheless, 65 K of difference is still a relative low value if compared with the total produced temperature, that is around 500 K.

This indicates a relative weaker influence of the characteristic plastic length on the developed thermal field if compared with the effects produced by different values of elastic characteristic lengths on the thermal fields.

Conclusion

Simulations of manufacturing processes can provide valuable information that help optimizing machined materials and machining benches. Furthermore, the knowledge related to the deformation history of the material is a key factor in addressing the post-processed status of the medium, for example in terms of residual stresses. In order to achieve more reliable simulations of these processes, we proposed the adoption of the Cosserat model as description of the medium behavior. A thermodynamically-consistent geometrically linearized formulation of the Cosserat theory was proposed in Section 2, where a thermoelasto-visco-plastic framework was provided for the material characterization through the definition of visco-plastic potential, Helmholtz free energy and dissipation potential. Using a purely-softening plastic behavior, the Cosserat medium was first proven to provide mesh-independent results, then the TANH flow-stress model was implemented, and it was used to describe an example of the calibration procedure of the elastic and plastic characteristic lengths using a hat-shaped specimen. The calibration of the elastic characteristic length was done by comparing the global load-displacement graph, and, for the specific plastic model used here, its values was found to be in the range of microns, that is comparable to grain size of the Titanium alloy used for the simulation [START_REF] Dong | In-situ measurement of Ti-6Al-4V grain size distribution using laserultrasonic technique[END_REF]. By fixing the value of the elastic characteristic length to be equal to 10 -2 mm, the calibration of the plastic characteristic length was performed once again by comparing the load-displacement graphs. The analysis suggested the existence of a range for the plastic characteristic length in which the contribution of plastic strain and plastic wryness were playing a comparable role. Outside these limits the plastic developments were fully dominated by the plastic components of one of the two deformation measures. However, finally the elastic characteristic length was found to have greater effects than the plastic ones.

Nonetheless, since the effects of the plastic characteristic length are highlighted at large plastic deformation, it is advisable to use a large deformation theory for the Cosserat medium. This remains the topic for the future research. For future research it is also foreseen the adoption of a physically-based material plastic law, ideally dislocation-based, and the validation of the Cosserat model for more complex manufacturing operations, such as machining or orthogonal cutting.

A Derivation of Ṫ

The derivation of the temperature rate can be derived starting from the definition of the entropy from Eq. ( 27) and assuming the Helmholtz free energy function as in Eq. ( 21): The one reported in Eq. ( 67) is an expression of the temperature rate which contains many terms whose magnitude is well below the magnitude of the larger terms as σ : : e : p for example, therefore assumptions were made on some of the quantities populating Eq. (67) in order to make it usable, and here follows the list of hypotheses:

η = - 1

Parameter

Order of Magnitude Table 3: Order of magnitudes of the terms in Eq. ( 67).

-q = r = 0;

Fig. 1 :

 1 Fig.1: Hat-Shaped Specimen[START_REF] Peirs | The use of hat-shaped specimens to study the high strain rate shear behaviour of Ti-6Al-4V[END_REF], geometry (a) and mesh with boundary conditions (b). Units in millimeters. In (b) the cross section along the shear band is indicated with red nodes.

Fig. 2 :

 2 Fig. 2: Cumulative plastic distribution for different mesh sizes using Classical CM (a) and Cosserat medium (b).

Fig. 4 :

 4 Fig. 4: Effects of the characteristic elastic length on the load-displ graph (a) and on the temperature distribution (b). Values of the characteristic lengths are given in millimeters.

Fig. 5 :lFig. 6 :

 56 Fig. 5: Temperature distribution in a Hat-Shaped specimen using the Cosserat framework with different values of β compared to the Classical CM result.

  them it into the heat Equation Eq. (33), we obtain:σ : : ė : p + µ : : k : pq • ∇ + r = ∂ψα ∂α

10 - 1 [J / m 3 •K 2 ] ∂ 2

 101322 ψα ∂T ∂p 10 -1 [J / m 3 •K]

:: | = 10 8

 8 = ΓI : = 10 -5 K -1 ; Γ : k = 0 : K -1 m -1 ; Pa;

Table 1 :

 1 Elastic and thermal material properties corresponding to the Titanium allot Ti6Al4V[START_REF] Peirs | The use of hat-shaped specimens to study the high strain rate shear behaviour of Ti-6Al-4V[END_REF].

	mm]

Acknowledgements This project has received funding from the European Union's Marie Sk lodowska-Curie Action (MSCA) Innovative Training Network (ITN) H2020-MSCA-ITN-2017 under the grant agreement Nº 764979.

Furthermore, by assuming the material elastic and plastic models to be the ones described in Tables 1 and2, the terms in Eq. ( 67) have the orders of magnitudes listed in Table 3.

From the comparison of the order of magnitudes of the terms in Eq. ( 67) that are presented in Table 3, the temperature flow rule can assume the following form: