Stéphanie Caillies

Pamela Gobin

Alexandre Obert

Sarah Terrien

Alexandre Coutté

Galina Iakimova

Chrystel Besche-Richard

Asymmetry of affect in verbal irony understanding: What about the N400 and P600 components?

Keywords: ironic criticism, ironic praise, emotional connotation, ERPs

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

Introduction

Verbal irony is a form of nonliteral language in which speakers can convey implied meanings that intentionally contradict the literal meanings (e.g., saying "He is bright!" about somebody who says stupid thing). When they use verbal irony, speakers simultaneously communicate literal information and a dissociative attitude toward this literal information, and thus toward the referent (event, person or object). This attitude can be manifested by counterfactual aspects of the context, or by the speaker's facial cues, body posture, or tone of voice [START_REF] Gibbs | Irony in language and thought: A cognitive science reader[END_REF]. For example, "That's a really good idea!" can be taken as either a compliment or a criticism, depending on the tone of voice, referred to as prosody in the literature.

Although there is no consensus as to which vocal cues are most indicative of verbal irony, research has confirmed that prosody can inform listeners about a speaker's ironic attitude. For example, [START_REF] Bryant | Recognizing verbal irony in spontaneous speech[END_REF] found that participants could correctly distinguish between spontaneously produced ironic versus nonironic utterances based on prosody. Consistent with this, [START_REF] Rockwell | Lower, slower, louder: Vocal cues of sarcasm[END_REF] showed that sarcastic irony can be detected when no context is provided, further emphasising the role of prosody in irony perception (see also [START_REF] Voyer | Subjective auditory features of sarcasm[END_REF]. Prosodic aspects of speech can therefore supplement or modify the meaning of the spoken sentence by providing valuable clues to the speaker's attitude. At least two kinds of prosody have been identified in the literature: emotional prosody and attitudinal prosody [START_REF] Mitchell | Attitudinal prosody: What we know and directions for future study[END_REF]. While emotional prosody refers to the emotions conveyed by the tone of voice, attitudinal prosody refers to the expression of a person's attitude towards an event, person or object, be it scepticism, doubt, enthusiasm, or boredom [START_REF] Mitchell | Attitudinal prosody: What we know and directions for future study[END_REF][START_REF] Wickens | What do you mean by that? An electrophysiological study of emotional and attitudinal prosody[END_REF]. Attitudinal prosody can drastically modify the literal meaning of verbal irony, by indicating the speaker's dissociative attitude, although emotional prosody can also be used [START_REF] Bryant | Prosodic contrasts in ironic speech[END_REF]. For example, a speaker can pronounce "That's just great" with an ironic intent and using an angry prosodic contour. In the present study, we therefore sought to investigate further the understanding of ironic intention based on the attitudinal prosody of utterances presented in isolation.

According to several theories, verbal irony is generally used to express a negative attitude towards a person or event [START_REF] Kumon-Nakamura | How about another piece of pie: The allusional pretense theory of discourse irony[END_REF][START_REF] Sperber | Irony and the use-mention distinction[END_REF]). An ironic sentence therefore takes the form of a positive sentence used to convey a negative meaning, as in the sarcastic comment "He is bright!" about somebody who says something wrong. However, less common, but still heard, is ironic praise, in which a negatively worded utterance conveys a positive message, as when someone says "You have a hard life!" to a friend going on a long vacation [START_REF] Schwoebel | Obligatory processing of the literal meaning of ironic utterances: Further evidence[END_REF].

The literature shows that there is an asymmetry in the processing of these two kinds of irony, in that ironic criticism is understood sooner and more easily than ironic praise [START_REF] Hancock | Children's comprehension of critical and complimentary forms of verbal irony[END_REF][START_REF] Harris | Children's perceptions of the social functions of verbal irony[END_REF]. [START_REF] Clark | On the pretense theory of irony[END_REF] called this effect the asymmetry of affect.

The present study specifically addressed this asymmetry of affect by asking whether the nature of the neurocognitive processes underlying the understanding of a speaker's ironic intention depends on the positive or negative connotation of the ironic remark. We examined these processes via event-related brain potentials (ERPs).

Two ERP components are of particular interest when studying language comprehension: the N400 and the P600. [START_REF] Brouwer | Getting real about semantic illusions: Rethinking the functional role of the P600 in language comprehension[END_REF] even assumed that language comprehension proceeds in biphasic N400/P600 cycles. The N400 is a negative component that occurs between 300 and 500 ms, with a peak at about 400 ms after stimulus onset, and is characterised by a centroparietal distribution. It was initially interpreted as being sensitive to semantic violations [START_REF] Kutas | Reading senseless sentences: Brain potentials reflect semantic incongruity[END_REF], but later found to be sensitive to semantic features at both word [START_REF] Holcomb | Auditory and visual semantic priming in lexical decision: A comparison using event-related potentials[END_REF][START_REF] Kutas | An electrophysiological probe of incidental semantic association[END_REF] and sentence levels (Kutas & Federmeier, 2007). N400 amplitude was originally assumed to reflect semantic integration [START_REF] Osterhout | Event-related brain potentials elicited by syntactic anomaly[END_REF][START_REF] Van Berkum | Semantic integration in sentences and discourse: Evidence from the N400[END_REF], but more recently a semantic retrieval hypothesis has gained ground (Kutas & Federmeier, 2000;[START_REF] Van Berkum | The "neuropragmatics" of simple utterance comprehension: An ERP review[END_REF]. According to this hypothesis, the N400 is a general index of the difficulty of retrieving stored conceptual knowledge associated with a meaningful stimulus, which is dependent on both the stored representation itself and the retrieval cues provided by the preceding context [START_REF] Van Berkum | The "neuropragmatics" of simple utterance comprehension: An ERP review[END_REF]. [START_REF] Brouwer | Getting real about semantic illusions: Rethinking the functional role of the P600 in language comprehension[END_REF] went as far as to postulate that the N400 component is modulated by the retrieval of lexical-semantic information, rather than by semantic integration or any other kind of compositional semantic processing. ERP findings on irony processing are not consistent regarding the N400, even though they all relate to ironic criticism. While some studies have reported a greater N400 amplitude for ironic statements than for literal ones [START_REF] Katz | Saying what you don't mean. Social influences on sarcastic language processing[END_REF], others have not [START_REF] Balconi | Isn't it ironic? An analysis on the elaboration of ironic sentences with ERPs[END_REF][START_REF] Regel | Isn't it ironic? An electrophysiological exploration of figurative language processing[END_REF][START_REF] Spotorno | What's behind a P600 ? Integration operations during irony processing[END_REF]. One possible explanation for this disparity of results in the literature is that the N400 can be modulated by the task demand and by the degree of familiarity of the ironic comment. For example, [START_REF] Cornejol | Electrophysiological evidence of different interpretative strategies in irony comprehension[END_REF] found an increased N400 for irony when participants were required to apply a holistic interpretative strategy ("if you can understand what the character means by the remark"), but not when they applied an analytic strategy focusing on the congruency of the final sentence of the story, thus supporting the semantic integration hypothesis. For their part, Filik, Leuthold, Wallington, and Page (2014) observed a less negative waveform in the N400 time window for unfamiliar ironic remarks than for literal ones, but there was no difference when the ironic remarks were familiar, consistent with the hypothesis that N400 modulation reflects semantic retrieval.

The P600 (also known as the late positive component, LPC 1 ) is commonly observed in the 500-900 ms time window, with a parietal topography. The cognitive processes reflected by this component were initially interpreted as involving the manipulation of syntactic information [START_REF] Hagoort | The syntactic positive shift (SPS) as an ERP measure of syntactic processing[END_REF], but were subsequently associated with the notion of conflict monitoring [START_REF] Bornkessel-Schlesewsky | An alternative perspective on "semantic P600" effects in language comprehension[END_REF][START_REF] Kuperberg | Neural mechanisms of language comprehension: Challenges to syntax[END_REF][START_REF] Van Petten | Prediction during language comprehension: Benefits, costs, and ERP components[END_REF]. [START_REF] Brouwer | Getting real about semantic illusions: Rethinking the functional role of the P600 in language comprehension[END_REF] postulated that P600 amplitude reflects integration compositional processes vested in interpretation. They further suggested that, rather than reflecting a single process, it subsumes many different subprocesses that can be differentiated and labelled on the basis of differences in onset, duration and scalp distribution. Consistent with an interpretative integration view, the P600 effect has frequently been observed in ERP studies of irony comprehension. For example, when [START_REF] Spotorno | What's behind a P600 ? Integration operations during irony processing[END_REF] exposed participants to sentences preceded by a substantial context biasing to either an ironic or a literal interpretation, they found that ironic sentences induced greater P600 amplitude than literal ones. Similarly, [START_REF] Regel | Isn't it ironic? An electrophysiological exploration of figurative language processing[END_REF] observed a P600 effect for irony processing regardless of modality (visually or auditory) and task (comprehension or passive reading). [START_REF] Regel | The communicative style of a speaker can affect language comprehension? ERP evidence from the comprehension of irony[END_REF] replicated the irony-related P600-like effect for visually presented materials, and demonstrated its dependency on the participant's knowledge about a character's typical (ironic vs. nonironic) communicative behaviour. Along the same lines, [START_REF] Filik | Testing theories of irony processing using eye-tracking and ERPs[END_REF] observed that both familiar and unfamiliar auditory ironic sentences elicited a greater P600 than literal sentences.

Although these ERP studies investigated the neurocognitive processes underlying the understanding of ironic versus literal statements, none of them focused on the asymmetry of affect. As a reminder, asymmetry of affect refers to the fact that ironic criticism ("He is bright!"), which is negatively connoted, is understood sooner and more easily than ironic praise ("You have a hard life"), which is positively connoted. The main aim of the present study was to examine how the neurocognitive processes underlying the understanding of irony, exclusively expressed through dissociative prosody, are modulated by the emotional connotation of the ironic remark (i.e., ironic criticism vs. praise).

We recorded the ERPs of 49 participants while they listened to positive (e.g., "These children are always smiling") or negative (e.g., "His son is very unfortunate") remarks pronounced with a sincere or ironic prosody. Participants had to decide whether or not the speaker was sincere. To understand the ironic remark presented in isolation and the speaker's underlying ironic intention, listeners have to integrate two divergent sources: the meaning of the expression and its intonation. Based on the findings of irony studies with a similar design and the requirement that participants adopt a holistic approach [START_REF] Cornejol | Electrophysiological evidence of different interpretative strategies in irony comprehension[END_REF], we expected the N400 to vary in amplitude as a function of prosody, with greater amplitude for an ironic versus sincere intonation, reflecting either the greater integrative cost of irony or the greater difficulty of retrieving stored knowledge associated with the words making up the ironic expression. Furthermore, given the asymmetry of affect, modulation might vary according to the emotional connotation of the ironic remark, with greater N400 amplitude for ironic praise versus criticism, reflecting the greater processing cost of noncanonical versus canonical irony. Regarding the P600, based on what has been repeatedly observed in the irony literature, we would expect to find a greater amplitude for ironic versus sincere prosody. The overall pattern of N400 and P600 modulations in the different conditions would shed light on the processes involved in the asymmetry of affect.

Method

Participants

Participants were 49 students from Reims University (37 women; mean age = 24.0 years, SD = 5.33, range = 18-42). They were all native French speakers, had normal or corrected-to-normal vision, and reported no history of psychiatric or neurological disorders. The study was approved by the relevant French ethics committee, and all participants gave their written informed consent after receiving a full description of the study.

Material

The experimental material consisted of 80 sentences (40 negative and 40 positive) created especially for the present study. All the sentences took the same form (subject/verb/adjective),

where the adjective determined the emotional valence. The verb used was "to be", with the exception of two cases for which the verb was "to seem". Examples of positive or negative sentences, as per the valence of the adjective, are provided in Table 1.

The emotional valence of the adjectives had been established in a previous study in which 204 native French speakers, none of whom took part in the ERP study, rated 103 adjectives on a 7-point emotional scale ranging from 1 (negative) to 7 (positive). Based on these ratings, we selected 40 highly positive (mean = 5.92, SD = 0.48) and 40 highly negative (mean = 2.21, SD = 0.43) adjectives, which were inserted as final words in our experimental sentences. It should be noted that the positive and negative adjectives had similar mean lexical frequencies of 11.92 (SD = 22.45) and 11.99 (SD = 22.9), F(1, 39) < 1 (lexique.org database; New, Pallier, & Ferrand, 2005).

In order to assess other potential differences between the material conditions, we asked independent participants to perform two tasks, one assessing the cloze probability of the adjectives, the other assessing the familiarity of the sentences. In the cloze-probability task, 191 participants aged 18-48 years (M = 20.81, SD = 5.28; 22.5% men) were shown a list of sentences minus the last word (i.e., adjective) in an online test, and were asked to type a word that plausibly completed the sentence. Sentences were divided into two lists of 40 sentences each. Participants were randomly assigned to one of the two lists. Results revealed that cloze probability was close to 0 for most sentences, and there was no significant difference between sentences ending with a positive adjective (.003) and sentences ending with a negative one sincere and an ironic intonation (see Table 1). The 160 pronounced sentences were divided into two lists, so that participants saw 40 positive and 40 negative sentences but each only in one version, sincere or ironic. It should be noted that in the case of the ironic intonation, the initial positive sentence "Their children are always smiling" was meant to be understood as negative (criticism), and the negative sentence "The spectators were disappointed" as positive (praise).

Table 1 about here

Acoustic analysis

The purpose of the acoustic analysis was to determine whether, for the same speech segment, the ironic prosody differed clearly from the sincere prosody. The analysis was carried out using PRAAT (Version 6.0. 2.0.; www.praat.org). Prosodies were assessed in terms of segment duration, fundamental frequency, and intensity. These three variables were chosen because they are strongly associated with perceptually recognisable prosodic qualities [START_REF] Voyer | Subjective auditory features of sarcasm[END_REF]. The results of the acoustic analysis are set out in Table 2.

Table 2 about here

We ran an analysis of variance (ANOVA) on each of the three measures. Consistent with previous results [START_REF] Laval | French-speaking children's understanding of sarcasm: The role of intonation and context[END_REF][START_REF] Regel | Isn't it ironic? An electrophysiological exploration of figurative language processing[END_REF] (ending with a positive adjective) being longer than negative ones (ending with a negative adjective). These results confirmed that ironic prosody differed in its perceptual features from neutral prosody.

Procedure

Participants were tested individually, seated in front of a monitor at a distance of nearly 60 cm.

They listened to either the list 1 or the list 2 of the materiel, each containing 94 sentences. Each trial began with the display of a fixation cross for 1000 ms. Then the auditory sentence was played during the presentation of a black screen for a variable duration. This was followed by a fixation cross for 500 ms and by a question mark inviting the participant to answer the question "Does the speaker think what he says?" The time between the display of the question mark and the following trial was fixed at 4000 ms. Before the experimental task, participants underwent a training session featuring six sentences that were not used in the experimental task.

Electroencephalographic recording and ERP analysis

Electroencephalograms (EEGs) were recorded using a 10-20 system electrode cap with 32 channels, connected to an amplifier (BrainAmp; Brain Products, Munich, Germany). The reference electrode was located midway between Fz and Cz, and the ground electrode between Fz and the prefrontal electrodes. Impedance was kept below 5 kΩ for all participants. The amplification gain was 1000. We recorded the electro-from two electrodes derived from FP1 and FP2 placed on and below the outer canthus of the right eye. EEG data were filtered with bandpass cut-offs of 0.1-100 Hz and a 30-Hz low-pass filter. A notch 50-Hz filter was also applied. Segmentation was then applied from 200 ms before the target onset to 1500 ms after.

We performed ICA decomposition, and electro-oculographic and other linear artefacts were identified using SASICA (Chaumont, Bishop, & Bush, 2015). The ERP signal was then rereferenced offline to the digital mean of the left and right mastoids. We carried out baseline correction for the 200-ms epoch before target onset. Segments containing residual artefacts were then checked visually and using the moving window peak-to-peak threshold function. A moving 200-ms window with a 100-ms step and 100-µV processing threshold were set. This led to the rejection of 4.79% of the data. The mean number and range of averaged trials were equivalent across all four conditions: negative adjective + ironic prosody (13.6 ± 4.20), positive adjective + ironic prosody (16.43 ± 3.26), negative adjective + sincere prosody (18.3 ± 2.10), and positive adjective + sincere prosody (16.1 ± 3.50).

Based on visual inspection and on previous ERP studies, we selected the following nine electrodes for the N400 (300-500-ms time window) and P600 (500-900-ms time window) statistical analyses: F3, Fz, F4, C3, Cz, C4, P3, Pz and P4. We also conducted exploratory analyses for the 900-1500 ms time window (positive post-P600 component). These analyses were justified by both the visual inspection of the waveforms and previous observations of late positive effects of emotionally impacting stimuli in similar time windows (e.g., [START_REF] Benning | Late positive potential ERP responses to social and nonsocial stimuli in youth with autism spectrum disorder[END_REF]Thiruchselvama, Blecherta, Sheppesa, Rydstromb, & Gross, 2011).

Statistical analysis

For the behavioural data, the error rate was submitted to generalised linear mixed models, because of the nature of the dependent variable, via restricted maximum likelihood estimation (see Table 3 for comparative and absolute fit statistics). First, the null model (Model 0), including participants -crossed with adjective valence and prosody -, and items -crossed with prosody -, as random terms to consider the specific design of our experiment (cf. [START_REF] Barr | Random effects structure for confirmatory hypothesis testing: Keep it maximal[END_REF], served as the point of comparison for fit statistics presented in Table 3. In the following step (Model 0b), we tested the effect of familiarity, as it differed significantly between our conditions. Then, the effect of prosody (ironic vs. sincere -withinparticipant factor) was tested (Model 1), before adding the main effect of adjective valence (positive vs. negative -within participant factor) (Model 2). Finally, we added to the model our particular effect of interest: the interaction between prosody and adjective valence (Model 3).

Concerning the ERP data, mean amplitudes for the N400, P600 and post-P600 components were separately submitted to linear mixed models (see Table 3 for detailed fit statistics). For each component, the null model (Model N0/Model P0/Model L0) integrated the two random effects structures (i.e., variance across participants, crossed with adjective valence and prosody, and items, crossed with prosody), and Model 1 (Model N1/Model P1/Model L1), the effect of familiarity. Then, the respective effects of prosody and adjective valence as well as their interaction were directly modelled in the following step (Model N2/Model P2/Model L2), as they were the three effects of interest. Statistics for each effect in these models were presented in Results section.

All statistical analyses were performed using R software (R Core Team, 2017), as well as the lme4 (Bates, Maechler, Bolker, & Walker, 2014), MuMIn and phia packages.

Results

Table 3 about here

Behavioural data

The effect of prosody on the error rate was reliable, indicating that participants made more errors when the prosody was ironic (0.24) than when it was sincere (0.11). The effect of adjective valence was not reliable. There also was a significant Prosody x Adjective valence interaction. As shown in Figure 1, this interaction showed that the difference in the error rate between the ironic and sincere prosodies was significant for sentences ending with a negative adjective (ironic praise) (d = 0.21), χ² = 57.46, p < .001, but not for sentences ending with a positive one (ironic criticism) (d = -0.01), χ² = 0.0003, p = .987. These behavioural results are consistent with the asymmetry of affect [START_REF] Clark | On the pretense theory of irony[END_REF] observed in the literature:

negative sentences with an ironic prosody (ironic praise) were harder to understand than positive sentences with an ironic prosody (ironic criticism).

Figure 1 about here

ERP data

ERP waveforms for the nine electrodes are shown in Figure 2, along with topographical images.

Figure 2 about here

300-500 ms -N400 component

The results of Model N2 revealed a significant effect of the Adjective valence x Prosody interaction, F(1, 27257.9) = 17.43, p < .001. For sentences ending with a negative adjective, mean N400 amplitude was greater for ironic prosody (i.e., ironic praise; -2.070 µV) than for sincere prosody (-1.579 µV), whereas for ones ending with a positive adjective, mean N400 amplitude was greater for sincere prosody (-1.408 µV) than for ironic prosody (i.e., ironic criticism; -0.894 µV). No other significant effect was observed (all ps > .10).

500-900 ms -P600 component

The results of Model P2 showed that the Adjective valence x Prosody interaction was also reliable, F(1, 27247.2) = 23.88, p < .001. For sentences ending with a negative adjective, mean P600 mean amplitude was less positive for ironic prosody (i.e., ironic praise; 0.172 µV) than for sincere prosody (1.034 µV), whereas for ones ending with a positive adjective, mean P600 amplitude was larger for ironic prosody (i.e., ironic criticism; 1.253 µV) than for sincere prosody (0.823 µV). No other significant effect was observed (all ps > .10).

900-1500 ms -post-P600 component

The results of Model L2 showed a significant effect of prosody, F(1, 46.8) = 5.02, p = .030, with a post-P600 component amplitude that was greater for ironic prosody (2.832 µV)

than for sincere prosody (1.887 µV). Moreover, the Adjective valence x Prosody interaction was significant, F(1, 27234.7) = 6.06, p = .014. The effect of prosody on post-P600 component amplitude was greater for sentences ending with a positive adjective (ironic prosody: 3.101 µV vs. sincere prosody: 1.764 µV; d = 1.337 µV) than for ones ending with a negative adjective (ironic prosody: 2.563 µV vs. sincere prosody: 2.144 µV; d = 0.419 µV). No other significant effect was observed (all ps > .10).

Discussion

The main goal of the current study was to investigate the neurocognitive processes behind the asymmetry of affect generally observed in irony comprehension. First, our behavioural data confirmed the existence of this asymmetry, as a sentence ending with a positive adjective pronounced with an ironic prosody (ironic criticism) allowed for a better understanding of what the speaker meant than a sentence ending with a negative adjective that was ironically pronounced (ironic praise). Thus, it was easier to infer what the speaker meant when he ironically said "This audience is very pleasant" than when he ironically said "The spectators were disappointed". These two types of irony (i.e., criticism and praise) both involve mock evaluations of circumstances with a valence that contrasts with the speaker's true appraisal, but are apparently sustained by different processes or knowledge. What do the N400 and P600 tell us about the processes responsible for this asymmetry?

ERP data did not show a modulation of the N400 according to either prosody or valence, but consistent with the asymmetry of affect, this component was modulated by their interaction. For sentences ending with a negative adjective, mean N400 amplitude was greater for ironic prosody (i.e., ironic praise) than for sincere prosody, whereas for ones ending with a positive adjective, mean N400 amplitude was greater for sincere prosody than for ironic prosody (i.e., ironic criticism). In other words, prosody differently affected the N400 as a function of emotional information. There are two possible interpretations, based on the main theories regarding the N400. From an integrative viewpoint (Kutas & Federmeier, 2000), we can assume that sentence-prosody inconsistency impeded meaning construction more when the adjective's valence was negative rather than positive. From a retrieval viewpoint [START_REF] Kutas | Psycholinguistics electrified II[END_REF], we can argue that the inconsistency impeded the retrieval of negative adjectives from memory more than that of positive adjectives from memory. It should be noted that our results did not appear to be compatible with the prediction derived from [START_REF] Brouwer | Getting real about semantic illusions: Rethinking the functional role of the P600 in language comprehension[END_REF]'s model, which posits that the N400 component is modulated by the retrieval of lexical-semantic information, rather than by semantic integration or any other kind of compositional semantic processing. Moreover, the observed effect of irony on the N400 appears consistent with the finding of a recent study (Baptista, Manfredi, & Boggio, 2017) in which ERP responses coupled with transcranial direct current stimulation (tDCS) were recorded during the visual presentation of verbal irony remarks, criticism and praise following a picture describing a counterfactual context. This study also demonstrated an asymmetry of affect, regardless of tDCS stimulation, with greater negativity in response to ironic language than in response to literal language, but only in the praise condition. Taken together, these results indicate that the N400 can be functionally modulated by the emotional connotation of ironic remarks, whether they are presented aurally or visually. Thus, our findings show that the N400 can be modulated by emotional information in addition to the task demand [START_REF] Cornejol | Electrophysiological evidence of different interpretative strategies in irony comprehension[END_REF] or degree of familiarity (Filk et al., 2014).

Results regarding the P600 confirmed previous ERP findings showing that irony processing induces an enhancement of the P600 component [START_REF] Filik | Testing theories of irony processing using eye-tracking and ERPs[END_REF][START_REF] Regel | Isn't it ironic? An electrophysiological exploration of figurative language processing[END_REF][START_REF] Spotorno | What's behind a P600 ? Integration operations during irony processing[END_REF], but only for canonical irony (i.e., ironic criticism). Consistent with the asymmetry of affect, results indicated that the P600 was modulated by the emotional connotation resulting from adjective valence and prosody, but contrary to what we observed for the N400 component, it was ironic criticism rather than ironic praise that elicited a greater P600. These results confirm that the P600 effect is the most recurrent outcome of pragmatic phenomena, reflecting the inference of the speaker's meaning. In addition to canonical irony, P600 effects are observed during the comprehension of metaphors [START_REF] Bambini | Disentangling metaphor from context: An ERP study[END_REF][START_REF] Obert | He catapulted his words from the dais: An ERP investigation of novel verbal metaphors[END_REF], idioms (frontal topography; Canal, Pesciarelli, Vespignani, Molinaro, & Cacciari, 2017), and presuppositions [START_REF] Domaneschi | N400 and P600 modulation in presupposition accommodation: The effect of different trigger types[END_REF]. However, the fact that the P600 was sensitive to ironic criticism, but not to ironic praise, merits further discussion. It supports one of the hypotheses put forward by [START_REF] Regel | Isn't it ironic? An electrophysiological exploration of figurative language processing[END_REF], whereby the P600 effect is a reflection of the processing of emotional information conveyed by canonical irony, insofar as it mainly expresses the speaker's disappointment at an event (saying things as they should be). We return to this point below.

The overall pattern of N400 and P600 modulations according to type of irony (criticism vs. praise) was quite complex. We did not observe greater N400 modulation for ironic criticism than for its literal counterpart, but greater P600 modulation, and found the opposite pattern for ironic praise. This differential effect of emotional information on N400 and P600 component modulations raises questions about the processes involved. In the irony literature, two main explanations have been put forward for the asymmetry of affect. Ironic praise may be harder to interpret because although ironic criticism tends to implicitly allude to conventions about politeness and saying nice things, ironic praise does not [START_REF] Kumon-Nakamura | How about another piece of pie: The allusional pretense theory of discourse irony[END_REF]. Within this framework, ironic praise (i.e., a sentence with a negative valence but a figurative positive meaning), which is literally negative, is a less effective reminder of our implicit positive expectations or norms, and is consequently harder to process than ironic criticism. Another possible explanation for this asymmetry of affect is that ironic praise involves a double negation that is more difficult to process. According to [START_REF] Giora | On irony and negation[END_REF], to understand ironic praise, the negative literal meaning has to be negated. However, as the ironic praise sentences in our study did not contain any negation markers (just negative adjectives), this theory appears less relevant here. Turning to allusional pretence theory [START_REF] Kumon-Nakamura | How about another piece of pie: The allusional pretense theory of discourse irony[END_REF], it can be argued that ironic praise is more incongruous than irony criticism because it is not usual to voice (literal) evaluative negative thoughts, making it hard to process such counternormative remarks and find figurative interpretations.

Looking afresh at our ERP results with this theory in mind, the greater N400 modulation we observed for ironic praise (i.e., a sentence with a negatively valenced adjective but a positive figurative meaning) could result from the unconventional feature of its inherent literal negative remark, namely saying bad things to people. Likewise, the greater P600 modulation observe for ironic criticism (i.e., a sentence with a positively valenced adjective but a negative figurative meaning) could reflect the fact that the figurative negative connotation was harder to process than the figurative positive connotation of the ironic praise. It seems that different processes were involved, as reflected by both the N400 and the P600 components, which exhibited asymmetry of affect modulations. It is as if processes, behind each ERP component, were modulated differentially by emotionality in the ironic condition, with a particular focus on the negative meaning. By nature, the negativity is at the level of the sentence valence for ironic praise while it is at the level of the connotation for ironic criticism.

Therefore, N400 was more modulated by the negativity of the adjective while P600 was more affected by the negativity of the connotation. It may therefore be harder to start processing the sentence meaning with the prosody for ironic praise than for ironic criticism, because of the nonconventional emotional feature of the literal remark, as reflected by the greater N400. By contrast, as illustrated by the greater P600, the processing may be more costly for ironic criticism because of its figurative negative connotation.

But how did negativity modulate these ERP components? As we did not include neutral utterances in our material, we could not specifically assess the absolute effect of negativity.

Further studies are therefore needed to answer this particular question. Nevertheless, our exploratory analyses for the 900-1500-ms window, based on visual exploration of the ERPs, gave us some additional information, specifically on the post-P600 component. Considering the LPC in its broadest sense, components occurring in late time windows have been shown to respond, among others, to the emotional content of verbal stimuli, and more specifically to their emotional valence. Their amplitudes usually appear larger for valenced stimuli than for neutral ones, although results are not consistent regarding a positive or negative advantage (for a review, see [START_REF] Citron | Neural correlates of written emotion word processing: A review of recent electrophysiological and hemodynamic neuroimaging studies[END_REF]. When we scrutinize our results, we can see that the post-P600 varied according to the asymmetry of affect. More specifically, we only found a greater modulation of this component for an ironic remark than for its literal counterpart when the former was negative (i.e., ironic criticism). In the literature, authors suggest that a greater LPC reflects the enhanced motivation and arousal that is experienced in response to affective stimuli [START_REF] Benning | Late positive potential ERP responses to social and nonsocial stimuli in youth with autism spectrum disorder[END_REF]. With this in mind, the post-P600 effect we observed could be linked to the greater emotional impact of ironic criticism, compared with ironic praise.

What are the cognitive processes behind the ERP components involved in irony understanding? Our proposal regarding a negativity bias raises the question of the expectation of negativity at both semantic and pragmatic levels. Current psycholinguistic theories emphasize the importance of prediction for language understanding (for a review, see [START_REF] Ferreira | Integration and prediction in language processing: A synthesis of old and new[END_REF]. Processing is viewed as fundamentally Bayesian, with people processing current information in the light of their expectations, leading them to predict the following word. In this framework, both the N400 and the P600 may be triggered by disconfirmation of active predictions, each at a specific level. As it is not usual to voice evaluative negative thoughts, we can assume that people predict hearing a literal positive adjective during irony, or a positive connotation when prosody and sentence converge. However, the fact that cloze probability did not differ between positive sentences (ending with a positive adjective) and negative sentences (ending with a negative adjective) in our study means that this interpretation cannot be confirmed. The functional distinction proposed by Hagoort, Baggio, and Willems (2009) could shed an interesting light on our results. These authors distinguished integration from unification processes, and suggested that integration allows for information to converge to a common memory representation, whereas semantic unification is a constructive process in which a semantic (or pragmatic) representation is built -and not retrieved. Based on this model, we can hypothesise that the sentence-prosody inconsistency of the ironic remark impeded integration more when the adjective was negatively rather than positively valenced, as reflected by the N400. Furthermore, it entailed more unification when the figurative meaning was negative, as indexed by the P600. In this framework, the greater N400 we observed for ironic praise may have reflected more difficult convergence to a common memory representation, owing to the negativity of the sentence, while the greater P600 for ironic criticism may have reflected a higher cost of unification, owing to the negativity of the figurative meaning. Finally, the post-P600 effect may an indicator of the final impact of irony understanding, insofar as it is a rhetorical device used to criticize while saying nice things.

Conclusion

Our study suggests that the processes underlying the understanding of a speaker's ironic intention depend on the emotional connotation of the ironic remark (praise vs. criticism). It sheds light on the behavioural asymmetry of affect, improving our understanding of the processes involved, as reflected by the biphasic N400/P600 component modulation, in addition to the post-P600 positive component, and highlighting the sensitivity of these ERP components to the emotional information conveyed by speakers' messages. Further studies including neutral sentences are needed to confirm our finding that N400 and P600 amplitudes are differentially affected by emotional information, and more generally to explore the impact of emotional connotation on ERPs (see [START_REF] Fischler | Event-related potential studies of language and emotion: Words, phrases, and task effects[END_REF].

(

  .008) (W = 762, p = .586). In the familiarity task, inspired by[START_REF] Filik | Testing theories of irony processing using eye-tracking and ERPs[END_REF]'s methodology, SD = 2.31; 16.7% men), were asked to indicate how familiar they were with each sentence used ironically on a 7-point Likert-like scale ranging from 1 (Not familiar) to 7 (Familiar). They were randomly assigned to one of two lists of 80 sentences each (40 experimental sentences and 40 fillers). Results indicated that sentences ending with a positive adjective (3.69) were rated as more ironically familiar than those ending with a negative one (2.33), t(78) = 8.53, p < .001.Each of the 90 experimental sentences was pronounced several times by a professional male actor who was a French native speaker, with both a sincere and an ironic intention. All utterances were digitally recorded and saved as individual audio files. Four judges evaluated the different voice recordings. Only consensual voice recordings were selected. The experimental material thus consisted of 40 negative sentences (ending with a negative adjective) and 40 positive sentences (ending with a positive adjective), each pronounced with a

Figure 1 .Figure 2a )Figure 2b )

 12a2b Figure 1. Percentage of errors according to prosody (ironic or sincere) and the valence

Table 1 .

 1 Examples of sentences ending with a positive or negative adjective

	Positive adjective	Negative adjective
	Ses enfants sont très souriants	Son fils est très malheureux
	(Their children are always smiling)	(His son is very sad)
	Cet auditoire est très plaisant	Les spectateurs ont été déçus
	(This audience is very pleasant)	(The spectators were disappointed)

1

Table 2 .

 2 Acoustic analysis of speech segments for differences in duration, fundamental frequency (Hz), and intensity (dB)

	Condition Valence Mean duration	Mean F0	Mean intensity
		(s)
		Positive 2.09	149.14	74.16
	Ironic	Negative 1.44	110.80	77.10
		Positive 1.90	147.41	74.11
	Sincere	Negative 1.45	115.85	77.78

Note. F0: fundamental frequency.

Acknowledgments

We would like to thank Fabien Gierski and Flavien Thuaire for their help.

FUNDING

This work was supported by the French National Research Agency (ANR) (grant no. ANR-11-EMCO-001 01).


Alain Cimetière
email: alain.cimetiere@univ-poitiers.fr

Alain Léger
email: alain.leger.1@u-bordeaux.fr

Elaine Pratt
email: pratt@lma.cnrs-mrs.fr

On the coupling of large deformations and elastic-plasticity in the mechanics of a simple system

Keywords: critical load, incremental plasticity, large deformations, nonsmooth non linearity, plastic buckling, simple model, tangent modulus

The buckling and the post buckling of a simple model involving both large deformations and elasticplasticity are studied. Plasticity is described by the classical incremental constitutive law involving in particular non regularized transition between plastic loading and elastic unloading. The kinematics is chosen in order to take into account any rotation, however large, without having to resort to approximations or asymptotic expansions. The work focusses essentially on the specificity of the coupling between these two types of nonlinearity. At small strains the global behavior is correctly given by Hutchinson's simple model involving the same elastic-plastic constitutive law and linearized deformations. But at very large strains, the response notably differs from that of Hutchinson's model. Indeed plasticity seems to have no effect and the behavior is that of a system involving the same nonlinear deformations but with a constant modulus. From the point of view of bifurcation analysis this result is qualitatively interesting due to the fact that the elastic-plastic constitutive law can be regarded as a strong, or non-smooth, nonlinearity, whereas large deformations is a smooth, we could say weaker, non linearity, so that the result contradicts the intuition that the strongest non linearity should be associated with the strongest bifurcation effect. The analysis is carried out using two different geometrical nonlinearities.

Introduction

It is well known in mechanics of structures that the buckling phenomenon results in particular from the nonlinearities that are involved when taking into account the kinematics at large strains. In the case of elastic structures the bifurcation diagram has been widely studied, it consists in general of the discrete spectrum of an operator obtained by the linearization of the kinematics on a trivial branch, together with the bifurcated branches at large, or very occasionally at very large deformations.

In the present paper we are interested in the elastic-plastic buckling of a class of mechanical systems, with the aim of not only determining the set of bifurcation points but also understanding their asymptotic behavior at very large post-buckling deformations, which, as far as the authors are aware, had never been studied. Simple mechanical models, made of springs and bars, are built for that purpose. The bars are rigid, possibly rigidly connected together or articulated, the springs are elastic-plastic.

For each mechanical model we first give the solution of the elastic buckling: plasticity is removed and we recall the effects of the nonlinear strains within linear elasticity. The systems are investigated within linearized strains in a first case then within two different models of geometrical non linearity allowing very large deformations. These elastic buckling problems are elementary exercises, but they will be useful for the asymptotic analysis. The same examples with the same kinematic choices are then carefully calculated within elastic-plasticity.

The essential results expected from this study concern the coupling between the two types of non linearity involved, one due to the kinematics, which is described by smooth functions, and the other arising from the constitutive law, which is nonsmooth and described by graphs or inequalities. All the systems at hand possess a trivial solution for any positive real value of the parameter which is the external loading, the expected behavior then deals with bifurcated branches. The qualitative results are such that close to the trivial solution, that means within the so-called incipient buckling or close post-buckling, the set of solutions results exclusively from plasticity, and in particular seems to be unchanged by the different cases of kinematics. But following the bifurcated branches involves a boundary between plastic loading and elastic unloading, and the evolution of this free boundary may be rather complicated even though the discrete system is very simple.

Let's now outline the main steps of the paper.

-After this short introduction, the first part of this paper deals with the basic statements of elastic-plastic buckling and post buckling which are concerned by the present work, and recalls the complete analysis of the case where the elastic-plastic constitutive law is coupled with geometrically linear kinematics. -The next part is completely new: studying the same type of mechanical device but just changing the mechanical links, the elastic-plastic constitutive law is now coupled with the geometrically non linear kinematics which had been introduced in [START_REF] Cimetière | Some problems about elastic-plastic post-buckling[END_REF]. -The third part aims at the same kind of analysis as the previous one but using another example of mechanical link leading to another type of geometrical non linearity. This is done in order to be closer to the kind of behavior that can be observed in the investigation of real mechanical structures. -The last section is a conclusion: at first it gathers and comments the results that arise from the three previous parts. Here the main conclusion consists in the discussion of the qualitative behavior either in the range of incipient buckling and close post buckling or in the range of far post buckling. Then remaining problems are suggested from the point of view of the role of the chosen geometrical non linearities and from the point of view of the extensions to continuous structures.

The three main parts contain several repetitions, in particular when presenting the constitutive law, the functions which appear in the equilibrium equations, etc... This just aims at facilitating the use of the results since the different parts could be considered independently.

Motivation

Some reminders

From a historical point of view, while the buckling load of an elastic structure had been given in some generic cases since the eighteenth century, the first correct result supported by a correct justification for elastic-plastic buckling dates only from the middle of the twentieth century and states that the first buckling load is given by the so-called critical load of the tangent modulus and occurs always under an increasing load, which ruled out the possibility of calculations by the perturbation method called adjacent equilibrium [START_REF] Shanley | Inelastic column theory[END_REF]. In the case of continuous media, this qualitative difference with respect to elastic buckling was stated in a complete general way ten years later in [START_REF] Hill | A general theory of uniqueness and stability in elastic/plastic solids[END_REF], but the second qualitative difference with respect to elastic buckling had been given only forty years later when it was established that the buckling diagram of an elastic-plastic structure generically involves not only a critical bifurcation load but a whole interval completely filled with incipient bifurcation directions [START_REF] Cimetière | Flambage naissant des plaques élastoplastiques minces[END_REF].

In the present work attention is restricted to a two-degree of freedom system. The simple model which is tackled all along this work, referred to as the Shanley-Hutchinson model, had been suggested by J. Hutchinson [START_REF] Hutchinson | Post-bifurcation in the plastic range[END_REF], and aimed at providing an analysis of the post buckling by the study of the extension of an unloading zone inside the structure. This system had been extended in [START_REF] Triantafyllidis | On the bifurcation and post-bifurcation analysis of elastic-plastic solids under general prebifurcation conditions[END_REF] by the addition of a supplementary degree of freedom in order to take into account general pre-bifurcation conditions, leading to the so-called tangent bifurcations. First obtained in the case of simple models, the results of the two or three degree of freedom systems were generalized to any structure [START_REF] Hutchinson | Plastic Buckling[END_REF], although lots of open problems remain for this generalization.

Shanley-Hutchinson

Shanley-Hutchinson's system is made of two rigid bars, rigidly connected to each other, making an inverted T shape as represented on Figure 1. Let u be the vertical displacement of the whole structure, and θ a possible rotation around the junction of the two bars. Without external force the reference configuration of the system is such that one bar is horizontal and connected to a rigid horizontal support by a continuous distribution of vertical springs, the other bar is vertical and designed to receive an external vertical load at the top. The horizontal bar is on the segment [-1, +1] of the horizontal Ox axis, the vertical bar has a length L, the behavior of the springs is given by a simple model of elastic-plasticity, with Young modulus E and tangent modulus E T . The external vertical force is denoted by λ. It is worth recalling that this structure was a continuous version of Shanley's model which involved only two springs, respectively at the ends x = ±1 of the horizontal bar, which complete bifurcation diagram had been given in [START_REF] Nguyen | Bifurcation and stability of the time independent standard dissipative systems[END_REF].

The results provided in [START_REF] Hutchinson | Post-bifurcation in the plastic range[END_REF] confirmed that the buckled solution appears at the critical load of the tangent modulus, namely λ T = 2E T 3L , and that a nontrivial branch actually exists given locally by a power expansion of the following form, in which τ stands for a measure of an unloading zone :

λ = λ T + µ, with µ = Coef. τ n + t.o.s. λ = λ T + λ 1 θ + λ 2 θ 1+β + ... with            λ 1 > 0, λ 2 < 0, β rational, 0 < β < 1. (1)
Due to the sign of λ 2 and to the value of β, it was claimed, and presented as a general characterization of plastic bifurcations, that the nontrivial branch, which started at increasing load due to the strict positivity of λ 1 , was non monotonic, with a maximal load close to the buckling point, and involved fractional exponents.

The case of linear kinematics

We now recall the analysis of the mechanical system at small strains, that is when the deformation of each spring is linearized. In fact, whether the compression of each spring and the rotation of the system is really small or not, we first look at the behavior of a model where the kinematics is linear and abstract (in fact it can be seen as a correct approximation of a physical realization of the system only at very small strains, which means here very

The reference configuration.

The buckling shape. 2. elastic-plastic constitutive law, referred to as incremental plasticity in the following:

σ = E T ε for plastic loading, -σ = E ε for elastic loading or unloading, where the notation ( . ) results from the definition of quasi-static plasticity and denotes a derivative with respect to the load λ; 3. linearized kinematics: ε(x, λ) = u(λ) + xθ(λ). The associated convention is ε(0, λ) > 0 when λ is positive, the positive vertical axis for the load λ being oriented downward.

Let us first assume that buckling occurs in the elastic range, that is σ = Eε for any x and any λ, then the result, which is here elementary, is that the complete bifurcated diagram is made of two branches

     θ = 0, λ ∈ R + , θ ∈ R + , λ = 2E 3L := λ E . (2)
In order to take into account the elastic-plastic equilibrium problem we now assume that, given the constitutive parameters, that is the Young modulus E, the tangent modulus E T and the limit of elasticity σy, the structure is such that the springs are all in plastic loading before buckling1 . Then Shanley's criterion [START_REF] Shanley | Inelastic column theory[END_REF] establishes that, although buckling occurs under increasing load, the stress no longer increases everywhere in the structure, but stops increasing at one point. This has been stated as a necessary condition for buckling, and Hutchinson's result was precisely based upon the fact that post-buckling involves an unloading zone which spreads out inside the structure, so that any nontrivial state becomes characterized by a boundary, say Y (λ) in the segment [-1, +1], between plastic loading and elastic unloading, this boundary being, as in any free boundary problem, an unknown of the problem. The very clever contribution of this model due to J. Hutchinson was that it allowed a continuous displacement of the boundary Y (λ) while it has only two degrees of freedom, namely u(λ) and θ(λ). The kinematics which is built with a rigid horizontal bar and the constitutive law given above imply the following continuity for the characterization of the boundary Y (λ):

ε(Y, λ) ≡ u + Y (λ) θ = 0. (3)
The starting point of the analysis was to consider the boundary Y (λ) as the main unknown of the equilibrium problem: introducing the constitutive law, including Equation (3) for the boundary Y λ), into the equations of the resultant and the moment, we obtain:

(P)    i) λ -βf (Y ) θ = 0, ii) λLθ + [Lλ -βg(Y )] θ = 0, (4)
where f (Y ) and g(Y ) are polynomials which read

       f (Y ) := - 1 2 Y 2 -αY - 1 2 , g(Y ) := - 1 6 Y 3 + 1 2 Y + α 3 , (5)
and where α := E + E T E -E T and β := E -E T are introduced for simplicity's sake.

Then, by showing that Equations (4) give, through an implicit function argument, the boundary as a function of the load λ and of the angle θ, the quasi-static equilibrium problem turned out to be an ordinary differential equation written as (P) :

     dλ dθ = βf (Y (θ, λ)), λ(0) = λ 0 . (6)
Dealing with the case of the linearized kinematics the results of this study had been given in [START_REF] Elkoulani | Flambage naissant et post-flambage d'un modèle simple en élastoplasticité[END_REF] and [START_REF] Elkoulani | Comportement post-critique des poutres élastoplastiques: existence et régularité des branches bifurquées[END_REF], and are essentially gathered in two theorems the proof of which is not recalled here as it can be found in [START_REF] Elkoulani | Flambage naissant et post-flambage d'un modèle simple en élastoplasticité[END_REF] or with more detail in [START_REF] Léger | Bifurcations et stabilité en mécanique des structures[END_REF]. The first one reads:

Theorem 1 Problem (P) does not have any transverse solution if λ 0 / ∈ [λ T , λ E ]. On the other hand for each λ 0 ∈ ]λ T , λ E [: 1) there exists a transverse solution, with initial unloading on [-1, Y 0 ], where Y 0 is the unique solution of the

equation λ 0 = β L g(Y 0 );
2) there exists a unique function λ(θ), defined on [0, +∞[, monotonic, i) strictly increasing if

Y 0 ∈ ] -1, Y R [, that is if λ 0 ∈ ]λ T , λ R [, ii) strictly decreasing if Y 0 ∈ ]Y R , +1[, that is if λ 0 ∈ ]λ R , λ E [, iii) constant if Y 0 = Y R , that is λ 0 = λ R .
iv) Moreover, for any λ 0 ∈ [λ T , λ E ] the transverse solutions satisfy the following asymptotic behavior:

lim θ→+∞ Y (θ) = Y R , lim θ→+∞ λ(θ) = λ R .
The point Y R appeared from the very beginning of the discussion of plastic buckling (see [START_REF] Engesser | Ueber die knickfestigkeit gerader stäbe[END_REF], [START_REF] Considère | Résistance des pièces comprimées[END_REF], [START_REF] Th | Discussion of Inelastic column theory[END_REF]). It is defined here as the unique zero in [-1, 1] of polynomial f (Y ). The value λ R is then given by Equation (4-ii) at

θ = 0 that is λ R = β L g(Y R
) and is referred to as the critical load of the reduced modulus [START_REF] Katchanov | Eléments de la théorie de la plasticité[END_REF]. It can be seen as a kind of average of λ T and λ E , the main property of which being associated with point iii of Theorem 1. The second result is the following:

Conclusion

: observing the coupling of incremental plasticity and linearized kinematics.

1. The fractional power expansion, obtained in [START_REF] Hutchinson | Post-bifurcation in the plastic range[END_REF] and recalled in Equation (1), does not concern the whole continuous set of bifurcated branches issued from any real number λ in [λ T , λ E ] at θ = 0, but only the branches starting from the ends of this interval and simply results from the loss of regularity at θ = 0. All the other branches, i.e., those starting everywhere in ]λ T , λ E [ are analytical. 2. We observe in addition, and in view of the objective of the present study it is the most important point, that all the branches are converging towards a "horizontal" asymptote which is precisely the branch starting from λ R . This branch is given by {[0, +∞[ × {λ R }}. According to Theorem 1 the position of the boundary is fixed and does not depend on θ, so that it is natural to say that it can be obtained by the same elementary calculation as that used for the result given by Equation ( 2), only using the adequate modulus.

3 Coupling incremental plasticity with nonlinear deformations: a first case

Model

This model is similar to Shanley-Hutchinson's one, except for the fastening of the springs. The structure is still the T-shape system made of two rigid bars, rigidly connected at a right angle and supported by a continuous distribution of elastic-plastic springs, but as it is supposed to describe large strains by a physically realistic mechanism, it is necessary to add some information concerning compression or traction deformations of the springs. In order to do this we now assume that the springs are free to slide on the horizontal support, while they are tied at fixed points on the initially horizontal bar of the T-shape structure, these fixed points being uniformly distributed on the segment [-1, +1] of this bar and sufficiently close to each others to be considered as an approximation of a continuous distribution. This new system is qualitatively represented on Figure 3, and we stress the fact that, contrary to the previous system which was abstract due to the assumption of linearity whatever the deformation, large or not, the present one can be manufactured in a laboratory 2 . We can see that the points where the springs are sliding are closer to each other on the horizontal support, which would mean in a continuous model that the density of springs becomes larger, as the structure is rotating, while the momentum of the reaction of each spring becomes smaller. Taking the full description of this system into account, and using the same notations as those of the previous section, elementary calculations lead to the following set of equilibrium equations, giving respectively the equilibrium of the resultant, the equilibrium of the moment and the kinematic assumption:

                   λ = 1 -1 σ(x)dx, λL sin θ = 1 -1 xσ(x) cos θdx, ε(x) = u + x sin θ. (7)
In the same way as for the geometrically linearized case, let us first assume that buckling occurs in the elastic range. The constitutive relation is then assumed to be σ = Eε everywhere on [-1, +1] and through elementary calculations the complete bifurcation diagram consists of the two following branches:

     θ = 0, λ ∈ R + , θ ∈ [0, π 2 [, λ = 2E 3L cos θ, (8)
2 Of course additional devices, not represented on Figure 3, would be necessary for an experimental realization in a laboratory: One device should prevent the structure from any global horizontal displacement; this could be a rigid axis normal to the plane of the structure at the intersection of the two bars and sliding without friction along a vertical guide. Another device should force the springs to remain in the initial vertical direction, for example by inserting the springs inside vertical tubes in which they could slide without friction while the structure is rotating.

which means, as opposed to the geometrically linear case, that function λ(θ) is no longer constant on the non trivial branch and that the load goes to zero as the rotation goes to π/2.

Let us now assume that the external load λ is smoothly varying in such a way that the load itself or any given nonzero smooth and monotonous function of the load can be taken as a kinematic parameter. As the foundation of quasi-static plasticity this function can be seen as a fictitious time, let τ be this fictitious time and (˙) be the derivative with respect to τ . Then the elastic plastic constitutive law, in the so-called simple model of elastic-plasticity with constant strain hardening, reads:

   i) σ = E ε if σ <
σy or σ < 0, i.e. for elastic loading or for unloading, ii) σ = E T ε if σ = σy and σ ≥ 0, i.e. for plastic loading, [START_REF] Hutchinson | Plastic Buckling[END_REF] where 0 < E T < E3 , and σy denotes the actual limit of elasticity of the springs which increases with the actual state.

the quasi-static equilibrium problem

After a differentiation with respect to τ , equations ( 7) lead to a quasi-static equilibrium problem which is now:

                   i) λ = 1 -1 σ(x)dx, ii) L λ sin θ + Lλ cos θ θ = cos θ 1 -1 σ(x)xdx -sin θ θ 1 -1 σ(x)xdx, iii) ε(x, λ) = u + x θ cos θ. ( 10
)
From the reference configuration the system may symmetrically rotate with a positive or with a negative angle θ. Let us choose the positive rotation. From an initial state where the springs are uniformly plastically loaded (i.e., assuming that condition 9-ii) holds everywhere in [-1, +1]), unloading appears at the point x = -1 since the horizontal bar is rigid. Let us then assume that for some strictly positive angle θ, there exists a boundary Y depending on λ, which seperates the domain [-1, +1] into an elastic part [-1, Y ] and a plastic part [Y, +1]. This specificity changes problem (7) from a classical equilibrium problem into a so-called free boundary problem, which means in the present case that in addition to the components u(λ) and θ(λ) of the displacement the problem involves a new unknown, namely the boundary Y .

Due to the constitutive law (10-iii) a continuity relation should be satisfied at the boundary Y , which gives the complementary equation that closes system [START_REF] Katchanov | Eléments de la théorie de la plasticité[END_REF]. It reads

ε(Y, λ) = 0 that is u = -Y θ cos θ. ( 11
)
Inserting Equations ( 9) and [START_REF] Léger | Bifurcations et stabilité en mécanique des structures[END_REF] into Equation [START_REF] Katchanov | Eléments de la théorie de la plasticité[END_REF], the quasi-static equation of the resultant is now

λ = (E -E T ) θ cos θ - Y 2 2 -Y E + E T E -E T - 1 2 . ( 12
)
The equation of the moment is a bit more complicated.

Step by step it reads:

L λ sin θ + Lλ θ cos θ = 1 -1 x σ(x) cos θdx - 1 -1 x θσ(x) sin θdx, (13)
where the second integral of the right hand side can be calculated through the static equation of the moment, which gives

L λ sin θ + Lλ θ cos θ = 1 -1 x σ(x) cos θdx -Lλ θ sin 2 θ cos θ , or L λ sin θ = 1 -1 x σ(x) cos θdx -Lλ θ 1 cos θ . ( 14
)
Now the right hand side is classically calculated through the constitutive relation, respectively in [-1, Y ] and in [Y, +1]. We get :

L λ sin θ = (E -E T ) θ cos 2 θ - Y 3 6 + Y 2 + E + E T 3(E -E T ) -Lλ θ 1 cos θ . ( 15
)
The quasi-static equilibrium problem described by Equations ( 12) and ( 15) involves the same constants α and β and the same functions f (Y ) and g(Y ) of the boundary as those of the geometrically linear case, and finally reads:

     i) λ -βf (Y ) cos θ θ = 0, ii) L λ sin θ -βg(Y ) cos 2 θ -λL 1 cos θ θ = 0, (16)
where we recall that

β = E -E T , α = E + E T E -E T , f (Y ) = - Y 2 2 -αY - 1 2 , g(Y ) = - Y 3 6 + Y 2 + α 3 .
It is now useful to recall some properties of functions f and g.

The function

f : Y ∈ [-1, +1] -→ f (Y ) = - Y 2 2 -αY - 1 2 :
is monotonically strictly decreasing in the whole interval [-1, 1] between f (-1) = 2E T E -E T > 0 and

f (1) = -2E E -E T < 0, -has consequently a single zero in ] -1, 1[ denoted by Y R with Y R = - 1 - E T E 1 + E T E , -1 < Y R < 0, -is strictly positive in [-1, Y R [ and negative in ]Y R , 1]. 2. The function g : Y ∈ [-1, 1] -→ g(Y ) = - Y 3 6 + Y 2 + α 3 : -is strictly increasing in ] -1, 1[ between 2E T 3(E -E T ) and 2E 3(E -E T )
.

is strictly positive in [-1, 1], with a zero derivative at Y = ±1.

We first observe that Problem [START_REF] Katchanov | Eléments de la théorie de la plasticité[END_REF] always has the solution θ ≡ 0 for any λ ∈ R + . Let us now come back to Equations [START_REF] Hill | A general theory of uniqueness and stability in elastic/plastic solids[END_REF]. It has just been stated that if the springs are all loaded in the elastic range the equilibrium problem is satisfied by two branches of solutions given by Equation [START_REF] Hutchinson | Post-bifurcation in the plastic range[END_REF] . It seems then natural that the same result would be obtained if the springs were all loaded in the plastic range, by just changing the notation and by replacing E by E T in Equation ( 8), but Shanley's argument states that this is impossible. Indeed a solution branch {λ = λ 0 , θ ∈ [0, π/2[} cannot exist if the springs are all in plastic loading due to the kinematic relation (10-iii), which implies that the springs will be elastically unloaded at the point x = -1 of the horizontal bar as soon as θ = 0. Consequently, only the solution branch {λ ∈ R + , θ = 0} exists if plastic loading is assumed to hold everywhere 4 , or in other words, a possible bifurcated branch involving θ = 0 from an initial state where all the springs were plastically loaded would necessarily imply elastic unloading in a part of the system. For the moment we keep in mind that θ ≡ 0 is solution for any λ, which means that looking for bifurcated solutions amounts to looking for solutions such that θ = 0 for θ = 0.

Let us now look at Equation (16-i):

The boundary Y is a bounded function since it belongs to [-1, +1] by definition, so that f (Y ) is bounded and dλ dθ is also bounded on any interval of [0, π/2[ in which the free boundary Y exists, which means that the equation of the quasi-static equilibrium is a differential equation with a bounded right hand side at least on a compact sub-interval of [0, π/2[. Using this result in Equation (16-ii) and recalling that g(Y ) is also bounded we get that function λ(θ) is also bounded on the same interval of [0, π/2[. Moreover, since θ = 0 Problem (16) leads to the following equation:

0 = βf (Y ) cos θ sin θ - β L g(Y ) cos 2 θ + λ cos θ , (17)
which is rewritten as:

Ψ (Y, θ, λ) ≡ βf (Y ) sin θ - β L g(Y ) cos θ + λ cos 2 θ = 0. ( 18
)
This algebraic equation gives

∂Ψ ∂Y = βf ′ (Y ) sin θ - β L g ′ (Y ) cos θ = -β(Y + α) sin θ - β 2 (1 -Y 2 ) cos θ.
where Only one root Y of the third degree algebraic equation (18) belongs to the interval [-1, +1] for θ = 0 so that non trivial branches are the solutions to the following differential equation:

Y
     dλ dθ = βf (Y (θ, λ)) cos θ, λ(0) = λ 0 , λ 0 ∈ [λ T , λ E ], (19)
and now the right-hand side of Equation ( 19) is not only bounded but is as smooth as function Y (θ, λ).

Incipient buckling and close post-buckling

The first step consists in the structure of the set of solutions close to the trivial branch. At θ = 0, Equation (17) becomes:

0 = β L g(Y ) -λ,
so that the buckling load and the boundary Y (λ, 0) are connected by the same relation as in the geometrically linear case 5 . In particular we have the following properties which follow immediately from the above study of polynomials f (Y ) an g(Y ):

-The relation between Y and λ is such that λ T is associated with Y = -1 and λ E is associated with Y = +1.

-When λ = λ T , there exists a strictly positive non trivial initial velocity dλ dθ , while a strictly negative non trivial initial velocity exists when λ = λ E .

-The value λ = λ R corresponds to the so called adjacent equilibrium, which in other words means bifurcation point with initial velocity equal to zero, or bifurcation at constant load. -Since Y must belong to the interval [-1, +1], there do not exist non trivial initial velocities for λ /

∈ [λ T , λ E ].
Let us introduce a small enough value θ 0 of the angle and study the solution branches for θ ∈ [0, θ 0 [, which may depend on λ 0 . Then:

Lemma 1 There exist bifurcated solutions to Problem (16), transverse to the trivial one, for any

λ 0 ∈ [λ T , λ E ], analytical if λ 0 ∈ ]λ T , λ E [, singular at θ = 0 if λ 0 = λ T or λ 0 = λ E ].
Proof: The proof of this Lemma involves two steps:

1. We first remark that due to implicit function theorem θ 0 is strictly positive. 17) cannot be inverted at the points {λ = λ T , θ = 0} and {λ = λ E , θ = 0}. Let us study one of these points and the argument will be adapted to the other. As presented in [START_REF] Elkoulani | Flambage naissant et post-flambage d'un modèle simple en élastoplasticité[END_REF] for the geometrically linear case, assume that for a given natural integer n, the boundary of the initial unloading zone at θ = 0 is given by

Y 0n = -1 + 1 n .
[×[λ T , λ E ].
Elementary techniques then give the existence of a solution branch λ∞(θ) starting from λ∞(0) ≡ λ T , as the uniform limit of the λn(θ) when n -→ ∞.

Intermediate and asymptotic post-buckling

The main qualitative feature of the evolution on the nontrivial branches is that the free boundary, which enters the domain [-1, +1] at θ = 0 according to the value of the initial data λ 0 , may be a non monotonic function of θ moving back to x = +1 for some θ > 0. In fact the first step consists in observing that all solution branches associated with any initial data λ 0 ∈ [λ T , λ E ] attain the boundary x = +1 for some positive value of the angle θ strictly smaller than π/2. More precisely, the numerical investigation of the behavior of functions Y (θ) gives the following: The free boundaries as functions of θ are plotted on Figure 4 As noticed in Figure 4, the admissible domains of functions Y (θ) are strictly included in [0, π/2[. In order to study the branches on the whole domain [0, π/2[, it is then necessary to understand the continuation of the branches up to π/2.

Lemma 2 i) Let

Theorem 3

The global bifurcation diagram satisfies the following statements: i) the incipient buckling is identical to the one of the geometrically linear case; ii) there exist bifurcated branches on [0, π/2[, non monotonous for

λ 0 ∈ [λ T , λ R [, decreasing for λ 0 ∈ [λ R , λ E ],
strictly decreasing for λ 0 > λ R ; iii) all these branches attain zero for a rotation θ depending on λ 0 but smaller than π/2.

Proof

Point i) simply recalls the result of the geometrically linear case given in Lemma 1.

The proof of points ii) and iii) need another lemma which completes what has been described in Lemma 2. Let λ 0 be the initial data of Problem (19), and λ λ0 (θ) its corresponding solution. Let also θ λ0 be the rotation at which the free boundary Y (θ) reaches x = +1, and denote λ θ0 := λ(θ λ0 ). The free boundary being at this end, the whole interval [-1, +1] is either elastically loaded or in elastic unloading since the deformation ε = u + x sin θ is maximum for x = +1. Although the external force associated with the quasi-static equilibrium is decreasing, the T-shape structure does not move back to its original position, as it would be the case in elasticity, because a part of the interval [-1, +1], i.e., a part of the springs, has been deformed in the plastic range.

The part of Point ii) concerning the branches where Y < +1 is a simple corollary of Lemma 2. As a matter of fact, assume λ 0 ∈ [λ T , λ R [, then Lemma 2 states that function Y (θ) is monotonically strictly increasing from Y 0 < Y R to +1. Since it is a smooth function, Y R is its single zero. Recalling Equation ( 19) and the properties of function f (Y ), we get immediately that as long as

Y (θ) ∈ ]Y 0 , Y R [, function f (Y ) is positive so that function λ(θ) is strictly increasing, while when Y (θ) ∈ ]Y R , +1[, function f (Y ) is negative and function λ(θ) is strictly decreasing. Since λ(θ) is smooth, it has a maximum at a point θ M interior to ]0, θ λ0 [.
Let us now consider the case where λ 0 ∈ ]λ R , λ E ], then Lemma 2 states that function Y (θ) is non monotonic, at first decreasing from Y 0 > Y R at θ = 0 to some value larger than Y R after which Y (θ) is increasing up to Y = +1, so that Y (θ) is greater than Y R everywhere. Function f (Y ) is consequently negative everywhere in ]0, θ λ0 [ and function λ(θ) is strictly decreasing in this domain.

Then for each λ 0 ∈ ]λ T , λ E ], only the continuation for θ > θ λ0 is needed to complete Point ii). Since each free boundary reaches the border of the domain at some rotation θ λ0 , we may assume that a free boundary no longer exists for any larger rotation. Actually, such a fully elastic continuation satisfies all conditions of the system, and, taking into account the match with the left limit of the elastic-plastic solution at θ λ0 , it reads:

λ(θ) = 2E 3L (cos θ -cos θ λ0 ) + λ θ0 , in ]θ λ0 , π/2[, (20)
where θ λ0 is defined as the value of the rotation depending on λ 0 for which Y = +1, and λ θ0 , also depending on λ 0 , is given by Equation (17) 6 .

The global bifurcation diagram of the quasi static evolution is represented on Figure 5. According with Remark 1, the load tends to zero whereas the rotation tends to π/2.

4 Coupling incremental plasticity with nonlinear deformations: a second case

The model

In the same way as in the elastic-plastic and geometrically non linear system represented on Figure 3, the structure which will be studied now differs from Shanley-Hutchinson's model only by the fastening of the springs. The structure is still the T-shape system made of two rigid bars, rigidly connected with a right angle and supported by a continuous distribution of elastic-plastic springs but in the present case the springs are tied at their bottom, at fixed points uniformly distributed on the interval [-1, +1] of the horizontal rigid support, and their upper end is assumed to slide without friction on the initially horizontal bar of the structure. In the same way as in the case represented on Figure 3 or in Shanley-Hutchinson's original model, the external load is assumed to remain strictly in the vertical direction while the structure buckles or undergoes vertical displacements. This new system is qualitatively represented on Figure 6. Again it is supposed to describe large strains by a physically realistic mechanism which could be manufactured in a laboratory 7 .

Using the same notations as those of the previous section, the above description of the system leads, through elementary calculations to the following set of equilibrium equations, giving respectively the equilibrium of the that would have been obtained by fixing Y (θ) at Y = +1 after θ λ 0 , although such a fixed position of the free boundary would also mean that elastic behavior holds everywhere. 7 Again, an additional device, not represented on Figure 6, should be added in an experimental realization. This time the structure cannot undergo global horizontal displacements contrary to the previous case, but it is crucial here again to force the springs to remain vertical by some mechanism while the system is rotating. This could be done by sufficiently many tubes clamped at their lower end in the upright position in which the springs could slide without friction.

resultant, the equilibrium of the moment and the kinematic assumption:

                   i) λ = +1 -1 σ(x)dx, ii) λL sin θ = +1 -1 xσ(x) 1 cos 2 θ dx, iii) ε(x) = u + x tan θ. (21)
In the same way as what has been given in the geometrically linearized case and in the previous geometrically non linear case, let us first assume that buckling occurs in the elastic range. The constitutive relation is then assumed to be σ = Eε everywhere on [-1, +1] and the complete bifurcation diagram is made of the two following branches

     θ = 0, λ ∈ R + , θ ∈ [0, π 2 [, λ = 2E 3L cos 3 θ , (22)
which again means that function λ(θ) is no longer constant on the non trivial branch, but in the present case tends to infinity as θ tends to π/2.

The elastic plastic constitutive law has been given in Equation ( 9) and reads:

   i) σ = E ε if σ < σy or σ < 0, i.e.
for elastic loading or for unloading,

ii) σ = E T ε if σ = σy and σ ≥ 0, i.e. for plastic loading,

where σy denotes the actual limit of elasticity of the springs and (˙) stands for the derivative with respect to the fictitious time τ introduced in the same way as in the previous section.

the quasi-static equilibrium problem

Obtained by a differentiation of problem (21) with respect to τ , the quasi-static equilibrium problem reads:

                       i) λ = 1 -1 σ(x)dx, ii) L λ sin θ + Lλ cos θ θ = 1 cos 2 θ 1 -1 σ(x)xdx + 2 sin θ cos 3 θ θ 1 -1 σ(x)xdx, iii) ε(x, λ) = u + x θ cos 2 θ . (24)
Again we choose without restriction the case of the positive rotation and Problem (24) is closed by a complementary equation which connects a free boundary Y (λ, θ) with the components u and θ of the displacement. Due to the constitutive law (24-iii) the boundary Y is given by:

ε(x, λ) = 0 that is u = -Y θ cos 2 θ . (25)
Recalling Equation (21-ii), Problem (24) together with relation (25) gives, after elementary integrations:

         λ = (E -E T ) 1 cos 2 θ - 1 2 Y 2 - E + E T E -E T Y - 1 2 θ, λ sin θ = E -E T L - 1 6 Y 3 + 1 2 Y + 1 3 E + E T (E -E T ) 1 cos 4 θ -λ cos θ -2 tan θ sin θ θ, (26)
Using notations already defined, and in particular introducing the polynomials f (Y ) and g(Y ) and the constitutive parameter β which are the same as in the previous cases, problem (26) is rewritten as: implies that function λ(θ) is also bounded in the same interval.

       i) λ = βf (Y ) 1 cos 2 θ θ, ii) λ sin θ = β L g(Y ) 1 cos 4 θ -λ cos θ -2 tan θ sin θ θ. ( 27
)
Moreover, since we are looking for solutions such that θ = 0 at θ = 0, Problem (27) leads to the following equation:

0 = -βf (Y ) sin θ cos 2 θ + β L g(Y ) 1 cos 4 θ -λ cos θ -2 tan θ sin θ . (28)
Let Φ(Y, λ, θ) be the right hand side of Equation ( 28). The derivative of Φ with respect to Y is easily obtained as

∂Φ ∂Y = -βf ′ (Y ) sin θ cos 2 θ + β L g ′ (Y ) 1 cos 4 θ = β(Y + α) sin θ cos 2 θ + β 2L (1 -Y 2 ) 1 cos 4 θ .
Then, function Φ can be inverted, giving Y as a function of λ and θ, in the same domain as in the previous case that is in:

V = θ ∈ ]0, π 2
[ , λ ∈ R + ∪ θ = 0 , λ ∈ ]λ T , λ E [ .
Only one root Y of the third degree algebraic equation (28) belongs to the interval [-1, +1] for θ = 0 so that the non trivial branches are the solutions to the following differential equation:

     dλ dθ = βf ( Y (λ, θ)) 1 cos 2 θ , λ(0) = λ 0 , λ 0 ∈ ]λ T , λ E [, (29)
where again the right hand side is not only bounded but is analytical in the domain where Equation (28) can be inverted. We shall now investigate the buckling and post-buckling behavior on the whole domain [0, π 2 [.

Incipient buckling and close post-buckling

The first part of the bifurcation analysis is exactly the same as the one of the previous case studied in Section 3, itself identical to the case of the geometrically linear case recalled at Section 2. The reason is that Equation (28) of the present geometrically non linear case taken at θ = 0 is the same as Equation ( 17) of the previous case also taken at θ = 0, both being 0 = β L g(Y ) -λ, which is also the result of the geometrically linear case given in Section 2. Then Subsection 3.3 gives a good approximation of the bifurcation diagram for the present case as long as θ is small enough.

Intermediate and asymptotic post-buckling

The evolution of the non trivial branches is such that the free boundary may be a non monotonic function of θ and may move back to x = -1 for some θ > 0. More precisely, the numerical investigation of the solutions of equation (28) gives: Lemma 4 i) All branches are such that the free boundaries move down to x = -1. ii) Let the initial data λ 0 of equation ( 29) belong to the interval [λ T , λ E ] of the trivial branch. Then the functions Y (θ) are defined on some interval [0, θ λ0 ] with 0 < θ λ0 < π/2. Moreover, the root Y (θ, λ) of equation ( 28) reaches x = -1 at θ λ0 8 . iii) Fonctions Y (θ) are non monotonic, with a maximum strictly smaller than

Y R if λ 0 ∈ [λ T , λ R [, monotonic strictly decreasing from Y (0) > Y R if λ 0 ∈ ]λ R , λ E ]
This result is close to the one obtained in the previous section, but there is nevertheless a qualitative difference since the curves Y (θ) intersect at some point θs, with 0 < θs < θ λ0 , as opposed to point iii) of Lemma 2. As a matter of fact, it is easily seen from Equation (28), that for all λ 0 ∈ ]λ T , λ E ], the curves Y (θ) intersect for θ satisfying cos θ -2 tan θ sin θ = 0 that is θ = 0.61548 whatever the value of λ. From symbolic calculations we get that Y = -0.8878 at this point 9 , which is consequently a point where equation (28) may be satisfied for any finite λ, or in other words a common position of the free boundary for the bifurcated branches. Functions Y (θ) are plotted on Figure 7. From the point of view of bifurcation theory, this point is singular in a particular way: the bifurcated branches do not intersect nor do they have a limit point, but the all the free boundaries associated with the branches intersect at the same point 10 . We can now state the global result of this section: Theorem 4 The bifurcation diagram involves the following properties: i) For any λ 0 , initial data of Equation (29) belonging to the interval ]λ R , λ E ] of the trivial branch, there exist non monotonic bifurcated branches defined on [0, θ λ0 ] having a minimum at a point θm with 0 < θm < θs. ii) For any λ 0 , initial data of equation (29) belonging to the interval [λ T , λ R ] of the trivial branch, there exist bifurcated branches defined on [0, θ λ0 ] monotonically increasing, strictly increasing if λ 0 ∈ [λ T , λ R [. iii) All bifurcated solutions tend to +∞ as θ tends to π/2. Proof 8 We keep the notation θ λ 0 already used in the previous section as no misunderstanding is possible. 9 In fact we obtain from equation ( 28), written at cos θ -2 tan θ sin θ = 0, that the third degree algebraic equation

-f (Y ) sin θ + g(Y ) 1 L cos 2 θ
= 0 has a single root Y = -0.8878 in the interval [-1, +1]. In addition, this algebraic equation shows that this result is not generic but depends on L. But numerical calculations give that this intersection point no longer exist in [0, π/2[ for very small L, which, recalling Shanley's model, would mean extremely short beams. 10 The question of whether this kind of singularity is a result of bifurcation theory for variational inequalities is very important and might be related with the fact that the free boundary itself results from variational inequalities.

Points i) and ii) deal with the solutions in [0, θ λ0 ], where the behavior is given by Equation (29). They are consequently direct corollaries of Lemma 4. In the same way as for Theorem 3 this simple result follows here from the quasi-static equations in which we insert βf (-1) = 2E T and β L g(-1) = 2E T 3L . It means that when the rotation is large enough the downward vertical displacement of the whole structure exceeds the upward displacement of the left end of the initially horizontal bar. This result for large rotations is interesting since the buckling phenomenon gave u + ζ < 0 for any smaller rotation. It implies in turn that all springs, everywhere in [-1, +1], are strictly compressed in the plastic range for large rotation.

-Let λ 0 ∈ ]λ R , λ E ],
According to Lemma 5 the springs are all in strict plastic loading in ]θ λ0 , π/2[. We may consequently assume that the continuation of the solution of Equation (29) up to π/2 is given by plasticity with no unloading, that is a formally elastic behavior but involving the tangent modulus 11 . Then for any λ 0 the branches are continued by the following functions:

λ(θ) = 2E T 3L 1 cos 3 θ - 1 cos 3 θ λ0 + λ θ0 , in ]θ λ0 , π/2[. (30)

QED

The global bifurcation diagram is represented on Figure 8 5 Concluding comments

The analysis which has been presented here aimed at understanding the coupling between two types of non linearity, one non smooth such as elastic-plasticity, and the other smooth such as large deformations in classical theories of the mechanics of structures.

The results can be summarized as follows:

-At small strains, that is in the case of incipient buckling or close post-buckling, only elastic-plasticity is involved in the bifurcation analysis. This result is emphasized by the fact that either linear strains or two different types of nonlinear strains all with the same model of plasticity lead to the same bifurcation result. We recall that contrary to elasticity of simple structures, the main qualitative property of elastic-plastic buckling is that the set of bifurcation points fills completely an interval, ranging on the real axis from the buckling load of the tangent modulus to the buckling load of the Young modulus. This interval does not depend on the geometrical non linearity. In the same way, the set of incipient buckling deflections only depends on the elastic-plastic behavior and does not depend on the geometrical non linearity.

-Quite the opposite, for very large rotations the asymptotic behavior is formally elastic, but in a specific sense which must be taken carefully: Using basic properties of elastic-plasticity, the whole analysis has been based upon the motion of a free boundary which separates the domain into a plastic loading part and an elastic unloading part, but asymptotically the free boundary is no longer involved. As a matter of fact, for sufficiently large rotations the free boundary attains the border of the domain either at one end, in which case the behavior is elastic everywhere with stresses strictly lower than the yield value for plasticity, or at the other end, in which case the behavior is plastic in strict loading everywhere. Obviously this comment is dependent on the fact that the domain is an interval, but let us look at the result that asymptotically no free boundary remains although we are dealing with incremental elastic-plasticity. Such a result could be interpreted qualitatively by saying that in some sense there remains no memory of the non smooth non linearity at very large smooth non linearity.

The present analysis has been performed in the case of a very simple discrete system having only two degrees of freedom. Although it needed a careful understanding of the mechanics, the mathematical tools were classical and simple. But extending the results to the mechanics of structures may involve important mathematical difficulties.

In elastic buckling problems it is well known that the bifurcated branches can be either subcritical or supercritical, tending asymptotically to zero or to infinity depending on the geometrical non linearity. The idea of the introduction of the two simple models of geometrical non linearity was precisely to represent qualitatively these behaviors. But in the case of elastic-plasticity, the free boundary is no longer a point in an interval but a curve or a surface which implies huge additional complexities and the problem has been carried out only in the case of a straight beam [START_REF] Elkoulani | Comportement post-critique des poutres élastoplastiques: existence et régularité des branches bifurquées[END_REF] and [START_REF] Léger | Bifurcations et stabilité en mécanique des structures[END_REF], .... which corresponds in the Kirchhoff-Love kinematics to the geometrically linear case of the simple models that have been studied here !! But although incomplete from the point of view of bifurcation in the mechanics of structures, the analysis presented here may have interesting engineering applications. As a matter of fact the important qualitative result is that the asymptotic behavior is obtained without any free boundary analysis. This means that if a constant modulus is used, the Young modulus everywhere or the tangent modulus everywhere according to the behavior of the elastic buckling of the original T-shape structure, we obtain a qualitative information of the behavior at very large strains, which is a strong simplification at least as a first approximation.

Fig. 1

 1 Fig. 1 Shanley-Hutchinson's model.

Theorem 2 Fig. 2

 22 Fig. 2 The bifurcation diagram of Shanley-Hutchinson's model.

Fig. 3 A

 3 Fig. 3 A geometrically non linear version of Shanley-Hutchinson's model.

  + α is always positive and as long as a boundary Y exists (1 -Y 2 ) is also positive since the boundary be such that |Y | ≤ 1 by definition. It is consequently readily seen that the derivative ∂Ψ ∂Y is negative or equal to zero for θ ∈ [0, π/2[ and λ ∈ R + , strictly negative for θ ∈ ]0, π/2[, and equal to zero for θ = 0 at λ = 2E T 3L := λ T and λ = 2E 3L := λ E . The implicit function theorem then asserts that Equation (18) can be inverted giving Y as a single function Y (λ, θ), for which we keep the notation Y (λ, θ) for simplicity, wherever ∂Ψ ∂Y is strictly negative. Moreover, for the free boundary to separate the interval [-1, +1] into two nonzero measure sub-intervals, Y should be strictly larger than -1 and strictly smaller than +1.

Fig. 4

 4 Fig. 4 Functions Y (θ) are defined for Y ∈ [-1, +1] and we observe that Y = +1 for 1.335 ≤ θ ≤ 1.383. The parameters of this computation are chosen as L = 1, E = 3E T = 200., i.e., Y R = -0.268.

Lemma 3 :Remark 1

 31 The free boundaries associated with the solution branches starting from any initial data λ 0 ∈ [λ T , λ E ] actually attain the border of the domain x = +1 for some sufficiently large rotation θ. Moreover, when Y (λ) reaches the point x = +1, the springs are all elastically unloaded and remain in elastic unloading for θ > θ λ0 .ProofLet δ be the part due to the rotation θ of the vertical component of the displacement at a point x > 0 of the initially horizontal bar, and recall that u stands for the vertical displacement of the middle point x = 0 of the same bar. Since the bar is rigid, the geometry gives δ = x sin θ. Under a variation λ of the external vertical load, the constitutive relation then leads to u -δ = 0 at Y = +1. Observe that βf (+1) = 2E and β L g(+1) = 2E 3L ; recall in addition that the quasi static equilibrium equation of the resultant (16-i) reads: λ = βf (Y ) cos θ θ, which becomes, for Y = +1 λ = 2E cos θ θ = 2E δ.On the other hand the constitutive law inserted into the equilibrium equation(10-i) gives the increment of the vertical displacement of the point x = 0 of the structure:λ = 2E u.QED This simple result is interesting: it means that the global motion down to the horizontal support is equal to the part of the vertical displacement of the right end of the initially horizontal bar due to the rotation.

Fig. 5 Fig. 6 A

 56 Fig. 5 The global bifurcation diagram in the {θ, λ}-plane of the geometrically non linear version of Shanley-Hutchinson's model represented on Figure 3. Three branches are plotted, issued respectively from λ 0 = λ T , λ 0 = λ R , λ 0 = λ E . The match between the left part and the right part of these curves depends on the initial data λ 0 , ranging from θ = 1.335 for λ 0 = λ E to θ = 1.383 for λ 0 = λ T .

  Y (θ) are defined on [0, 0.67]. b -Functions Y (θ) intersect at θ = 0.61548.

Fig. 7

 7 Fig. 7 From Equation (28), fonctions Y (θ) belong to [-1, +1] for θ ∈ [0, 0.67]. But the interval [0, 0.67[ includes a crossing point of all functions Y (θ).

Lemma 5

 5 then Lemma 4 states that function Y (θ) is monotonically strictly decreasing from Y 0 > Y R at θ = 0 to Y = -1 at some θ < θ λ0 . Since it is a smooth function, Y R is its single zero. Recalling the properties of function f (Y ) and equation (29) we get immediately that as long asY (θ) ∈ ]Y 0 , Y R [, function f (Y ) is negative so that function λ(θ) is strictly decreasing, while when Y (θ) ∈ ]Y R , -1[, function f (Y ) ispositive and function λ(θ) is strictly increasing. Since λ(θ) is smooth, it has a minimum at a point θm strictly included in ]0, θs[. Moreover, the minimum tends to zero as Y 0 tends to Y R . -Let us now consider the case where λ 0 ∈ ]λ T , λ R ], then Lemma 4 states that function Y (θ) is non monotonous, increasing from Y 0 ≥ -1 to some value strictly smaller than Y R , decreasing after, down to -1. Recalling the properties of function f (Y ) and equation (29) we conclude that function f (Y ) is always positive so that function λ(θ) is increasing. -Only the continuation in ]θ λ0 , π/2[ remains to be examined in order to complete the proof of Theorem 4. This requires the following lemma: Let ζ be the part due to the rotation θ of the vertical component of the displacement of the initially horizontal bar, and recall that u stands for the vertical displacement of the middle point x = 0 of the same bar. Since the bar is rigid, the geometry gives ζ = x tan θ. Under a variation λ of the external vertical load, the constitutive relation then leads to u + ζ = 0 at the end x = -1

Fig. 8

 8 Fig.8The global bifurcation diagram in the second geometrically nonlinear case. Each branch is made of two parts, the left one from θ = 0 to θ such that 0.65 < θ < 0.66 where Y = -1, which involves the free boundary Y (θ, λ), and the right part from this rotation to π/2 where the behavior is plastic loading everywhere.

  the initial data λ 0 of equation (19) belong to the interval [λ T , λ R [ of the trivial branch. Then the functions Y (θ) are monotonic, strictly increasing from an initial value Y 0 , associated with λ 0 by Lemma 1, to Y = +1. ii) Let the initial data λ 0 of equation (19) belong to the interval [λ R , λ E ] of the trivial branch. Then the functions Y (θ) are non monotonic with a minimum larger than Y R , after which they increase up to Y = +1 for some angle θ strictly smaller than π/2. iii) Let Y 01 and Y 02 with Y 01 < Y 02 be two initial positions of the free boundary and Y 1 and Y 2 the curves Y (θ) on the corresponding solutions branches. Then the curves Y 1 and Y 2 do not intersect and reach x = +1 respectively for some θ 1 and θ 2 with θ 1 > θ 2 .

  Again it is clear that Problem (21) always has the solution θ ≡ 0 for any λ ∈ R + . The study of solutions such

	that θ = 0 at θ = 0 starts from Problem (27): As a matter of fact, Y is bounded since it belongs to [-1, +1] by
	definition, then f (Y ) is bounded, so that βf (Y ) boundary Y (λ) exists, and so is consequently dλ dθ	1 cos 2 θ from Equation (27-i). Introducing this result in Equation (27-ii) is bounded on any compact of [0, π/2[ in which the free

In the case of compressed beams, this assumption simply means that the beam is short enough not to buckle in the elastic range; in the present case it means that L is small enough.

Here the tangent modulus E T is strictly positive, which is the mechanical meaning of hardening, it is assumed to be constant in the present study. Recall that E T = 0 would specify the case of the so-called perfect plasticity, and E T < 0, the mechanical meaning of which is softening, is usually associated with damage analysis

Actually, if the springs are all assumed to remain in plastic loading for any λ, then θ = 0, θ = 0 satisfies Problem[START_REF] Katchanov | Eléments de la théorie de la plasticité[END_REF] where the equation of the resultant gives λ = 2E T u; the equation of the moment reduces to 0 = 0, and the kinematics is simply ε = u

This result is quite natural, since the case with linear deformations can be seen precisely as a linearization for small θ of the present one.

It is important to stress that the free boundary does not exist in ]θ λ 0 , π/2[, the force in each spring being strictly lower than the yield value of plasticity, which means that the effective elastic continuation is completely different from the one

We have observed that such a fully plastic behavior implies θ = 0 at θ = 0, but of course θ = 0 is possible, and compatible with all conditions of the quasi-static problem, when θ = 0.

Acknowledgement The authors would like to express their warmest thanks to Professor Bernadette Miara, Université Paris-Est at Noisy-le-Grand, for valuable discussions all along this work.