Johanna Calderon
email: johanna.calderon@inserm.fr

| Nathalie Angeard

| Charlotte Pinabiaux

| Damien Bonnet

Isabelle Jambaqu

Facial expression recognition and emotion understanding in children after neonatal open-heart surgery for transposition of the great arteries

Transposition of the great arteries ToM Theory of mind TREFE Test of Facial Recognition of Emotions for Children AIM Theory of mind impairments are part of the cognitive morbidities associated with transposition of the great arteries (TGA). We sought to assess core components of social cognition in school-aged children with TGA. METHOD Thirty-eight children with neonatal corrected TGA (27 males, 11 females; mean age 7y 3mo, SD 1y 2mo) and a comparison group (n=31; 24 males, 7 females; mean age 7y 4mo, SD 1y 1mo) participated in this study. All children completed measures of facial expression recognition, emotion comprehension, and second-order cognitive and affective false-belief tasks. The association of medical pre-, intra-, and postoperative variables with cognitive outcomes was explored. RESULTS After controlling for potential covariates, children with TGA performed significantly less accurately in the mental category of the emotion comprehension battery (p=0.002) and on second-order affective false-belief tasks (p<0.05). Preoperative variables including an associated ventricular septal defect (p=0.02), a younger age at open-heart surgery (p=0.03), and a prenatal diagnosis of TGA (p=0.02) were significantly associated with better outcomes.

INTERPRETATION School-aged children with TGA display significant impairment on complex affective mental state understanding even though facial expression recognition was generally preserved. Preoperative factors may be important determinants for long-term outcomes after cyanotic congenital heart disease.

[ToM] tasks) as they had significant difficulties on making correct inferences on the false-beliefs held by others and on understanding and predicting their behaviour. [START_REF] Calderon | Executive function and theory of mind in school-aged children after neonatal corrective cardiac surgery for transposition of the great arteries[END_REF][START_REF] Calderon | Impact of prenatal diagnosis on neurocognitive outcomes in children with transposition of the great arteries[END_REF] Adolescents with TGA also had significantly worse performances on a test of complex inner state attribution (reading the mind in the eyes test) and exhibit more autistic traits than expected in the general population. [START_REF] Bellinger | Adolescents with d-transposition of the great arteries corrected with the arterial switch procedure: neuropsychological assessment and structural brain imaging[END_REF] These findings opened a new window for the study of neurocognitive consequences associated with CHD; however, important questions remain concerning the extent and generalization of these difficulties to the broad domain of social cognition.

Social cognition is a multifaceted concept that includes elementary processes such as the recognition of facial expressions of emotion and more complex abilities referring to the comprehension of mental and affective states. [START_REF] Adolphs | Neural systems for recognizing emotion[END_REF] In typical development, early improvements on basic facial emotion recognition are reported in preschool-aged children. [START_REF] Vicari | Recognition of facial expressions of emotions in school-age children: the intersection of perceptual and semantic categories[END_REF] This progress parallels the comprehension of some affective states, such as the understanding that people may Term infants born with cyanotic congenital heart disease (CHD), including transposition of the great arteries (TGA) have been reported to have significant disturbances in cerebral oxidative metabolism [START_REF] Miller | Preoperative brain injury in newborns with transposition of the great arteries[END_REF] and mild to moderate white matter injuries primarily in the form of periventricular leukomalacia before and after neonatal open heart surgery. [START_REF] Licht | Brain maturation is delayed in infants with complex congenital heart defects[END_REF][START_REF] Petit | Preoperative brain injury in transposition of the great arteries is associated with oxygenation and time to surgery, not balloon atrial septostomy[END_REF] Several medical and patient-related risk factors including pre-, intra-, and postoperative variables may contribute to their neurological status and long-term neurocognitive outcomes. [START_REF] Wernovsky | Central nervous system outcomes in children with complex congenital heart disease[END_REF] Despite overall normal general intelligence, [START_REF] Bellinger | Neurodevelopmental status at eight years in children with dextrotransposition of the great arteries: the Boston Circulatory Arrest Trial[END_REF] longterm cognitive impairments as well as a high prevalence of attention-deficit-hyperactivity disorder, [START_REF] Shillingford | Inattention, hyperactivity, and school performance in a population of school-age children with complex congenital heart disease[END_REF] 'externalizing' problems (e.g. aggressive behaviour), and 'internalizing' difficulties (i.e. social withdrawal) [START_REF] Bellinger | Are children with congenital cardiac malformations at increased risk of deficits in social cognition?[END_REF] have been reported. Recent studies have also demonstrated significant impairments in understanding other people's inner emotional states, suggesting deficits in social cognition. [START_REF] Bellinger | Behaviour at eight years in children with surgically corrected transposition: the Boston Circulatory Arrest Trial[END_REF] Compared with children with typical development, children with TGA have been reported to display impairments on mental state attribution (first-and second-order theory of mind hold different desires [START_REF] Wellman | Scaling of theory-of-mind tasks[END_REF] or that emotions may have external causes (e.g. if somebody gets a present for their birthday, how would they feel?). [START_REF] Pons | Emotion comprehension between 3 and 11 years: developmental periods and hierarchical organization[END_REF] Cognitive mental state attribution also undergoes significant progress during this period as children can recognize first-order false-beliefs. [START_REF] Carlson | Individual differences in executive functioning and theory of mind: an investigation of inhibitory control and planning ability[END_REF] Furthermore, in middle childhood, at around 7 to 8 years, children are able to understand the possibility of hiding emotions (e.g. showing a happy face but feeling sad inside), [START_REF] Pons | Emotion comprehension between 3 and 11 years: developmental periods and hierarchical organization[END_REF] and they come to understand second-order falsebeliefs with affective contents (e.g. Paul thinks that Mary feels) followed by epistemic contents (Paul thinks that Mary thinks). [START_REF] Parker | John thinks that Mary feels..' False belief in children across affective and physical domains[END_REF][START_REF] Miller | Children's understanding of second-order mental states[END_REF] Finally, at around 9 to 11 years, they begin to master mixed emotions (e.g. feeling happy and sad at the same time) and some aspects of morality (e.g. feeling guilt after telling lies). [START_REF] Pons | Emotion comprehension between 3 and 11 years: developmental periods and hierarchical organization[END_REF] From a neurofunctional point of view, emotion recognition and comprehension in typical development involves networks including the amygdala [START_REF] Somerville | Behavioral and neural representation of emotional facial expressions across the lifespan[END_REF] and the prefrontal cortex. [START_REF] Brink | The role of orbitofrontal cortex in processing empathy stories in 4-to 8year-old children[END_REF] Furthermore, concomitant deficits in facial expression recognition and emotion understanding as well as in affective and cognitive ToM have been found in 6-to 8-year-old children after traumatic brain injury [START_REF] Tonks | Reading emotions after childhood brain injury: case series evidence of dissociation between cognitive abilities and emotional expression processing skills[END_REF] and in children with attentiondeficit-hyperactivity disorder. [START_REF] Fonseca | Emotion understanding in children with ADHD[END_REF] Despite the growing concern regarding systematic neurocognitive screening and remediation strategies for children with CHD, [START_REF] Marino | Neurodevelopmental outcomes in children with congenital heart disease: evaluation and management: a scientific statement from the American Heart Association[END_REF] no data concerning the extent and nature of social cognition impairments including emotional processing are available.

The objective of this study was to assess core components of social cognition including elementary aspects such as basic facial expression recognition (perceptual identification and labelling), as well as main milestones of emotion comprehension organized hierarchically according to their developmental path of acquisition (e.g. from desires to moral feelings comprehension). In addition, as school-aged children with TGA have been reported to have significant delays on the litmus test of ToM (cognitive false-belief attribution), [START_REF] Calderon | Executive function and theory of mind in school-aged children after neonatal corrective cardiac surgery for transposition of the great arteries[END_REF] we sought to investigate if delays would also be observed for same-structured tests with an affective component (second-order false-belief tasks with a positive or a negative emotional outcome). In typical development, affective components in second-order false-belief understanding seem to facilitate children's performances. [START_REF] Parker | John thinks that Mary feels..' False belief in children across affective and physical domains[END_REF] We sought to assess if children with TGA would also benefit from this affective effect. Finally, we investigated potential neonatal medical risk factors including pre-, intra-, and postoperative variables associated with emotional and cognitive outcomes.

METHOD Participants

All eligible children born between 2003 and 2005 at the French referral centre for complex congenital heart disease were enrolled in a prospective study of neurocognitive outcomes after neonatal open-heart surgery for D-transposition of the great arteries. No magnetic resonance imaging data were collected. Inclusion criteria included a diagnosis of TGA with or without a ventricular septal defect (VSD) requiring one arterial switch operation under fullflow cardiopulmonary bypass (CPB) support (mean CPB duration 132min; SD 22min) with no requirements for deep hypothermic circulatory arrest (mean age at operation 7d). Exclusion criteria were birthweight less than 2500g, the presence of genetic syndromes, complex associated cardiovascular anomalies including aortic arch reconstruction or extracardiac pathologies as well as additional surgical procedures. All exclusion criteria information was extracted from medical records. Among children who met one or more of these criteria, two were excluded because of the presence of a 22q11 deletion. Genetic screening was conducted if clinical concerns were reported. Normal cardiac condition, parent's consent to participation, and geographic location (Paris region) were also taken into account for children's enrolment in the study. Forty-five eligible children were initially enrolled and first examined in 2009 (see Calderon et al. [START_REF] Calderon | Impact of prenatal diagnosis on neurocognitive outcomes in children with transposition of the great arteries[END_REF] for further information on this cohort), of which seven were lost to follow-up for the present study. Thirty-eight children with TGA participated in this study (11 females, 27 males; mean age 7y 3mo, SD 1y 2mo). Medical related variables were examined. These included time of diagnosis (prenatal vs postnatal), presence or absence of a ventricular septal defect, birthweight, gestational age, Apgar score at 5 minutes, preoperative metabolic acidosis (pH<7.2), the need for balloon-atrial septostomy, age at arterial switch operation, total bypass time, total cross-clamp time, and total postoperative intensive care unit stay. No significant differences were found between participants and non-participants for all medical related variables (p>0.05).

Forty-five comparison children were also initially enrolled and first examined at a mean age of 5 years, of whom 14 were lost to follow-up for the present study. Comparison children were recruited from public kindergartens and elementary schools in the same geographic area as children with TGA over the same time period. One or two classes per school were randomly chosen and exclusion criteria were a known developmental or physical impairment, confirmed or suspected intellectual disability, and/or mother language other than French. Parents of all children meeting the inclusion criteria received a letter of information and a consent form. A ratio of one female to three males was applied to match TGA sex ratio by means of simple randomization among females in each classroom. Eighty per cent of eligible families gave consent for participation. Data presented here are from the last evaluation of What this paper adds

• Children with TGA do not display significant deficits in facial emotion recog- nition.

• Children with TGA display significant delays on age-expected measures of emotion comprehension.

• Affective false-belief understanding is also altered for a significant percent- age of children with TGA.

• Preoperative variables may play a significant role in the development of so- ciocognitive abilities after TGA.

scenario. A short story is read about the depicted character (s), whose face was left blank, and children were asked to attribute an emotion to the character by pointing to the correct outcome below. Participants' responses were nonverbal and closed. No feedback was given. The general level of emotion comprehension is measured by the number of correct answers for all components (range 0-9), which are also grouped into three main categories according to their level of complexity. The first category 'external emotions' includes the recognition of facial expressions, understanding of the causes of emotions, and the impact of external events on emotions. The second category 'mental emotions' includes the comprehension of desires, affective first-order false belief and the distinction between outwardly shown and inner emotions. The third and more complex category 'reflective emotions' focuses on the comprehension of concurrent mixed feelings, strategies for emotional regulation, and morality. For each category, scores can range from 0 (all components are failed) to 3 (all components are passed).

False-belief tasks

One first-and three-second-order false-belief tasks were used. All tasks were mimed using small puppets in a standardized scenario. The two most common versions of the standard false-belief task (first-and second-order) [START_REF] Wimmer | Beliefs about beliefs: representations and constraining function of wrong beliefs in young children's understanding of deception[END_REF][START_REF] Perner | John thinks that Mary thinks that…': attribution of second-order beliefs by 5-to 10year-old children[END_REF] were proposed to evaluate children's ability to attribute cognitive mental states to a character. Two second-order affective false-belief tasks adapted from the standard version were selected in order to assess the children's ability to attribute mental emotional states. [START_REF] Parker | John thinks that Mary feels..' False belief in children across affective and physical domains[END_REF] In these tasks, two stories involving a positive and a negative affective outcome for the main character were presented. Children had to infer the character's false belief about a second character's emotions (How does Peter think Mary feels now?). For all tasks, control questions and one test question per story were asked. Responses were non-verbal for the cognitive false-belief tasks where children pointed to one of the two possible locations, and forced-choice verbal for the emotional false-belief tasks where two possible outcomes were given (e.g. He thinks she feels happy or sad?). Test responses were coded in a binary fashion for all tasks (0 for failure, 1 for success). No feedback was given.

Data analyses

Group (children with TGA vs comparison children) differences were analyzed using Student's t-test for independent measures or non-parametric v 2 tests for numerical and categorical variables. Statistical analyses were performed separately on scores on the TREFE labelling and matching tasks, using a mixed analysis of variance (ANOVA) with group as a between-participants factor, and emotion (happiness vs fear vs anger vs disgust vs sadness vs neutral) as a within-participant factor. A one-way ANOVA was computed to elicit the effect of group on the number of categories achieved on the TEC. Analyses of covariance (ANCOVAs) were then used to determine the possible our cohort at a mean age of 7 years. A total of 31 comparison children were re-evaluated (seven females, 24 males; mean age 7y 4mo, SD 1y 1mo). Parents of all comparison children included in this study provided information on children's health status, medical antecedents, and demographic variables. No abnormal data were reported for any child in the comparison group. Parental educational level and socio-economic status (SES) were recorded for all children. The hospital ethics committee approved this study and parental consent was obtained for participation.

Neuropsychological assessment

Global Non-verbal IQ was evaluated using the Columbia Mental Maturity Scale [START_REF] Burgemeister | Columbia Mental Maturity Scale[END_REF] and language was explored with the comprehension subtest from the Neuropsychological Assessment Battery (NEPSY). [START_REF] Korkman | NEPSY: A Developmental Neuropsychological Assessment Manual[END_REF] Facial expression recognition was assessed with the Test of Facial Recognition of Emotions for Children (TREFE). [START_REF] Golouboff | Impaired facial expression recognition in children with temporal lobe epilepsy: impact of early seizure onset on fear recognition[END_REF][START_REF] Pinabiaux | Impaired emotional memory recognition after early temporal lobe epilepsy surgery: the fearful face exception?[END_REF] The Test of Emotion Comprehension (TEC) [START_REF] Pons | Emotion comprehension between 3 and 11 years: developmental periods and hierarchical organization[END_REF][START_REF] Pons | Longitudinal change and longitudinal stability of individual differences in children's emotion understanding[END_REF] was used to assess different components of emotion understanding with gradual complexity. Affective false-belief was evaluated with a positive and a negative outcome task. [START_REF] Parker | John thinks that Mary feels..' False belief in children across affective and physical domains[END_REF] Cognitive false-belief was evaluated with a standard first-and second-order falsebelief task. [START_REF] Wimmer | Beliefs about beliefs: representations and constraining function of wrong beliefs in young children's understanding of deception[END_REF][START_REF] Perner | John thinks that Mary thinks that…': attribution of second-order beliefs by 5-to 10year-old children[END_REF] All tests are commonly used in clinical and experimental settings with typical and atypical populations and have age-standardized scores based on scores of the general population reported in the literature. The first author (JC), a trained neurodevelopmental psychologist, conducted all assessments. Details are presented in Appendix S1 (online supporting information).

Test of Facial Recognition of Emotions for Children (TREFE)

The TREFE [START_REF] Golouboff | Impaired facial expression recognition in children with temporal lobe epilepsy: impact of early seizure onset on fear recognition[END_REF][START_REF] Pinabiaux | Impaired emotional memory recognition after early temporal lobe epilepsy surgery: the fearful face exception?[END_REF] is composed of 102 faces, 17 per emotional expression (happiness, fear, anger, disgust, sadness, neutrality). The tasks began with six training trials where children received positive or negative feedback depending on their accuracy. No feedback was given during the further trials. Assessment order remained fixed across participants and no time limit was set. First, during the labelling task, each face is presented one by one to the participant, who is asked to label the emotion expressed by the face. Then, a verbal matching task is proposed to control for correct visuo-perceptive processing of facial emotional expressions. The participant is asked to choose the faces that express the three emotions enunciated successively by the experimenter. Six faces (one per emotion, three targets and three foils) are presented at a time. For both tasks, accuracy is evaluated by the number of correct answers per emotion (/8) and a total score of emotion recognition is computed (/48) for the labelling task and the verbal matching task separately.

Test of Emotion Comprehension (TEC)

The TEC [START_REF] Pons | Emotion comprehension between 3 and 11 years: developmental periods and hierarchical organization[END_REF][START_REF] Pons | Longitudinal change and longitudinal stability of individual differences in children's emotion understanding[END_REF] is administered by using a picture book with cartoon scenarios in the upper section and four possible facial emotional expressions among five possible outcomes (sad, angry, scared, happy, and neutral) beneath each confounding effect of language comprehension, age, SES, and sex on the group effects for both TREFE and TEC measures. Post-hoc comparisons were conducted using the Bonferroni test with a p value of <0.05 threshold. Proportions of children who succeeded or failed on the falsebelief tasks were compared between groups using non-parametric v 2 tests. Multiple logistic regression analyses for group differences in false-belief tasks controlled for language, age, SES, and sex. McNemar v 2 tests for paired groups were used to compare success rates between affective and non-affective second-order false-belief tasks. Influences of medical variables including all pre-, intra-, and postoperative factors were analyzed using multivariate regression analysis and multiple logistic regression analysis depending on the nature of the score analyzed (numeric for the TREFE and TEC, dichotomic for the ToM tasks). We reported non-standardized effect sizes for mean differences, 95% confidence intervals (CIs) for all group comparisons and standard errors (SE) for linear regression analyses. Finally, when data were missing, a method of case-wise deletion was used. Statistical analyses were performed with Statistical software (version 9.1; StatSoft Inc., Tulsa, OK, USA).

RESULTS

Demographic characteristics, IQ, and language scores

As shown in Table I, children with TGA did not significantly differ from comparisons for SES scores, father's educational level, and mother's educational level. Age was not significantly different between groups and neither was sex. Non-verbal IQ levels were within normal ranges for all children and did not significantly differ between groups. Raw language comprehension scores were significantly lower for children with TGA. When converted to normalized scores reported by the NEPSY for 7-year-olds, however, children with TGA had a mean standard score of 11, which is considered within the normal range.

Test of facial emotion recognition (TREFE)

Analyses of variance indicated that children with TGA had a significantly lower total score (mean [SD]) on the TREFE labelling task (41.5 [2.58]) than comparison children (42.8 [2.02]; F 1,66 =5.35, p=0.02, mean difference À0.27; 95% CI À2.4 to À0.13). However, when confounding variables (age, SES, sex, and language scores) were included in an ANCOVA, this difference between the groups was no longer significant (F 1,51 =1.80, p=0.18, 95% CI À0.51 to 0.10). Moreover, there was no significant difference between groups on the TREFE matching task for the group with TGA (45.4 [2.75]) and for comparisons (45.9 [1.98]; F 1,66 =1.01, p=0.32, mean difference À0.50, 95% CI À1.68 to 0.68).

Results show that accuracy on the recognition of negative emotions such as sadness and disgust was significantly worse for both groups at both parts of the TREFE for the labelling task (F 5,330 =62.59, p<0.001) and for the matching task (F 5,330 =11.02, p<0.001, Tukey HSD p<0.01) indicating that these two negative emotions were generally more difficult to identify for all children.

Test of emotion comprehension (TEC)

Results indicate that the general level of emotion comprehension as measured by the mean number of components (0-9) correctly achieved is significantly lower in the group of children with TGA (6.7 [1.22]) than in the comparison group (7.4 [0.91]; F 1,67 =5.4, p=0.02, mean difference À0.61, 95% CI À1.14 to À0.08). ANCOVA controlling for age, language scores, SES, and sex showed a significant effect of language (F 1,54 =7.79, p=0.008, 95% CI 0.03-0.24) but no significant effect of age (F 1,54 =0.80, p=0.37, 95% CI À0.17 to 0.44), SES (F 1,54 =0.11, p=0.73, 95% CI À0.33 to 0.23), sex (F 1,54 =1.05, p=0.30, 95% CI À0.15 to 0.48), and no differences between the groups (F 1,54 =2.47, p=0.12, 95% CI À0.57 to 0.06).

Components of the TEC were grouped into three categories of gradual complexity. Results from the ANOVA with post-hoc testing (Bonferroni test) indicated that performances of children with TGA (2.4 [0.60]) are significantly lower than comparisons' (2.9 [0.37]) only for the mental emotions category (F 3,65 =4.22, p=0.002, mean difference À0.39, 95% CI À0.63 to À0.14). This difference persisted in an ANCOVA after controlling for age, language scores, SES, and sex (F 1,54 =6.19, p=0.01, 95% CI À0.34 to À0.03). Results for all three categories are shown in Figure 1.

False-belief tasks

All children from the comparison group succeeded at the first-order ToM task compared with (30 children) of the children with TGA (v 2 (1)=7.38, p=0.006; see Fig. 2).

There was no significant difference in success rates between the groups for the cognitive second-order falsebelief task. The percentage of children succeeding at the positive affective second-order false-belief task was significantly lower in the group with TGA (v 2 (1)=4.18, p=0.04, 95% CI 0.02 to 1.0). Similarly, in the negative affective second-order false-belief task, the percentage of children with TGA succeeding was significantly lower (v 2 (1)=6.29, p=0.01, 95% CI 0.14 to 1.14). Logistic regression analyses also revealed a significant effect of language comprehension score on performances on the first-order task (v 2 (1)=4.13, p=0.04, 95% CI À1.16 to À0.14) and on the negative affective false belief (v 2 (1) =6.59, p=0.01, 95% CI À0.63 to À0.08) tasks. Age, SES, and sex did not have a significant effect on performances on any ToM task. Finally, success rates on the positive and negative affective second-order false-belief tasks were significantly higher compared with the higher order cognitive false-belief task for both groups (respectively McNemar test (1)=8.59, p=0.003 and McNemar test (1)=9.5, p=0.002).

Associated medical factors

Multivariate regression analysis including all medicalrelated variables indicated that the presence of a VSD (b=0.48, SE=0.20, p=0.02, R²=0.32) and a younger age at the arterial switch operation were significantly associated with better scores at the TREFE labelling task (b=À0.59, SE=0.25, p=0.03, R²=0.56). Multiple logistic regression analysis showed that prenatal diagnosis of TGA is significantly associated with better performance at the positive emotion false-belief task (v 2 (1)=5.22, p=0.02, 95% CI À3.83 to À0.29). No other significant associations were found between cognitive scores and medical factors.

DISCUSSION

In recent times there has been a growing concern about social cognition impairments after CHD, [START_REF] Marino | Neurodevelopmental outcomes in children with congenital heart disease: evaluation and management: a scientific statement from the American Heart Association[END_REF][START_REF] Bellinger | Theory of mind deficits in children with congenital heart disease[END_REF] yet no research to date has investigated the extent of these deficits. The aim of the current study was to assess the recognition of facial expressions and the gradual comprehension of emotions, as well as the understanding of affective and cognitive false-belief. When compared with the comparison group, children with TGA had significantly lower scores on the test of facial expression labelling despite normal identification of the same emotions when the verbal label was explicitly given. When a series of potential covariates (age, language scores, SES, and sex) was controlled, however, this difference was no longer significant. In typical development, the visuo-perceptive recognition of emotion facial expressions precedes the capacity to produce and correctly retrieve verbal labels for these emotions. [START_REF] Vicari | Recognition of facial expressions of emotions in school-age children: the intersection of perceptual and semantic categories[END_REF] The current results suggest that facial expression recognition generally may be preserved after TGA. Moreover, children with TGA did not exhibit an atypical pattern of performance according to the type of emotion. As observed in typical early development, [START_REF] Gao | A happy story: developmental changes in children's sensitivity to facial expressions of varying intensities[END_REF] negative emotions are less accurately recognized than positive ones and this also applies to our cohort with TGA.

Results demonstrated that children with TGA achieved significantly fewer components in the TEC than comparison children. However, when language comprehension scores, age, SES, and sex were controlled, this group difference disappeared. Indeed, it has been reported that language comprehension levels may explain a significant portion of variance of this test in normal development. [START_REF] Pons | Individual differences in children's emotions understanding: effects of age and language[END_REF] When children's performances were compared according to each category specifically, however, the group with TGA performed significantly less well than the comparison group for the 'mental emotions' category even after controlling for all covariates mentioned including language scores. Children with TGA exhibit significantly lower scores on components such as dealing with desires, affective first-order false-belief or understanding concealed emotions, normally acquired at around 6 to 7 years of age. [START_REF] Pons | Longitudinal change and longitudinal stability of individual differences in children's emotion understanding[END_REF] According to the gradual complexity of the TEC, our cohort displayed normal scores on the most elementary category taxing external emotions usually achieved at around 5 years of age. [START_REF] Pons | Longitudinal change and longitudinal stability of individual differences in children's emotion understanding[END_REF] This developmental pattern indicates that basic processing of emotion is not impaired and difficulties apply to age-expected milestones. Thus, their performances were not significantly lower than those observed in the comparison group for the 'reflective emotions' category, which is expected to be acquired later in development. This suggests that children with TGA do not necessarily have an atypical early onset of complex emotion comprehension as their performances were similar to comparison children in the reflective category at age 7. Instead, it could be hypothesized that developmental lags may become apparent once these abilities have consolidated enough to age-expected levels, even if it is still unclear how these weakness may evolve in the long term. Bellinger et al. [START_REF] Bellinger | Adolescents with d-transposition of the great arteries corrected with the arterial switch procedure: neuropsychological assessment and structural brain imaging[END_REF] reported that 16-year-olds with TGA are significantly impaired in 'reading the mind in the eyes' test, suggesting that the processing of complex cognitive or affective mental states may have long lasting difficulties. Our data showed that success rates on first-order cognitive false-belief and second-order affective (positive and negative) false-belief were significantly lower for children with TGA and are in accordance with previous findings showing significant impairments in cognitive first-and second-order false-belief understanding in 7-year-old children with TGA. [START_REF] Calderon | Executive function and theory of mind in school-aged children after neonatal corrective cardiac surgery for transposition of the great arteries[END_REF] Taken together, these results add to the evidence for a potential generalization of impairments to emotional and affective components of social cognition. It is worth noting, however, that success rates on affective secondorder false-belief understanding were significantly higher than those on a cognitive, yet similarly structured, secondorder false-belief task. This could mean that the emotional outcome of the task facilitates inner states attribution for children with TGA, comparable to what is observed in typical development. [START_REF] Parker | John thinks that Mary feels..' False belief in children across affective and physical domains[END_REF] The success rate of our comparison children at the cognitive second-order task (29%) was lower than that reported in a previous study on typical development (in Parker et al. [START_REF] Parker | John thinks that Mary feels..' False belief in children across affective and physical domains[END_REF] 78% at age 7) and lower than our previous cohort of typical 7-year-olds. [START_REF] Calderon | Executive function and theory of mind in school-aged children after neonatal corrective cardiac surgery for transposition of the great arteries[END_REF] This could partially be explained by the inclusion of 6-year-olds in the current study, who systematically failed at this task. It is commonly admitted that typical children can pass second-order false-belief tests between the ages of 7 years and 9 years. [START_REF] Perner | John thinks that Mary thinks that…': attribution of second-order beliefs by 5-to 10year-old children[END_REF] It is also important to note that second-order ToM tasks, especially standard cognitive versions, [START_REF] Perner | John thinks that Mary thinks that…': attribution of second-order beliefs by 5-to 10year-old children[END_REF] have been reported to entail high language comprehension and information processing requirements. Failure at this task could reflect not only difficulties in ToM conceptual understanding per se, but also deficits in decoding and integrating this information as a result of limited linguistic and/or executive abilities. [START_REF] Miller | Children's understanding of second-order mental states[END_REF] An important step in dissociating these deficits would be to test children with a linguistically facilitated version of second-order tasks that have been reported to elicit better performances in children with typical development. [START_REF] Coull | Simplifying secondorder belief attribution: what facilitates children's performances on measures of conceptual understanding[END_REF] Neural correlates of emotion processing including decoding facial emotion expressions and emotional ToM in typical development have been reported to involve cortical and sub-cortical structures including the amygdala, [START_REF] Somerville | Behavioral and neural representation of emotional facial expressions across the lifespan[END_REF] the orbitofrontal cortex, 20 medial prefrontal cortex, and the right temporo-parietal junction. [START_REF] Saxe | Brain regions for perceiving and reasoning about other people in school-aged children[END_REF] These studies have reported an increased neural specialization of these structures [START_REF] Saxe | Brain regions for perceiving and reasoning about other people in school-aged children[END_REF] and, more importantly, a crucial role of neural connectivity as white matter volumes progressively increase with age during childhood. [START_REF] Durston | Anatomical MRI of the developing human brain: what have we learned?[END_REF] The functional relationships between regions underlying higher-order cognitive skills such as social cognition may require the integrity of association white matter tracts, as has been proposed in adult models. [START_REF] Philippi | Damage to association fiber tracts impairs recognition of the facial expression of emotion[END_REF][START_REF] Carrington | Are there theory of mind regions in the brain? A review of the neuroimaging literature[END_REF] Term infants with cyanotic CHD requiring neonatal interventions have been reported to present white matter anomalies and delays in myelination, similar to what can be observed in preterm infants. [START_REF] Miller | Preoperative brain injury in newborns with transposition of the great arteries[END_REF][START_REF] Licht | Brain maturation is delayed in infants with complex congenital heart defects[END_REF] No study to date has established an association between neonatal white matter disturbances and long-term neurocognitive outcomes in cardiac patients. It is critical to understand the impact that these early anomalies could have on the development of complex cognitive skills known to rely on intact brain connectivity throughout childhood.

Our data showed that among the pre-, intra-, and postoperative variables examined, preoperative factors only (presence of a VSD, younger age at the arterial switch operation, and a prenatal diagnosis of TGA) were significantly associated with better cognitive results on facial expression recognition and affective false-belief. An associated VSD plays a crucial role in cyanosis tolerance in the newborn infant as it allows for mixing of the arterial and pulmonary circulations and, therefore, improvement of oxygen saturation before cardiac surgery. [START_REF] Bonnet | Detection of transposition of the great arteries in fetuses reduces neonatal morbidity and mortality[END_REF] Younger age at the arterial switch operation has been associated with a reduced risk for white matter injury in the form of periventricular leukomalacia in newborns with TGA and overall preoperative brain injury seems related to severity and duration of exposure to hypoxaemia in these patients. [START_REF] Petit | Preoperative brain injury in transposition of the great arteries is associated with oxygenation and time to surgery, not balloon atrial septostomy[END_REF] Finally, our results are in accordance with previous data showing a beneficial impact of prenatal diagnosis on neurocognitive outcomes in preschool-aged children with TGA. [START_REF] Calderon | Impact of prenatal diagnosis on neurocognitive outcomes in children with transposition of the great arteries[END_REF] Indeed, prenatal diagnosis of cyanotic CHD is associated with reduced neonatal morbidity including better haemodynamic stability and neurological status as it reduces the odds of prolonged cyanosis before cardiac surgery. [START_REF] Bonnet | Detection of transposition of the great arteries in fetuses reduces neonatal morbidity and mortality[END_REF] Data from this study may suggest concomitant vulnerabilities in all core aspects of social cognition including ageexpected emotion-processing abilities. Nevertheless, several limitations should be considered. As we sought to limit potential medical confounding variables including multiple cardiac operations and chronic cyanosis, we focused on patients who had TGA as they constitute a homogeneous group with cyanotic CHD and no associated genetic syndromes. Our results cannot be generalized to all children with CHD as variations in haemodynamic instability and operative management should be noted. Our study does not allow for investigation of the neuropathology related to cognitive dysfunction because no brain imaging was available for our patients. In addition, as ToM is also dependent on language development, [START_REF] Astington | Why Language Matters for Theory of Mind[END_REF] future evaluations should systematically test this factor not only as a control variable but also including less demanding ToM tasks in terms of linguistic abilities. Moreover, observational studies (contrary to randomized controlled trials) may be subject to biases, notably participant selection and multiple confounding variables. We provided traditional statistical methods of adjustment for baseline covariates in all our models. Covariance analyses, however, should be conducted with caution and future studies in this population should consider the use of propensity score methods for optimal bias reduction. [START_REF] D'agostino | Tutorial in biostatistics, Propensity score methods for bias reduction in the comparison of a treatment to a non-randomized control group[END_REF] Finally, future studies should determine the association of social cognition and emotion processing impairments assessed with neuropsychological testing to more ecological parental reports of children's functioning. This is particularly relevant as children with heart disease may display high levels of behavioural and emotional problems, [START_REF] Bellinger | Behaviour at eight years in children with surgically corrected transposition: the Boston Circulatory Arrest Trial[END_REF][START_REF] Spijkerboer | Long-term behavioural and emotional problems in four cardiac diagnostic groups of children and adolescents after invasive treatment for congenital heart disease[END_REF] indicating that daily life social functioning may be vulnerable in these patients. Recently, a scientific statement from the American Heart Association [START_REF] Marino | Neurodevelopmental outcomes in children with congenital heart disease: evaluation and management: a scientific statement from the American Heart Association[END_REF] categorized children requiring open-heart surgery as at high risk for developmental disorders and recommended formal evaluations for these children. Social cognition screening should also be routinely conducted in all patients with CHD at high-risk, as deficits could affect social and emotional outcomes. Remediation or preventive interventions based on formal cognitive training such as 'thought-bubble ToM training' including pictures to depict mental states visually [START_REF] Paynter | Further evidence of benefits of thought-bubble training for theory of mind development in children with autism spectrum disorders[END_REF] or interventions based on reinforcement of executive functions, [START_REF] Benson | Individual differences in executive functioning predict preschoolers' improvement from Theory-of-Mind training[END_REF][START_REF] Diamond | Preschool program improves cognitive control[END_REF] may help reduce social cognition deficits after CHD.

Figure 1 :Figure 2 :

 12 Figure 1: Mean number of components achieved in each category of emotions (external, mental, and reflective) for children with transposition of the great arteries (TGA) versus the comparison group. *p<0.05.

Table I :

 I Demographic characteristics of children with transposition of the great arteries (TGA) and comparison individuals

		Group with TGA	Comparison
	Variable	(n=38)	group (n=31)	p a	Mean difference (95% CI)
	Age, y:mo	7:3 (1:2)	7:4 (1:1)	0.72	À0.08 (À0.56 to 0.39)
	Sex, male/female, n	27/11	24/7	0.54	-
	Father's educational level b	3.37 (1.56)	3.83 (0.97)	0.20	À0.46 (À1.18 to 0.25)
	Mother's educational level b	3.38 (1.67)	3.16 (1.50)	0.58	0.21 (À0.56 to 0.99)
	SES c	2.15 (0.91)	2.09 (0.97)	0.78	0.06 (À0.39 to 0.51)
	Non-verbal IQ	113 (8.4)	116 (9.6)	0.22	À2.73 (À7.21 to 1.73)
	Language comprehension d	21.1 (1.87)	22.8 (2.6)	0.01	À1.71 (À3.13 to À0.28)

Values are mean (SD) apart from sex. a Independent samples t-tests and v 2 tests; b Parental educational level was scored according to the French National Education Diploma classification (0=primary school education, middle education; 1=high school diploma; 2=college undergraduate; 3=college graduate⁄masters degree; 4=postgraduate⁄doctoral degree) c SES was scored according to the index from the French National Demographic Statistical Institute. Scores were grouped into four categories: 'very low socio-economic status (SES)=0', 'low SES=1', 'intermediate SES=2', and 'high SES=3'; d Language was tested using the Neurological Assessment Battery language comprehension subtest. Raw scores are reported here (range 0-28).

ACKNOWLEDGEMENTS

We are grateful to all families who participated in this study. This study was supported by the Association pour la Recherche en Cardiologie du Foetus a l'Adulte (ARCFA) and by a university doctoral fellowship to JC.

SUPPORTING INFORMATION

The following additonal material may be found online:

Appendix S1: Details of assessments used in the study.


F Auricchio

A Constantinescu

C Menna

G Scalet

A shakedown analysis of high cycle fatigue of shape memory alloys

Keywords: Shape memory alloys, Lifetime prediction, Dang Van fatigue criterion, Shakedown, High cycle fatigue

Shape memory alloys (SMAs) are exploited in several innovative applications, experiencing up to millions of cycles, and thus requiring a fully understanding of material fatigue and fracture resistance. However, experimental and methodological descriptions of SMA cyclic response are still incomplete. Accordingly, the present paper aims to investigate the cyclic response of SMAs under macroscopic elastic shakedown and to propose a criterion for the high cycle fatigue of SMAs. A multiaxial criterion based on a multiscale analysis of the phase transformation between austenite and martensite and using the rigorous framework of standard generalized materials is proposed. The criterion is an extension of the Dang Van high cycle fatigue criterion to SMAs. The criterion is applied to uniaxial experimental data taken from the literature. It distinguishes run out from failure tests in the infinite lifetime regime. The sound structure of the underlying concepts permits a novel insight into the development of a general multiaxial failure criterion for SMA materials.

INTRODUCTION

Shape memory alloys (SMAs) possess unique properties, known as shape memory effect and pseudoelasticity. These properties result from reversible diffusionless solid-solid transformations (known as martensitic transformations) between a relatively ordered parent phase, called austenite (A), and a less ordered product phase, called martensite (M).

Shape memory effect and pseudoelasticity are successfully exploited in many fields, e.g., structural engineering, automotive, aerospace, micro-electromechanical, robotics, and biomedical industry [START_REF] Jani | A review of shape memory alloy research, applications and opportunities[END_REF][START_REF] Lagoudas | Shape Memory Alloys: Modeling and Engineering Applications[END_REF]. In particular, a wide segment is covered by SMA actuation systems [START_REF] Mertmann | Design and application of shape memory actuators[END_REF] and by innovative devices for the control of civil structures [START_REF] Asgarian | Seismic response of steel braced frames with shape memory alloy braces[END_REF]. Within the biomedical industry, self-expandable vascular stents represent a considerable part of SMA applications for mini-invasive techniques [START_REF] Petrini | Biomedical Applications of Shape Memory Alloys[END_REF]. The question of lifetime prediction and of the improvement of the alloys with respect to this aspect is a major topic in the field [START_REF] Chluba | Ultralow-fatigue shape memory alloy films[END_REF][START_REF] James | Taming the temperamental metal transformation[END_REF].

The rather complex micromechanical behaviour of SMAs also induces unusual fracture and fatigue responses when compared with polycrystalline metallic alloys [START_REF] Mahtabi | Fatigue of nitinol:the state-of-the-art and ongoingchallenges[END_REF][START_REF] Robertson | Mechanical fatigue and fracture of Nitinol[END_REF]. It has already been discussed, for instance by [START_REF] Tabanli | Mean stress effects on fatigue of NiTi[END_REF][START_REF] Tabanli | Mean strain effects on the fatigue properties of superelastic NiTi[END_REF], that classical fatigue criteria cannot be directly applied, due to the uncertain role of the phase transformation under cyclically varying deformations and of the stress and/or thermally-induced microstructural evolution of the different phases [START_REF] Robertson | Mechanical fatigue and fracture of Nitinol[END_REF]. Indeed, transformations between austenitic and martensitic phases, moving martensite interfaces, accumulation of dislocations are believed to play an important role in the fatigue lifetime of SMAs [START_REF] Barney | Impact of thermomechanical texture on the superelastic response of nitinol implants[END_REF][START_REF] Pelton | Nitinol Fatigue: A Review of Microstructures and Mechanisms[END_REF].

The prediction of crack initiation and growth under thermo-mechanical cyclic loading is an essential requirement for the design of novel SMA components [START_REF] Rahim | Bending rotation HCF testing of pseudoelastic Ni-Ti shape memory alloys[END_REF], since fatigue failure has emerged as one of the main design issues [START_REF] Azaouzi | Optimization based simulation of self-expanding Nitinol stent[END_REF][START_REF] Hibbert | Coronary stent fracture[END_REF]. As an example, SMA actuators are subjected to thermal cycling and are expected to undergo at least 10 4 -10 5 cycles during their service life [START_REF] Karhu | Long-term behaviour of binary Ti49.7Ni (at.%) SMA actuators -the fatigue lives and evolution of strains on thermal cycling[END_REF][START_REF] Strittmatter | Long-time stability of ni-ti-shape memory alloys for automotive safety systems[END_REF]. For SMA cables used as damping prevention in stay cable, suspension, and prestressed concrete bridges, fatigue life is usually taken into account considering the frequency range of vibration on real scale bridges, i.e. 5-20 Hz, and a number of working cycles up to 5 • 10 6 [START_REF] Menna | Chapter 13 -applications of shape memory alloys in structural engineering[END_REF]. In the majority of the biomedical applications, stents are permanently implanted in the human body and experience millions of in-vivo cycles due to blood pressure; stents should survive at least for 10 years without exhibiting failure, which translates into 4 • 10 8 service cycles (US Food and Drug Administration, 2010).

In order to prevent premature failure of SMA components, it is firstly necessary to verify whether they will shakedown elastically or by alternating phase transformation, or will fail by alternating plasticity. The shakedown regions in a typical fatigue diagram are usually associated with high and low cycle fatigue regime [START_REF] Constantinescu | A unified approach for high and low cycle fatigue based on shakedown concepts[END_REF]. Figures 1(a Several experimental investigations and fatigue methodologies have analyzed both SMA structural fatigue (component failure) and functional fatigue (the evolution of shape memory effect and pseudoelasticity under repeated thermo-mechanical cycles); see [START_REF] Robertson | Mechanical fatigue and fracture of Nitinol[END_REF] as review article.

Experimental investigations are generally coupled with observations to track the nucleation and evolution of martensite and austenite during mechanically unstable regimes with the final aim of characterizing the material fatigue response on a microscopic and even macroscopic level [START_REF] Brinson | Stress-induced transformation behavior of a polycrystalline niti shape memory alloy: micro and macromechanical investigations via in situ optical microscopy[END_REF][START_REF] Creuziger | Fracture in single crystal NiTi[END_REF][START_REF] Gall | Effect of microstructure on the fatigue of hot-rolled and cold-drawn NiTi shape memory alloys[END_REF][START_REF] Gloanec | Fatigue crack initiation and propagation of a TiNi shape memory alloy[END_REF][START_REF] Kim | Martensite strain memory in the shape memory alloy Nickel-Titanium under mechanical cycling[END_REF][START_REF] Lackmann | Highresolution in-situ characterization of the surface evolution of a polycrystalline NiTi SMA-alloy under pseudoelastic deformation[END_REF][START_REF] Merzouki | Coupling between measured kinematic fields and multicrystal SMA finite element calcula-tions[END_REF][START_REF] Pelton | Nitinol Fatigue: A Review of Microstructures and Mechanisms[END_REF][START_REF] Treadway | Tensile and fatigue behavior of superelastic shape memory rods[END_REF][START_REF] Yin | Effect of deformation frequency on temperature and stress oscillations in cyclic phase transition of NiTi shape memory alloy[END_REF]. Experimental observations have also inspired a series of fatigue approaches aimed to estimate the lifetime, as a macroscopic crack initiation criterion. Most of the studies focus on stress-or strain-life SMA fatigue approaches for different types of uniaxial tensile loading, e.g., [START_REF] Gupta | High compressive pre-strains reduce the bending fatigue life of nitinol wire[END_REF][START_REF] Kang | Whole-life transformation ratchetting and fatigue of super-elastic NiTi Alloy under uniaxial stress-controlled cyclic loading[END_REF][START_REF] Maletta | Fatigue of pseudoelastic NiTi within the stress-induced transformation regime: a modified Coffin-Manson approach[END_REF][START_REF] Pelton | Rotary-bending fatigue characteristics of medical-grade Nitinol wire[END_REF][START_REF] Wang | Phase transformation behavior of pseudoelastic NiTi shape memory alloys under large strain[END_REF], while only few focus on the torsional fatigue loading of SMAs, e.g., [START_REF] Predki | Cyclic torsional loading of pseudoelastic NiTi shape memory alloys: Damping and fatigue failure[END_REF][START_REF] Runciman | An equivalent strain/Coffin-Manson approach to multiaxial fatigue and life prediction in superelastic Nitinol medical devices[END_REF].

Concerning available failure criteria, although uniaxial ones may fail to accurately predict the lifetime of devices under multiaxial loading conditions, only few multiaxial fatigue criteria exist for SMAs. It is worth mentioning the works by [START_REF] Moumni | Fatigue analysis of shape memory alloys: energy approach[END_REF][START_REF] Moumni | Cyclic behaviour and energy approach of the fatigue of Shape Memory Alloys[END_REF] and [START_REF] Morin | Cyclic behavior and fatigue design of shape memory alloy devices[END_REF] who firstly proposed an energy approach, where the dissipated energy of the pseudoelastic hysteresis cycle was used as a parameter for lifetime estimation. Recently, [START_REF] Hartl | Three-dimensional constitutive model considering transformation-induced damage and resulting fatigue failure in shape memory alloys[END_REF] proposed a constitutive model describing SMA behaviour undergoing a large number of cycles, coupled with a continuum theory which includes an internal damage evolving into final failure. These approaches focus on the cyclic alternating phase transformation behaviour of SMAs. Only few works have been proposed to extend the shakedown theorems for elasto-plastic materials to SMA structures, see, e.g., [START_REF] Feng | Shakedown analysis of shape memory alloy structures[END_REF][START_REF] Peigney | Shakedown theorems and asymptotic behaviour of solids in nonsmooth mechanics[END_REF][START_REF] Peigney | Shakedown theorems and asymptotic behaviour of solids in nonsmooth mechanics[END_REF][START_REF] Peigney | On shakedown of shape memory alloys structures[END_REF][START_REF] Pham | On shakedown theory for elasticeplastic materials and extensions[END_REF][START_REF] Wu | Modeling phase fraction shakedown during thermomechanical cycling of shape memory materials[END_REF]. To the authors' knowledge, no works address the fatigue analysis of SMA elastic shakedown, even though such a loading condition is very frequent in various applications [START_REF] Robertson | Mechanical fatigue and fracture of Nitinol[END_REF]. Motivated by the above considerations, the present paper focuses on the cyclic response of SMAs, under the elastic shakedown regime, and proposes a multiaxial criterion for the high cycle fatigue of SMAs. The derivation starts from the following considerations: such criterion should (i) predict high cycle fatigue crack initiation;

(ii) be based on a multiscale analysis taking into account the complexity of the phase transformation between austenite and martensite; (iii) be multiaxial.

The Dang Van-Papadopoulos criteria (Dang Van et al., 1989;[START_REF] Papadopoulos | Fatigue polycyclique des métaux: une nouvelle approche[END_REF] belong to the class of fatigue criteria fulfilling the above conditions. Their merit comes from the underlying fundamental concepts of shakedown and standard generalized materials applied at the grain level in metallic polycrystals. Such criteria have have been successfully applied to both infinite and finite lifetime in the high cycle fatigue regime (Ferjani et al., 2011a;[START_REF] Papadopoulos | Long life fatigue under multiaxial loading[END_REF][START_REF] Van | On some recent trends in modelling of contact fatigue and wear in rail[END_REF]Wackers et al., 2010b). The homogenization assumptions relating the grain scale plasticity with the macroscopic behavior have been discussed in [START_REF] Bertolino | A multiscale approach of fatigue and shakedown for notched structures[END_REF][START_REF] Hofmann | Numerical exploration of the dang van high cycle fatigue criterion: application to gradient effects[END_REF]. They have permitted to extend consistently the criterion to finite lifetime [START_REF] Bosia | Fast time-scale average for a mesoscopic high cycle fatigue criterion[END_REF]Morel, 2000a) as well as to explain the scatter of fatigue experiments and to explore the effect of loading path [START_REF] Guerchais | Micromechanical study of the loading path effect in high cycle fatigue[END_REF] to include the presence of stress concentrations created by defects [START_REF] Guerchais | Micromechanical study of the loading path effect in high cycle fatigue[END_REF]. Moreover, the coherent thermodynamic foundation permitted to relate plastic dissipation with self-heating [START_REF] Charkaluk | Dissipative aspects in high cycle fatigue[END_REF][START_REF] Luong | Fatigue limit evaluation of metals using an infrared thermographic technique[END_REF] and it opened innovative techniques for lifetime predictions [START_REF] Doudard | Identification of the scatter in high cycle fatigue from temperature measurements[END_REF][START_REF] Doudard | Determination of an hcf criterion by thermal measurements under biaxial cyclic loading[END_REF][START_REF] Poncelet | Probabilistic multiscale models and measurements of self-heating under multiaxial high cycle fatigue[END_REF].

The aim of this paper is to derive a similar thermodynamically consistent framework for the analysis of fatigue in polycrystalline SMAs. Generally, polycrystalline SMAs present a complex microstructure made of austenite and/or martensite, which appear in the form of plates, inclusions, or grains. Therefore, three different length scales can be considered, corresponding to different systems: (i) the microscopic scale of microstructures (pellets, inclusions, bands, etc.) formed in each grain; (ii) the mesoscopic scale of individual grains; and (iii) the macroscopic scale of the polycrystalline material, consisting in several grains [START_REF] Peigney | A non-convex lower bound on the effective energy of polycrystalline shape memory alloys[END_REF]. Here, we consider classical homogenization assumptions to evaluate the mesoscopic mechanical quantities from the macroscopic ones and we perform the shakedown analysis by using the recent theorems by [START_REF] Peigney | Shakedown theorems and asymptotic behaviour of solids in nonsmooth mechanics[END_REF][START_REF] Peigney | Shakedown theorems and asymptotic behaviour of solids in nonsmooth mechanics[END_REF][START_REF] Peigney | On shakedown of shape memory alloys structures[END_REF], obtained for a large class of SMA constitutive laws within the generalized standard materials framework.

The shakedown and multiscale analysis is then applied to a series of uniaxial fatigue experiments taken from literature [START_REF] Pelton | Nitinol Fatigue: A Review of Microstructures and Mechanisms[END_REF]. A constitutive law of the Souza-Auricchio type [START_REF] Auricchio | A three-dimensional model describing stresstemperature induced solid phase transformations: solution algorithm and boundary value problems[END_REF]Souza et al., 1998a,b) is identified on the specific material and the complete mechanical behaviour is computed for all the test cases. The proposed fatigue criterion successfully predicts failure or run out of the tested specimens.

The analysis leads to a new insight in the analysis of the fatigue phenomena in SMA materials and it opens a new path for the development of a general multiaxial failure criterion for this class of materials and the manufacturing of new alloys.

The present paper is organized as follows. Section 2 presents a review of the thermodynamic framework of the constitutive laws and the shakedown theorems for SMAs.

Then, Section 3 introduces the Dang Van type criterion for SMAs. Section 4 presents the results of its application to experiments from the literature. Conclusions and summary are finally given in Section 5. A scheme of the numerical algorithm for the computation of the fatigue criterion is provided in Appendix A.

THERMODYNAMIC FRAMEWORK AND SHAKEDOWN THEOREM FOR

SMAs

This section reviews the constitutive laws and the shakedown theorem for SMAs in the framework of generalized standard materials [START_REF] Halphen | Sur les materiaux standards generalises[END_REF].

Thermodynamic framework

Assuming a small strain regime, the initial configuration of the local SMA material state is described by the total strain " " ", the temperature ✓, and an internal variable ↵.

The variable ↵ represents the inelastic strain and can include the description of several physical phenomena characterizing SMA behaviour, ranging from permanent plasticity and phase transformations, up to void generation and fracture [START_REF] Peigney | Shakedown theorems and asymptotic behaviour of solids in nonsmooth mechanics[END_REF].

According to [START_REF] Halphen | Sur les materiaux standards generalises[END_REF], the constitutive model is defined in terms of the free energy w and the pseudo-potential of dissipation , from which the stress tensor and the thermodynamic force A associated to ↵ are derived.

The free energy is assumed to depend on the total strain " " ", the temperature ✓, and the internal variable ↵, i.e., w = w (" " ", ✓, ↵), as follows:

w = 1 2 (" " " ↵) : C : (" " " ↵) + f (1)
where f = f (✓, ↵) is a positive differentiable function, describing the energy contribution associated to the internal variable and temperature variations. Here and in the following, the stiffness tensor C is assumed to depend on the component of the internal variable describing phase transformation.

The free energy describes the relation between the state and internal variables " " " and ↵ and their conjugate quantities and A, as follows:

8 > > < > > : = @w @" " " = C : (" " " ↵) A = @w @↵ = 1 2 (" " " ↵) : dC d↵ : (" " " ↵) @f @↵ (2)
The evolution of the internal variable ↵ is described in terms of the time derivative

↵ by using the pseudo-potential of dissipation = ( ↵), which is a positive convex functional vanishing at the origin.

It is then convenient to introduce the complementary pseudo-potential of dissipation ⇤ = ⇤ (A), which is the Legendre-Fenchel transform of . Such a step requires the introduction of the indicator function I P (A):

⇤ = I P (A) = 8 > < > : 0 if A 2 P +1 otherwise (3)
where

P = {A | F (A)  0}
is the set of admissible thermodynamic forces, described in terms of the limit function F . In classical plasticity F represents the yield limit, while it describes phase transformation and/or permanent plasticity in the case of SMAs.

The evolution equation for ↵ in terms of ⇤ is derived as [START_REF] Frémond | Non-smooth Thermomechanics[END_REF][START_REF] Peigney | Shakedown theorems and asymptotic behaviour of solids in nonsmooth mechanics[END_REF]:

↵ 2 @ ⇤ (A) = @I P (A) (4)
while the evolution equation for A in terms of is obtained as follows:

A 2 @ ( ↵) + @I T (↵) (5)
The internal variable ↵ is constrained to take values in a convex and closed subset T , with the term @I T (↵) representing the subdifferential of the indicator function I T (↵), defined to enforce the constraint on ↵, as follows:

I T (↵) = 8 > < > : 0 if ↵ 2 T +1 otherwise (6)
Then, the thermodynamic force A associated to ↵ is defined as follows:

8 > > > > < > > > > : A = A d + A r A d 2 @ ( ↵) A r 2 @I T (↵) (7)
where A d represents the dissipative force and A r the non-dissipative force reacting to the internal constraint on ↵.

Shakedown analysis

In the reviewed modeling of the thermodynamic framework, the concept of shakedown is essential for systems undergoing a given cyclic loading history. Recently, [START_REF] Peigney | Shakedown theorems and asymptotic behaviour of solids in nonsmooth mechanics[END_REF] has addressed the asymptotic behaviour of non-smooth mechanical systems and provided the necessary and sufficient conditions ensuring elastic shakedown. This was accomplished by bounding the mechanical dissipation, which is actually the extension of the original line of thought of [START_REF] Koiter | General problems for elastic solids[END_REF][START_REF] Nguyen | On shakedown analysis in hardening plasticity[END_REF] to non-smooth mechanics. It is worth pointing out that shakedown theorems reported by [START_REF] Peigney | Shakedown theorems and asymptotic behaviour of solids in nonsmooth mechanics[END_REF] are general and do apply to all SMA models entering the class of generalized standard materials. Within this family, we can cite the models proposed by Let us consider a SMA structure under cyclic loading. Its response is described in terms of state and internal variables (" " ", ✓, ↵) and the conjugate quantities ( , A). If the structure attains an elastic shakedown state at time t > t 0 , then ↵ (t) and A (t) are constant, meaning that the response of the structure is elastic around a fixed residual stress field triggered by a fixed inelastic strain characterizing the phase transformation and/or permanent plasticity. It is therefore convenient to consider the elastic response of the structure, defined by the fictitious elastic stress hystory e .

The theorem by [START_REF] Peigney | Shakedown theorems and asymptotic behaviour of solids in nonsmooth mechanics[END_REF] establishes the following result:

Shakedown theorem. If there exists a positive coefficient m > 1, a time t 0 , and a time-independent field A r ⇤ such that

F (m e (t) A r ⇤ )  0 for all t > t 0 (8)
then there is elastic shakedown, whatever the initial condition is.

Here, e = C" " " e is the fictitious elastic response of the system and A r ⇤ the residual stress or eigenstress generated by the inelastic processes of phase transformation and/or irreversible plasticity.

The shakedown theorem actually states that the structure will attain a shakedown state under a given cyclic loading, provided that the fictitious elastic stress path can be included in the limit domain defined by the limit function F , translated by a residual stress field A r ⇤ . For the numerical computations of Section 4 we will employ, as SMA constitutive model, the three-dimensional phenomenological model originally presented by Souza et al. (1998a,b) and then investigated by Auricchio and[START_REF] Auricchio | A three-dimensional model describing stresstemperature induced solid phase transformations: solution algorithm and boundary value problems[END_REF]Auricchio et al. (2009b) (denoted in the following as the Souza-Auricchio model). The model assumes a deviatoric second-order tensor representing the transformation strain associated to transformations between austenite and martensite as internal variable, i.e.

↵ = e tr . Such an internal variable is constrained to satisfied the inequality ke tr k  " L , where " L is a parameter related to the maximum transformation strain reached at the end of the transformation during a uniaxial test. We further define the martensite volume fraction as z = ke tr k/" L . Indeed, z varies between 0 (fully austenite) and 1 (fully martensite).

The free-energy and the dissipation pseudo-potentials are thus defined as:

8 > > > > > < > > > > > : w = 1 2 (" " " e tr ) : C : (" " " e tr ) + h✓ ✓ ⇤ i ke tr k + 1 2 hke tr k 2 = Rk ėtr k ⇤ = I P (A) (9)
Here, h is a positive parameter related to material hardening during phase transformation; a positive material parameter related to the dependence of the critical stress on temperature; ✓ ⇤ the temperature below which only martensite phase is stable; the notation h•i denotes the positive part function, while || • || denotes the Euclidean norm.

The set of admissible thermodynamic forces P = {A | F (A)  0} is described in terms of the limit function F , taken in the form:

F = ||A|| R (10)
with R the positive radius of the elastic domain. Recall that the thermodynamic force A is the work-conjugate to the deviatoric transformation strain and it is thus indicated as deviatoric transformation stress.

For the Souza-Auricchio model, the shakedown theorem (see inequality ( 8)) becomes:

Shakedown theorem (Souza-Auricchio model). If there exists a positive coefficient m > 1, a time t 0 , and a time-independent field A r ⇤ such that

||ms e A r ⇤ ||  R for all t > t 0 ( 11
)
then there is elastic shakedown, whatever the initial condition is.

Here, s e is the deviatoric part of e .

FATIGUE CRITERION POSTULATION FOR SMA INFINITE LIFETIME

Macro-meso passage

In high cycle fatigue, dissipation can be considered negligible and, therefore, it is accepted that only few grains of the SMA material undergo inelastic deformations whilst most of the material remains elastic. Hence, a multiscale analysis is applied to relate the macroscopic with the microscopic variables, i.e. between the scale of the structure and that of the material grains (Dang [START_REF] Van | On a New Multiaxial Fatigue Limit Criterion: Theory and Application[END_REF]Morel, 2000b;[START_REF] Papadopoulos | Long life fatigue under multiaxial loading[END_REF]; this situation is schematically represented in Figure 2 for SMAs.

The material point at the macroscopic scale is considered as a representative elementary ) , ( ε σ

ε σ, ε σ, V V MACROSCOPIC SCALE MESOSCOPIC SCALE ) , ( ε σ A" M"
A+M" The loading of the REV by macroscopic stress , strain " " ", and inelastic strain ↵ is computed through standard continuum theory (macroscale). To evaluate the mesoscopic state, i.e. mesoscopic stress ˆ , strain " " ", and inelastic strain ↵, several homog-

A ≈ M

A + M enization techniques have been considered in the literature for metallic structures [START_REF] Bui | Relations entre grandeurs microscopiques et macroscopiques[END_REF][START_REF] Cano | Critere de fatigue polycyclique pour des materiaux anisotropes: application aux monocristaux[END_REF]; Dang [START_REF] Van | On a New Multiaxial Fatigue Limit Criterion: Theory and Application[END_REF]. In order to obtain a simple closed formula for the fatigue criterion, we adopt a Lin-Taylor's homogenization scheme [START_REF] Lin | Analysis of elastic and plastic straines of a FCC crystal[END_REF][START_REF] Taylor | Plastic strain in metals[END_REF], based on the equality of macroscopic and mesoscopic strains:

" " " = " " " (12)
From the expressions of the elastic Hooke's law at the macroscopic and mesoscopic scale, we deduce [START_REF] Cano | Critere de fatigue polycyclique pour des materiaux anisotropes: application aux monocristaux[END_REF]; Dang [START_REF] Van | High-cycle metal fatigue in the context of mechanical design[END_REF]:

ˆ = A : A : C : ( ↵ ↵) (13)
where the fourth-order tensor A is the localization tensor. As commonly accepted in fatigue problems (Dang [START_REF] Van | Sur la résistance á la fatigue des métaux[END_REF][START_REF] Van | High-cycle metal fatigue in the context of mechanical design[END_REF]; Dang [START_REF] Van | On a New Multiaxial Fatigue Limit Criterion: Theory and Application[END_REF][START_REF] Maitournam | A multiscale fatigue life model for complex cyclic multiaxial loading[END_REF]Morel, 2000b;[START_REF] Papadopoulos | Long life fatigue under multiaxial loading[END_REF], it is assumed A = I, I being the fourth-order identity tensor. As stated previously, the macroscopic stiffness tensor C and consequently the mesoscopic stiffness tensor Ĉ depend on the components of the internal variables ↵ and ↵ describing phase transformation of SMA. A similar dependence is also assumed, respectively, for the macroscopic L and mesoscopic L compliance tensors.

The Lin-Taylor scheme is known to provide a stiff response in the homogenization theory [START_REF] Zaoui | Taylor-type modelling of polycrystal plasticity[END_REF]. However, since the results are here used to justify measures of the stress path and not intrinsic values, it will not affect the obtained fatigue predictions, as shown in the examples discussed in the following. Such a choice has already been tested in case of polycrystalline metals [START_REF] Hofmann | Numerical exploration of the dang van high cycle fatigue criterion: application to gradient effects[END_REF].

Fatigue criterion

In high cycle fatigue, when the loading is close to infinite lifetime, a commonly accepted assumption is that each point material is in an elastic shakedown state. Therefore, ↵ and ↵ are constant during the cyclic loading (see Section 2.2). Then, according to the shakedown theorem (see Eq. ( 8)), a time-independent field A r ⇤ can be defined such that:

ˆ (t) = (t) A r ⇤ with A r ⇤ = C : ( ↵ ↵) (14)
As it can be observed, given the computed (t), the mesoscopic stress tensor ˆ is known at each time if the residual stress A r ⇤ is known. The residual stress A r ⇤ can not be directly computed without the exact knowledge of the microstructure and its evolution up to the shakedown state. As in the formulation of the Dang Van criterion by [START_REF] Papadopoulos | Fatigue polycyclique des métaux: une nouvelle approche[END_REF], we shall compute the smallest hypersphere encompassing the path of the deviatoric stress s. Then, A r ⇤ is the center of such hypersphere, that is the solution of the following min-max problem:

A r ⇤ = min A r 1 max t ks (t) A r 1 k (15)
The mesoscopic shear stress ⌧ (t) can be calculated once computed ˆ from Eq. ( 14):

⌧ (t) = ˆ I (t) ˆ III (t) 2 (16)
where ˆ I , ˆ II , ˆ III are the principal stresses, with ˆ I ˆ II ˆ III . Similarly, the mesoscopic hydrostatic stress can be computed as follows:

ˆ h (t) = 1 3 tr( ˆ (t)) (17)
In the case of ↵ deviatoric (as in the case of the Souza-Auricchio model), it yields h = ˆ h (Dang [START_REF] Van | On a New Multiaxial Fatigue Limit Criterion: Theory and Application[END_REF].

Since it is well known that the hydrostatic part of the stress plays an important role in crack opening, evolution of damage, and implicitly in fatigue lifetime [START_REF] Schijve | Fatigue of Structures and Materials[END_REF], we propose to consider a fatigue criterion, defined by: Dang Van (DV) fatigue criterion for SMAs. Let us consider a structure subjected to cyclic loading and resulting in an elastic shakedown state at both macroscopic and mesoscopic scale. If

max t {⌧ (t) + a (↵) ˆ h }  b (↵) (18)
for all points of the structure, then fatigue crack initiation will not occur.

The safety domain in the stress space ( h , ⌧ ) is delimited by a straight line, denoted next as DV line. Figure 4 shows the splitting of the stress space in the infinite and finite lifetime by the DV line (red). The lifetime of a component will be infinite if the stress path lies below the DV line or will be finite if at least one of the points lies above the same line (see the grey dotted curve of Figure 4); if any point of the loading path is located above the DV line, life will be finite and fatigue crack initiation will occur (see the black curve of Figure 4).

Parameters a and b, introduced in Eq. 18 and defining the DV line, are material constants usually identified on classical uniaxial fatigue experiments in tension and/or torsion (Dang [START_REF] Van | High-cycle metal fatigue in the context of mechanical design[END_REF]. As a and b are computed from experiments at N cycles, we shall use the notation a N and b N , where subscript N stands for the number of run-out cycles of the experiments used to calibrate the model (Ferjani et al., 2011a,b;Wackers et al., 2010a). Moreover, such parameters are assumed to depend on the internal variable ↵, as verified by the numerical simulations presented in Section 4. This allows to obtain a set of DV lines in the stress space ( h , ⌧ ) for fixed number of cycles N . Each line is defined by an internal variable ↵. As an illustrative example, a set of DV lines, each defined for a fixed value of the martensite volume fraction 0 < z < 1 and number of cycles N , is represented in Figure 5.

To evaluate the mesoscopic stress path ( h , ⌧ ) we employ an algorithm similar to that employed for the classical DV criterion [START_REF] Bernasconi | Efficient algorithms for calculation of shear stress amplitude and amplitude of the second invariant of the stress deviator in fatigue criteria applications[END_REF]. In particular, the criterion is computed as post-processing of the mechanical fields obtained by the simulation described in Section 4 and the post-process is performed with the Matlab implementation of the optimization code SDPT3 [START_REF] Tütüncü | Solving semidefinite-quadratic-linear programs using SDPT3[END_REF]. Details about the algorithm are given in Appendix A.

EXAMPLE OF SMA APPLICATIONS

Experimental data and model calibration

The papers by Pelton and coworkers [START_REF] Pelton | Nitinol Fatigue: A Review of Microstructures and Mechanisms[END_REF][START_REF] Pelton | Fatigue testing of diamond-shaped specimens[END_REF][START_REF] Pelton | Fatigue and durability of Nitinol stents[END_REF] report detailed experimental results from fatigue data collected on NiTi microdogboneand diamond-shaped specimens, as well as on stents. For the sake of completeness of tubing, subsequently expanded and thermally shape-set into their final dimensions. The design and manufacturing conditions led to an austenite finish (A f ) temperature of 20 C, i.e. comparable to the A f temperature of other NiTi self-expanding peripheral stents [START_REF] Pelton | Nitinol Fatigue: A Review of Microstructures and Mechanisms[END_REF]. Such uniaxial fatigue experiments in tension will allow to calibrate the DV parameters a and b.

τ! $↓ℎ ! ^! h 0 < z i < 1! fixed N!
The Souza-Auricchio model is herein adopted to simulate the experimental tests and Table 1 reports the model parameters, identified according to the procedure reported in (Auricchio et al., 2009a). The experimental monotonic pseudoelastic stressstrain curve at 37 C (i.e. at the temperature of the considered fatigue tests) is reported in Figure 6 (red dotted line), superimposed on the predicted one (blue line).

With regard to fatigue test conditions, the microdogbone specimens were prestrained at 37 C to 9% (beyond the stress plateau) and then fatigued from the unloading plateau. Particularly, the specimens were cycled, either up to fracture or up to 10 7 cycles (run-out), with various combinations of mean strain and strain amplitude. For all the mean strain-strain amplitude cases reported in Figure 7(a), we first try to establish which data are possibly related to the elastic shakedown condition, i. From the numerical simulations we also obtain that the several combinations of mean strain and strain amplitude give rise to different values of the martensite volume fraction z. As it can be observed from Figure 7(b), the formation of martensite is responsible for an increasing of the elastic shakedown domain between 1 and 6% mean strain. Table 2 reports the computed values of z and along with the Young's modulus E = E(z) for the fatigue cases numbered in Figure 7(b).

Fatigue analysis

We start by considering the cases of Table 2 showing a fully martensitic transformation (z = 1), i.e. fatigue loading is applied on a fully transformed material. In particular, we consider the experimental points numbered from 7 to 17 in Figure 7(b).

As an example, Figures 8 and 9 report some representative loading curves in terms of macroscopic stress and strain.

The DV criterion is then implemented to predict failure for the considered cases.

Figure 10 represents the DV diagram in terms of mesoscopic shear stress, ⌧ , and hydrostatic stress, h , where the loading paths generated by the simulations of the experimental specimens are represented. The obtained V-shape is a consequence of the simulation of the considered uniaxial tests. As it can be observed in Figure 10, material behavior in the DV diagram is similar to that classically observed for metals. In particular, by increasing the strain amplitude at constant mean strain, an increase of the only mesoscopic shear stress is determined; see, e.g., cases 7, 8, and 9 in Figure 10. Similarly, by increasing the mean strain at constant strain amplitude determines the increase of the only hydrostatic stress; see, e.g., cases 7, 10, and 13 in Figure 10.

The calibrated DV line (green line) is also represented in Figure 10. The DV parameters, a 10 7 and b 10 7 , are calibrated by using the loading paths related to cases 11 It should be remarked that the calibrated DV parameters refer to 10 7 cycles (i.e. N = 10 7 ).

Once determined the DV line, we can verify the other fatigue experimental results

by [START_REF] Pelton | Nitinol Fatigue: A Review of Microstructures and Mechanisms[END_REF]. As it can be observed in Figure 10, cases 7, 8, 9, 12, and 17 do not 2, showing a fully martensitic transformation (z = 1).

Macroscopic stress-strain curve for 7% mean strain and 0.6% strain amplitude (case 7); 8% mean strain and 0.4% strain amplitude (case 11); 9% mean strain and 0.05% strain amplitude (case 17). 2, showing a fully martensitic transformation (z = 1).

Macroscopic stress-strain curve for 0.6% strain amplitude and different mean strains: 8% for case 10 and 9% for case 13.

result in failure, while cases 10, 11, 13, 14, 15, and 16 fail. The predicted results are in agreement with experiments of Figure 7(a). Now, we consider the cases reported by [START_REF] Pelton | Nitinol Fatigue: A Review of Microstructures and Mechanisms[END_REF] which show the coexistence of austenitic and martensitic phases (0 < z < 1), as reported in Table 2, i.e. referring to fatigue loading acting on the unloading plateau. In particular, we consider the experimental points numbered from 1 to 6 in Figure 7 2 showing a fully martensitic transformation (z = 1).

two representative loading curves in terms of macroscopic stress and strain. 2 showing the coexistence of austenitic and martensitic phases (0 < z < 1). Macroscopic stress-strain curve for 3% mean strain and 0.4% strain amplitude (case 2); 6% mean strain and 0.6% strain amplitude (case 4).

Figure 12 represents the DV diagram in terms of mesoscopic shear stress, ⌧ , and hydrostatic stress, h , where the loading paths generated by the simulations of the experimental cases are represented. As it can be observed, in such a mixed domain, material behavior in the DV diagram presents significant differences compared to that classically observed for metals. In particular, by increasing the strain amplitude at constant mean strain, it triggers the increase of both mesoscopic shear and hydrostatic stresses and the decrease of martensite fraction (see Table 2); e.g., cases 2 and 3 in Figure 12. Increasing the mean strain at constant strain amplitude determines the decrease of both mesoscopic shear and hydrostatic stresses and the increase of martensite fraction (see Table 2); see, e.g., cases 1, 3, and 6 or 2, 5 in Figure 12.

The DV line (green line) is also represented in Figure 12. The line is assumed to have the same slope of that obtained for the range 7-9% mean strain, i.e. a 10 7 does not depend on z. On the contrary, b 10 7 parameter depends on z, which ranges between 0 and 1. This means that fatigue limit varies with z, but not with triaxiality. Such an assumption (standard for metallic alloys [START_REF] Fares | Determining the life cycle of bolts using a local approach and the dang van criterion[END_REF]Ferjani et al., 2011a)) is arbitrary due to the absence of more experimental data (failed specimens) and will need to be investigated in future works. In fact, experimental data reveal that no fracture is attained for mean strains between 1 and 6% and strain amplitudes between 0.2 and 0.6%, thus indicating that there are insufficient data for a complete analysis within this range to provide the b 10 7 value.

Figures 13(a) and (b) show the dependency of the mesoscopic shear stress ⌧ and hydrostatic stress h on the martensite fraction z, respectively, for the cases 1-6. Note that ⌧ and h show the same dependence on z: cases with equal strain amplitude (1, 3, and 6 or 2 and 5) show a decrease of ⌧ and h with increasing mean strain and z;

cases with equal mean strain (2 and 3, or 4, 5, and 6) show a decrease of ⌧ and h with decreasing strain amplitude and increasing z.

The obtained results are in accordance with experimental observations by [START_REF] Pelton | Nitinol Fatigue: A Review of Microstructures and Mechanisms[END_REF], demonstrating that, between 1 and 7% mean strain, NiTi-based SMAs can accommodate larger strain amplitude, for a given fatigue life. Therefore, above a mean strain of 1% and within elastic shakedown, the fatigue life appears to increase with increasing mean strain, thus indicating that the formation of stress-induced martensite may be responsible for such a behavior [START_REF] Pelton | Nitinol Fatigue: A Review of Microstructures and Mechanisms[END_REF]. This implies that microstruc- 2 showing the coexistence of austenitic and martensitic phases (0 < z < 1).

tural effects due to formation of stress-induced martensite (whose volume fraction increases as mean strain increases) or the effects due to the lower moduli in the stressinduced transformation regime (i.e. decreased hysteresis energy) can lead to longer fatigue lives, than the presence of fully austenite or martensite. The work by [START_REF] Ono | Plastic deformation governed by the stress-induced martensitic transformation in polycrystals[END_REF] shows in fact that few selected variants of stress-induced martensite reduce the internal strains due to the transformation. On the contrary, above 7% mean strain (approximately fully martensitic phase at 37 C), the constant life data exhibit a negative slope. No experimental data are available for the case of fully austenitic transformation (z = 0) for the considered SMA material.

CONCLUSIONS AND PERSPECTIVES

This paper has investigated the elastic shakedown behaviour of SMA materials and has presented the extension of the DV high cycle fatigue criterion to SMAs. The proposed formulation is general and suitable to several SMA constitutive laws, combining both plastic and transformation strains as well as thermal and mechanical cycling. The where ˆ h = h .

  ) and (b) show two possible cases of SMA uniaxial pseudoelastic response under cyclic loading, i.e. under alternating phase transformation and elastic shakedown, respectively.

Figure 1 :

 1 Figure 1: Two examples of SMA pseudoelastic response under cyclic loading with strain-stress paths denoted by red lines: (a) alternating phase transformation; (b) elastic shakedown.

  [START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF];[START_REF] Leclercq | A general macroscopic description of the thermomechanical behavior of shape memory alloys[END_REF];[START_REF] Moumni | Theoretical and numerical modeling of solidsolid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Sedlák | Thermomechanical model for NiTi-based shape memory alloys including R-phase and material anisotropy under multi-axial loadings[END_REF];Souza et al. (1998b);[START_REF] Stupkiewicz | A robust model of pseudoelasticity in shape memory alloys[END_REF];[START_REF] Zaki | A 3D model of the cyclic thermomechanical behavior of shape memory alloys[END_REF].

  volume (REV) at the mesoscopic scale. Such a volume may contain a large number of grains of austenite (A), martensite (M), or of both austenite and martensite (A+M). A representative situation of this schematic view is suggested by SEM observations of a NiTi sample, see Figure3. At the local scale, blue boundaries highlight austenite grains (A), red boundaries martensite grains (M), and red-blue boundaries mixed austenitemartensite grains (A+M).

Figure 2 :

 2 Figure 2: Schematic representation of the macroscopic scale of the SMA specimen and the mescopic scale of the RVE. The RVE consists of a set of austenite (A), martensite (M) or mixed grains containing martensite and austenite (A + M) in the form of bands.

Figure 3 :

 3 Figure 3: SEM image of a NiTi sample: local material scale. Blue boundaries highlight austenite grains (A), red boundaries martensite grains (M), and red-blue boundaries mixed austenite-martensite grains (A+M).

Figure 4 :

 4 Figure 4: Illustration of the Dang Van (DV) criterion in the ( h , ⌧ ) plane. The stress space is split in the infinite and finite lifetime by the DV line (red).

Figure 5 :

 5 Figure 5: Set of Dang Van (DV) lines in the stress space ( h , ⌧ ) for N fixed. Each line is defined by a fixed value of the martensite volume fraction 0 < z < 1.

Figure 6 :

 6 Figure6: Experimental (red dotted line)[START_REF] Pelton | Nitinol Fatigue: A Review of Microstructures and Mechanisms[END_REF] and numerical (blue line) monotonic stress-strain curves related to a NiTi microdogbone specimen tested at a constant temperature of 37 C.

Figure 7

 7 Figure 7(a) reports the experimental results obtained by Pelton (2011) as constant-life diagram, where the different conditions of mean strain and strain amplitude are displayed. The specimens that survived 10 7 cycles are shown as blue triangles, whereas specimens that fractured are shown as red circles.

Figure 7 :

 7 Figure 7: Constant-life diagram. (a) Experimental data by Pelton (2011). Conditions that survived the 10 7 cycles testing are shown as blue triangles, whereas cyclic conditions that fractured as red circles. (b) The green curve distinguishes between the elastic and phase transformation shakedown subspaces. The experimental data considered in the present work are numbered (red numbers refer to specimens that fractured). The color map indicates martensite formation.

Figure 8 :

 8 Figure 8: Representative loading cases of Table2, showing a fully martensitic transformation (z = 1).

Figure 9 :

 9 Figure 9: Representative loading cases of Table2, showing a fully martensitic transformation (z = 1).

FailureFigure 10 :

 10 Figure 10: Calibrated DV line (green) in the hydrostatic-mesoscopic stress plane and loading paths of cases 7-17 of Table2showing a fully martensitic transformation (z = 1).

Figure 11 :

 11 Figure 11: Representative loading cases of Table2showing the coexistence of austenitic and martensitic

Figure 12 :

 12 Figure 12: Calibrated DV line (green) in the hydrostatic-mesoscopic stress plane and loading paths of cases 1-6 of Table2showing the coexistence of austenitic and martensitic phases (0 < z < 1).

Figure 13 :

 13 Figure 13: Dependency (a) of the mesoscopic shear stress ⌧ and (b) of the hydrostatic stress h on martensite fraction z.

Table 1 :

 1 Calibrated parameters of the Souza-Auricchio model in a one-dimensional setting; seeAuricchio et al. (2009a,b).

	Parameter description		Symbol Value	Units
	Initial elastic modulus			E i	38000	MPa
	Final elastic modulus			E f	11000	MPa
	Maximum transformation strain			" L	4.49	%
	Stress-strain slope measure during transformation	h	290	MPa
	Elastic domain radius			R	148	MPa
	Slope of ✓ ⇤ with respect to temperature		9.50	MPa/ C
	Reference temperature			✓ ⇤	278.55	C
	Regularization parameter			10 12
		500
			EXPERIMENTAL T=37°C
		400	MODEL T=37°C
	s [MPa]	200 300
	Stress	100
		0
		0	2	4	8
			Strain e [%]

Table 2 :

 2 Summary of the experimental cases by Pelton (2011), considered in the present work and numbered in Figure 7(b). Computed values of the martensite volume fraction z and Young's modulus E = E(z).

	Case	Experiments from Pelton (2011)	Numerical simulations
		✏ a [%] ✏ m [%]	Fatigue life	z	E(z) [MPa]
	1	0.2	1	Run-out	0.08	31730
	2	0.4	3	Run-out	0.41	18917
	3	0.2	3	Run-out	0.45	18109
	4	0.6	6	Run-out	0.92	11666
	5	0.4	6	Run-out	0.96	11357
	6	0.2	6	Run-out	0.99	11064
	7	0.6	7	Run-out	1	11000
	8	0.4	7	Run-out	1	11000
	9	0.2	7	Run-out	1	11000
	10	0.6	8	Failure	1	11000
	11	0.4	8	Failure	1	11000
	12	0.2	8	Run-out	1	11000
	13	0.6	9	Failure	1	11000
	14	0.4	9	Failure	1	11000
	15	0.2	9	Failure	1	11000
	16	0.1	9	Failure	1	11000
	17	0.05	9	Run-out/Failure	1	11000
	and 17, that fractured, and result:
			8 > < > :	b 10 7 = 137.770 MPa a 10 7 = 0.808		(19)

as the proposed approach is clearly phenomenological, further investigation involving observations of the evolution of SMA microstructure during such a type of cyclic loading and its relation to fatigue are also needed. This will provide additional insight into microscopic deformation mechanisms and into factors influencing both crack initiation and growth. Finally, methods for the fast identification of fatigue limits, like the infrared thermography measurements (generally applied to steels), should be employed also for SMAs.

Acknowledgement

This work is partially funded by Ministero dell'Istruzione, dell'Università e della Ricerca through the Project no. 2010BFXRHS and by the Italian-French University (UIF-UIF) through the 'Bando Galileo 2013-2014' Grant no. 148-30174T J. C. Menna acknowledges Programma Star of University of Naples, Federico II -Sostegno Territoriale alle Attività di Ricerca, Linea d'Intervento 2 -Mobilità Giovani Ricercatori.

Appendix A. Algorithmic scheme for the computation of the DV fatigue criterion

The general algorithmic scheme employed for the computation of the DV fatigue criterion is composed of the following steps [START_REF] Bernasconi | Efficient algorithms for calculation of shear stress amplitude and amplitude of the second invariant of the stress deviator in fatigue criteria applications[END_REF][START_REF] Van | On a New Multiaxial Fatigue Limit Criterion: Theory and Application[END_REF]:

1. calculate the stabilized macroscopic stress (t) in each point of the structure; 2. split the macroscopic stress (t) in its hydrostatic and deviatoric part, defined respectively as: 8 > < > :

3. calculate the center A ⇤ r of the smallest hypersphere circumscribing the stress path through the following min-max problem: