Dr Philippe G Sarrazin
email: philippe.sarrazin@ujf-grenoble.fr

Damien P Tessier

Luc G Pelletier

David O Trouilloud

Julien P Chanal

Teachers' Expectations about Students' Motivation 1 Running Head: Teachers' Expectations about Students' Motivation The Effects of Teachers' Expectations about Students' Motivation On Teachers Autonomy-Supportive and Controlling Behaviors

Keywords: expectations effect, self-determination theory, motivation, differential treatment, autonomy-support, control, teaching behaviors

Previous studies in both educational and sport settings have examined the relationship between teachers' and coaches' expectations and behaviors towards students and athletes. The purpose of the present study was to extend this line of research by examining the effects of teachers' expectations about students' motivation on the frequency of controlling and autonomy-supportive behaviors. Following the assessment of teachers' expectations about students' motivation and the assessment of students self-determined motivation at the end of the first lesson, the interactions between PE teachers from 9 classes and 172 students were videotaped and systematically coded during the 6 following gymnastics courses. Analyses revealed that (1) teachers' expectations were not related to students' initial self-determined motivation, (2) overall teachers interacted less frequently with students that they expected to be more motivated, and (3) teachers were much more controlling with students that they believed to be not motivated. The results are discussed in terms of the consequences of this differentiated treatment for students' autonomous motivation.

The Effects of Teachers' Expectations about Students' Motivation On Teachers Autonomy-Supportive and Controlling Behaviors A considerable amount of research in the last 20 years has examined the implications of being intrinsically or extrinsically motivated in the classroom. More specifically, research guided by self-determination theory (SDT; see [START_REF] Deci | Motivation in education: The self-determination perspective[END_REF][START_REF] Deci | Intrinsic motivation and self-determination in human behavior[END_REF], 2000;[START_REF] Reeve | Self-determination theory applied to educational settings[END_REF]Ryan & Deci, 2000[START_REF] Reeve | Self-determination theory applied to educational settings[END_REF] for reviews), has shown that the extent to which students behaviors are intrinsically motivated and autonomous (i.e., fully volitional, freely pursued and wholly endorsed by the self) as opposed to extrinsically motivated and controlled (i.e., pressured and directed by external or internal forces leaving people feeling like they have no choice), have important consequences for the quality of students' learning, performance, and well-being.

According to SDT, the central concept that could explain the relationship between the students' motivation and their experience in the classroom, is the degree to which their behaviors are self-determined. SDT proposes different types of motivation [START_REF] Deci | Intrinsic motivation and self-determination in human behavior[END_REF], 2000;[START_REF] Ryan | Overview of Self-Determination theory: An organismic dialectical perspective[END_REF] which can be ordered along a continuum ranging from high (intrinsic motivation, integrated and identified regulation) to low (introjected, external regulation, and amotivation) levels of self-determination. Intrinsic motivation (i.e., when people do an activity for the pleasure it provides or for its own sake) is highly autonomous and represents the prototype of self-determination, whereas amotivation (i.e., when individuals do not perceive contingency between their actions and subsequent outcomes, do not value the activity, or felt incompetent) is the least self-determined type of motivation.

External regulation (i.e., when people behave in order to attain a desired consequence such as tangible rewards or to avoid a threatened punishment), introjected regulation (i.e., when individuals do behaviors in order to fulfill an inner strength like to avoid anxiety or to experience ego-enhancing pride), identified regulation (i.e., when people choice fully decide to engage in behavior that is not interesting per se, but nevertheless important, because it helps them reach valued personal goals), and integrated regulation (i.e., when activity is fully assimilated to the self because it has been evaluated and brought into congruence or harmony with other aspects of an individual's values and identity) 1 are four different forms of motivation ordered from lower to higher levels of self-determination, that fall between intrinsic motivation and amotivation. Much research now supports the self-determination continuum in the realms of education (e.g., [START_REF] Vallerand | On the assessment of intrinsic, extrinsic, and amotivation in education: Evidence on the concurrent and construct validity of the Academic Motivation Scale[END_REF], sport (e.g., [START_REF] Pelletier | Toward a new measure of intrinsic motivation, extrinsic motivation, and amotivation in sports: The sport Motivation Scale (SMS)[END_REF] or exercise (e.g., [START_REF] Li | The exercise motivation scale: Its multifaceted structure and construct validity[END_REF].

Because it is postulated that higher levels of self-determination are associated with better psychological functioning, SDT proposes that the extent to which an individual is selfdetermined is strongly reflected in the quality of his or her experience. For example, it has been shown that higher levels of self-determined motivation are related to several positive outcomes, such as effort, academic achievement, engagement, quality of conceptual learning, preference for optimal challenge, creativity, rates of retention (see [START_REF] Reeve | Self-determination theory applied to educational settings[END_REF]Ryan & Deci, 2000, for reviews). Although research within the sport or physical education (PE) context are more scarce, they also confirm the results observed in the education domain (see [START_REF] Vallerand | Intrinsic and extrinsic motivation in sport and exercise: a review using the Hierarchical Model of Intrinsic and Extrinsic Motivation[END_REF], for a review). For example, studies have shown that self-determined motivation is a significant predictor of long-term perseverance in handball [START_REF] Sarrazin | Motivation and dropout in female handballers: A 21-month prospective study[END_REF] or in swimming [START_REF] Pelletier | Associations among perceived autonomy support, forms of self-regulation, and persistence: A prospective study[END_REF]. It also predicts positively the intention of being physically active in the future or to play sport (e.g., [START_REF] Ntoumanis | A self-determination approach to the understanding of motivation in physical education[END_REF][START_REF] Pelletier | Toward a new measure of intrinsic motivation, extrinsic motivation, and amotivation in sports: The sport Motivation Scale (SMS)[END_REF][START_REF] Standage | A model of contextual motivation in physical education : using constructs from self-determination and achievement goal theories to predict physical activity intentions[END_REF] and self-reported effort in PE [START_REF] Ntoumanis | A self-determination approach to the understanding of motivation in physical education[END_REF] or in sport [START_REF] Pelletier | Toward a new measure of intrinsic motivation, extrinsic motivation, and amotivation in sports: The sport Motivation Scale (SMS)[END_REF], and negatively with boredom in PE [START_REF] Ntoumanis | A self-determination approach to the understanding of motivation in physical education[END_REF].

The Determinants of Motivation

According to SDT, the type of motivation people manifest within a particular context is strongly related to the satisfaction of basic psychological needs within that context. That is, the more factors in the social environment support the needs for autonomy (i.e., feeling like the 'origin' and not the 'pawn' of one's actions), competence (i.e., feeling effective in one's ongoing interactions), and relatedness (i.e., feeling connected to others, to caring for and being cared for by those others), the more people will be intrinsically and autonomously motivated [START_REF] Deci | Intrinsic motivation and self-determination in human behavior[END_REF], 2000;[START_REF] Ryan | Overview of Self-Determination theory: An organismic dialectical perspective[END_REF]. More specifically, in the educational context, research reveals that the degree to which teachers are autonomy supportive versus controlling has a particularly important effect on students' motivation [START_REF] Deci | Facilitating internalization: The self-determination theory perspective[END_REF][START_REF] Reeve | Autonomy-supportive teachers: How they teach and motivate students[END_REF][START_REF] Vallerand | Toward a hierarchical model of intrinsic and extrinsic motivation[END_REF]see Reeve, 2002;Ryan & Deci, 2000, for reviews). In essence, autonomy-supportive teachers are responsive (e.g., spend time listening, acknowledge the student's feelings and perspective), supportive (e.g., praise the quality of performance), explicative (e.g., provide a rationale for tasks and limits), and provide choice and opportunities for initiative taking and independent work. In opposition, controlling teachers essentially take charge (e.g., hold the instructional materials, use directives/commands), shape students toward a right answer (e.g., give solution), and motivate through pressure (e.g., threats, criticisms and deadlines) (see [START_REF] Reeve | Self-determination theory applied to educational settings[END_REF]Ryan &Deci, 2000, and[START_REF] Mageau | The coach-athlete relationship: A motivational model[END_REF][START_REF] Mageau | The coach-athlete relationship: A motivational model[END_REF], for reviews respectively in educational and sport domains).

The Determinants of Teachers Interpersonal Behaviors

Given the importance of these dimensions of teaching behaviors for student's motivation and subsequent educational outcomes, it is important to understand why some teachers provide more autonomy support while others are primarily controlling. Few studies have investigated this issue. For example, it has been shown that some teachers' characteristics like their general orientation toward behaving with others in a controlling versus autonomy supportive way (e.g., [START_REF] Deci | An instrument to assess adults' orientations toward control versus autonomy with children: Reflections on intrinsic motivation and perceived competence[END_REF], their own motivational orientation (e.g., [START_REF] Pelletier | Pressure from above and pressure from below as determinants of teachers' motivation and teaching behaviors[END_REF][START_REF] Wild | Perceiving others as intrinsically or extrinsically motivated: Effects on expectancy formation and task engagement[END_REF], and (c) their lay theories regarding ways of optimizing intrinsic motivation (e.g., [START_REF] Boggiano | Use of maximal-operant principle to motivate children's intrinsic interest[END_REF] represent predictors of their teaching behaviors. Research has also demonstrated that some contextual factors have an influence on teachers' interpersonal behaviors. Among those factors, two important sources of influence have been identified: the pressure from above and a pressure from below [START_REF] Pelletier | Pressure from above and pressure from below as determinants of teachers' motivation and teaching behaviors[END_REF]. The pressure from above relates to pressures that are placed on teachers by demands in the school organization. For example, when higher authorities impose restrictions about a curriculum (e.g., [START_REF] Pelletier | Pressure from above and pressure from below as determinants of teachers' motivation and teaching behaviors[END_REF] or when teachers are responsible for their students being able to perform up to standards (e.g., [START_REF] Deci | The effects of performance standards on teaching styles: The behavior of controlling teachers[END_REF], then teachers tend to become more controlling with their students. The pressure from below relates to the influence that students may have on teachers. For example, the students' performance and their behavior at school can entail more or less controlling behaviors among teachers. More specifically, students who are listless or disruptive create tension for teachers, which can easily push them to become controlling with these students.

The Effects of Teachers Expectations

An interesting aspect of the influence of students' characteristics on teacher behaviors is that this influence is not only determined by the actual students' characteristics but also by the teachers' perceptions or beliefs about those characteristics. For instance, a considerable amount of studies on self-fulfilling prophecies have shown that teachers' beliefs about their students' abilities or motivation had an influence on their behaviors towards their students. [START_REF] Merton | The self-fulfilling prophecy[END_REF] first introduced the term of self-fulfilling prophecy to refer to circumstances in which people's belief or expectation about something could lead them to Teachers' Expectations about Students' Motivation 7 behave in ways that cause the belief to come true even if the belief was initially incorrect. In educational context, the phenomenon is better known as "Pygmalion effect" [START_REF] Rosenthal | Pygmalion in the classroom: Teacher expectation and student intellectual development[END_REF]). In the last thirty years, this topic has generated many studies in the education domain (see, [START_REF] Good | Looking in classrooms[END_REF]Jussim, Smith, Madon, & Palumbo, 1998, for reviews), in physical education classes or in competitive sport context (see, [START_REF] Horn | The self-fulfilling prophecy theory: When coaches' expectations become reality[END_REF][START_REF] Martinek | The psycho-social dynamics of the Pygmalion phenomenon in physical education and sport[END_REF][START_REF] Martinek | Psycho-social dynamics of teaching physical education[END_REF]for reviews). In all of these contexts the researchers (e.g., [START_REF] Good | Looking in classrooms[END_REF][START_REF] Horn | The self-fulfilling prophecy theory: When coaches' expectations become reality[END_REF][START_REF] Jussim | Teacher expectations[END_REF][START_REF] Martinek | The psycho-social dynamics of the Pygmalion phenomenon in physical education and sport[END_REF] generally agree on several steps by which expectancies can be self-fulfilling: (a) the perceiver (e.g., a teacher) adopts particular beliefs (e.g., future achievement) about a target (e.g., a student), (b) the perceiver treats the target differently (qualitatively and quantitatively) according to these expectations (e.g., he/she is more supportive, gives clearer and more positive feedback, pays more attention and provides more opportunities for learning difficult subject matter to the high expectation), (c) this differential treatment influences the target's behavior who, in turn, confirms the perceiver's initial expectancies (e.g., the students who were more supported and who had more opportunities for learning have more chances than the others to carry out good performances), (d) this behavioral confirmation reinforces the target's original expectation, and the process continues.

The majority of the studies in the educational domain (including sport and PE) have examined teachers' expectations concerning students' potential ability or competence (e.g., [START_REF] Trouilloud | The influence of teacher expectations on students achievement in physical education classes: Pygmalion revisited[END_REF]. However, it is highly probable that teachers elaborate beliefs related to other students' characteristics such as the students' capacity to make efforts, to spontaneous engage in an activity, to be disruptive, which are things that "count" for a teacher, as shown by works on educational attitudes (e.g., [START_REF] Wolfe | Dimensions of opinion about teacher-pupil relations[END_REF][START_REF] Yee | Factor content of the MTAI[END_REF]. For instance, teachers seem to show clear preferences for students who work hard and make efforts, in spite of limited abilities (e.g., [START_REF] Covington | It's best to be able and virtuous too: Student and teacher evaluative responses to successful effort[END_REF]. Similar results were found in PE by [START_REF] Biddle | Effort is virtuous: Teacher preferences of pupil effort, ability and grading in physical education[END_REF]. In this study, teacher clearly preferred to interact with motivated and hard-working students, whatever their level of ability or achievement.

It is also highly probable that teachers develops beliefs and expectations about the specific nature of their students' motivation. [START_REF] Pelletier | Supervisors' beliefs and subordinates' intrinsic motivation: A behavioral confirmation analysis[END_REF] have examined more specifically whether a supervisor's beliefs about a subordinate's intrinsic (or extrinsic) motivation could induce the supervisor to support autonomy (or to be controlling) with the subordinate, which in turn, would cause the behavior of the subordinate to confirm the supervisor's beliefs. In two experimental studies, the authors observed that when "teachersubjects" were led to believe that the "student-subject" they were about to teach was extrinsically motivated, rather than intrinsically motivated, they were very controlling toward the students, which in turn led the students to display low levels of intrinsic motivation toward the puzzles. On the other hand, teachers who were led to believe that they were interacting with intrinsically motivated students were more autonomy supportive and their student showed high levels of intrinsic motivation. Thus, the teachers' beliefs about the student's motivation actually created their own reality.

Two other studies examine whether such effects could occur in naturalistic settings [START_REF] Jussim | Teacher expectations: Self-fulfilling prophecies, perceptual biases, and accuracy[END_REF]. [START_REF] Skinner | Motivation in the classroom: Reciprocal effects of teacher behavior and student engagement across the school year[END_REF] examined the effects of three dimensions of teacher behavior (involvement, structure, and autonomy support) on students' engagement across a school year as well as reciprocal effect of student's engagement on teacher's behavior. Correlational and path analyses revealed that students' engagement (measured in the spring) was associated with the three dimensions of teacher's behavior (as measured in the fall). More interestingly, reciprocal effects of student motivation on teacher behavior were also found. Teachers' perceptions of students' engagement predicted teachers' interactions with students across the school year. The more students were engaged, the more teachers were autonomy supportive. However, teaching behavior was assessed by teachers self-report of their interactions with each child in their classrooms. Consequently, students' engagement and autonomy supporting behavior shared systematic error variation that may have resulted in an overestimation of the effects. Moreover, the correlations between a teacher's behavior and students' perceptions of them were low. Because of those limitations, it seemed important to have an objective (independent) measure of those variables to check if the results reported by the authors were not a methodological artifact. Sarrazin, Trouilloud, Tessier, Chanal, and Bois (in press) designed a similar study to address specifically those issues. More exactly, in that study the teaching sessions were videotaped and rated by independent coders to assess the frequency and the style (i.e., controlling vs. autonomy-supportive) of interactions between teachers and students. Results

showed that (1) PE teachers used preferentially a controlling style (in 95% of the interactions), and (2) teacher's positive expectations about their students' motivation were positively related to an autonomy-supportive style. Nevertheless, this study suffered also from few methodological problems. First, students of the various classes were taught different sports (volleyball, table tennis, badminton and indoor football). The specificity of each sport could modify the nature of the teacher-student relationship. Secondly, the high versus low expectations conditions were created by splitting the sample at the median. This created a loss of information insofar as all the full variability of the answers was not used. It was likely that a student who obtained a score of 1 (on a 7-points scale) were not completely similar to the one who obtains a score of 3 (even if the both are below the median split). Thirdly, all the teachers' behaviors were classified in two categories (controlling vs. autonomy-supportive behaviors). Thus each category included a great variety of different behaviors (e.g., manner of asking a question, manner of holding instructional materials and organizational instruction, type of feedback, and so on) (see [START_REF] Mageau | The coach-athlete relationship: A motivational model[END_REF][START_REF] Reeve | Autonomy-supportive teachers: How they teach and motivate students[END_REF]Reeve, 2002, for reviews). A more refined analysis of the relationship between teachers' expectations and specific teaching behaviors could undoubtedly provide a more complete understanding of how the teachers' expectations are related to students' motivation. Finally, in order to take into account the variability of behaviors related to the personal style of each teacher (independently of the expectations they hold for each of their students) an horizontal weighting was carried out which consisted in converting each category of behavior to a proportion by dividing the number of individual communications received by a student by the total number of individual communications received by that student (see for a similar coding, Sinclair & Vealey, 1989). As a consequence, the total number of received communications was expressed in percentages which resulted in a loss of information about the total number of communications (e.g., a student who receives 3 controlling behaviors on a total of 4 obtained the same score, 75%, as another student who received 15 on a total of 20). Also, this led to categorization of autonomy supportive and controlling behaviors as exclusive (i.e., if a student received 33% of autonomy-supportive behavior, he/she automatically obtained a controlling behavior score of 66%). It was then impossible to determine how the two dimensions of teachers behaviors fluctuated in function of the beliefs about each student's motivation.

In sum, so far few studies have examined how teachers' beliefs or expectations about students' motivation are related to teachers' interpersonal behaviors in the laboratory and in naturalistic contexts. According to these studies, teachers who expect their students to be intrinsically motivated or self-determined are more likely to behave in an autonomysupportive way toward their students while teachers that expect their students to be extrinsically motivated or non self-determined are more likely to behave in a controlling way with their students. Although this research have shown once again that teachers expectations could influence their behaviors, we do not know with precision what are the specific autonomy supportive or controlling behaviors that are related to teachers' expectations and how those behaviors relate to each other. In other words, it is difficult to determine if teachers become more autonomy supportive when they expect to interact with a selfdetermined student or if they become controlling when they expect to interact with a non selfdetermined student, and what are the specific autonomy supportive or controlling behaviors affected by those expectations.

Purpose of the Study and Hypotheses

The purpose of the present study was to answer those questions by replicating and extending the Sarrazin et al's study. A sample of 172 students and their teachers were observed in PE classes where the same sport activity was taught (gymnastics). Several categories of controlling and autonomy-supportive teaching behaviors were coded and related with the teachers' expectations identified at the beginning of an 8-weeks training program.

Finally, a new data coding was used which takes into account both the inter-teacher variability and the quantity of the overt teaching behaviors.

According to self-fulfilling prophecy and SDT we made the general hypothesis that teachers would treat students differently qualitatively and quantitatively according to their expectations regarding their students' motivation. We defined expectations about students' motivation in function of its current meaning in the educational field, namely the teachers' beliefs about the students' capacity to provide efforts and to become involved spontaneously in the school tasks. More precisely, we hypothesized first, concerning the frequency of interactions, that teachers would initiate more interactions when they had "high" than "low" expectations towards their students, because those expectations correspond more to their definition of the "good" student (e.g., [START_REF] Biddle | Effort is virtuous: Teacher preferences of pupil effort, ability and grading in physical education[END_REF][START_REF] Covington | It's best to be able and virtuous too: Student and teacher evaluative responses to successful effort[END_REF].

Second, concerning the type of interactions, we hypothesized that teachers would behave in an autonomy-supportive way when they had "high" versus "low" expectations about their Teachers' Expectations about Students' Motivation 12 students' motivation. By contrast, they would become more controlling when they have "low" versus "high" expectations about their students' motivation [START_REF] Pelletier | Supervisors' beliefs and subordinates' intrinsic motivation: A behavioral confirmation analysis[END_REF]Sarrazin et al., in press; [START_REF] Skinner | Motivation in the classroom: Reciprocal effects of teacher behavior and student engagement across the school year[END_REF]. However, we did not have specific hypotheses about the precise nature of the autonomy supportive or controlling behaviors that teachers would emit.

Finally, we hypothesized that teachers' behaviors should be primarily influenced by their "beliefs" or the expectations they have about the students' motivation independently of the students "true" motivation. In other words, teachers should be more controlling when they believe or expect that their students are less motivated, uninterested or amotivated even if their behavior is in fact not so different from the one of the other students.

Method

Participants Seven PE teachers (that ranged in age from 33 to 46 years) and their 172 students (98 females and 74 males, 6 th -to 9 th -grades students; M age = 13.14 years, SD = 1.47, range = 11-16 years) from 9 classes in a single junior high school situated in south east of France volunteered to participate to the study.

Procedures

In France, physical education is a compulsory subject for all high school students.

Generally, physical education teachers teach each of several physical and sporting activities in 8-weeks cycles (i.e., 8 lessons of 2 hours). The study was conducted during gymnastics cycles in scheduled physical education lessons. Prior to the initiation of the research, teachers, parents, students and school administrators were asked to participate in an observational study. All participants were guaranteed anonymity. Because of the nature of the investigation, the teachers were not told the exact purpose of the study or the specific variables under investigation. Rather they were told that the researchers were only interested in different types of student behavior exhibited during PE courses without alluding to the "Pygmalion effect". This was a precautionary measure taken to prevent the Hawthorne effect (e.g., [START_REF] Adair | Placebo, Hawthorne, and other artifact controls: Researchers' opinions and practices[END_REF] from influencing teachers' interactions with their students. Because partial deception was employed, appropriate debriefing was carried out following the data collection.

At the end of the first lesson, teachers' expectations about students' motivation and students' self-determined motivation were assessed. Next, the teacher-student interactions were videotaped during the 6 following gymnastics courses with a digital camcorder. The teachers were equipped with a small microphone fixed on the collar of their cloth, with a transceiver allowing a precise recording of the content of the communications and the synchronization between the pictures and the sound. In order to not perturb the teacher and the students, the camcorder was situated in a fixed spot with a large angle, but at a sufficient distance to identify the student concerned by the interactions.

Measures

Teacher's Expectations about Students' Motivation. At the end of the first lesson, the teachers filed a questionnaire assessing for each student their expectations about effort (e.g., "according to you, will this student provide efforts during this gymnastic session?") and autonomy (e.g., "according to you, will this student be able to work in an autonomous way during this gymnastics session?"). Responses were made on a 7-points frequency scale anchored by 1 (never) and 7 (always). These two items were strongly correlated (r = .78, p <.001), and their average has been calculated to form a single dimension called "expectancies of motivation".

Students' self-determined motivation. Motivation toward gymnastics was assessed at the beginning of the teaching cycle, with an adapted French version of the Sport Motivation Scale [START_REF] Pelletier | Toward a new measure of intrinsic motivation, extrinsic motivation, and amotivation in sports: The sport Motivation Scale (SMS)[END_REF]. This tool is a 28-items inventory that assesses the multifaceted motivational regulations proposed by SDT. These types of motivation include the intrinsic motivation to know (IMK; e.g., "I go in gymnastics for the fun of discovering new skills/techniques"), the intrinsic motivation toward accomplishments (IMA; e.g., "…for the satisfaction I experience while I am perfecting my abilities"), the intrinsic motivation to experience stimulation (IMS; e.g., "…for the excitement I feel when I am really involved in the activity"), identified regulation (IDR; e.g., "…because what I learn in PE will be useful later"), introjected regulation (INR; e.g., "…because I must do PE to feel good about myself"), external regulation (EXR; e.g., "…because I will be marked"), and amotivation (AM; e.g., "I don't know why I go in gymnastics, if I could, I will get exempted"). Responses were made on a 7-points scale ranging from 1 (strongly disagree) and 7 (strongly agree).

Each subscale had an adequate internal consistency (α = .87, .87, .85, .85, .78, .68, and .83 respectively for IMK, IMA, IMS, IDR, INR, EXR and AM). In order to have a composite indicator of the students' motivation, the self-determination index was used (see [START_REF] Grolnick | Autonomy in children's learning: An experimental and individual difference investigation[END_REF][START_REF] Vallerand | Toward a hierarchical model of intrinsic and extrinsic motivation[END_REF]. This index was calculated by giving each subscale a specific weight according to its respective place on the self-determination continuum, in accordance with the following formula: [(2*(IMK+IMA+IMS)/3)+IDR] -[((INR+EXR)/2)+(2*AM)]. In the previous studies this index showed good reliability and predictive validity (see, [START_REF] Vallerand | Toward a hierarchical model of intrinsic and extrinsic motivation[END_REF].

Videotape Analyses. Each class was videotaped during six lessons of two hours. Only the teaching behaviors that were directed towards individual students were coded (i.e., all the communications addressed to a group or the whole class were not assessed). Given that the most motivated students are more likely to seek interactions with their teacher (and thus to increase artificially the number of feedbacks they receive), only the teacher's behaviors initiated by the teachers were taken into account (see [START_REF] Horn | Expectancy effects in the interscholastic athletic setting: Methodological concerns[END_REF] for a similar analysis). The student "targeted" has been identified during the coding session with a passport photo. Both the frequency and the type scores were computed for each student based on the number of communications given to that student from the teacher. Fifteen categories of verbal interactions derived from previous works in school [START_REF] Reeve | Self-determination theory applied to educational settings[END_REF][START_REF] Reeve | Autonomy-supportive teachers: How they teach and motivate students[END_REF] and sport [START_REF] Mageau | The coach-athlete relationship: A motivational model[END_REF] domains have been coded: organizational communications (told in an autonomy-supportive vs. controlling vs. neutral way), technical or tactical hints (told in an autonomy-supportive vs. controlling vs. neutral way), questions (asked in an autonomy-supportive vs. controlling vs. neutral way), praises, encouragements, perspectivetalking statements, negative communications (related to the student's social behavior vs. student' s work), criticisms (see Table 1 for the operational definitions and examples of these 15 categories). Given the complexity to classify some teaching behaviors either as controlling or autonomy supportive, a "neutral" category was added for some communications.

Coding Reliability and Data Analysis

Teachers' behaviors were coded by two coders trained with this type of analysis and blind of the teachers' expectations. Five lessons randomly selected were separately coded by the two observers, to estimate inter-rater reliability. Relative frequency scores within the categories were then compared through correlational analyses to determine the degree of agreement between raters. To estimate intra-rater reliability, two coders scored the same random sample of lessons again 2 weeks later. Analyses revealed a good homogeneity interrater (between .70 and .95; M = .80) and intra-rater (between .75 and 98; M = .85) overall categories of utterances.

A Principal-components factor analysis with Oblimin rotations was carried out on 12 utterances (the 3 "neutral" categories were not included in the analyses). Four factors emerged (based on the number of factors with eigenvalues greater than 1). The first factor extracted accounted for 28% of the variance and contained 5 utterances (with a weight greater than .45) relating to the teachers' controlling behaviors: organizational communications told in a controlling way (.68), hints told in a controlling way (.69), controlling questions (.60), negative communications related to the student's social behavior (.46) and negative communications related to the student's work (.56). The second factor accounted for 14% of the variance and contained 3 utterances relating to the teachers' autonomy-supportive behaviors: organizational communications told in an autonomy supportive way (.72), questions asked in an autonomy supportive way (.51) and hints told in an autonomy supportive way (.58). The third factor accounted for 13% of the variance and contained 2 utterances: encouragements (.47) and perspective-talking statements (.73). Finally, the fourth factor accounted for 9% of the variance and contained only one utterance: criticisms (.67).

Praises loaded on none of the factors. The behaviors of the first two factors were summed and named controlling style and autonomy-supportive style respectively 2 .

In order to take into account the few absences of certain students to some courses (generally for health reasons), the data were calculated in mean frequency per lesson (i.e., 2 hours). Otherwise, a considerable controversy exists in the literature concerning the appropriate unit of analysis to be used in examining instructional behavior (see, [START_REF] Horn | Expectancy effects in the interscholastic athletic setting: Methodological concerns[END_REF].

Given the variability of teaching behaviors which exists between teachers (e.g., [START_REF] Good | Looking in classrooms[END_REF] and given that teacher-student interactions are not independent of the setting in which they occur, it seems unsuitable to use the individual student as the unit of analysis without taking into account the specificity of each class. Therefore, the methodology proposed by [START_REF] Horn | Expectancy effects in the interscholastic athletic setting: Methodological concerns[END_REF] was used. The mean frequency of each utterance for each student were converted to standard scores which reflected the number of standard deviations that each student was from the class mean on each variable. For example, if David received an average of 5 hints told in a controlling way per lesson, in a class where the teacher gives on average 2.5 hints of this type with a standard deviation of 1.5. David will obtain then the score of 1.67

(5 -2.5/ 1.5); this means that he is 1.67 standard deviation above the mean of his class concerning this hints. On another hand, if Richard received an average of 5 controlling hints, Teachers' Expectations about Students' Motivation 17 in a class where the teacher gives on average 4.5 hints of this type with a standard deviation of 2. Richard will obtain then the score of 0.25 (5 -4.5/ 2); that is, he is 0.25 standard deviation above the mean of his class concerning this hints). Thus these standardized scores allowed the comparison of the students between them but also recognized the dependency which occurs between teacher-student interactions and the general setting in which they occur. The same standardization was made for the expectation variable.

Results

The observational procedures described above resulted in the collection of 6,369 teaching behaviors, as summed across all teachers. Descriptive statistics (see Table 2) revealed that the average number of teaching utterances received by an individual student in a two hours lesson was 12.52, with considerable variation among students (from 0 to 99; SD = 11.92). In terms of average frequency across all teachers and students, the results showed a prevalence of controlling behaviors (37.22%), followed-up by hints told in a neutral way (23.44%) and praise (11.81%). The autonomy-supportive behaviors only represented 4.6% of the communications (see Table 2).

To test the hypotheses, we relied on partial correlations between teachers' expectations and teaching behaviors while controlling for differences in students' self-determined motivation. Because the students self-determined motivation was moderately correlated with the teachers' expectations (r=.30, p<.001) and it had little effect on the teacher's behavior (Mr = .09), so the magnitude of the partial correlations were very similar in all cases to the magnitude of the zero-order correlations shown in Table 2.

Overall, the results showed that the frequency of communications was negatively correlated with the teachers' expectations (pr= -.30, p<.001), that is when teachers had expectations of high motivation they were initiating less interactions with their students.

Otherwise, teachers' expectations correlated significantly with 9 of 15 utterances. When Teachers' Expectations about Students' Motivation 18 teachers had expectations of low motivation, they were more likely to give organizational communications in a controlling way (pr = -.44, p<.001), provide hints in a controlling way (pr = -.18, p<.05) or in a neutral way (pr = -.21, p<.001), to ask controlling questions (pr = -.35, p<.001), to encourage (pr = -.15, p<.05), to criticize (pr = -.20, p<.01), to give negative communications related to the student's social behavior (pr = -.26, p<.001) or work (pr = -.20, p<.01) and they were less likely to ask autonomy-supportive questions (pr =.15, p<.05).

When an α <.003 (to protect against making a Type 1 error) was used (based on [START_REF] Hays | Statistics[END_REF] formula: α test /number of tests, or .05/15), teachers' expectations were still significantly correlated with four categories of behaviors (organizational communications told in a controlling way, controlling questions, negative communications related to the student's social behavior or work). Finally, when the different categories of behaviors were summed to create an overall category of controlling behaviors and an overall category of autonomy supportive behavior, only the controlling teaching category was significantly correlated with the teachers' expectations (pr =-.40, p<.001).

Discussion

Research on self-fulfilling prophecies has generated a considerable amount of information in the school context in general and in the sport and PE context in particular. This research shows generally that teachers/coaches' expectations have a strong influence on the way that teachers or coaches treat the students/athletes. Most of those studies have specifically focused on the effects of expectations about students' or athletes' potential, ability or competence. The present study extended this line of research by examining the effects of a specific type of expectations about students' motivation on the extent to which teachers supported autonomy or became controlling towards the student. Our decision to focus on those specific interpersonal behaviors was justified by the vast amount of information generated by two decades of empirical work on the application of self-determination theory in the educational context. Essentially, that research has shown that autonomous or self-determined students thrive in educational settings, and that high levels of students' self-determined motivation are observed when teachers support the students' autonomy [START_REF] Deci | Motivation in education: The self-determination perspective[END_REF][START_REF] Reeve | Self-determination theory applied to educational settings[END_REF]Ryan & Deci, 2000, for reviews). Thus, it seemed important to know more about the determinants of such behaviors. In agreement with past research on self-fulfilling prophecy and SDT, we made the general hypothesis that teachers would treat students differently both quantitatively and qualitatively according to their expectations concerning their students' motivation. More specifically, we hypothesized that teachers would initiate more interactions with students that were perceived as being more motivated, that they would show more autonomy support with those students, and that they would be more controlling with the students that were perceived as being less motivated.

Our results offered partial support for the hypotheses. Although teachers supported more the autonomy of students that were perceived as being motivated and they were more controlling with students that perceived as being less motivated, overall our results have shown that teachers initiated more communications and interactions when they held the expectation that a student had a low level of motivation. This last result is rather in contradiction with previous work carried out in the sport context (e.g., [START_REF] Horn | Expectancy effects in the interscholastic athletic setting: Methodological concerns[END_REF]Sinclair & Vealey, 1989;[START_REF] Solomon | The self-fulfilling prophecy in college basketball: Implications for effective coaching[END_REF], which has shown that coaches initiated more interactions when they had "high" expectations towards their athletes. This difference in the results observed may be attributed to the fact that previous studies in the sport area observed coaches' behaviors in competitive contexts where the main purpose of the interactions was the victory and the optimization of the team performance. Because winning in a competitive context is a very important goal, it is comprehensible that coaches may focus their interactions on the athletes that increase the chances on achieving that goal. By contrast the educational context may be associated with a different type of goals where teachers are probably more centered on the development of skills and the socialization of all the students. This may lead teachers to spend more time with the students who are perceived as lacking motivation. As our results suggest, teachers seem to be preoccupied by the students that were perceived as being less motivated. However, what was even more interesting was the fact that teachers were mainly controlling in their interactions with the students, more specifically with the students that were perceived as being less motivated.

In agreement with past studies on the influence of teachers' beliefs about students' motivation on teachers' interpersonal behaviors (e.g., [START_REF] Pelletier | Supervisors' beliefs and subordinates' intrinsic motivation: A behavioral confirmation analysis[END_REF]Sarrazin et al., in press;Skinner & Belmond, 1993), we were expecting a more balanced effect where teachers would be more autonomy supportive with students that were perceived as being more motivated, and more controlling with students that were perceived as being less motivated.

Our results partially confirmed this hypothesis. They showed that teachers were particularly controlling (i.e., by giving organizational communications in a controlling way, providing hints in a controlling way, asking controlling questions, and providing negative communications related to the student's social behavior or work) with all students but more specifically with students that were perceived as being less motivated. This effect could be explained by the possibility that the teachers were globally expecting to interact with students that had low levels of motivation because the course is mandatory for all the students. An examination of the mean for the self-determination index through the classes tends to show that the students' self-determination level was relatively low (M = 2.21) when we consider that the possible range for that index (-18 to +18). It is also possible, as shown by Pelletier et al. (2002), that, overall, the teachers' behaviors were affected by other factors such as their implicit theory about the way teachers should interact with their students in physical education classes, or the pressure to achieve a certain level of performance that could be coming from other sources (ex., the school's direction, the teachers' colleagues, or the parents). Because those factors were not measured in the present study, it is difficult to determine with precision why the teachers were apparently so controlling with their students.

Despite the fact that teachers were more controlling than autonomy supportive, it was still unexpected to find such a low level of autonomy support with the students perceived as more motivated. In fact, although the correlation between the teachers' expectations and the autonomy-supportive style is in agreement with the hypothesis, the level of significance of the correlation is marginal (pr=.12, p=.12). Only one category of autonomy supportive behaviors tends to be significant (to ask autonomy-supportive questions, pr=.15, p=.049). Nevertheless, it should be noted that the average frequency of the autonomy-supportive behaviors is weak (less than five percent of the interactions). It seemed that the PE teachers involved in the present study, like the Sarrazin et al.'s (in press) study, are more inclined to use, or more at ease with, controlling strategies than autonomy-supportive ones. Consequently, this teaching style does not seem to constitute (at least in this study) a meaningful indicator of differential treatment between students.

When we consider globally the results of the present study, a question remains: why are teachers more controlling with students that were perceived as being less motivated? Some laboratory studies have tried to answer that question (see [START_REF] Deci | Intrinsic motivation and self-determination in human behavior[END_REF], for a review). First, student passivity is aversive. It may make a teacher feel incompetent or unliked by the student. In addition, passivity can be interpreted as lack of "internal" motivation, which leads teachers to apply "external" pressure to participate in classroom activities.

Unfortunately, former researches (see, [START_REF] Deci | Motivation in education: The self-determination perspective[END_REF]Ryan & Deci, 2000, for reviews) showed that ironically, such teaching behaviors tend to lead to the opposite effects than the one pursued, which is to increase the students motivation. As shown by research on SDT, contexts that are controlling undermine intrinsic motivation and impair internalization, resulting in more controlled types of regulation or no motivation at all.

It is interesting to note that teachers that expected to interact with students that were less motivated did not rely solely on controlling strategies, they also provided more encouragements (pr=-.15 pr<.05), perhaps because teachers considered that as a good strategy to motivate students that are not motivated. It was not entirely clear if that strategy could be classified as an autonomy-supportive or controlling strategy. In the present research, like previous research in education [START_REF] Reeve | Autonomy-supportive teachers: How they teach and motivate students[END_REF], encouragements loaded on a distinct factor than the ones representing the controlling and the autonomy supportive behaviors.

Along the same lines, giving praises did not load on any factors and it was not significantly related to teachers' expectancies (r=-.01; see Table 2). Past research has shown that the provision of positive competence feedback is an important determinant of individuals' intrinsic motivation (see [START_REF] Deci | A meta-analytic review of experiments examining the effects of extrinsic rewards on intrinsic motivation[END_REF], for a review). However, this research also suggests that providing positive feedback is a more complex than it might appear [START_REF] Henderlong | The effects of praise on children's intrinsic motivation: a review and synthesis[END_REF], insofar as a feedback can be perceived as either informational (i.e., provides the person with information about his/her competence) or as controlling (i.e., it incites the person into reemitting the behavior). If the informational aspect is salient then positive feedback enhances people's intrinsic motivation. By contrast, if the controlling aspect is salient (when the teacher says for example "You did very well, just as you should") then positive feedback undermines people's intrinsic motivation. In the present study, our grid of analysis did not allow us to distinguish these two types of praise. It is perhaps why this type of utterance was not connected to the teachers' expectancies nor with a particular teaching style.

It should be noted that these results were observed, after controlling for the students' self-determined level of motivation. This variable was practically not correlated with any teaching behaviors, and moderately correlated with the teachers expectations (r =.30, p<.001).

In other words, it is a teacher's perception of a student's motivation, more than the student's own motivation, that had an impact on the teacher interpersonal behaviors. As shown by [START_REF] Pelletier | Supervisors' beliefs and subordinates' intrinsic motivation: A behavioral confirmation analysis[END_REF], this could lead some students perceived as not motivated but that were in fact self-determined, to become less motivated because of the way they are treated by controlling behaviors teacher.

Nevertheless, it is necessary to put the role of teacher's expectations as a determinant of teachers' behaviors in perspective. Indeed, the moderate correlations (prs ≤-.44) between teachers' expectations and their teaching behaviors (specifically their controlling behaviors) suggests that this variable can not explain by itself the strong proportion of controlling behaviors adopted by the PE teachers (see Table 2). In other words, the teachers' expectations are not the only significant determinant of the teaching interpersonal behaviors. Some teachers' characteristics like their ideologies, their lay theories or they own motivation for their work are also significant determinants that may affect their teaching style.

Moreover, contextual and/or institutional characteristics -such as the pressure from higher authorities, parents, and the characteristics of the taught activity (e.g., the constraints related to the security management of the students) -can lead teachers to become more controlling. It would be interesting in future studies to examine how all those determinants relate to the different categories of teaching behaviors as measured in the present study. Multilevel analyses (e.g., [START_REF] Goldstein | Multilevel Statistical Models[END_REF] with a large sample of students and classes seem particularly suited to this kind of question insofar as they allow the partition of variance associated with different effects into components associated with the individual student (e.g., teachers' expectations) and the class (e.g., teacher's lay theories, general motivational orientation, taught activities, and so on) levels.

Teachers' Expectations about Students' Motivation 24

In conclusion, this study provides more support for the idea that teachers' beliefs about their students' capacity to provide efforts and to become involved spontaneously in the school activities influence their teaching behaviors. More specifically, it seems that their beliefs about the lack of motivation among some students may lead them to become more controlling. As suggested by past studies on the consequence of such a style in an educational context (see [START_REF] Reeve | Self-determination theory applied to educational settings[END_REF]Ryan & Deci, 2000;[START_REF] Vallerand | Intrinsic and extrinsic motivation in sport and exercise: a review using the Hierarchical Model of Intrinsic and Extrinsic Motivation[END_REF], for reviews), these controlling behaviors would, ironically, foster more compliance and less selfdetermined motivation among students, which would eventually confirm the teachers initial beliefs.

Lastly, given the motivational benefits of adopting an autonomy-supportive style, additional studies remain to be carried out on the reasons why PE teachers seem primarily controlling (is it a specific context of teaching?), and on the means of encouraging teachers to support more students' autonomy [START_REF] Reeve | Autonomy support as an interpersonal motivating style: Is it teachable?[END_REF].

Footnotes 1-This regulation was not assessed in the present study because pilot data collected during the development of the SMS [START_REF] Pelletier | Toward a new measure of intrinsic motivation, extrinsic motivation, and amotivation in sports: The sport Motivation Scale (SMS)[END_REF] revealed that this regulation did not emerge as a perceived reason for participation in the physical domain. Furthermore, this type of motivation seems more often encountered among adults rather than children [START_REF] Vallerand | Toward a hierarchical model of intrinsic and extrinsic motivation[END_REF].

2-Neutral categories were excluded of the factor analysis because they did not fit really with any SDT categories. Nevertheless, the factor analysis carried out with all the 15 categories (i.e., including the 3 neutral categories) reveals relatively identical results. Five factors with eigenvalues greater than 1 emerged. Two are rigorously identical to the first two factors found in the former factorial analysis (i.e., the ones named "controlling style" and "autonomysupportive style" respectively), a third factor gathers the three neutral categories with praises, and encouragements. On the last two factors only one item loaded significantly: perspectivetalking statements on the first and criticisms on the second. Given the difficulty to named these last three factors our strategy to gather only the items on an identifiable factor do not seems affected by the results of this factor analysis.

Teachers' Expectations about Students' Motivation 30 -Frequency of technical or tactical statements for which the tone is neither controlling nor autonomy-supportive, the intention is above all to make the student progress.

-Frequency of suggestions that encourage pupils to take initiatives and to solve problems independently "Extend the arms, I have told you that 10 times" "Bend your leg at the reception of the jump you will succeed better"

"Maybe you could try different positions to jump over this obstacle and choice the best" 3. Questions asked -in a controlling way -in a neutral way -in an autonomy-Supportive way -Frequency of directives posed as a question.

-Frequency of questions for which the tone is neither controlling nor autonomy-supportive.

-Frequency of questions that provide choices to the pupil "What have I just said, Paul?" "Is it your last try ?" "Which exercise do you want to start with?" 4. Praises Frequency of verbal approvals of the student's performance "Well done!" "Good job!" 5. Encouragements Frequency of pep-talk statements to boost the student's effort "Now you're getting the hang of it; let's go!" 6. Perspective-talking statements Empathic statements reflecting an understanding of the student's perspective "I can see that you are starting to be tired"

7.

  Negative communications related to -the student's social behavior -the student's work -Frequency of directives intended to restore discipline into the classroom.-Frequency of directives meant to emphasize the lack of efforts and which could be sarcastic. "Shut up Paul!" "Do not do too much, you will wear away the apparatus!"8. CriticismsFrequency of hurtful statements "You are completely numskull!".

Table 1

 1 Types of verbal interactions which were coded in the study Frequency of the organizational commands that the student must, has to, or ought to do something.-Frequency of organizational statements for which the tone is neither controlling nor autonomy-supportive.-Frequency of statements that provide a choice in the organization of the material.

	Types of verbal interactions:	Definitions:

Table 2

 2 Descriptive statistics for all dependent measures and their correlation with teachers' expectancies of motivation Each behavior is coded in mean frequency in a two hours lesson. 1 Partial correlation between teachers' expectancies and behaviors (each variables being standardized as a function of the class), controlling for student's self-determined motivation. *p<.05, ** p<.01, *** p<.001.

	Variables	M SD	Range Percentage	r with expectancy 1
	Utterances: 1. Organizational communications told in a controlling way 2. Organizational communications told in a neutral way 3. Organizational communications told in an autonomy-supportive way 4. Hints told in a controlling way 5. Hints told in a neutral way 6. Hints told in an autonomy-Supportive way 7. Controlling questions 8. Neutral questions 9. Autonomy-supportive questions 10. Praises 11. Encouragements 12. Perspective-talking statements 13. Negative communications related to the student's social behavior 14. Negative communications related to the student's work 15. Criticisms	1.35 1.22 0.16 2.11 2.93 0.18 0.56 0.72 0.24 1.48 0.86 0.24 0.24 0.41 0.03	1.72 1.19 0.27 2.59 3.57 0.36 0.91 0.91 0.37 2.24 1.11 0.37 0.55 0.74 0.16	0 to 10 0 to 6 0 to 1.5 0 to 18.5 0 to 35 0 to 2.75 0 to 5 0 to 6 0 to 2 0 to 18 0 to 6 0 to 2 0 to 4 0 to 26 0 to 2	10.79 9.72 1.24 16.82 23.44 1.42 4.43 5.80 1.94 11.81 6.83 1.94 1.88 3.31 0.27	-.44*** -.14 .05 -.18* -.21** -.07 -.35*** -.12 .15* -.01 -.15* -.06 -.33*** -.26*** -.20**
	Teaching style: Controlling (1+4+7+13+14) Autonomy-supportive (3+6+9)	4.66 0.57	5.26 0.64	0 to 31 0 to 4	37.22 4.60	-.40*** .12
	Number of utterances Note.	12.52 11.92	0 to 99		-.30***


Ginot Malo

Castanié Bruno

Bouvet Christophe

Mahuet Nicolas

STRUCTURAL TESTING OF LOCAL BUCKLING FOR THE DESIGN OF SANDWICH AIRCRAFT STRUCTURES

Keywords: Local buckling, Wrinkling, Sandwich structures, Structural testing, Light aircraft

The design of light aircraft sandwich structures is often driven by a local buckling phenomenon named "wrinkling" which can be the primary cause of failure of such structures. Structural tests using the VERTEX Methodology were performed to access the wrinkling behaviour of technological sandwich specimens representative of the technology of Elixir Aircraft. The operating principle of the VERTEX machine is the use of four hydraulic actuators to load a rectangular box structure and therefore, sandwich panels were tested under compressive and shear loading. Wrinkling is observed thanks to Digital Image Correlation (DIC) and a highspeed camera. At this level of the test pyramid for certification of aeronautic structures, the observation of wrinkling on sandwich panels remains rare in literature and may lead to interesting future investigations on the experimental VS numerical finite elements model's dialogue.

Introduction

Thanks to their high specific bending stiffness, sandwich structures have been widely used in aviation for more than a hundred years [START_REF] Castanié | Review of composite sandwich structure in aeronautic applications[END_REF]. Recently, they are increasingly used for primary structures for lightweight aviation (like the Elixir from Elixir Aircraft®, certified by the EASA in June 2020). Local buckling phenomenon named "wrinkling" can be the primary cause of failure of such structure [START_REF] Castanié | Review of composite sandwich structure in aeronautic applications[END_REF]. An effective methodology for wrinkling sizing is still under discussion [START_REF] Ginot | Benchmark of wrinkling formulae and methods for presizing of aircraft lightweight sandwich structures[END_REF], and experimental studies could discriminate between existing models. However, experimental tests on the wrinkling phenomenon in the literature are still scarce. Previous works have shown, at beam scale (most experimental studies in literature), the high sensitivity of the phenomenon to boundary conditions and initial geometrical defects leads to difficulties in correlating experiments and models (3)(4) [START_REF] Koissin | Effect of physical nonlinearity on local buckling in sandwich beams[END_REF]. An interesting approach consist in studying the upper-stage of the "pyramid of tests" called "technological", in other words, switching from beam scale to panels. This, to characterize the true behaviour of a structure and reduce the influence of boundary conditions. Therefore, this paper aims to offer protocols/methods with results and analysis of sandwich panels, tested under compressive and shear loading, where wrinkling type failure is observed. For this purpose, the VERTEX test bench (Figure 1

Materials and methods

The specimens are sandwich plates with a tapered area to close the sandwich at the centre. As a result, edges are monolithic and can be drilled to bolt the specimen to the VERTEX test bench. The specimen is positioned as shown in Figure 1 and bolted on its 4 sides with 128 screws. The external dimensions are 558x536mm², and the sandwich area with the tapered regions is 390x390mm² and 21 mm thick.

Specimen design

2.1.1 Geometry: The lower skin of the sandwich plate is not accessible as it faces the test bench box structure (Figure 1). The observable surface is the upper skin, it is thus imperative to localise the wrinkling mode on it. In the case of compressive loading, the tapered asymmetric geometry of the plate allows the upper skin to be more loaded than the lower skin and therefore to buckle. This mismatch between the load introduction axis and the mean geometric plane of the sandwich structure leads to a local bending moment, which induces an additional compressive load in the upper skin and a tensile load in the lower skin (blue arrows in Erreur ! Source du renvoi introuvable.). This phenomenon was already explored in asymmetric sandwich structures by Castanié et al [START_REF] Castanié | Theoretical and experimental analysis of asymmetric sandwich structures[END_REF].

Materials and process:

The materials used for the specimens are ROHACELL HERO® 51 kg/m3 foam for the core and prepreg epoxy/carbon woven fabric for the skin. An autoclave following a "one-shot/co-cured" process is used. The trimming and drilling are carried out using a CNC (computer numerically controlled) machine.

Figure 3: Skin definition

To localise the wrinkling in the centre of the plate, a high-density foam frame is used to reinforce the tapered area of the sandwich where local stresses could be relatively high. For the specimen under compressive loads (H51_Comp), the nominal area is 260x260 mm (dotted frame in Figure 3(a)). For the specimen under shear loads (H51_shear), the nominal area (dotted frame in Figure 3(b)) is lozenge-shaped of 172x172 mm to have perpendicular edges to the main stresses at 45°. For the same reasons, skins in the tapered area are reinforced (deep brown area in Figure 3). For the specimen under compressive loads, the monolithic is reinforced in the specimen height (dark area in Figure 3(a)), to avoid bolt bearing. In the width, the monolithic is left with a relatively thin laminate (grey area in Figure 3(a)) so that this area does not carry all the loads at the expense of the sandwich zone. For the specimen under shear load, the nominal skin (light brown area in Figure 3(b)) is lozenge-shaped. It should be noted that skin reinforcement and core reinforcement are not in the same areas. We call nominal area (where wrinkling is expected) the area where the upper skin stacking sequence corresponds to two +/-45° ply woven fabrics for H51-Shear specimen and a +/-45° and 0°/90° ply for H51_Comp.

Instrumentation

Because of numerous structural redundancies in the VERTEX test bench, nowadays, there is still no direct transfer function that links actuator forces with stress flows entering the specimen. An in-situ strains measurement method had to be used to calculate the stresses entering the specimen. Digital Image Correlation (DIC) has been shown to be a very reliable method, and already been used on previous VERTEX tests (7)(8) [START_REF] Trellu | Combined loadings after medium velocity impact on large CFRP laminate plates: Tests and enhanced computation/testing dialogue[END_REF]. Moreover, stereo-DIC allows to measure out-of-plane displacements and reconstruct the 3D shape evolution. This is particularly useful for the wrinkling observation which manifests as short out of plane waves in the skins. To achieve this DIC, a speckled pattern is made on the upper faces of the specimens. A high-speed camera (7000fps) is then used to observe a potential explosive failure. In lower skin, "Rosette" gauges are used.

Results

Deformation fields: stereo-correlation

Figure 4 shows in-plane deformation fields obtained by DIC of the specimen's upper faces, just before failure. In the nominal area, the deformations are mainly uniform and follow typical expectations. For specimen H51_Comp, shear is observed at the corners, but is nearly zero in the central nominal area (see 𝜺 𝒙𝒚 's field of H51_Comp). For specimen H51_shear, the average principal compressive strain direction is about 38° from the x-axis. Some traction is shown by the non-null xcomponents (see positive 𝜺 𝒙𝒙 's field of H51_Shear), a consequence of an inherent machine behaviour when operated for this solicitation. Local gradients (see 𝜺 𝒙𝒙 's field of H51_Comp and 𝜺 𝒙𝒚 's field of H51_Shear) at the edge of the nominal area indicate local bending induces by wrinkling waves.

Failure scenario

For both specimens, the failure behaviour is similar (Figure 5). The specimens fail by wrinkling the upper skin, which validates the design of the specimens. The wrinkling wave appears at the interface between the nominal HERO 51 foam and the denser foam. The core is crushed, and the wave then spreads over the width. Finally, the core fails in tension and the wave moves on a large part of the nominal area. This lasts about 2 milliseconds. For specimen H51_shear the wrinkling wave appears at -45° direction, induced by the compression along the +45° direction associated with shear loading. Skin static failure is observed from the edge of the specimen after the wrinkling event. Note that failure scenarios could be accurately reconstructed thanks to the high-speed camera. ). This can be confirmed by looking at in-plane deformation curves (Figure 7(c)). 𝜺 𝒙𝒙 at P1 and P2 first decreases linearly with load, until the apparition of the wrinkling wave (around 80% load). It then becomes non-linear, a consequence of a local bending due to the wrinkling wave. The slopes at P1 and P2 differ from the one of the average deformation E0. This is a consequence of the out-of-plane effects of the initial conditions. Since geometrical defects localise wrinkling, it has occurred near the interface with the densified foam, where an initial slight skin depression was observed thanks to DIC. This phenomenon is local, thus the non-linearity does not influence the average deformation E0 which remains linear until failure.

H51_Shear: In this specimen, the global bending behaviour presented previously is not present because the tension and compression at 45° compensate for each other. Wrinkling waves are observed in a major part of the nominal area. The maximum wave's amplitudes are observed at the interface between the nominal HERO 51 foam and the denser foam (Figure 8

Conclusion

This paper presents the materials and methods with the associated results and analysis, of two sandwich plates bolted on 4 sides, tested respectively in compression and shear using the VERTEX test bench. Wrinkling type failure was observed using both DIC and a high-speed camera. Such specimen geometry, material definition and measurements tools have demonstrated their effectiveness to observe wrinkling and insulate the failure scenario from the influence of the boundary conditions. It's appeared that wrinkling is driven by initial geometrical defects which localise the failure. 3D shapes and local deformations rates have been quantified and should be compared with existing models. This will be the object of future works.

  ) developed by Castanié et al. and first used by Serra et al. (6)(7)(8) was used. The operating principle of the VERTEX machine consists of four hydraulic actuators used to load a rectangular box structure. The tested panel closes the upper part of the central box. Actuators 1&2 can push or pull symmetrically to bend the box structure of the bench, thus the tested panel is locally loaded in tension or compression. Similarly, actuators 3&4 can push to twist the centre of the box structure, thus the tested panel is locally loaded in shear.

Figure 1 :

 1 Figure 1: Vertex test rig and sandwich plate specimen

Figure 2 :

 2 Figure 2: Bending effect on asymmetric sandwich structures.

Figure 4 :

 4 Figure 4: Deformation fields obtained by stereo-correlation.

Figure 5 :

 5 Figure 5: Failure scenario schemes

Figure 6 :

 6 Figure 6: Specimen H51_Comp out of plane behaviour

Figure 7 :

 7 Figure 7: Specimen H51_Comp local behaviour A wrinkling wave is observed at the interface between the nominal HERO 51 foam and the highdensity foam (Figure 7(a) and (b)). It appears at the end (>80%) of the test (Figure 7(b)). This can be confirmed by looking at in-plane deformation curves (Figure 7(c)). 𝜺 𝒙𝒙 at P1 and P2 first

  (a) and (b)), where an initial slight skin depression was observed thanks to DIC.

Figure 8 :

 8 Figure 8: Specimen H51_shear local behaviour Similar to the compressive test, 𝜺 𝒙𝒙 first follows a linear trend with loading, followed by a nonlinear regime at the end of the test for inspect points P1 and P2.Again this can be attributed to the local buckling onset that has happened around 80% load (Figure 8(c)).

/ 9 ©2022 1 st Author et al. https://doi.org/ 10.5075/978-X-XXX-XXXXX-X published under CC BY-NC 4.0 license

/ 9 ©2022 1 st Author et al. https://doi.org/ 10.5075/978-X-XXX-XXXXX-X published under CC BY-NC

4.0 license

/ 9 ©2022 1 st Author et al. https://doi.org/ 10.5075/978-X-XXX-XXXXX-X published under CC BY-NC 4.0 license

/ 9 ©2022 1 st Author et al. https://doi.org/ 10.5075/978-X-XXX-XXXXX-X published under CC BY-NC 4.0 license

/ 9 ©2022 1 st Author et al. https://doi.org/ 10.5075/978-X-XXX-XXXXX-X published under CC BY-NC 4.0 license

/ 9 ©2022 1 st Author et al. https://doi.org/ 10.5075/978-X-XXX-XXXXX-X published under CC BY-NC 4.0 license

/

©2022 1 st Author et al. https://doi.org/

10.5075/978-X-XXX-XXXXX-X published under CC BY-NC 4.0 license

/ 9 ©2022 1 st Author et al. https://doi.org/

[START_REF] Trellu | Combined loadings after medium velocity impact on large CFRP laminate plates: Tests and enhanced computation/testing dialogue[END_REF].5075/978-X-XXX-XXXXX-X published under CC BY-NC 4.0 license