domain: sdu phys

text:
L Stefanelli

G Metris

RESONANCES DUE TO THIRD BODY PERTURBATIONS IN THE DYNAMICS OF MEOS

Keywords: Medium Earth Orbits, resonances, stability

The dynamics of Medium Earth Orbits (MEO) sees nowadays a renewed interest because of the development of satellites constellations (GNSS), that raises the problem of parking orbits for satellites at end of life. Numerical evidence shows that the resonances related to the presence of a third body can affect the stability of MEO orbits. The goal of our work is to study the effects of resonances on the stability of MEO orbits, over long or very long term (hundreds of years). For orbits above 20,000 km altitude, the perturbation due to a third body (the Moon or the Sun) is not fully negligible, so we take into account the third body perturbation on the secular evolution of the angular variable of the satellite orbits, and thus of the resonant angle. We estimate numerically the evolution of the resonant inclination. We study the stability of some resonances analytically and numerically, and in particular the resonance associated to the operational inclination of the Galileo satellites.

Introduction

The region of space hosting the Medium Earth Orbits (MEO) is becoming more and more crowded, especially because of the increasing presence of the satellites constellation, like the GPS and GALILEO families. The lifetime of these satellites is limited, and they have to be replaced regularly. This rises the problem of positioning the inactive satellites on "safe" parking orbits, in order to avoid the classical problems with spatial debris: they can become a danger for the active satellites and can cause the multiplication of the debris by collision. For this reason, understanding the stability properties of MEO orbits for long times can guide the choice of suitable parking orbits. Among the various factors, the study of this stability includes an analysis of the behaviour of the orbits in presence of resonances, which can be a possible source of instability or, on the contrary, can help to keep the orbit stable.

The long term effects of the resonances on the stability of MEO orbits is not yet well understood. Moreover, in the litterature there are only few articles about the case of MEO orbits. In Rossi (2008) several numerical experiments concerning the interactions between resonances and MEO orbits are presented. For the theory of the resonances, the main references are the two papers Hughes (1980) and Hughes (1981), mainly dedicated to list the resonant angular combinations. Some analytical studies about the resonances dependent only on the eccentricity can be found in Breiter (1999) and Breiter (2001), but only for the case of low earth orbits.

In this paper, we deal with resonances due to the presence of a third body, that can be either the Sun or the Moon. These resonances involve linear combinations of the angles of the satellites and of the third body which appear in the development of the potential due to this third body. Such combinations write

ψ = (n -2p)ω + mΩ + (n -2p + q)M + (n -2k)ω 3b + mΩ 3b + (n -2k + j)M 3b (1.1)
where ω, Ω, l are the argument of the perigee, the longitude of the ascending node and the mean anomaly of the satellite and ω 3b , Ω 3b , M 3b are the same angles associated to the third body. A resonance occurs when ψ ≈ 0 .

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While the frequencies associated to the third bodies (Sun and Moon) are well fixed, the frequencies associated to the satellite strongly depend of its orbit. Hughes (Hughes (1980) and Hughes (1981)) has carefully studied the relation between the possible resonances and the characteristics of the orbit. He has in particular evidenced a class of resonances related to combinations of the form αg + βh which involve only secular variations of the satellite angles. These variations are mainly due to the Earth's oblateness (related to the J 2 coefficient of the geopotential) and in this case, the resonance condition is only function of the inclination. One of these resonances occurs for an inclination of about 56 • which is very close of the inclination of the Galileo orbits. The study of this particular resonance is thus of practical interest. This paper is a first attempt in this direction.

The article is organized as follows. In section 2, we set up the model for the hamiltonian, including the Earth's gravity potential reduced to its monopolar and quadrupolar terms and the third body gravity potential; moreover we average the hamiltonian with respect to every non resonant angles in order to keep only the driven terms for the very long term evolution (over tens of years). Section 3 contains the results of the study of this averaged system for the case of one selected resonance.

The model of resonance

Our fundamental variables will be the Delaunay canonical variables l, g, h, L, G, H but we will also use for convenience the classical orbital elements a, e, i, Ω, ω, M . The two sets of variables are related by the relations

l = M, g = ω, h = Ω, L = √ µa, G = L 1 -e 2 , H = G cos i .
Our model takes into account the main part of the Earth's gravity potential as well as the perturbation of a third body, i.e. the Sun or the Moon, which is the source of the resonances, the main subject of this paper. The Hamiltonian of the system is decomposed as

J = J Kep + J J2 + J 3b ; (2.1)
J Kep is the Hamiltonian corresponding to the keplerian motion, J J2 is the perturbation due the Earth's oblateness and J 3b is the perturbation due to the third body. The full expression of these well known perturbations can be found, for example, in [START_REF] Stefanelli | Proceedings of the 64 th IAC[END_REF]. In particular, in this paper, we will focus on the Sun perturbation (so 3b = ).

Since we want to study the orbital evolution over very long time spans (typically several tens of years), we will discard the short periodic terms related to the fast variables l and M . This is achieved by means of a classical canonical (time dependent) change of variables not detailed here, so that the new hamiltonian no longer depends on l and M .

It is well known that in the J 2 problem, the elimination of the mean anomaly is accompanied, at the first order of the transformation, by the elimination of the perigee argument g (Brouwer (1959)). Thus, we obtain the new Hamiltonian (in principle, we should use new (primed) variables but we will keep the same notations for the sake of simplicity)

J J2 = 1 4 J 2 µ a R e a 2 1 η 3 1 -3 cos 2 i , (2.2)
where µ = Gm e , and R e are the Earth's gravitational constant and equatorial radius, respectively, J 2 is the coefficient of the second order zonal armoniccs of the Earth's gravitational potential and η = √ 1 -e 2 . Concerning the Sun perturbation involving linear combinations of the angles, removing l and M is equivalent to select the appropriate angular combinations [START_REF] Stefanelli | Proceedings of the 64 th IAC[END_REF]):

J = -µ a 2 a 3 H 2,1,0 (e ) 2 m=0 (2 -δ 0m ) (2 -m)! (2 + m)! 2 p=0 F2,m,p (i) F2,m,1 ( )G 2,p,2p-2 (e) cos ((2 -2p)g + mh) , (2.3)
where is the obliquity, Fn,m,p denotes the modified Allan inclination functions (Allan (1965)) and G n,p,q , H n,p,q are the eccentricity special functions related to the classical Hansen coefficients (Kaula (1966)). Note that J does no longer depend on Ω , ω , M and thus on time. Now the averaged Hamiltonian takes the form

J = J Kep (-, -, -, L, -, -) + J J2 (-, -, -, L, G, H) + J (-, g, h, L, G, H) .
(2.4)

In this paper we will deal with the resonances due to the presence of the Sun. Potentially, a large class of resonances of the form α ġ + β ḣ ≈ 0 exists [START_REF] Stefanelli | Proceedings of the 64 th IAC[END_REF], Hughes (1980)) but we focus on the resonance, that we will call 2:1, 2 ġ + ḣ ≈ 0 .

(2.5) Indeed, this resonance has one of the largest amplitude and it affects orbits with inclination close to 56 • or 110 • , concerning in particular the Galileo orbits.

To this end, we perform a suitable canonical change of variable (g, h, G, H) → (σ, ξ, Σ, Ξ) so that the resonant angle σ = 2g + h becomes one of the new angles; one possible choice is the following:

σ = 2g + h , Σ = H , ξ = g , Ξ = G -2H .
(2.6)

In terms of these new variables, the phase entering in J becomes :

(

2 -2p)g + mh = mσ + 2(1 -p -m)ξ .
The next step consists in eliminating the non resonant angles from the Hamiltonian. In other words, we select in the sum (2.3) only terms either free of angular variables or containing only the resonant angle σ:

J = -µ a 2
a 3 H 2,1,0 (e ) F2,0,1 (i) F2,0,1 ( )G 2,1,0 (e) + F2,1,0 (i) F2,1,1 ( )G 2,0,-2 (e) cos (σ) .

(2.7)

In the general case, the new Hamiltonian can be decomposed as

J = J (σ, Σ, Ξ; L) = J Kep + J J2,sec + J ,sec + J ,res ; (2.8)
where J Kep (L) = J Kep = J Kep and J J2,sec (Σ, Ξ; L) = J J2 ; we have split the perturbation due to the Sun into a secular part J ,res (Σ, Ξ; L), free of angular variables, and a resonant part J ,res (σ, Σ, Ξ; L).

3 Study of the resonance 2:1

The main idea of this section is to study the properties of a resonance by means of the study of the equilibrium points of the Hamiltonian system in the new canonical variables (σ, ξ, Σ, Ξ).

After performing the averaging and the change of coordinates described in the previous section, we will now study the new Hamiltonian, labelled K, function of the new canonical variables (σ, ξ, Σ, Ξ), given by

K = K(σ, ξ, Σ, Ξ) = K Kep + K J2,sec + K ,sec + K ,res .
(3.1)

The equations of motion associated to this Hamiltonian system are:

σ = ∂K ∂Σ , Σ = - ∂K ∂σ , (3.2) ξ = ∂K ∂Ξ , Ξ = - ∂K ∂ξ . (3.3)
Since the Hamiltonian does not depend on ξ, the action Ξ is constant and the value and time evolution of ξ are irrelevant, we take into account only the equations (3.2) that, in view of (3.1), are now:

σ = ∂K J2 ∂Σ + ∂K ,sec ∂Σ + ∂K ,res ∂Σ , Σ = - ∂K ,res ∂σ , (3.4)
with Ξ becoming a parameter set by the initial conditions. It can be shown that the system (3.4) takes the form σ = A(a, e, i) + B(a, e, i) cos σ , Σ = e 2 C(a, e, i) sin σ .

(3.5)

In particular, A is such that A(a, e, i) = f (a, e)P (cos i; a, e), where P (cos i; a, e) is a polynomial of degree 2 in cos i (with coefficients dependent on eccentricity and semimajor axis) admitting two roots i * 1 ≈ 56 and i * 2 ≈ 110 for each fixed couple of parameters a, e.

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Looking for the equilibrium points means to solve σ = 0, Σ = 0 ;

(3.6) note that in oder to have Σ = 0 we have three possibilities.

• e = 0: since B ∼ O(e), the equation σ = 0 reduces to A(a, 0, i) = 0 that admits the two aforementioned solutions (or resonant inclinations) i * 1 and i * 2 . Note that these are equilibrium points regardeless to the value of σ.

• If sin σ = 0, or equivalently σ = 0 or π, the first equation reduces to A(a, e, i) ± B(a, e, i) = 0 ;

(3.7)

for fixed a and e, this gives resonant inclinations i * * 1 and i * * 2 . Since the amplitude of the resonant Sun perturbation (related to the function B) is small, i * * 1 and i * * 2 are just slightly displaced w.r.t. the resonant inclinations i * 1 and i * 2 of the previus case. To resume, we have four equilibrium points, σ = 0 or π and i = i * * 1 or ı * * 2 and we can show numerically, plotting the phase portraits (figures 1 and 2), that the points such that σ * = 0 are two stable points (centers) and σ * = π are two unstable points (seddles). • Finally, if i = 0 or π we have C(a, e, i) = 0 and the first equation is satisfied if σ = ± arccos A(a, e, i = 0) B(a, e, i = 0) and the same for i = π. The stability of this class of equilibrium point will be investigated in a future work.

Conclusions

The aim of this work is to understand the effects of lunisolar resonances on the stability of Medium Earth Orbits over long periods (of the order of more centuries). The source of resonance is the presence of a third massive body, which can be the Sun or the Moon; this kind of resonance is due to combinations of frequencies of the satellite and of the third body. ) . Right: Time evolution of the eccentricity for seven orbits with initial conditions in a neighbourhood of the seddle point.

On the other hand, the orbit of the satellite, and thus its frequencies, is affected by many perturbations: mainly from the Earth's gravity potential, but also, at intermediate altitudes, by the potential of the third body itself; in this work we considered the Sun. Thus, beside the Earth oblateness, our model includes the secular terms of the third body potential and a resonant perturbation related to a selected resonance, 2ω + Ω. This seems to be among the most important lunisolar resonances and the associated resonant inclination is very close to the operational inclinations of the Galileo orbits.

We combined an analytical approach (especially in the case taking into account only the J 2 secular effect and the case of circular orbits) with direct numerical integrations.

The analytical approach, making use of an Hamiltonian model, allowed to study in detail this resonance, looking at the (linearized) equilibrium points of the averaged systems. This method is quite general and could be also extended to other resonances in the future. This allowed to charachterize the stabilty of equilibrium points for the simplest case (only J 2 secular effect) and ensure that, when adding the Sun perturbation, the equilibrium points are just slightly displaced. The numerical integration of the model states that the equilibrium points preserve their stability character and allows to study the time evolution of the other orbital elements of resonant orbits; in particular, the perspective is to focus on the behaviour of the eccentricity, which is of fundamental importance to ensure that orbital crossings do not occur.

LS is grateful to the French space agency (CNES) for her postdoctoral fellowship.

Fig. 1 .

 1 Fig.1. Initial eccentricicty e0 = 0.1. Left: Phase portrait in a neighborhood of one of the centres, (σ, i) = (0, i * * 1 ). Right: Time evolution of the resonant angle for seven orbits with initial conditions in a right neighbourhood of the centre point.

Fig. 2 .

 2 Fig.2. Initial eccentricicty e0 = 0.1. Left: Phase portrait in a neighborhood of the seddle point (σ, i) = (π, i * * 1 ) . Right: Time evolution of the eccentricity for seven orbits with initial conditions in a neighbourhood of the seddle point.

Geoazur, Université de Nice Sophia-Antipolis, Observatoire de la Côte d'Azur,

rue Albert Einstein, 06150 Valbonne (France) c Société Francaise d'Astronomie et d'Astrophysique (SF2A) 2013


Macarena Flores González
email: macarena.flores-gonzalez@cyu.fr

Fabrice Vandebrouck
email: vandebro@univ-paris-diderot.fr

Laurent Vivier
email: laurent.vivier@univ-paris-diderot.fr

A classic

Keywords: Mathematical work, Paradigm, mathematical activities, Activity Theory, control

come

Introduction

Many studies have characterized the significant transition between secondary school and university in mathematical education. This transition brings both continuities and discontinuities in teaching and learning [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF][START_REF] Gueudet | Transitions in Mathematics Education[END_REF]Winsløw et al., 2018;[START_REF] Gueudet | Secondary-tertiary transition in mathematics education[END_REF]Monaghan et al., 2019). In calculus, functions, limits, derivatives and real numbers are mathematical objects that have been widely studied, and are known to be problematic for students [START_REF] Thomas | Key Mathematical Concepts in the Transition from Secondary School to University[END_REF][START_REF] Bressoud | Teaching and Learning of Calculus[END_REF][START_REF] Oktaç | Conversion, Change, Transition[END_REF]. However, despite their importance, only a few studies have focused on sequences of the form 𝑢 !"# = 𝑓(𝑢 ! ) [START_REF] Boschet | Cours sur les suites numériques dans le premier cycle de l'enseignement supérieur[END_REF][START_REF] Weigand | Iteration sequences and their representations[END_REF][START_REF] Weigand | Sequences-Basic elements for discrete mathematics[END_REF][START_REF] Ghedamsi | Enseignement du début de l'analyse réelle à l'entrée à l'université : Articuler contrôles pragmatique et formel dans des situations à dimension a-didactique[END_REF][START_REF] Ghedamsi | Étude de l'évolution des images de la convergence de suites lors d'un enseignement ordinaire[END_REF]. These objects are interesting because they draw upon many notions found in calculus that students find difficult. Moreover, in science degrees, recursive sequences are often used to solve problems in science disciplines other than mathematics [START_REF] Krainer | Recursive sequences in first-year calculus[END_REF]. Studies such as [START_REF] Sierpinska | Some remarks on understanding in mathematics[END_REF], [START_REF] Mamona-Downs | Letting the Intuitive bear on the Formal; A Didactical Approach for the Understanding of the Limit of a Sequence[END_REF] and [START_REF] Przenioslo | Images of the limit of function formed in the course of mathematical studies at the university[END_REF][START_REF] Przenioslo | Introducing the Concept of Convergence of a Sequence in Secondary School[END_REF] highlight the importance of sequences, and their convergence, in the teaching and learning of calculus. In Germany, [START_REF] Weigand | Iteration sequences and their representations[END_REF][START_REF] Weigand | Sequences-Basic elements for discrete mathematics[END_REF] noted the role of representations as high school students discover the properties of recursive sequences. He also emphasized the need to train students in the interpretation of graphical representations with the help of computers. Our study focuses on the recognition of the properties of recursive sequences using representations given by technological tools, reflecting the transition from high school to university in France. The work we present here is a continuation of a previous study, which confirms that high school teaching of recursive sequences is insufficient in preparing students for higher education. At university, undergraduates are expected to work autonomously and master new knowledge (including formal notions). Although tasks remain in the domain of calculus, they are less clear-cut and give fewer indications. Students are expected to be able to recognize correct methods, but there is no help to check their results. In spite of its potential, and the competencies acquired by students in high school, a calculator is often kept aside or forbidden at university. In that context, how can we prepare students to recognize and control their work, in using both their theoretical knowledge and their instrumental competencies with calculator? In section 2, we discuss the importance of the recursive sequences in the curriculum, and outline the main difficulties encountered by students. In Section 3, we present the theoretical tools used in our study, which are mainly drawn from Activity Theory in Didactic of Mathematics [START_REF] Vandebrouck | Activity Theory in French Didactic Research[END_REF]. Our methodology is presented in Section 4. Section 5 presents the a priori analysis of our experimental exercise, while Section 6 outlines our a posteriori analysis.

Learning about recursive sequences

From advanced mathematics to the curriculum

Recursive sequences are of great interest in many mathematical problems at university level. They are used to find solutions to equations 𝑔(𝑥) = 0, using for instance Newton's numerical method that ask to study the convergence of a sequence (𝑢 ! ) defined by 𝑢 !"# = 𝑓(𝑢 ! ) with a new function 𝑓(𝑥) = 𝑥 + 𝛼(𝑥)𝑔 (𝑥). Examples also include the Cauchy Lipschitz (Picard-Lindelöf) method for solutions of differential equations, the implicit function theorem, multivariable analysis, differential geometry, fractal and dynamical systems theory and so on. Moreover, the structure of ℕ, and specially the notion of successor, is crucial for the definition of recursive sequences (and induction). Recursive sequences cannot be interpreted only as functions between two sets. Given their importance in the mathematical universe, in France, recursive sequences appear both at the end of secondary school, and at the start of university. First of all, in high school, sequences constitute an important part of calculus: in 12th grade, it is almost a quarter of the calculus curriculum in the textbook [START_REF] Barbazo | Mathématiques Tle Spécialité[END_REF]. Calculus starts with two chapters on sequences; one is dedicated to recursiveness. This concerns the students who choose the spécialité mathématique option (taken by 41% of the students in 2020 1 ). In 2019, at the baccalaureate2 , there was a study of a recursive sequence with a parameter. High school exercises generally present the task as follows. First, study the function that generates the sequence: determine any fixed points (𝑓(𝑥) = 𝑥), variations3 and, in some cases, the stability of an interval. Then, study the sequence: look for monotony, bounds (often with a proof by induction), convergence (using the theorem: "any monotone and bounded sequence is convergent") and the value of the limit. Thus, at high school, the study of recursive sequences is carried out with algebraic techniques and, in some cases, they propose either a graphical representation or the use of the calculator. On the other hand, in the first year at university, students begin with some preliminary exercises (limits, continuity and differentiability of functions), before moving to tasks based on recursive sequences of the form 𝑢 !"# = 𝑓(𝑢 ! ). As an example, on the exercise sheet about real numbers and sequences for the first semester at the University, one fifth of the exercises directly or indirectly concern such recursive sequences, without any graphical representation and without the use of the calculator. At this stage, students are expected to use a wider range of theorems and methods. In addition to the methods and theorems studied in high school, students work with a formal definition of convergence, monotonic, adjacent and extracted sequences, the Bolzano-Weierstrass theorem, the extension of broad inequalities, and the two theorems: 1) if lim

$→& 𝑓(𝑥) = 𝑙 and if lim 𝑢 ! = 𝑘 ⟹ lim 𝑓(𝑢 ! ) = 𝑙; 2) if for all sequences (𝑢 ! ) convergent to 𝑘 one has lim 𝑓(𝑢 ! ) ⟶ 𝑙, hence lim $→&
𝑓(𝑥) ⟶ 𝑙. Moreover, neither the proof by induction nor the study of the recursive sequences are explicitly addressed in the lecture course.

Students' difficulties in the high school-university transition

From a didactic perspective, recursive sequences, together with the concepts of convergence and limit, bring difficulties for students. These topics combine two objects: sequences and functions, which are not yet mastered by students at the end of high school and the beginning of university [START_REF] Rousse | Discret et continu au lycée. Enjeux de ces notions à travers l'étude de l'enseignement de l'[END_REF]. Concerning semiotic representations, representations such as the "cobweb diagram" (Figure 1), frequently used in teaching, are not understood by students and are not at all helpful [START_REF] Weigand | Iteration sequences and their representations[END_REF]. (Krainer, 2015, p. 301).

In our previous study [START_REF] Flores González | Suites définies par récurrence dans la transition lycée-université : activité et travail mathématique[END_REF], we compared some assessment tasks at the end of high school and in the first year of university in France about recursive sequences, and we analysed students' responses at both levels of education. Success rate was low, especially at the university level. Students easily confuse exercises that are based on techniques from the domain of sequences and those from the domain of functions. In spite of few similarities in both domains, overall for functional sequences of type 𝑓(𝑛) and sometimes with subtility like in "if 𝑓 is increasing then 𝑢 !"# = 𝑓(𝑢 ! ) is monotone", most calculus techniques are different; derivative computation or proof by induction are specific to, respectively, functions and sequences. There is no link between the limits of 𝑓 and the limit of recursive sequence (𝑢 ! ), and the existence of a recursive sequence is a specific problem, even if it remains implicit in teaching. Some students did not recognize recursive sequences as specific sequences, and treated them as 𝑢 ! = 𝑓(𝑛) (see Figure 2). Thus, many students were unclear about the distinction between sequences and functions and were unable to produce coherent mathematical work. Finally, the task did not use a numerical tool to explore the recursive sequence in either the last year of high school or the first year of university. In this paper, we design and experiment a task at high school level aiming at preparing future students for the recognition and mathematical control activities in the field of recursive sequences.

Theoretical tools

Activity Theory in Didactic of Mathematics

Adapted from Activity Theory, Activity Theory in Didactic of Mathematics (ATDM) was developed about 20 years ago [START_REF] Vandebrouck | Activity Theory in French Didactic Research[END_REF]. ATDM models epistemological and cognitive aspects by differentiating between task and activity [START_REF] Rogalski | Theory of Activity and Developmental Frameworks for an Analysis of Teachers' Practices and Students' Learning[END_REF]. Drawing on the tools that ATDM uses to describe and interpret the student's activity, we identify both the mathematical knowledge needed to solve the task and the way it must be used in terms of adaptations of knowledge. Examples of adaptations of knowledge are: mixtures of knowledge; the use of intermediaries; a change of mathematical domain; the introduction of steps; the introduction of results from previous questions, etc. [START_REF] Horoks | Tasks Designed to Highlight Task-Activity Relationships[END_REF]. The direct application of a theorem does not need any adaptation. Adaptations of knowledge leads us to the concept of individual mathematical activities among students. This emphasis on the cognitive dimension of the student's activity can explain the process of internalisation [START_REF] Vygotsky | Mind in society. The development of higher psychological processes[END_REF] from a socio-constructivist point of view, providing, as Simon et al. (2018, p. 2) note, "a complement to sociocultural theory".

Our study expands on complex tasks (those that require adaptations of knowledge), their context, the mediations, and the student's activities. We focus especially on two critical mathematical activities that are characteristic of complex tasks [START_REF] Robert | Proximités en acte mises en jeu en classe par les enseignants du secondaire et ZPD des élèves : analyses de séances sur des tâches complexes[END_REF]):

• Recognizing activities mainly occur in a context where students have to recognize the mathematical knowledge they can use to solve the task they are given. They may also be asked to recognize how they can apply or adapt this knowledge. Students can also recognize a method and that various steps in their reasoning can be connected. • Control activities are found when students must highlight that their mathematical reasoning is coherent, by introducing several check points. They also ensure that the answer produced corresponds to the intended goal of the activity.

Tasks of recursive sequences are complex, due to the mix of knowledge about functions and sequences and the need for students to adapt this knowledge. These tasks lead specifically to recognizing and control activities4 . Recognizing activities are developed when students autonomously choose the method and the associate knowledge that will help to solve the task: how to determine the variation of a sequence, to calculate the sign of 𝑢 !"# -𝑢 ! or compare

' !"# ' !
with 1, to use a proof by induction or not, to find out whether the sequence is bounded above or below, to select a possible theorem among all the theorem about sequences, to recognize the need of the results of the previous questions etc. Of course, mediations can help students recognize the method to be applied. Their form depends on the context (i.e., it may come from the problem statement or from the teacher). However, mediation often reduces the cognitive demand. On the other hand, control activities are seen when students relate different elements and properties of the recursive sequence to their results.

Paradigms of analysis

The transition from school to university is marked by significant differences in mathematical work. For instance, calculators are widely used in high school, but are often not allowed in French universities.

In the following, we refer to the MWS notion of paradigms [START_REF] Houdement | Paradigmes géométriques et enseignement de la géométrie[END_REF]. This notion allows us to identify different types of work that are internally coherent. In the domain of analysis, Montoya Delgadillo and Vivier (2016, pp. 742-743) distinguish three paradigms:

• Arithmetic/geometric analysis [A1]. This supports interpretations that draw upon implicit assumptions based on geometry, arithmetic calculations or the real world.

Although argumentation plays a role, work is grounded on the visualization of signs, possibly produced by software or a calculator. Visualizing the curve of a function or a table of values is admissible as a proof in [A1].

• Calculation analysis [A2]. The rules of the calculation are defined more-or-less explicitly, and are applied independently of any reflection on the existence and nature of the objects in question. Calculations are often based on an algorithmic approach, along with formal expressions that have a representative role. These routines are executed without being aware of the nature of the mathematical objects. Work is oriented toward the production of proofs and demonstrations, using the properties of the objects and theorems, in the mathematical tradition. Visualizing the curve of a function or a table of values is not admissible in [A2]. However, tools, such as computer algebra system software may be used (to compute a derivative, for instance).

The third paradigm appears at university level, especially when students begin to manipulate the epsilon definitions. Our study on control activities focuses on paradigms [A1] and [A2], leaving the third paradigm for further investigations.

In the context of a recursive sequence of the form 𝑢 !"# = 𝑓(𝑢 ! ), paradigm [A1] consists in working with the representation of the first terms of the sequence, for example in a table of values or graphically, and visualizing, for instance, monotony or convergence. The work can be controlled, for example, by calculating more terms or by zooming in or out on a graph. This is similar to how a mathematician studies a new object for which no procedure is available. On the other hand, work in the [A2] paradigm consists in working with algebraic calculations and theorems. For example, showing by induction that the sequence is bounded by 1, that 𝑢 !"# -𝑢 ! is positive, therefore the sequence is increasing, to conclude, by theorem, that it is convergent.

It can be difficult to evaluate a student's work in terms of a single paradigm, and the paradigms given above should not be seen as a way to categorize students. Rather, they offer a way to understand how the work is expected by the institution and the work actually done. Moreover, paradigms are not isolated from each other and the links between them can promote mathematical control. For example, paradigm [A1] can be used to develop a conjecture and, thus, to guide the work in paradigm [A2] which helps to prevent errors and promote a mathematical coherence (control). Consequently, mathematical work is based on a dialectic between paradigms; the same idea is seen in geometry, where few would attempt to make a demonstration without preparing a figure.

However, in the French context, the teaching of recursive sequences at the transition from high school to university focuses on the [A2] paradigm which guides the expected work. Work in the [A1] paradigm appears mainly at the beginning of the teaching in high school to introduce new notions and to make (the correct) conjectures before switching to a work in [A2], without going back to an [A1] work. Only a little part of the [A1]/[A2] dialectic is used and there is almost nothing about control, which causes difficulties for students at university (Flores González, 2021).

Hypotheses and research question

The study aims to deepen the understanding of the transition from high school to university in the domain of calculus. In France, recursive sequences play an important role in the curriculum of both institutions with specific difficulties for students at university level. In this sense, the research is based on the following two hypotheses: (1) in order to be better prepared for the mathematical work which awaits them at university, students need to develop recognizing and control activities for recursive sequences [START_REF] Flores González | Suites définies par récurrence dans la transition lycée-université : activité et travail mathématique[END_REF]; (2) the dialectic between [A1] and [A2] paradigms seems to have a good potential for the development of these two activities in the field of calculus.

Then, focusing on high school students, we wonder to what extent does a [A1]/[A2] dialectic paradigms support the development of the recognizing and control activities expected in the study of recursive sequences?

4 Method: the design of an experimental task

In order to answer our research question, taking into account our previous study, we design an experimental task devoted to promote a dialectic between [A1] and [A2] paradigms aiming at improving the students' recognizing and control activities. We pay particular attention to the calculator, an emblematic tool of the transition that enables students to produce numerical or graphical representations on which a work in [A1] is possible.

In our previous study [START_REF] Flores González | Suites définies par récurrence dans la transition lycée-université : activité et travail mathématique[END_REF], we studied an exercise of first year of a university in the Paris region, on the recursive sequence

𝑢 !"# = # ()' ! with 𝑢 * = 𝛼 in [0,1].
This exercise was complex, above all because of the parameter 𝛼. In this case, the sequence and the function 𝑓(𝑥) = # ()$ were interlinked and required students to be able to recognize, and control, the mathematical objects they are working with. In a baccalaureate exercise, this kind of task was divided into two parts: the first related to the function, and the second to the recursive sequence. With these two exercises, we designed the experimental complex task in adopting the structure of the baccalaureate, in two parts, with the sequence of the university exam. Finally, and consistent with our research question, we introduced a part that ask for a work in the [A1] paradigm to promote a dialectic between the [A1] and [A2] paradigms. Then the exercise was divided into three parts: 3, below) which properties are needed to study the sequence. Technical questions are stated like in the baccalaureate.

• Part A concerned the function 𝑓(𝑥) = # ()$ (Figure
• Part B was directly related to the study of the sequence 𝑢 !"# = 𝑓(𝑢 ! ) in the [A1] paradigm. After using their calculator to calculate the first terms of the sequence, we asked students to develop conjectures about the properties of the sequence. • Part C was dedicated to proving the properties in the [A2] paradigm.

Hence, part B (paradigm [A1]) prepares students for part C (paradigm [A2]

). Part C is designed to reflect undergraduate exercises; the questions are more open than in the baccalaureate task, and require undergraduates to develop recognizing activities. For instance, proof by induction is not explicitly stated, and the question about the variation of the sequence is less directive than in the baccalaureate -students are asked to study it once they have worked on part B.

Our aim is to identify students' ability to recognize the sequence as a recursive one 𝑢 !"# = 𝑓(𝑢 ! ), especially the clear distinction between the properties of the function and the properties of the sequence. For instance, 𝑓 is increasing on [0,1] and this property only implies that the sequence is monotone. From the point of view of control activities, our aim is to identify examples of such activities or, on the other hand, inconsistencies indicative of a lack of control in part C. For this, part B includes an explicit instruction for using a calculator. By experimenting in this order (paradigm [A1] in part B for conjectures, and paradigm [A2] in part C for proving), we want to promote control and recognizing activities relying on the [A1]/[A2] dialectic. This order is known to high school students, but only at the beginning of the teaching and not, as in our experimentation, after the teaching where [A2] is usually predominant.

Our experiment was run with a 12th grade mathematics class in a general public high school in Paris. It took place at the beginning of the school year, after the teaching of limits of sequences and functions. Thus, the students had already worked on proof by induction, finite and infinite limits of a sequence, general theorems on the limits of sequences and functions (operations and comparisons), and limits of usual functions (polynomial, exponential). The teacher regularly used the scientific calculator with the students.

The class was composed of 30 students, named E1 to E30, who worked individually with their calculators (Texas Instruments TI 83 and CASIO graph) without any interaction with the teacher. A researcher was in the classroom and took some photos when she noticed interesting work. We analysed the students' written answers.

A priori analysis of the designed task and students awaited activities

In this section, we present the three parts of the designed tasks, detailing the possible In part A (Figure 3), work can be done in either of the two paradigms. But we do not present an analysis of part A, as we focus only on parts B and C.

Part B

Part B begins (Figure 4) with three tables to be filled in, by computing values of 𝑢 # to 𝑢 + for three values of 𝛼, 0, 0.5 and 0.9 (Table 1).

*macarena.flores-gonzalez@cyu.fr, vandebro@univ-paris-diderot.fr, laurent.vivier@univ-paris-diderot.fr The purpose of question B1 was to identify and become familiar with the sequence using explicit examples for three values of 𝛼. In theoretical terms, we expected students to recognize the recursive sequence 𝑢 !"# = # ()' !

(not explicitly given in the problem statement). This recognition was expected to lead to a control of the terms of the sequence using the calculator which here was essential: computing 𝑓(𝑛) leads to a problem of definition of the sequence for 𝑛 = 2 and the three tables, for different value of 𝛼, are the same.

Note that the calculator could give two numerical representations: decimal (Table 1) and fractional. The former (the one we were hoping to see) is more helpful in answering the following questions (B2 and B3). Although the latter is less useful, it does help in conjecturing the algebraic expression of the sequence 𝑢 ! = ! !"# . The choice of representation can either be conscious, or 'decided' by the calculator's configuration. If conscious, it could help students to either develop a control activity (using the expression of 𝑢 ! or the calculator).

The following two questions (Figure 5) asked students to develop conjectures based on the table of values. Questions B2 and B3 asked the student to develop conjectures for the variation, convergence and limit of the sequence based on their observation of its first six terms, for each given value of 𝛼. While the monotony of the sequence is quite clear from the first terms (possibly supported by a control using the calculator to compute more terms), this is not so easy for the convergence and the limit, which are a local property at infinity and cannot be clearly determined from the information of B1 5 . Thus, some non-responses to the question could come from a good understanding of the values given in the table and the notion of convergence.

We anticipated that students would recognize (increasing) monotonicity, convergence, and that the limit of the sequence is 1. The recognizing activities initially rely on the interpretation of signs, by visualization. Then, students develop their mathematical discourse and the conjecture that it is an increasing, convergent sequence, and that its limit is possibly 1.

In order to stimulate control of the mathematical object and successful recognition, we added the question "How did you use the calculator?". The aim of this question was to make the student aware of how they used their calculator (and of the possibility to use it!) and stimulate them to recognize the coherence (or non-coherence) of their approach. These elements were expected to encourage them to control their mathematical work by, for example, programming an algorithm, developing a graphical representation, or calculating more terms of the sequence.

Part C

In part C, we expected students to recognize theorems, procedures and proofs related to each of the questions. The task brings many recognizing activities which are characteristic of the university level. Mathematical work was anchored in the [A2] paradigm in each of the questions with possible controls grounded in their work in part B. Moreover, some students may work in paradigm [A1] based on their answers to part B.

For the sequence (𝑢 + ), defined in part B.

C1) Show that for all natural numbers 𝑛, 𝑢

+ is in [0,1].

C2) What is the variation of the sequence (𝑢 + )?

5 It is difficult to move from a punctual perspective based on a few values, to a local perspective: can it be established that the sequence ln(𝑛) converges to 3, given the 20 first values? Or that the harmonic series converges to 4 from the 30 first values? *macarena.flores-gonzalez@cyu.fr, vandebro@univ-paris-diderot.fr, laurent.vivier@univ-paris-diderot.fr C3) Show that the sequence (𝑢 + ) is convergent. C4) We denote 𝑙 as a limit of the sequence (𝑢 + ), explain why 𝑙 = 𝑓(𝑙). C5) Given 𝑙 = 𝑓(𝑙), compute the value of 𝑙. In question C1, we expected to see a recognition activity, with a demonstration by induction of the steps to be taken (as in the university exam). We interpret that students have a control activity if they explicitly relate their proofs to the conjectures of part B.

In question C2, we expected to see recognition of a method to determine the increasing monotonicity of the sequence, as in the original university exam. There are various options.

Students can either continue to use a proof by induction (showing that for every 𝑛, 𝑢 ! ≤ 𝑢 !"# ), using directly -or not -the growing property of the function 𝑓 from part A, or analyse the variation of the sequence with algebraic expressions 𝑢 !"# -𝑢 ! or

' !"# ' !
. For the former strategy, we expect them to recognize the identity

(𝑢 ! -1) ( in the numerator of 𝑢 !"# -𝑢 ! = ' ! $ )(' ! "# ()' ! ,
to conclude, using 𝑢 ! ≤ 1, that 𝑢 !"# -𝑢 ! is positive and, then, that the sequence is increasing.

In the latter case, the student must be able to recognize the quadratic polynomial in the numerator and use their answer to question C1 (i.e. that 𝑢 ! is in [0,1]). A lack of coherence with the part B answers is seen as a lack of, or absence of, controls (notably when the students develop an interpretation without having recognized the method).

Finally, question C3 only required the students to recognize the monotonic convergence theorem. Here, students can use their results from previous questions. From their answer to C1 they can interpret that the sequence (𝑢 ! ) is bounded by 1; from C2 they can interpret that the sequence (𝑢 ! ) is increasing; therefore, by using the theorem, it is convergent. We also expected students to check their work (control activities) by linking their answer to question C3 with the conjectures developed in part B.

C4 and C5 are quite technical and we do not analyse them, except the value of limit 𝑙 with respect to conjecture of part B.

In this way, the design of the task and its a priori analysis are directly related to the hypotheses outlined in section 3.3.

A posteriori analyses of the part B and C

We gave students' work one of the three statuses: 1) correct mathematical work (the mathematical activities were the expected ones); 2) incomplete or not totally correct mathematical work (discourse and written assertions that were mathematically true, but only partially justified); and 3) non-correct mathematical work (when the discourse and written assertions were false or did not correspond to what was expected). In the analyses below, this status of work is combined with an analysis in terms of mathematical activities and paradigms.

As previously mentioned, we do not analyse part A. We give a report about part B, the original part of our study, then we focus on part C and the influence of work in part B.

Part B

Question B1: recognizing of the recursive sequence

First, we distinguished between students who correctly recognized the recursive sequence as it is defined,

𝑢 !"# = 𝑓(𝑢 ! ) = # ()' !
, and those who did not. In particular, we identify the students who incorrectly recognized the sequence as 𝑢 !"# = 𝑓(𝑛) = # ()! . About half of the students succeeded in recognizing the correct recursive sequence (17 out of 30 students): fourteen students correctly used the calculator, which led them to produce correct tables of values, and the other three produced incomplete tables. For the other thirteen students: nine confused 𝑓(𝑢 ! ) with 𝑓(𝑛), three did not answer the question, and one filled the table with zeros.

The answers given by student E22 (Figure 7) illustrate the typical confusion between 𝑓(𝑢 ! ) and 𝑓(𝑛), with three identical tables, together with an explanation of the shift 𝑛 -1 (however, the student calculates # ("! ). This work in paradigm [A1] shows that many students found it difficult to correctly distinguish between recursive and functional sequences. As this distinction requires theoretical knowledge about recursive versus functional sequences, it is unlikely that they will produce correct work in paradigm [A2]. Consequently, in the following analysis, we focus only on the seventeen students who correctly recognized the sequence even if their work was non-correct.

An example of control, possible with the use of the calculator

The following figures exemplify one type of control. The student E8 uses their calculator to calculate the values of the sequence. E8 is confronted with a systematic error message (Figure 8 shows the message "ERREUR" on the calculator's screen). We can make the hypothesis that E8 has a sufficient level of control to be able to understand the lack of coherence. E8 reviews its calculations. The whole process of E8 is not clear, but E8 may start by computing 𝑢 # = 𝑓(1) = 1 and wrote "ERROR" on the sheet as on the calculator screen (Figure 8). Hence, starting with a 𝑓(𝑛) interpretation, the calculator helped E8 by giving feedback that allowed E8 to produce the correct table (Figure 9). Moreover, it should be noted that this student is the one who performed best during the whole task. This raises the question of whether some students correctly recognize the sequence because they are explicitly asked to use a calculator to compute the first terms, and thus receive the ERROR feedback of the calculator for a 𝑓(𝑛) interpretation.

Questions B2 and B3: recognizing growth, convergence and limit

Among the seventeen students who successfully recognize the recursive sequence in B1, fifteen answered B2 (Table 2). For the twelve students (12-C) who had produced correct tables in B1: seven recognized the growth, the convergence and the limit of the sequence (7-C); one did not recognize convergence, one found that the limit was 0.9 and one only recognized the growth (3-I); finally, two found that the limit was 0 (2-NC). Students are better able to recognize growth compared to convergence or the value of the limit.

For the three students who had produced incomplete tables of values in B1 (3-I): two recognized that the sequence was growing, and one of them also recognized a convergence towards 0.9 (2-I); and one recognized both convergence and divergence (1-NC).

For the two students who consider the limit as 0.9, there is a lack of control. Indeed, calculating additional terms would enable to identify the mistake. Maybe there is a lack of control also for students who answer "limit is 1", but there is no evidence.

Among these fifteen students, ten gave an answer to B3, and only five with correct work. Indeed, recognition of growth, convergence and the limit are much more successful for the three fixed values of α (B2) than for the generalisation (B3). As this question asks the student to develop a conjecture (or the generalisation of another conjecture) based on their answer to B2, the falling success rate from B2 to B3 is unsurprising.

Note that only one student worked with the fractional representation of the terms of the sequence given by the calculator: their work was correct for B2 and incomplete for B3.

For the following, we decided to focus on these fifteen students who answered B2 (since very few students were able to answer B3) and recognized, even partially, the properties of growth, convergence and the limit of the sequence.

Part C

Question C1: the sequence is bounded

Table 3 presents a summary of the work in question C1 done by the fifteen previous students. This table indicates the paradigms, the recognition of the proof by induction and the correctness of the work. We expected that these students would check the coherence of their discourse in part C with their answers in part B.

From the students of table 2, we analyze separately: the seven students who produced a correct solution to B1 and B2; the three who produced correct tables in B1 and an incomplete solution to B2; the two students who produced incomplete work both in B1 and B2; and finally, we regroup the three students who did not produce a correct solution to B2 (two B1-C and one B1-I). Of the fifteen students who answered B2, five could not give any answer to C1 (NR), and only one of these five have produced a correct work in B2. Correct work in paradigm [A1] (part B) therefore seems to be a good precursor for the work that is expected in paradigm [A2] (part C). We can hypothesize that the more students are able to work in part B, the more they are prepared to address question C1.

It should be noted, however, that among the seven students who produced correct work in B2 (and who answered question C1), four spontaneously continued to work in the [A1] paradigm.

Although working in the [A1] paradigm seems to be good preparation for switching to [A2], making the switch appears to be an obstacle for some students.

Five students worked in the [A1] paradigm: four produced incomplete work but they were able to explicitly control their work in relation to part B. For example, student E15 argues that the sequence converges to 1 but does not reach it, so it is in [0,1]. Then, they say that they can use the calculator's table function.

One student (E4), who had produced an incomplete work in B1 and B2, gives an intermediate work between [A1] and [A2]. They base their arguments on the graph of the function 𝑓 (Figure 10). Here again, although their answer is not in line with expectations, there is a control based on the graph of 𝑓 and they produce an incomplete work. Four students worked in the expected [A2] paradigm. While two provided correct work to B1 and B2, the remaining two correctly recognized the growth of the sequence but did not give correct work to the convergence question (notably, a value of 0.9 for the limit). These four students started with a proof by induction: one provided a correct work, two provided an incomplete work, and the fourth (E10) switched to the model 𝑢 ! = 𝑓(𝑛). Finally, among the fifteen students, only one student produced a non-correct work (E14) and only one produced fully correct work (E8). Eight students did incomplete work, in paradigms [A1] or [A2] (4 with an attempt of proof by induction).

Question C2: the sequence is increasing

All of the ten students who answered C1 also answered C2. This is consistent with the idea that working in paradigm [A1], even in question C1, supports students' work up to question C2.

Table 4 presents a summary to question C2. Data only refers to the ten students who answered both questions C1 and C2 (and who recognized the recursive sequence in B1 and answered B2). It shows the paradigms, the two most-used strategies 6 and the correctness of the work. (where they had worked with the 𝑢 ! = 𝑓(𝑛) model) to [A1] in question C2, this time with a generalisation, saying that "the sequence seems to be increasing because 𝑢 * < 𝑢 # < 𝑢 ( … 𝑢 !"# ". They probably relied on their correct tables of values produced in part B, however, their work is incomplete in C2.

For the tenth student considered (E4), it is difficult to identify a paradigm: they argue that the sequence is increasing by referring to the increasing curve of the function. The theorem-in-act [START_REF] Vergnaud | La théorie des champs conceptuels[END_REF] "𝑓 is increasing therefore (𝑢 ! ) is increasing" that they seem to use may either hide a confusion between sequences and functions, suggesting a model of the type 𝑢 ! = 𝑓(𝑛), or indicating a transfer of the properties of 𝑓 to the sequence in the case of the model 𝑢 !"# = 𝑓(𝑢 ! ). Four of the students in class applied this theorem-in-act.

Question C3: the sequence is convergent

We recall that fifteen students provided at least partially correct answers to questions B1 and B2. Ten of these provided answers to questions C1 and C2. Of these, only seven managed to answer to C3.

Table 5 summarizes answers to C3 given by the seven students: paradigms, recognition of the theorem and the correctness of the work. ] to answer all of the questions in part C. In particular, one of them worked with both references to the tables of values and the use of a calculator (Figure 13).

According to the calculator, the sequence converges towards 1 but doesn't reach it Yes, with the table

Non-correct work in [A1] (part B) does not produce a correct work in [A2] (part C)

Among the nine students who used 𝑢 ! = 𝑓(𝑛) in B1, 5 do not answer Part C. Of the remaining four, with respect to C1: two work with induction ([A2] paradigm), but they do not necessarily follow the rules of the proof (1 NC and 1 I). One still works in the [A1] paradigm (I) and one only wrote "For 𝑛 ∈ ℕ, 0 ≤ 𝑢 ! ≤ 1" (I) ([A2] paradigm). In C2, two of these four students make the table of variation to show the growth of the sequence (NC); only one states the strategy 𝑢 !"# -𝑢 ! (I), and one student writes that the sequence is growing but provides no proof (I).

For the answers to C3, 2 do not answer, one writes that the sequence is diverging (NC), and the last one that the sequence converges to 0 (NC) (relative to their working paradigm which we cannot tell). These observations are consistent with our choice in section 6.2 to focus only on the fifteen students who answered B2 and recognized, even partially, the properties of growth, convergence and the limit of the sequence.

Conclusion

Our work aims to deepen the understanding of the transition from high school to university with respect to recursive sequences 𝑢 !"# = 𝑓(𝑢 ! ).

Considering our hypotheses and our research question, the task sought to determine the extent to which the [A1]/[A2] dialectic could be used to develop recognition and control activities in students. We start by summarizing the results for parts B and C. Then we answer, partially, our research question.

Recognizing and control activities in [A1] and [A2] paradigms

In part B, fewer than half of the students correctly recognized the recursive sequence, its growth and limit. Hence, the earlier practical work in the [A1] paradigm did not help these students to better recognize the form of the sequence: this recognizing activity is not obvious for many students. This pleads for more work in the [A1] paradigm.

However, some students have already used some control activities in [A1], like E8 in B1 (section 6.1.2) whose control allows him to avoid the error of considering 𝑓(𝑛)… and to do very good work up to C3.

In part C, some students continued to work in paradigm [A1] partly based on their work in part B, with controls activities, and their work was incomplete. It seems that working in paradigm [A1] was helpful for these students.

Students who worked in paradigm [A2] recognized the need for a proof by induction; this recognition seems to be a strong indicator of the switch to paradigm [A2] to study recursive sequences. However, some students who worked in the [A2] paradigm produced incorrect solutions and did not relate their work to part B (no control).

We saw paradigm shifts between questions C1 and C2 (from [A1] to [A2] and from [A2] to [A1]). A work in the [A2] paradigm required students to recognize an adequate method, but these students were unable to produce correct work. Moreover, it seems that they cannot control their work.

Only one student answered all the questions correctly. They are the only one to answer C1 correctly and also the only one to recognize the correct theorem in C3. This student appeared to be able to control their work based on part B, stating that "the sequence converges towards 1 and not 0.9" (his conjecture).

What can be said about the research question

In our previous study [START_REF] Flores González | Suites définies par récurrence dans la transition lycée-université : activité et travail mathématique[END_REF], we concluded that a dialectic between paradigms could be interesting to help students improve their understanding of sequences 𝑢 !"# = 𝑓(𝑢 ! ). In this new research, we wonder to what extent does a [A1]/[A2] dialectic paradigm support the development of the recognizing and control activities expected in the study of recursive sequences. With this aim, we designed a task that was given to students of one class at the end of high school.

Obviously, the results are limited to the sample, and further investigations would be required if we wanted to extend them. However, we want to point out the following conclusions:

-Some students recognized the adequate knowledge, whether in [A1] or in [A2] paradigms, sometimes with control;

-The use of calculator favors control, even if it is not automatic;

-A lot of students did not make any connection between paradigms [A1] and [A2], therefore blocking any fruitful control relying on the dialectic [A1]/[A2].

If the two first points prove the potential of the task with respect to the development of recognizing and control activities, it did not succeed in engaging the majority of students in this way. The third point is clearly a constraint, but it is worth mentioning that developing such activities is a ongoing work in progress.

Other important findings

In French high school, working in paradigm [A1] is asked mainly at the beginning of a chapter when new notions are introduced. There is an implicit hypothesis that students naturally connect 'elementary' work in [A1] and 'serious' work in [A2]. But this is not so easy. Our study clearly shows that [A1] and [A2] are disconnected for most of the students. Are they aware that they are dealing with the same mathematical object?

This has a consequence on the understanding that students might develop. Indeed, the recursive sequence is not recognized by a lot of students even for 'elementary' computations. What representations therefore of such a mathematical object do these students have? If they cannot make this recognition in [A1] paradigm, what is the meaning for them when working in the expected [A2] paradigm? It is paradoxical to think that such students would be able to correctly work in paradigm [A2], which is more abstract, when it appears that they are not even able to use the relevant recognition and control activities in paradigm [A1].

Figure 1 :

 1 Figure 1: Examples of cobweb diagrams(Krainer, 2015, p. 301).

Figure 2 :

 2 Figure 2: First year university student who does not recognize the correct sequence.

  (and expected) recognizing and control activities both in paradigm [A1] (part B) and paradigm [A2] (part C) as well as the control activities based on the [A1]/[A2] dialectic (going back to part B when working in part C).

Figure 3 :

 3 Figure 3: Part A devoted to the study of 𝑓.

Figure 4 :

 4 Figure 4: First question in part B.

Figure 5 :

 5 Figure 5: Questions B2 and B3, [A1] paradigm.

Figure 6 :

 6 Figure 6: Questions in part C, [A2] paradigm. For each question, students were asked if thecalculator could be used and, if so, to explain how.

Figure 7 :

 7 Figure 7: E22's answers to B1.

Figure 8 :

 8 Figure 8: E8's answers before control.

Figure 9 :

 9 Figure 9: E8's answers given after control.

Figure 10 :

 10 Figure 10: Solution provided by E4 to C1.

  With no anormal variation in the bound [0,1] and as the curve is increasing, then for any whole number n, 𝑢 % is in [0,1].

Four

  out of ten students worked in the [A2] paradigm. Three of these had already worked in [A2] to answer C1 and one after working in [A1] in question C1 (E14). Among the first three, one produced a correct work and one produced an incomplete work (both by using the strategy 𝑢 !"# -𝑢 ! ). One student (E25, Figure11) produced a non-correct work without control (because of the confusion 𝑢 ! = 𝑓(𝑛)) -while an incomplete proof by induction in C1 was accompanied by the correct expression of the recursive sequence.

Figure 11 :

 11 Figure 11: E25's answer to C2.

Figure 12 :

 12 Figure 12: E14's answer to C2.Five students worked in the [A1] paradigm. Four had worked also in [A1] to solve question C1: two produced an incomplete work based on tables of values or the graph given by the calculator, and two produced a non-correct work. The fifth (E29) switched from [A2] in question C1 (where they had worked with the 𝑢 ! = 𝑓(𝑛) model) to [A1] in question C2, this time with a generalisation, saying that "the sequence seems to be increasing because 𝑢 * < 𝑢 # < 𝑢 ( … 𝑢 !"# ". They probably relied on their correct tables of values produced in part B, however, their work is incomplete in C2.

Figure 13 :

 13 Figure 13: E15's answer to C3.

Figure 14 :

 14 Figure 14: E14's answer to C3.

Table 1 :

 1 Expected answers to question B1, in decimal register. 𝜶 𝟏 = 𝟎 Case 𝜶 𝟐 = 𝟎, 𝟓 Case 𝜶 𝟑 = 𝟎, 𝟗

	Case 𝑢 $	0	𝑢 $	0,5	𝑢 $	0,9
	𝑢 %	0,5000	𝑢 %	0,6666	𝑢 %	0,9090
	𝑢 &	0,6666	𝑢 &	0,7500	𝑢 &	0,9166
	𝑢 '	0,7500	𝑢 '	0,8000	𝑢 '	0,9230
	𝑢 (	0,8000	𝑢 (	0,8333	𝑢 (	0,9285
	𝑢 )	0,8333	𝑢 )	0,8571	𝑢 )	0,9333
	𝑢 *	0,8571	𝑢 *	0,8750	𝑢 *	0,9375

Table 2 :

 2 Summary of the work done by fifteen students at B2 who recognized the recursive sequence

	Class	Tables	Recognition in B2	Paradigm		Correctness
	number in	for B1
	B2
			Growth Convergence	Limit	C	I	NC

*macarena.flores-gonzalez@cyu.fr, vandebro@univ-paris-diderot.fr, laurent.vivier@univ-paris-diderot.fr

Table 3 :

 3 Summary of the work done by fifteen students at C1.

Table 4 :

 4 Summary of the work done by 10 students at C2.

	Class number:		Paradigm	𝒖 𝒏'𝟏 -𝒖 𝒏	𝒇 increasing	Correctness
	Categories from C1	[A1] [A2]	?	⇒ (𝒖 𝒏 ) increasing	C	I	NC
	[A2]: 4 students	1	3	2	1	1	2	1
	[A1]: 5	4	1	1	1		2	3
	Other: 1 student			1	1		1

Table 5 :

 5 Summary of the work of seven students to C3.Three students worked in [A1] paradigm and provided incomplete work. It is remarkable that all of them worked in [A1

	Categories from C2		Paradigm	Theorem		Correctness
		[A1]	[A2]	?	C	I	NC
	[A1] in C1 and C2: 3 students	3				3
	[A2] in C1 and C2: 2 students		2	1	1		1
	Different paradigms in C1 and		1	1			2
	C2: 2 students

https://www.education.gouv.fr/la-rentree-2020-les-eleves-de-terminale-precisent-leur-choix-de-parcours-

The final exam of high school required to enter at university.

With respect to functions, it is usual to summarize variations using a table with arrows on each interval where the function is increasing or decreasing.

Note that sometimes we omit the word "activity", and simply use "control" or "recognize".


Laurent Daudeville

Hélène Carré

THERMAL TEMPERING SIMULATION OF GLASS PLATES: INNER AND EDGE RESIDUAL STRESSES

Narayanaswamy's model is sc» to describe the thermomechanica/ behavior of glass. It includes both scws relaxation (to take into account the viscous aspect of glass) and st1Uctural relaxation (to take into account the st1Ucture state of glass). The necessary thennal and mechanical characteristics are given for the fl oat soda-lime silicate glass.

The thermal tempering of thin glass plates is simulated. Transient and residual stresses are given for the inner part of the plate. Computalional results are compared with experimental results of previous works. This comparison validates Naranoswamy's model associated with material characteristi cs given previously.

The edge ef f (var i ation of stresses close to the edges) is described for thin plates.

The thermal tempering of thick plates is simulated, and computational results are validated with optical measurements and a fractographic analysis.

The objective of this work is to provide a reliable computational method for the simulation of the thermal tempering of scpda-lime silicate glass plates. Quenching consists of heating glass to a temperature above 600°C and cooling it brutally by air casts. Such thermal treatment induces residual stresses in the plate at the end of the cooling, thereby increasing the mechanical strength of glass parts.

In previous works, the simulation of an infinite glass plate has been carried out experimentally and numerically. Both inner and edge effects are analyzed in this article. The knowledge of residual stresses in the vicinity of edges is very important in the case of elements loaded in their plane. The main application of these glass components concerns building structures (beams or columns).

Glass is. described as a viscoelastic material. Its behavfor varies with time and temperature. Narayanaswamy's model, which includes both stress and structural relaxation (influence of the state of the glass structure), is used in this work.

The tridimensional residual stress state is computed using the finite element method (FEM). First, the model is checked by comparing the residual stresses in the inner part of thin plates from simulations and literature.

difficult to obtain reliable data for the concerned material. No material datum was identified, and a sensitivity analysis to material parameters was conducted.

Then. , the analysis of transient and residual stresses in the neighboring edges of tempered glass plates is presented. Inner and edge residual stresses were measured optically on several tempered glass plates. The heat transfer coefficients are temper process parameters. A comparison of two optical measurements with simulation results facilitates the identification of a priori unknown heat transfer coefficients.

The tempered glass plates were tested up to failure. A fractograpliic analysis gives the residual stress on the plate edge that cannot be measured optically.

The edge effect analysis is validated by a comparison of computed stresses with fractographic and optical information.

THERMOMECHANICAL MODELING OF GLASS Generalized Maxwell's Model

For a tridimensional solicitation, the stress response (u(t)} of a viscoelastic material can be separated into its volume and deviatoric parts: the deviatoric and volume parts, respectively [l, 2]. Relaxation shear and bulk moduli are described with instantaneous and deferred moduli and Prony's series.

For the deviatoric part

where G 8 =instantaneous shear modulus, and Ge =deferred shear modulus= 0.

For the volume part

where K 8 =instantaneous bulk modulus, and Ke= deferred bulk modulus.

Stabilized Glass

When its structure is stabilized, glass can be considered a thennorheologically simple material. It means that the mechanical behavior is the same for different temperatures; only the rate of microstructural mechanisms changes. The relaxation function can be determined at any temperature if it is known at the reference temperature (T,.r). It consists of a shift of the relaxation function in a time logarithm scale [START_REF] Schwarz | Ti me-Temperature Dependence Qf Linear Viscoelaslic Behavior[END_REF].

The influence of the actual temperature T can be intro duced by means of reduced time ( � ), which is defined as

i = 1,2 (4)
Weights w1; and w2;, defined in G(t) and K(t), are constant with tem perature. An

Arrhenius relation allows a correct description of the times of relaxation changes with temperature [4, 5):

11 Trcf 1' t(t,T) = ( ' ) dt ' = <l>(T(t))dt' 0 T T,I 0 where ln(<l>(T)) = ----- H ( 1 1)
R Teer T H = energy of activation, and R = perfect gas constant.

Structural Relaxation

(5)

Quenching is a brutal thermal treatment. During cooling, the glass structure may not be stabilized. Since g lass is sensible to the thermal loading rate, the structure state must be taken into account. That is Narayanaswamy's model [4, 6, 7].

The influence of the thermal loading rate can be shown by varying t'he specific volume with temperature (Figure 3). That phenomenon essentially concerns tem peratures close to the transition range between the "glass" and "liquid" states. When gfass is heated to a temperature T2 (point A on Figure 3) below the transition range, the structure of glass evolves to a stable state characteristic of the liquid glass (point B). This evolution is tlhe direct structural relaxation. If the temperature is far below the transformation range, this evolution is too slow to be visible.

Specific volume

T2.� T'If

If glass is heated brutally to T1 above T8, the glass state goes to a point below the "liquid" straight line (point C). This unstable configuration changes to a stable one; the "liquid" glass state (point D). This process is called reverse structural relaxation.

There are several possible structural glass states for the same temperature depending on the cooling rate.

In 1946, Tool [START_REF] Tool | Relation Between Inelastic Deformability and Thermal Expansion of Glass in Its Annealing Range[END_REF] intro duced the fictive temperature (Tr) to take into account the structure of glass. Variations of the specific volume with temperature are used to describe variations of Tr•

• Tc = T if T is above the transition range.

• Tc = the intersection between the "liquid" straight line and the parallel to the "glass" straight line in the transition range.

• Tc= the intersection between the "liquid" straight line and the "glass" straight line helow the transition range.

A response function (Mv) is defined by varying the specific volume with tempera ture as

V(t)-V.. 2
Mv(t)= V. -V. ' 0.2 eo,2 (6)
where T1 -T2 = step of temperature, Tc = fictive temperature at T2, V = instantaneous specific volume, V0.2 =volume just after the temperature change, and V.. .

Tr(t) = T(t) -lo Mv[ s(t) -s(t')] � dt' (7)
By analogy with viscous relaxation, the response function can be described with a Prony's series as [START_REF] Tool | Relation Between Inelastic Deformability and Thermal Expansion of Glass in Its Annealing Range[END_REF] The fictive temperature variations are calculated with Markovsky and Soules' algorithm [START_REF] Markovsky | An Efficient and Stable Algorithm for Calculating Fictive Temperatures[END_REF]10]

Structural relaxation times (A;) are considered proportional to shear relaxation times. A dilatometric curve facilitates obtaining their ratio, which has been found

to be very close to 9 (11 ].

The fictive temperature allows one to take into account the structu:re state of glass. All characteristics vary with the actual temperature but also with the fictive temperature. The fictive temperature is introduced in the reduced time expression for the variations of viscoelastic characteristics by modifying Eq. 5 where t(t,T,T,)= 1 ' (Trcf ' ) dt ' = 1 1 a(T(t),T,(t))dt'

0 T T,t 0 ln(a(T, T1)) = -------- H( 1 x 1-x ) R T,., T T r (10)
x =constant (0 <x < 1), H =energy of acti vation, and R =perfect gas constant.

Introduction of Fictive Temperature

The dependency of viscosity on the structure state induces very small variations in residual stresses. But, the dependency of the density on the structure state has a much more significant effect on residual stresses (Figure 4). Variations of viscosity in relation to t he fictive temperature are not considered in our simulations. The in f luence of the structural state on the density is intro duced by varying the thermal expansion coefficient in relation to temperature where f3, = thermal expansion coefficient of solid glass, {31 = thermal expansion coefficient of liquid glass, T0 = initial temperature, and f3 =equivalent the rmal expans ion coefficient.

£1h = {38(T(1) -TrCt)) + {31(T1(t) -T0) = {3(t)(T(t) -T0) (11)

FINITE ELEMENT MODELING

The thermal tempering of glass con sists of using air casts to cool very rapidly the plate that was previously heated.

The proble m that must be solved is tim e depen dent . At each moment mn time, the temperature is comp uted; then the nonlinear mechanical problem is solved. The nonlinear aspect is due to the viscous feature of the glass behavior.

The temperature of the glass plate is assumed to be uniform at the beginning of the computation (at a temperature above 600°C). In fact, the surface tempera ture of the glass plate decreases between the time it is removed from the oven and the beginni111 g of cooling.

Heat Exchange Boundary Conditions

The co oling is modeled by a forced convection to each face of the plate. It is characterized by a heat transfer coefficient (h) and air t emp eratur e (TCJ<). For a g lass plate, three planes of symmetry and three forced convections define the cooli ng (Figure 5).

The exchange boundary condition by forced convection is written as

(12)
where q =heat flow, h = thermal transfer coefficient, T s =surface t emperature, and Texi = room temperature.

x y Fi,gurc S. Displacement boundary condi.

tions.

Radiation is accounted for in forced convection via the thermal transfer coefficient. The radiation boundary condition can be written as (by equivalence with convection) q = ll"sue(T� -T e�t)

= u5ije(Ts,' + T{Tc xl + T5T., ;, + T/' ,)(Ts -T,, 11)

where q =heat flow, ifss =Stefan-Boltzmann's coefficient, e = em1ss1VJty, T s = surface temperature, Text = room temperature, and h.,q(asB• e, 15, I'ex1) =equivalent thermal transfer coefficient. Thermal transfer coefficients are assumed to be constant during tempering.

Mesh and Displacement Boundary Conditions

Interesting results are the stress state in the middle of the plate and in the current JYclrt of the edges. The problem that needs to be solved does not depend on z because the plate is considered infinite in this direction. This is a plane problem since it depends only on x and y. Nevertheless, tridimcnsional elements are used to account for thermal deformation along z.

The modeling of an infinite strip implies quite complicated boundary condi tions. Then, the tempering of an eighth Qf plate is simulated with the finite element code MARC. The mesh is refined in zones where stresses present high varia tions-in the thickness of the plate (along x) and close to the edge (along y ).

The edge effect is not studied along z because it is similar to that along y. The displacement boundary conditions that are introduced are symmetry conditions;

that is, there is no displacement perpendicular to the plane of symmetry (Figure 5).

Glass Characteristics

Glass is a very old material that has been studied for a long time. But mechanical and thermal characteristics are not well known because they depend on the glass composition and are difficult to measure experimentally (in particular, for the viscoelastic characteristics and the thermal expansion coefficient of the liquid glass).

The six mechanical characteristics of Narayanaswamy's model that are used in simulations are as follows.

• Elastic characteristics at room temperature [1]

Young modulus Poisson ratio

• Vi scoelastic characteristics [I] (Table 1),

E = 7.1010 Pa II= 0.22 • Structural characteristics for the calculation of Tr [12] (Table 2).
The energy of activation is not a well-known parameter. The dependence of glass viscosity on temperature was determined in a previous work The thermal conductivity and the specific heat are the thermal characteristics.

They vary with temperature [12].

• Thermal conductivity (W /m.K)

.A = 0,975 + 8,58.10-4 T where T in

°C

• Specific heat (J/kg.K) [START_REF] Guillemet | Viscosity, Configurational Entropy and Structural Relaxation of Silica-Soda-Lime Olass[END_REF] (with T in K)

liquid glass (T > T g

= 850 K)
solid glass (T < T g )

INNER EFFECT

cp.1 = 1433 + 6.5 10-3 T

Cp.s = 893 + 0.4T-l.8 10-7 /T2

Experimental studies give the residual stress state in the inner part of thin plates (0.61-cm thick) [4,6]. The temperin g simulation of these glass plates is presented and compared to previous works for the validation of the model [START_REF] Carre | Finite Element Calculation of Stresses in Tempered Gla[END_REF][START_REF] Carre | Numerical Simulation of Soda-Lime Silicate Glass Temperi ng[END_REF].

Results

During tempering, the temperature decreases more rapidly at the surface than in the inner part (Figure 6). A thermal gradient appears in the thickness (which can be greater than 150°C); it implies transient and residual stresses.

At the beginning of cooling, the surface contracts more rapidly than the inner part, which leads to a compression state at the core and a tensile state at the surface by equilibrium. The transient tensile stress on the surface can induce failure at the onset of cooling.

When the surface temperature is below the transition range, the surface congeals. The core, still liquid, continues to cool and contract. The surface is then in compression and the core in tension. After several tens of seconds, stresses are stabilized (Figure 7).

In the middle of the plate, the residual tractions are equal for directions y and z. Stresses are parabolic in the thickness. The surface compressive stre:ss is twice higher than the tensile midplane stress (Figure 8). This stress state gives plates a better mechanical strength.

The characteristic parameter values of the model were obtained frorri different works. A large discrepancy in the values can be observed in the literature, possibly due to the difficulty involved in measuring some characteristics and to sensibility to the glass chemical composition.

In order to study the influence of characteristic parameter values, a sensibility analysis is carried out by varying the mechanical and thermal parameters: Young's modulus (£), the Poisson ratio ( v ), the solid and liquid glass thermal expansion coefficients ( f3g and {31), reference temperature (T,.r ), specific heat (C P ), and thermal conductivity (,\). The parameters are modified ( + 20%, + 10%, -10% , -20%) from a current case: T0 = 650°C and h1 = 280.1 W /m2.K.

Variations of residual stresses in the surface and in the midplane are observed.

The Poisson ratio, the solid glass thermal expansion coefficient, the specific heat, and the thermal conductivity are parameters of less importance than the others.

The variations of the residual stresses are quasi-proportional to the variations of Young's modulus and of the liquid glass expansion thermal coefficient. The reference temperature is a parameter of great importance because it controls the viscoelastic behavior (Figures 9 and10).

Comparisons with Experimental Works

Experimental results issued from [4, 6) are compared with FEM results obtained with the previous glass characteristics. The variation of the traction Cun.) in the thickness is compared with optical measurements from [6] in a particular case of quenching ( T 0 = 738°C and h 1 = 221.5 11).

W /m2.K) (figure
Jn F;gscºc 12, experimental [START_REF] Gardon | Thermal Tempering of Glass[END_REF] and numerical results are gi ven. They show the influence of the cooling rate (h) and the initial temperature {T0) on the residual tensile stress in the core of the plate. The higher h, the higher this residual tractfon. They also reveal a dependence on the initial temperature: the residual stress increases with the initial temperature until 680°C, where it becomes quasi constant with T0. This satisfying comparison between computational and experimental results validates Narayanaswamy's model given the chosen data.

Stress (MPa)

EDGE EFFECT

The goal is the determination of tridimensional transient and residual stresses on the edge of glass plates where fracture may occur. The inner effect of tempering was presented for thin pfates. The edge .effect is now studied, on one hand, to the same thin plates and, ora another hand, to thick plates (19-mm thickness) for a comparison with optical measurements.

The tempering parameters are the initial temperature (easily determined during the industrial process) and the thermal transfer coefficients (unknown). Thermal transfer coefficients can be calculated by an inverse analysis.

Results

As for the inner effect, the stress sign changes during tempering on the edges. The transient traction (u11) is very high at the comer (point 4). At the end of the cooling p•hase, the three points on the surface (points 2, 3, 4) are in compression even though the core is in tension (point I) (Figure 13).

In the middle of the plate, the stresses are parabolic along the thickness (Figure 14). The stress profile in the thickness on the edge depends on the ratio h1/h2, where h1 is the thermal transfer coefficient of convection on the plane and h2 is the thermal transfer coefficient on the edge (Figure 5). If h2 is below h1, the traction along z (u,,) is more important at the corner than in the middle of the edge. If h2 is far above h 1, it is the contrary.

so •SO

•100

-ISO \ \ \ T0=650°C h 1=280.l W/m2.K hi= 100 Wlm1.K JOO 200 _ __ -... _ _ 300 400 --------------------------------- \ \ \' -nodel \� ' ,,, -node 2 ', ---node 3 �--------- ---node4 --------------��- --- --------
Fi gure 13. Variations of stress u,. with time in the inner part and at the edge.

Influence of the Chamfer

The chamfer introduced (or the computations of thin glass plates is 0.5-mm wide.

The thermal transfer coefficient is assumed to be the same in the broken corner as in the edge. The chamfer has no influence 011 the residual stresses (Figure 15) but can decrease tbe value of rhe maximum transient surface tension stress (Figure 16).

Case of a Thick Plate Optical measurements. Tempered glass is birefringent. The photoelasticimetry technology allows stress measurements [START_REF] Novo�ny | Relation entre Jes contraintes au voisinage de la surface des verres trempes, determinees a l'aide d'un epibiascopc, leur degre de trcmpe et leur destruction aprcs Casse[END_REF]. Two kinds of these measurements an:

realized on thick tempered glass plates.

First, the Epibiascope [16, 17} is an apparatus that is used to obtain the surface stress. lt is measured in several points in the inner part of the plate (o;.>' = ozz) the edge), at the limit of the chamfer: ---•� .::.: ._-.:.::.--•::.:.•: .:.. :.•::.:._-.:.::.-�:: _ -_ • -Figure 16. Variations of �tr escq 1r:, with time in the middle and at sce edge with and wisco ut chamfer.

Several measures at different distances from the edge were carried out. The variations of the integral in the thickness of the difference of principal stresses with the distance from the edge are given (Figure 17). The initial temperature and the temperature of blown air can be determ:ined quite easily dur�ng the industrial process. For studied thick plates, their values are 620°C and 20°C, respectively. 111 and h2 are very difficult to measure. These coefficients will be identified by a comparison of optical measurements with calculation results.

Since the surface residual stress values in the inner part of the plate are only governed by h1, the Epibiascope measurements allow one to determine h1

The edge stresses are controlled by both h1 and h2• h2 can be identified with the Babinet com pensator measurement at the limit of the chamfer (point A on Figure ValidJJJion by optical measurements In Fig ure 17, only one point was used to determine the thermal transfer coefficient h2• Figure 18 shows a comparison of optical measurements and •COmputationa• results from the edge to the inner part.

Validation by /ractographic analysis Thick tempered glass plates were tested to failure under four-point bending. The local failure stress is obtained with the ultimate load using beam theory.

A semicircular shiny zone (mirror of failure) can be observed in the failure pattern of annealed glass. The mirror radius, corresponding to a crack branching (CB), can be linked to the tensile failure stress by [START_REF] Conway | Use of Crack Branching Data for Measuring Near Surface Residual Stresses in Tempered Glass[END_REF]

M (15)

where r m = mirror radius and M = 1.85 MPa.m 112 • Th.e previo us relation is ob ta i ned using linear elastic fracture mechanics theory.

For tempered glass, the failure mirror is also observed (Figure 19). The residual stress state must be taken into account in the fractographic analysis. The stress intensity factor can be written [START_REF] Shand | Breaking Stress of Glass Determined from Dimensions of Fracture of Mirrors[END_REF][START_REF] Kerper | uderi, Relation of Strength of Thennally Tempered Glass to Fracture Mirrors Size[END_REF] as [START_REF] Aben | Phot�lasticity of Glass[END_REF] where a= crack radius, Yf(B) = shape factor associated with bending, and YR(O) ...,. shape factor associated with residual stress= 1. The apparent failure stress (a A ). the residual compression stress ( aR), and the mirror radius (r m) can be Jinked (K1 =Kea and a= rm) via The edge effect can be analyzed using the variations of stresses close to the edges. The residual stress shape in the edge depends on the thermal transfer coefficient of forced convection applied in the edge. Two kinds of experiments are done to validate residual stress values close to the edge: optical measurements and fractograp1hic analysis.

An experimental campaign was carried out to identify the fracture features of glass [START_REF] Carre | Determination de la resistance du verre trempe pour son utilisation en genie ci vil[END_REF]. The lifetime of glass components in building structures can be analyzed with the presented approach.

  {s ;/x,t) = J' G(tt') �e; i (x,t')dt' ,t) = K(tt')-, E(x,t') dt ' where u, i = sii + u8;i' c;i = e;i + £8;;• G =relaxation shear modulus, and K = relaxation bulk modulus. The generalized Maxwell's model is a rheological discrete model. It is an association of several Maxwell's models (spring and dashpot in series). The proposed rheological model of glass is il lu stra ted in Figures I and 2 for f'lgu� I. Generalized Maxwell's model for the deviatoric part.

Figure 2 .

 2 Figure 2. Generalized MascËell's model for lhe volume part.

Figure 3 .

 3 Figure 3. Variations of specific volume with temperature for two cooling rates.

2 =

 2 equilibrium volume at T2• The dependence of the response function on temperature is given by reduced time f M/t) can be consii dered the volume relaxation function and can define variations of the fictive temperature I dT(t')

Figure 4 .

 4 Figure 4. Influence of the fictive temperature on residual stresses (5).

  [l]. lt is an empirical relation of Vogel-Fulcher-Tammann B Log(17) =A+ -The value of the ratio H / R scÈ identified by comparing the previous relation with the function <l>(T) of Narayanaswamy's model H/R = 55,000 K • Thermal expansion coefficient [12] solid glass (room temperature) 13 , = 9.10-60c-1 liquid glass (high temperature) {3 1 = 2s.10-60c-1 Table l Cha£acteristics of shear and volume moduli Devi atoric part (T,c1 -869 K) Volume part {K,/K, = 0.18) (T,d -869 K)

-Figure 6 .

 6 Figure 6. Varialions of 1empera1ure with lime.

.Figure 7 .

 7 Figure 7. Variations of stress with ti.me.

�PFigure 9 .Figure J 0 .

 90 Figure 9. Sensi bility analysi s for residual stress in the midplanc.

Figure 12 .

 12 Figure 12. Varia1ions of residual tension stress with quenching parameters ..

5 Figure 14 .

 514 Figure 14. Variation o( stress if,, in the thicknc:>:> io the middle and at the edge for several values of hz.

mean = 120, 4

 4 Figure lS. Str-cs.\ c:r,, in sce thickness in the middle and at the edge of the plate with and without charufer .

mean = 72, 0 --node 3

 03 MPa standard deviation = 3 ,3 MPa with precision of measure = ± 1,2 MPa Stress zz (MPa) without chamfer •• --node 4 without chamfer node J with chamfer ---node 6 with chamfer ---• node 5, with chamfer Time (s) 500 '�... . ... .... ___ ... -'�• ----------------------- .... -: � -_. : : -_• -•-

  Determination of thermal transfer coefficients Uniform tempering of a glass plate is characterized by four parameters: T0 = initial temperature h1 = thermal transfer coefficient of convection on the plane h2 = thermal transfer coefficient of convection on the edge T c,1 = temperature of blown air

Integral 20 Figure 17 .Figure 18 .

 201718 Figure 17. Variations of the integral of the difference of principal stresses with the distance from the edge.

( 17 )

 17 where 1/10 = 2.41 MPa.ml/2 and Yr(O) = Krn/M = 2.33/l.85 = 1.26. Measure ments were taken for two plates: radius l = 2.5 mm radius 2 = 1.19 mm crA1 = ll8.6 MPa <TAz = 135.1 MPa Fii:ure 19. View of failure mirror of a thick terupercd glass plate. "R = lOl .2 MPa <TR = 100.4 MPa 20 glass behavior depends on the chemical glass composition, and a slight change in the reference temperature implies great changes in the residual stresses.

These values of residual stresses must be compared to the mean value of the compression stress on the edge obtained with calculations: 101.1 MPa (maximum = 108.8 MPa and minimum = 93.8 MPa). The deviation is only about 1 %.

Residual stresses. The residual stresses in the inner part of the plate are parabolic along the th�ckness, as seen earlier for thin plates. The residual stress value on the edge of the plate is quasi-constant and slightly inferior to the surface residual stress in the inner part of the plate (Figure 20).

CONCLUSION

The FEM of modeling glass tempering was presented. The calculation of the tridimensional residual stress state in the edge regions is necessary for a possible failure analysis. The use of Narayanaswamy's model to describe the glass thermo mechanical behavior is validated thanks to comparisons between our calculated results and the experimental results of previous works. The sensibility analysis has shown the foreseeable importance of Young's modulus and the liquid glass thermal expansion coefficient values on the residual stress state. Among the other material parameters of our model, the reference temperature is the more sensitive. The