Thibault Dassonville
email: thibault.dassonville@u-pem.fr

Martin Poncelet

Nicolas Auffray

Toward a homogenizing machine

Keywords: Architectured materials, Experimental testing, Homogenisation, Boundary Conditions, Full-field measurement

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

Introduction

Purpose

Expectations on materials in terms of multi-functional properties and mass saving are constantly increasing. For instance for industries as automotive or aerospace, gains in mass not only imply matter saving but also lead to significant enhancing the overall performance of the designed devices.

One of the possible answers to lighten structures is the use of architectured materials. For such materials, the matter is organised at different intermediary scales between the microstructure and the macroscopic shape. Their inner structure can be various, ranging from randomly distributed phases to a perfectly organized architectured. In the last case, we are in the realm of periodic materials and the mesoscopic scale is composed of a basic cell repeated according to a regular lattice. Intermediate configurations are possible, for instance in biological architectured materials, in which almost periodic structures or perturbed lattices are often encountered [START_REF] Barthelat | Architectured materials in engineering and biology: fabrication, structure, mechanics and performance[END_REF].

In addition, and beyond simply sparing matter, the creation of an inner architecture opens up new possibilities for designing multifunctional materials. This design is halfway between material design and structure engineering, indeed length scales, composition and geometry are controllable at the same scale [START_REF] Barthelat | Architectured materials in engineering and biology: fabrication, structure, mechanics and performance[END_REF][START_REF] Bouaziz | Heterogeneous and Architectured Materials: A Possible Strategy for Design of Structural Materials[END_REF].

If the organisation of the matter at intermediate scales paves the way for new possibilities [START_REF] Ashby | Designing hybrid materials[END_REF][START_REF] Brechet | Architectured materials: Expanding materials space[END_REF][START_REF] Fleck | Microarchitectured materials: past, present and future[END_REF], it also poses new challenges in modelling and simulation. Indeed, due to the complex internal geometry of architectured materials, Direct Numerical Simulations (DNS, i.e. when the geometry of the inner structure is explicitly meshed in simulations) are often limited to few unit cells, or few Representative Volume Elements (RVEs). In other words, the full-field computation of a structure made out of an architectured material can be prohibitively expensive, especially in the design phase in which the structure is optimized.

To solve these technical difficulties, a possible way is to substitute the actual material by a homogeneous one having equivalent effective properties.

In the context of linear elasticity, to which this paper sticks to1 , the goal is to define the apparent fourth-order elasticity tensor C

E ∼ = ε ∼ (x) = 1 V Ω ε ∼ (x) dV ; Σ ∼ = σ ∼ (x)
with V the volume of Ω. This approach is particularly useful for material and structure optimiza-tion, in which the model needs to be evaluated a numerous amount of times. The use of a homogenized approach makes it possible to localize "quickly" the vicinity of an optimal solution. In this neighborhood, the return to a more detailed model can become relevant again [START_REF] Donders | Homogenization method for topology optmization of struc-tures built with lattice materials[END_REF][START_REF] Ranaivomiarana | Concurrent optimization of material spatial distribution and material anisotropy repartition for two-dimensional structures[END_REF].

The recent development of additive manufacturing techniques induces a gain of interest in the field of architected materials. Intricate inner structures that were, up to recently, restrained to theoretical discussions can now be produced and tested [START_REF] Li | Progress in additive manufacturing on new materials: A review[END_REF]. However, despite important progresses that have recently been made, the material state and the geometric characteristic obtained by those techniques remain poor and variable according to the printing technique [START_REF] Juneja | Accuracy in dental surgical guide fabrication using different 3-D printing techniques[END_REF][START_REF] Tan | Comparison of solid, liquid and powder forms of 3d printing techniques in membrane spacer fabrication[END_REF], the characteristics of the machine [START_REF] Barclift | Examining variability in the mechanical properties of parts manufactured via polyjet direct 3d printing[END_REF], the heating energy strategy [START_REF] Parry | Understanding the effect of laser scan strategy on residual stress in selective laser melting through thermo-mechanical simulation[END_REF][START_REF] Williams | In situ thermography for laser powder bed fusion: Effects of layer temperature on porosity, microstructure and mechanical properties[END_REF], the orientation [START_REF] Tymrak | Mechanical properties of components fabricated with open-source 3-D printers under realistic environmental conditions[END_REF] and even the position on the printing table [START_REF] Tymrak | Mechanical properties of components fabricated with open-source 3-D printers under realistic environmental conditions[END_REF].

These disparities not only affect the elastic properties of the produced structures, but also their life expectancy.

Hence, in order to fully characterize a sample of architectured material the numerical studies have to be supplemented by experimental testings. But these testings are not standard since what has to be measure is not the behaviour of a constitutive material, but the effective elasticity of an architectured material.

If homogenization is a well-known topic both in theoretical and numerical mechanics [START_REF] Geers | Multi-scale computational homogenization: Trends and challenges[END_REF], its experimental counterpart has to be built. It is worth being noted that if many homogenization procedures are available in the literature [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF][START_REF] Feyel | Multiscale FE2 elastoviscoplastic analysis of composite structures[END_REF][START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1 : Matériaux aléatoires élastiques et milieux périodiques[END_REF][START_REF] Allaire | Optimisation topologique par méthoded'homogénéisation[END_REF][START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF], few of them can be adapted to an experimental setting. For instance, if applying complex volume loadings, and full-field measurements of displacement field are heavy to carry out but feasible, direct stress measurement is not possible. Further, the approaches requiring too complex mechanical loads or accessing too much information can not be envisaged with a reasonable cost. The strategies which seem the most likely to be adapted to the experimental framework are those based on the control of boundary conditions. Those approaches are known, in the field of numerical homogenization, as Kinematic Uniform Boundary Conditions (KUBC) and Static Uniform Boundary Conditions (SUBC) or Periodic Boundary Conditions (PBC) [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF][START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1 : Matériaux aléatoires élastiques et milieux périodiques[END_REF]. The last one is maybe the more appealing since PBC provides exact results [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF][START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1 : Matériaux aléatoires élastiques et milieux périodiques[END_REF], however they are not simple to apply in practice. Uniform boundary conditions are simpler to use, but they rather measure apparent properties of the sample instead of effective properties of the considered material, KUBC overestimating overall rigidity while SUBC underestimating it [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF][START_REF] Huet | Application of variational concepts to size effects in elastic heterogeneous bodies[END_REF][START_REF] Sab | On the Homogenization and the Simulation of Random Materials[END_REF][START_REF] Hazanov | Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume[END_REF]. The result depends on the number of unit cells in the sample, but converges while increasing the number of unit cells to the effective properties of the architectured material [START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF]. Hence, to ensure that the measured properties are those of the architectured material (effective properties) and not of the structure (apparent properties), a preliminary convergence study is required to size the sample to be tested.

The step-by-step transition from a perfectly theoretical situation to an amenable one for experimental testing is depicted in the diagram Figure 1 where going from one step to another step implies the weakening of a hypothesis.

All along this paper, plane strain will be be assumed, and the approach is hence mostly bidimensionnal. When required, assumptions justifying this hypothesis will be indicated in the core of the text. The aim of the present paper is thus to investigate the technological feasibility of such a transition from theoretical homogenization to an experimental rig. As mentioned before, this transition will be here considered in the context of linear elasticity. Needless to say that the development of an experimental system capable of applying complex boundary conditions can find applications broader than homogenization of linear elasticity of composites. First it can be used for the homogenization of more complex behavior, like viscoelasticity and elastoplasticity. We believe, for instance, that the device we set up will be of great interested for understanding the anisotropic viscoelastic proper-ties of (real) architectured materials. This question is of prime importance for the modelling of polymer architectured materials (such as the one obtained by 3D printers), or for biological materials. Second, and for non-linear materials, it can be used to study the change in their effective properties when strained (e.g. acoustoelasticity). Last, it can be used to study the evolution of local phenomena, like instabilities or crack propagations, when the embedded material is submitted to a remote biaxial strain/stress state (a sort of Experimental Eshelby Problem). 2The paper is organized as follows: first, in section 2, the description of the problem and the principle of the homogenization by KUBC or SUBC is explained. This leads to the question of the technical possibility of a mechanical system able to perform KUBC, or SUBC, or both. For simplicity's sake, the route of displacement controlled experiments is followed, leaving aside the study of the feasibility of a SUBC system for forthcoming investigations. By the end of this section the important question of the measurement of the average stress in the sample is asked and solved. It opens the way for studying the technological aspects of the mechanism in section 3, and a candidate architecture for the testing device is proposed. In section 4, a Virtual Testing Device (VTD in the following) based on the choices previously made is implemented. The aim of the VTD is to validate the operating principle of the considered device architecture, optimize its design and evaluate its capabilities. Hence, after describing aspects of its numerical implementation, the VTD is ultimately validated on some elementary simulations representing critical testing situations. Finally, some general conclusions and perspectives are given in section 5.

• • • • • • • • • • • • • • • • • •

Theoretical homogenization

Experimental homogenization

Finite size

Loading discretization

Material state

Notations Space and basis:

Throughout this paper, the physical space is modelled on the Euclidean space E 2 with E 2 its associated vector space. Once an arbitrary reference point is chosen, those spaces can be associated and P = {e 1 , e 2 } will denote an orthonormal basis of E 2 . When needed tensor components are specified with respect to P.

For forthcoming needs, let also defined M = {ê 1 , ê2 , ê3 } the orthonormal canonical basis of R 3 , in the present context M will be referred to as the Mandel basis. Vectors of M are obtained from second orders tensors of P according to the following construction. Consider the following linear application φ : S 2 (R 2 ) → R 3 :

ê1 = φ(e 1 ⊗ e 1 ), ê2 = φ(e 2 ⊗ e 2 ), ê3 = φ( √ 2 2
(e 1 ⊗ e 2 + e 2 ⊗ e 1 )).

(

) 1
With respect to M, we can define the vector t image of t ∼ by φ.

Tensor products: Tensors of order 0, 1, 2 and 4 are denoted respectively by

• , • , • ∼ , • ≈ .
The matrix expression of the quantity ( • ) in the basis β is symbolized by [ • ] β . The simple, double and fourth-order contractions are written • , :, :: respectively. In components, with respect to P, these notations correspond to

u • v = u i v i a ∼ : b ∼ = a ij b ij (A ≈ : B ≈ ) ijkl = A ijpq B pqkl (A ≈ :: B ≈ ) = A pqrs B pqrs
where the Einstein summation on repeated indices is used. The order on which components are summed has been chosen in such a way that the full contraction between same order tensors is a scalar product. ⊗ stands for the classical product.

Miscellaneous notations:

• [∇] P = ∂ ∂x i e i the nabla operator; • ∂ K Ω the restriction of ∂Ω on which kinematic conditions are imposed;

• ∂ S Ω the restriction of ∂Ω on which static conditions are imposed;

• ( • ) imposed quantity; • • the jump of ( • ).
2. Transposition of an homogenisation scheme . . . In this section, a methodology for determining the apparent elastic behaviour of a sample of architectured material with an experimental setup is exposed. First, the mechanical problem is defined. Next, an equivalence criterion between micro and macro scales is made explicit, which leads to the choice of a homogenisation scheme which will then be detailed. The end of this section is devoted to proposing technological solutions to transpose this scheme into an experimental context.

Description of the sample

The sample of the architectured material we intent to determine the homogenized behaviour of is supposed to occupy a bounded domain Ω of R 2 with boundary denoted by ∂Ω (fig 2 ) and Ω it's closure. Points within Ω will be referred to as material points and, upon the choice of a spatial reference point O defining R a frame of reference of E 2 , material points can be designated by their position vector x within R. O is taken at the bottom left corner, position and normal vector on the boundary are defined in table 1.

Ω is supposed to be filled by two elastic materials thereafter referred to as phase α and β. The boundary between the phases is symbolized by ∂(α, β). Let's denote by L, l and λ the characteristic lengths of the macrostructure, mesostructure and microstructure respectively (fig 3). The microstructure and its characteristic length λ is assumed to be well separated with respect edge to l, so, although intrinsically present in the real material, it will not be investigated. The scale separation ratio can thereby be defined:

1 2 3 4 n -1 0 1 0 0 -1 0 1 x 0 y L y x 0 x L
= l L
The scales are said separated when → 0, which is the situation of classical homogenization. The physical properties of those elastic phases are supposed to be defined by two, possibly anisotropic, elasticity tensors c ≈ α and c ≈ β . The elasticity tensor at a given point x of Ω is described by the function:

Sample

c ≈ (x) = χ(x)c ≈ α + (1 -χ(x))c ≈ β
in which χ denotes the characteristic function of phase α, i.e.

χ(x) := 1, if x ∈ α 0, if x ∈ α
In classical homogenization, c ≈ must be positive definite. However, for our needs we will allow ourselves c ≈ β → 0 in order to represent pores. This situation belongs to the category of high contrast homogenization [START_REF] Abdoul Anziz | Homogénéisation de composites élastiques périodiques à fort contraste : Conception de métamatériaux de second gradient[END_REF]. This hypothesis might present theoretical and numerical problems, indeed the homogenized properties depend on the way the asymptotic method is conducted ( [START_REF] Abdoul Anziz | Homogénéisation de composites élastiques périodiques à fort contraste : Conception de métamatériaux de second gradient[END_REF]) and [START_REF] Tartar | Nonlocal Effects Induced by Homogenization[END_REF] show that non local effect can be obtained. However none of these should be a problem in the case of experimental homogenization. No body forces are considered. Hence the local equations for a pure Dirichlet problem are:

       ∇ • (c ≈ (x) : ε ∼ (x)) = 0, for x ∈ Ω σ ∼ (x) • n = 0, for x ∈ ∂(α, β) u = u on ∂Ω
with the kinematic compatibility condition3 :

ε ∼ (x) = 1 2 (u(x) ⊗ ∇ + ∇ ⊗ u(x)) , for x in Ω 2.

Hill-Mandel condition

Before the homogenization process the considered material is heterogeneous and the fields vary locally with the heterogeneities. After the homogenization process it is replaced by a homogeneous one and the fields vary globally. It is important to define an equivalence between these two descriptions of the same medium. Different equivalence criteria are possible, among which the Hill-Mandel energy criterion is chosen [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF]:

σ ∼ (x) : ε ∼ (x) = Σ ∼ : E ∼ (2)
Let's denote by KA(Ω) the set of kinematically admissible displacement fields over Ω and by SA(Ω) the set of statically admissible stress fields over Ω. Neither body force, nor rigid body motion are considered.

KA(Ω) = { u | u (x) = u(x), ∀x ∈ ∂ K Ω} SA(Ω) = σ ∼ | ∇ • σ ∼ (x) = 0, ∀x ∈ Ω || σ ∼ (x) • n = t(x), ∀x ∈ ∂ S Ω
u(x) and t(x) being the kinematic and static boundary conditions applied on

∂ K Ω and ∂ S Ω respectively. Considering u (x), ε ∼ (x) and σ ∼ (x) such as:    ε ∼ (x) = 1 2 (u (x) ⊗ ∇ + ∇ ⊗ u (x)) , with u ∈ KA(Ω) σ ∼ (x) ∈ SA(Ω)
By extension a strain field will be said KA if it is the symmetrised gradient of a displacement u ∈ KA(Ω). In such case and with a slight abuse of notation, we will note ε ∼ ∈ KA(Ω).

For any u , σ ∼ ∈ KA × SA the following quantities can be defined:

       Σ ∼ = σ ∼ (x) E ∼ = ε ∼ (x) U = E ∼ • x ⇒        δσ ∼ (x) = σ ∼ (x) -Σ ∼ δε ∼ (x) = ε ∼ (x) -E ∼ δu (x) = u (x) -U
in which U is the average displacement field, while δu , δε ∼ and δσ ∼ are, respectively, the displacement, strain and stress fluctuations around the means. In this analysis, it is not necessary to make the assumption of a given constitutive model linking σ The mechanical energy associated to these fluctuations has the following expression: δσ

∼ (x) : δε ∼ (x) = σ ∼ (x) : ε ∼ (x) -Σ ∼ : E ∼
Complying with Hill-Mandel condition (eq (2)) amounts to have a null average work of the fluctuations.

Since

δε ij (x) = ε ij (x) -E ij = 1 2 δu i,j (x) + δu j,i (x)
and due to the symmetry of δσ

∼ δσ ij (x) δu i,j (x) = δσ ij (x) δu j,i (x)
the work of the fluctuation can be expressed as:

δσ ∼ (x) : δε ∼ (x) = 1 V Ω δσ ij (x) δu i (x) ,j dV - Ω δσ ij,j (x) δu i (x) dV
The vanishing of body forces

δσ ij,j (x) = ∇ • σ ∼ (x) -Σ ∼ = 0 leads to δσ ∼ (x) : δε ∼ (x) = 1 V ∂Ω t(x) -Σ ∼ • n • u (x) -E ∼ • x dS
So, among others, the following conditions satisfied eq (2):

t(x) = Σ ∼ • n ∀x ∈ ∂Ω u(x) = E ∼ • x ∀x ∈ ∂Ω (3)
with

Σ ∼ and E ∼ uniform.
The first set of Boundary Conditions in eq (3) is known as Static Uniform Boundary Conditions (SUBC) while the second defined Kinematic Uniform Boundary Conditions (KUBC) over Ω. These two boundary conditions ensure the Hill-Mandel criterion. In the rest of the article, only the method of homogenization by average fields in KUBC is studied.

Homogenization scheme

Fulfilling the homogenization of a linear elastic material by a KUBC method is looking for C ≈ such as:

Σ ∼ = C ≈ : E ∼ .
To introduce the method lets first consider the matrix expressions of

E ∼ and Σ ∼ in P. [E ∼ ] = E 11 E 12 E 12 E 22 P and [Σ ∼ ] = Σ 11 Σ 12 Σ 12 Σ 22 P
Using Mandel transformation defined in section 1.2, these can be turned into vectors in R 3 :

[ Ê] =   Ê1 = E 11 Ê2 = E 22 Ê3 = √ 2 E 12   M ; [ Σ] =   Σ1 = Σ 11 Σ2 = Σ 22 Σ3 = √ 2 Σ 12   M
The principle of mean field homogenization is as follows. Consider a mean state of strain Ê and stress Σ over Ω, and decompose them on M Ê = Êi êi , Σ = Σi êi Suppose that Σ is related to Ê by a linear relationship, that is by a constant positive definite fourth-order elasticity tensor

Ĉ ∼ [ Ĉ ∼ ] =   C 1111 C 1122 √ 2C 1112 C 1122 C 2222 √ 2C 2212 √ 2C 1211 √ 2C 1222 2C 1212   M =   Ĉ11 Ĉ12 Ĉ13 Ĉ12 Ĉ22 Ĉ23 Ĉ13 Ĉ23 Ĉ33   M
By mean of linearity, Σ can be written

Σ = Ĉ ∼ • Ê = Êi Σ1 i , with Σ1 i = Ĉ ∼ • êi
in which Σ1 i denotes the mean stress field associated to the unit strain field êi . This amounts to decompose Σ over { Σ1 i } 1≤i≤3 , which is a base of R 3 due to the positive definiteness assumption. Thus

Σ = Σj êj = Êj Σ1 j Hence Σi = Σ • êi = Êj Σ1 j • êi which means that Ĉij = êi • Σ1 j or, more explicitly [ Ĉ ∼ ] =   Σ1 1 • ê1 Σ1 2 • ê1 Σ1 3 • ê1 Σ1 1 • ê2 Σ1 2 • ê2 Σ1 3 • ê2 Σ1 1 • ê3 Σ1 2 • ê3 Σ1 3 • ê3   M (4)
Hence the theoretical determination of a column of [ Ĉ ∼ ] is a three-step process:

1. imposing a mean strain state êi over Ω;

2. measuring the resulting stress field σ1 (x);

3. computing the average stress field Σ1 and the associated dot products.

So, the application of a KUBC homogenization approach in an experimental setting presents two main difficulties:

• applying kinematic uniform boundary conditions on a sample;

• determining the value of the macroscopic stress Σ ∼ . The end of this section is devoted to proposing solutions to these complications. First the global concept of a device capable of applying KUBC is discussed. Then, a method to compute the average stress field in an experimental setting with realistic measuring means is presented.

Machine type

Up to authors best knowledge, no machine designed for imposing KUBC have been studied, or designed yet. It is therefore necessary to conceive one (almost) from scratch. Theoretically KUBC are continuous, and a fine discretization is mandatory to perform corresponding numerical simulations. However experimentally, continuous conditions might be hard to obtain, and thereby, will not be searched at all costs.

In a practical way, architectured materials will be tested so that edges will be divided into several subsets corresponding to the various constituents and/or to the voids. The sample will be fastened to the testing device through its most rigid phase. The number of rigid subsets on the boundary is directly linked to the number of cells in the sample (fig 4). In order to achieve a good scale separation for the architectured material ( → 0), a minimum of 10 cells along each dimension is required [START_REF] Rosi | On the validity range of strain-gradient elasticity: A mixed static-dynamic identification procedure[END_REF]. Here, one will only investigate the Cauchy-elasticity so that this condition must be satisfied otherwise overall classical elasticity may no longer be sufficient to describe the static behaviour of the material [START_REF] Abdoul-Anziz | Strain gradient and generalized continua obtained by homogenizing frame lattices[END_REF][START_REF] Poncelet | An experimental evidence of the failure of Cauchy elasticity for the overall modeling of a non-centro-symmetric lattice under static loading[END_REF]. Thus we will consider a square sample with 10 rigid subsets on each edge. Each surface will be mechanically fastened and assimilated to a control point or a lattice node with 2 degrees of freedom (dof). A few examples of both types are found in the literature (e.g. [START_REF] Eberhardsteiner | Biaxial testing of orthotropic materials using electronic speckle pattern interferometry[END_REF] and [START_REF] Summer | Structural Testing in Nuclear Engineering[END_REF]) for local type; [20] and Accupull c device for global type). The pros and the cons related to each option will be discussed.

Two types of machines with multiple control points are conceivable (Figure 4): local type, with independent control points; or global type, with A A A A A A A A A A A A A A A A

Local type

The idea of this approach is to independently command each control point using as many actuators as necessary: one or more per point depending on the desired kinematics. In our case, which involves 10 control points with 2 dof on 4 edges, this represents 80 actuators. This type of machine allows to apply any discretized KUBC or SUBC and to measure the dual quantity on the boundary of the specimen. With such a configuration, all the information necessary to the homogenization scheme is measurable.

The cost of such a system is roughly equivalent to the cost of a global type one (numerous actuators and parts on one side, less actuators but more parts on the other). However, having a large number of actuators also presents technical problems. To a small extent, controlling simultaneously and precisely eighty actuators is possible but requires a lot of equipment. But the main problem is bulkiness. Indeed, for the considered size and load range it is hard to find matching actuators. These two problems are accentuated by the addition of load cells.

Global type

The idea of this approach is to have a limited number of actuators and a mechanical device between them and the specimen that carry out the repartition of the displacement or force over the different control points. If one imposes load on the boundary, a kinematic measurement has to be taken.

Conversely if one imposes a displacement to the boundary, load cells will be necessary on each dof to know the load repartition and the same difficulties as before would appear.

This approach requires to control less actuators but the error of each control point due to the machine defects can not be corrected by compensating the command send to the common actuator. A global force or displacement measurement is conceivable but it might not be possible to achieve a homogenization scheme with so little information.

The two approaches seems reasonable and both have strengths and weaknesses, but the global one seems more elegant to our eyes. Only this idea will be considered thereafter.

Feasibility of the experimental homogenization

Before going further, it is necessary to ensure the feasibility of determining the average stress state by realistic experimental means. Indeed, the field σ(x) has to be known and if this is easy to achieve in a numerical situation, generally speaking this information is not measurable in an experimental setting. It has therefore to be determined with information poorer than the knowledge of the field in every point. As it will be shown, this is however possible by making some assumptions. First, the average operation can be projected on the edges:

V Σ ∼ = Ω σ ij dV = Ω σ ik δ kj dV = Ω σ ik x j,k dV = Ω (σ ik x j ) ,k dV - Ω σ ik,k x j dV = ∂Ω (σ ∼ • n) ⊗ x dS
Because of the absence of body force, ∇ • σ ∼ in the former relation vanishes, while the first one is transformed into a surface integral according to the divergence theorem. In our case, the boundary ∂Ω is divided into 2 subsets corresponding to the two phases: material and void (stress free) (fig 4). The material subset is divided into several fastened surfaces. So the integral can be split as follows with 4N the number of fasteners on ∂Ω and ∂Ω (p) the restriction of ∂Ω to one fastener:

V Σ ∼ = 4 N p=1 ∂Ω (p) σ ∼ (p) • n (p) ⊗ x dS (5)
in which ( • ) (p) denotes the restriction of ( • ) to ∂Ω (p) . Now supposing that the fasteners are sufficiently narrow so that the fields are locally linear:

σ ∼ (p) (x) σ ∼ 0(p) + y (p) (x) • e (p) σ ∼ 1(p)
with σ ∼ 0(p) and σ ∼ 1(p) two constant tensors while y (p) and e (p) are the local position vector and the basis vector collinear to the edge, respectively:

y (p) = x -x (p)
In this last expression x (p) represents the center of the fastener (p), that is the control point on fig 4c.

Starting from eq [START_REF] Ashby | Designing hybrid materials[END_REF], and after some algebra detailed in Appendix A, the following expression for the average stress is obtained:

V Σ ∼ = 4 N p=1 L S (p) σ ∼ 0(p) • n (p) ⊗ x (p) L + 1 12 l (p) σ ∼ 1(p) • n (p) l (p) L in which S (p) = l (p)
H is the surface of a fastener (l (p) being its width and H the sample out-of-plane thickness), and L is the length of a side of the specimen. The first term corresponds to the mean value of the stress over a fastener while the second corresponds to a first order approximation for the variation of the stress. On can now compare the order of magnitudes of these two contributions:

             O x (p) j L = 1 ; inherently O l (p) L = 10 -2 ; by hypothesis
The order of magnitude of the second term vary with l (p) L , l (p) being the in-plane size of the fasteners, which are very narrow by hypothesis. As a consequence the second term can be neglected with regard to the first and will not be considered afterwards. Our assumption leads ultimately to:

V Σ ∼ = 4N p=1 S (p) σ ∼ 0(p) • n (p) ⊗ x (p) (6)
16

We then split down the previous sum (eq [START_REF] Barclift | Examining variability in the mechanical properties of parts manufactured via polyjet direct 3d printing[END_REF]) on all four edges.

V Σ ∼ = 4 s=1 N p=1 S (sp) σ ∼ (sp) • n (s) ⊗ x (sp)
Or, in matrix form:

V Σ ∼ =        N p=1 L f (2p) x -x (3p) f (3p) x + x (4p) f (4p) x N p=1 -y (1p) f (1p) x + y (2p) f (2p) x + L f (4p) x N p=1 L f (2p) y -x (3p) f (3p) y + x (4p) f (4p) y N p=1 -y (1p) f (1p) y + y (2p) f (2p) y + L f (4p) y        P ( 7
)
in which f

(sp) j = S (sp) σ (sp) jk n (s)
k the j-th component of the force on the (p)-th fastener of the edge (s). The expression of Σ ∼ obtained here is not symmetrical following the projections because of the choice of the origin on a corner of the domain. However, when determining the experimental values, a symmetric expression is expected.

The value of these sums remains to be determined. The terms contributing to the overall stress are of two kinds:

           N p=1 L f (sp) j = L F (s) j , j = {x, y}, s = [1; 4] N p=1 X (sp) f (sp) j , X = {x, y}, j = {x, y}, s = [1; 4]
The first is easy to compute in an experimental setting since the information provided by the measurement of the global forces on one edge is sufficient. At first glance, the computation of the terms of the second kind requires the implementation of force sensors on each of the fasteners. However for cost and bulkiness reasons it is best, if possible, to avoid this addition. To determine this sum more simply, consider the equilibrium of the mechanical device associated with one of the edges. The case of edge 1 (s = 1) is represented and analyzed below, the other cases being similar.

• • • U imp x U imp y u (1) x u (2) x u (n) x u (1) y u (2) y u (n) y F (0) x F (0) y F (1) x F (1) y f (1) x f (2) x f (N ) x f (1) y f (2) y f (N ) y
x (1) x (2) L y

x Everything being considered linear, they can be solved separately.

1. In the first case N p=1

x (p) f (p) y is simply given by the angular momentum equation of the static equilibrium:

N p=1 x (p) f (p) y = F (1) y L (8)
2. In the second case, one can evaluate the work of the external forces W . Since the material is linear elastic, each effort is proportional to the displacement of the application point. Assuming that the loading device does not store energy, from the Clapeyron's theorem the elastic energy has the following expression.

W el = N p=1 1 2 f p x u p x + 1 2 F 1 x U imp x
The application of Castigliano's theorem provides the following relation:

∂W el ∂U i = F i (9)
in which U i and F i denotes the displacement and force associated to the same dof. The application of this relation to the imposed displacement leads

∂W el ∂U imp x = F 1 x ( 10
)
which can be developed as:

∂W el ∂U imp x = N p=1 1 2 f p x ∂u p x ∂U imp x + 1 2 F 1 x ( 11
)
The expression for the term ∂u p x ∂U imp

x is known from the specific kinematics imposed by the KUBC, i.e.,

∂u p x ∂U imp x = x p L (12)
Finally, by combining eq [START_REF] Bouaziz | Heterogeneous and Architectured Materials: A Possible Strategy for Design of Structural Materials[END_REF], eq [START_REF] Donders | Homogenization method for topology optmization of struc-tures built with lattice materials[END_REF] and eq [START_REF] Eberhardsteiner | Biaxial testing of orthotropic materials using electronic speckle pattern interferometry[END_REF] the following relation is obtained:

N p=1 x (p) f (p) x = F (1) x L (13)
Each quantity involved in eq [START_REF] Barthelat | Architectured materials in engineering and biology: fabrication, structure, mechanics and performance[END_REF] is therefore measurable from the knowledge of global forces on each side. As an important consequence, in the situation of KUBC, the average stress Σ ∼ can be computed in a realistic experimental setup. The final formula is given in eq [START_REF] Feyel | Multiscale FE2 elastoviscoplastic analysis of composite structures[END_REF].

Σ ∼ = 1 L H F (2B) x + F (2C) x -F (3B) x + F (4C) x -F (1D) x + F (2C) x + F (4C) x + F (4D) x F (2B) y + F (2C) y -F (3B) y + F (4C) y -F (1D) y + F (2C) y + F (4C) y + F (4D) y (14)
Where F (2C) y

is the effort over y on the edge 2 applied at the point C. Points and edges are reminded on fig 6. Once again, the expression of Σ ∼ is not symmetrical due to the choice of the origin on a corner of the domain. Actually, the expression of sigma given here is very dependant on the choice of the origin. However, when determining the experimental values, a symmetric expression is expected. To conclude this preliminary analysis of the necessary measurement technique, the reader must keep in mind that a global-type solution seems reasonable since global force are sufficient. The force required to deform the device must be negligible compared to that of the sample or at least of the same order of magnitude so as not to mask the latter. It is then necessary to be able to determine the force due to the device alone in order to decouple the two contributions measured by the global load cells, which is the main goal of section 4.

Last, it is also noteworthy that even if the device uses a 'global' force measurement for homogenization, one may use full-field measurement techniques to obtain the pantograph strain, and by use of its model, assess the forces transmitted at the specimen interface. The homogenization by KUBC being the main purpose of the device, the pantographs are designed to be stiff in comparison to the specimen, and this secondary load measurement technique might be not optimal, but this principle must be kept in might for future applications. displacement accuracy at small strain scale is not of utmost importance and only bi-axial tension is considered. In [20] the frame is intended to apply also shear loads with a possibility to rotate the principal axes which corresponds to KUBC. However it was not conceive in the scope of homogenization and thus does not fulfil all the present requirements. The table 2 summarizes the specifications of these devices and the ones of the intended device. In both examples the deformable frame consists of four pantographs assembled around the sample. An example of such a system is given in fig 8. This mechanism is constituted of rigid arms and pivots joints. The longitudinal strain of the structure is a soft mode: no strain energy is associated to it. All other modes involve the stiffness of the bars, so that imposing the kinematic of two joints amounts to impose the kinematic of the whole A pantographic based solution thus seems to be well-adapted to our needs [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF] (fig 9) since it gives exactly a discrete kinematic uniform condition.

Technological choices

Now that the main idea of the mechanism has been defined the technological implementation has to be detailed. In order to minimize the errors in the imposed kinematics, clearances should be avoided as much as possible. Similarly, all sources of dissipation will be circumvent if possible. Dissipation sources and clearance will be called indeterminate later.

Two technological solutions for the design of the pantograph will be discussed below:

1. Bearing-based solution;

2. Flexible Link (FL)-based solution.

Ball-bearing based solution

A ball-bearing based solution minimises the energy losses but the clearance problem remains. For a centimetric bearing, the clearance with normal adjustment is around c = 5 µm, which gives for a pantographic structure:

• Axial error: A pantograph is composed of a series of assembled elementary cells. Each cell causes an elementary error that will proportionally accumulate along the pantograph. This implies an error of 2nj for a n elementary cells pantograph making 100 µm here.

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Sample Imposed BC • fastener
• Transverse error: Due to the bending of the pantograph, the spurious transverse displacement is much bigger than the axial one. A simple geometric construction gives e bend = n √ 2 1 + n(n-1) 2 j > 2.5 mm with a small angle hypothesis. The transverse guiding error can be reduced to about 0 by constraining the pantograph displacement on rails. This solution is adopted by [20] and Accupull c .

Since the a total displacement should remains below 120 µm, this solution involving 100 µm error without taking into account the deformation of the component parts is not suitable. In addition, even if the dissipation of a bearings are very low, the large number of bearings and the very strong hyperstatism of the mechanism may generate significant energy losses. We will therefore consider another solution. Given that the in-plane strain range is quite low, a solution based on flexible links is possible. The principle of a flexible link is to provide a pivot connection by elastic deformation of a thin part (fig 10). Such a connection has no clearance and no energy dissipation as long as the link remains in its elastic domain. Stiffness and maximum rotation angle of the pivot depend on the mechanical properties of the material (Young modulus and elastic limit) and on the geometry of the thin part R, h and b (fig 10). However, these links are delicate: they can only withstand low rotation and low shear load. They have to be carefully sized. First, have a look to the global mechanism under load to determine the static equilibrium and the forces in the links (fig 11). It should be noted that even if the mechanism seems to be a lattice, the beams have three pivot links and then do not work only in tension/compression. A significant misalignment of forces occurs at the ends of the pantograph (fig 12b). To overcome the problem of shear of the FL that this causes, three FL will be placed on each axis of rotation at 0, +30 and -30 degrees with respect to the pantograph axis (fig 13). The maximum loading of the links Cell number link force / applied force

Flexible link based solution

f f f f f f f f f T (n) B(n) C(n)
||T (n)|| N f ||B(n)|| N f ||C(n)|| N f
(a) Magnitude according to the total effort applied nf Cell number Angle of the loading

( T (n), x) ( B(n), x) ( C(n), x)
(b) Orientation with respect to pantograph axis gives their minimal section according to the yield strength of the material.

A variable section beam study [START_REF] Konieczka | De la rigidité d'une liaison élastique calculée par la théorie des poutres[END_REF] gives the maximum angle of rotation of flexible connections according to material and geometric parameters:

ω max = 2 C R h R h R e E with C R h a monotonic function only dependent on R h .
A Modulus -Strength chart from [START_REF] Ashby | Material property charts[END_REF] gives elastomers and polymers as best results for the flexible links according to this criterion. However all the pantograph will be made in this material and the arms must be rigid enough to impose their kinematic to the sample. Moreover they are materials with hysteric mechanical behavior, which one must avoid to ensure a bijection between force and displacement. For these reasons they will not be retained. The next choices are titanium alloys, magnesium alloys and aluminium alloys. For cost reasons we will choose 7000 series aluminium alloys for constitutive material.

Thus, taking into account the desired space requirement, the following set of parameters is obtained for a 7075 aluminium alloy

         R e 430 MPa E = 72 GPah = 0.7 mm b = 100 mm R = 3 mm
This configuration with a simple pantograph does not seem viable. Actually, to machine a thin of 0.7 mm through a 100 mm aluminium sheet is not conceivable. Thereby a multi rows pantograph is considered as shown on fig 14 The sample will take place between the two stages (fig 15d) to ensure an in-plane loading. The three layers are the result of the choice of three orientations of flexible links. In such a situation the static equilibrium can not be written easily and a numerical resolution is mandatory.

Overall geometry

The geometries of the arms are obtained by maximizing their global rigidity while maintaining the functional surfaces necessary for operation (fig 16).

It is worth noting that the relation between the kinematic of the arms and their geometry is not straightforward, the pivot connections seeming located at the screw bores while they are in fact at the FL neck. The kinematics of the assembly is shown on fig [START_REF] Hazanov | Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume[END_REF]. Now that the main shapes of the arms have been determined, the geometrical parameters have to be optimized. However a global computation is not conceivable because of the very thin parts that tremendously increase the cost of calculation. To avoid mesh dependency, at least 500 000 dof per arm are necessary, leading to more than 3 × 10 8 dof in the case of fig 15d. The associate numerical cost is prohibitive and justifies the creation of a specific numerical tool.

Creation and validation of the VTD

A multiple rows pantograph can be seen as a lattice with specific macroelements: each element represents an arm with multiple anchors. Thus, to simulate the behaviour of the pantograph, it is sufficient to determine the behaviour of the arms and then to make a lattice with the adequate elements. Two types of arms will be considered: flexible and rigid, the flexible links being included in the flexible arms. Each node of the macro-elements possesses all the in-plane dof: two translations u x , u y and one rotation ω z .

As can be seen in fig 15d the complete device is composed of 2 stages of 4 pantographs with 3 layers of multiple arms. This composition will guide the main steps in the development of the numerical code. [k]

[K]

[K]

[K]

[K]

[κ] Arm 1 layer

layers

Global device

Inter-layer

Corner link

The same symbolism will be used for displacements and forces (table 3).

The reader must pay attention to the notion of subscript. Here, subscripts refer to the nodes and therefore to 3 dof, consequently [ • (ij) ] is not a scalar but a 3x3 matrix.

Elementary matrix calculation

Since it is necessary to control the pantograph by the axes of the pivots to impose the sought kinematics, it is simpler to express the elementary stiffness matrix [k] at the center of the flexible links (kinematic centres) rather than at their ends (geometric centres) (fig 16).

'scale' stiffness matrix displacement force subscript arm To do so, ones modifies the geometry so that the mechanical assemblies are the same as the kinematic assemblies, leading to the design depicted on fig [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF]. The number of flexible links N F on each arm depends on the number of rows of the pantograph. The elementary stiffness matrices are computed row by row by computing the reaction associated to each elementary dof:

[k] [u] [f ] i, j 1 layer [K] [U ] [F ] I, J 3 layers [K] [U] [F] α, β global device [K] [U] [F] Γ, ∆
[f ] = [f x 1 , f y 1 , m z 1 ] , . . . , f x j , f y j , m z j , . . . , f x N F , f y N F , m z N F
and the results are concatenated:

[k] = f x1 . . . f xj f yj m zj . . . m z N F u x1 u y1 ω z1 . . . u xi u yi ω zi . . . u x N F u y N F ω z N F                         [k (ij) ]                        with [k (ij)
] the (3 × 3) matrix to nodes i and j of the same arm. Since the complete device is made of two types of arms (R,F), having 3 specific orientations (U,M,D) the elementary matrices are of 6 different type.

k F U k F M k F D k RU k RM k RD
where the top letters are used to describe the arm geometry:

Building of a layer

The local matrices have to be assembled two by two to obtain the stiffness of a pantograph. Since the flexible and rigid arms that have the same orientation create a layer, three layers are obtained:

K X With X = {U, M, D}
Each layer's stiffness matrix expresses as follow and is illustrated on fig [START_REF] Huet | Application of variational concepts to size effects in elastic heterogeneous bodies[END_REF].

K X (IJ) = χ R (I, J) k RX (ij) + χ F (I, J) k F X (ij)
Where

I

Tying the layers

Having assembled all the elements has lead to four series (sides) of three layers of pantographs. It is then necessary to connect the three layers of the same side. One can not simply impose an identical kinematics on the stacked nodes, because the centres of the flexible links of the three layers do not strictly have the same displacement. The points with the same displacement from one layer to another are the centers of the holes (geometric centres on fig 16a andfig 16b). However these points do not belong to the lattice, so we can not impose conditions on.

To overcome this problem, an other macro-element, of stiffness matrix [k], corresponding to the rigidity between the layers is introduced. This makes it possible to make the link between the nodes of the different layers. The detail of its determination is given in Appendix B

The stiffness matrix between three superposed nodes of three layers [k] can be split in [k ab ] depicting the stiffness between the nodes of the layer a and b.

[k] =       k (U U ) k (U M ) k (U D) k (M U ) k (M M ) k (M D) k (DU ) k (DM ) k (DD)      
This allows to link the layers and build the stiffness matrix of the three-layer pantographs [K]:

[K] =   [K] [K] [K]   + [K]
with

K (αβ) = a={U,M,D} b={U,M,D} δ l αa δ l βb δ c αβ k (ab)
where

               δ l αa = 1 if α ∈ layer a δ l αa = 0 otherwise δ c αβ = 1
if α and β have the same coordinates δ c αβ = 0 otherwise δ l being a Kronecker like symbol that defines the belonging to a layer and δ c a Kronecker like symbol for the coordinates of the nodes.

Imposing BC

Finally, it remains to connect the 4 sides. It is desired that the translations be transmitted between the sides but that the rotations are not in order to impose shear on the specimen. The rotations of the sides must therefore be independent. To impose it numerically, a method by penalization is applied:

[K] =     [K] [K] [K] [K]     + [κ] with [κ] = χ pena (Γ, ∆) k where                χ pena (Γ, ∆) = 1, if Γ = ∆ Γ ∈ LP D χ pena (Γ, ∆) = -1, if Γ = ∆ (Γ, ∆) ∈ LP D χ pena (Γ, ∆) = 0, otherwise
LP D is the List of Penalized couples of Degrees of freedom. This amounts to hook a spring of stiffness k between the dofs. k = 10 p+q/2 with p order of magnitude of the larger term of the matrix q number of significant digits with which the software works

The value of k is determined so as not to deteriorate the conditioning of the overall stiffness matrix while ensuring a good bond of the dofs [START_REF] Felippa | Introduction to finite element methods[END_REF]. Unlike a substitution method or a Lagrange multipliers method, the penalization method is not exact. However, it is much simpler to implement and generates smaller errors than those due to the numerical determination of the local stiffness matrices.

It is thus possible to simulate the behaviour of a complex set of pantographs and to deduce the displacements of each of the nodes (kinematic centers).

To achieve the sizing of the arms it is necessary to know the displacement of the nodes (to minimise the maximum error of the device) but also to ensure its mechanical resistance. To do so, the mechanical actions transiting through each FL are calculated by using the kinematical results and the elementary stiffness matrices of the arms

[k]. [f ] = [k] [u]
Where [u] and [f ] are respectively the displacement and mechanical actions on the nodes of a macro-element and so on the FL of an arm.

[f ] = [f x 1 , f y 1 , m z 1 ] , . . . , f x j , f y j , m z j , . . . , f x N F , f y N F , m z N F
The stress state in each FL can then be obtained thanks to the knowledge of the mechanical actions transiting through it [f x i , f y i , m z i ] by superposition of previously calculated elementary stress states: 3 FE calculations are conducted on a FL, on each the link is subjected to an elementary loading corresponding to a degree of freedom of the plane.

[f x i , f y i , m z i ] = [1, 0, 0] ⇒ σ ∼ 0 x [f x i , f y i , m z i ] = [0, 1, 0] ⇒ σ ∼ 0 y [f x i , f y i , m z i ] = [0, 0, 1] ⇒ σ ∼ 0 z where σ ∼ 0 X
(with X = {x, y, z}) are the elementary stress state used to determine real stress state:

σ ∼ i = X={x,y,z} f X i σ ∼ 0 X where σ ∼ i
is the stress state of the i-th link of the arm. Knowing the stress state in the link, an equivalent Von Mises stress can be computed to ensure that the links remain in their elasticity domain.

Validation

To control the validity of the proposed numerical model, full-field simulations of the device are conducted. In order to be as close as possible to the real device, the simulation presented in fig 20 is considered. A three layer pantograph with one row and 5 fasteners is studied. However the number of nodes, layers, ... are lower than on the complete device, all the technical difficulties are represented:

• special arms are used to impose the kinematic of the pantograph (arms with green cylinders);

• important out-of-plane dimension because of the 3 layers and the anchors while the reduced model is in-plane; represents a numerical cost of less than 1.10 2 dofs, with 13 preliminary calculations of the elementary matrices (13 different arms including the special ones used to impose the kinematic and the static conditions), with around 1.10 5 dofs each. For such a computation the benefit of the reduced model is not very important, however for the complete device the number of nodes associated to the reduced model does not increase significantly while the one associated to the full-field simulation is multiplied by more than 30 (table 4 The figure 22 presents the displacement disparities in µm between the fullfield computation (reference) and the reduced model for the nodes (centres of FL) on which the sample will be tied. A maximum of 8 µm of difference are observed, that represent 0.3% of the solicitation (3 mm). Finally, the model gives the possibility to simulate a multi rows and multi layers pantograph with kinematic and static discretized conditions and return the displacements and an evaluated maximum Von Mises criterium on the flexible links.

Sizing

The specifications given in table 2 are considered : a maximum of 120 µm of error on the imposed kinematic is allowed, representing 2% of the solicitation range under maximal static loading. Computations on the reduced model are conducted to obtain the optimized geometrical parameters (fig 23).

The thickness and radius of the flexible links are chosen in order to stay in the elastic domain and to ensure the solicitation range (6 mm).

The thickness of the arms, the number of rows and the global geometry of the arms are chosen in order to ensure kinematic precision and realistic machining process.

Conclusion and perspectives

An adaptation of mean field homogenization by KUBC to an experimental framework has been exposed. This adaptation is possible thanks to, two major points:

• a method for determining the average stress field over the specimen by realistic measuring means (global measurements on the edges of the specimen); • an innovative design of an experimental device allowing the application of discretized Kinematic Uniform Boundary Conditions on the edges of a 300 mm square specimen. The use of pantographs for the repartition of the displacement on the edge allows the use of a small number of actuators [START_REF] Ashby | Designing hybrid materials[END_REF] to control the displacement of all the points on the boundary. The realization of the pantographs by flexible links permits to eliminate all the clearance and the major part of non-linear dissipations of the pantograph.

A reduced model of the device has been proposed by assimilating the arms of the pantograph to macro-elements. This reduced model has been used to size the device and will later be used:

• before the tests to predict and adapt test conditions;

• after the test, by using this model in an inverse approach it will be possible to determine the loading repartition on the boundary of the specimen by measuring the displacement of the device (Digital Image Correlation).

Subsequently, the manufacture of the machine and the validation of its characteristics will have to be finalized before carrying out homogenization tests. These tests will be coupled with a numerical study in order to highlight

≈∼

  which, over a RVE, relates the second-order volume-average strain tensor (E ∼ ) to the second order volume-average stress tensor (Σ The averaged strain and stress tensors are defined over the domain Ω from the local fields ε ∼ (x), σ ∼ (x) :

Figure 1 :

 1 Figure 1: Step-by-step transition from a perfectly theoretical situation to an amenable one for experimental testing.

Figure 2 :

 2 Figure 2: Definition of the considered domain Ω

Figure 3 :

 3 Figure 3: Characteristic length scales of the mechanical problem

Figure 4 :

 4 Figure 4: Schematic diagram of local and global type devices

Figure 5 :

 5 Figure 5: Equilibrium of the mechanical device associated to one of the edges

Figure 6 :

 6 Figure 6: Reminder of notations Ω

3 .

 3 . . . to an experimental context 3.1. Mechanism Some devices using the idea of a linearly deformable frame have already been proposed, for research context as the 1γ2ε device [20] (fig 7a) or in an industrial context by Accupull c (fig 7b) for example. The machine by Accupull c uses this idea as a way to distribute normal loading in a very large strain range in the context of plastic injection mold studies. Consequently

  (a) 1γ2ε device -[20] (b) Biaxal pulling machine by accupull c

Figure 7 :

 7 Figure 7: Testing machines using pantographs to apply rich edge conditions

Figure 8 :

 8 Figure 8: Schematic diagram of a pantograph

Figure 9 :

 9 Figure 9: Field application device by pantographic frame and the three elementary applicable strain states

Figure 10 :

 10 Figure 10: Description of a flexible link and its minimum section

Figure 11 :

 11 Figure 11: Loading of the pantograph

Figure 12 :Figure 13 :

 1213 Figure 12: Efforts transiting through the FL

  . The desired device (fig 15d) will then be constituted of two stages (fig 15c), each composed of three layers (fig 15b) of four pantographs (fig 15a) with multiple rows. ascending arm descending arm

Figure 14 :

 14 Figure 14: 1 and 2 rows pantograph

Figure 15 :

 15 Figure 15: Composition of the device

Figure 16 :Figure 17 :

 1617 Figure 16: Shape of the arms -u

1 . 4 .

 14 Stiffness matrices of the arms [k] are determined by Finite Element (FE) simulation. 2. Matrices of single-layer pantographs [K] are build with the elementary ones. 3. Three-layer pantographs matrices [K] are obtained by tying the [K] matrices together by [K]. Global stiffness matrix of the device [K] is constructed by binding the [K] matrices by their corners with [κ].

Figure 18 :

 18 Figure 18: FE simulation for elementary matrix determination

FU:

  Upwards link (30 deg.) M : Mid link (0 deg.) D : Downwards link (-30 deg.)

  and J are the indices of the nodes in the layer i and j are the corresponding indices in the arm with χ R (I, J) = 1, if I and J are nodes of the same rigid arm χ R (I, J) = 0, otherwise χ F (I, J) = 1, if I and J are nodes of the same flexible arm χ F (I, J) = 0, otherwise Stiffness matrix of a layer :

Figure 19 :

 19 Figure 19: Filling of the stiffness matrices

•

  complementary parts, called anchors, are fixed on the arms of the pantograph (through the orange faces on fig 20) to make the link with the specimen. The static conditions are imposed on the anchors, on the blue areas (fig 20).

Figure 20 :

 20 Figure 20: Assembly used for the validation of the first step.-green cylinders : imposed kinematics -orange faces : fixed to the anchors -blue area : imposed statics

Figure 21 :

 21 Figure 21: Reduced model. -green diamond : imposed kinematics -blue dots : imposed forces.

Figure 22 :

 22 Figure 22: Displacement disparities between full-field computation and reduced model

Figure 23 : 3

 233 Figure 23: Evolution of bending error (µm) with the numbers of rows for the maximum loading (200 N/fastener)

Figure 24 :Figure B. 26 :Figure B. 27 :

 242627 Figure 24: CAD overview of the design and a specimen (in red) → to actuator

Table 1 :

 1 Values of n and x on the different edges

Table 2 :

 2 Comparison of existing and wanted testing device

	Testing device	1γ2ε-device	accupull c	wanted device
	sample size max extension	650 × 550 mm 2 0.2%	58 × 58 to 106 × 106 mm 2 700%	300 × 300 mm 2 2%
	max compression	0.2%	0%	2%
	max shear angle	20 o	0 o	2 o
	load capa/fastener	100 kN	0.2 kN	0.2 kN
	fasteners by side	5	7	10
	max displ error	unknown	> 5 000 µm	120 µm

Table 3 :

 3 Symbolism used in the description of the construction of the global stiffness matrix

Table 4 :

 4 ). Calculations costs associated to the complete device

		preliminary calculations global simulation
	full-field	0	3.10 8 dfos
	reduced model	13 × 2.10 5 dofs	4.10 3 dofs

Table 5 :

 5 General characteristics of the device

To avoid any misunderstanding, the approach we follow here is not restricted to linear elasticity and can be extended to more general constitutive behaviours such as: finite elasticity, viscoelasticity, plasticity,...

The context of homogenization in linear elasticity is interesting for designing a machine since the theoretical framework is clear and well-understood, for instance, 1) the control parameters are clearly identified as the quantities to measure for reconstructing the overall behaviour, 2) analytical/numerical solution to validate the design of the machine are easy to obtain.

All the equations are provided here under the assumption that u(x)⊗∇ 1. This hypothesis makes it possible to confound the initial and distorted configurations, and to bring the equilibrium study on the initial configuration. This hypothesis, which is legitimate in view of the applications considered in this paper, can nevertheless be weakened, and the equations adapted accordingly.

Appendix A. Detailed calculation of the mean stress

Starting from eq [START_REF] Ashby | Designing hybrid materials[END_REF] :

• n (p) ⊗ x dS and using the linear decomposition of the stress

and the position

we have

y (p) • e (p) y (p) dS with S (p) = l (p) h the surface of a fastener.

Having assembled all the has lead to four series (sides) of three layers of pantographs. • the anchor parts of the FLs, so the points with strictly the same displacement;


F Cheynis

N Rougemaille

R Belkhou

J.-C Toussaint

O Fruchart
email: olivier.fruchart@grenoble.cnrs.fr

X-ray photoelectron emission microscopy in combination with x-ray magnetic circular dichroism investigation of size effects on field-induced Néel-cap reversal

X-ray photoelectron emission microscopy in combination with x-ray magnetic circular dichroism is used to investigate the influence of an applied magnetic field on Néel caps (i.e., surface terminations of asymmetric Bloch walls). Self-assembled micron-sized Fe(110) dots displaying a moderate distribution of size and aspect ratios serve as model objects. Investigations of remanent states after application of an applied field along the direction of Néel-cap magnetization give clear evidence for the magnetization reversal of the Néel caps around 120 mT, with a ±20 mT dispersion. No clear correlation could be found between the value of the reversal field and geometrical features of the dots.

Magnetization reversal using an external magnetic field as a driving parameter is a major topic in magnetism. Most studies focus on the reversal of magnetic domains by rotation and/or domain wall propagation. Few reports consider domain walls as object whose inner magnetic structure can be manipulated by a magnetic field. In thin films of soft magnetic material, internal wall structures have been extensively studied by Lorentz microscopy 1,2 and Kerr effect microscopy. [START_REF] Hubert | Magnetic Domains: The Analysis of Magnetic Microstructures[END_REF] However, these systems are easily saturated and a minute in-plane tilt in the magnetic field orientation makes the domain walls propagate. Magnetic nanostructures are systems of growing interest as they constrain geometrically magnetic objects which can be in turn manipulated. Recent reports concern submicron magnetic disks, in which magnetic vortices, i.e., essentially one-dimensional objects, are stabilized. Such vortices have been reversed using a strong out-of-plane magnetic field [START_REF] Okuno | [END_REF]5 or an in-plane magnetic field of smaller amplitude, either pulsed 6 or oscillating. 7 Here, we consider an essentially two-dimensional magnetic object: an asymmetric Bloch wall of finite length. [START_REF] Hubert | Magnetic Domains: The Analysis of Magnetic Microstructures[END_REF]8,9 The wall was studied in micron-sized elongated self-assembled Fe(110) dots, where a so-called Landau state occurs. 10 We evidence that the direction of magnetization in the Néel caps can be reversed by applying a magnetic field around 120 mT along the magnetization of the caps. This has been observed at remanence using x-ray photoelectron emission microscopy in combination with x-ray magnetic circular dichroism (XMCD-PEEM) and reproduced to a good agreement by numerical simulation. In this paper, we put some emphasis on the reversal field distribution, ±20 mT. No clear correlation could be evidenced in relation to the size or geometry of the dots.

The investigation has been performed on self-assembled Fe(110) dots grown by pulsed laser deposition under UHV. 11 These single-crystalline submicron-sized structures are obtained by epitaxial growth on a Mo(110) or W(110) buffer layer deposited on a (11-20) sapphire substrate at typically 500 °C and exhibit atomically flat facets [Fig. 1(a)]. Previous studies 12 have demonstrated both numerically and experimentally that as-grown Fe(110) dots with heights ≥100 nm exhibited the Landau state.

In this configuration, the magnetization vector lies essentially in the plane of the sample and displays two main domains along the dot length, antiparallel with each other and thus separated by a 180° Bloch wall. Inside the wall, the magnetization is perpendicularly oriented We used the French X-PEEM instrument installed at the nanospectroscopy beamline of the synchrotron Elettra (Trieste, Italy) in combination with x-ray magnetic circular dichroism (XMCD-PEEM). Mirror electron microscopy (MEM) has been used to determine the topography of dots. XMCD-PEEM technique enables, with high spatial resolution (≈20 nm), to image the surface magnetization distribution along the incident x-ray beam. Using this technique, top surface Néel caps of similar Fe(110) dots have been evidenced in a previous study. 12 However, as the PEEM technique is based on the collection of electrons with low energy, only very moderate in-plane magnetic field could be applied. To overcome this limitation, a statistical approach has been adopted. We have applied a transverse field ex situ, introduced the sample under UHV, and imaged it at remanence. This manipulation has been repeated for each value of the applied field.

Typically, 30-40 dots have been imaged for each field value to determine with a sufficient statistics the ratio of each top Néel-cap population (black or white in our images) to the total Néel-cap population.

4

Starting from an as-grown assembly of Fe(110) dots, the ratio of the white Néel caps to the total number of dots imaged at remanence increases with the applied field until it reaches ~95% for an applied field of 150 mT [Fig. 2(c)]. The mean reversal field is 120±20 mT. We have also checked that applying a magnetic field of -150 mT yielded a population inversion of the Néel caps (ratio of black Néel caps of ≥95%). We have thus evidenced that an internal component of a domain wall can be controlled by a magnetic field. As XMCD-PEEM gives access to the top surface component of the magnetization only, three-dimensional micromagnetic simulations are required to understand the details of the reversal. This is reported in Ref. 13. Here, we will repeat only the information needed to follow the main topic of the article, which is the investigation of the distribution of reversal field. We used a custom-made micromagnetic code based on the temporal integration of the Landau-Lifschitz-Gilbert equation, i.e., in a continuous medium approximation. These simulations are performed with the finite difference method. In this framework, the sample is discretized into prisms of regular size. The micromagnetic simulations predict that at remanence, the Néel caps are always antiparallel to each other, which means that during the reversal process, both caps reversed their magnetization. This is accomplished by the reversal of the cap with magnetization antiparallel to the field for rising field and by the reversal of systematically the top cap for decreasing field. The cap observed by XMCD-PEEM therefore has its magnetization opposite to the applied field. The experimental and numerical reversal fields are in good agreement.

Let us now investigate whether the reversal field depends on geometrical parameters.

Several arguments could be put forward. First, we could expect that increasing the nanostructure height would result in decreasing the dipolar coupling between the two Néel caps which would in turn decrease the field value needed to reverse the Néel caps. Second, the reversal field may also depend on the vertical aspect ratio (height/width). Indeed, if the vertical aspect ratio increases, then the transverse demagnetizing coefficient also increases, so that the magnetization in the two longitudinal domains rotates less under a given transverse field, and finally the field value at which the two caps become parallel is increased. The reason is, following the variational model first proposed by Labonte, 8 that the cost in exchange energy of the Néel cap antiparallel to the field, i.e., for which a rotation larger than 180° is required, is lowered as the angle between the longitudinal domains is also reduced. Thus, an increase of the reversal field of the cap would be expected.

For each applied field, we have studied the total Néel cap population (black and white) and the population of white Néel caps as a function of height, lateral aspect ratio (length/width), and vertical aspect ratio (height/width). The length and width of both top and bottom surfaces have been determined from MEM images (Fig. 3). Each distance has been measured using intensity profiles and defined at half-height. For dots displaying asymmetry, maximum projected lengths of both surfaces have been considered. As the angle between the (110) base plane and the (010) planes that define the longest sides of a dot equals 45°, the height of a dot is computed as the half-difference between the bottom and top surface widths. Typical relative uncertainty on distances is 7 nm. These overlaps entail that, in the present study, no clear correlation between reversal fields and geometrical parameters can be evidenced. This means that either the reversal field is independent of geometrical considerations or our statistics was not sufficient to evidence this effect. In the former case, effects connected with the three-dimensional nature of the magnetic configuration, such as the magnetization reversal at edges, may be involved.

In conclusion, using self-assembled Fe(110) dots as model systems, we have shown with XMCD-PEEM microscopy that the orientation of internal features of an asymmetric Bloch wall, known as Néel caps, can be reversed by applying a magnetic field transverse to the dot. A mean reversal field of 120 mT has been evidenced, in good agreement with numerical simulations. The reversal field distribution of ±20 mT could not be correlated to any geometrical parameter. Its origin is still an open question.

[

  FIG. 1. (Color online) (a) Typical AFM image of an elongated Fe(110) dot. [(b) and (c)] Micromagnetic simulations of the Landau state in a facetted Fe(110) dot (1000×500×100 nm 3 ). In (b), regions with a strong perpendicular component of the magnetization (|m z |≥0.5) are displayed, showing the Bloch wall. In (c) and (d), only the transverse magnetization component is represented, highlighting the antiparallel Néel caps.

FIG. 2 .

 2 FIG. 2. (Color online) [(a) and (b)] Typical views of Fe(110) dots: (a) MEM images (topography) and (b) XMCD-PEEM images (magnetism). In (b), the white arrow indicates the x-ray beam direction. (c) Evolution of the white Néel cap (NC) ratio as a function of the transverse magnetic field.

FIG. 3 .

 3 FIG. 3. (Color online) Illustration of distance measurements. (a) MEM image of an Fe(110) dot displaying asymmetry. The bottom surface length is defined between the two vertical black lines. (b) Intensity profile corresponding to the black dotted line in (a). The bottom surface width is defined at half-height (black vertical dotted lines) of the intensity profile.

Figure 4 7 FIG. 4 .

 474 Figure4illustrates the data that we have obtained as a function of the height (left) and the vertical aspect ratio (right) of nanostructures (similar results have been obtained for the lateral aspect ratio, not shown here). In the case of a dependence of the reversal field on these parameters, we would have expected, for increasing field values, the onset of an asymmetry in Figs. 4(a)-4(c) and different saturation fields (i.e., field value at which all imaged dots are white) in Fig.4(d). However, due to statistical uncertainties, the different curves overlap. To determine these uncertainties, we have supposed that all white Néel-cap histogram bin values are random


Markos Dallas
email: markos.dallas@uia.no

Towards an interactional perspective on argumentation in school mathematics

Keywords: Socio-mathematical norms, participation, mathematical argumentation, mathematical communication, mathematics classroom interaction

This paper discusses theoretical and methodological considerations which have emerged from reviewing the literature related to my PhD research project which adopts an interactional perspective on the development of the argumentation process in primary school mathematics. This led me to distinguish the factors involved in mathematics classroom interaction during the development of the argumentation process, as well as to examine the possible relations and interrelations of these factors. The outcome of this process was, first, to clarify theoretical aspects and, second, to create a preliminary model, called the "Mathematics Classroom Interactional Model" (MCIM), positing two levels of classroom interaction and the possible relations within each of them and between them.

Introduction

In the field of Mathematics Education, according to the related literature review, the notions of argument and of mathematical argument can be found in two different strands. The first strand relates to socio-mathematical norms and participation in the learning environment (e.g. [START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF][START_REF] Wood | What does it mean to teach mathematics differently[END_REF], mostly based on situated learning theory in the research (e.g. [START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF][START_REF] Greeno | Theories and practices of thinking and learning to think[END_REF][START_REF] Boaler | Mathematics from another world: Traditional communities and the alienation of learners[END_REF] and the second strand relates to mathematical argumentation and classroom interaction (e.g. [START_REF] Steinbring | The construction of new mathematical knowledge in classroom interaction: An epistemological perspective[END_REF][START_REF] Stylianides | Proof and proving in school mathematics[END_REF]. However, so far, the literature does not seem to have made any clear attempt to develop a coherent theoretical framework and methodological tools emerging from the two strands. Thus, in my study, I focus on the development of the argumentation process in primary mathematics classrooms, considering the norms, the interactions and the role of the participants in learning and teaching practices. In this paper, I discuss aspects related to mathematical argument, argumentation and participation from the two strands of the literature to formulate the preliminary central research questions of the study, as well as to discuss the coherence -both compatibility and complementarity-of the two strands. On this basis, I develop a preliminary model, called the "Mathematics Classroom Interactional Model" (MCIM), of the concepts and their possible relations that will guide the scope of the study. [START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF] describe learning as a social phenomenon that is constituted in the real world through a process of legitimate peripheral participation in communities of practice which are in development. This means that pupils are members of wider worlds that are socially and culturally formed, so developing links that cross the identities of ethnicity, gender, religion, etc., acts in classrooms, schools and communities and the practices that govern all these environments. The word "knowledge" has been replaced by the word "knowing" that declares an action. This fundamental shift indicates that activities cannot be considered independent of the context. The "practice" is mainly characterized by the terms "discourse" and "communication" which implies that a pupil should be regarded as a person interested in participation in certain types of activities not only in the accumulation of knowledge [START_REF] Sfard | On two metaphors for learning and the dangers of choosing just one[END_REF].

Socio-mathematical norms and participation in the learning environment

While the learning process in the current study is regarded in terms of the participation metaphor [START_REF] Sfard | On two metaphors for learning and the dangers of choosing just one[END_REF], what is important is the person's participation in activities influenced by the context. The underlying theory is that of [START_REF] Vygotsky | Mind in society: The development of higher psychological processes[END_REF] socio-cultural approach, considering learning as an outcome of interaction with others, while theoretical approaches in situated learning are preeminent in contextualizing and describing classroom communities. These approaches, according to [START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF] and [START_REF] Bransford | Designs for environments that invite and sustain mathematical thinking[END_REF], have led to situated learning theories, in which knowledge is situated in particular forms of experience that arise in specific situations, and are understood in a relational way as something shared between people, activities and environments rather than as a fixed, individual characteristic [START_REF] Boaler | Mathematics from another world: Traditional communities and the alienation of learners[END_REF]. Hence, mathematical knowledge, as a dynamic process of mathematization, is "still being open and not fixed in advance of the learning and acquisition processes" (Steinbring, 2005, p. 48).

According to [START_REF] Greeno | Theories and practices of thinking and learning to think[END_REF], many researchers based on situated perspectives study the development of classroom activities which involve pupils participating "in the discourse of the subject matter, including formulating and evaluating hypotheses, conjectures, arguments, evidence, examples, and conclusions" (Hatano & Lambert, 1990;Inagaki, 1991;Cobb et al., 1993;Cohen et al., 1993;Schoenfeld, 1994as cited in Greeno, 1997, p. 99). A situated view suggests that activities of different practices are important, for instance involving pupils in classroom discussions is a way of pupils learning not only the content knowledge but also to participate in discourse practices (Greeno & MMAP, 1998as cited in Boaler, 2000). Pupils learn not only methods and processes in the mathematics classrooms, but they are trained in mathematics, and the learning of content knowledge cannot be separated from the classroom interaction, as they are two reciprocal components [START_REF] Boaler | Mathematics from another world: Traditional communities and the alienation of learners[END_REF]. One question posed in the situated learning theory is whether the pupil's pattern of participation can be a potential obstacle to his/her membership of the classroom community. [START_REF] Wood | What does it mean to teach mathematics differently[END_REF] states that classroom culture consists of a set of social norms, a specific structure of participation, as well as characteristic forms of discourse that support both social norms and the structure of participation. A participatory structure refers to the specific characteristics of the classroom that affect pupils' participation in the classroom: who is involved, when and how. Wood recognizes three types of culture that characterize a classroom of inquiry and can lead to different patterns of participation. The first relates to the development of alternative resolution strategies, the second relates to a culture of exploration of the strategies developed by their classmates, and the third to a culture of argumentation where social norms require pupils to justify or defend the methods of solution they choose. [START_REF] Wood | Children's mathematical thinking in different classroom cultures[END_REF] investigated primary mathematics classroom interactions and the development of mathematical thinking. One of the most important results was the finding that only in an inquiry/argument classroom culture were there opportunities for all children to be involved in meaning making and shared understanding. Nevertheless, as [START_REF] Klein | A poststructuralist analysis of mathematics inquiry in a year 6 classroom: The rhetoric, realisation and practical implications[END_REF] points out, participation in cultures such as the above may be problematic for pupils who either have not conquered the tools of defending or challenging ideas through discussion (e.g. language or norms such as what constitutes a different answer) or lack the self-confidence or self-image expected by an apprentice working in a collaborative learning environment.

Many researchers attach importance to the role of classroom culture, providing cooperative learning opportunities and, in particular, developing the intellectual autonomy of pupils. They focus on sociomathematical norms and argumentative skills in the constitution of mathematical meaning in the classroom. [START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF], investigating the role of communication as a cultural tool, concluded that social norms (e.g. explanation and justification of a solution) directly affect the patterns of participation, a conclusion also supported by [START_REF] Sfard | On two metaphors for learning and the dangers of choosing just one[END_REF], while socio-mathematical norms (e.g. which answer is considered mathematically different) provide equal opportunities to all pupils in that particular structure and regulate mathematical arguments. Finally, [START_REF] Kazemi | Promoting conceptual thinking in four upper-elementary mathematics classrooms[END_REF] defined and described four categories of social norms and socio-mathematical norms respectively. The authors, in their discussion, emphasize the need for future research with longitudinal data that may reveal other norms, how socio-mathematical norms are created and sustained, and how they influence pupils' mathematical understanding.

The discussion so far led me to consider the connection between social and socio-mathematical norms relating to argumentation in the mathematics classroom. It seems that the mediator in this connection is the specific structure of the patterns of participation that allows (or not) the social norms to be transformed, created and sustained as socio-mathematical norms. Thus, the first preliminary central research question which emerged is: "what social and socio-mathematical norms relating to argumentation are established in the mathematics classroom, and how are these expressed in terms of patterns of participation?".

Mathematical argumentation and classroom interaction

In the field of Mathematics Education, [START_REF] Krummheuer | Argumentation and participation in the primary mathematics classroom. Two episodes and related theoretical abductions[END_REF][START_REF] Krummheuer | Methods for reconstructing processes of argumentation and participation in primary mathematics classroom interaction[END_REF] started off using [START_REF] Toulmin | The uses of argument[END_REF] argumentation scheme to analyse classroom-based mathematical arguments. However, [START_REF] Krummheuer | Argumentation and participation in the primary mathematics classroom. Two episodes and related theoretical abductions[END_REF][START_REF] Krummheuer | Methods for reconstructing processes of argumentation and participation in primary mathematics classroom interaction[END_REF] used a reduced version (conclusion, data, warrants and backings) of Toulmin's full scheme of argumentation (conclusion, data, warrants, backings, modal qualifier and rebuttal). Many researchers (Yackel, 2001;Hoyles & Küchemann, 2002;Evens & Houssart, 2004;Cabassut, 2005;Pedemonte, 2005;Weber & Alcock, 2005) as cited in Inglis, Mejia-Ramos and Simpson (2007) appear to have followed Krummheuer in using the reduced scheme. While [START_REF] Inglis | Modelling mathematical argumentation: The importance of qualification[END_REF] concluded that without using Toulmin's full scheme of argumentation it may be difficult to accurately formulate the full range of mathematical arguments, on the other hand Mariotti, Durand-Guerrier and [START_REF] Mariotti | Argumentation and proof[END_REF] mention that difficulties of pupils to organize arguments in a deductive chain in the form of proof cannot be fully explained by Toulmin's model.

Despite the widespread use and proven usefulness of Toulmin's scheme of argumentation over the last two decades, researchers in Mathematics Education do not use this scheme in a consistent way. Besides different emerging interpretations, limitations can be identified. Thus, I studied further the related literature to find a model created in the field of Mathematics Education that could serve both aspects of mathematical argumentation and classroom interaction. [START_REF] Stylianides | Proof and proving in school mathematics[END_REF] developed a theoretical framework about proof and proving in the context of K-mathematical claim" (Stylianides, 2007, p. 191). Any given argument can be broken down into three major components: the set of accepted statements, the modes of argumentation and the modes of argument representation. The distinction between base arguments and ensuing arguments could provide the context in which instructional analysis and instructional interventions by teachers influence classroom interactions and vice versa. In terms of this distinction, it is worth to be mentioned that in everyday mathematics classrooms situations, where the learning process is considered in terms of a participation metaphor [START_REF] Sfard | On two metaphors for learning and the dangers of choosing just one[END_REF], the participants could produce a range of arguments, for example, "relatively sophisticated arguments" or "explications of elements of an argument" (Krummheuer, 2015, p. 53). Thus, the basic assumption is that mathematical argumentation can only emerge through interaction and mathematical communication within the classroom culture. [START_REF] Steinbring | The construction of new mathematical knowledge in classroom interaction: An epistemological perspective[END_REF] developed an analytic framework to examine the relation between mathematical knowledge and mathematical communication. While, "language is the central medium for the creation of possible connections between communication and consciousness" (Steinbring, 2005 p. 53), proof is the communication medium of invisible mathematical objects and the mediator between communication and consciousness (Heintz, 2000 as cited in [START_REF] Steinbring | The construction of new mathematical knowledge in classroom interaction: An epistemological perspective[END_REF][START_REF] Steinbring | The construction of new mathematical knowledge in classroom interaction: An epistemological perspective[END_REF].

Instructional school-mathematical interaction is expected to contribute to introducing individuals into mathematical communication practice, and thus to increase these individuals' ability to participate in (mathematical) communication in the society. (Steinbring, 2005, p. 74) This interaction could be understood by the term "situational".

My perspective is situational, meaning here a concern for what one individual can be alive to at a particular moment, this often involving a few other particular individuals and not necessarily restricted to the mutually monitored arena of a face-to-face gathering. (Goffman, 1974, p. 8) According to [START_REF] Krummheuer | Argumentation and participation in the primary mathematics classroom. Two episodes and related theoretical abductions[END_REF], the term "situational" refers not only to a particular situation that could be characterized as "situated", but to anything that can happen in the interaction between people. Thus, for example, if during a lesson the pupils solve an activity on their own it may be a "situated learning" process [START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF] which is shaped by the pre-knowledge that allows them to face similar activities. The action changes into a "situational" process if the pupils take initiatives to act with their classmates. [START_REF] Levinson | Putting linguistic on a proper footing: Explorations in Goffman's concepts of participation[END_REF] extended the ideas of [START_REF] Goffman | Footing. In Forms of talk[END_REF] and [START_REF] Krummheuer | Argumentation and participation in the primary mathematics classroom. Two episodes and related theoretical abductions[END_REF][START_REF] Krummheuer | Methods for reconstructing processes of argumentation and participation in primary mathematics classroom interaction[END_REF] by adapting the concepts of participants' (speakers') roles in Mathematics Education: "author", "relayer", "ghostee" and "spokesman". Although [START_REF] Krummheuer | Argumentation and participation in the primary mathematics classroom. Two episodes and related theoretical abductions[END_REF][START_REF] Krummheuer | Methods for reconstructing processes of argumentation and participation in primary mathematics classroom interaction[END_REF] approach to participation in argumentation offers insights on the way that participation is performed in mathematical argumentation, the mechanism of being in one role or another is not obvious and cannot explain the obstacles in pupils' participation in the developing of mathematical arguments, something also claimed by [START_REF] Cramer | Participation in argumentation[END_REF]. [START_REF] Cramer | Participation in argumentation[END_REF] highlight the importance of participation in mathematics classroom argumentation, considering the discursive and social processes that affect argumentation. Thus, participation is not only a discourse but the practice of constructing arguments through the social order of participants' interactions in the classroom which can be constructed, maintained and transformed. [START_REF] Cramer | Participation in argumentation[END_REF] mention an interesting case of pupils' implicit participation (pupil's initially spoken idea developed further in the classroom discourse but the pupil's voice disappeared), to describe possible obstacles for participation in argumentation and possible interventions by a teacher.

The discussion so far led me to consider the connection between mathematical argumentation and classroom interaction. It seems that the mediator in this connection is the specific role of the participant that regulates the participation in mathematical argumentation. Thus, the other two preliminary central research questions which emerged are: "how do pupils' interactions contribute to the development of the base arguments?" and "how do teachers' instructional interventions influence pupils' activity in the developing of ensuing arguments?".

Discussion

The ideas about socio-mathematical norms and participation in the learning environment and those about mathematical argumentation and classroom interaction seem to be related. Socio-mathematical norms seem to be a major factor that regulates classroom interaction in developing arguments. Especially, pupils' and teachers' roles in developing base and ensuing arguments are related to sociomathematical norms that could foster (or not) mathematical argumentation and participation in the classroom. Thus, I decided to create the preliminary MCIM model, of the basic concepts and their relations as considered in the previous two sections and expressed in terms of the three preliminary central research questions.

In this model, Figure 1, I posit two levels of classroom interaction. The first (basic) level of classroom interaction is defined by the relations among the social norms, the socio-mathematical norms and the classroom culture. These are the predominant factors that characterize classroom interaction and are defined fully through the review of the first strand of the literature. The socio-mathematical norms established in the classroom culture are affected by social norms. The second (advanced) level of classroom interaction is defined by the relations among the participation, the mathematical argumentation and the participants' roles. This level includes, A: the three factors as a structural unit (participation, mathematical argumentation, participants' roles) and B: three sub-structures: 1) sociomathematical structure: participation-mathematical argumentation, 2) argumentation structure: mathematical argumentation-participants' roles, 3) social structure: participants' roles-participation. Each of these (sub)structures presupposes the connection of the factors at the basic level. Thus, the common ground of the structures at the advanced level is the connection among the three factors at the basic level. When a researcher or a teacher in the classroom would like to understand, investigate and further develop the advanced level, they should firstly understand, consider, and develop the connection of the social norms, socio-mathematical norms and the classroom culture, the factors at the basic level.

According to the related literature, the main connection between the two levels rests on social norms and socio-mathematical norms that regulate mathematical arguments, defining participation in the classroom culture (e.g. [START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF][START_REF] Wood | What does it mean to teach mathematics differently[END_REF][START_REF] Wood | Children's mathematical thinking in different classroom cultures[END_REF]. Therefore, in order to define this model with consistency, I consider theoretical and methodological frameworks to get access to mathematical argumentation and classroom interaction. [START_REF] Stylianides | Proof and proving in school mathematics[END_REF] framework could serve as an analytic tool in order to examine the development of mathematical argumentation and teachers' actions, through the processes of instructional analysis and instructional intervention. Especially, the distinction between base and ensuing arguments and the possible differentiation between them and in each of them could be related with the patterns of participation which emerge through the socio-mathematical norms. Nevertheless, it seems that the focus of the framework is from the teachers' perspectives rather than on pupils' and teachers' interactions and their roles in the classroom culture. Stylianides uses the notion of classroom community in the definition of proof with a perspective different from that usually found in the literature, and from the one I have taken in my study. He regards the pupils as the main members of the classroom community, giving the teacher a special membership status and distinct role, while in my study I focus on the dynamic of the classroom interactions and the power of the relationships that shape the participation in the classroom community. Thus, I elaborate [START_REF] Steinbring | The construction of new mathematical knowledge in classroom interaction: An epistemological perspective[END_REF] framework of mathematical knowledge and communication where the interactions and communication among participants are presented as predominant in the classroom. In this context, the reference to the role of the language in mathematical communication and argumentation in the classroom is very interesting and this leads me to consider frameworks through which I could get insights on the utterance of the mathematical argumentation through the participants' roles. [START_REF] Levinson | Putting linguistic on a proper footing: Explorations in Goffman's concepts of participation[END_REF] categorization of speakers' roles seeks to offer insights on the patterns of participation. However, [START_REF] Krummheuer | Argumentation and participation in the primary mathematics classroom. Two episodes and related theoretical abductions[END_REF][START_REF] Krummheuer | Methods for reconstructing processes of argumentation and participation in primary mathematics classroom interaction[END_REF] does not seem to undertake or investigate the limitations described by [START_REF] Levinson | Putting linguistic on a proper footing: Explorations in Goffman's concepts of participation[END_REF], where multiplicity and alteration of the participants' roles in some utterance events, as well as new roles, could be recognized, defined and re-defined. Moreover, Cramer's and Knipping's (2018) evidence of implicit participation is an interesting aspect related to participants' roles in mathematical argumentation and could be investigated further, either as forms of participation or non-participation. Considering the roles of the participants and interactions among pupils and teachers in the classroom culture, it could be possible to examine pupils' interactions in developing base arguments and explain teachers' actions in influencing pupils' activity to develop ensuing arguments.

Finally, the way that the socio-mathematical norms and the social order, related to the development of mathematical arguments and argumentation, are created, sustained and transformed could provide the context of an interactional perspective on argumentation in school mathematics through the MCIM model. However, this model has still to be considered and defined fully through empirical research to be conducted, developing the methodological context and the research protocols that fulfil the goal of the study.

Figure 1 :

 1 Figure 1: Mathematics Classroom Interactional Model (MCIM)