Marc Rébillat
email: rebillat@lms.polytechnique.fr

Xavier Boutillon

A method to measure elastic and dissipative material properties of sandwich structures and its numerical validation

Keywords:

come

A method to measure elastic and dissipative material properties of sandwich structures and its numerical validation

Marc Rébillat, Xavier Boutillon

Introduction

Sandwich structures are often used in the industry because they can be made light and stiff. However, mechanical properties of sandwich plates may be difficult to predict accurately, particularly if damping is considered. Measurements on isolated samples of the constituents or of whole panels are often needed.

The problem of the estimation of solely the elasticity parameters of a homogeneous material using plate vibrations has been widely addressed using a thick-plate (see for example [START_REF] Soares | Identification of material properties of composite plate specimens[END_REF][START_REF] Ayorinde | Elastic constants of thick orthotropic composite plates[END_REF][START_REF] Frederiksen | Experimental procedure and results for the identification of elastic constants of thick orthotropic plates[END_REF][START_REF] Cugnoni | Identification by modal analysis of composite structures modelled with FSDT and HSDT laminated shell finite elements[END_REF]) and provides in-plane and out-of-plane information about the materials. The problem of estimating the elasticity and damping parameters by using point measurements [START_REF] Mcintyre | On measuring the elastic and damping constants of orthotropic sheet materials[END_REF][START_REF] Devisscher | Identification of the damping properties of orthotropic composite materials using a mixed numerical experimental method[END_REF][START_REF] Talbot | The vibration damping of laminated plates[END_REF][START_REF] Qian | A vibration method for measuring mechanical properties of composite, theory and experiment[END_REF] has retained some attention, but in a thinplate context only. In a thick-plate context, only methods involving full-field measurements are currently available [START_REF] Pagnacco | Inverse strategies for the identification of elastic and viscoelastic material parameters using full-field measurements[END_REF][START_REF] Matter | Numerical-experimental identification of the elastic and damping properties in composite plates[END_REF][START_REF] Giraudeau | An alternative to modal analysis for material stiffness and damping identification from vibrating plates[END_REF]: they are very time-consuming or need sophisticated equipment.

In this paper, a previously presented method for estimating some complex moduli of elasticity of the constituents of sandwich structures [START_REF] Rébillat | Identification of elastic and damping properties of sandwich structures based on high resolution modal analysis of point measurements[END_REF][START_REF] Rébillat | Measurement of elastic and damping material properties in sandwich structures[END_REF] is numerically validated. The proposed procedure yielding in-plane and out-of-plane elasticity and dissipation parameters of the constituents of the sandwich is schematically presented in Fig. 1. The analytical model of the sandwich panel is presented in Sec. [START_REF] Rébillat | Identification of elastic and damping properties of sandwich structures based on high resolution modal analysis of point measurements[END_REF]. Based on this model, the numerical modal frequencies f Num n and dampings α Num n are derived by means of an extended Rayleigh-Ritz procedure (Sec. 3). In the validation procedure, the modal frequencies f FEM n and dampings α FEM n replace the experimental values; they are given by a finite-element model analysis of a known virtual plate (Sec. 4). Given the numerical and virtual experimental data, the optimisation procedure that estimates the elasticity and damping parameters of the constituents of the sandwich material is detailed in Sec. 5. The estimation results and a sensitivity analysis are shown in Sec. 6 and 7 respectively.

Mechanical model of sandwich panels

The sandwich panel consists in two identical skins and a core (Fig. 1). In the following, "panel" designates the physical structure whereas "plate" refers to the idealised structure made out of an equivalent homogeneous material. Parameters pertaining to the core, the skin, and the homogeneous material constitutive of the equivalent plate are respectively denoted with c, s, and H indexes. The thicknesses of the core, skins, and panel are h c , h s , and h = h c + 2h s respectively. The skin and core materials are considered as homogeneous, orthotropic in the x and y directions, and viscoelastic. The standard hysteretic model (which is frequency-independent) has been retained for describing the viscoelastic behaviour, with complex moduli E = E(1+ jη). The stress-strain relationship σ γ (ǫ γ ) in each γ-material (γ =s, c, or H) involves 7 complex numbers and can be written, to first order in η γ as:

σ γ =       E γ x (1 + jη γ x ) ν γ yx E γ x [1 + j(η γ ν yx + η γ x )] 0 0 0 ν γ xy E γ y [1 + j(η γ ν xy + η γ y )] E γ y (1 + jη γ y ) 0 0 0 0 0 G γ xz (1 + jη γ xz ) 0 0 0 0 0 G γ yz (1 + jη γ yz ) 0 0 0 0 0 G γ xy (1 + jη γ xy )       ǫ γ
(1) The symmetry of the strain/stress relation implies

ν γ xy E γ y = ν γ yx E γ x and η γ ν = η γ ν xy + η γ y = η γ ν yx + η γ
x , leaving 12 independent real parameters to be identified for each material (24 altogether).

The following hypotheses are made on the sandwich panel:

-The sandwich panel is symmetric with respect to its mid-plane.

-Compared to the core, skins are thin enough to ensure that shear stress in the skin can be ignored:

h s G s xz,yz ≪ h c G c xz,yz . -The core is much softer than the skins (E c x ≪ E s x , E c y ≪ E s y , G c xy ≪ G s xy ).
Given the generic expression of the moduli of the homogeneous equivalent material

E H = h c h 3 E c + 1 - h c h 3 E s ,
this ensures that all in-plane stress in the plate are entirely due to those in the skins. According to these hypotheses, there is no stress associated with E c,s,H

z , ν c,s,H xz , ν c,s,H yz , G s xz , G s yz , E c x , E c y , G c
xy , ν c xy , ν c yx which are ignored in what follows. Under the hypothesis given in section 2, the sandwich panel behaves in the low frequency range like a homogeneous thick-plate [START_REF] Nilsson | Prediction and measurement of some dynamic properties of sandwich structures with honeycomb and foam cores[END_REF]. The thickness of the plate is chosen to be h. Its mechanical properties are given in Eq. (2) as functions of the mechanical and geometrical properties of the skins and the core.

               E H x (1 + jη H x ) = E s x (1 + jη s x ) 1 - h c h 3 E H y (1 + jη H y ) = E s y (1 + jη s y ) 1 - h c h 3 ν H xy (1 + jη H ν xy ) = ν s xy (1 + jη s ν xy ) G H xy (1 + jη H xy ) = G s xy (1 + jη s xy ) 1 - h c h 3 G H xz (1 + jη H xz ) = G c xz (1 + jη c xz ) G H yz (1 + jη H yz ) = G c yz (1 + jη c yz ) (2)
The 12 independent real parameters

{E H x , η H x , E H y , η H y , G H xy , η H xy , G H xz , η H xz , G H yz , η H yz , ν H xy , η H ν }
are to be estimated. Their knowledge yields the elastic and dissipative properties of each layer of the sandwich panel provided that the system of 12 real equations formed by Eqs. ( 2) is invertible. Given the previous hypotheses, a sufficient condition is:

η c x E c x E s x ≪ η s x η c y E c y E s y ≪ η s y η c xy G c xy G s xy ≪ η s xy ( 3
)
This condition is not satisfied only if the η c -coefficients are several orders of magnitude larger than the η s -ones.

3 An extended Rayleigh-Ritz procedure for obtaining modal dampings

Modal representation

In the present study, it is assumed that structural damping is dominant over damping due to acoustical radiation and boundary conditions. The honeycomb sandwich panel is considered here as a system with N degrees of freedom q = {q m }, where q is any set of generalised displacements. The equation of its free motion is [START_REF] Geradin | Mechanical Vibrations: Theory and Applications to Structural Dynamics[END_REF]:

M q + C q + Kq = 0 (4)
where M , C, and K are the mass, damping, and stiffness matrices. The link between C and the dissipative model presented previously will be made further. The associated conservative system (corresponding to C = 0) has N normal modes with modal shapes ξ n and normal frequencies f n . If the complete, nonconservative, system is lightly damped, it can be shown [START_REF] Geradin | Mechanical Vibrations: Theory and Applications to Structural Dynamics[END_REF] that, to first order, its so-called natural modes have the same modal shapes ξ n and natural frequencies are f n + jα n .

Potential, kinetic and dissipated energies in the equivalent thick-plate

Let U n be the potential energy associated with the n th natural mode of the dissipative plate for a maximum vibrational amplitude of 1 on the plate. It varies in time as exp(-2α n t). Since the conservative and non-conservative systems have the same modal shapes, the same potential energy is associated with their n th mode, equal to the kinetic energy T n of the n th normal mode of the conservative system. The energy lost by this mode during one cycle ∆U n is therefore:

∆U n = -2 α n f n U n = -2 α n f n T n (5)
Within the frame of the first order Reissner-Mindlin theory [START_REF] Cremer | Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies[END_REF]Chap. 3], the displacements {u, v, w} in the {x, y, z}-directions respectively can be written within a good approximation (see below) as: u(x, y, z,t) = -zΦ x (x, y,t) v(x, y, z,t) = -zΦ y (x, y,t) w(x, y, z,t) = w 0 (x, y,t) (

The potential energy of the plate is:

U = 1 2 S D x ∂Φ x ∂x 2 + D ν
D x = E H x h 3 12(1 -ν xy ν yx ) D ν xy = ν xy E H y h 3 6(1 -ν xy ν yx ) D y = E H y h 3 12(1 -ν xy ν yx ) D xz = 2κ 2 xz hG H xz D yz = 2κ 2 yz hG H yz D xy = G H xy h 3 6 (8)
The shear correction factors κ 2 yz and κ 2 xz account for the fact that Eq. ( 6) is an approximation: the (functional) angles Φ x and Φ y depend lightly on z and sections of the plate do not remain plane in the flexural deformation. The values κ yz = κ xz = 1 have been chosen according to the recommendations of [START_REF] Birman | On the choice of shear correction factor in sandwich structures[END_REF] for sandwich panels.

By definition and based on the material model (Sec 2), the fraction of energy ∆U lost during one cycle T is: The kinetic energy T of the system is:

∆U = -T V (σ H ) T ∂ǫ H ∂t dτ dt = -π S η H x D x ∂Φ x ∂x 2 + η H ν D ν
T = ρ H ω 2 2 (V ) u 2 + v 2 + w 2 dτ = ρ H ω 2 2 (S ) h 3 12 (Φ 2 x + Φ 2 y ) + hw 2 0 dxdy ( 10
)
where ρ H is the density of the equivalent homogeneous thick plate: hρ H = h c ρ c + 2h s ρ s .

Derivation of the modal dampings α Num n

In order to derive the modal dampings α n from Eqs. 5, 9, and 10, analytical expressions of the Φ x -, Φ y -, and w 0 -modal fields are needed. This is achieved by a Rayleigh-Ritz procedure for the normal modes of the associated conservative system since they are the same as those of the natural modes.

The generalised-displacement fields Φ x (x, y), Φ y (x, y), and w 0 (x, y) are projected on the elements of an orthonormal polynomial basis of order Q satisfying partially the free-free boundary conditions [START_REF] Bhat | Natural frequencies of rectangular-plates using characteristic orthogonal polynomials in Rayleigh-Ritz method[END_REF]:

Φ x (x, y) = ∑ i, j L i j p i (x)p j (y) Φ y (x, y) = ∑ i, j M i j p i (x)p j (y) w 0 (x, y) = ∑ i, j N i j p i (x)p j (y) (11)
This procedure generates a new set of N = 3 Q 2 generalised displacements L i j , M i j and N i j . The next step consists in writing the kinetic and potential energies T and U. The Hamilton principle reads as:

∀(i, j) ∈ [0, Q -1] 2 : ∂(T -U) ∂L i j = 0 ∂(T -U) ∂M i j = 0 ∂(T -U) ∂N i j = 0 ( 12
)
The above system of 3Q 2 linear equations can be re-written as [K -4π 2 f 2 M ]q = 0. The expressions of the partial derivatives of U with respect to L i j , M i j , N i j yield K while the partial derivatives of T with respect to L i j , M i j , N i j yield M . The resolution of this eigenvalue problem gives a straightforward access to the modal frequencies f n and modal shapes ξ n .

Introducing the modal coefficients ξ n (expressed in the {L i j , M i j , N i j } system of coordinates) into Eqs. 11 yields the desired analytical expressions for the Φ x , Φ y , and w 0 modal fields and also for their x-and y-derivatives. For each of the N modes, the potential, lost, and kinetic energies can be written by introducing these expressions into Eqs. 7, 9, and 10:

∀n ∈ [1, N] : T n = 4π 2 f 2 n t n U n = 6 ∑ k=1 D k u n k ∆U n = -π 6 ∑ k=1 η k D k u n k ( 13
)
where the subscripts {1, 2, 3, 4, 5, 6} stand for {x, ν, y, xz, yz, xy} respectively. The coefficients t n and u n k depend on the geometry and mass parameters of the plate, and are quadratic in modal shapes ξ Num n . The expression [START_REF] Rébillat | Measurement of elastic and damping material properties in sandwich structures[END_REF] of the modal dampings α Num n can be deduced from the Eq. ( 5) and the last two expressions of (13):

α Num n = - f n ∆U NC n 2T C n = 1 8π f Num n t n 6 ∑ k=1 η k D k u n k (14) l x (m) l y (m) h (m) ρ (

Modal frequencies and dampings of a virtual plate

Finite-element modelling and the associated computations have been performed using Cast3M [START_REF]Cast3M[END_REF], a free software developed by the French Centre for Atomic Energy (CEA). The model consists of 8node quadratic 2D-thick-shell elements and assumes the Reissner-Mindlin hypothesis. Each node has 6 degrees of freedom. Elements are placed on a regular mesh of N = 60 elements per side of the plates. The chosen sandwich plate is made of 3 homogeneous layers and is symmetrical with respect to its mid-plane. Geometrical, mechanical and mass parameters of the plate are given in Tab. 1 and in Fig. 2 (numbers).

Modal frequencies and dampings of the virtual plates are computed in two steps. At first, the modal frequencies of the conservative system f C n are computed by solving the eigenvalue problem M q + Kq = 0. In the second step, the non-conservative system is described according to the constitutive model of the material (section 2) which implies that K is a complex matrix K * = K + jK ′ . The dynamic equation of the dissipative system becomes M q + C q + Kq = 0 with C = K ′ /ω. The "light damping hypothesis" has been retained. The damping α n is obtained as the real part of the n-th eigenvalue of this new problem where C is taken as K ′ /(2 π f n ). Unlike in the usual modal analysis, the eigenvalue problem must be re-written (and solved) with a new parameter (C) as many times as the number of complex eigenvalues to be found.

For the 3-layer virtual sandwich plate virtual plate, increasing the number of elements above 60 elements per side results in less than a 1 % relative variation of the 35 first modal frequencies (conservative and non conservative cases) and in less than 0.4 % of the 35 first modal dampings. Thus, 60 elements per side are enough to ensure the desired precision on the analysis of the first 35 modes of the two plates.

Optimisation procedure

This section describes how to derive, in two steps, the complex moduli of elasticity of the homogenised equivalent material of the sandwich plate

{E H x , η H x , E H y , η H y , G H xy , η H xy , G H xz , η H xz , G H yz , η H yz ,
C f = N ∑ n=1 f FEM n -f Num n f FEM n 2 (15)
A simplex search method [START_REF] Lagarias | Convergence properties of the nelder-mead simplex method in low dimensions[END_REF] (function "fminsearch" in Matlab TM ) based on the rigidities {D x , D ν xy , D y , D xz , D yz , D xy } has been chosen. Estimation results obtained by this methods are known to be dependent on the initial values of the parameters. To minimise the influence of the starting point, the following initialisation strategy for the rigidities has been chosen:

1. The initial values of in-plane rigidities D x , D ν xy , D y and D xy are the most influential; they were derived from the three lowest modal frequencies of the panel, as proposed in [START_REF] Mcintyre | On measuring the elastic and damping constants of orthotropic sheet materials[END_REF]. As can be seen in Eq. ( 14), modal dampings depend linearly on the loss factors

{η H x , η H y , η H xy , η H xz , η H
yz , η H ν } once the rigidities have been found. The estimation of the loss factors is therefore much easier than that of the elasticity parameters. A simple least-square optimisation procedure is sufficient to estimate the loss factors from the modal dampings. A cost-function similar to (15) has been chosen. The optimisation procedure is not iterative and needs no particular initialisation.

Estimation results for the 3-layer virtual sandwich-plate

The 3-layer virtual sandwich-plate has been used to validate the estimation procedure described in Fig. 1. Based on the first 35 modal frequencies given by the FEM and using a Rayleigh-Ritz order q = 16, the estimated values of the elasticity parameters are compared to the original values given to the FEM. The loss factors have been estimated with 28 modes and a model order q = 18. The estimated mechanical parameters are presented in Fig. 2 for each layer of the sandwich.

The residual mismatch between the results of estimation and the original values is discussed here.

The mean absolute value ∆ f n f n of the relative difference between experimental and numerical modal frequencies is 2.6 %. For the dampings, the residual mismatch ∆α n α n is 21.6 % in average but widely different between coefficients. These orders of magnitude suggest that the assumption that a 3layer sandwich plate can be modelled as a simple homogeneous thick plate is correct in the frequency range under study.

It can be seen in Fig. 2 that the agreement between estimated and original parameters is globally very good. In-plane elasticity parameters of the skins and out-of-plane elasticity parameters of the core are estimated with a mean absolute relative error of 10.2 %. Principal in-plane loss-factors η x and η y are estimated with a comparable accuracy of 7.5 %. The imaginary part of ν H xy is largely overestimated while the imaginary part of G H xy is underestimated. However, the overestimation of one parameter may be the result of the underestimation of the other, by compensation. The imaginary parts of G H xz and G H yz are assigned zero values by the estimation process. This underestimation is due to the fact that only a marginal part of the total energy-loss per cycle is dissipated through the mechanical couplings described by G H xz and G H yz . As a consequence, modal dampings factors are not very sensitive to the loss factors described by their imaginary parts.

Sensitivity analysis

The sensitivities of the modal frequencies f n or dampingsα n to the coefficients {E H

x , ν : if the X parameter is increased by 1%, the n-th modal value β n is increased by S β n (X)%. The sensitivities reflect the information contained in a modal value relatively to a given material parameter.

Results on modal frequencies with regard to the rigidities D x , D ν xy , D y , D xy (which depend on the elastic parameters as given by Eqs. 8) are presented in the left frame of Fig. 3 whereas the sensitivities to the out-of-plane rigidities D xz and D yz are presented in the right frame. Since the modal frequencies are very little sensitive to the Poisson coefficients, the corresponding sensitivities have not been represented. As expected, it turns out that modes of the form (0, i) or ( j, 0) convey a lot of information relatively to E x and E y respectively (left part of Fig. 3). Since the thick-plate model differs from the thin-plate model in the high-frequencies only, it is normal that there is almost 10 times more information relative to G xz and to G yz in the higher modes than in the lower ones (right part of Fig. 3). The lower sensitivity of the modal frequencies to G xz than to G yz is simply due to the aspect ratio of the plate (l x < l y ).

Results on the sensitivities of modal dampings to the dissipative parameters of the fictious material of the equivalent homogenised plate (η H

x , η H ν xy , η H y , and η H xy ) are given in Fig. 4. Modal dampings are sensitive to all in-plane parameters whereas out-of-plane parameters η H xz and η H yz were found to be irrelevant.

Conclusion

An easy-to-implement method to measure all relevant elastic and dissipative properties of the constituents of a sandwich structure has been described. Its experimetnal implementation has been presented elsewhere [START_REF] Rébillat | Identification of elastic and damping properties of sandwich structures based on high resolution modal analysis of point measurements[END_REF]. For the sake of numerical validation, a finite-element model replaces the experimental modal analysis. The sensitivity analysis shows that, within the homogenisation hypotheses, only half of all elastic and dissipative parameters are relevant for the vibration of thick plates. The method appears as reliable for their determination. The method does not give access to the other parameters but on the other hand, they are of little practical importance as far as the vibration of sandwich plates is concerned. Compared to the method by De Visscher et al. [START_REF] Devisscher | Identification of the damping properties of orthotropic composite materials using a mixed numerical experimental method[END_REF], it gives access to much higher frequencies and thus, to out-of-plane parameters. It is much simpler to implement and faster to perform than the method by Pagnacco et al. [START_REF] Pagnacco | Inverse strategies for the identification of elastic and viscoelastic material parameters using full-field measurements[END_REF] and to that of Matter et al. [START_REF] Matter | Numerical-experimental identification of the elastic and damping properties in composite plates[END_REF], based on measurements of the vibration on the whole panel.

Figure 1 :

 1 Figure 1: Left: geometry of the sandwich plate. Right: exeprimental and FEM-validation procedures.

Figure 2 :

 2 Figure 2: Comparison between the values of the mechanical parameters used in the FEM analysis ("FEM input data"), and their estimated values ("Estimation results") for each constituent (Skins, Core) of the virtual sandwich plate.

Figure 3 :

 3 Figure 3: Sensitivities of the modal frequencies to the real parts of the rigidities given by Eqs. 8. Each column is the sum of the sensitivities to D x , D ν xy , D y , and D xy (left frame) and to D xz and D yz (right frame). The total may exceed 0.5 since some sensitivities are negative. the different involved parameters.

Figure 4 :

 4 Figure 4: Sensitivities of the modal dampings to the dissipative parameters of the fictious material of the equivalent homogenised plate η H x , η H ν xy , η H y , and η H xy .

Table 1 :

 1 Geometry and constituent densities of the virtual sandwich-plate.

	kg/m 3 )

  ν H xy , η H ν } from the virtual and numerical values of the modal frequencies and dampings f FEM H xy } is done by comparing the experimental and numerical modal frequencies. The estimation problem to solve is non-linear and several orders of magnitude are involved in the properties values. The following cost-function was used:

	α Num n The estimation of the elasticity parameters {E H . x , E H y , G H xy , G H xz , G H yz , ν	n	, f Num n	, α FEM n	, and

The initial values of out-of-plane rigidities D xz and D yz are less critical; homogenisation theory proposed by Gibson[START_REF] Gibson | Cellular Solids: Structure and Properties[END_REF] for honeycomb core sandwich panels is used. This theory requires a value for the elasticity moduli of the material composing the honeycomb core. The first estimation was based on static tests.


M David
email: david@lms.polytechnique.fr

J.-J Marigo
email: marigo@lms.polytechnique.fr

C Pideri
email: pideri@univ-tln.fr

9-13 Mai 2011, Presqu'île de Giens (Var) Homogenenized interface model describing defects periodically distributed on a surface

Keywords: Homogenization, Separation of scales, Matched asymptotic developments

We undertake the homogenization of a three-dimensional elastic medium, which contains heterogeneities localized on a surface. These can be either reinforcements, like steel reinforcements in concrete, or defects, like microcracks periodically distributed. We propose a simple model describing the effective behaviour of these heterogeneities. This model, formulated in an energetic framework, combines an elastic interface behaviour with that of a membrane. Its interest is discussed through several applications.

Introduction

Let us consider a three-dimensional elastic medium, which contains heterogeneities periodically distributed on a surface. The size of these heterogeneities is much smaller than the size of the overall structure, at least by a factor 5 or 10. These heterogeneities can be for example steel reinforcements in concrete, or microcracks or cavities localized on a surface. In the context of finite element simulations, this kind of problem is very difficult to treat. Indeed, modeling each one of the small heterogeneities requires a very fine mesh, which gives a huge numerical cost. On the other hand, these heterogeneities have an influence on the behaviour of the overall structure. It is therefore interesting to construct a simplified model, which reproduces the effective behaviour of the heterogeneities, while being much easier to compute.

For this purpose, we will combine the methods of separation of scales and matched asymptotic expansions. The former was used by Sanchez-Palencia, Léné and Suquet in the eighties to homogenize the behaviour of three-dimensional periodic materials [START_REF] Léné | Contribution à l'étude des matériaux composites et de leur endommagement[END_REF][START_REF] Michel | Effective properties of composite materials with periodic microstructure : a computational approach[END_REF]. By separating the so-called micro-and macro-scales, this method enables to identify an effective behaviour. The latter was introduced in this context by Nguetseng and Sanchez-Palencia [START_REF] Nguetseng | Stress concentration for defects distributed near a surface, Local Effects in the Analysis of Structures[END_REF]. Other simplified approaches have been made on this problem, by studying a thin homogeneous layer, whose behaviour is different from the surrounding volume. This problem has been first analyzed by Huy and Sanchez-Palencia [START_REF] Huy | Phénomène de transmission à travers des couches minces de conductivité élevée[END_REF], followed by Caillerie, Licht and Michaille, Geymonat, Benveniste and Bessoud [START_REF] Benveniste | A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media[END_REF][START_REF] Benveniste | Imperfect soft and stiff interfaces in two-dimensional elasticity[END_REF][START_REF] Bessoud | Multi-materials with strong interface : variational modelings[END_REF][START_REF] Bessoud | Plate-like and shell-like inclusions with high rigidity[END_REF][START_REF] Geymonat | Mathematical analysis of a bonded joint with a soft thin adhesive[END_REF][START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF]. These authors made the assumption that the layer is very soft (i.e. scales with the thickness of the layer), or very stiff (i.e. scales with its inverse). These two hypothesis lead respectively to an elastic interface and a membrane model.

Here we propose a more general approach, where the layer is heterogeneous, while its stiffness does not depend a priori on the thinness of the layer. This work is the sequel of previous papers made on this subject [START_REF] Abdelmoula | Comportement asymptotique d'une interphase élastique mince[END_REF][START_REF] Abdelmoula | The effective behavior of a fiber bridged crack[END_REF][START_REF] Marigo | The effective behavior of elastic bodies containing microcracks or microholes localized on a surface[END_REF]. The peculiarity of this work is that we propose an energetic formulation for the behaviour of the homogenized interface. This leads to an interesting interpretation of the behaviour of the interface, and will help us analyze the stability of our model. The paper is organized in the following way. In the first section of the paper, we detail the setting of the problem, and apply the method of separation of scales to our problem. The second one is devoted to constructing an energetic formulation describing the homogenized behaviour of the interface. Finally, the third part illustrates the application of our method to steel reinforcements in concrete.

Separation of scales 2.1 Setting of the problem

The problem we want to study is the following. Consider a deformable three-dimensional body, with natural reference configuration Ω, a regular connected open subset of R 3 . The behaviour of the body is linear elastic, characterized by the stiffness tensor A. We also assume that the body is homogeneous, except near a plane Γ. In the vicinity of this surface, the elastic tensor takes the form of a two-dimensional periodic network, with periodic vectors denoted by t 2 and t 3 . These vectors are small compared to the size of the overall structure. For the sake of simplicity, the array is assumed orthogonal, i.e. t 2 .t 3 = 0. The projected surface of a periodic pattern t 2 ⊗ t 3 on Γ is denoted by Y .

The boundary ∂Ω of the solid is separated into two parts ∂ F Ω and ∂ u Ω. Forces are imposed on ∂ F Ω, while the displacement of the body is prescribed on ∂ u Ω. The body is also subjected to volume forces. In this context, the behaviour of the body is governed by the following equations.

σ = A : ε(u) div(σ) + f = 0 σ.n = F d on ∂ F Ω u = u d on ∂ u Ω
which are respectively the behaviour equation, the equation of equilibrium and the boundary conditions.

To simplify the following developments, we set the characteristic size of the structure to 1, which corresponds to a simple change of scale. We also introduce a small parameter η, which is defined by

η = ||t 1 || ||t 2 ||
and which characterizes the size of a periodic pattern.

Coordinates systems and variables

We now need to define the coordinate systems for our problem. At the global scale, we build an cartesian coordinate system (x 1 , x 2 , x 3 ). x 1 gives the (signed) distance to the plane Γ, while x 2 and x 3 parameterize the tangent plane, and are parallel to the periodic vectors t 2 and t 3 . We also introduce microscopic coordinates (y 1 , y 2 , y 3 ) to describe the heterogeneities. They are simply derived from the previous ones near the interface by using the relation y = x/η.

The complete material can be decomposed into several parts. The heterogeneous part is restricted to the vicinity of the surface Γ, while the homogeneous region is the volume Ω\Γ. In the external region, the displacement and stress fields are denoted by u(x) and σ(x). They do not depend on y since the volume is homogeneous. Similarly, we denote the displacement and stress fields near the interface by v(y, x 2 , x 3 ) and τ(y, x 2 , x 3 ).

Equations

Following the method of separation of scales, the fields are decomposed in powers of η

u = u 0 + ηu 1 + η 2 u 2 + ... (1)
σ = σ 0 + ησ 1 + η 2 σ 2 + ... (2)
v = v 0 + ηv 1 + η 2 v 2 + ... (3)
τ = τ 0 + ητ 1 + η 2 τ 2 + ... (4)
The equilibrium and constitutive equations near the interface are also modified in the following way

div x (τ) + 1 η div y (τ) + f = 0 τ = A : ε x (v) + 1 η ε y (v)
By introducing the expansions (1-4) in the previous equations, we obtain a serie of equations which will be solved in the following section. We finally need to take into account matching conditions between the homogeneous volume and the heterogeneous interface. These matching conditions for the displacement read

u 0 | Γ ± = lim y 1 →±∞ v 0 u 1 | Γ ± = lim y 1 →±∞ v 1 -y 1 ∂u 0 ∂x 1 Γ ±
with u| Γ ± respectively the limit of u on the positive and negative side of the interface. We also obtain similar matching conditions for the stresses.

Calculations

Micro problem of order 0

Using the previous equations, we can solve the whole problem in an iterative way, starting by the order 0, and studying alternatively the local problem near the interface, and the global problem in the external volume. Near the interface, the equation of behaviour of order -1 can be written

A : ε y (v 0 ) = 0 v 0 is
therefore a rigid body motion. Since v 0 is also periodic in y 2 and y 3 , v 0 can only be a translation, and does not depend upon y.

Macro problem of order 0

Inside the homogeneous volume, we have the constitutive and equilibrium relations of order 0

σ 0 = A : ε x (u 0 ) div x (σ 0 ) + f = 0 with the usual limit conditions σ 0 .n = F d on ∂ F Ω u 0 = u d on ∂ u Ω
The matching conditions on each side of the interface show that the jumps of displacement and stress are zero

[[u 0 ]] = 0 on Γ [[σ 0 .e 1 ]] = 0 on Γ
In conclusion, the defects do not have any influence on the global structure at order 0. This result is well known from previous studies. It may be understood with simple arguments : when the thickness of the heterogeneous layer goes to zero, its own rigidity becomes negligible. We therefore have to push our analysis until order 1 to identify the effective behaviour of the interface.

Micro problem of order 1

The constitutive and equilibrium equations read

τ 0 = A : (ε x (v 0 ) + ε y (v 1 )) div y (τ 0 ) = 0 v 1
and τ 0 are related to the macroscopic fields through the matching conditions

u 1 | Γ ± = lim y 1 →±∞ v 1 -y 1 ∂u 0 ∂x 1 σ 0 | Γ ± = lim y 1 →±∞ τ 0
These relations suggest to introduce the correctors v1 and τ0 , defined by

v 1 = y 1 ∂u 0 ∂x 1 + v1 τ 0 = σ 0 + τ0
The constitutive equation then reads

σ 0 + τ0 = A : ε y (v 1 ) + A : ε y y 1 ∂u 0 ∂x 1 + A : ε x (v 0 )
and since σ 0 = A 0 :

ε x (u 0 ) τ0 = A : ε y (v 1 ) + (A -A 0 ) : ε x (u 0 )
with A 0 the homogeneous behaviour of the volume. Since σ 0 is homogeneous at the scale of a pattern, the equation of equilibrium reads div(τ 0 ) = 0 while the matching conditions for the stress of order 0 become lim y 1 →±∞ τ0 = 0 From the previous results, one can write the complete set of equations at the scale of a pattern τ0 = A : ε y (v 1 ) + (A -A 0 ) : ε x (u 0 ) div(τ 0 ) = 0 lim y 1 →±∞ τ0 = 0 v1 and τ1 are periodic This problem is an elastic problem with prestress in infinite medium. It is well posed, even though rigid translations must be restricted. The loading parameter is ε x (u 0 ), which is obtained from the global problem of order 0. Since this problem is linear, the superposition principle may apply. There are six different modes of sollicitations corresponding to the six independant components of ε x (u 0 ). It is thus sufficient to solve six elementary problems at the scale of a pattern to characterize the behaviour of the interface.

Macro problem of order 1

The next step is to analyze the consequences of the response of the interface at the global scale. At order 1, the behaviour equation reads

σ 1 = A : ε x (u 1 )
and the equilibrium equation div x (σ 1 ) = 0 with the homogeneous boundary conditions

σ 1 .n = 0 on ∂ F Ω u 1 = 0 on ∂ u Ω
To complete the set of equations, we need to identify the jump conditions across the plane Γ. As before, we use the matching conditions between the volume and the interface

u 1 | Γ ± = lim y 1 →±∞ v1 σ 1 | Γ ± = lim y 1 →±∞ τ 1 -y 1 ∂σ 0 ∂x 1
The jump of displacement is therefore given by

[[u 1 ]] = lim y 1 →∞ ( v1 (y 1 , y 2 , y 3 ) -v1 (-y 1 , y 2 , y 3 )) = d(ε 0 )
where d(ε 0 ) is the jump of displacement extracted from the microscopic problem. The jump of stress is similarly given by

[[σ 1 ]] = lim y 1 →∞ τ1 (y 1 , y 2 , y 3 ) -τ1 (-y 1 , y 2 , y 3 ) -2y 1
∂σ 0 ∂x 1 which may be put in the following form

[[σ 1 ]] .e 1 + div x (σ I ) = 0 with σ I (ε 0 ) = Z Y ×R
τ0 dΩ σ I may be seen as the stress going through the interface. One can show that it only contains membrane components, that is σ I .e 1 = 0. We have shown that the main influence of the heterogeneities on the global structure is given by the jump of displacement and membrane stress d and σ I . Since these values are extracted from the local problem of order 1, they are linear functions of the deformation ε x (u 0 ). The complete set of equations of order 1 for the global problem is therefore

σ 1 = A : ε x (u 1 ) div x (σ 1 ) = 0 σ 1 .n = 0 on ∂ F Ω u 1 = 0 on ∂ u Ω [[u 1 ]] = d [[σ 1 ]] .e 1 + div x (σ I ) = 0 3 Energetic formulation 3.

Reformulation of the problem

Solving independantly the order 0 and 1 for the macroscopic problem is not very relevant in practice. It is much more efficient to solve the whole problem in one step. The purpose of this section is to construct an energetic formulation, whose solutions are accurate until order 1 in η. By adding order 0 and 1, we obtain the following problem

σ = A : ε x (u) div x (σ) + f = 0 σ.n = F d on ∂ F Ω u = u d on ∂ u Ω [[u]] = ηd(ε x (u)) [[σ]] .n = -ηdiv x (σ I (ε x (u)))
where we made approximations which only impact on order 2. Note also that ε is the average of the (possibly discontinuous) deformation on each side of the interface.

Formulation in an energetic framework

To recast the set of equations in an energetic framework, it is necessary to define an interface energy, describing the energy which is carried by the interface Γ. The behaviour of the interface is currently defined by the linear functions d(ε x (u)) and σ I (ε x (u)), which are related to the jump of displacement [[u]] and membrane stress σ m through

[[u]] σ m = η d σ I
The behaviour of the interface is obtained through the resolution of 6 elementary problems at the scale of a pattern. These elementary problems enable to identify the following linear functions, which completely define the behaviour of the inhomogeneities at first order.

[[u]] σ m = η A B C D . ε.e 1 ε m
where A, B,C and D are 3 × 3 matrices. This formulation is not satisfactory, because it mixes primal and dual components. We therefore construct a better formulation by performing several matrices operations and obtain the following relation

σ.e 1 σ m = η -1 K dd K dε K εd ηK εε . [[u]] ε m
This formulation of the behaviour of the interface is much more interesting, because it allows us to define the following mechanical energy

E I = 1 2 [[u]] ε m T . η -1 K dd K dε K εd ηK εε . [[u]] ε m
This energetic formulation has several advantages, which will be drawn in the following.

Analysis of the formulation

In this formulation, one can recognize very common behaviours for an interface. K dd corresponds to a jump of displacement [[u]] that is proportional to the stress applied on the interface σ.e 1 , which is typical of an elastic interface. This behaviour is coherent with the so-called soft model described in the literature. The prefactor η -1 indicates that the rigidity of the interface goes to infinity as the size of the heterogeneities goes to zero : in other words, when the heterogeneities become smaller, the jump of displacement vanishes. On the other hand, K εε corresponds to a membrane stress σ m that is proportional to the membrane deformation ε m , which is typical of a simple membrane. This behaviour is in turn coherent with the so-called stiff model. For this behaviour, the prefactor η shows that the rigidity of the membrane goes to zero when the size of the inhomogeneities goes to zero. In general, the behaviour of the interface combines membrane and elastic interface behaviour, with a coupling between the two given by the non-diagonal term K εd .

Notice however that the rigidity matrix is not positive definite in general. This will appear clearly in the illustrative example in the following. For now, let us just emphasize that the membrane behaviour K εε is generally negative when the inhomogeneities are softer than the surrounding volume, while the interface energy K dd is negative when the inhomogeneities are stiffer. The global problem is therefore not convex anymore, which can lead to spurious oscillations and instabilities, depending on the algorithm used to solve the problem. This problem is however not detrimental, because it can be solved by considering a finite thickness homogenized behaviour. Furthermore, as will be underlined in the following section, the complex general behaviour can be reduced to simpler models depending on the applications, which can help restore the convexity of the problem. limitation, the concrete is generally reinforced by steel bars, which have a strong resistance to traction. These bars are much smaller than the size of the overall structure, with a diameter of several centimeters. This large difference of scale results in a huge numerical cost if one tries to model each one of the reinforcement bars in three dimensions. It is therefore interesting to construct a simplified model, which reproduces the influence of these reinforcements on the whole structure. We consider a set of unidirectional steel bars, whose diameter is 2 cm, and separated by 20 cm. The steel has a Young modulus of 200 GPa, and a Poisson ratio of 0.3. The reinforcements are surrounded by concrete, whose Young modulus and Poisson ratio are 31 GPa and 0.22. The elementary calculations at the scale of a pattern give the following homogenenized behaviour This homogenized behaviour shows a membrane and an elastic interface behaviour. The latter is unstable, because the diagonal terms are negative. However, we see that the energy terms involved in the elastic interface behaviour are very large, resulting in a very small jump of displacement. This justifies that we restrict the behaviour of the reinforcements to the following membrane behaviour

K =         -1.
K m =  
26.6 1.9 1.9 6.1 5.6

  10 7 J/m
This membrane behaviour can be seen as the effective behaviour of the reinforcements at the global scale. It is very satisfactory, because it is easy to implement in a finite element code by using membrane elements. We can also analyze the behaviour of the reinforcements by comparing the different terms in the stiffness matrix. As expected, the rigidity of the reinforcements is especially large in the direction of the bars, but the rigidity in the transverse direction is not negligible. It is about a fifth of the main rigidity. Interestingly, since the membrane behaviour is positive definite, we get rid of the instability of the general model. This ensures that the global problem is convex, thus avoiding spurious oscillations of the stress field. This membrane behaviour can be compared with simple models used by engineers to take into account the reinforcements in large simulations. These simple models are generally made of bars and shell elements. Since our model was rigorously derived by the method of separation of scales, it can be used as a reference model for simpler approaches.

Conclusion and perspectives

We have presented a new approach to identify the effective behaviour of periodic heterogeneities localized on a surface. This approach is very general, and is not restricted to stiff or soft interfaces, nor to homogeneous layers. We described in particular an energetic formulation which enables to analyze the major influence of these defects on the whole structure. This extensive model couples elastic interface and membrane behaviour, and is able to represent all possible kinds of behaviours of the interface. This model can obviously be reduced to simpler models when it is relevant. We have also emphasized the fact that this model may be unstable, in particular for elastodynamics applications. This particular point will be the subject of future works.

This approach can be easily extended to all kinds of linear differential equations of second order. This includes thermal diffusion, electrostatic, magnetism and so on. Other subsequent works may include generalisations of this study. We have made two major hypothesis : the array of defects is orthogonal, and the materials on each side of the interface are the same. These hypothesis are not always relevant, and this study may be modified to take into account such cases.

Références

ηFIG. 1 -

 1 FIG.1-Unidirectional steel reinforcements.

Application of the model to reinforcements in concreteWe can finally apply our homogenenized model to steel reinforcements in concrete. The concrete used in civil engineering has a major drawback : it is not very resistant to traction. To overcome this


Muriel Ney

Clément Maisch

Patricia Marzin

LEARNING IN THE LABORATORY: AN INTERACTIONAL, FACTUAL AND CONCEPTUAL EXPERIENCE

When the teacher acts in a particular way, how does the student probably experience his act? What is the nature of these experiences and how teachers may influence them? Our starting point was a three-dimensional model (subject, object, project) defining three domains of experience: the interactional, perceptual domain, the conceptual, mental domain and the factual, concrete domain. We adapted this theoretical framework to the context of learning in the laboratory. It is assumed that all three domains are present in any learning experience but that some laboratory sessions may be dominated by one or the other domain. We present the consequences of this model for the design of laboratory sessions. Three types of laboratory situations are presented that lead to different natures of student experience and that are possibly connected to different learning outcomes. Using the framework, we derive the problems involved in using only one type of laboratory. Finally, we argue that if several types of laboratory calling on different domains of human experience are proposed, laboratory work becomes a privileged activity in which these domains can be used for learning purposes.

Background and questions

.

. For some researchers and teachers, the added value of labwork is that it enhances

motivation, and stimulates excitement by providing unusual objects and events -a contrast with the usual learning experience of sitting still and listening or doing exercises. Moreover, the laboratory is said [START_REF] Hofstein | The laboratory in science education: foundations for the twenty-first century[END_REF][START_REF] White | The link between the laboratory and learning[END_REF] to be a place where personal experience can be linked to a scientific way of reflecting on that experience (using scientific representations of the world such as models). The students' personal experiences are either their direct observations of a phenomenon through hands-on manipulation or their recollection of past experiences (e.g. in a Mechanics course a teacher could ask students: "What did you feel when the elevator started or stopped?"). Understanding student's personal experience more effectively is essential in order to create a learning situation that encourages and challenges students to develop experimentally-based inquiry skills. Laboratory work involves thinking, feeling and doing, and the lab is a hub of interactions between these three aspects. To elaborate on this, we will first review the literature on discussions and evidence as to how feeling and doing provoke learning in the laboratory. This paper addresses the following questions: What is the nature (the different domains) of students' experience during a laboratory session? What are the teachers' acts that lead to different natures of student experience? In the following parts of this paper, we will firstly propose a framework derived from a theory (with a brief introduction to that theory), which suggests an answer to our first question. We will then present empirical studies to answer to our second question. These studies have enabled us to build tools from the proposed framework and to illustrate the concepts introduced by means of concrete examples. Finally, the discussion raises the consequences for the design of labwork.

Literature review

A range of different tools and frameworks have been proposed to categorize and study different learning domains, and the goal here is merely to provide a rapid overview. One of the earliest and most widely used systems is the Bloom taxonomy, with its three learning domains: the cognitive, the affective and the psychomotor. Bloom worked on the cognitive domain (1956), while the affective domain was later developed by Krathworhl et al. (1964). Simpson and Harrow ultimately provided a classification of the third domain, the psychomotor domain: see e.g. [START_REF] Bergendahl | Development of competence in biochemical experimental work -Assessment of complex learning at university level[END_REF], or [START_REF] Forehand | Bloom's taxonomy: Original and revised[END_REF] for a historical review.

There are a number of definitions concerning the affective domain, which accordingly comprises several sets of variables, see e.g. [START_REF] Gungor | The relationship of freshmen's physics achievement and their related affective characteristics[END_REF]. A wide range of affective factors have been studied, including interest, motivations, feelings, emotions, values, or attitude toward science. In a review on learning and teaching in the affective domain, [START_REF] Miller | Teaching and Learning in Affective Domain[END_REF] adopts the view that this domain covers attitudes, motivations and values. She argues that there is a need for theory, and proposes several frameworks from psychology to educational sciences. [START_REF] Nieswandt | Student affect and conceptual understanding in learning chemistry[END_REF] considers interest, self-concept (perceptions and belief about oneself based on past experiences) and attitudes as the main affect components, in a study of the relationship between affective and cognitive variables in high school students taking chemistry for the first time. Several authors in the literature of science education acknowledge the fact that learning is influenced by affective factors (see e.g. the special issue of the Journal of introduced by Alsop and Watts 2003). However, there are few empirical studies on how it may do so in this field [START_REF] Hofstein | The laboratory in science education: foundations for the twenty-first century[END_REF], although many authors stress that the laboratory provides a complex perceptual experience (e.g. [START_REF] Gooding | The procedural Turn or, why do thought experiment work[END_REF]. [START_REF] Alsop | Facts and feelings: exploring the affective domain in the learning of physics[END_REF] explore the issue of students' feelings about a topic (in this case, radioactivity, which aroused mostly negative feelings) to determine whether or not these feelings influenced their approach to learning. In their view, the aim underlying the introduction of non-purely cognitive learning is to depart from impersonal and "antiseptic" science [START_REF] Alsop | Facts and feelings: exploring the affective domain in the learning of physics[END_REF]. In a study of relationships between achievement and affective characteristics in physics freshmen, [START_REF] Gungor | The relationship of freshmen's physics achievement and their related affective characteristics[END_REF] put forward two arguments for the importance of affective factors for learning. The first of these is the impact of affective factors on achievement, and the second is the fact that affective factors are more important for life in society than cognitive factors and should be given priority. The latter is related to the loss of interest in science: "students think that science is not related to them" [START_REF] Gungor | The relationship of freshmen's physics achievement and their related affective characteristics[END_REF]. [START_REF] Prain | Representation and learning in science from a second generation cognitive science perspective[END_REF] shows that when a perceptual approach is used when learning science, students make sense of phenomena by drawing on perceptual links. He considers that a perceptual approach increases the construction of the individual path to meaning. For [START_REF] Coquidé | Le rapport expérimental au vivant[END_REF] the connection with reality has at least two effects. Firstly, effective and authentic practices allow the discovery of techniques and of their constraints, and contribute to a choice of positive vocational guidance. Secondly, they lead the learners to be aware of the need to manage the materiality of sciences, an essential element of any scientific or technical culture. As stated by [START_REF] Gooding | The procedural Turn or, why do thought experiment work[END_REF], the dominance of head over hand leads to an intellectual view of science, and the way in which kinesthetic activities provoke learning in the laboratory is therefore still unclear, and challenged. Several authors discuss the relationship between doing and learning in the laboratory, for instance between practical actions and reflection on scientific theories behind these actions. This relationship can make things more concrete by illustrating abstract phenomena and concepts and by making the facts visible. [START_REF] Millar | The role of practical work in the teaching and learning of science, High School Science Laboratories: Role and Vision[END_REF] defines practical work as: any teaching and learning activity which involves at some point the students in observing or manipulating real objects and materials. [START_REF] Millar | The role of practical work in the teaching and learning of science, High School Science Laboratories: Role and Vision[END_REF] wants to promote "knowledge in action", and a more explicit, reflexive and declarative knowledge. He claims (Millar, 2004 p.2) that "as Piaget argued, it is by acting on the world that our ideas about it are formed and develop. Students need to have experiences of acting on the world, in the light of a theory or model, and seeing the outcomes".

It is not easy to separate thinking from doing or feeling and study their impact on learning. Because these three domains are always present in any learning experience and particularly important in those that take place in the laboratory, there is a need to define them in a single framework. However, few authors have worked towards establishing consistency between these three domains, particularly in research in science teaching.

Proposed theoretical framework

We would first like to state explicitly our position on both learning and knowledge. Our position on learning comes from [START_REF] Dewey | Expérience et éducation (translation of Experience and Education[END_REF][START_REF] Dewey | Expérience et éducation (translation of Experience and Education[END_REF], who considers that research on learning needs a theory of experience. Furthermore, our position on knowledge of the natural world is summarized by [START_REF] Bachelard | The new scientific spirit (translation of Le nouvel Esprit Scientifique[END_REF]Bachelard ( /1985, p. 15), p. 15), who expresses the notion of a three-dimensional reality as follows: the subject cannot reflect on the object without considering the project. Let us take an example to illustrate the notion of a project: I (subject) use a pen (object) to write here and now (project), but I can have many other projects for this pen, for instance using it later to dial a number on a phone. This experience of interaction with an object, the pen, is fully described only through the project. From Bachelard's point of view, reality is not reduced to an interaction between subject (a scientist or a community) and object (devices, materials and phenomena of science). There is a third dimension, which is the project that allows considering the spatial and temporal deployment of the reality under consideration.

The aim of our second research question was to understand the student's experiences as organized by a teacher in the laboratory, that is, to infer the students' learning experiences from what the teacher does and asks them to do. To analyze teachers' discourse and writings, we used a framework that is coherent with the two positions mentioned above. We found a way to elaborate on Bachelard's view of reality by using the Theory of Sense and Human Coherences [START_REF] Nifle | Au coeur du sujet : la théorie de l'instance et des cohérences[END_REF]. This theory is presented in a French philosophical essay, [START_REF] Nifle | Au coeur du sujet : la théorie de l'instance et des cohérences[END_REF], and used mostly in non academic situations (but see a recent study on teaching modeling, [START_REF] Ney | Biology and classifying formal models based upon the way they work[END_REF]. Sense is a specific view of the world, a specific logic, held by an individual, a community or an institution, according to this theory. This view reveals itself in attitudes, behaviors and representation, giving an orientation and a coherence between thought and action. The theory is based on three assumptions: ( 1) reality (what one can consider, apprehend) is always a life experience (2) reality is an instance, a product, of a shared Sense, and (3) reality is described by three variables or dimensions, subject-object-project (figure 1).

Figure 1. Coherentiel.

A paper presented at the European Science Education Research Association 2009 Conference

Istanbul, Turkey August 31 -September 1, 2009 2 From these assumptions, [START_REF] Nifle | Au coeur du sujet : la théorie de l'instance et des cohérences[END_REF], and see also [START_REF] Ney | Biology and classifying formal models based upon the way they work[END_REF], introduces a three-dimensional structure called a coherenciel and formed by -the subject dimension (pointing upwards in figure 1) is the expression of an intention, the desire to consider an object or a phenomenon. This intention can have many forms: desire, motivation, tendency, aspiration. Indeed, any reality is that of a particular person (or community or institution) -the object dimension (pointing downwards, left) is what can be distinguished from its context and also from ourselves by the attention that we pay to it. It can include things, models, people and other resources.

-the project dimension (pointing downwards, right) is the vector product of the other two dimensions and represents reality deployed and ordered in time and space. Reality continuously evolves through the interaction of subject and object.

Pairs of dimensions form planes (see figure 1) that will subsequently be referred to as domains of experience: interactional, factual and conceptual [START_REF] Nifle | Au coeur du sujet : la théorie de l'instance et des cohérences[END_REF]. The plane formed by the object and subject dimensions is the domain of affect and interactions: interactions between subject and objects. It is called the interactional and perceptual domain of experience. The plane formed by the project and subject dimensions is the domain of concepts and representations, and is called the conceptual and mental domain of experience. The plane formed by the project and object dimensions is the domain of actions and facts, called the factual and concrete domain of experience.

We will now answer our first question: What is the nature (the different domains) of students' experience during a laboratory session? A teacher who organizes a predominantly interactional experience may ask students to familiarize themselves with the apparatus, to contextualize the problems involved, to situate themselves (who does what, how do I feel about this), and to get a "feel" for a phenomenon. These are sources of perceptual and interactional experiences. This plane (the interactional domain) is orthogonal to the project dimension (figure 1), since it is not necessary for students to plan actions and organize tasks according to an objective. Next, with respect to the conceptual domain, students are asked to construct their own view and their own ideas, either in order to anticipate the analysis before carrying out an experiment, or to represent their results; they may spend most of the laboratory time debating on models and concepts. These are mostly mental or conceptual experiences. This plane (the conceptual domain) is orthogonal to the object dimension (figure 1), which reflects the fact that students do not need to interact directly with all the practical components of the experiment (roles of teacher and students, timing, apparatus, labsheet available, material constraints); they merely need to have an idea or image of these components. Finally, focusing on the factual domain, students are asked to observe phenomena, to perform the experiments and execute a protocol. They have to ascertain scientific facts, and experience that things happen. This is what we mean by factual or concrete experiences. This plane (the factual domain) is orthogonal to the subject dimension (figure 1), and indeed students do not need to appropriate the scientific issue under consideration in the laboratory session in order to execute a protocol or make observations. They can be guided in their observations and experiments. Although we have just considered them two by two (i.e. each two-dimensional plane), the three dimensions are to be found in every reality, or human experience. A human experience such as that of the student in a laboratory session can thus be deployed on this three-dimensional structure. Nevertheless, the teacher may choose not to consider all three and to emphasize one or two dimensions for learning, and thus focus on a single plane or domain of experience.

To get a feel for the three domains of learning experience as defined in this framework, let us take an example: imagine that you are a student in a laboratory session entitled "the intern anatomy of a frog". It can be a mostly interactional or perceptual experience for you if you are asked to touch and feel the frog, open it and freely draw what you see. You will probably remember your aversion to or curiosity for the experience. It does not necessitate any experimental planning: the interactional domain can be seen as the absence of the project dimension (figure 1). By contrast, it may be a factual, concrete experience if you are asked to follow dissection instructions. You will remember things such as the fact that you need gloves for this dissection, that forceps are useful, that frogs 2 have no necks, that males are smaller than females, etc. Moreover, simply following the "recipe" does not require that you understand why: the factual domain can be seen as the absence of the subject dimension (figure 1). Finally, it can be a mostly conceptual or mental experience for you, if you are provided with a diagram of an open frog, a typical biological representation. You are told that the aim of the dissection session is really to identify and label each organ in the diagram. It is not absolutely necessary to dissect the frog, and you memorize the image of a diagrammatic frog rather than the body of the frog: the conceptual domain can be seen as the absence of the object dimension. In fact, in any of these lab sessions, you have a three-dimensional learning experience combining (figure 1) the factual, interactional and conceptual domains, as it is not possible to separate one from the other. However, our point is that each of these imaginary lab sessions would be dominated by one domain or another, depending on the instructions given by the teacher.

Other frameworks emphasize similar domains of experience. As we saw in the previous section, a group of researchers led by B. Bloom identified three domains of educational activities: cognitive (mental skills), affective (feelings or emotional areas) and psychomotor (manual or physical skills). By elaborating on this framework, Hostorm and Ottander (2005) used these three domains to analyze teachers' discourses on the aims of their labwork. Using another framework, [START_REF] Gooding | The procedural Turn or, why do thought experiment work[END_REF], when studying the way in which scientists construct their production, referred to the interaction of concepts, percepts and objects (the latter can be understood as perceived events or objects, [START_REF] Tiberghien | An analysis of labwork tasks used in science teaching at upper secondary school and university levels in several European countries[END_REF], that is similar to the domains presented above. These works differ from ours in that here we present an integrated view of the three domains (figure 1) that reveals them in a new light. We define a domain as a plane determined by only two of three dimensions.

Empirical studies

We conducted two empirical studies. The aim was double: one the one hand, to illustrate how to use the proposed framework, and on the other to build tools to analyze labwork according to the three learning domains. In short, the data and the focus of their analysis are the following: (Study 1) 8 semi-structured interviews of teachers (practices in a lab session,) and (Study 2) content analysis of 15 labsheets (tasks given to the students). Both studies enabled us to answer our second question, since both provided us with teachers' acts in the form of discourse on practices (study 1) and written instructions (study 2). These teachers' acts were analyzed in terms of the domain of the students' experience. The question asked was: When the teacher acts in a particular way what does the student probably experience? Is this experience mostly conceptual, factual or interactional?

In study 1, after selection of the eight courses, we interviewed the teacher of each course and first agree on a particular laboratory session to be discussed. Such sessions usually lasted four hours. The interview, using the 'standin' technique [START_REF] Clot | Méthodologie en clinique de l'activité. L'exemple du sosie[END_REF], introduced a setting in which the interviewee (here a teacher) was told to imagine that the interviewer (here a researcher) was going to stand in for him or her the next day in the lab. The teacher was thus interviewed about the reality of the laboratory session for him/herself and the reality of the laboratory for the student as seen by the teacher. The interview was semi-structured, and included ad hoc questions that made it possible to determine what students had to do (according to the labsheet), what they did, the time allocated to the different parts of the lab, the difficulties encountered by the students and how the teacher might help them. In study 2, the content of 15 labsheets was analyzed. The labsheet was chosen as material for analysis because it is the students' first contact with the labwork and because it shows how they will engage in the lab activity (assuming that the teacher does not intervene excessively from the beginning). [START_REF] Tiberghien | An analysis of labwork tasks used in science teaching at upper secondary school and university levels in several European countries[END_REF] underline that written instruction materials are the ideal medium for studying regular practices as they embody the decisions and perspectives that shape that practice. There can, of course, be considerable variation in the way the labsheet is used by the teacher, but we did not have access to that information in this particular study. We used materials from another study in the same research project, namely the CoPEX project that explores learning by designing an experimental procedure, (Girault et al. in preparation, Marzin, d'Ham, & Sanchez, 2007 ;[START_REF] Wajeman | Analysing experimental design tasks in scientific labwork[END_REF]. We used their selection of labsheets from the two first years at university, i.e. five in physics, five in biology and five in chemistry.

In both studies, the question asked is whether the labwork engages the students in a learning experience that is predominantly interactional, factual or conceptual, in cases where predominance can be identified. We used the coherenciel (figure 1) to categorize excerpts of teacher discourse (study 1) and paragraphs in labsheet (study 2). We ended up with a grid of analysis dividing each labwork into three parts: (1) the introduction, where students are given questions or tasks to engage them in the lab;

(2) the central part, where they are given information on how to collect data and carry out experiments; and (3) the conclusion of the labwork, where they are given indications as to how to conclude (see table 1). One goal of our studies was to improve our tool to analyze current practices in labwork. Table 1 is the final version of this tool.

Table 1. Proposed grid of analysis.

The results of study 2 are presented in table 2. It shows that, from beginning to end (i.e. the three parts), 7 labsheets were predominantly factual, that only one was conceptual (in physics) and none interactional. Furthermore, our results (table 2) show that: the interactional domain is very seldom predominant in any of the three disciplines (4/45 parts in biology or physics labsheets) and the conceptual domain is rarely predominant (14/45 parts) elsewhere than in physics (10/14).

Table 2. Results from the 15 labsheet analysis.

The labsheet analysis and the teachers interviews lead to new questions and assumptions. First, some lab sessions seem to be organized in such a way that the students' experience is predominantly factual, interactional or conceptual. Our studies suggest two hypotheses: (1) the interactional domain is rarely predominant in any of the three disciplines and (2) the conceptual domain is rarely predominant elsewhere than in physics. It is to be noted that some lab sessions were not classified in the same domain from beginning to end, a session sometimes starting as factual and ending as conceptual, and another starting as interactional and ending as factual. At the level of a laboratory course, one may ask whether successive sessions should (depending on aims) go from a more interactional experience to a more factual one, to end with a conceptual part. We address this issue in the following section.

Discussion and Implications

The first question in this research concerned the nature of the learning experience that is organized by teachers in the laboratory. Our starting point was the three-dimensional structure of human experience based on the Theory of Sense and Human Coherences [START_REF] Nifle | Au coeur du sujet : la théorie de l'instance et des cohérences[END_REF] and our previous work [START_REF] Ney | Biology and classifying formal models based upon the way they work[END_REF]. We then adapted this framework to the context of learning in the laboratory. This led us to characterize three domains of experience: interactional, conceptual and factual. It is assumed that all three domains are present in any learning experience but that some laboratory sessions may be dominated by one or the other domain. The second research question focused on teachers' acts that lead to different types of experience in the laboratory. This question was first answered on the basis of eight interviews of teachers and 15 labsheet analysis. In summary, in these analyses we found typical features that could be attributed to each of the three domains of experience. These analysis allowed us to illustrate how to use the proposed framework, and to refine tools to analyze labwork according to the three learning domains.

Moreover, this work led to a proposition presented in this section. We define three types of laboratory that have different functions. One of these proposes interactional/perceptual experiences, another factual/concrete experiences and yet another conceptual/mental experiences. We indicate the features of each type of laboratory, and also the possible consequences of focusing only on a particular type of laboratory. This is to argue in favor of series of laboratory sessions that target different experiences and do not focus on one of them alone.

The "interactional laboratory". We call an interactional laboratory a first type of session where students are typically asked to make comparisons and to ask themselves questions. They do so by exploration, by familiarizing themselves with apparatus, living things (in biology) or phenomena. They can use their senses to ascertain differences between their beliefs from past experiences and what they observe. The method they work with are not formal, and their actions are based mainly on their feelings and intuitions about the phenomena. The teacher uses this type of laboratory to enable students build an initial representation of the phenomena, and ask questions, which will later be elaborated into scientific questions. This laboratory makes it possible to build a common experiential ground shared by all the students in a class. The role of the teacher is to encourage students to make comparisons, to ask them questions, to introduce doubts, and to help them to formulate their own questions. The problem when focusing exclusively on this type of laboratory is the absence of the project dimension (figure 1). Students could have an image of science that lacks organization into scientific methods and procedures and clear objectives.

The "factual laboratory". This second type of laboratory is defined as follows: students make measurements, ascertain scientific facts; they follow experimental procedures that appear as a series of operations. Their confrontation with phenomena is guided. They note that certain things happen under certain circumstances. The teacher guides them to make observations and perform manipulations. He/she helps them to relate facts, to identify sources of measurement error, and to see causalities (when I do that, I get this). Some attention is given to bibliography, since students should know other experimental results. They learn about known procedures and methods that work. The problem when focusing exclusively on this type of laboratory is the absence of the subject dimension (figure 1); that is, students might not appropriate the question and might not relate the scientific questions to the procedure itself (Why do I proceed this way? How should I answer this type of question?).

The "conceptual laboratory". In this laboratory, students elaborate a model, verify a law, test the usefulness of a law or a rule. They may elaborate a concept or a representation. The experimental procedure plays a minor role in the session. It is a tool that serves the conceptual work or the modeling. The experiment may even be simply alluded to, with students working directly on data. The data analysis plays a larger part in the session. The experimental procedure comprises both theoretical elements (equations, diagrams, etc) and operations, and students are asked to switch back and forth from theory to experimental facts. They are asked to build a theoretical or conceptual image. The problem when focusing exclusively on this type of laboratory is the absence of the object dimension (figure 1): students might not relate the resulting models to material conditions and constraints.

To conclude, we have emphasized the problems involved in using only one type of laboratory. Hence, we propose to use all three types and to integrate the three into a series of laboratory sessions. For instance, especially for students who are not yet expert experimenters, a series could be designed as follows:

-an interactional laboratory session to get a feel for phenomena, to discover, and to explore, -a factual laboratory session to make measurements, to ascertain facts, and to learn about known procedures, -a conceptual laboratory session to conceptualize, to model, and to construct representations. Interestingly, in a work based on several definitions of the term "experience", [START_REF] Coquidé | Face à l'expérimental scolaire[END_REF] proposed three similar modes of teaching in the laboratory. She defines an experience as either "experientiation", an "experiment" or a "validation-experience" leading to three corresponding modes of teaching. The difference here is that we focus on the domains of experience and characterize these domains in terms of a theoretical framework. By organizing several types of laboratory calling on different domains of experience, each with its own logic and its own demands, each requiring its specific form of teacher guidance, the teacher can tackle different learning issues. If contrasting situations are proposed, laboratory work becomes a privileged activity in which the different domains of human experience can be used for learning purposes. An open question remains for future work: what is the impact of the different types of laboratory on learning?

Table 2 . Results from the 15 labsheet analysis.

 2

	Scientific field	Introduction	Experiment	Conclusion
	Animal biology	F / P	F	F
	Genetics	F	F	F
	Physiology	C	F	F
	Bacteriology	F	F	F
	Plant biology	F	F	-
	Biochemistry	F	F	F / C
	Electrochemistry	F	F	F / C
	Inorganic chemistry	F	F	F
	Organic chemistry	F	F	F
	General chemistry	F	C	F
	Fluid Mechanics	C	F / C	C
	Electricity	F	F	C
	Optics	P	P	P / F
	Mechanics	C	C	C
	Magnetism	C	F / C	C

August 31 -September 1, 2009