Zakariya Boufaida

Julien Boisse
email: julien.boisse@univ-lorraine.fr

Stéphane André
email: stephane.andre@univ-lorraine.fr

Laurent Farge

Mesoscopic strain field analysis in a woven composite using a spectral solver and 3D-DIC measurements

Keywords: Woven composite materials, Spectral solver, Digital Image Correlation, Textured strain distributions, CraFT

Introduction

Industry's widespread use of composite materials remains of paramount importance especially in the field of transport for evident reasons linked to structure alleviation, fuel economy and environmental concerns. Nowadays composite materials are not just used for the fabrication of accessories (appearance parts) for vehicles but enter into the design and production of structural parts (car industry) and high performance parts (aeronautics). This shift brings additional constraints which requires continued research efforts in order to produce better predictive calculations. These efforts are also enforced by the multiple possibilities of selecting the basic materials for the building blocks of a composite material:

• Different reinforcement materials -either technical or bio-sourced,

• Different armor architectures (e.g. types of fabrics, laminated or 3D textile structural composites...),

• Wide range of possible materials for the matrix, especially in the present context of replacing thermoset materials by thermoplastics for recycling purposes,

• Different techniques to modify the reinforcement material-matrix adhesion which is a key point for the global mechanical performance.

To theoretically analyse the influence of the internal structure of the material on the overall mechanical properties, it is essential to take into consideration both averaged and full local fields. For instance, damage and fracturing are heavily dependent on the local details of stress or strain fields. For such analysis, it is essential to possess reliable and efficient tools which are also able to produce calculations in a systematic way, moving easily from one type of architecture to another while correctly taking the structure at different scales into account. Modelling issues in this field are therefore increasingly important. In this paper we used a spectral method to model the mechanical behaviour of a 2D textile composite at the mesoscopic scale. We show that this type of approach, which had never been used previously for such an application, allows the above mentioned requirements to be met.

Widely used 2D textile composites are periodically structured at the mesoscopic (fabric) scale. Accordingly, analytical homogenization approaches over the periodic fabric RVE (Representative Volume Element) were initially favoured and further developed to predict macroscopic equivalent material properties [START_REF] Ishikawa | Stiffness and strength of woven fabric composites[END_REF][START_REF] Naik | An analytical method for plain weave fabric composites[END_REF][START_REF] Hallal | Review and comparative study of analytical modeling for the elastic properties of textile composites[END_REF]. The joint development of powerful computers and softwares made it possible to develop numerical simulations allowing the computation of stress and strain distributions over the fabric RVE. The overwhelming majority of these studies were performed through discretization methods of the Finite Element (FE) type [START_REF] Searles | Micro-and mesomechanics of 8-harness satin woven fabric composites: I -evaluation of elastic behaviour[END_REF][START_REF] Yu | Non-orthogonal constitutive equation for woven fabric reinforced thermoplastic composites[END_REF][START_REF] Page | Finite element simulation of woven fabric composites[END_REF][START_REF] Barbero | Finite element modeling of plain weave fabrics from photomicrograph measurements[END_REF][START_REF] Lomov | Meso-FE modelling of textile composites: Road map, data flow and algorithms[END_REF][START_REF] Komeili | The effect of meso-level uncertainties on the mechanical response of woven fabric composites under axial loading[END_REF][START_REF] Dixit | Modelling techniques for predicting the mechanical properties of woven textile composites: a review[END_REF][START_REF] Liu | Lightweight design of carbon twill weave fabric composite body structure for electric vehicle[END_REF][START_REF] Jia | Nonlinear viscoelastic multi-scale repetitive unit cell model of 3D woven composites with damage evolution[END_REF]. Without doubt, FE modelling of the local stress fields in the periodic RVE of textile composites has allowed significant advances in our understanding of the mechanical behaviour of these materials at the mesoscopic scale: localization of damage [START_REF] Zako | Finite Element Analysis of damaged woven fabric composite materials[END_REF][START_REF] Daggumati | Local damage in a 5 -harness satin weave composite under static tension: part ii -meso -FE modelling[END_REF], influence of the nesting effect on the mechanical properties in the case of a multi-layered composite [START_REF] De Carvalho | Numerical modelling of woven composites: Biaxial loading[END_REF], etc. FE modelling remains however a challenging task given that a correct meshing for the textile RVE is difficult to achieve The problem is related to the contact surface regions between the yarns, where to avoid the generation of spurious voids and interpenetrations between the yarn surface meshes, the location of the nodes associated to the meshes have to coincide. For a particularly clear illustration, the reader is referred to figure 1 of the article written by Grail et al. [START_REF] Grail | Consistent Finite Element mesh generation for meso-scale modelling of textile composites with preformed and compacted reinforcements[END_REF]. A lot of strategies can be considered to circumvent properly this drawback of FE approaches [START_REF] Lomov | Meso-FE modelling of textile composites: Road map, data flow and algorithms[END_REF][START_REF] De Carvalho | Numerical modelling of woven composites: Biaxial loading[END_REF][START_REF] Kuhn | Modeling of plain weave fabric composite geometry[END_REF][START_REF] Woo | Effect of fibre tows misalignment on the engineering properties of plain weave composite textile[END_REF] which unfortunately bring additional troubles [START_REF] Grail | Consistent Finite Element mesh generation for meso-scale modelling of textile composites with preformed and compacted reinforcements[END_REF][START_REF] Doitrand | Comparison between voxel and consistent meso-scale models of woven composites[END_REF]. Related to this mesh step, obtaining highly resolved calculations through FE based techniques requires very thin meshing in multiple areas, which results in very high computational costs.

It seems then desirable to assess the potential of mesh-free methods for studying the mesoscopic-scale mechanical properties of textile composites. To our knowledge, such an analysis has only been carried out by the Wen and Aliabady group [START_REF] Wen | Mesh-free micromechanical model for woven fabric composite elastic moduli[END_REF][START_REF] Li | Meshfree modeling and homogenization of 3D orthogonal woven composites[END_REF]. In these studies, a radial basis function coupled with moving interpolations were used for the shape function constructions, and the formulation of the discretized system was obtained through the well-known Galerkin scheme. However, the interpolation was performed using a relatively small number of nodes which makes it impossible to obtain highly-resolved stress or strain fields.

The so called "spectral solvers" are alternative meshless methods that permit computation of the local mechanical response of heterogeneous materials. These codes rely on the use of 3D Fourier Transforms and are well-suited to the case of materials with periodic microstructure. They give results with very good spatial resolution, since the fields of interest (stress or strain) are calculated at every pixel of the microstructure image. The use of spectral methods is growing markedly and is gaining increasing interest especially with the recent application of phase field approaches to metallurgical transformations [START_REF] Chen | An integrated fast Fourier transform-based phase-field and crystal plasticity approach to model recrystallization of three dimensional polycrystals[END_REF].

In this study, we show for the first time that a spectral solver can successfully be used to model the mesoscopic scale mechanical behaviour of a textile composite. The outcomes of the simulation in terms of strain fields are validated at both the macroscopic and mesoscopic scales by measurements obtained with 3D Digital Image Correlation (DIC). The pixel size with which the RVE is digitalized for the simulation corresponds roughly to the fibre diameter size. This means that the stress and strain fields are provided with the best possible spatial resolution in the framework of a simulation performed at the mesoscopic scale.

The paper is structured as follows. In section 1, in relation with the considered application, the characteristics of the spectral solvers will be briefly summarized. In Section 2 we present the idealized representation of the plain-weave composite under study in terms of geometry and mechanical material properties. Section 3 is mainly devoted to the experimental validation of the results obtained by simulation: the composite effective mechanical properties and the strain fields obtained at the mesoscopic scale are compared to their experimental counterparts. We also show that the use of a spectral solver allows the details of the strain field in very specific regions of the composite microstructure to be obtained, in particular in the core of the material where it is difficult to envisage direct measurements to be performed. Finally, we summarize our findings in the conclusion section.

Main characteristics of spectral solvers

The details of the mathematical framework on which the spectral solvers are based is well-documented in the context of solid mechanics and can be found in one of our previous studies [START_REF] Boufaida | A mechanical study of a glass fabric-thermoplastic resin composite: 3D-DIC and X-Ray tomographic observations explained by numerical simulations based on a spectral solver[END_REF]. In the following description, we will merely recapitulate the main characteristics of these approaches, bearing in mind the specific objectives of the present study. Namely, we will describe modelling of the mechanical behaviour of a textile composite at the mesoscopic scale. The use of a spectral method for solving the partial differential equation system that results from the strain compatibility conditions and from the equilibrium equations was introduced by Moulinec and Suquet two decades ago [START_REF] Moulinec | A fast numerical method for computing the linear and nonlinear mechanical properties of composites[END_REF]. The efficiency of this approach is based on the Fourier series development of strain and stress fields when considering periodic RVE with strong heterogeneities in their material properties. Thanks to the repetitive use of Fast Fourier Transforms (FFT), the iterative solution algorithm operates alternatively in the Fourier and in the real spaces (see the algorithm in Moulinec and Suquet [START_REF] Moulinec | A fast numerical method for computing the linear and nonlinear mechanical properties of composites[END_REF]). In particular, the local equilibrium that involves partial derivatives in real space is expressed in a very efficient manner in terms of computational costs in the Fourier space using the Green operator. In this way, except for calculating the FFT, a given iteration of the algorithm only involves very fast elementary operations limited to each individual pixel of the real or Fourier space. For a given iteration, the limiting step is the FFT computation, which is nowadays obtained with highly optimized algorithms. The crucial point is that the spectral solvers directly use the microstructure numerical images. No meshing or coarser additional image discretization is needed. The stress and strain fields are obtained on a grid that simply corresponds to each pixel of the initial image, which corresponds to the highest possible resolution for a given digitalized image. In the case of comparisons performed with the same spatial resolution for the stress and strain fields, the spectral solvers are proved to be much faster than the FE codes in the linear elastic domain [START_REF] Michel | Effective properties of composite materials with periodic microstructure: a computational approach[END_REF], as well as in plastic or viscoplastic ranges [START_REF] Eisenlohr | A spectral method solution to crystal elasto-viscoplasticity at finite strains[END_REF]. The method was first applied to the framework of elasticity and later extended to the case of materials with arbitrary phase contrast, elasto-plastic natures, and linear hardening behaviour [START_REF] Michel | A computational scheme for linear and non-linear composites with arbitrary phase contrast[END_REF]. It remains nevertheless restricted to small transformations (kinematic framework of infinitesimal strains).

There are currently two spectral solvers in free access dedicated to the modelling of the mechanical behaviour of solid materials: DAMASK (Düsseldorf Advanced MAterial Simulation Kit) developed by Roters et al. [START_REF] Roters | DAMASK: the Düsseldorf Advanced Material Simulation Kit for studying crystal plasticity using an FE based or spectral numerical solver[END_REF] and CraFT (Composite response and Fourier Transforms) developed by Moulinec and Suquet [START_REF] Moulinec | A fast numerical method for computing the linear and nonlinear mechanical properties of composites[END_REF][START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF]. Both were used in a benchmark test case [START_REF] Boufaida | A mechanical study of a glass fabric-thermoplastic resin composite: 3D-DIC and X-Ray tomographic observations explained by numerical simulations based on a spectral solver[END_REF] to secure our results, those being presented in this paper having been obtained by the latter one.

Modelling information

The composite material under study is modelled at the mesoscopic scale, i.e. the scale corresponding to the structure of the fabric. In all following figures, the (XY) or [START_REF] Jia | Nonlinear viscoelastic multi-scale repetitive unit cell model of 3D woven composites with damage evolution[END_REF] planes correspond to the plane of the plain weave lamina (Fig. 1). Thus the through-thickness (or stacking) directions Z or 3 are the same in both frameworks. The [START_REF] Jia | Nonlinear viscoelastic multi-scale repetitive unit cell model of 3D woven composites with damage evolution[END_REF] directions refer to the Cartesian framework defined by the averaged warp (1) and weft (2) tow directions. The (XY) directions refer to the testing machines system of axis (X=tensile direction).

Microstructure generation

The plain weave microstructure was ideally reproduced using Solidworks® with the general characteristics of the real material (Fig. 1) as provided by the weaving company or as measured. The 2D plane view of the millimetric periodic RVE is a square of size

mm T 94 . 7 =
and consists of four intertwined identical tows surrounded by the matrix. Optical microscopy was used to characterize tows that have a cross-section with an elliptic shape of Two additional layers (one-pixel size) on the bottom and up surfaces were considered as having null mechanical properties. These "air" layers were added to impose the stress free boundary conditions on these two surfaces of the periodic RVE. Each ply was divided into 9 pixels along direction 3 which gives 83 pixels for the whole discretization along this direction. In the (12) plane, a RVE was made up of 443 443 × pixels which conserves the pattern's real aspect ratio and gives an in-plane resolution of about:

px µm / 18 443 / 94 . 7 =
. It can be noted that it roughly corresponds to the fibre diameter size, which means that the resolution is the best possible in the framework of a study carried out at the mesoscopic scale.

Once this geometrical description of the RVE has been made, the second step of spectral modelling just requires assigning mechanical properties to each "phase" of the material.

2.2

The mechanical properties at the meso-scale.

Most of the experimental features and results on this material (including fatigue analysis and mechanical spectroscopy) have been explained and reported elsewhere [START_REF] Boufaida | Influence of the fibre/matrix strength on the mechanical properties of a glass fibre/thermoplastic matrix plain weave fabric composite[END_REF] and are only used here as inputs or validation outputs for the numerical computation. Only the elastic behaviour is considered for each phase and Table 1 lists the values of all relevant moduli. The acrylic resin was a newly referenced thermoplastic (Elium® -Arkema) considered to be homogeneous and isotropic and whose properties were characterized experimentally (Dynamic Mechanical Analysis-DMA at 1 HZ was used in the three points bending mode for the elastic modulus determination) or from a well-established data base (Poisson coefficient for acrylic material). E-Glass fibres with well-known mechanical properties were used to perform the tows. Each tow was modelled as a unidirectional composite whose mechanical behaviour can therefore be described with 5 independent elastic constants: G . However, we checked that the main conclusions obtained during our study remain qualitatively and quantitatively true even if other micromechanical models were used to evaluate the elastic constants in the tows. The variations of elastic constants expressed in the (1, 2, 3) coordinate system due to the fibre undulation effect were not taken into account. To check the validity of this approximation, we modelled the tow undulation using a sinusoidal function and found the variations of the local value (along the tow) of 11 E to be less than 5%.

2.3

In-plane shear loading.

On woven composites showing weak undulation, it is well-known that tensile tests performed with either the warp or weft directions aligned along the tensile axis can be correctly modelled considering an equivalent cross-ply laminate of UniDirectional (UD) composites [START_REF] Berthelot | Composite Materials[END_REF]. This means that in that case, the mesoscopic scale organization has no strong incidence on the general behaviour. In order to obtain results at a macroscopic scale that depends strongly on the material organization at the mesoscopic scale, the tensile tests considered in this analysis were made for a force applied at 45 ± ° from the warp (1) and weft (2) fabric directions (in-plane shear loading, ASTM D3518 or NF ISO 14129).

In-plane shear loading conditions introduce very harsh conditions being applied to the fibre/matrix interface and one interesting perspective of this work rests upon the analysis of the mesoscopic stress fields calculated under these loading conditions. In the (XY) plane, a tensile load was applied in the X-direction, at 45 ± ° from the warp

(1) and weft (2) fabric directions. The following plane stress tensor was therefore applied:

( ) ( ) 2 , 1 , 2 2 2 2 0 0 0 - - = = L L L L Y X L σ σ σ σ σ σ (1)
where L σ was considered to be the average loading stress imposed on the whole RVE in CraFT software. The average stress applied over the whole thickness along the X-direction, is MPa 50

. Due to the presence of the two one pixel air layer, the average stress actually applied to the composite material is therefore:

MPa L 2 . 51 50 81 / 83 = × = σ .
We will prove in section 3.2.2 that the stress field computed at the mesoscale are perfectly in line with this uniform constant stress at the macroscale.

Typically, the loading stress is imposed with a ramp of 20 steps. Each step requires less than 300s to perform stress and strain computations over a 3D grid of 16.8

Millions of Voxels (standard PC, Intel® Xeon CPU W3520 @ 2.67GHz with 4 threads).

Results and discussion

In this section, all quantities with upperscripts M and m will denote values corresponding respectively to the Macroscale and the meso-scale. This section is organized into two parts. First we validate CraFT modelling, mainly by comparing the simulation results to surface measurements obtained by 3D DIC on the main and lateral specimen faces. In the second part, we used our CraFT simulations to analyse certain characteristics of the stress and strain fields in the core of the laminated composite. A modelling approach is inevitable in that case, because in the composite core region, far from the specimen-free surfaces, no simple way exists to perform strain field measurements. In particular, we studied the variations of the transverse normal strain in the resin region that extends throughout the thickness of the composite (region R in Fig. 1). Taking into account the small size of these resin pockets, such a study is only possible using highly resolved strain fields such as those obtained with a spectral solver like CraFT.

Experimental validation of the numerical model

Experimental data of whole-field strain measurements were obtained using the 3D DIC Aramis® device provided by the GOM-Instruments® company. These measurements were performed on the so-called main and lateral specimen faces. A random pattern was applied to the specimen with an air brush. We used 30x30 pixels facets (or subsets) and a 7 pixels step-size (distance between the centres of two adjacent facets). On the surface studied, one pixel represents 16 µm in size. The spatial measurement resolution is thus DIC_SR=44pixels 700 µm [START_REF] Boufaida | Influence of the fibre/matrix strength on the mechanical properties of a glass fibre/thermoplastic matrix plain weave fabric composite[END_REF][START_REF] Eriksen | Towards high velocity deformation characterization of metals and composites using digital image correlation[END_REF]. The two cameras are placed so that both the main and lateral specimen faces are visible in the recorded images. On the main face ( XY plane, Fig. 2b), the measurement of the mesoscopic strains m XX ε and m YY ε are obtained. Simultaneously, on the lateral face ( XZ plane, Fig. 3b), the mesoscopic strains m XX ε and m ZZ ε are measured.

From this mesoscopic scale experimental information, the validation of our spectral computations will be firstly assessed through recalculated properties at the macroscale, and secondly by a close comparison of local strain fields.

Macroscopic scale: Composite effective elastic moduli

The strain fields m XX ε at the mesoscopic scale obtained by CraFT simulation are shown in Fig. 2a:

( ) Y X m XX , ε
on the top of the RVE (specimen main face), and in Fig. 3a:

( ) Z X m XX , ε
along the cross-section indicated in Fig 2a . In Figures 2b and3b we show the strain fields ( )

Y X m XX , ε and
( ) Z X m XX , ε
measured by 3D DIC on the specimen main face and edge, respectively. The variation of the strain field over the RVE will be analysed in the following section and here we will only focus on the composite macroscopic elastic moduli that can be obtained from the average strain distributions over the RVE simulation in response to the macroscopic applied stress: Three macroscopic elastic constants of the composite can be evaluated:

• the longitudinal modulus: . The averaged values obtained in this strain range are indicated in table 2. For evaluating the longitudinal modulus M XX E , it was found that the stress-strain curve obtained experimentally during °± 45 tensile testing with strains measured using 3D-DIC was not linear [START_REF] Boufaida | Influence of the fibre/matrix strength on the mechanical properties of a glass fibre/thermoplastic matrix plain weave fabric composite[END_REF], even in a very small strain range as that in which the moduli needed to be evaluated according to the ASTM D3518 standard. This suggests that the resin has a noticeable visco-elastic effect in this kind of stress state.

As a result, the elastic behaviour needs to be looked for at very early times of the tensile test which is unfortunately when the machine response can be smoothed by the feedback loop resulting in a biased estimation of the longitudinal Young Modulus. This fact was identified when measuring the instantaneous modulus of semi-crystalline polymers -a visco-elastic physical model needs to be applied to the whole tensile curve to obtain a good estimation of the modulus [START_REF] Blaise | Advantages of a 3-parameter Reduced Constitutive Model for the Measurement of Polymers Elastic Modulus using Tensile Tests[END_REF]. Therefore, to avoid this drawback, the experimental measurement of X X E Μ provided in Table 2 was obtained through DMA performed on a Bose 3300N fatigue machine in the same way as the Young modulus of neat PMMA resin was evaluated. DMA enables the measurement of the "true" elastic modulus of a visco-elastic material (real part of the complex modulus). The experimental value for X X E Μ is shown in table 2.

In table 2, the agreement between the experimental values for M XX E , M XY ν and M XZ ν compared to those obtained through simulation for the moduli is fairly good. Many factors can explain the few % of discrepancies between the experimental results and those obtained through simulation (see table 2). The shape of tows may be more complex than that used for CraFT simulation. Measurement errors can be made when evaluating the elastic constants by 3D DIC and the fibre volume fractions tow f V and f V .

The volume fraction tow f V may also not be constant within a tow. The choice of the micro-mechanical model for 22 E and 12 G (see table 1) can also affect the results given by the CraFT simulation for the composite effective moduli. A parametric study has shown that the most influential tow elastic constant affecting the results for

M XX E is 12 G .
In table 1, it can be seen that the scattering of the 12 G values given by the different micro-mechanics models is of order of 25%. Increasing the micro-mechanical parameter 12 G by 10% results in a 6% increase of M XX E , which is larger than the discrepancy between the experimental value and that obtained through modelling (see Table 2). In what follows, we will not focus anymore on the RVE averaged strain, but we will compare local strain variations over the RVE as provided by CraFT simulation to those measured by 3D DIC.

Qualitative analysis

Qualitatively, on the main face, it is possible to compare the mesoscopic strain field m XX ε obtained by CraFT simulation (Fig. 2a) and by experimental DIC measurements (Fig. 2b), see also Fig. 4a. It is clear that the exact same pattern is obtained for strain distribution which displays a strong correlation with the fabric geometry. In short, the squared areas of high strains were found to be located in the small regions containing only resin throughout the thickness of a layer (region R in Fig. 1), those of small strains, in the warp and weft superimposed regions (regions F in Fig. 1). The areas with intermediate strains (bands surrounding the interlaced regions and forming a cross pattern) correspond to the undulation nodes of the tows where the resin forms strong junctions between the laminates. On this qualitative comparison, we can also highlight the significant difference in spatial strain resolution between the simulation (highly resolved spatial resolution set as 1/Np×RVE size=18 µm where Np=443 is the number of pixels chosen for the RVE size) and the DIC measurements. As previously mentioned, the best resolution achieved for the latter was about 700 µm or about one twelfth the RVE size. The ratio of simulation versus experimental resolution was found to be within a factor of 40 (see Fig. 4a for a clear illustration of the resolution difference). In brief, to compare experimental and simulated data, one should bear in mind that the experimental strain distribution is averaged by the measurement process appear very noisy but it is possible to check that the measurement is fairly reproducible: same curve shape with roughly identical minimum and maximum values at same locations. Because of the difference in resolution between the simulation and the measurement, it was impossible to capture experimentally the strong strain variations shown by the simulations in the frontier region between the rigid tows and the softer resin material (see the blue curve in Fig 4b). As a result, the highly deformed zones associated with the resin pockets appeared to be magnified to some extent. A strong quantitative validation of both results can be obtained nevertheless by looking at the profiles of Fig. 4b. The 8 experimental strain profiles were found to have similar features as the theoretical one. In particular, the ratio between the strains in the fibre rich and resin rich regions (respectively regions F and R in Fig. 1) was found to be approximately 3 both experimentally and by simulation (Fig. 4a). However, the theoretical profiles exhibit two secondary peaks at the interface between tow and resin (see Fig. 4b), which were not present on the experimental curve. Below, we will show that because of the measurement spatial filtering, these narrow secondary peaks were blurred as far as not appearing on experimental profiles. The size of the strain concentration regions associated with the secondary peaks is of the order of 100 or 200 µm. This justifies the need to obtain results with high spatial resolution ( 18 µm/pixel), to capture these details of the strain field. In an attempt to reproduce the experimental results more closely, we considered the highly resolved calculated strain patterns to be real objective measurements and tried to reproduce the spatial filtering operated when using DIC. In short, the highly resolved strain field was integrated to obtain the displacement field (in the small displacement-gradient theory framework). The detail of the procedure can be found in Boufaida et al. [START_REF] Boufaida | A mechanical study of a glass fabric-thermoplastic resin composite: 3D-DIC and X-Ray tomographic observations explained by numerical simulations based on a spectral solver[END_REF]. For this, the heterogeneous component of the strain (considered as periodic over the RVE) was transformed in the Fourier domain to produce the Fourier transform of the displacement which was backtransformed in the real domain. From this highly-resolved displacement field, we implemented an averaging procedure identical to that performed by the Aramis® software to obtain the transformation gradient tensor at a smaller resolution corresponding to that of the real measurement. Fig. 5 shows the new strain pattern obtained numerically after lowering the resolution and the corresponding numerical profile along direction X. On the same plot we also show one of the strain profiles measured by 3D-DIC (blue curve in Fig. 4b). The secondary peaks found at the resintow interfaces can be seen to have disappeared while the strain profiles now match the experimental ones much better (apparent enlargement of the peak bottom).

Analysis of strain and stress fields obtained using CraFT

As was previously mentioned, with a spectral solver the strain (or stress) fields are obtained on the same grid as the microstructure, which allows for a highly detailed analysis of the local stress or strain variations caused by material heterogeneity. To ε in the very small zones corresponding to the neat resin regions that cover the whole composite thickness (region R in Fig. 1). In particular, we highlight the complex influence of the surrounding tows on the strain state in these zones. In a second part, we show and briefly analyse the stress fields obtained through CraFT simulation in the specimen core region.

Variations of the transverse normal strain in the resin pockets

The transversal normal strain ( ) x -axis) taken at points A through D i.e. that are situated strictly in the resin region (Fig. 6a). In Fig. 6b, the corresponding profiles are denoted A,B,C, and D accordingly. Point A (x 1 =333 pixels, x 2 =109 pixels, x 3 =38 pixels) is taken at the centre of the resin pockets and, using the digitalized microstructure image, it was checked that the closest point belonging to a tow is situated at 18 pixels from Line A. Points B (326 pixels,116pixels, 38 pixels), C (320 pixels,122 pixels, 38 pixels), and D (316 pixels,126 pixels, 38 pixels) are taken along the Y-axis (bisector to the 1 x and 2 x axes, see Fig. 6a). The closest points belonging to a tow are respectively situated at 11 pixels, 5 pixels and 1 pixel from lines B, C and D. On the profiles of Fig. 6b, it can be seen that the dependence of m 33 ε on 3

pixels x x x m 38 , , 3 2 1 33
x shows a strong quasi-periodic modulation resulting from the ply alternation with maxima located in the frontier regions between the layers and minima in the middle of the plies. The quasi-periodic variation of the ( ) positive values in small zones situated at the ply frontiers. As will be seen in the following subsection, this is due to the complex tri-axial stress state to which the material is subjected. For the sake of clarity we have only shown four curves in Fig. 6b but, thanks to the resolution of the strain fields obtained through CraFT simulation, 17 curves could have been plotted to show the progressive evolution from profile A to profile D.

Stress fields in RVE

In Fig. 7, we show the maps associated with the 6 stress tensor components:

( ) 6b). On the other hand, in the tows (region F in Fig. 1), it can be checked that the material is under compression along the 3-axis.

2 1 11 , x x m σ , ( )
2 1 22 , x x m σ , ( )
2 1 33 , x x m σ , ( )
2 1 12 , x x m σ , ( ) 2

Conclusion

The mechanical behaviour at the mesoscopic scale of a woven composite was modelled for the first time with a spectral solver, a mesh-free method particularly adapted to materials with periodic microstructure. We show that the stress and strain details over RVE can be obtained in very short computation time, with excellent spatial resolution.

All the data necessary for simulations were provided by independent characterizations )

  resin burning-off method.It is then easy to check that the volume fraction of tows in the composite is 67 . 0 . It should be noted that the plain weave balanced fabric creates a periodic distribution of full resin pockets (denoted R in Fig.1) of approximately square shape that fill the whole ply thickness. The larger fibre-rich region, with also approximately square shape and where warp and weft tows are superimposed is denoted F in Fig1.The 3D modelling of the RVE in a ply based on Solidworks® is shown in Fig.1.It clearly shows the distribution of materials within sections at different depths of the fabric lamina following the tow undulations. The tows are shown here with the same 'color' but in the material description under CraFT, two sets of properties are given with same mechanical properties but in different orders, so as to take into account their different orientations. The real laminate composite is simulated as a lay-up of 9 plies of thickness mm 44 . 0 corresponding to a total composite thickness of mm 4 .

•

  the in-plane effective Poisson's ratio: through CraFT simulation are summarized in Table 2. The experimental values of M XY ν and M XZ ν are obtained in the same way but using averaged mesoscopic strain fields measured by 3D DIC on the specimen main face :

22

 22

3. 1 . 2

 12 Mesoscopic scale analysis of the strain fields

  the CraFT simulation on the top of the central ply (ply 5) of the woven laminate is shown in Fig.6a. To analyse the spatial variations of this strain component in the resin pockets (region R in Fig.1), we plotted ( ) -section profiles (parallel to the 3

  moving away from the resin pocket centre (Profile A) to the resin pocket extremity (Profile D). In profile D, it can be seen that in the composite material core region, the strain component m 33 ε , although transverse to the tensile axis, gets

33 σ

 33 . 7, taken in the specimen core, it is worth noting that locally m can reach significant values. It is positive in the neat resin pockets (region R in Fig.1): the material there is under traction along the 3the corner of these resin pockets, the normal transverse stress reaches its highest values ( be neglected compared to the in-plane stresses ( be taken into account to study damage initiation in textile composites. This explains the positive values found for m 33 ε in the composite core at the ply frontiers (see Profile D in Fig.

Figure 1 .

 1 Figure 1. 3D numerical RVE in a ply corresponding to the real woven and

Figure 4 . 34 Figure 5 .

 4345 Figure 4. a) Numerical and experimental strain patterns of m XX ε

Figure 6 Figure 7 .

 67 Figure 6. a)

table 1

 1

	11 E , 22 E	12 ν ,

mentioned micromechanical model. The RVE is taken within a tow cross section at the micrometric scale. An average stress state is applied to the RVE as an input for CraFT simulation. The averaging of the corresponding output full field strains allows calculating the five independent elastic constants of the tow. Moreover, the determined elastic constants are the mean value resulting from repeated simulations over 10 different random fibre configurations of the tow RVE. The final results are given in the last line of table 1 with the resulting standard deviations indicated in brackets. The standard deviations obtained are smaller than 5%, which shows that the RVE we considered with 64 fibres is large enough to describe the statistics of the fibre and matrix arrangement.

For all the micro-mechanic models and for CraFT simulation, 11 E -and to a lesser extent 12

ν -are nearly the same and are very close to the well-known mixture rule (see

Table 1 :

 1 Mechanical properties of the different phases of the studied composite (1 = average tow direction).

	E (GPa)	ν

Table 2

 2 Composite effective moduli obtained by experience and modelling

		X X E Μ (MPa): DMA	M XY ν : DIC (measurement on	M ν : DIC XZ (measurement on
			the main Face)	the specimen edge)
	Experimental	14.0	0.49	0.206
	CraFT Simulation	13.5	0.45	0.198
	Discrepancy	4 %	8.5%	4%

, xx

Acknowledgments:

The authors wish to thank the Région Lorraine for financial support including a PhD grant, Alain Gérard a technician at LEMTA who provided technical assistance during the experiments, Hervé Moulinec (LMA) for a fruitful introduction to CraFT and guidance in using it.

over zones of non-negligible size and that the strain measurements are affected by significant noise.

On the lateral face, for the numerical (Fig. 3a) and experimental (Fig. 3b) results, it can be seen that the high strain regions (R) form approximately a continuous band that is oriented nearly normally to the specimen main surface. For the simulated case with "in-phase" ply configuration, it can be checked that the distance between two consecutive bands is 2 / T . In the experimental case (Fig. 3b), the coincidence of regions R throughout consecutive layers can be visually appreciated as well as the estimated resulting distance between bands of strong strain values which is also

. It shows that the fabric of the different layers was placed approximately in an "in-phase configuration" and explains why this case was chosen for CraFT simulation (Fig. 3a).

Quantitative analysis

In Fig. 4b we compare on the same plot the m XX ε strain profile (solid line)

theoretically obtained along the X direction in the RVE to 8 experimental curves (dashed lines) corresponding to the same quantity measured by 3D-DIC in 8 different RVE taken on the specimen surface (see Fig. 2b). However, to allow the reader to visualize easily the aspect of a given individual experimental profile, one of these curves was drawn in blue solid line in Fig. 4b. As expected, the experimental curves performed by the authors and introduced, in some cases, into micromechanical models to determine properties at the mesoscopic scale. Despite the complex geometry and strong heterogeneous properties of RVE (acrylic resin, Glass fibres arranged in warp and weft tows at 90°), the results serve here as an explanation for in-depth experimental observation in terms of strains (direct comparison with full-field DIC measurements).

The high potential of this tool has also been illustrated in view of well-controlled industrial developments of new composites taking into consideration the complex stress distribution at the mesoscale.


Leila Amgoud

Florence Dupin De Saint-Cyr

Extracting the core of a persuasion dialog to evaluate its quality *

In persuasion dialogs, agents exchange arguments on a subject on which they disagree. Thus, each agent tries to persuade the others to change their minds. Several systems, grounded on argumentation theory, have been proposed in the literature for modeling persuasion dialogs. It is important to be able to analyze the quality of these dialogs. Hence, quality criteria have to be defined in order to perform this analysis.

This paper tackles this important problem and proposes one criterion that concerns the conciseness of a dialog. A dialog is concise if all its moves are relevant and useful in order to reach the same outcome as the original dialog. From a given persuasion dialog, in this paper we compute its corresponding "ideal" dialog. This ideal dialog is concise. A persuasion dialog is thus interesting if it is close to its ideal dialog.

Introduction

Persuasion is one of the main types of dialogs encountered in everyday life. A persuasion dialog concerns two (or more) agents who disagree on a state of affairs, and each of them tries to persuade the others to change their minds. For that purpose, agents exchange arguments of different strengths. Several systems have been proposed in the literature for allowing agents to engage in persuasion dialogs (e.g. [START_REF] Amgoud | Modelling dialogues using argumentation[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF][START_REF] Dunne | Two party immediate response disputes: Properties and efficiency[END_REF][START_REF] Gordon | The pleadings game[END_REF][START_REF] Parsons | Games that agents play: A formal framework for dialogues between autonomous agents[END_REF][START_REF] Prakken | Coherence and flexibility in dialogue games for argumentation[END_REF][START_REF] Zabala | Beliefs, reasons and moves in a model for argumentative dialogues[END_REF]). A dialog system is built around three main components: i) a communication language specifying the locutions that will be used by agents during a dialog for exchanging information, arguments, etc., ii) a protocol specifying the set of rules governing the welldefinition of dialogs such as who is allowed to say what and when? and iii) agents' strategies which are the different tactics used by agents for selecting their moves at each step in a dialog. All the existing systems allow agents to engage in dialogs that obey to the rules of the protocol. Thus, the only properties that are guaranteed for a generated dialog are those related to the protocol. For instance, one can show that a dialog terminates, the turn shifts equally between agents in that dialog (if such rule is 1 specified by the protocol), agents can refer only to the previous move or are allowed to answer to an early move in the dialog, etc. The properties inherited from a protocol are related to the way the dialog is generated. However, the protocol is not concerned by the quality of that dialog. Moreover, it is well-known that under the same protocol, different dialogs on the same subject may be generated. It is important to be able to compare them w.r.t. their quality. Such a comparison may help to refine the protocols and to have more efficient ones.

While there are a lot of works on dialog protocols (eg. [START_REF] Johnson | When are two protocols the same?[END_REF]), no work is done on defining criteria for evaluating the persuasion dialogs generated under those protocols, except a very preliminary proposal in [START_REF] Amgoud | Measures for persuasion dialogs: a preliminary investigation[END_REF]. The basic idea of that paper is, given a finite persuasion dialog, it can be analyzed w.r.t. three families of criteria. The first family concerns the quality of arguments exchanged in this dialog. The second family checks the behavior of the agents involved in this dialog. The third family concerns the dialog as a whole. In this paper, we are more interested by investigating this third family of quality criteria. We propose a criterion based on the conciseness of the generated dialog. A dialog is concise if all its moves (i.e. the exchanged arguments) are both relevant to the subject (i.e. they don't deviate from the subject of the dialog) and useful (i.e. they are important to determine the outcome of the dialog). Inspired from works on proof procedures that have been proposed in argumentation theory in order to check whether an argument is accepted or not [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF], we compute and characterize a sub-dialog of the original one that is concise. This sub-dialog is considered as ideal. The closer the original dialog to its ideal sub-dialog, the better is its quality. All the proofs are in [START_REF] Amgoud | Extracting the core of a persuasion dialog to evaluate its quality[END_REF].

The paper is organized as follows: Section 2 recalls the basics of argumentation theory. Section 3 presents the basic concepts of a persuasion dialog. Section 4 defines the notions of relevance and usefulness in a dialog. Section 5 presents the concept of ideal dialog founded on an ideal argumentation tree built from the initial dialog.

Basics of argumentation systems

Argumentation is a reasoning model based on the construction and the comparison of arguments. Arguments are reasons for believing in statements, or for performing actions. In this paper, the origins of arguments are supposed to be unknown. They are denoted by lowercase Greek letters. In [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], an argumentation system is defined by: Definition 1 (Argumentation system) An argumentation system is a pair AS = A, R , where A is a set of arguments and R ⊆ A × A is an attack relation. We say that an argument α attacks an argument β iff (α, β) ∈ R.

Note that to each argumentation system is associated a directed graph whose nodes are the different arguments, and the arcs represent the attack relation between them.

Since arguments are conflicting, it is important to know which arguments are acceptable. For that purpose, in [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], different acceptability semantics have been proposed. In this paper, we consider the case of grounded semantics. Remaining semantics are left for future research.

Definition 2 (Defense-Grounded extension) Let

AS = A, R and B ⊆ A. • B defends an argument α ∈ A iff ∀ β ∈ A, if (β, α) ∈ R, then ∃δ ∈ B s.t. (δ, β) ∈ R.
• The grounded extension of AS, denoted by E, is the least fixed point of a function F where

F (B) = {α ∈ A | B defends α}.
When the argumentation system is finite in the sense that each argument is attacked by a finite number of arguments, E = i>0 F i (∅). Now that the acceptability semantics is defined, we can define the status of any argument. As we will see, an argument may have two possible statuses: accepted or rejected.

Definition 3 (Argument status) Let AS = A, R be an argumentation system, and E its grounded extension. An argument α ∈ A is accepted iff α ∈ E, it is rejected otherwise. We denote by Status(α, AS) the status of α in AS.

Property 1 ([10]

) Let AS = A, R , E its grounded extension, and α ∈ A. If α ∈ E, then α is indirectly defended1 by non-attacked arguments against all its attackers.

Persuasion dialogs

This section defines persuasion dialogs in the same spirit as in [START_REF] Amgoud | Modelling dialogues using argumentation[END_REF]. A persuasion dialog consists mainly of an exchange of arguments between different agents of the set Ag = {a 1 , . . . , a m }. The subject of such a dialog is an argument, and its aim is to provide the status of that argument. At the end of the dialog, the argument may be either "accepted" or "rejected", this status is the output of the dialog. In what follows, we assume that agents are only allowed to exchange arguments.

Each participating agent is supposed to be able to recognize all elements of arg(L) and R L , where arg(L) is the set of all arguments that may be built from a logical language L and R L is a binary relation that captures all the conflicts that may exist among arguments of arg(L). Thus, R L ⊆ arg(L) × arg(L). For two arguments α, β ∈ arg(L), the pair (α, β) ∈ R L means that the argument α attacks the argument β. Note that this assumption does not mean at all that an agent is aware of all the arguments. But, it means that agents use the same logical language and the same definitions of arguments and conflict relation.

Definition 4 (Moves)

A move m is a triple S, H, α such that:

• S ∈ Ag is the agent that utters the move, Speaker(m) = S

• H ⊆ Ag is the set of agents to which the move is addressed,

Hearer(m) = H • α ∈ arg(L) is the content of the move, Content(m) = α.
During a dialog several moves may be uttered. Those moves constitute a sequence denoted by m 1 , . . . , m n , where m 1 is the initial move whereas m n is the final one. The empty sequence is denoted by . These sequences are built under a given protocol. A protocol amounts to define a function that associates to each sequence of moves, a set of valid moves. Several protocols have been proposed in the literature, like for instance [START_REF] Amgoud | Modelling dialogues using argumentation[END_REF][START_REF] Prakken | Coherence and flexibility in dialogue games for argumentation[END_REF]. In what follows, we don't focus on particular protocols.

Definition 5 (Persuasion dialog) A persuasion dialog D is a non-empty and finite sequence of moves

m 1 , . . ., m n s.t. the subject of D is Subject(D) = Content(m 1 )
, and the length of D, denoted |D|, is the number of moves: n. Each sub-sequence m 1 , . . ., m i is a sub-dialog D i of D. We will write also D i ⊏ D.

To each persuasion dialog, one may associate an argumentation system that will be used to evaluate the status of each argument uttered during it and to compute its output.

Definition 6 (AS of a pers. dialog)

Let D = m 1 , . . ., m n be a persuasion dialog. The argumentation system of D is the pair AS D = Args(D), Confs(D) such that: -Args(D) = {Content(m i ) | i ∈ 1, n } -Confs(D) = {(α, β) | α, β ∈ Args(D) and (α, β) ∈ R L }
In other words, Args(D) and Confs(D) return respectively, the set of arguments exchanged during the dialog and the different conflicts among those arguments.

Example 1 Let D 1 be the following persuasion dialog between two agents a 1 and a 2 .

D 1 = a 1 , {a 2 }, α 1 , a 2 , {a 1 }, α 2 , a 1 , {a 2 }, α 3 , a 1 , {a 2 }, α 4 , a 2 , {a 1 }, α 1 .
Let us assume that there exist conflicts in R L among some of these arguments. Those conflicts are summarized in the figure below.

α 3 α 2 α 1 α 4
Here, Args(D 1 ) = {α 1 , α 2 , α 3 , α 4 } and Confs(D 1 ) = {(α 2 , α 1 ), (α 3 , α 2 ), (α 4 , α 2 )}.

Property 2 Let D = m 1 , . . ., m n be a persuasion dialog. ∀D j ⊏ D, it holds that Args(D j ) ⊆ Args(D), and Confs(D j ) ⊆ Confs(D).

The output of a dialog is the status of the argument under discussion (i.e., the subject):

Criteria for Dialog quality

In this paper, we are interested in evaluating the conciseness of a dialog D which is already generated under a given protocol. This dialog is assumed to be finite. Note that this assumption is not too strong since a main property of any protocol is the termination of the dialogs it generates [START_REF] Torroni | A study on the termination of negotiation dialogues[END_REF]. A consequence of this assumption is that the argumentation system AS D associated to D is finite as well. In what follows, we propose two criteria that evaluate the importance of the moves that are exchanged in D, then we propose a way to compute the "ideal" dialog that reaches the same outcome as D.

In everyday life, it is very common that agents deviate from the subject of the dialog. The first criterion evaluates to what extent the moves uttered are in relation with the subject of the dialog. This amounts to check whether there exists a path from a move to the subject in the graph of the argumentation system associated to the dialog.

Definition 8 (Relevant and useful move)

Let D = m 1 , . . ., m n be a persuasion dialog. A move m i , i ∈ 1, n , is relevant to D iff
α 1 α 3 β 1 β 2 Suppose that Subject(D 2 ) = α 1 . It is clear that the arguments α 3 , β 1 are rele- vant, while β 2 is irrelevant. Here β 1 is useful, but α 3 is not. Property 3 If a move m is useful in a dialog D, then m is relevant to D.
On the basis of the notion of relevance, one can define a measure that computes the percentage of moves that are relevant in a dialog D. In Example 2, Relevance(D 2 ) = 3/4. It is clear that the greater this degree is, the better the dialog. When the relevance degree of a dialog is equal to 1, this means that agents did not deviate from the subject of the dialog. The useful moves are moves that have a more direct influence on the status of the subject. However, this does not mean that their presence has an impact on the result of the dialog, i.e., on the status of the subject. The moves that have a real impact on the status of the subject are said "decisive". Definition 9 (Decisive move) Let D = m 1 , . . ., m n be a persuasion dialog and AS D its argumentation system. A move

m i (i = 1, . . . , n) is decisive in D iff Status(Subject(D), AS D ) = Status(Subject(D), AS D ⊖ Content(m i ))
where

AS D ⊖ Content(m i ) = A ′ , R ′ s.t. A ′ = Args(D)\ {Content(m i )} and R ′ = Confs(D)\ {(x, Content(m i )), (Content(m i ), x) | x ∈ Args(D)}. Property 4 If a move m is decisive in a persuasion dialog D then m is useful in D.
From the above property, it follows that each decisive move is also relevant. Note that the converse is not true as shown in the following example.

Example 3 Let D 3 be a dialog whose subject is α 1 and whose graph is the following:

α 1 α 3 α 2 α 4 α 5
The set {α 1 , α 3 , α 5 } is the only grounded extension of AS D3 . It is clear that the argument α 4 is relevant to α 1 , but it is not decisive for D 3 . Indeed, the removal of α 4 will not change the status of α 1 which is accepted.

Example 4 Let D 4 be a dialog whose subject is α 1 , and whose graph is the following:

α 2 α 3 α 1
In this example, neither α 2 nor α 3 is decisive in D 4 . However, this does not mean that the two arguments should be removed since the status of α 1 depends on at least one of them (they are both useful).

On the basis of the above notion of decisiveness of moves, we can define the degree of decisiveness of the entire dialog as the percentage of moves that are decisive.

Computing the ideal dialog

As already said, it is very common that dialogs contain redundancies in the sense that some moves are uttered but these are useless for the subject, or have no impact on the output of the dialog. Only a subset of the arguments is necessary to determine the status of the subject. Our aim is to compute the subset that returns exactly the same status for the subject of the dialogue as the whole set of arguments, and that is sufficient to convince that this result holds against any attack available in the initial dialog. That subset will form the "ideal" dialog. In what follows, we will provide a procedure for finding this subset and thus the ideal dialog.

A subset of arguments that will be convenient for our purpose contains those arguments that belong to a proof tree leading to the status of the subject. This is due to the fact that a proof tree contains every necessary argument for obtaining the status of the subject. When the subject is accepted, the proof tree contains defenders of the subject against any attack. When the subject is rejected, the proof tree contains at least every non attacked attacker. Hence, proof trees seem adequate to summarize perfectly the dialog. However, it is important to say that not any proof theory that exists in the literature will lead to the ideal dialog. This is due to the fact that some of them are not concise. In [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF], a comparison of proof theories for grounded semantics shows that the one used here is the most concise.

Canonical dialogs

Let us define a sub-dialog of a given persuasion dialog D that reaches the same output as D. In [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF], a proof procedure that tests the membership of an argument to a grounded extension has been proposed. The basic notions of this procedure are revisited and adapted for the purpose of characterizing canonical dialogs.

Definition 10 (Dialog branch) Let D be a persuasion dialog and AS D = Args(D), Confs(D) its argumentation system. A dialog branch for D is a sequence α 0 , . . . , α p of arguments s. t. ∀i, j ∈ 0, p

1. α i ∈ Args(D) 2. α 0 = Subject(D) 3. if i = 0 then (α i , α i-1 ) ∈ Confs(D)
4. if i and j are even and i = j then α i = α j 5. if i is even and i = 0 then (α i-1 , α i ) ∈ Confs(D) 6. ∀β ∈ Args(D), α 0 , . . . , α p , β is not a dialog branch.

Intuitively, a dialog branch is a kind of partial sub-graph of AS D in which the nodes contains arguments and the arcs represents inverted conflicts. Note that arguments that appear at even levels are not allowed to be repeated. Moreover, these arguments should strictly attack2 the preceding argument. The last point requires that a branch is maximal. Let us illustrate this notion on examples.

Example 5

The only dialog branch that can be built from dialog D 2 is depicted below:

α 1 β 1
Example 6 Let D 5 be a persuasion dialog with subject α whose graph is the following:

α
The only possible dialog branch associated to this dialog is the following: α α

Property 5 A dialog branch is non-empty and finite.

This result comes from the definitions of a dialog branch and of a persuasion dialog. Moreover, it is easy to check the following result: Property 6 For each dialog branch α 0 , ..., α k of a persuasion dialog D there exists a unique directed path α k , α k-1 , ..., α 0 of same length 3

(k) in the directed graph associated to AS D .
In what follows, we will show that when a dialog branch is of even-length, then its leaf is not attacked in the original dialog.

Theorem 1 Let D be a persuasion dialog and α 0 , . . . α p be a given dialog branch of D. If p is even, then ∄β ∈ Args(D) such that (β, α p ) ∈ Confs(D).

Let us now introduce the notion of a dialog tree.

Definition 11 (Dialog tree) Let D be a persuasion dialog and AS D = Args(D), Confs(D) its argumentation system. A dialog tree of D, denoted by D t , is a finite tree whose branches are all the possible dialog branches that can be built from D.

We denote by AS D t the argumentation system associated to

D t , AS D t = A t , C t s.t. A t = {α ∈ Args(D) s.t. α appears in a node of D t } and C t = {(α, β) ∈ Confs(D) s.t. (β, α) is an arc of D t }.
Hence, a dialog tree is a tree whose root is the subject of the persuasion dialog.

Example 7

Let us consider D 6 whose subject is α 1 and whose graph is the following:

α 11 α 10 α 8 α 4 α 9 α 7 α 6 α 2 α 5 α 3 α 1 α 0
The dialog tree associated to this dialog is depicted below:

α 1 α 2 α 4 α 3 α 5 α 6 α 7 α 9 α 8 α 10 α 11
Note that the argument α 0 does not belong to the dialog tree.

Property 7 Each persuasion dialog has exactly one corresponding dialog tree.

An important result states that the status of the subject of the original persuasion dialog D is exactly the same in both argumentation systems AS D and AS D t (where AS D t is the argumentation system whose arguments are all the arguments that appear in the dialog tree D t and whose attacks are obtained by inverting the arcs between those arguments in D t ).

Theorem 2 Let D be a persuasion dialog and AS D its argumentation system. It holds that Status(Subject(D), AS D ) = Status(Subject(D), AS D t ).

In order to compute the status of the subject of a dialog, we can consider the dialog tree as an And/Or tree. A node of an even level is an And node, whereas a node of odd level is an Or one. This distinction between nodes is due to the fact that an argument is accepted if it can be defended against all its attackers. A dialog tree can be decomposed into one or several trees called canonical trees.

Definition 12 (Canonical tree)

Let D be a persuasion dialog, and let D t its dialog tree. A canonical tree is a subtree of D t whose root is Subject(D) and which contains all the arcs starting from an even node and exactly one arc starting from an odd node.

It is worth noticing that from a dialog tree one may extract at least one canonical tree. Let D c 1 , . . . , D c m denote those canonical trees. We will denote by AS c 1 , . . . , AS c m their corresponding argumentation systems. It can be checked that the status of Subject(D) is not necessarily the same in these different systems.

Example 8

From the dialog tree of D 6 , two canonical trees can be extracted:

α 1 α 2 α 4 α 1 α 2 α 4 α 3 α 5 α 3 α 6 α 7 α 9 α 8 α 10 α 11
It can be checked that the argument α 1 is accepted in the argumentation system of the canonical tree on the left while it is rejected in the one of the right.

The following result characterizes the status of Subject(D) in the argumentation system AS c i associated to a canonical tree D c i .

Theorem 3 Let D be a persuasion dialog, D c i a canonical tree and AS c i its corresponding argumentation system.

• Subject(D) is accepted in AS c

i iff all the branches of D c i are of even-length. • Subject(D) is rejected in AS c i iff there exists a branch of D c i of odd-length.

The following result follows immediately from this Theorem and Theorem 1.

Corollary 1 Let D be a persuasion dialog, D c i a canonical tree and AS c i its corresponding argumentation system. If Subject(D) is accepted in AS c i , then all the leaves of D c i are not attacked in D.

An important result shows the link between the outcome of a dialog D and the outcomes of the different canonical trees.

-Output(D) 4 is accepted iff ∃ i ∈ 1, m s.t. Status(Subject(D), AS c i ) is accepted. -Output(D) is rejected iff ∀j ∈ 1, m , Status(Subject(D), AS c j ) is rejected.
This result is of great importance since it shows that a canonical tree whose branches are all of even-length is sufficient to reach the same outcome as the original dialog in case the subject is accepted. When the subject is rejected, the whole dialog tree is necessary to ensure the outcome.

Example 9

In Example 7, the subject α 1 of dialog D 6 is accepted since there is a canonical tree whose branches are of even length (it is the canonical tree on the left in Example 8). It can also be checked that α 1 is in the grounded extension {α 1 , α 4 , α 5 , α 8 , α 9 , α 11 } of AS D .

So far, we have shown how to extract from a graph associated with a dialog its canonical trees. These canonical trees contain only useful (hence relevant) moves:

Theorem 5 Let D c
i be a canonical tree of a persuasion dialog D. Any move built on an argument of D c i is useful in the dialog D.

The previous theorem gives an upper bound of the set of moves that can be used to build a canonical dialog, a lower bound is the set of decisive moves.

Theorem 6 Every argument of a decisive move belongs to the dialog tree and to each canonical dialog.

The converse is false since many arguments are not decisive, as shown in Example 4. Indeed, there are two attackers that are not decisive but the dialog tree contains both of them (as does the only canonical dialog for this example).

The ideal dialog

In the previous section, we have shown that from each dialog, a dialog tree can be built. This dialog tree contains direct and indirect attackers and defenders of the subject. From this dialog tree, interesting subtrees can be extracted and are called canonical trees. A canonical tree is a subtree containing only particular entire branches of the dialog tree (only one argument in favor of the subject is chosen for attacking an attacker while each argument against a defender is selected). In case the subject of the dialog is accepted it has been proved that there exists at least one canonical tree such that the subject is accepted in its argumentation system. This canonical tree is a candidate for being an ideal tree since it is sufficient to justify the acceptance of the subject against any attack available in the initial dialog. Among all these candidate we define the ideal tree as the smallest one. In the case the subject is rejected in the initial dialog, then the dialog tree contains all the reasons to reject it, hence we propose to consider the dialog tree itself as the only ideal tree.

Definition 13 (ideal trees and dialogs) If a dialog D has an accepted output -then an ideal tree associated to D is a canonical tree of D in which Subject(D) is accepted and having a minimal number of nodes among all the canonical graphs that also accept Subject(D) -else the ideal tree is the dialog tree of D.

A dialog using once each argument of an ideal graph is called an ideal dialog. Given the above definition, an ideal dialog contains exactly the same number of moves that the number of nodes of the ideal graph.

Property 8 Given a dialog D whose subject is accepted. An ideal dialog ID for D is the shortest dialog with the same output, and s.t. every argument in favor of the subject in ID (including Subject(D) itself) is defended against any attack (existing in D).

This property ensures that, when the subject is accepted in the initial dialog D, an ideal dialog ID is the more concise dialog that entails an acceptation. In other words, we require that the ideal dialog should contain a set of arguments that sumarize D.

Note that the ideal dialog exists but is not always unique. Here is an example of an argumentation system of a dialog which leads to two ideal trees (hence it will lead to at least two ideal dialogs).

α 3 α 2 α 1 α 1 α 2 α 3 α 4 α 1 α 2 α 4
So far, we have formally defined the notion of ideal dialog, and have shown how it is extracted from a persuasion dialog. It is clear that the closer (it terms of set-inclusion of the exchanged arguments) to its ideal version the dialog is, the better the dialog.

Conclusion

In this paper, we have proposed three criteria for evaluating the moves of a persuasion dialog with respect to its subject: relevance, usefulness and decisiveness. Relevance only expresses that the argument of the move has a link with the subject (this link is based on the attack relation of the argumentation system). Usefulness is a more stronger relevance since it requires a directed link from the argument of the move to the subject. Decisive moves have a heavier impact on the dialog, since their omission changes the output of the dialog.

Inspired from works on proof theories for grounded semantics in argumentation, we have defined a notion of "ideal dialog". More precisely, we have first defined a dialog tree associated to a given dialog as the graph that contains every possible direct and indirect attackers and defenders of the subject. From this dialog tree, it is then possible to extract sub-trees called "ideal trees" that are sufficient to prove that the subject is accepted or rejected in the original dialog and this, against any possible argument taken from the initial dialog. A dialog is good if it is close to that ideal tree. Ideal dialogs have nice properties with respect to conciseness, namely they contain only useful and relevant arguments for the subject of the dialog. Moreover for every decisive move its argument belongs to all ideal trees.

From the results of this paper, it seems natural that a protocol generates dialogs of good quality if (1) irrelevant and not useful moves are penalized until there is a set of arguments that relate them to the subject (2) adding arguments in favor of the subject that are attacked by already present arguments has no interest (since they do not belong to any ideal tree). By doing so, the generated dialogs are more concise (i.e. all the uttered arguments have an impact on the result of the dialog), and more efficient (i.e. they are the minimal dialogs that can be built from the information exchanged and that reach the goal of the persuasion).

Note that in our proposal, the order of the arguments has not to be constrained since the generated graph does not take it into account. The only thing that matters in order to obtain a conclusion is the final set of interactions between the exchanged arguments. But the criteria of being relevant to the previous move or at least to a move not too far in the dialog sequence could be taken into account for analyzing dialog quality. Moreover, all the measures already defined in the literature and cited in the introduction could also be used to refine the proposed preference relation on dialogs and finally could help to formalize general properties of protocols in order to generate good dialogs.

Furthermore, it may be the case that from the set of formulas involved in a set of arguments, new arguments may be built. This give birth to a new set of arguments and to a new set of attack relations called complete argumentation system associated to a dialog. Hence, it could be interesting to define dialog trees on the basis of the complete argumentation system then more efficient dialogs could be obtained (but this is not guaranteed). However, some arguments of the complete argumentation system may require the cooperation of the agents. It would mean that in an ideal but practicable dialog, the order of the utterance of the arguments would be constrained by the fact that each agent should be able to build each argument at each step.

Definition 7 (

 7 Output of a persuasion dialog) Let D be a persuasion dialog. The output of D, denoted by Output(D), is Status(Subject(D), AS D ).

Theorem 4

 4 Let D be a persuasion dialog, D c 1 , . . ., D c m its different canonical trees and AS c 1 , . . . , AS c m their corresponding argumentation systems.

Example 10

 10 An ideal Dialog for Dialog D6 (on the left) has the following graph (on the right):

An argument α is indirectly defended by β iff there exists a finite sequence a 1 , . . . , a

2n+1 such that α = a 1 , β = a 2n+1 , and ∀i ∈ 1, 2n , (a i+1 , a i ) ∈ R, n ∈ I N * .

An argument α strictly attacks an argument β in a argumentation system A, R iff (α, β) ∈ R and (β, α) ∈ R.

The length of a path is defined by its number of arcs.

Recall that Output(D) = Status(Subject(D), AS D ).