Experimental characterization and modeling of GFPP commingled yarns tensile behavior

Keywords: Commingled yarns, experimental protocol, tensile behavior, weaving damage, specimen length, strain rate

3D fabrics of commingled yarns offer the possibility of a low-cost and fast manufacturing of complex shaped composite parts. Textile-reinforcement behavior during the forming processes is very important since the appearance of defects (for example wrinkles, yarn misalignment or breakage) can significantly affect the mechanical properties of the final part.

Experimental characterization of the mechanical behavior of textile-reinforcements is expensive, time consuming and a large scattering of results is often observed. To overcome this, meso-scale modeling is an interesting method to study and understand the textile behavior at the unit cell level. To perform realistic simulations, an accurate modeling, and therefore knowledge of the yarn mechanical behavior is needed. In this paper, a simple protocol is proposed and validated in order to investigate the tensile behavior of commingled polypropylene/glass yarns. Influence of specimen length and strain rate are highlighted. A comparison of tensile behavior of yarns before and after weaving is carried out in order to evaluate the weaving damage effect. Finally, a model describing the commingled yarn behavior is proposed. The parameters of the model are defined. Their dependency to strain rate, specimen length and weaving damage are highlighted.

Introduction

There is a growing interest in automotive, aeronautic and leisure industries for the use of textile reinforced thermoplastic composites as lightweight and high performance structural components. Indeed, this kind of product offers interesting advantages compared to classical thermoset composites, especially recycling and the possibility to reshape the part by reheating [START_REF] Bernet | Commingled yarn composites for the rapid processing of complex shapes[END_REF][START_REF] Hufenbach | Polypropylene/glass fiber 3D-textile reinforced composites for automotive applications[END_REF][START_REF] Svensson | Manufacturing of thermoplastic composites from commingled yarns-A review[END_REF] . Thermoplastic matrices also offer advantages in terms of mechanical behavior, with better fracture toughness properties and easiest storage contrary to thermoset ones.

However, the high viscosity of thermoplastic matrices is among the main drawbacks because it makes difficult the reinforcement impregnation. One solution can rely on the use of hybrid yarns where the reinforcement and thermoplastic fibers are mixed together [START_REF] Torun | Advanced manufacturing technology for 3D profiles woven performs[END_REF] . Different technologies have been developed to produce such yarns that can then be used to produce highly drapeable fabrics [START_REF] Hufenbach | Polypropylene/glass fiber 3D-textile reinforced composites for automotive applications[END_REF] . The final composite part is obtained by a thermoforming process, offering the possibility of low-cost and short time cycle manufacturing [START_REF] Bernet | Commingled yarn composites for the rapid processing of complex shapes[END_REF][START_REF] Hufenbach | Polypropylene/glass fiber 3D-textile reinforced composites for automotive applications[END_REF][START_REF] Svensson | Manufacturing of thermoplastic composites from commingled yarns-A review[END_REF] , which is an important criterion when selecting a manufacturing process in the automotive industry for example. For the production of thick parts, the use of 3D fabrics offers advantages like reducing handling of layers and therefore manufacturing times. 3D textile composites also offer mechanical advantages over classical 2D fabrics: they exhibit, thanks to the use of zbinder better delamination properties, impact and fatigue resistance than conventional composites [START_REF] Guénon | Toughness properties of a three-dimensional carbon-epoxy composite[END_REF][START_REF] Brandt | Mechanical performance of composites based on various three-dimensional woven-fibre preforms[END_REF][START_REF] Thanomsilp | Penetration impact resistance of hybrid composites based on commingled yarns fabrics[END_REF] . Within this study, which is part of the European project 3D light-Trans, advantages of both technologies are combined in a 3D fabric of commingled yarns.

Manufacturing of the textile-reinforced composite parts requires the forming of the reinforcements into the desired shape. The fabric behavior during this forming process is very important since the appearance of defects (wrinkles, yarn misalignment) can affect the mechanical properties and quality of the final part [START_REF] Allaoui | Experimental preforming of highly double curved shapes with a case corner using an interlock reinforcement[END_REF][START_REF] Thije | A multi-layer triangular membrane finite element for the forming simulation of laminated composites[END_REF][START_REF] Lightfoot | Defects in woven preforms: Formation mechanisms and the effects of laminate design and layup protocol[END_REF] . Experimental studies have shown the good drapeability of commingled yarn made fabrics [START_REF] Hufenbach | Polypropylene/glass fiber 3D-textile reinforced composites for automotive applications[END_REF] . But their potential is still poorly understood and numerous choices of weaving architectures or yarns are possible, that is why further investigation of their properties is needed.

Experimental characterization of mechanical behavior of textile materials is difficult, and a large scattering of results is often observed [START_REF] Cao | Characterization of mechanical behavior of woven fabrics: Experimental methods as benchmark results[END_REF] . To avoid costly trial and error development, numerical approaches can be used. Numerical approaches at the micro [START_REF] Durville | Simulation of the mechanical behavior of woven fabrics at the scale of fibers[END_REF] and meso [START_REF] Boisse | A mesoscopic approach for the simulation of woven fibre composite forming[END_REF][START_REF] Badel | Simulation and tomography analysis of textile composite reinforcement deformation at the mesoscopic scale[END_REF][START_REF] Badel | Rate constitutive equations for computational analyses of textile composite reinforcement mechanical behavior during forming[END_REF][START_REF] Hivet | Consistent 3D geometrical model of fabric elementary cell. Application to a meshing preprocessor for 3D finite element analysis[END_REF] scales have been developed to investigate the mechanical behavior and formability of woven fabrics.

In meso-scale simulations, CAD models of the fabric unit cell where yarns are modeled as a continuum are used. For roving fabrics, yarns are considered as transversely isotropic material [START_REF] Boisse | A mesoscopic approach for the simulation of woven fibre composite forming[END_REF][START_REF] Badel | Simulation and tomography analysis of textile composite reinforcement deformation at the mesoscopic scale[END_REF][START_REF] Tavana | Meso/macro-scale finite element model for forming process of woven fabric reinforcements[END_REF] with null or roughly estimated lateral contraction during loading (zero longitudinal Poisson's ratio).

To perform realistic meso-scale simulations on fabrics made of commingled yarns, it is important to investigate and model accurately the mechanical behavior, and in particular the tensile behavior, of commingled yarns.

In this paper, a simple protocol to perform yarn tensile tests is proposed and validated. The tensile behavior of commingled polypropylene/glass yarns is then investigated. Influence of specimen length and strain rate are evaluated. Previous studies have shown the influence of weaving damage on the yarn mechanical behavior [START_REF] Lee | Effect of weaving damage on the tensile properties of three-dimensional woven composites[END_REF] . To consider the real mechanical behavior of yarns inside the fabric, a comparison of yarn tensile behavior before and after weaving (yarns extracted from the fabric) is carried out in order to analyze and quantify the effect of weaving damage. Based on these experimental results, a model describing the specific commingled yarns tensile behavior is proposed. Parameters of this model are identified, paving the way to meso-scale simulations on commingled yarns made fabrics.

Materials and methods

Materials

Different technologies have been developed for the production of hybrid yarns. Thermoplastic can be integrated in the yarn under different forms like powder, staple fibers or filaments. The first main objective when producing hybrid yarns is to obtain a good mixture of components that will lead to a good repartition of matrix and low porosity levels in the composite [START_REF] Svensson | Manufacturing of thermoplastic composites from commingled yarns-A review[END_REF] . The second important point is that hybrid yarns must be usable in weaving looms to produce fabrics.

Commingling is one of the technologies used to produce hybrid yarns. It relies on the use of air jet texturizing machines [START_REF] Torun | Advanced manufacturing technology for 3D profiles woven performs[END_REF] . The commingled yarn is obtained using reinforcement and matrix yarns as input. These yarns are opened and their filaments are mixed in the air nozzle.

The use of overdelivery (delivery speed of input yarns more important than the take out speed of the commingled yarn) is necessary to perform the commingling. Still, the overdelivery value must be as low as possible especially for reinforcement fibers to avoid significant loss of orientation [START_REF] Torun | Advanced manufacturing technology for 3D profiles woven performs[END_REF] . This manufacturing process is cost effective and provides a good mixture of matrix and reinforcement filaments in the yarn [START_REF] Svensson | Manufacturing of thermoplastic composites from commingled yarns-A review[END_REF] . It enables producing yarns with different reinforcement and matrix materials and different fiber volume fraction. The obtained commingled yarns are flexible and can be woven, braided or knitted [START_REF] Svensson | Manufacturing of thermoplastic composites from commingled yarns-A review[END_REF][START_REF] Guénon | Toughness properties of a three-dimensional carbon-epoxy composite[END_REF][START_REF] Brandt | Mechanical performance of composites based on various three-dimensional woven-fibre preforms[END_REF] . However, the main drawbacks are that the commingling process can damage fibers and that high counts yarns cannot be produced effectively.

The commingled yarn structure is also highly influenced by the manufacturing process. The use of pressured air leads to the appearance of opened areas and nips (compact areas) along the yarn length (Figure 1). Different kinds of nip structures, corresponding to various repartitions of the reinforcement and matrix filaments, can be observed [START_REF] Kravaev | Influence of process parameters on filament distribution and blending quality in commingled yarns used for thermoplastic composites[END_REF][START_REF] Alagirusamy | Development and characterization of GF/PET, GF/Nylon, and GF/PP commingled yarns for thermoplastic composites[END_REF][START_REF] Ogale | Tensile properties of GF-polyester, GF-nylon, and GFpolypropylene commingled yarns[END_REF] . The types of nips, their stability, the degree of interlacing (total length of nips divided by the yarn length), the commingling damage and the filaments distribution in the yarn are highly influenced by the chosen commingling parameters: speed, pressure, jet design and overdelivery values.

Consequently, the tensile behavior of commingled yarns depends on properties of the used input yarns but also to a large extent on the yarn structure and therefore of the commingling process [START_REF] Kravaev | Influence of process parameters on filament distribution and blending quality in commingled yarns used for thermoplastic composites[END_REF][START_REF] Alagirusamy | Development and characterization of GF/PET, GF/Nylon, and GF/PP commingled yarns for thermoplastic composites[END_REF][START_REF] Ogale | Tensile properties of GF-polyester, GF-nylon, and GFpolypropylene commingled yarns[END_REF] . Investigation of commingled yarns tensile behavior is hence necessary since models developed and used for other types of yarns are not relevant.

Yarns used in this study were manufactured using a 300 tex E-glass fiber yarn (referred to as GF yarn in the following) and 3*32 tex polypropylene (PP) yarns. Overdelivery (OD) used was of 2% for GF and 5% for PP. More details concerning the commingling process can be found in the study of Fazeli et al. [START_REF] Fazeli | High-performance lightweight multifunctional composites based on 3D-shaped multilayered woven fabrics[END_REF] . Yarns are similar to the ones studied by Torun et al. [START_REF] Torun | Advanced manufacturing technology for 3D profiles woven performs[END_REF] Compared to rovings with roughly rectangular cross section at the end of the yarn production process [START_REF] Saad | Geometrical modelling and characterization of 3D warp interlock composites and their on-line structural health monitoring using flexible textile sensors[END_REF] , the commingled yarn section is rather circular or elliptic [START_REF] Kravaev | Influence of process parameters on filament distribution and blending quality in commingled yarns used for thermoplastic composites[END_REF][START_REF] Van West | The consolidation of commingled thermoplastic fabrics[END_REF] . Yarn diameter can vary from less than 1 mm in the nips to more than 3 mm in opened areas (see Figure 1(a)).

Using the input materials densities (2.54 for GF and 0.91 for PP) and the overdelivery value, the average material section in the GFPP yarns can be estimated to 0.23 mm². This value will be used to calculate yarn's stresses from measured forces in the following.

Within

Yarn tensile testing

The goal of this section is to propose and validate a simple protocol to perform reliable yarn tensile tests in order to investigate the commingled yarn's behavior. To the authors' knowledge there are no specific standards for tensile testing of commingled yarns. However, the standard ISO 3341 [START_REF]Textile glass--Yarns--Determination of breaking force and breaking elongation[END_REF] giving recommendations for the determination of the breaking force and elongation of textile glass yarns can be used as a basis. This standard is applied for rovings of less than 2000 tex. The use of 5mN/tex preload, 500 mm specimen length and 200 mm/min testing speed are recommended. During testing, samples must break at 10 mm away from the grips. Reproducibility of results must be investigated by performing at least 10 tests.

In the literature, various specimen length and strain rates have been used. The results on different types of yarns have shown that they both can have significant effect on the measured modulus and breaking strength of yarns [START_REF] Wang | The effects of strain rate on the mechanical behavior of Kevlar fibre bundles: an experimental and theoretical study[END_REF][START_REF] Chudoba | Effect of twist, fineness, loading rate and length on tensile behavior of multifilament yarns (A multivariate study)[END_REF][START_REF] Zhou | Tensile behavior of carbon fiber bundles at different strain rates[END_REF][START_REF] Ghosh | Spun yarn strength as a function of gauge length and extension rate: a critical review[END_REF] . The goal of these tests is to estimate what will be the yarn behavior within the fabric. In this sense, it has been shown 30 that determining yarn's strength at short specimen length is more appropriate. Moreover, during forming, yarns will be subjected to various strain rate conditions. Experimental investigation of the influence of strain rate and specimen length on the commingled yarns tensile behavior is therefore required.

For this purpose, tests on commingled yarns were performed using a 500 mm specimen length and three different crosshead speeds of 5 mm/min (strain rate ߝሶ ൌ 1.67 × 10 ିସ s ିଵ ), 50 mm/min (ߝሶ = 1.67 × 10 ିଷ s ିଵ ) and 500 mm/min (ߝሶ = 1.67 × 10 ିଶ s ିଵ ) to investigate the strain rate effect. Tests were also performed with a 200 mm specimen length at same strain rates (i.e with testing speeds of 2, 20 and 200 mm/min) to evaluate the influence of specimen length on measured parameters. The 1.67 × 10 ିସ s ିଵ strain rate was chosen to investigate the quasi-static tensile behavior of yarns. The 1.67 × 10 ିଶ s ିଵ strain rate corresponds to what could be observed in industrial conditions for composite parts manufacturing while the 1.67 × 10 ିଷ s ିଵ strain rate was chosen as an intermediary value.

For the sake of simplicity, tests were performed using a universal testing machine (INSTRON 4507) with conventional jaws. Samples were prepared by fixing yarns between thin aluminum plates with glue. A uniform pressure was applied on the plates during at least 24 hours (time of drying). Aluminum tabs were then directly fixed between the machine jaws for the testing.

At least 10 tests were performed in each testing configuration and more when a significant scattering of tensile behavior between samples was highlighted. A 5mN/tex preload was used.

When performing tensile tests it is not always possible to rely on displacement data given by the testing machine: when load is applied, machine components or other pieces like aluminum tabs or glue used to fix the samples can deform. As a consequence, displacement given by the machine is not the one to which the sample is subjected. This error can be significant if high loads are reached or if the sample's deformation is low. Another important issue is the clamping: bad clamping conditions can lead to the premature breaking of the samples near the clamping areas due to stress concentration phenomenon. To investigate these issues, optical measurements with marker tracking method were used. Figure 3 shows black markers 50, 100 and 150 mm spaced that were placed on 200 mm long yarn samples. These tests were performed on GF yarns (the 300 tex yarn used as an input for the manufacturing of commingled yarns), since it is not possible to use the marker tracking method with commingled yarns because of their particular structure described in the previous section. Positions of markers were registered during testing using a CCD camera (resolution 1280 × 1024 pixels) and deformations were calculated by the Deftac software [START_REF] Bretagne | Development of the marks tracking technique for strain field and volume variation measurements[END_REF] . In Figure 4 an example of stress-strain curves of a sample reconstructed using machine displacement and optical measurements (100 mm spaced markers) shows the very good agreement between the two methods in the loading phase. Average Young's modulus obtained on seven samples using the machine displacement and the 3 couples of markers in this linear phase are shown in Table 1.

Fig. 4. Tensile curves of a GF yarn obtained using machine and optical measurements Results of optical measurements show that the deformation is homogeneous along the yarn length during the loading, and a very good agreement (difference of less than 2%) between Young's modulus calculated using the machine displacement and the marker tracking method is highlighted. Moreover, the linear behavior in the loading phase and the measured modulus of 73 GPa are in good agreement with the literature for E-glass rovings. These results show that the machine compliance is negligible for GF yarns and therefore that machine displacement can be used directly to calculate yarn strain. Since GFPP yarns have a lower stiffness compared to GF yarns (as testing results will show in the next section), machine compliance can also be neglected with 200 mm long GFPP yarns, and, as displacements will increase with specimen length, machine displacement can also be used with the 500 mm long samples.

When fibers begin to break inside the yarn, markers progressively disappear and it becomes more difficult to obtain coherent optical measurements. This can be observed in Figure 4 when, after reaching the maximum load, optical and machine strain measurements start to mismatch. Still, observations on samples tend to show that deformation remains homogeneous along the yarn length during the breaking phase and that fiber breakages are not localized near the clamps but homogeneously distributed along the GF yarn length. Due to the more brittle breaking behavior of GFPP yarns, it is possible to observe directly where their breaking occurred (see Figure 5). In most cases, breaking is not localized near clamps, tending to prove that yarns don't break prematurely due to stress concentration phenomenon.

Fig. 5. GFPP yarn after breaking

Results of tensile tests performed on ten GF yarn samples in the same testing configuration (200 mm specimen length, 2 mm/min testing speed) are shown in Figure 6. A very good reproducibility of results in the loading part is highlighted with coefficients of variation of less than 3% for the measured moduli and breaking stress between the different samples.

These results show that the scattering that could be introduced by the testing protocol and manufacturing process of samples is low. Fig. 6. Repeatability of GF tensile test As mentioned, longitudinal Poisson's ratio considered in models describing roving tensile behavior are often roughly estimated or supposed null. In our case, because of the particular structure of commingled yarns it is interesting to see if modifications of the yarn structure under tensile loading conditions can be observed. To do so, pictures were registered using a camera (AVT manta, 2452*2048 pixels and 50 mm zoom lens) during tensile testing (see The presented testing protocol gives, from the preparation of samples to the analysis of testing results, a simple and reliable method to characterize the GFPP yarns tensile behavior. Use of optical measurements and of a dedicated program for yarn contour detection also enables analyzing changes of yarn section during testing.

Experimental results

In this section, experimental results of tests performed on commingled yarns are presented.

These results are discussed and interpreted in details in the section "modeling and discussion". There is a significant scatter of tensile properties of GFPP yarns in terms of modulus and breaking strength (Figure 9(a)). The coefficient of variation is more than 10%. Average breaking stress is of 340 MPa corresponding to a breaking load close to 80 N. From preload to maximum strength value, the breaking strain is approximately 1.5%. Average Young's modulus (measured in the linear part of the stress-strain curves) is approximately 30 GPa. A correlation appears to exist between the modulus of a particular yarn and its breaking strength (see Figure 9(b)); yarns with a higher modulus tending to break at higher loads. It can also be noticed that the ߝ ே value seems to decrease with increasing modulus (see Figure 9(c)). The calculated Spearman's rho coefficients (assessing how well the relationship between two variables can be described using a monotonic function) for both couples of variables are respectively of 0.88 and -0.93. Linear regressions and coefficients of determination are also reported in the two figures. Effects of strain rate and specimen length Average values and standard deviation of measured moduli, breaking strength and strain of GFPP samples for the two specimen lengths and the three strain rates are shown in Table 2. For comparison results obtained for GF yarns in the same testing configurations are also reported. Average GFPP tensile curves for the 200 mm specimen length at the 3 strain rates are shown in Figure 12. The first parts of the stress-strain curves are almost superposed. A pronounced increase of breaking strength with strain rate is highlighted. For the two specimen lengths, the average breaking strength at 1.67 × 10 ିଶ s ିଵ is more than 60% higher than the one measured at 1.67 × 10 ିସ s ିଵ (Table 2). For GF yarns this increase is only of 20%. For the 1.67 × 10 ିସ s ିଵ strain rate, influence of specimen length on breaking strength appears to be quite similar for GF and GFPP yarns, average breaking strength being approximately 6% higher with the 200 mm specimen length compared to the 500 mm specimen length. This difference remains the same for the three strain rates in the case of GF yarns but seems to increase slightly with strain rate for GFPP (13% difference at 1.67 × 10 ିଶ s ିଵ strain rate).

Young's modulus of GFPP yarns is close to 30 GPa. It increases slightly with strain rate and decrease slightly with specimen length. For GF yarns it remains approximately the same in all testing configurations. With high strain rates, a progressive decrease of stiffness of GFPP yarns is observed before yarn break. In some cases, load is not taken up by PP fibers after the breaking of glass fibers (total break of the yarns), this was mainly observed for the 500 mm specimen length at high strain rates. 73.9 ± ଵ.ଵ 73.9 ± ଵ. 74.1 ± ଵ.ହ 72.9 ± ଶ.ଵ GFPP ૢ. ± .ૢ . ± . . ૡ ± .ૢ ૡ. ± .ૢ ૢ. ± . . ± .

GF Breaking strain [%]

(strain 0 at preload) 1.35 ± .

1.54 ± .ଷ 1.61 ± .ହ 1.24 ± .

1.39 ± .ଷ 1.51 ± .ଷ GFPP . ૠ ± . . ૢ ± .ૡ . ± .ૠ . ± . . ૡ ± . . ૠ ± .

GF

Breaking stress

[MPa] 852 ± ଶଶ 948 ± ଶଶ 1024 ± ଷଶ 819 ± ଶଽ 901 ± ଶ 968 ± ଵ GFPP ± ± ૡૢ ± ૢ ± ૡ ૡ ± ± GF Breaking load [N] 102.2 ± ଶ.
113.8 ± ଶ. 122.9 ± ଷ.଼ 96.7 ± ଷ.ସ 106.4 ± ଶ.ଷ 114.2 ± ଵ.ଽ GFPP ૠૡ. ± .ૡ . ± .ૠ . ± ૠ. ૠ. ૠ ± . ૢ. ± .ૠ ૡ. ૠ ± ૢ.ૢ Figure 13 presents the evolution of breaking strength for the two specimen lengths as a function of the strain rate logarithm for GF and GFPP. Breaking load of GFPP yarns is of the same order of magnitude as GF one. Still it can be observed that for quasi-static loading conditions, GFPP yarn's breaking load is lower than GF ones but becomes greater at high strain rates. 3 gives the measured moduli, breaking strengths and strains for the 3 groups of yarns and their variations compared to values obtained for yarns before weaving. The average tensile curves of yarns are compared in Figure 14. A decrease of breaking strength is observed for the three groups of yarns. Results show that weft yarns are the most damaged, with a decrease of 28% of breaking strength and also a decrease of 17% of modulus. A more significant scattering of weft yarn's breaking strength and modulus is also observed. Damage seems to vary significantly from one yarn to the other.

In this sense, observations on weft yarns show that on some samples a non-negligible number of fibers are broken. It has been shown [START_REF] Lee | Effect of weaving damage on the tensile properties of three-dimensional woven composites[END_REF] that fibers can be broken during the weft-insertion The modulus of warp and binding yarns is not modified; they are similarly damaged by the weaving process, with a decrease of breaking strength of respectively 15 and 22%. Their average tensile behavior is very similar (Figure 14).

As apparent from Figure 14, the initial nonlinear behavior (ߝ ே value) is less pronounced for warp and binding yarns as compared to undamaged yarns. During the weaving process, warp and binding yarns pass through a tensioning device to provide adequate tension for the shedding stage [START_REF] Lee | Effect of weaving damage on the tensile properties of three-dimensional woven composites[END_REF] . They are therefore subjected to several tensioning cycles. To a lesser extent, during the insertion stage, weft yarns can also be subjected to tension. It is in this sense interesting to investigate the tensile behavior of GFPP yarns under cyclic loading. For this purpose, cyclic loading tests have been performed on undamaged yarns (testing configuration 200 mm, 2 mm/min), and an example of result is shown in Figure 15. After the first loading cycle, a permanent deformation is highlighted. The nonlinear phase of the second loading is reduced compared to the one of the first loading and Young's modulus observed during the second loading also appears slightly more important. An increase of Young's modulus of up to 10% after few loading cycles is observed: on the testing result example presented in Figure 16, the maximum modulus reached during the first loading is 26.7 GPa, it increases in the next two loadings to stabilize at a value of 29.3 GPa. This phenomenon can be attributed to the progressive alignment of fibers along the loading direction by the first loadings. If a high number of cycles are performed, it can lead to the premature breaking of the yarn (see Figure 16). A possible explanation could be that damage by inter-fiber friction occurs in the yarn because of the fiber interlacing and complex stress repartition in the nip areas. Stress MPa

Strain %

During the weaving process, two effects have therefore to be considered. The first one is abrasion between yarns and weaving machine components leading to fiber damage and fiber breakage (observed for weft yarns) that can explain the decrease of yarn's breaking strength and stiffness. On the other hand, tensioning of yarns during the weaving process can lead to a smaller initial nonlinear behavior, an increase of modulus and a decrease of breaking strength.

This mechanism can explain the smaller initial nonlinear behavior of warp and binding yarns.

Still, no increase of their Young's modulus was observed (Table 3), maybe because loads experienced by these yarns during the weaving process are less significant than those used for cyclic testing. The possible increase of modulus can also have been compensated by fiber breakage caused by abrasion with components like for weft yarns. Both fiber damage and breakage due to abrasion with machine components and multiple tensioning of yarns during weaving can explain the breaking strength decrease of warp and binding yarns.

Weaving induces a significant decrease of yarn's breaking strength and also a stiffness decrease for weft yarns. This can lead to yarn breakage during the forming process and decreased mechanical properties of the final composite parts. An accurate knowledge of tensile stresses to which yarns are subjected during weaving, and an investigation of their tensile properties after each weaving step would help to better understand the effects of weaving. One solution to avoid fiber damage could rely in the use of hybrid yarns where matrix filaments are placed around the reinforcement fibers to protect them from abrasion. On the other hand, due to the poor mixture of filaments in such yarns, the impregnation of reinforcement fibers by matrix could be more difficult [START_REF] Torun | Advanced manufacturing technology for 3D profiles woven performs[END_REF] .

The main consequences of these results is that, even if only just one type of yarn is used to weave the fabric, due to the non-equivalent weaving damage effect on warp, binding and weft yarns, their tensile behavior once inside the woven fabric will not be the same. It is therefore important to perform tensile tests on yarns extracted from the fabrics to estimate the influence of weaving in order to take into account this aspect in models describing the yarn tensile behavior.

Modeling and discussion

The goal of this part is to propose a model to describe correctly the GFPP stress-strain tensile behavior and to identify the influence of specimen length, strain rate and weaving damage on parameters of this model. Few results can be found in the literature concerning this topic.

Torun [START_REF] Torun | Advanced manufacturing technology for 3D profiles woven performs[END_REF] has proposed a model to estimate Young's modulus of commingled yarns, still there are not known results to describe the whole stress-strain behavior.

First goal is to explain the initial nonlinear behavior of commingled yarns. Main contribution to the yarn's tensile behavior can be attributed to GF fibers; their modulus being much higher than PP fibers modulus. As detailed in section 2.a, the commingled yarn structure is composed of nips and opened areas. In opened areas, some fibers are initially almost straight, while others are buckled. Different mechanisms can explain the initial nonlinear behavior.

Torun [START_REF] Torun | Advanced manufacturing technology for 3D profiles woven performs[END_REF] , made the assumption that all GF overdelivery is consumed in nips and therefore that glass fibers are straight in opened areas, whereas PP overdelivery is consumed in both nips and opened areas. It can be supposed that initially, it is not quite true and that glass fibers can be slightly buckled in opened areas. Moreover, a fiber buckled in a particular opened area can possibly be straight in another opened area, as schematized in Figure 17. If a fiber starts to undergo significant tension in a particular portion of its length (where the fiber is straight) but is not loaded in another one (where the fiber is buckled), it can be supposed that the fiber could move relatively to the others to equilibrate the carried load on its whole length. Of course this mechanism will be strongly linked to the intermingling of fibers in nips and to the friction between fibers. Moreover, structure of the commingled yarn in nips is supposed similar to the one observed in twisted yarns [START_REF] Torun | Advanced manufacturing technology for 3D profiles woven performs[END_REF] . Results presented by Chudoba [START_REF] Chudoba | Effect of twist, fineness, loading rate and length on tensile behavior of multifilament yarns (A multivariate study)[END_REF] tend to show that once twisted, yarns have a more extended initial nonlinear behavior than untwisted. This can be attributed to the reorganization of the twisted yarn structure when subjected to tensile loading (decrease of yarn section and increase of yarn packing factor). Cyclic testing results highlighting a permanent deformation after the first loading and a significant decrease of nonlinear behavior in the next loadings tend to confirm these phenomena of reorganization of the yarn structure. been proposed and validated by Chudoba [START_REF] Chudoba | Random properties within the cross-section and size effect[END_REF] . The slack of filament ݅ being noted ߠ , when a roving of length ܮ is subjected to a displacement ݑ the deformation of fiber ݅ can be written [START_REF] Chudoba | Random properties within the cross-section and size effect[END_REF] :

ߝ ൌ ିఏ ଵାఏ (1)
With ݁ ൌ ௨ the yarn deformation. Assuming a linear-elastic and brittle response of fibers of modulus ܧ and a large number of fibers in the roving, the stress-strain behavior can be expressed by:

ߪ ఏ ሺ݁ሻ ൌ ܧ  ିఏ ଵାఏ ఏ ܩ݀ ఏ ሺߠሻ (2)
With ܩ ఏ ሺߠሻ the cumulative density of delayed activation that can be extracted from experimental results by constructing the normalized derivative of the stress-strain curve.

It is interesting to see if this model can also describe the tensile behavior of commingled yarns since the different reorganization mechanisms of yarn structure presented above can be supposed to act like as slack of fibers. To do so, it is necessary to evaluate the cumulative density ܩ ఏ ሺߠሻ in the case of commingled yarns. It is clear that predicting the shape of this curve from direct analyze of the very disordered GFPP yarn structure is a very difficult issue.

For the sake of simplicity, a predefined shape of curve is proposed and fitted on the experimental curve. Modulus of the commingled yarn is referred to as ܧ ୋ . Evaluation of this parameter will be discussed in the next paragraph.

The simplest model can be formulated by supposing a uniform activation density:

ܩ ఏ ሺߠሻ ൌ ఏ ఏ ೌೣ 1 [,ఏ ೌೣ [ ሺߠሻ + 1 [ఏ ೌೣ ,ஶ[ ሺߠሻ (3)
With

1 [,[ ሺߠሻ ൌ ቄ 1 if ܽ ≤ ߠ < ܾ 0 else
The stress-strain tensile behavior can then simply be expressed as 32

ߪ ఏ ሺ݁ሻ ≈ ா ృూౌౌ ఏ ೌೣ [ሺ݁ + 1ሻ lnሺ1 + ݁ሻ -݁] 0 ≤ ݁ ≤ ߠ ௫ (4)
ߪ ఏ ሺ݁ሻ ≈ ா ృూౌౌ ఏ ೌೣ [ሺ݁ + 1ሻ lnሺ1 + ߠ ௫ ሻ -ߠ ௫ ] ݁ ≥ ߠ ௫ (5)
In Figure 18 the cumulative density of delayed activation obtained from the average experimental curve in configuration 200 mm-2 mm/min for undamaged yarns is shown. From an initial value close to 0.3 at the preload state, cumulative density increases quite linearly up to 0.8 and then more slowly to finally stabilize. Experimental values are shifted so that they fit with the uniform activation density model.

Advantage is that in addition to ܧ ୋ value, only one parameter ( ߠ ௫ ) needs to be identified. But the experimental curve is not perfectly described, especially when the cumulative density come close to 1 as seen in Figure 18. Still, the experimental and modeled stress-strain curves (identified value ߠ ௫ ൌ 0.008) are in quite good agreement (Figure 19).

Of course it is possible to better describe the cumulative density activation using for example piecewise linear function [START_REF] Chudoba | Random properties within the cross-section and size effect[END_REF] . As mentioned above, initial nonlinear behavior is only slightly influenced by both specimen length and strain rate. Parameters identified on one testing configuration can be used on other testing configuration just taking into account the slight variations of ܧ ୋ value with strain rate and specimen length. Using Equation 3, initial value of the cumulative density of activation is supposed null and therefore, the initial slope of the stress-strain curve is also null.

It seems appropriate to model results presented in section 3.b (tensile behavior of yarns obtained after the commingling process) but a non-null initial modulus should be considered for yarns that have been subjected to previous cyclic loadings and therefore possibly to yarns after the weaving process.

A simple rule of mixture can give an estimation of the Young modulus ܧ ୋ of a GFPP yarn in which all fibers would be parallel and perfectly aligned along the yarn direction. With a modulus of 73 GPa for GF as identified with tensile tests and a modulus of 1.5 GPa for PP (according to literature); given volume fraction values of GF and PP, the maximum modulus ܧ ௫ ≈ 38 GPa could be expected for such yarns. Actually, even if it can be supposed that after the nonlinear behavior phase glass fibers are straight and aligned following the yarn direction in opened areas, they remain intermingled in nips areas. ܧ ୋ is therefore less than ܧ ௫ , as shown by experimental results highlighting an average modulus of 30 GPa (see Table 2).

Torun [START_REF] Torun | Advanced manufacturing technology for 3D profiles woven performs[END_REF] has proposed a model to estimate Young modulus of commingled yarns. The corresponding mechanical model is schematized in Figure 20.

Fig. 20. Mechanical model of GFPP yarn

In this model, assumption is made that glass fibers are straight in opened areas and structure of the commingled yarn in nip areas is supposed similar to the one observed in twisted yarns.

Influence of PP on the behavior is neglected. The total length of the yarn is:

ܮ ൌ ܮ ୭୮ୣ୬ୣୢ ୟ୰ୣୟ + ܮ ୬୧୮ ൌ ሺ1 -ܮሻݔ + ܮݔ (6)
With ݔ the degree of interlacing.

In nips, twisting of fibers is taken into account by introducing a function ݂ሺߛሻ (with ߛ the twist angle) which describes the loss of modulus due to twisting. ܣ ீி is the glass fiber material section in the yarn. Stiffness ݇ of the commingled yarn is deduced using

ଵ ൌ ଵ భ + ଵ మ
and its Young's modulus is ܧ ீி ൌ with ܣ the total material section in the yarn.

Therefore, the final expression of the Young modulus is:

ܧ ୋ ≈ ா ౣ౮ ଵା௫ቀ భ ሺംሻ ିଵቁ (7) ߛ angle is obtained with 4 ߛ ൌ ݏܿ ିଵ ቆ ଵ ଵା ೀವ ృూ ೣ ቇ (8)
In the study of Torun 4 , assumption is made that all GF overdelivery has been consumed in nips. But, as discussed earlier, part of the GF overdelivery can also be consumed in opened area. Only the part consumed in nips, which will be noted ܦܱ ୋ ୬୧୮ should be taken into account in Equation 8. Torun [START_REF] Torun | Advanced manufacturing technology for 3D profiles woven performs[END_REF] used ݂ሺߛሻ ൌ ݏܿ ଶ ሺߛሻ and obtained a value of ܧ ୋ only slightly lower than ܧ ௫ . The difference between experimental results and value obtained using the model is explained by the influence of commingling damage. But, in our case, observations on yarns after the commingling process (Figure 1(b)) have shown that only a small number of fibers appeared to be broken. Moreover, small modifications of the model could lead to a better approximation of the modulus:

-Especially for high twist angles, the ݏܿ ଶ model can underestimate significantly the influence of twist [START_REF] Rao | A modeling and experimental study of the influence of twist on the mechanical properties of high-performance fiber yarns[END_REF] .

-In twisted yarns, the twist angle ߠ of a fiber depends on its radial position [START_REF] Rao | A modeling and experimental study of the influence of twist on the mechanical properties of high-performance fiber yarns[END_REF] : ݊ܽݐሺߠሻ ൌ ଶగ , with ݎ the radial position of the fiber in the yarn and ℎ the yarn length for one turn. Models like the ݏܿ ଶ one are applied with angle ߙ corresponding to the twist angle of fibers at the most outer layer (surface twist angle) of the twisted yarns. This angle is given by ݊ܽݐሺߙሻ ൌ ଶగோ , with R the yarn radius. The average fiber twist angle ߠ in the yarn is therefore less than ߙ and is given by [START_REF] Madsen | Hemp yarn reinforced composites-I. Yarn characteristics[END_REF] :

ߠ ൌ ߙ + ఈ ௧ మ ఈ - ଵ ୲ୟ୬ ఈ (9)
Angle ߛ calculated with Equation 8 corresponds in fact to the average twist angle of fibers due to overdelivery i.e. to the ߠ value. Angle used in Equation 7 shouldn't therefore be ߛ but rather angle ߙ calculated using Equation 9with ߠ ൌ ߛ.

Main difficulty comes from an accurate evaluation of parameters used in the model. From hypothesis of this model it can be supposed that scattering of GFPP yarn's properties shown in Figure 9(a) can be explained by the variability of yarn structure due to the commingling process: the number, length and types of nips along the yarn length can vary from one sample to the other leading to variations of ݔ and ܦܱ ୋ ୬୧୮ values, and as a consequence to a scattering of size of the initial nonlinear behavior and modulus.

The decrease of yarn section in opened areas during loading can be attributed to the alignment of fibers along the yarn as they start to take load. In nips, twist of fibers will also leads to a decrease of yarn section [START_REF] Pan | Prediction of statistical strengths of twisted fibre structures[END_REF] , which is experimentally confirmed in Figure 11. Results presented in Figure 11 can be used to estimate the average decrease of yarn section but further work would be necessary to investigate more precisely transverse behavior in nips and opened areas. In meso-scale models, where yarns are modeled as a continuum, this transverse behavior should be taken into account by considering a non-null longitudinal Poisson's ratio.

No models that could allow estimating the commingled yarn's breaking strength and failure mechanisms have been found in the literature, but results and models are available for twisted yarns [START_REF] Rao | A modeling and experimental study of the influence of twist on the mechanical properties of high-performance fiber yarns[END_REF] and rovings [START_REF] Weibull | A statistical distribution of wide applicability[END_REF][START_REF] Chi | Determination of single fibre strength distribution from fibre bundle testings[END_REF][START_REF] Van Der Zwaag | The concept of filament strength and the weibull modulus[END_REF] . Results on twisted yarns show that, depending on the material, a low yarn twist leads to an increase of breaking strength compared to an untwisted yarn, the maximum strength being reached for angles close to 7°. This effect can be attributed to the fragmentation process i.e. a fiber can break in the yarn but its fragments will continue to carry load [START_REF] Pan | Prediction of statistical strengths of twisted fibre structures[END_REF] . When the angle is more important, typically more than 15°, yarn's breaking strength decreases compared to the one of untwisted yarns. Twist angle calculated in nip areas using the model described above is greater than 15°. Therefore, neglecting the influence of PP, in quasi-static loading conditions, GFPP yarns should break in the nip areas with a lower breaking force than the GF roving one. In this sense, results presented in Figure 9(b) show that yarns with an important modulus (so with a smaller number of nips or an average twist angle in nips less than the average) are also those that have the highest breaking strength in quasi-static loading conditions. Presence of nips results in the fact that deformation is not homogeneous along the yarn length. Considering again the mechanical model described in Figure 20, and using the same values of ݂ሺߙሻ and ,ݔ if the commingled yarn is subjected to a displacement speed ݑሶ , displacement speeds in the opened areas (subscript 1) and in nips (subscript 2) are:

ݑሶ ଵ ൌ ௨ሶ మ భ ା మ ≈ 0.73ݑሶ (10)
and

ݑሶ ଶ ൌ ௨ሶ భ భ ା మ ≈ 0.27ݑሶ (11)
Corresponding yarn's strain rates in these zones are therefore:

݁ሶ ଵ ൌ ௨ሶ భ భ ≈ .௨ሶ .ଽ ≈ 0.81݁ሶ (12)
and

݁ሶ ଶ ൌ ௨ሶ మ మ ≈ .ଶ௨ሶ .ଵ ≈ 2.7݁ሶ (13)
In opened areas, as fibers are straight, fiber's strain rate will be ߝሶ ଵ ൌ ݁ሶ ଵ ൌ 0.81݁ሶ . In nips, strain rate for a fiber with twist angle ߠ will be ߝሶ ଶ ሺߠሻ ൌ ݁ሶ ଶ ݏܿ ଶ ሺߠሻ [START_REF] Rao | A modeling and experimental study of the influence of twist on the mechanical properties of high-performance fiber yarns[END_REF] . Using the average twist angle value of 30° calculated above, average fiber's strain rate ߝሶ ଶ ≈ 2݁ሶ is obtained in nips.

Therefore, when ݁ሶ is increased, breaking strength of GFPP increases more rapidly than the one of GF, because, in nips areas where the GFPP yarn is the weakest, the local fiber's strain rate is more important than the global strain rate to which is subjected the yarn. When high strain rates are reached, yarn could start to break preferentially in opened areas, where strain rate is lower than in nips areas. As there is no twist in opened areas, fragmentation process can possibly explain that the breaking force of GFPP becomes even greater than the one of GF at high strain rates. The presence of PP could also play a role.

The model described above allows a good description of commingled yarn's tensile behavior up to linear-behavior part. In quasi-static loading conditions, it is possible in first approximation to describe the full stress-strain curve of a particular GFPP yarn using the previously proposed model and assuming a brittle failure of the yarn (see Figure 21(a)), load being then taken up by PP. Still, the stress-strain curve peak is not well described and this model is not adapted for high strain rates where a progressive loss of linearity can be observed before yarn breaking.

It has been shown [START_REF] Weibull | A statistical distribution of wide applicability[END_REF][START_REF] Chi | Determination of single fibre strength distribution from fibre bundle testings[END_REF][START_REF] Van Der Zwaag | The concept of filament strength and the weibull modulus[END_REF] that roving's tensile behavior can be described in first approximation using Weibull model by:

ߪሺ݁ሻ ൌ ݁ܧ × ݔ݁ ିቀ బ ቁ ( 14
)
Where ܮ is the roving length, ܧ is the fiber modulus and ݁ and ݉ are respectively the scale and shape parameters of the Weibull distribution. This expression is obtained under the weakest link hypothesis based on the assumption that failure of fibers is due to the presence of defects. With increasing fiber length ,ܮ the probability of presence of severe defects along the fiber increases, leading to a decrease of its breaking stress.

It can be shown that, slightly modifying this model, it is possible to reproduce the average GFPP yarn's stress-strain tensile curve. The following expression is proposed:

ߪሺ݁ሻ ≈ ߪ ఏ ሺ݁ሻ݁ݔ ିቀ బ ቁ + ߪ ቆ1 -ݔ݁ ିቀ బ ቁ ቇ (15)
Where ߪ ఏ ሺ݁ሻ is the previously identified model which describes stress-strain curve up to the linear behavior. The second part of the equation describes the fact that load is taken up by PP fibers when a GFPP yarn breaks.

Conclusion

In this paper, a simple protocol to perform yarn tensile tests is proposed and validated. Results obtained on commingled GFPP yarns highlight a nonlinear behavior. Influence of both specimen length and strain rate on the yarn's tensile behavior are investigated. A significant increase of breaking strength with increasing strain rate is observed. Young modulus of GFPP yarns is close to 30 GPa. It increases slightly with strain rate and decrease slightly with specimen length. Tests performed on yarns extracted from a 3D fabric show the influence of weaving on the stress-strain behavior of yarns: a significant decrease of breaking strength for yarns in both weft and warp direction is highlighted. Weft yarns were found to be the most damaged with also a decrease of Young modulus of 17%. The extent of nonlinear behavior is slightly reduced on yarns extracted from the fabric compared to undamaged yarns. This can be attributed to realignment of fibers within the yarns induced by cyclic tensile loadings experienced by yarns during weaving. Using experimental results and taking into account the particular structure of commingled yarns, a model is proposed to describe the GFPP yarn's tensile behavior. After identification of few parameters physically meaningful, the model allows describing accurately the full stress-strain curve. Dependency of model parameters to loading rate, specimen length and weaving damage is highlighted. Further work remains necessary to identify the internal characteristic length relative to the behavior of yarns inside the fabric and to investigate more precisely the transverse behavior of commingled yarns.

Fig. 1 .

 1 Fig. 1. Commingled yarn: a. Global structure; b. Nip detail

  the study, yarns have been used to produce a 3D fabric. The weaving architecture of the fabric is shown in Figure 2. It is composed of five layers of yarns in the weft direction and four in the warp direction. Binding yarns are vertically oriented and interlock just 3 layers of weft yarns. The weaving density of fabric is 240 yarns / 10 cm for weft yarns and 200 yarns / 10 cm for both warp yarns and binding yarns. All yarns used in the fabric are the previously described GFPP yarns.

Fig. 2 .

 2 Fig. 2. Weaving architecture of the 3D fabric (grey: weft yarns, red: warp yarns, other colors: binders)

Fig. 3 .

 3 Fig. 3. Schematic representation of a yarn tensile sample with markers for optical acquisition

Figure 7 (

 7 Figure 7(a)). Obtained images were analyzed using a Matlab program to detect the yarn

Fig. 7 .

 7 Fig. 7.a. Tensile testing of GFPP yarn with picture registration; b. Result of yarn contour detection; c. Evolution of yarn diameter with applied tensile load.

  Position along yarn length mm tensile load : 2N tensile load : 10N tensile load : 50N Quasi-static tensile behavior of commingled yarns Results presented are those obtained with the 200 mm specimen length and 2 mm/min testing speed. Figure 8(a) shows the typical quasi-static tensile behavior of a GFPP yarn. At the beginning of loading, a nonlinear behavior is observed. Yarn stiffness gradually increases and the behavior progressively becomes linear, ߝ ே value corresponds to the strain for which behavior becomes linear. Little drops of force corresponding to the opening of unstable nips can be observed (see in black ellipsis in Figure 8(a)). Nips opening might also explain the yarn rotation that can be observed during testing. The behavior remains almost linear up to the brutal break of glass fibers, then the load is taken up by the PP 21 (see Figure 8(b)).

Fig. 8 .

 8 Fig. 8. GFPP tensile behavior: a. First part of tensile behavior and effect of the opening of an unstable nip; b. Global tensile behavior.

Fig. 9 .Fig. 11 .

 911 Fig. 9. a. Scattering of GFPP yarns tensile behavior; b. Breaking stress-Young's modulus correlation; c. Size of nonlinear behavior (ߝ ே value) -Young's modulus correlation

Fig. 12 .

 12 Fig.12. Average stress-strain curves of GFPP yarns for the 3 strain rates (200 mm gauge length)

Torun 4

 4 performed tensile tests on quite similar yarns with a 500 mm specimen length and a 25 mm/min testing speed. Results show a modulus of less than 20 GPa and a breaking force of approximately 45 N. These values are quite low compared with our results. Origin of these differences could come from the different commingling process of yarns or from the different tensile testing protocol.

Fig. 13 .

 13 Fig.13. Influence of strain rate and gauge length on the breaking load of GF and GFPP yarns

  pass through a guide. This damaging can explain the decreased breaking strength and modulus observed on weft yarns.

Fig. 15 .

 15 Fig. 15. Cyclic testing of a GFPP yarn

Fig. 16 .

 16 Fig. 16. Premature breaking after multiple cyclic loadings

Fig. 17 .

 17 Fig. 17. Simplified representation of the yarn structure

Fig. 18 .Fig. 19 .

 1819 Fig. 18. Cumulative density of delayed activation

Fig. 21 .

 21 Fig. 21. Model/experimental results comparison: a. Quasi-static behavior of a GFPP yarn; b. Average tensile behavior, undamaged yarns 200 mm-2 mm/min

Table 1 .

 1 GF yarns Young's modulus calculated by the different methods

		Modulus [GPa]
	Optical 50 mm	73.8 ± 1.3
	Optical 100 mm	73.5 ± 1.8
	Optical 150 mm	73.3 ± 1.2
	Machine	72.6 ± 0.6

Table 2 .

 2 Results in all testing configurations for GF and GFPP yarns.

		Gauge length [mm]	200	500
	GF	Testing speed [mm/min] Strain rate ݏ[ ିଵ ] Young's modulus [GPa]	2 1.67 × 10 ିସ 1.67 × 10 ିଷ 1.67 × 10 ିଶ 1.67 × 10 ିସ 1.67 × 10 ିଷ 1.67 × 10 ିଶ 20 200 5 50 500 73.6 ± .ଽ 73.1 ± ଵ.

Table 3 ]

 3 Fig. 14. Average tensile behavior of damaged and undamaged GFPP yarns

	Breaking strength N	GF 200 mm GF 500 mm GFPP 200 mm
		GFPP 500 mm
	2 -3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6
	log(strain rate)

Table 3

 3

	. Mechanical parameters of damaged yarns and comparison with undamaged yarns
	values in parenthesis
	Breaking strain [%] Breaking strength [N] Breaking stress [MPa] Modulus [GPa] Binding yarns 1.24 ±.ଵହ (-16 %) 66.9 ±଼. (-15 %) 290 ±ଷ଼ (-15 %) 29.4 ±ଶ. (+ 1 %) Warp yarns 1.21 ±.ଵଷ (-17 %) 61.4 ±଼.ଵ (-22 %) 266 ±ଷହ (-22 %) 28.2 ±ଶ. (-3 %) Weft yarns 1.31 ±.ଵହ (-11 %) 56.3 ±ଶ.ଽ (-28 %) 244 ±ଽ (-28 %) 24.2 ±଼.ଷ (-17 %)

  Calculation were made using ݔ ൌ 0.1 and ܦܱ ୋ ୬୧୮ ൌ 1.5%. An average twist angle of 30° is

	obtained with Equation 8, corresponding to a surface twist angle ߙ ≈ 42°. Few experimental
	results investigating the decrease of modulus with such high angles are available, still
	extrapolating average results presented by Rao 33 , value ݂ሺߙሻ ≈ 0.3 can be estimated. Using Equation 7 a value of E ୋ ≈ 31 ܽܲܩ is obtained giving a good estimation of the
	commingled yarns modulus.

Acknowledgments

The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement n° 263223.


Didier Dubois
email: dubois@irit.fr

Emiliano Lorini
email: emiliano.lorini@irit.fr

Henri Prade
email: prade@irit.fr

Nonmonotonic desires -A possibility theory viewpoint

If an agent desires that ϕ and desires that ψ, this agent often also desires that ϕ and ψ hold at the same time (ϕ ∧ ψ). However, there are cases where ϕ ∧ ψ may be found less satisfactory for the agent than each of ϕ or ψ alone. This paper is a first attempt at modeling such nonmonotonic desires. The approach is developed in the setting of possibility theory, since it has been recently pointed out that guaranteed (or strong) possibility measures are a good candidate for modeling graded desires. Although nonmonotonic reasoning has been studied extensively for knowledge, and that preferential nonmonotonic consequence relations can be faithfully represented in the possibilistic setting, nonmonotonic desires appear to require a different approach.

Introduction

In a recent work [START_REF] Dubois | Bipolar possibility theory as a basis for a logic of desires and beliefs[END_REF], we have advocated the idea that if an agent is satisfied with ϕ ∨ψ, it should be also the case that the agent is satisfied with ϕ and that the agent is satisfied with ψ. This claim is justified if by "the agent is satisfied with ϕ" we mean: "in all situations where ϕ holds". This has led us to propose the modeling of desires by means of guaranteed possibility measures ∆ that obey axiom ∆(ϕ ∨ ψ) = min(∆(ϕ), ∆(ψ)).

This approach leads us to consider that ϕ is desirable for the agent, it is still the case in any context χ, i.e. saying that having ϕ true is satisfactory forces us to consider that having χ ∧ ϕ true is satisfactory as well, whatever χ. Even if it may be generally the case, it may happen that when an agent specifies that it has the desire to have ϕ true, it may not specify that in the abnormal / rare case where χ is true, it rather desires having ¬ϕ true. Another worth considering option is when, in the context where χ is true, it becomes indifferent which of ϕ or ¬ϕ is true. This means that although the agent generally desires to have ϕ true, there may exist some particular circumstances where this desire no longer exists, or even that an opposite desire may take place.

Such a concern sounds like a nonmonotonic reasoning issue [START_REF] Brewka | Nonmonotonic Reasoning. Essays Celebrating its 30th Anniversary[END_REF], where a set of explicit desires may express some apparent logical contradiction ("I definitely desire ϕ, but not necessarily in any circumstances", in particular "I definitely desire ¬ϕ if χ occurs"). It is not clear that nonmonotonic reasoning approaches, which have been developed for handling default knowledge in the presence of incomplete information, still apply to computing a preference rank-ordering between situations. The paper intends to discuss this situation, and to provide an approach suitable for handling non monotonic desires, a problem that has been mentioned by philosophers [21], but not apparently considered for its specific features in artificial intelligence until now.

The paper is organized as follows. We first provide a short background on the possibility theory framework and its suitability for representing desires. We then recall how reasoning with default knowledge in the presence of incomplete information can be properly handled in the possibilistic reasoning setting. Then we examine the problems arising in the treatment of nonmonotonic desires, propose and discuss an approach.

Background on possibility theory and the modeling of desires

Let π be a mapping from a set of worlds W to [0, 1] that rank-orders them. Note that this encompasses the particular case where π reduces to the characteristic function of a subset E ⊆ W . The possibility distribution π may represent a plausibility ordering (E the set of situations considered not impossible) when modeling epistemic uncertainty, or a preference ordering (E is then the subset of satisfactory worlds) when modeling preferences. Let us recall the complete system of the 4 set functions underlying possibility theory [START_REF] Dubois | Possibility theory: qualitative and quantitative aspects[END_REF] and their characteristic properties:

i) The (weak) possibility measure (or potential possibility) Π(A) = max w∈A π(w) evaluates to what extent there is a world in A that is possible. When π reduces to E, ). ∆, ∇ are decreasing set functions, while the (weak) possibility and (strong) necessity measures are increasing. A modal logic counterpart of these 4 modalities has been proposed in the binary-valued case (things are possible or impossible) [START_REF] Dubois | Knowledge-driven versus data-driven logics[END_REF]. There is a close link between Spohn functions and (weak) possibility / (strong) necessity measures [START_REF] Dubois | Epistemic entrenchment and possibilistic logic[END_REF].

Π(A) = 1 if A ∩ E = ∅,

Possibility theory as basis for a logical theory of desires

The possibility and necessity operators Π and N have a clear epistemic meaning both in the frameworks of possibility theory, and of Spohn's uncertainty theory [START_REF] Spohn | Ordinal conditional functions: a dynamic theory of epistemic states[END_REF] (also referred to as 'κ calculus', or as 'rank-based system' and 'qualitative probabilities'). Differently from the operators Π and N , the operators ∆ and ∇ are less employed to model epistemic attitudes. In fact while the epistemic use of N models the idea of "knowing at least", the one of ∆ accounts for the idea of "knowing at most". Combining both modalities provides a representation of the idea of "only knowing" [START_REF] Banerjee | A simple logic for reasoning about incomplete knowledge[END_REF][START_REF] Dubois | Reasoning about uncertainty and explicit ignorance in generalized possibilistic logic[END_REF]. In epistemic logic, the notion of strong belief put forward by Battigalli and Siniscalchi [START_REF] Battigalli | Strong belief and forward induction reasoning[END_REF], corresponds to the inequality ∆(A) > Π(A) (which provably implies N (A) > N (A) = 0).

Here we advocate the idea that ∆ and ∇ can be viewed as operators modeling motivational mental attitudes such as goals or desires. 1 In particular, we claim that ∆ can be used to model the notion of desire, whereas ∇ can be used to model the notion of potential desire. 2 According to the philosophical theory of motivation based on Hume [START_REF] Hume | A Treatise of Human Nature[END_REF], a desire can be conceived as an agent's motivational attitude which consists in an anticipatory mental representation of a pleasant (or desirable) state of affairs (representational dimension of desires) that motivates the agent to achieve it (motivational dimension of desires). In this perspective, the motivational dimension of an agent's desire is realized through its representational dimension. For example when an agent desires to be at the Japanese restaurant eating sushi, he imagines himself eating sushi at the Japanese restaurant and this representation gives him pleasure. This pleasant representation motivates him to go to the Japanese restaurant in order to eat sushi.

Intuitively speaking, with the term potential desire, we refer to a weaker form of motivational attitude. We assume that an agent considers a given property ϕ potentially desirable if ϕ does not conflict with the agent's current desires. In this sense, ϕ is potentially desirable if it is not incompatible with the agent's current desires. Following ideas presented in [START_REF] Dubois | Bipolar possibility theory as a basis for a logic of desires and beliefs[END_REF], let us explain why the operator ∆ is a good candidate for modeling the concept of desire and why ∇ is a good candidate for modeling the idea of potential desire.

Mental states

We define an agent's mental state as a tuple M = (E, D) where:

-E ⊆ W is a non-empty subset of the set of all worlds, and -D ⊂ W is a proper subset of the set of all worlds.

The set E defines the set of worlds not ruled out by the agent (i.e., the maximal set of worlds that the agent considers possible), whereas D is the set of desirable worlds for the agent. Let M denote the set of all mental states. We here assume for every mental state M there exists a world with a minimal degree of desirability 0 corresponding to indifference (this is why D = W ). This type of normality constraint for guaranteed possibility distributions is usually assumed in possibility theory. More generally, a graded mental state is a pair M = (π, δ) where:

π : W → L is a normal possibility distribution over the set of all worlds, and δ : W → L is a function mapping every world w to its desirability (or pleasantness) degree in L, with δ(w) = 0 for some w ∈ W .

-L is a bounded chain acting as a qualitative scale for possibility and desirability, that make these notions commensurate. Let us stress the point that while δ(w) = 1 expresses complete desirability, δ(w) = 0 expresses indifference, rather than repulsion. The condition δ(w) = 0 for some w ∈ W accounts for the claim that desire presupposes not everything is desired.

Modeling desire using ∆ function We here assume that in order to determine how much a proposition ϕ is desirable an agent takes into consideration the worst situation in which ϕ is true. In other words, ∆(||ϕ||) = α means that the agent desires all situations where ϕ holds to at least level α. Thus, denoting by ||ϕ|| the set of situations where ϕ is true, for all graded mental states M = (π, δ) and for all propositions ϕ, we can interpret ∆(||ϕ||) = min u∈||ϕ|| δ(u) as the extent to which the agent desires ϕ to be true. Let us justify the following two properties for desires:

∆(||ϕ ∨ ψ||) = min(∆(||ϕ||), ∆(||ψ||)) and ∆(||ϕ ∧ ψ||) ≥ max(∆(||ϕ||), ∆(||ψ||)).
According to the first property, an agent desires ϕ to be true with a given strength α and desires ψ to be true with a given strength β if and only if the agent desires ϕ ∨ ψ to be true with strength equal to min(α, β). Note that the copula "and" in "desiring ϕ and desiring ψ" corresponds to performing the disjunction of the propositions, as we must perform the set-union of all situations where ϕ is true and all those where ψ is true (these propositions are not viewed as events referring to a single real world, but as collections of situations). Clearly, in the case of representing uncertainty, this property would not make any sense because the plausibility of ϕ ∨ ψ should be clearly at least equal to the maximum of the plausibilities of ϕ and ψ. For the notion of desires, it seems intuitively satisfactory to have the opposite, namely the level of desire of ϕ ∨ ψ should be at most equal to the minimum of the desire levels of ϕ and ψ. Indeed, we only deal with here with "positive"3 desires (i.e., desires to reach something with a given strength). Under this proviso, the level of desire of ϕ ∧ ψ cannot be less than the maximum of the levels of desire of ϕ and ψ. Here ϕ ∧ ψ does not refer to the disjunction "desiring ϕ or desiring ψ", but to the idea of desiring ϕ and ψ simultaneously. According to the second property, the joint occurrence of two desired events ϕ and ψ is more desirable than the single occurrence of one of the two events. This is the reason why in the right side of the equality we have the max. The latter property does not make any sense in the case of epistemic attitudes like beliefs, as the joint occurrence of two events ϕ and ψ is epistemically less plausible than the occurrence of a single event. On the contrary the opposite inequality makes perfect sense for motivational attitudes like desires.

By way of example, suppose Peter wishes to go to the cinema in the evening with strength α (i.e., ∆(||goToCinema||) = α) and, at the same time, he wishes to spend the evening with his friend with strength β (i.e., ∆(||stayWithFriend ||) = β). Then, according to the preceding property, Peter wishes to to go the cinema with his friend with strength at least max{α, β} (i.e., ∆(||goToCinema ∧stayWithFriend ||) ≥ max{α, β}). This is a reasonable conclusion because the situation in which Peter achieves his two desires is (for Peter) at least as pleasant as the situation in which he achieves only one desire. A similar intuition can be found in [START_REF] Casali | A graded BDI agent model to represent and reason about preferences[END_REF] about the min-decomposability of disjunctive desires, where however it is emphasized that it corresponds to a pessimistic view.

From the normality constraint of δ, we can deduce the following inference rule:

Proposition 1. For every M ∈ M, if ∆(||ϕ||) > 0 then ∆(||¬ϕ||) = 0.
This means that if an agent desires ϕ to be true -i.e., with some strength α > 0 -then he does not desire ϕ to be false. In other words, an agent's desires must be consistent.

The operator ∆ satisfies the following additional property:

Proposition 2. For every M ∈ M, if ||ϕ|| = ∅ then ∆(||ϕ||) = 1.
i.e. in the absence of actual situations where ϕ is true, the property ϕ is desirable by default.

Modeling potential desire using ∇ As pointed out above, we claim that the operator ∇ allows us to capture a concept of potential desire (or non-incompatibility with desire): ∇(||ϕ||) represents the extent to which an agent considers ϕ a potentially desirable property or, alternatively, the extent to which the property ϕ is not incompatible with the agent's desires. An interesting situation is when the property ϕ is maximally potentially desirable for the agent (i.e., ∇(||ϕ||) = 1). This is the same thing as saying that the agent does not desire ϕ to be false (i.e., ∆(||¬ϕ||) = 0). Intuitively, this means that ϕ is totally potentially desirable in as much as the level of desire for ¬ϕ is 0. In particular, given a graded mental state M = (π, δ), let D = {w ∈ W : δ(w) > 0} be the set of somewhat satisfactory or desirable worlds in M . Then, we have ∇(||ϕ||) = 1 if and only if D ∩ ||¬ϕ|| = ∅, i.e., ¬ϕ is consistent with what is not desirable, represented by the set D.

Another interesting situation is when the property ϕ is maximally desirable for the agent (i.e., ∆(||ϕ||) = 1). This is the same thing as saying that ¬ϕ is not at all potentially desirable for the agent (i.e., ∇(||¬ϕ||) = 0). It is worth noting that if an agent desires ϕ to be true, then ϕ should be maximally potentially desirable. This property is expressed by the following valid inference rule which follows straightforwardly from the previous one and from the definition of ∇(||ϕ||) as 1 -∆(||¬ϕ||):

Proposition 3. For every M , if ∆(||ϕ||) > 0 then ∇(||ϕ||) = 1.
Let us now consider the case in which the agent does not desire ϕ (i.e., ∆(||ϕ||) = 0). In this case two different situations are possible: either ∆(||¬ϕ||) = 0 and ϕ is fully compatible with the agent's desires (i.e., ∇(||ϕ||) = 1), or ∆(||¬ϕ||) > 0 and then ϕ is not fully compatible with the agent's desires (i.e., ∇(||ϕ||) < 1).

Some valid inference rules for desires

The following is a valid inference rule for ∆-based logic, see [START_REF] Dubois | Knowledge-driven versus data-driven logics[END_REF] for the proof:

Proposition 4. For every M ∈ M, if ∆(||ϕ ∧ ψ||) ≥ α and ∆(||¬ϕ ∧ χ||) ≥ β then ∆(||ψ ∧ χ||) ≥ min(α, β).
Therefore, if we interpret ∆ as a desire operator, we have that if an agent desires ϕ∧ψ with strength at least α and desires ¬ϕ∧χ with strength at least β, then he desires ψ ∧ χ with strength at least min(α, β). This seems a reasonable property of desires. By way of example, suppose Peter is planning what to do in the weekend. He has two concomitant desires. On the one hand, Peter desires to go to the contemporary art museum on saturday afternoon and to have dinner at a japanese restaurant on saturday evening with strength at least α. On the other hand, Peter desires not to go the contemporary art museum on saturday afternoon but to go to the sea on sunday morning with strength at least β. Then, it is reasonable to conclude that Peter desires to have dinner at a japanese restaurant on saturday evening and to go to the sea on sunday morning with strength at least min(α, β).

Nonmonotonic reasoning with incomplete knowledge

Nonmonotonic reasoning has been extensively studied in AI in relation with the problem of reasoning with rules having exceptions under incomplete information [START_REF] Group | Reasoning under Incomplete Information in Artificial Intelligence: A Comparison of Formalisms Using a Single Example[END_REF], or for dealing with the frame problem in dynamic worlds [START_REF] Brewka | Nonmonotonic Reasoning. Essays Celebrating its 30th Anniversary[END_REF]. In the following, we recall the possibilistic approach [START_REF] Benferhat | Practical handling of exception-tainted rules and independence information in possibilistic logic[END_REF], which has been proved [START_REF] Benferhat | Nonmonotonic reasoning, conditional objects and possibility theory[END_REF] to provide a faithful representation of the postulate-based approach proposed by Kraus, Lehmann and Magidor [START_REF] Kraus | Nonmonotonic reasoning, preferential models and cumulative logics[END_REF], and completed in [START_REF] Lehmann | What does a conditional knowledge base entail?[END_REF].

A default rule "if ϕ then ψ, generally", denoted by ϕ ❀ ψ, is then understood formally as the constraint

Π(ϕ ∧ ψ) > Π(ϕ ∧ ¬ψ) (1)
on a possibility measure Π describing the semantics of the available knowledge. It expresses that in the context where ϕ is true, there exist situations where having ψ true is strictly more likely than any situation where ψ is false in the same context. This constraint can be shown to be equivalent to N (ψ | ϕ) > 0, when Π(ψ | ϕ) is defined as the greatest solution of the min-based equation

Π(ϕ ∧ ψ) = min(Π(ψ | ϕ), Π(ϕ)) (2)
and the duality N (ψ | ϕ) = 1 -Π(¬ψ | ϕ) holds. There also exists a product-based definition instead of (2). But only the min-based conditioning is used in the following. Let us consider the following classical example with default rules d1: "birds fly", d2: "penguins do not fly", d3: "penguins are birds", symbolically written

d1 : b ❀ f ; d2 : p ❀ ¬f ; d3 : p ❀ b.
The set of three defaults is thus represented by the following set C of constraints:

b ∧ f > Π b ∧ ¬f ; p ∧ ¬f > Π p ∧ f ; p ∧ b > Π p ∧ ¬b.
Let Ω be the finite set of interpretations of the considered propositional language, generated by b, f, p in the example. If this language is made of the literals ϕ 1 , . . . , p n , these interpretations correspond to the possible worlds (i.e., the completely described situations) where the conjunctions * p 1 ∧ . . . ∧ * p n are true, where * stands for the presence of the negation sign ¬ or its absence. In our example, Ω = {ω 0 : ¬b∧¬f ∧¬p,

ω 1 : ¬b ∧ ¬f ∧ p, ω 2 : ¬b ∧ f ∧ ¬p, ω 3 : ¬b ∧ f ∧ p, ω 4 : b ∧ ¬f ∧ ¬p, ω 5 : b ∧ ¬f ∧ p, ω 6 : b ∧ f ∧ ¬p, ω 7 : b ∧ f ∧ p}.
Any interpretation ω thus corresponds to a particular proposition. A possibility distribution π is a mapping from Ω to [0, 1]. It induces a ranking of Ω according to the level of normality of each situation. Let > Π denote a ranking of Ω, such that ω > Π ω ′ iff π(ω) > π(ω ′ ) on Ω. π is indeed the restriction to propositions describing complete situations of a possibility measure Π defined by Π(ϕ) = max ω|=ϕ π(ω), where ω |= ϕ means ω is an interpretation which makes ϕ true. For instance Π(b ∧ f ) = max(π(ω 6 ), π(ω 7 )).

Then the set of constraints C on interpretations is:

C1 : max(π(ω 6 ), π(ω 7 )) > max(π(ω 4 ), π(ω 5 )), C2 : max(π(ω 5 ), π(ω 1 )) > max(π(ω 3 ), π(ω 7 )), C3 : max(π(ω 5 ), π(ω 7 )) > max(π(ω 1 ), π(ω 3 )).
Any finite consistent set of constraints of the form ϕ i ∧ψ i > Π p i ∧¬ψ i , representing a set of defaults ϕ i ❀ ψ i induces a partially defined ranking > Π on Ω, that can be completed according to the principle of minimal specificity, e.g. [START_REF] Benferhat | Nonmonotonic reasoning, conditional objects and possibility theory[END_REF]. This principle assigns to each world ω the highest possibility level (in forming a well-ordered partition of Ω) without violating the constraints. This defines a unique > Π .

The well ordered partition of Ω which is obtained in the example is

{ω 0 , ω 2 , ω 6 } > Π {ω 4 , ω 5 } > Π {ω 1 , ω 3 , ω 7 }.
Let E 1 , . . . , E m be the obtained partition in the general case. A numerical counterpart to > Π can be defined by π(ω) = m+1-i m if ω ∈ E i , i = 1, . . . , m. In our example we have m = 3 and π(ω 0 ) = π(ω 2 ) = π(ω 6 ) = 1; π(ω 4 ) = π(ω 5 ) = 2/3; π(ω 1 ) = π(ω 3 ) = π(ω 7 ) = 1/3. Note that it is purely a matter of convenience to use a numerical scale, and any other numerical counterpart such that π(ω) > π(ω ′ ) iff ω > Π ω ′ will work as well. Namely the range of π is used as an ordinal scale.

From this possibility distribution π, we can compute for any proposition ϕ its necessity degree N (ϕ). For instance,

N (¬p ∨ ¬f ) = min{1 -π(ω)|ω |= p ∧ f } = min(1 -π(ω 3 ), 1 -π(ω 7 )) = 2/3, while N (¬b ∨ f ) = min{1 -π(ω) | ω |= b ∧ ¬f } = min(1 -π(ω 4 ), 1 -π(ω 5 )) = 1/3 and N (¬p ∨ b) = min(1 -π(ω 1 ), 1 -π(ω 3 )) = 2/3.
The default rule-base can then be encoded in possibilistic logic [START_REF] Dubois | Possibilistic logic[END_REF]. The method consists in turning each default ϕ i ❀ ψ i into a possibilistic clause (¬ϕ i ∨ ψ i , N (¬ϕ i ∨ ψ i )), where N is computed from the greatest possibility distribution π induced by the set of constraints corresponding to the default knowledge base, as already explained. Then we apply the possibilistic inference machinery for reasoning with the defaults together with the available factual knowledge. In our example, we obtain the possibilistic logic base K = {(¬p ∨ ¬f, 2/3), (¬p ∨ b, 2/3), (¬b ∨ f, 1/3)}. This encodes the generic knowledge embedded in the default rules. Suppose that all we know about the factual situation under consideration is that "Tweety" is a bird, which is encoded by (b, 1). Then we apply the possibilistic logic resolution rule [START_REF] Dubois | Possibilistic logic[END_REF]

(¬a ∨ b, α), (a ∨ c, β)⊢(b ∨ c, min(α, β)).
Then, we can check that K ∪{(b, 1)}⊢(f, 1/3), i.e., we conclude that if all we know about "Tweety" is that it is a bird, then it flies. If we are said that "Tweety" is in fact a penguin, encoded by (p, 1), then K ∪ {(b, 1)} ∪ {(p, 1)}⊢( ⊥, 1/3), which means that K augmented with the available factual information is now inconsistent (at level 1/3).

However, the conclusions which can be obtained with a certainty level strictly greater than the level of inconsistency are safe (the level of inconsistency of a possibilistic logic base is the greatest weight with which ⊥ can be derived from the base, applying the resolution rule repeatedly). Namely, here, we have K ∪ {(b, 1)} ∪ {(p, 1)}⊢(¬f, 2/3). Thus, knowing that "Tweety" is a penguin, we now conclude that it does not fly (since 2/3 > level of inconsistency(K ∪ {(b, 1)} ∪ {(p, 1)}) = 1/3). Roughly speaking, the most specific rules w.r.t. a given context remain above the level of inconsistency.

Nonmonotonic desires

As recalled in Section 2, expressing that ϕ true is desired can be arguably represented by ∆(ϕ) > 0. Still ∆(ϕ) > 0 means that any situation where ϕ is true is somewhat desirable, which may appear quite strong, since even if ϕ true is indeed desired in general, there may exist quite particular situations where this is no longer the case. Thus, one may need to make the modeling nonmonotonic.

This raises the question of defining conditioning for the ∆ function, an issue that has been only briefly discussed in [START_REF] Benferhat | Bipolar possibilistic representations[END_REF].

Conditional desires Conditioning can be also defined for guaranteed possibility measures. Since they are decreasing, it should work in a reversed way w.r.t. Π. Namely, conditioning obeys the following equation

∆(ϕ ∧ ψ) = max(∆(ψ|ϕ), ∆(ϕ)).
Since ∆(ϕ) = min(∆(ϕ ∧ ψ), ∆(ϕ ∧ ¬ψ)), it expresses that either the minimal desirability level over the models of ϕ is reached on ϕ ∧ ψ, so, ¬ψ is preferred in the context ϕ and ∆(ψ|ϕ) should be small enough (e.g. 0, for normalisation purposes), or that ∆(ϕ) = ∆(ϕ ∧ ¬ψ) < ∆(ϕ ∧ ψ) and then ∆(ψ|ϕ) should be equal to ∆(ϕ ∧ ψ).

Thus, we have

∆(ψ|ϕ) = ∆(ϕ ∧ ψ) if ∆(ϕ) < ∆(ψ ∧ ϕ) = 0 if ∆(ϕ ∧ ¬ψ) ≥ ∆(ϕ ∧ ψ) = ∆(ϕ)
It means that ϕ and ψ are simultaneously desired because ϕ is so, and in context ϕ, ψ is desired as well.

It can be checked that if ∆(ψ|ϕ) = 0 then ∆(ϕ ∧ ψ) = max(∆(ϕ), ∆(ψ)), i.e., it is the case when the agent desires ϕ and ψ simultaneously at least as much as ϕ or ψ separately.

Moreover, as recalled in the previous section, we have

N (ψ|ϕ) > 0 iff Π(ϕ ∧ ψ) > Π(ϕ ∧ ¬ψ) iff N (ϕ → ψ) > N (ϕ → ¬ψ),
where → denotes material implication. It expresses that ψ is somewhat certain in context ϕ iff ϕ ∧ ψ is strictly more plausible than ϕ ∧ ¬ψ. An analogous relation holds for guaranteed possibility:

∆(ψ|ϕ) > 0 iff ∆(ϕ ∧ ψ) > ∆(ϕ ∧ ¬ψ) iff ∇(ϕ → ψ) > ∇(ϕ → ¬ψ).
This points out the fact that the definition of conditioning enforces the following normalization condition: min(∆(ψ|ϕ), ∆(¬ψ|ϕ)) = 0. It means that everything (i.e. ψ and ¬ψ) cannot be simultaneously desired in any context, and the fact that ψ is desired in context ϕ means that it is more desired than ¬ψ in this context. Nonmonotonic constraints: an example We are now in a position to investigate nonmonotonic constraints expressed by means of ∆ functions. Consider the example in the abstract, where ϕ and ψ are supposedly logically independent. Assume we have several apparently conflicting pieces of desire ϕ and ψ are separately desired (rather than ¬ϕ, ¬ψ), which translates into ∆(ϕ) > ∆(¬ϕ) and ∆(ψ) > ∆(¬ψ);

-in context ψ, ¬ϕ is desired rather than ϕ, and likewise for context ϕ, which translates into ∆(¬ϕ ∧ ψ) > ∆(ϕ ∧ ψ); ∆(ϕ ∧ ¬ψ) > ∆(ϕ ∧ ψ).

Letting ∆(ϕ ∧ ψ) = x, ∆(ϕ ∧ ¬ψ) = y, min(∆(¬ϕ ∧ ψ) = z, ∆(¬ϕ ∧ ¬ψ) = t, and since ∆(ϕ) = min(∆(ϕ ∧ ψ), ∆(ϕ ∧ ¬ψ)), we get the system of constraints min(x, y) > min(z, t); min(x, z) > min(y, t); z > x.

y > x

This is equivalent to z > x > t and y > x > t. We can reason assuming that a situation is never desired beyond what it is explicitly claimed (this is also called the maximal specificity principle), we obtain desirability values z = y > x > t = 0, i.e.,

∆(¬ϕ

∧ ψ) = ∆(ϕ ∧ ¬ψ) > ∆(ϕ ∧ ψ) > ∆(¬ϕ ∧ ¬ψ) = 0.
As can be seen, we solve the apparent conflict resulting from desiring ϕ and ψ separately but not at the same time, by giving the highest desirability to ¬ϕ ∧ ψ and ϕ ∧ ¬ψ, while preserving the ones of ϕ and ψ; interestingly, it implies that ¬ϕ ∧ ¬ψ is not desired at all, while ϕ ∧ ψ remains slightly desirable (otherwise, it would contradict the claims that ϕ is desirable on the one hand and ψ too on the other hand).

Inferring what is desired Given a set of default desires modelled by

D = {∆ d (ϕ i ∧ ψ i ) > ∆ d (ϕ i ∧ ¬ψ i ) | i = 1,
m} and a formula χ describing a context of interest, the problem is to infer what is desirable in this context.

Note that the input information is here of two distinct kinds: i) conditional desires encoded in terms of a guaranteed possibility measure ∆ d , and ii) a given context described by the set of situations where a certain formula χ is true.

The inference procedure is then the following:

Step 1. Compute the maximally specific possibility distribution δ d , solution of the set of constraints D by introducing the minimal number of distinct levels necessary for satisfying the set of constraints. If the set of constraints is consistent, this qualitative distribution δ d exists and is unique. Compute the corresponding ∆ d (ϕ i ∧ ψ i ) associated with δ d , and encode them as a set T of possibilistic ∆-formulas

[ϕ i ∧ ψ i , ∆ d (ϕ i ∧ ψ i )] [4].
Step 2. Move the variables pertaining to the description of the context of each rule to the weight part of the possibilistic ∆-formulas, thus defining T * . Namely,

[ϕ i ∧ ψ i , α i ] semantically entails [ψ i , min(v(ϕ i ), α i )] where v(ϕ i ) = 1 is true, and v(ϕ i ) = 0 otherwise [4].
Step 3. Project T * on the context χ of interest. Namely if χ p i , then v(ϕ i ) is set to 1, otherwise it is set to 0. We thus obtain T * χ .

Concluding remarks

In this note we have outlined an approach for handling the nonmonotonic behavior of desires, a problem which has apparently not be much considered in AI. Starting with the fact that desires can be appropriately described by means of guaranteed possibility measures in the sense of possibility theory, and keeping the lesson of the encoding of knowledge-oriented nonmonotonic reasoning in the setting of possibility theory, we have proposed a possibilistic approach to the nonmonotonic handling of desires. This approach, based on the notion of conditional guaranteed possibility measures, somewhat parallels the nonmonotonic handling of default knowledge under incomplete information, but with some noticeable differences due to the decreasingness of guaranteed possibility measures wrt entailment, and the need of distinguishing between desires and contexts of interest in which we consider them.

Although the outlined approach is promising, it remains preliminary in many respects. Wee still need a formal proof that the deduction method in possibilistic ∆-based logic outlined above does compute then inference of ∆(ϕ|χ) > 0 from the conditional desire base. A postulate-based approach to reasoning about conditional desires in a way that would parallel Kraus, Lehmann and Magidor postulates for reasoning with default knowledge, is a natural objective as well. The nonmonotonic handling of graded desires might also parallel the approach used in [START_REF] De Saint-Cyr | Handling uncertainty and defeasibility in a possibilistic logic setting[END_REF] for dealing with more or less certain pieces of default knowledge.

  which expresses the consistency of the event A with E, and Π(A) = 0 otherwise. Possibility measures are characterized by the following decomposability property:Π(A ∪ B) = max(Π(A), Π(B)).ii) The dual (strong or or actual) necessity measureN (A) = min w ∈A 1π(w) = 1 -Π(A)evaluates to what extent it is certain (necessarily true) that all possible worlds are in A. When π reduces to E, N (A) = 1 if E ⊆ A, which expresses that E entails event A (when E represents evidence), and N (A) = 0 otherwise. The duality of N w. r. t. Π expresses that A is all the more certain as the opposite event A is impossible. Necessity measures are characterized by the following decomposability property: N (A ∩ B) = min(N (A), N (B)). iii) The strong (or actual, or "guaranteed") possibility measure ∆(A) = min w∈A π(w) evaluates to what extent all situations in A are possible. When π reduces to E, ∆(A) = 1 if A ⊆ E, and ∆(A) = 0 otherwise. Strong possibility measures are characterized by the following property: ∆(A ∪ B) = min(∆(A), ∆(B)). iv) The dual (weak) (or potential) necessity measure ∇(A) = max w ∈A 1π(w) = 1 -∆(A) evaluates to what extent there is a situation outside A that is impossible. When π reduces to E, ∇(A) = 1 if A∪E = U , and ∇(A) = 0 otherwise. Weak necessity measures are characterized by property: ∇(A ∩ B) = max(∇(A), ∇(B)

Step 4 .

 4 Compute the level of ∆-inconsistency inc(T * χ ) of T * χ , where inc(T * χ ) = max{α|T * χ [⊤, α]}. For doing it, use the cut rule associated with ∆-formulas [14], namely [ϕ ∧ ψ, α], [¬ϕ ∧ φ, β] [ψ ∧ φ, min(α, β)] As a consequence [ϕ, α], [¬ϕ, β] [⊤, min(α, β)].Step 5. Infer the desires [ψ i , α i ] such as α i > inc(T * χ ).Example:D = {∆ d (ϕ) > ∆ d (¬ϕ), ∆ d (ψ ∧ ¬ϕ) > ∆ d (ψ ∧ ϕ)} We get T = {[ϕ, β], [ψ ∧ ¬ϕ, α]}, with α > β. Then T * = {[ϕ, β], [¬ϕ, min(v(ψ), α)]} Assume χ ≡ ⊤, one can only infer [ϕ, β], since T * ⊤ = {[ϕ, β]}, (indeed [¬ϕ, 0], equivalent to ∆ d ≥ 0, is a trivial formula which can be deleted, since v(ψ) is set to 0). Now suppose χ ≡ ψ, then v(ψ) = 1 and T * ψ = {[ϕ, β], [¬ϕ, α]}. Since T * ψ [⊤, min(α, β)] = [⊤, β], only [¬ϕ, α]is safe from inconsistency, as α > inc(T * ψ ) = β. We thus get the expected nonmonotonic behavior.

We use the term 'motivational' mental attitude (e.g., a desire, a goal or an intention) in order to distinguish it from an 'epistemic' mental attitude such as knowledge or belief.

Here, the word potential does not refer to the idea that ϕ would be desired by the agent as a consequence of his mental state, but the agent has not enough deductive power to become aware of it. It is more the idea that the agent has no reason not to desire ϕ. Another possible term is desire admissibility or desire compatibility.

The distinction between positive and negative desires is a classical one in psychology. Negative desires correspond to state of affairs the agent wants to avoid with a given strength, and then desires the opposite to be true. However, we do not develop this bipolar view here.