Frédérique Trivaudey

Vincent Placet
email: vincent.placet@univ-fcomte.fr

Violaine Guicheret-Retel

M Lamine Boubakar

Nonlinear tensile behaviour of elementary hemp fibres. Part II: Modelling using an anisotropic viscoelastic constitutive law in a material rotating frame

Keywords: Hemp fibres, B. Mechanical properties, C. Computational Modelling, C. Micro-mechanics elastic behaviour MFA: cellulose microfibrils angle MFR: cellulose microfibrils reorientation VE: viscoelastic behavior Cauchy true stress tensor 𝜀 : total strain tensor 𝜀 𝑒 : elastic strain tensor 𝜀 𝑣𝑒 : viscoelastic strain tensor 𝑆: elastic compliance tensor 𝑆 𝑣𝑒 : viscoelastic compliance tensor Elastic parameters

A 3D viscoelastic model is presented, based on the use of finite element method for the study of the nonlinear tensile behaviour of hemp fibres. On the basis of an experimental investigation, Part I of this study (Placet, Cisse and Boubakar, Composites Part A 56:319-327, 2014) proposed a scenario based on several physical mechanisms, in order to explain the nonlinear behaviour of such fibres. These mechanisms included viscoelastic strain, cellulose microfibril reorientation, and shear strain-induced crystallisation of the amorphous paracrystalline components. The second part of this ongoing study proposes to implement such mechanisms and the associated constitutive laws in a simplified 3D model, in order to evaluate the contribution of each mechanism to the macroscopic tensile behaviour of the fibre.

The results show that the proposed anisotropic viscoelastic constitutive law, describing finite transformations through a material rotating frame formulation, is able to accurately simulate the shape of the experimentally observed tensile curves. This model is also used to investigate the influence of dislocation areas on the tensile behaviour and stress fields.

INTRODUCTION

Cellulose fibres from wood and plants represent a promising class of reinforcement materials, for use in high-performance polymer matrix composites. Because natural fibres are characterised by the strong variability of their mechanical properties and their high sensitivity to temperature and humidity, the design of reliable structure-based composites containing such natural fibres is a significant challenge for engineers, who are accustomed to using the consistent and accurate data available in the case of man-made fibres. These last ones are inherently produced in highlycontrolled conditions, unlike natural fibres.

In addition to their highly scattered mechanical properties, numerous fibres are characterised by nonlinear tensile behaviour. The understanding of this particular behaviour is of great importance, in view of the development of composites reinforced with plant fibres, since the behaviour of these fibres could have an impact on their performance when they are used to reinforce composite materials. Several authors have recently observed nonlinear behaviour in unidirectional composites reinforced with flax fibres [START_REF] Liang | A comparative study of fatigue behaviour of flax/epoxy and glass/epoxy composites[END_REF][START_REF] Shah | Can flax replace E-glass in structural composites? A small wind turbine blade case study[END_REF][START_REF] Shah | Determining the minimum, critical and maximum fibre content for twisted yarn reinforced plant fibre composites[END_REF][START_REF] Shah | The tensile behavior of off-axis loaded plant fiber composites: An insight on the nonlinear stress-strain response[END_REF][START_REF] Baets | Determination of the optimal flax fibre preparation for use in UD flax-epoxy composites[END_REF]. Some authors have also mentioned an increase in stiffness under cyclic loading, a behaviour which has been observed at the scale of individual fibres [START_REF] Baley | Analysis of the flax fibres tensile behaviour and analysis of the tensile stiffness increase[END_REF].

Many hypotheses have been proposed in the literature to explain the nonlinear tensile behaviour of isolated wood or plant fibres, and the fibres' stiffness recovery or improvement after loading beyond the yield point. These hypotheses were discussed in Part I of this study [START_REF] Placet | Nonlinear tensile behaviour of elementary hemp fibres. Part I: Investigation of the possible origins using repeated progressive loading with in situ microscopic observations[END_REF]. In the case of wood tracheids, Page et al. [START_REF] Page | Behaviour of Single Wood Fibres under Axial Tensile Strain[END_REF] confirmed a relationship between the nonlinear shape of the tensile curve and the onset of cell wall buckling. They clearly showed that the yield point (i 1 ) corresponds to the onset of wall buckling. More recently, Eder et al. [START_REF] Eder | The fracture behaviour of single wood fibres is governed by geometrical constraints: in situ ESEM studies on three fibre types[END_REF] also showed that thick-walled fibres are highly resistant to tension buckling, and that this phenomenon can therefore explain the nonlinearity of the stress-strain curve only, in the case of a thin-walled fibre. Since hemp fibres generally have a thick wall, tension buckling therefore does not explain the nonlinearity of the stress-strain curve in the case of such fibres. Other authors have also attributed nonlinear behaviour to irreversible reorientations of the cellulose microfibrils with respect to the fibre axis in particular in the S 2 layer, when they are submitted to axial loading. S 2 is the main sub-layer of the cell wall in terms of its thickness and therefore contributes mainly to the global mechanical behaviour of the fibre. A linear fit was established between the microfibril angle (MFA) and strain for coir fibres, by Martinschitz et al. [START_REF] Martinschitz | Changes in microfibril angle in cyclically deformed dry coir fibers studied by in-situ synchrotron X-ray diffraction[END_REF]. Other authors have attributed this nonlinear behaviour to shear deformations in the noncrystalline region, which can partially damage the cell wall, or provoke a stick-slip phenomenon [START_REF] Keckes | Cell-wall recovery after irreversible deformation of wood[END_REF]. This stick-slip mechanism proposed by Keckes et al. (11) operates like a 'molecular Velcro'.

According to these authors, beyond the yield point (i 1 ) the shear stress could provoke a viscous flow of the matrix. When the stress is released there would be no back-flow of the matrix, but a lock-in phenomenon associated with immediate bond re-formation in the fibrils' new position. This could explain irreversible strain without significant damage. This model supposes the presence of noncovalent interactions between hemicellulose chains attached to adjacent cellulose microfibrils.

Altaner and Jarvis ( 12) proposed an alternative model in which hemicellulose chains bridging continuously from one microfibril aggregate to the next. However, the mechanism is different; the computed tensile curves are similar to those of the original model.

Nonlinear tensile behaviour was modelled by Nilsson and Gustafsson (13) for hemp fibres, by introducing defects into the helical structure of the cellulose microfibrils and assuming that the hemicellulose has an elasto-plastic constitutive relation. Navi and Sedighi-Gilani [START_REF] Sedighi-Gilani | A micromechanical approach to the behavior of single wood fibers and wood fracture at cellular level[END_REF][START_REF] Sedighi-Gilani | Experimental observations and micromechanical modeling of successivedamaging phenomenon in wood cells' tensile behavior[END_REF] proposed a model for wood fibres with an elasto-plastic behaviour for amorphous polymers, based on the assumption of a helical, non-uniform distribution of cellulose microfibrils in the fibre and damage to the amorphous constituents after yielding. The origin of the dissipative behaviour of elementary fibres is thus attributed to plasticity by these authors.

In the Part I of this study, using experimental investigations we proposed a scenario centred on several physical and microstructural mechanisms, in order to explain the nonlinear behaviour (type 3) of such fibres, including viscoelastic strain, cellulose microfibril reorientation and shear straininduced crystallisation of the amorphous cellulose. In the present, second part of the study, we propose to implement such mechanisms, together with the associated constitutive laws, into a 3D model which can be used to evaluate the contribution of each physical mechanism to the macroscopic tensile behaviour of type 3of the fibre.

MODELING 3D model

The macroscopic simplified 3D model used for this study is based on a previously developed model, described in detail in [START_REF] Placet | Diameter dependence of the apparent tensile modulus of hemp fibres: A morphological, structural or ultrastructural effect?[END_REF]. The elementary hemp fibre is idealised as a single-layered, hollow, thick-walled cylinder made of an orthotropic material having a helical orientation (corresponding to the cellulose microfibril orientation). The possible variation in cross-section shape and size along the fibre length, non-unity transverse aspect ratios (i.e. non-circular crosssection) and their induced effects on the fibre mechanical behaviour (stress concentration, fibre rotation…) are therefore not addressed in the model. The transverse anisotropy introduced by nonunity aspect ratios is also in consequence ignored in this model. The wall is modelled as if it were a long fibre-reinforced composite material, made with a mixture of three different polymers. The wall properties are determined using a homogenisation technique [START_REF] Placet | Diameter dependence of the apparent tensile modulus of hemp fibres: A morphological, structural or ultrastructural effect?[END_REF]. The values obtained for its elastic properties are summarized in Tab. 1. The fibres were considered to be free to rotate, and the influence of the boundary conditions on the computed value of the apparent modulus was studied in Part I of this paper. It was shown that although the testing device prevented the fibre from rotating during tensile tests, the experimentally determined value of the apparent E-modulus was more closely related to the numerical value determined using a free rotation. This was explained by the relatively strong hypothesis of cellulose microfibril continuity formulated when constructing the model.

Hygro-mechanical coupling was not considered in this version of the model.

The external diameter of the fibres is 13.3 µm, and that of the lumen is 4.46 µm. These nominal values were determined experimentally, and are consistent with the dimensions typically reported in the literature.

The model for the fibre wall was developed assuming conditions of anisotropic viscoelasticity, in order to correctly represent the fibre's behaviour, when it is observed under experimental conditions. Microstructural variations as a function of loading level, such as MFA and cellulose crystallinity, were also taken into account using the appropriate constitutive laws, which were then implemented in the finite element code Abaqus ®.

Dislocation model

It is well known that plant fibres have wall irregularities, which are commonly referred to as dislocations, nodes or kink bands. For hemp fibres, these dislocation areas are located mainly in the S 2 layer [START_REF] Thygesen | Visualisation of dislocations in hemp fibres: A comparison between scanning electron microscopy (SEM) and polarized light microscopy (PLM)[END_REF] and represent from 12% to 21% of the total fibre wall [START_REF] Thygesen | The effects of growth and storage conditions on dislocations in hemp fibres[END_REF]. They are induced mainly by axial compression during the growth phase of the plant, and by the fibre separation process [START_REF] Thygesen | Quantification of dislocations in hemp fibers using acid hydrolysis and fiber segment length distributions[END_REF].

Although the mechanisms responsible for these dislocations are not well understood, it is generally accepted that the cellulose microfibrils inside the dislocations are oriented differently than in the surrounding fibre wall. Using polarised Raman microspectroscopy, Thygesen and Gierlinger [START_REF] Thygesen | The molecular structure within dislocations in Cannabis sativa fibres studied by polarised Raman microspectroscopy[END_REF] have recently shown that the MFA ranges from approximately 5° to 10° in the bulk fibre wall, and is approximately 30° in the central part of the dislocation. On each side of the dislocation, two regions were found with opposite winding directions and an MFA of approximately 10°-15°.

On the basis of this observation, as well as the results from some of our measurements, an MFA of 11° was used in our numerical simulations for the bulk regions of the fibre, and an MFA of 30° was used for the dislocation areas.

As some of these simulations showed that the regions on either side of the dislocation (with opposite winding directions) have no significant influence on the fibre's tensile behaviour, these areas were not taken into account in the model presented here.

The dislocations were also considered to extend all the way across the fibre, and to make up 25% of the fibre wall. Dislocations are distributed along the fibre at approximately regular intervals. Again, since the numerical simulations showed that the fibre's behaviour is not significantly influenced by the distribution of these dislocations, these were modelled as if they were grouped together in the same area.

Constitutive laws

Visco-elasticity

Viscoelastic model

Assuming that the local state of the material is completely defined by the elastic strain tensor 𝜀 𝑒 and a set of second order tensors 𝜉 𝑖 (𝑖 ∈ ℕ) corresponding to elementary mechanisms of viscoelastic flow, the state law leads to the following instantaneous constitutive equation :

ℰ ̇-ℰ v𝑒 ( 𝜉 𝑖 ; 𝜎) = 𝑆𝜎, ( 1
)
where 𝜎 is the Cauchy true stress tensor, 𝑆 the elastic compliance tensor corresponding to a transverse isotropic behaviour with respect to the microfibrils direction, 𝜀 𝑣𝑒 the viscoelastic strain tensor and 𝜀 the total strain tensor such as 𝜀 = 𝜀 𝑒 + 𝜀 𝑣𝑒 .

Following [START_REF] Boubakar | A meso-macro finite element modelling of laminate structures: Part II: time-dependent behaviour[END_REF], the viscoelastic flow rule is given by :

𝜀̇𝑣 𝑒 = ∑ 𝜉 ̇𝑖 𝑛 𝑖=1 = ∑ 1 𝜏 𝑖 𝑛 𝑖=1 (𝜇 𝑖 𝑆 𝑣𝑒 𝜎 -𝜉 𝑖 ), (2)
where 𝑆 𝑣𝑒 is a viscoelastic compliance tensor whose general form is derived by considering an isotropic transverse behaviour with respect to the microfibrils direction, which is assumed to be purely elastic :

𝑆 𝑣𝑒 = ( 0 0 0 0 0 0 0 𝛽 𝑇 𝐸 𝑇 - 𝛽 𝑇𝑇 𝜈 𝑇𝑇 𝐸 𝑇 0 0 0 0 - 𝛽 𝑇𝑇 𝜈 𝑇𝑇 𝐸 𝑇 𝛽 𝑇 𝐸 𝑇 0 0 0 0 0 0 𝛽 𝐿𝑇 𝐺 𝑇 0 0 0 0 0 0 𝛽 𝐿𝑇 𝐺 𝑇 0 0 0 0 0 0 𝛽 * 𝐺 𝑇𝑇 ) ; 𝛽 * 𝐺 𝑇𝑇 = 2 ( 𝛽 𝑇 𝐸 𝑇 + 𝛽 𝑇𝑇 𝜈 𝑇𝑇 𝐸 𝑇 ) (3)
Such a tensor requires, in addition to the classical elastic parameters, the identification of three new parameters (𝛽 𝑇 , 𝛽 𝑇𝑇 , 𝛽 𝐿𝑇 ) related to the viscoelastic behaviour.

Also in equation 2, for a given i value, 𝜇 𝑖 and 𝜏 𝑖 represent respectively the relative weight and the corresponding release time.

A triangular spectral model is used in this work to define a distribution function of 𝜇 𝑖 with respect to 𝑙𝑜𝑔(𝜏 𝑖 ) under the following condition:

∑ 𝜇 𝑖 𝑛 𝑖=1 = 1 (4)
This kind of spectrum is defined by its centre 𝑧𝑛 𝑐 and its half-width 𝑧𝑛 0 as shown in Fig. 1 where Δ depends on both these quantities and on the number n of elementary mechanisms. For the sake of accuracy, thirty-one elementary mechanisms are used.

Parameter identification

Two scalar independent creep functions have to be experimentally determined to characterize anisotropic time-dependent behaviour of hemp fibres. Laborious experimentations work has to be performed to determine these functions, in particular to determine the transverse creep function. In this study, we are only working with the experimentally determined axial creep function. The theoretical axial viscoelastic compliance is expressed as follows:

0 31 1 0 ) 1 ( ) ( ) ( 1 mod            i t ve ZZ i ve ZZ ZZ e S t t J ; 𝑆 𝑧𝑧 𝑣𝑒 = 𝛽 𝑇 𝐸 𝑇 cos 4 (𝜓) + 𝛽 𝐿𝑇 𝐺 𝐿𝑇 cos 2 (𝜓)sin 2 (𝜓) (5)
 is the supplementary angle of the MFA (90°-𝑀𝐹𝐴) and 0

 the constant axial stress.

On the basis of the axial viscoelastic compliance, only two (𝛽 𝑇 , 𝛽 𝐿𝑇 ) of the three viscoelastic parameters can be identified (Eq. 5). These two viscoelastic parameters were identified from a creep test on a single hemp fibre, combined with the use of the aforementioned model and a hybrid minimisation algorithm. The results of a broad experimental campaign, made with elementary hemp fibres in order to accurately characterize their creep behaviour, will be presented in a forthcoming paper. For this study, a creep curve recorded on a single hemp fibre at a relative humidity of 50 ± 5% and a temperature of 21 ± 1 °C was used to identify the viscoelastic parameters. This test was made using a tensile machine (DMA Bose Electroforce 3230) with a clamping length of 10 mm and a constant load of 30 mN. The axial strain was determined as the fibre elongation divided by its initial length, and was recorded as a function of time for a period of 3000 seconds.

In order to avoid convergence towards local optima in the case of multimodal test functions, the Levenberg-Marquardt algorithm was combined with a heuristic method based on a genetic algorithm. This hybrid algorithm is described in [START_REF] Carbillet | Contribution aux calculs fiabilistes sur des structures composites[END_REF][START_REF] Richard | Identification du comportement et évaluation de la fiabilité des composites stratifiés[END_REF], and involves minimisation of the objective function f (Eq. 6).

n J J J f n k k ZZ k ZZ k ZZ 2 ) max( 2 1 exp mod exp             (6)
where k ZZ J exp is the k th value of the experimental compliance data, k ZZ J mod is the k th computed value using the aforementioned model, and n is the number of elements in the dataset.

Fig. 2 shows that by combining this spectral model with a suitable identification technique, a relatively good fit can be achieved with the experimental data. A broad study on viscoelastic spectral models will soon be submitted for publication. It will show that a better fit can be achieved with the experimental data when using Gaussian or parabolic spectra. For this study, the timedelayed response identified with the simple triangular spectral model is sufficient to ensure a suitable representation of the experimental behaviour. The identified parameters are summarized in Tab. 1.

Evolution law of MFA

Although the relationship between the initial MFA and the initial or tangent apparent modulus of In a previous study using WAXS, we measured a slight decrease in MFA (approximately 3°, from 10.8° to 7.8°, for an axial strain varying from 0% to 2%) with a bundle of hemp fibres, mainly in the first portion of the tensile curve [START_REF] Placet | Suivi par DRX des réarrangements microstructuraux induits par sollicitations mécaniques dans les fibres végétales tirées du chanvre. 20ème Congrès Français de Mécanique[END_REF]. These results, recorded at the scale of a fibre bundle, cannot be generalized to the case of a single fibre. In the first part of this study, using in situ polarised light microscopy (PLM) on single hemp fibres, it was also observed that some of the microfibrils became realigned. This type of quantitative approach should be interpreted with considerable caution, since PLM allows mainly the microfibrils located on the fibre wall surfaces to be observed, and not those of the S 2 layer.

As a consequence of the difficulties encountered in determining a law to provide an accurate description of the variations observed in the MFA, in the present study we propose a consistent modelling based on kinematic and material considerations. To the best of the authors' knowledge, this is the first time a theoretical model has been proposed to describe this phenomenon.

The proposed approach assumes that the fibre is locally subjected to finite transformations which involve large rotations and small elastic-viscoelastic strains. In the context of an incremental computation, the previous constitutive framework (Eqs. 1, 2, 3) remains valid in a material rotating frame whose one of the axis coincides with the microfibrils direction.

In this study, the evolution law of the cellulose microfibrils is derived from the intrinsic expression of the deformation gradient 𝑭 in the case of a plane kinematics with respect to the radial direction.

Hence, by using a lagrangian frame (𝒆 ⃗⃗ 𝟎𝟏 , 𝒆 ⃗⃗ 𝟎𝟐 , 𝒆 ⃗⃗ 𝟎𝟑 ) and a material one (𝒆 ⃗⃗ 𝟏 , 𝒆 ⃗⃗ 𝟐 , 𝒆 ⃗⃗ 𝟑 ) such as 𝒆 ⃗⃗ 𝟎𝟏 and 𝒆 ⃗⃗ 𝟏 define the same material line during the transformation and 𝒆 ⃗⃗ 𝟎𝟐 (respectively 𝒆 ⃗⃗ 𝟐 ) remains in the plane perpendicular to the radial direction 𝒆 ⃗⃗ 𝟎𝟑 = 𝒆 ⃗⃗ 𝟑 (see Fig. 3), the deformation gradient takes the following form :

0 i i e e F F F F F            (7)
Let 𝜒 ⃗ 0 be the microfibrils direction prior to transformation, and let the two directions 𝑒 ⃗ 1 0 and 𝜒 ⃗⃗⃗⃗ 0 be separated by an initial reference angle 𝜓 0 = 90°-𝑀𝐹𝐴. When the rotation 𝑄 𝑓 is considered, the new microfibrils direction becomes:

𝜒 ⃗ = 𝑄 𝑓 𝜒 ⃗⃗⃗⃗ 0 (8)
and the direction 𝑎 ⃗ 1 = 𝑄 𝑓 𝑒 ⃗ 1 0 are still separated by the angle 𝜓 0 .

Hence :

𝑄 𝑓 = 𝜒 ⃗ 𝑖 ⨂𝜒 ⃗ 𝑖 0 = 𝑄 𝑖𝑗 𝑒 ⃗ 𝑖 ⨂𝑒 ⃗ 𝑗 0 (9)
where 𝑄 11 = cos (𝜓 -𝜓 0 ), 𝑄 12 = -𝑠𝑖𝑛 (𝜓 -𝜓 0 ), 𝑄 21 = sin (𝜓 -𝜓 0 ) and 𝑄 22 = cos (𝜓 -𝜓 0 ).

Since the microfibrils direction is a material one, the direction 𝜒 ⃗ can also be computed by the following expression :

𝜒 ⃗ = 𝐹 𝜒 ⃗⃗⃗ 0 ‖𝐹 𝜒 ⃗⃗⃗ 0 ‖ (10)
From Eq.8 and Eq. 10, it comes finally:

0 12 0 11 0 22 sin cos sin tan     F F F   (11)

Strain-induced crystallisation law

In addition to the change in MFA under tensile loading, we also take into account any possible crystallisation of the paracrystalline polymers. Indeed, using in situ X-ray scattering on dry flax fibres, Astley and Donald [START_REF] Astley | The tensile deformation of flax fibres as studied by X-ray scattering[END_REF] have shown that strain-induced crystallization of the cellulose microfibrils can occur when the fibre is subjected to tensile strain. The synchrotron measurements performed by these authors provide evidence that non-crystalline cellulose chains are initially oriented. The existence of non-cellulosic, highly ordered paracrystalline domains has also been revealed by proton spin relaxation time calculations, made by Duchemin et al. [START_REF] Bt | Ultrastructure of cellulose crystallites in flax textile fibres[END_REF]. Using an X-ray diffraction technique, Toba et al. [START_REF] Toba | Crystallization of cellulose microfibrils in wood cell wall by repeated dry-and-wet treatment, using X-ray diffraction technique[END_REF] also showed that cyclic sorption can promote the crystallization of cellulose microfibrils in wood cell walls, regardless of heating. In fact, it is well known that heat treatment can lead to the degradation of the amorphous cellulose surrounding the cellulose microfibrils, thus inducing a change in their crystallinity. These authors concluded that non-crystalline cellulose progressively forms hydrogen bonds with the cellulose at the surface of the crystalline region during sorption cycling, thus increasing the size of the crystal.

To the best of the present authors' knowledge, there is currently no evidence in the literature of strain-induced crystallization in hemp fibres, when these are subjected to tensile loading. However, when measurements performed on flax fibres and wooden materials are considered, the hypothesis proposed in Part I of this study, according to which crystallization could occur under mechanical or hygro-thermal loading of the highly ordered paracyrstalline components, appears to be relevant.

In view of the lack of existing experimental data (both quantitative and phenomenological), it is proposed, in the present study, to consider an arbitrary linear variation of the crystallinity ratio, as a function of shear strain. It is assumed that this ratio increases linearly from 65% to 85%, when the in-plane strain shear increases from 0.5% to 4%.

FE Modelling

Several 3D (8-or 20-node) meshes were tested using the FE Abaqus® Code to represent the fibre, using the model described above (see [START_REF] Placet | Diameter dependence of the apparent tensile modulus of hemp fibres: A morphological, structural or ultrastructural effect?[END_REF] for the corresponding numerical modelling). When the elastic axial, hoop and shearing stresses aligned in the radial direction of the fibre are compared with the analytical elastic solution [START_REF] Placet | Diameter dependence of the apparent tensile modulus of hemp fibres: A morphological, structural or ultrastructural effect?[END_REF], and the modelling and computation time requirements are compared, a one-8nodes-element mesh in the direction of the fibre radius appears to provide a good compromise. In order to follow the fibre's curved geometry, 40 elements are used to represent its circumference. A total of 80 elements are used in the axial direction, to accurately model any defects present along the fibre's length. Fig. 3 provides an example of an undeformed, and a deformed mesh. The micro-fibril direction is defined with respect to this direction (𝜓 0 = 79°), rather than the axial direction (MFA = 11°) used for the FE calculations. The calculations are made using either an elastic or a visco-elastic behaviour, with or without MFA updating, and with or without the assumption of the crystallinity ratio increasing as a function of fibre deformation. Any experimentally observed defects along the length of the fibre are modelled by an area in which the mechanical behaviour is the same as in the bulk area of the fibre, but the MFA is smaller than in the fibre (𝜓 0 = 60°). The initial angle in the zone having a defect was tested for different MFA values, over the range (50°-70°), with a corresponding percentage of defects lying in the range between 25% and 75%.

The loading is applied by imposing displacements at the both ends of the sample; a global axial strain is obtained by computing the ratio of the displacement to the length of the sample. The strain rate is of the same order of magnitude as the experimentally observed value, i.e. approximately 5.10 -4 s -1 .

In order to compare these calculations with our experimental results, the global axial stress was computed by summing the reactions -to the imposed displacement -at all nodes belonging to the end surface. This force was then divided by the surface area of the sample, to obtain the axial stress.

For each load increment, the tangent of the apparent Young's modulus was obtained by computing the ratio between the stress increment and the strain increment. In order to correlate the simulations with the experimentally determined values, the fibre's mechanical properties were determined using its initial values of length and radius.

RESULTS AND DISCUSSION

FE simulations were carried out using different constitutive laws, using the elastic and viscoelastic parameters listed in Tab. 1. The results are summarised in Figs. 4 and show the change in apparent tangent Young's modulus as a function of tensile strain. Two experimental curves selected among a database collected during an extensive experimental campaign on elementary hemp fibres, are also plotted in these figures. They reveal the strong discrepancies observed during the experimental campaign. Using the described model, we propose to assess the degree to which each physical mechanism governs the tensile response of the fibre.

Elastic behaviour (E)

When elastic behaviour is considered for the homogenized materials comprising the fibre wall, a quasi-linear curve is observed in the stress-strain plot. Fig. 4.b. shows that the apparent rigidity increased slightly when the strain was increased, from an initial modulus of approximately 40 GPa to a tangent modulus of 47 GPa at 4% axial strain. This stiffening is due to the structural effect resulting from the fibre's cylindrical geometry, the helical organization of the cellulose microfibrils, and the change in cross-sectional area as a function of tensile loading. The structural effect was extensively analysed in a previous study [START_REF] Placet | Diameter dependence of the apparent tensile modulus of hemp fibres: A morphological, structural or ultrastructural effect?[END_REF]. It can be seen that the simulated behaviour is quite different to that observed experimentally.

Cellulose Microfibril Reorientation (MFR)

The variations in MFA as a function of axial strain were determined using the proposed theoretical model. During the tensile test, the microfibrils rotated from their initial angle of 11° to an angle of 5.7°, when placed under the greatest value (4%) of axial strain (Fig. 6). These computed values are in very good agreement with those measured on hemp bundles using the WAXS technique. Using the WAXS technique on bundle of hemp fibres, a decrease in MFA, from 10.8° to 7.8°, was measured when the axial strain varied from 0 to 2%. The computed values of strain, ranging from 11° to 8°, thus lie in the same range. Fig. 4.a. clearly shows that when this progressive decrease in MFA is taken into account during the tensile test, the resulting curve has a slightly parabolic shape, which is similar to that obtained in part III of the experimental tensile curve. The final value of the apparent modulus (64 GPa), corresponding to an axial strain of 4%, is approximately 1.6 times higher than the initial value (40 GPa), see Fig. 4.b. This degree of stiffening is consistent with the experimental results obtained by the present authors with Repeated Progressive Loading [START_REF] Placet | Characterization of the thermo-mechanical behaviour of Hemp fibres intended for the manufacturing of high performance composites[END_REF][START_REF] Placet | Tensile behaviour of natural fibres. Effect of loading rate, temperature and humidity on the "accommodation[END_REF], and is of the same order as that observed by Baley [START_REF] Baley | Analysis of the flax fibres tensile behaviour and analysis of the tensile stiffness increase[END_REF] and Silva [START_REF] Fda | An experimental investigation of the fatigue behavior of sisal fibers[END_REF], with flax and sisal fibres, respectively.

The simulated behaviour is nevertheless quite different to that observed experimentally.

Viscoelasticity

Viscoelasticity in the case of small strain hypothesis (VE)

The viscoelastic law was identified during an earlier experimental campaign based on a creep test with an elementary hemp fibre. The identified parameters are summarized in Tab. 1. On the basis of the axial viscoelastic compliance, only two (𝛽 𝑇 , 𝛽 𝐿𝑇 ) of the three viscoelastic parameters were identified. A sensitivity analysis on the  TT parameter was performed. Our numerical simulations showed that the axial viscoelastic compliance is not sensitive to this parameter  TT when its value is comprised between 1 and 25. An additional work based on a parametric identification with the 3D FE model is underway to identify the three viscoelastic parameters from experimental creep function. The results will be published in an upcoming article.

Using the viscoelastic law, significant nonlinearity was revealed in the stress-strain plot (Fig. 4.a., VE, red curve).

The apparent modulus decreased from 40 GPa to approximately 20 GPa during the first 2% of strain, and then had a slight tendency to increase beyond 2% of strain. Although the first part of the curve is in relatively good agreement with the upper experimental curve, the stiffening observed after the inflection point is not sufficient to ensure a well-adjusted fit with the experimental results. This result is nevertheless interesting, since the viscoelastic behaviour can to a certain extent explain the nonlinear tensile behaviour observed in wood or plant fibres, in particular the stiffening which occurs following an initial decrease in rigidity.

Viscoelasticity in the case of finite transformations (VE/MFR)

When the viscoelastic behaviour of the homogenised material of the fibre wall is combined with the MF re-alignment phenomenon associated with tensile loading, a strongly nonlinear curve is obtained in the stress-strain plot (Fig. 5.a., VE/MFR). The shape of the simulated tensile curve (VE/MFR) is very close to that observed experimentally (Type 3). The first part of the curve is linear until the first yield level, beyond which a strong decrease in rigidity is observed. A second inflection point (corresponding to the experimentally observed point i 2 ) appears at a higher strain level, and is followed by a parabolic increase in rigidity up until final failure. This result clearly shows that the nonlinear tensile behaviour of plant or wood fibres can be described and simulated by a viscoelastic law for finite transformations, and that the main source of dissipative behaviour is certainly viscoelasticity, rather than plasticity as proposed by some authors [START_REF] Nilsson | Influence of dislocations and plasticity on the tensile behaviour of flax and hemp fibres[END_REF][START_REF] Sedighi-Gilani | A micromechanical approach to the behavior of single wood fibers and wood fracture at cellular level[END_REF]. This result is in good agreement with our experimental observations, the time-dependent mechanisms in particular, and tends to validate the scenario proposed in the first part of this study.

In a previous study, we also observed that the proportion of curves of type 3 increases significantly as a function of the fibre's water content [START_REF] Placet | Influence of environmental relative humidity on the tensile and rotational behaviour of hemp fibres[END_REF]. It was assumed that, by plasticizing amorphous components of the cell wall and thereby activating their viscoelasticity, water could facilitate the realignment of cellulose microfibrils. Our numerical simulations clearly reveal that MF realignment is increased by the viscosity ratio. MF re-orientations occur more rapidly and have higher angular values for a given value of axial strain, when the material is assumed to be viscoelastic. As shown in Fig. 6, the MFA decreased from an initial value of 11° to 4.2° in the case of an axial strain of 3%, whereas it reached a value of only 6.7° in the case of purely elastic behaviour.

It should be noted that the parabolic increase in rigidity which can be observed at high values of strain on the tensile curve (Fig. 5a) is lower than that observed for the upper experimental curve. In this figure, it can be seen that a good fit is obtained (VE/MFR/C, light green curve) when the potential shear-induced crystallisation is taken into account. Fig. 7 shows the stress-strain and apparent tangent modulusstrain curves for different strain rates from 5.10 -5 to 5 .10 -3 s -1 . As expected and as for all polymeric materials, the fibre response depends directly on the strain rate. Our results show that the degree of nonlinearity decreases with an increase in strain rate.

Influence of dislocation areas on the tensile behaviour

The influence of dislocation areas on the tensile behaviour of hemp fibres has not been fully characterised. In living plants, it is speculated that these have no detrimental effect, and could perhaps be beneficial [START_REF] Thygesen | The molecular structure within dislocations in Cannabis sativa fibres studied by polarised Raman microspectroscopy[END_REF]. In the case of isolated fibres, no clear tendency has been demonstrated in the literature. At present, a small number of published results suggest that the fibre's strength is not affected by an increase in the proportion of dislocation areas [START_REF] Thygesen | Dislocations in single hemp fibres -investigations into the relationship of structural distortions and tensile properties at the cell wall level[END_REF]. However, experimental investigations are particularly difficult to implement and interpret, due to the widely scattered results. In the present study, it is proposed to analyse the influence of dislocations on the mechanical behaviour of single fibres, using the proposed model. The results of these numerical simulations shed new light on this topic.

If a dislocation proportion of 25% is assumed, in association with an MFA of 30°, a strong decrease in apparent tangent modulus is observed (VE/MFR/C/D Fig. 5.a.). The initial modulus is approximately 27 GPa, and decreases to 14 GPa at 1% axial strain, following which it increases gradually up to 25.5 GPa at 3% axial strain. The shape of this simulated tensile curve is well adjusted to the lower experimental curve. In view of these numerical results, when they are compared with the behaviour of bulk fibre having a constant and relatively small MFA along its length, it can be clearly understood that the presence of dislocations leads to a significant decrease in the apparent stiffness of the fibres. It can thus be concluded that in the case of hemp fibres, an increase in the proportion of dislocations contributes to a decrease in the fibre's apparent rigidity. Fig. 8 also shows that the amplitude of the reorientations is almost twice as strong in the dislocation area as in the bulk area. The decrease in MFA, corresponding to an axial strain ranging from 0 to 4%, is respectively 14° and 7.5° in these two areas. The difference in MFA in these two areas, for the higher strain values, is thus significantly reduced. This result is in agreement with the in situ PLM observation described in Part I of this study, which showed that the dislocations gradually disappear during tensile testing, as a result of MFA homogenization along the length of the fibre.

When a purely elastic behaviour is considered, a significant stress concentration (Figs. 9.a. and 10.a.), together with an increase in the axial stress gradient within the thickness of the cell wall (Fig. 9.b.), are observed in the dislocation area. It thus seems clear that these areas could be preferential failure initiation zones. Conversely, when viscoelastic behaviour is assumed for large transformations, the axial stress level is found to be relatively constant along the length of the fibre (Fig. 9.a. and Fig. 10.b.), and the axial stress gradient is nearly the same in both areas. Although this constitutive law accurately reflects the underlying physical mechanism, it is clear that, contrary to all expectations, dislocation areas do not induce significant stress concentrations. This numerical result could provide an explanation for the absence of any statistically significant influence of dislocations on the experimentally observed behaviour of the fibre's strength [START_REF] Thygesen | Dislocations in single hemp fibres -investigations into the relationship of structural distortions and tensile properties at the cell wall level[END_REF].

CONCLUSIONS

The results presented in this paper show that a 3D hollow model with a fibre-reinforced viscoelastic constitutive law for large transformations is able to simulate the nonlinear tensile behaviour of single hemp fibres. Very good agreement in the curve shape is found with the experimental results. This study also highlights the influence of dislocation areas on the mechanical behaviour of single fibres, in particular on their apparent rigidity. Both initial and tangent apparent rigidity up to failure are influenced by the presence of dislocations. The results also show that when elastic behaviour is assumed, the induced stress concentration and gradient in the dislocation area, associated with viscoelastic behaviour, are cancelled. This could explain why, contrary to all expectations, dislocation areas are not preferential areas of failure initiation.

The next parts of this paper series on the tensile behaviour of plant fibre will be focused on the modelling of other tensile responses, such as Type 2, and also to the modelling of tensile unloading with irreversible residual strains. The implementation of damage laws and visco-plastic mechanisms constitutes an interesting track to explore this subject. It could also be interesting to evaluate the influence of morphological aspects (such as non-unity transverse aspect ratios, discontinuities in the cross-section shape and size along the fibre length and so on) on the tensile response, the ultimate goal being to use the real 3D morphology of a single isolated fibre.

  fibres placed under tensile loading has been widely investigated, both experimentally and theoretically, very little research has been devoted to the study of MFA variations in the same fibre during tensile tests. From the experimental point of view, this type of study is particularly difficult to perform and requires the use of synchrotron radiation when an X-ray scattering technique is used to measure the MFA. Contradictory results have been published in the literature. Contrary to the results of Kölln et al.[START_REF] Kölln | Mechanical properties of cellulose fibres and wood. Orientational aspects in situ investigated with synchrotron radiation[END_REF],Astley and Donald (25) did not observe a change in the distribution of flax fibre orientations when they were placed under strain.

  5, where Figs. 4.a and 5.a. show the computed tensile stress variations as a function of tensile strain, and Figs. 4.b. and 5.b

  The model is able to successfully bound the batch of experimental curves. As hypothesised in Part I of this study, based on experimental investigations, the numerical results confirm the dominant role of viscoelastic phenomena in cellulose MF re-orientations, and the associated degree of nonlinearity.

Figure 1 :Figure 2 :

 12 Figure 1: Spectral model for the release times

Figure 3 :Figure 4 .

 34 Figure 3: Theoretical variation of the microfibril's orientation after loading.

Figure 6 :Figure 7 .

 67 Figure 6: Computed MFA as a function of axial strain for two constitutive laws: elastic (E) and viscoelastic (VE).

Figure 8 :Figure 10 :

 810 Figure 8: Computed MFA as a function of axial strain for two fibre areas: bulk fibre and dislocation area, using the VE constitutive law.

Table 1 : Values of the model parameters used in the FE computations.

 1

	Elastic parameters	Viscoelastic parameters
	E L	75 000 MPa	𝛃 𝐋𝐓	12.25
	E T	11 000 MPa	𝛃 𝐓	1.5
	ν LT	0.153	zn c	2.45
	G LT	2 520 MPa	zn 0	1.9
	ν TT	0.2


Jade Fawaz

Maxime Barat

Sébastien Gaujoux
email: sebastien.gaujoux@aphp.fr

Sébastien Pr

MD ; PhD Gaujoux

DISTAL PANCREATECTOMY WITH CELIAC AXIS RESECTION -WITH POSTERIOR RAMPS APPROACH -FOR LOCALLY ADVANCED PANCREATIC ADENOCARCINOMA WITH CELIAC TRUNK INFILTRATION (MODIFIED APPLEBY PROCEDURE) (WITH VIDEO)

Keywords: Adenocarcinoma, Pancreas, Celiac trunk infiltration, Appleby procedure: Distal pancreatectomy, Celiac axis resection

Distal Pancreatectomy with Celiac Axis Resection-with

Posterior Ramps Approach-for Locally Advanced Pancreatic Adenocarcinoma with Celiac Trunk Infiltration (Modified Appleby Procedure) (with Video)

Locally advanced pancreatic cancer (LAPC) of the body/tail of the pancreas involving the celiac axis, with excellent clinical, biological and radiological response to neoadjuvant treatment, can be eligible to a distal pancreatectomy with celiac axis resection (modified Appleby procedure). Besides accurate patient's selection and surgical expertise, this procedure also requires experienced radiologist, oncologist, intensive care anesthesiologist and should be performed in high-volume centers [START_REF] Klompmaker | Distal Pancreatectomy with Celiac Axis Resection (DP-CAR) for Pancreatic Cancer. How I do It[END_REF][START_REF] Latona | Modified Appleby Procedure with Arterial Reconstruction for Locally Advanced Pancreatic Adenocarcinoma: A Literature Review and Report of Three Unusual Cases[END_REF][START_REF] Wagner | CT evaluation after neoadjuvant FOLFIRINOX chemotherapy for borderline and locally advanced pancreatic adenocarcinoma[END_REF] . This didactic video presents a distal pancreatectomy with celiac axis resection in a 68year-old woman, with a LAPC. Clinical symptoms started with abdominal pain radiating in the back. Abdominal CT scan and MRI showed a 4 cm pancreatic tumor of the isthmus, with complete infiltration of the celiac axis and proximal splenic artery, abutting the left adrenal gland. After 7 courses of neoadjuvant FOLFIRINOX completed with 50,4 Gy radiotherapy, she had a complete clinical response with total disappearance of pain, a decrease of Ca 19-9 from 1800 to 35 UI/ml and a RECIST stability on CT and MRI with a regression of tumorvein contact. Surgery including distal pancreatectomy with celiac axis and adrenal resection according to a posterior RAMPS approach was decided in a dedicated tumor boards including specialists from surgery, medical oncology, radiation oncology, radiology, genetics and pathology. One month before surgery, the patient had preoperative arterial embolization of the splenic artery and proximal common hepatic artery (figure 1), to improve collateral flow and reduce postoperative liver ischemia, even if there is no clear evidence in the literature indicating a beneficial effect of preoperative embolization on the need for unplanned arterial reconstruction. Through a bi-subcostal laparotomy, surgical procedure began with a complete abdominal exploration. A large colo-epiploic detachment, a Kocher maneuver and opening of the right diaphragmatic pillar allowed a good exposure of the celiac trunk and the origin of the superior mesenteric artery which is released on its first centimeters. This is followed by lymph node dissection of the hepatic pedicle preserving the right gastric artery and control of

the common hepatic artery. The pancreas is then sectioned on the left edge of the superior mesenteric artery, with a careful preservation of its right-side collaterals. Adequate backflow of the gastroduodenal artery in the hepatic vasculature is first checked with intraoperative ultrasonography, and then the common hepatic artery is ligated with 3/0 polypropylene (Prolene ® ) and sectioned just upstream the origin of the gastroduodenal artery (figure 2).

Then the coeliac trunk (8-9 mm in diameter) is ligated with 2/0 polypropylene (Prolene ® ) and sectioned at its origin (figure 3). The pancreatectomy is continued according to a posterior RAMPS (Radical Antegrade Modular Pancreatosplenectomy) approach with en bloc resection of left adrenalectomy for oncological reason. At the end of the procedure, the presence of an intrahepatic arterial flow is once again checked by liver doppler ultrasound.

Postoperative course was marked by a grade B chylous ascites and no pancreatic fistula. The final pathological analysis finds a 23 mm pancreatic ductal adenocarcinoma with a 90 % tumor regression, classified ypT2N0, with R0 margin.


Boutaib Dahhou
email: dahhou@laas.fr

Anas Kamoun

A Comparative Analysis of Two Formulations

The task of FDI is a chain of processing blocks as shown in figure 1 The proposed approach to FDI relies on residual generation method using adaptive observers.

i i i i T i i u i i u i j i j j i x C y ) x ( Pg ) x x ( 2 ) x x ( H ) x ( g u ) x ( g ) x ( f dt x d m i 1 H P P H Q PH P H T Q i y i r y y ) t ( r i i
In the proposed scheme, we detect the presence of a fault and at the same time we identify its value. We will use this identification for the fault isolation.

For the fault detection and identification we will develop a bank of adaptive observers, with m i , the number of the system actuators. The adaptive observers form will be as in (3), only that we will consider that all actuators are faulty:

i i i T i i u i j , i u i j j i x C y ) x ( Pg ) x x ( 2 ) x x ( H ) x ( g ) x ( f dt x d m i 1
where i

x is the state vector of the th i observer; the appropriate gain matrices H and P will have been chosen in order to have a good fault estimation. We will apply to the residual i r for the fault detection of the form:

m i , dt y y ) t ( r i i
These residuals allow us to calculate fault estimations by using the following relation:

m i ), r max( ) r sgn( i i i -Fault isolation
Once fault is detected, it needs to be isolated. So the fault isolation requires the creation of m banks of m adaptive observers where we will use these m estimations i from estimation vector. The adaptive observer form will be as in (3) where all the banks will be identical except from the system input value:

A Comparative Analysis of Two Formulations for Actuator Faults... et al.

k i i i T k i k i u k i k i u i k j i j j k i x C y ) x ( Pg ) x x ( 2 ) x x ( H g v ) x ( g ) x ( f dt
m] , [1, i k, , y y s k i i , k
where k i y i is the estimated output of the th i observer of the th k bank.

All of the residual of all the bank of adaptive observers rest in this initial value zero the whole period until the time when the fault has been entered, where its leave zero for a short time period. Thus we can detect the fault and the structural residual ki s associated to the isolation bank will indicate us the faulty actuator and consequently the fault estimation value.

Fault detection and isolation using parameter interval approach

In the parameter interval based method, each actuator parameter is divided into a certain number of intervals. After occurrence of a fault, the value of the faulty actuator parameter must be in one of these intervals. After checking each interval whether it contains the fault or no, the faulty actuator parameter value is found, the isolation is therefore realized. In this section, we quickly describe the principle of the parameter interval based fault isolation method.

IJ-STA, Vol. 7, N°1, April, 2013.

Nonlinear dynamic system and fault modeling

Considering the nonlinear dynamic system (2), it can be described as:

Cx y ) u , , x ( f x i u (12)
where: i u is the constant parameter vector of the actuator such as 0 i u is its nominal value. For actuator fault isolation each component of the parameter vector i u is divided into certain number of intervals.

This method is based on the monotonous characteristic of w function. In [15] the studies only focus on the faults of the dynamic part of the system. In this paper, this method is extended to actuator fault isolation problem for nonlinear dynamic systems. f is a monotonous function of the parameter i u at any considered point x in the state space.

There is a fault in this dynamic system (12), if the dynamic difference:

) u , , x ( f ) u , , x ( f ) u , , , x ( f 0 i u i u 0 i u i u (13)
between the system and its nominal model caused by the difference of actuators parameter vectors

0 i u i u i u is significant.

Fault detection and isolation

After the fault occurrence, the fault isolation task is followed by the fault detection procedure. For fault detection, an existing method [14] is used. The time of the fault occurrence and the time when it is detected are considered as the same which is noted as f t .

We assume that the considered faults are caused by the change of single actuator parameter. For m actuators parameters 1 u , 2 u m u we partition the possible domain of each parameter into a certain number of intervals. For example, the parameter j u is partitioned as p intervals, the bounds of th i interval are . After fault occurrence, the faulty actuator parameter value must be in one of the parameter intervals. To verify if an interval contains the faulty value, an actuator parameter filter is built for this interval. A parameter filter consists of two isolation observers which correspond to two bounds of the interval.

The actuator fault detection and isolation scheme

For the model ( 12), the parameter filter with respect to actuator fault can be described with the isolation observers given below:

) ij ( a h h ) ij ( a ) ij ( a ) ij ( a ) ij ( a ) i ( oba j u ) ij ( a ) ij ( a y y x C y ) y y ( k ) u , , x ( f x (14) ) ij ( b h h ) ij ( b ) ij ( b ) ij ( b ) ij ( b ) i ( obb j u ) ij ( b ) ij ( b y y x C y ) y y ( k ) u , , x ( f x (15)
where:

m ) i ( oba j u R , m ) i ( obb j u
R are the parameter vectors of the observers corresponding to actuator parameter vector;

R ) ij ( a , R ) ij ( b are the estimation errors ; h y is the th h component of y ; ) ij ( a h y and ) ij ( b h y are the th h component respectively of ) ij ( a y ) ij ( a y ) ij ( a x and ) ij ( b
x have converged to the system state x , so:

0 ) t t ( ) t t ( f ) ij ( b f ) ij ( a since 0 j u f ) i ( obb j u f ) i ( oba j u ) t t ( ) t t (
But at the time f t , when the fault is occurred the th s actuator parameter changes:

IJ-STA, Vol. 7, N°1, April, 2013.

f t t 0 l u f l u f u 0 s u f s u (16)
and the th j parameter of the observers change in order to isolate the fault:

f ) i ( a j u f 0 j u ) i ( oba j u t t , t t , ) t ( ; j l , t , ) t ( 0 l u ) i ( oba l u (17) f ) i ( b j u f 0 j u ) i ( obb j u t t , t t , ) t ( ; j l , t , ) t ( 0 l u ) i ( obb l u (18)
Where:

) i ( a j u et ) i ( b j u
are the bounds of the th i interval of th j actuator parameter.

Our index of isolation is:

)) t ( sgn( )) t ( sgn( ) t ( v ) ij ( b ) ij ( a ij
, there are two cases [16]:

-For the case where the interval contains the faulty parameter value it will be:

)) t ( sgn( )) t ( sgn( ) ij ( b ) ij ( a (19)
-For the case where the interval does not contain the faulty value, it exists f e t t that:

)) t ( sgn( )) t ( sgn( e ) ij ( b e ) ij ( a (20)
The increasing pace of industrialization, urbanization and population growth that our planet has faced over the last century has considerably increased environmental pollution and habitat destruction, and it negatively affected water, air and soil qualities. In this context, wastewater treatment has become one of the most important environmental issues, as it reduces or prevents pollution of natural water resources promotes sustainable water re-use, protects the aquatic environment and improves the status of aquatic ecosystems.

During the operation of a biological wastewater treatment process, many disturbances and faults can occur. The nature of these changes can be either sudden or slow and they can be related to normal or faulty process operation, provoking real or apparent deviations from the normal operation. This biochemical process is highly complex system, with a great rpose. In this system, all A Comparative Analysis of Two Formulations for Actuator Faults... et al.

components are related in a complex manner, which means that a fault in one component can often cause the failure of the entire system. To prevent this event, it is essential to detect faults immediately in order to enable the controlling system to take actions, so that the system can still fulfill its purpose. In the last decades, the biological treatment processes has proven to be an effective way to deal with polluted wastewater. The activated sludge process (Fig. 2) is the most generally applied biological wastewater treatment method [9]. All the details about the system can be found in [6]. The FDI scheme will monitor the four

actuators in Q , L Q , r Q and w Q .
In this section, the FDI is applied to a wastewater treatment process model using these two methods.

IJ-STA, Vol. 7, N°1, April, 2013.

Synthesis of the observer using the first method

The process model is a nonlinear system with the same form as in (1):

O S S L O in , O r in O 2 1 H rec , H r r H in , H r in H 3 2 1 X S rec , S r r S in , S r in S 2 1 X I rec , I r r I in , I r in I 3 1 H S in , S r in S I in , I r in I X V Q Q X V Q Q X X V Q Q X V Q Q X X V Q Q X V Q Q X Y Y 1 ) S C ( C Q ) S S ( V Q S ) X X ( V Q ) X X ( V Q X ) f 1 ( ) X X ( V Q ) X X ( V Q X f ) X X ( V Q ) X X ( V Q X Y 1 ) S S ( V Q S ) S S ( V Q S where: rec , S rec , H O H S I S I T X X S X X X S S x (22) w r L in T Q Q Q Q u (23) O H S I S I T S X X X S S y (24)
As we will indicate later on, the algorithm for this model is constituted by a bank of four adaptive observers for monitoring these four actuators for the case of a simple fault [20].

The faulty model for the first actuator ( in Q ) is:

(25) ) x ( g 1 ) x ( g 2 ) x ( g 3 ) x ( g 4
are the four columns of the matrix ) x ( g , the corresponding observer is given by: (21)

A Comparative Analysis of Two Formulations for Actuator Faults... et al.

) x ( C y ) x ( Pg ) x x ( 2 ) x x ( H ) x ( g Q ) x ( g Q ) x ( g Q ) x ( g ) x ( f x 1 1 1 1 1 u 1 1 u 1 w 4 r 3 L 2 1
Where 1

x is the estimation of the state vector and 1 u is the fault estimation for the first observer. The residual 1 r is given by:

y y ) t ( r 1 1 (27)
The three other observers 2 3 and 4 ) have the same form.

In the case of multiple faults, firstly we should create a bank of four adaptive observers for the fault detection and identification. Secondly, to isolate the fault, we crate four banks of four adaptive observers where we use these four estimation i from estimation vector.

Synthesis of the observer using the second method

We will treat the fault that can occur at the one of the four actuators of the system

( in Q L Q r Q and w Q ).
Each of these actuators is divided into 5 parameter intervals, for each of them, a parameter filter is built. The values of the parameter filters for in Q

, L Q , r Q and w Q are shown in the following tables:

Q ( 0 in Q =2500 l/h) No 1 2 3 4 nominal a in Q b in Q Table.2. The values of the parameter filter of L Q ( 0 L Q =43 l/h) No 1 2 3 4 nominal a L Q b L Q (26)
IJ-STA, Vol. 7, N°1, April, 2013.

Q ( 0 r Q =1800 l/h) No 1 2 3 4 nominal a r Q b r Q Table.4. The values of the parameter filter of w Q ( 0 w Q =600 l/h) No 1 2 3 4 Nominal a w Q b w Q Let 1 j corresponds to parameter in Q . The isolation observer for the th i interval [ ) i ( a j u ) i ( b j u
of the st 1 actuator parameter is given by:

) S S ( k X V Q Q X V Q X ) S S ( k X V Q Q X V Q X ) S S ( k X V Q Q X V Q X ) S S ( k Y Y 1 ) S C ( C Q ) S S ( V S ) S S ( k ) X X ( V Q ) X X ( V X ) S S ( k ) f 1 ( ) X X ( V Q ) X X ( V X ) S S ( k f ) X X ( V Q ) X X ( V X ) S S ( k Y 1 ) S S ( V S ) S S ( k ) S S ( V S
To show in detail the fault isolation algorithm we have chosen the example where the

faulty actuator parameter is h / l 2420 Q f in . L Q , r
Q and w Q are maintained at their nominal value.

We have applied a fault at time days 50 t f in the first actuator in Q . In figure (4)

we presented the default effect on the six process outputs and the four residuals i r associated to the four observers. At the beginning the four residuals needs a short time period to converge. From the figure, we see that all residuals leave zero at days 50 t but after a very short period, ) t ( r 1 that corresponds to the input in Q return to its initial value.

While we observe two possible situations for the three others residuals: that is to stabi- that results are similar with the case without noise, so we can say that the fault s effect on outputs is independent of the noise vector.

The noise that occurred on the system have a influence on residuals, but the effect can not be inhibited us to detect the fault. Therefore, we conclude that we have isolated a fault in the first actuator by using the same method that is developed later.

Case1: Single fault

We have applied a fault at time days 50 t f in the first actuator in Q Figure 9 shows the default effect on the six process outputs. Figure 10 show that the filter of the 1 st interval does not send the non containing signal. This is the case where j s and the interval contains the faulty parameter value.

Therefore the fault is on in Q and in the first interval.

It shows also that after f t , the signals of two observers estimation errors are always l and we can assume that the parameter in Q is the faulty actuator parameter. adaptive observers. Though it is not so accurate as the detection and isolation results based on the nd 2 method, but it requires less computation and it is effective for nonlinear systems diagnosis.

The use of an interval notion contributes to the fault detection speed in a positive way and it is also fits large kind of nonlinear dynamics systems. The only required conditions for the type of the nonlinear system is that the dynamic of the system is a monotonous function with respect to the considered parameter. This method does not need any parameter identification procedure. It is proven that if the parameter intervals are small enough the isolation speed will be fast enough. But it can not solve the problem of multiple faults. This problem consist the interest of our future works

Conclusion

Fault detection and isolation for nonlinear dynamics systems is the subject of this paper. The objective is to compare two methods based on the model. Experimental results show that the two detection and isolation methods are both effective and more accurate than others methods.

The first method using adaptive observers and the isolation can be carried out for the single and the multiple actuator faults, but the isolation speed is not ideal. However the second one which is based on parameter intervals can solve this problem but only for the single actuator fault. Some simulation results illustrate these advantages.

In our work we only focus on the faults of the actuator parameter, that is why one interesting future research direction is to extend this 2 nd method firstly for multiple actuator faults and secondly to sensor fault isolation problem for nonlinear dynamic systems.

Fig

  Fig 1. The FDI based model scheme

  Then we will create m banks of adaptive observers, one for each actuator, who will have the same form as in (9). The corresponding residual of the fault isolation is given below:

Fig. 2 .

 2 Fig. 2. The conventional activated sludge schemeIn the activated sludge process, a bacterial biomass suspension is responsible for the removal of pollutants. The fundamental phase of the mathematical modeling for the processes of water treatment by activated sludge consists in determining the reaction rates of the macroscopic variables of the system to know the rate of: biomass growth, substrate degradation and dissolved oxygen uptake. These variables, as well as inputs and outputs, are collected in mathematical expressions constituting the model of the process. The mathematical model[17] of the activated sludge process is based on the equations, resulting from mass balance considerations, carried out on each of the reactant of the process:

Fig. 4 .A

 4 Fig. 4. Outputs process and residuals i r (single fault)

Fig. 5 .

 5 Fig. 5. Outputs process and residuals i r with output noise (single fault)

Fig. 6 .

 6 Fig. 6. Residuals i r for to the detection and identification bank

Fig. 8 .

 8 Fig. 8. Outputs process (No fault)

Fig. 9 .

 9 Fig. 9. Outputs process (single fault)

Figure 11

 11 Figure 11 presents the results of the nd 2 parameter filter of L Q . Since the fault is not on this parameter, so the sign of the prediction errors ) t ( a

Fig

  Fig. 10.

Fig. 11 .

 11 Fig. 11. tion of the 2 nd interval

Table . 1

 . . The values of the parameter filter of in

Table . 3

 . . The values of the parameter filter of r

Table . 5

 . . The values of the isolation time.IJ-STA, Vol. 7, N°1, April, 2013.

						50.2
	Faulty actuator	2410	2420 2435	2460	2480
	parameter
	Isolation time (days) (1st	7	5.5	3	4	5.1
	meth)
	Isolation time (days) (2nd	1.3	0.20 0.20	0.1	0.1
	meth)
	So we can also do a comparison by using	L Q , r Q or w Q and we can conclude that
	these experimental results based on parameter intervals are faster than those based on

A Comparative Analysis of Two Formulations for Actuator Faults... et al.

Simulation and comparison results

In this section, we will give the results from the two developed methods for fault actuator and visualize the process outputs, the residuals and the fault estimation.

Initially, we will give the results without fault, and then we will observe the case of a simple and multiple actuator faults.

The first method: Adaptive observer Case1: No fault