Taha Benhaddou
email: taha.benhaddou@gmail.com

Pierre Stephan

Alain Daidie

Feras Alkatan

Clement Chirol

Jean-Baptiste Tuery

Effect of axial preload on aerospace fastened joints durability

Keywords: axial preload, process for durability, fastened joints, fatigue life enhancement, finite elements analysis

The main aim of the present work is to study the effect of axial preload on the durability of aerospace fastened joints. For that, experimental and numerical approaches have been adopted to demonstrate the beneficial effect of axial preload on the fatigue life of representative aerospace fastened joints. In the experimental approach, an innovative tightening technique, based on iterative tightening, has been used to reduce the generated scatter induced on preload when using conventional torque tightening technique. The main benefit besides the use of this alternative tightening technique is to allow accurate preload introduction without geometry or stiffness modification, thus allowing to determine precisely the preload effect. The results shows the effect of preload on static resistance, fatigue life and failure mode of fastened joints. In the numerical approach, a 3D finite element model has been developed and the analysis of its behavior confirms and explains experimental observations. Analysis of longitudinal stresses through critical paths shows that the effect of high preload is beneficial due to the reduction of hydrostatic pressure and shear stress amplitude, leading to higher fatigue lives. Application of SWT and Crossland multiaxial fatigue criteria allows to correlate to experimental observations.

1-Introduction

Joining technologies are essential to the design and production of most mechanical structures. In particular, and when the mechanical structures are large as for aero-structures, it is often necessary, for technical and / or economic reasons, to conceive them in the form of a set of smaller dimensions parts that can be joined together. Fastened joining technique is widely used because of its ease of use, its efficiency and its relative low cost. However, fastening requires drilling holes in joined plates, leading to stress concentration areas and promoting fatigue cracks initiation from hole edges.

Several design and manufacturing processes can be used to improve the fatigue resistance of double lap bolted joints:

1-Interference radial mounting, consisting of a negative fit between the bolt shank and the bore of the plates holes (Duprat et al. [START_REF] Duprat | Fatigue life prediction of interference fit fastener and cold worked holes[END_REF]). 2-Cold expansion, consisting of radial expansion of the plate's holes by pulling a tapered mandrel, pre-fitted with a lubricated split sleeve, through the plate's holes, as shown by Yongshou et al. [START_REF] Yongshou | Analytical and experimental investigation of fatigue and fracture behaviors for anti-double dog-bone riveted joints[END_REF] for rectangular plates and Huang et al. [START_REF] Huang | Improving the fatigue life of 2297-T87 aluminum-lithium alloy lugs by cold expansion, interference fitting, and their combination[END_REF] for lugs. 3-Axial preload, commonly introduced through torque application (Minguez and Vogwell [19]).

For these three joining processes, the stresses distribution around the plate's hole is modified with the introduction of axial and/or radial compressive residual stresses, making it possible to improve fatigue strength.

Several works deal with improving the fatigue strength of assemblies as a function of the interference, expansion and / or preload applied to both single-lap and double-lap bolted joints. Duprat et al. [START_REF] Duprat | Fatigue life prediction of interference fit fastener and cold worked holes[END_REF] showed a significant improvement in fatigue life in the case of interference mounting with an optimal fatigue life obtained around an interference value of 1.5%. The shown improvement is related to the residual tensile stresses field near the hole leading to a significant reduction of the cyclic stresses amplitude.

A comparative study conducted by Lam [START_REF] Lam | A comparative study on the effects of interference fit and cold expansion on the fatigue life of cracked holes[END_REF] on the fatigue behavior of cracked holes using either interference mounting or cold expansion processes showed that for a low loading ratio and one low starting crack, interference is not as effective as cold expansion to extend the lifetime. On the other hand, the interference is less sensitive to the loading ratio and to the initial crack size.

The effect of torque tightening on static and fatigue resistance of single and double lap bolted joints has been addressed through many experimental research studies. Several authors had shown the positive effect of high torque on such joints. Shankar and Dhamari [START_REF] Shankar | Fatigue behaviour of aluminium alloy 7075 bolted joints treated with oily film corrosion compounds[END_REF] showed this beneficial effect on double lap joints and Chakherlou et al. [START_REF] Chakherlou | Experimental and numerical investigation of the effect of clamping force on the fatigue behaviour of bolted plates[END_REF] on simple preloaded plates. The reported increase can be very significant, and reaches an asymptote after which the fatigue life stays constant or can even decrease. Indeed, Wagle and Kato [START_REF] Wagle | Ultrasonic detection of fretting fatigue damage at bolt joints of aluminum alloy plates[END_REF] have shown that fatigue life of double lap shear joints decreases if the applied torque is too high, due to fretting fatigue. Therefore, determining optimal preload value and its allowable scatter is of major interest to permit maximal fatigue resistance of the joint. In addition, the allowable scatter determination should allow to select the most appropriate tightening method, in relation to the industrial constraints.

The improvement of fatigue resistance, due to application of high torque, is leading to the fatigue crack initiation position modification: for low torque values, the crack initiates from hole edges. However, when high torque is applied, the crack initiates in the gross section, outside of compressed area (Schijve [START_REF] Schijve | Fatigue of Structures and Materials[END_REF]). A comparable modification has been reported for joints with a high degree of interference or expansion percentages (Hahn et al. [START_REF] Hahn | Structural shear joints: Analyses, properties and design for repeated loading[END_REF]).

Preload control is of great importance to guarantee the mechanical strength of single and multimaterial fastened joints, whatever the type of external solicitation. In particular, under cyclic transverse loading, an insufficient level of preload does not allow to exploit the load transfer capacity of the joint interfaces. A very high level of preload is not desirable either, since it generates fretting-fatigue phenomena or excessive fastener underhead over-pressure which can result a detrimental effect on the fatigue life of joints and structures. Therefore, preload controlled tightening produces and maintains a balance between the modes of load transfer mechanisms, making it possible to fully exploit their potential and to obtain an optimum clamping of the joint (Benhaddou et al. [START_REF] Benhaddou | Pre-tensioning effect on fatigue life of bolted shear joints[END_REF]). The control of preload-linked parameters such as the nature of the joint interface, the fasteners stiffness's and the radial fitting nature could lead to perform further optimization of fastened joints. In the case of linear or non-linear axial loadings, a controlled preload level improves the service life of the joint thanks to an external load filtering effect which reduces the severity of external loads, especially in stress concentration areas (Bickford [3]).

In the industrial side, different methods exist to control the preload of a threaded fastener. The most commonly used is torque tightening. The nominal torque necessary to tighten the bolt to a given preload is calculated using a relationship between torque and the resulting bolt pretension. However, a variation of 30% in installed preload is common with torque tightening methods (Bickford [3]).This torque-preload relationship is significantly affected by the used materials, the geometry of the assembly, the surfaces finishing, the lubrication and the surface treatment, the heat treatment of the fastener, the threads type, the tightening speed ... In particular, friction coefficients variability within the assembly plays a predominant role in the definition of the relationship between torque and preload and justifies the scatter commonly generated on preload when using torque tightening. Consequently, using alternative tightening techniques to reduce the scatter, will allow reducing the number of fasteners or their diameter whilst using the maximal capacity of the bolt material. Using more cost-effective bolt material with a lower tensile strength can also be considered.

In addition, friction coefficient between joined plates plays also a significant role in fatigue resistance of bolted joints. Interfay sealant application between joined plates have a negative effect on their fatigue resistance due to the decrease of global friction coefficient (Boni and Lanciotti [START_REF] Boni | Fatigue behaviour of double lap riveted joints assembled with and without interfay sealant[END_REF]).

The combined influence of interference mounting and torque tightening (Abazadeh and al. [START_REF] Abazadeh | Fatigue life estimation of bolt clamped and interference fitted-bolt clamped double shear lap joints using multiaxial fatigue criteria[END_REF]) or cold expansion and torque tightening (Chakherlou et al. [START_REF] Chakherlou | Effect of cold expansion and bolt clamping on fretting fatigue behavior of Al 2024-T3 in double shear lap joints[END_REF], extended by Taghizadeh et al. [START_REF] Taghizadeh | Prediction of fatigue life in cold expanded fastener holes subjected to bolt tightening in Al alloy 7075-T6 plate[END_REF]) has been studied and showed an additional enhancement of fatigue resistance when the two processes are used. In the case of torque effect, numerical simulations that were performed shows that stress distribution around the plate's holes is greatly affected by the torque or preload value. In keeping with the reported literature, understanding preload effect is the main aim of the present study. Indeed, the dissociation of preload and torque is the focal point of the present research. Thus, preload effect, obtained by accurate tightening mean instead of torque tightening technique, can be rigorously assessed.

2-Experimental approach 2.1-Introduction

In order to demonstrate the beneficial effect of preload on double lap fastened joints, an extensive testing campaign was conducted. For the considered double lap joint, two distinct load transfer mechanisms are interacting for this kind of joint. The first one is the friction mode, allowing to transfer load through the adherence between plates, due to applied preload. The second one is the bearing load, for which load is transferred through the contact between the fastener shank and the plates holes.

2.2-Tightening operation

In order to achieve application of a precise preload, a dedicated tightening mean based on iterative tightening technique was used. As stated in the introduction, tightening torque induces a fairly dispersive preload within the joint (which can be as much as +/-30% uncertainty over the final preload value). The applied technique within the experimental campaign is based on the iterative technique and reduces this uncertainty to a measured value of + 10% on the final value, allowing to apply precise preload whilst avoiding modifying the joint geometry and stiffness through the use of strain or tension indicating washers. The principle of the iterative tightening is to apply a succession of tightening and untightening operations in order to better evaluate the torque generated by friction between the surfaces in contact (nut / plate, screw / plate and screw / nut), thus allowing to achieve a less scattered preload. Figure 1 shows an overview of the tightening mean and an example of the tightening sequence, representing the evolution of preload and torque in relation to time and describing the different steps of iterative tightening: overcoming locking torque then determining joint stiffness through geometric and friction parameters. Indeed, once the establishement torque exceeded (accomodation of the mechanical and contact gaps, and locking torque), the torquepreload relation is linear. A sequence of tightening and loosening is applied to guarantee an accurate preload. In fact, exploring the linear tightening-loosening curve makes it possible to estimate the preload values in a gradual continuous manner: the preload can be monitored as a function proportional to the difference between the torque applied on tightening, especially the maximum torque at the stopping point, and the torque applied on loosening, especially the maximum loosening torque, the difference being divided by (p/π), a quantity proportional to the fastener pitch p (Chastel and Rey [START_REF] Chastel | Procede pour le controle et l'asservissement de la tension d'un organe filete[END_REF]). The choice of preload levels has been performed in relation to the Ultimate Tensile Strength (UTS) of the considered fastener-nut association. This value is equal to the minimum required of the two separate components, which is the UTS limit of the fastener (23.5 kN).Therefore, to illustrate the effect of preload on fatigue life of double lap fastened joints, the tested configurations correspond to low (25%UTS = 5.9kN), medium (50%UTS = 11.7kN) and high (75%UTS=17.6kN) preload levels. Furthermore, long term preload relaxation was not measured, but it is assumed to be very limited in metallic joints with self-locking nuts (Bickford [3]).

2.3-Static tests procedure

Prior to fatigue tests, static tests were carried out on an 100kN Instron machine equipped with a Zwick strain gauge with a test speed of 1mm / min and a measured distance given by a Zwick macro extensometer following NASM 1312-4. A repeatability of three specimens was applied per test. The determination of the elastic load is based on the determination of the secondary module (following NASM1312-4 method).

2.4-Fatigue test procedure

Fatigue tests consisted of the application of a sinusoidal monotonic load type, of a ratio R = σ min / σ max = 0.1. The frequency was set to 10Hz and the tests were carried out at ambient temperature on hydraulic machines (capacity: ± 100kN to ± 250kN). The value of the external force was chosen in order to characterize the fatigue strength of the joint in the following fatigue life range: 10 4 to 2.10 6 cycles. Static tests that were previously conducted helped to determine the values of the initial external load to apply in fatigue tests. Finally, fatigue tests were performed according to NASM1312-11 standard.

2.5-Description of the joint:

The considered joints are composed of representative aerospace joints, the dimensions and the overview of the mounting are described in Given the architecture of double lap joints, aluminum doublers (Ref 5, Figure 2) were included to align the directions of external forces. The length of the fastener shaft was designed to ensure the integral transmission of the bearing force. The plate's hole diameter was manufactured to allow a very low clearance radial adjustment (between 0 and 20 μm). A chamfer has been applied to the hole sides dedicated to receive the fastener head to allow correct seating of the fastener. Finally, the surfaces of the plates were treated to incorporate oxide layer by the means of chromic acid anodizing process and paint primary protection has been applied to be as close as possible to industrial applications (Skorupa and Skorupa [START_REF] Skorupa | Riveted lap joints in Aircraft fuselage[END_REF]).

2.6-Static Tests results:

Figure 3 shows the load-displacement curves obtained from static tests, performed on low, medium and high preload loaded specimens. It is noted that the resulting force-displacement curves are quite similar, with a more pronounced preload effect on the yield load than on the ultimate load. Thus, the preload value has a neglected impact on the ultimate load and the ultimate displacement of the joint. The observed failure mode is the external plate net section failure, in accordance with the analytical and experimental observations of Bickford [START_REF] Bickford | Introduction to the Design and Behavior of Bolted Joints, Fourth Edition: Non-Gasketed Joints[END_REF] and extended by Guillot [START_REF] Guillot | Calcul des assemblages vissés Assemblages de pièces planes de faibles épaisseurs[END_REF]. This failure mode indicates that the strength of the joint is mainly determined by the working section of the joined plates. The results of these tests are therefore in agreement with the conclusions drawn by Dang Hoang et al. [START_REF] Hoang | On failure mode analysis in a bolted single lap joint under tension-shearing[END_REF] and Chessa et al. [START_REF] Chessa | A Verified and Validated Study of Quasi-Static Bolted Lap Joints[END_REF] concerning the influence of axial preload on the static behavior of the joint. The slight increase in elastic limit is mainly due to the difference in stiffness between the adhesion phase, i.e. when the external loading is transferred entirely by friction, and the sliding phase, i.e. when the bearing capacity of the fastener contributes to the load transfer, given that the transition between these two phases is directly related to the preload force applied. Finally, concerning the negligible effect of preload on the ultimate load of the joint, once the macro-slipping exceeded, the bearing capacity takes gradually over the loading transmitted through friction interfaces, leading to plasticization of fastener shaft and complete preload loss.

2.7-Fatigue Tests results:

S-N data and Wöhler curves resulting from fatigue tests are given in Figure 4 and Table 1 summarizes the evolution of the admissible stress for 10 5 cycles as a function of the applied preload. It is noted that the improvement of the fatigue life is very significant during the transition from 5.9 to 11.7kN, and a little less significant when increasing preload from 11.7 to 17.6kN. One can also notice that the result's dispersion is more significant at low preload (5.9kN), which is probably related to the tooling that has been used to enable the iterative tightening process: indeed, the limit of the screwdriver used is estimated to be about 5.5kN.

Contrary to the negligible preload effect on the static behavior of the reference joint, the observed results show a beneficial effect of the preload on fatigue life of fastened joints. Therefore, the results are showing the same tendencies that the ones highlighted by the work of Minguez and Vogwell [START_REF] Minguez | Effect of torque tightening on the fatigue strength of bolted joints[END_REF] as well as Shankar and Dhamari [START_REF] Shankar | Fatigue behaviour of aluminium alloy 7075 bolted joints treated with oily film corrosion compounds[END_REF]. The failure modes are also affected by the level of preload introduced. It is found that the fatigue crack initiation point migrates from the specimen' net section for low preload levels to the specimen' gross section for high preload levels (Figure 5). This change in failure mode indicates that the force distribution and the fastener-plate interactions are affected by preload and shows that a high level of preload allows to protect the stress concentration area and to initiate the failure outside of the compression zone, created through axial preload. Subsequent to fatigue tests, observation of the interface between plates using optical microscopy means shows that the tangential contact generated through preload has differently affected the overall contact zone (Figure 5). Indeed, a relatively wide contact zone can be observed in which it is possible to note a degradation of the initial surface state which is manifested by a paint stripping, which is synonymous to a high contact pressure conjugate to the cycles of tangential micro slip imposed by the external loading.

3-Numerical approach

Subsequent to the experimental findings reported in the previous section, the numerical approach, based on finite elements method, has been developed as a complementary analysis to assess, confirm and predict preloaded joints behavior that has been described up to now.

3.1-Modeling set-up

A numerical model, representative of experimental specimens, has been developed to quantify in a global way the effect of preload. Figure 6 presents a cross-sectional view of the reference joint. Essential geometrical details have been integrated into the numerical models such as the head radii, plate chamfer and the nut recess in order to ensure a good distribution of the preload in the joint and a correct estimation of the mechanical stresses in high stress concentration areas (namely under-head area and first engaged threads). A 3D numerical model has been developed under Abaqus and solved using the implicit solver. The static loading history consisted of three steps: a first step dedicated to preload application, applied through the bolt-load feature. The second step transforms this preload into an imposed displacement, and the third step that allows to apply external loading to the gross section of the internal plate while encastring the extremity of the external plate. Adequate boundary conditions have been imposed to match with the experimental tests: the axial displacement of a defined area of external and internal plates has been fixed, reflecting the action of machine jaws. Only a half joint has been considered with a plane symmetry conditions, thus allowing to reduce the calculation time required to perform the structural analysis. Aluminum material test data has been used to model the behavior of plate's materials using linear isotropic hardening law (plastic data were extracted from MMPDS [START_REF]Metallic Materials Properties Development and Standardization (MMPDS) Handbook[END_REF]). The materials of the other parts were modeled with an elastic behavior whose characteristics are detailed in Table 2.

Joint components

3.2-Contact and interactions

Figure 6 shows the different interactions that has been considered between the different elements of the joint to account for the contact conditions. The interactions I1, I4 are reflecting the contact conditions between the fastener/nut and the external plates. A tangential and normal behavior has been considered, with a friction coefficient of 0.2. The same type of contact interaction has been considered for interactions I3/I4, representing the contact between plates, but with a friction coefficient of 0.4. The interaction I5 stands for the normal contact between the plate's bores and the fastener shank, and the constraint T1 representing the fastener/nut contact has been considered as a tie constraint. This constraint couples all the degrees of freedom of the two surfaces in contact to guarantee the integral transmission of the movements [START_REF]Dassault Systèmes Simulia[END_REF]. The study performed by Montgomery [START_REF] Montgomery | Methods for Modeling Bolts in the Bolted Joint[END_REF] shows that a tie constraint condition between the fastener and the nut allows to correctly represent the distribution and the amplitude of the generated stresses by preload, and is substantially equivalent to the case where the full geometry, including threads representation and contact interactions between the fastener and the nut is considered.

The surface-to-surface discretization method was used because it limits the penetration between the surfaces in contact and allows to obtain more accurate results than the node-tosurface formulation. This observation was also shown by McCarthy et al. [START_REF] Mccarthy | Three-dimensional finite element analysis of single-bolt, single-lap composite bolted joints: part I-model development and validation[END_REF] who pointed out that the node-to-surface discretization method generated an erroneous radial deformation field. In general, the master surfaces have been assigned to the surfaces of the most rigid instances in order to exclude penetration slave surface into master surface. The contact surfaces have been decomposed into several surfaces to ensure continuity of surface normals.

The considered mesh consists essentially of C3D8R elements type: 8-node 3D continuum elements, with reduced integration. The choice of this element type was motivated by various numerical studies (Maximov et al. [START_REF] Maximov | Enhancement of fatigue life of net section in fitted bolt connections[END_REF]) and given the fact that the accuracy/calculation time of this element type is suitable for the study. Particular attention has been paid to the partitioning of the parts to have a mesh set and to have a continuity of the stress fields from one part to the other.

3.3-Mesh convergence

Given the interest in determining preload effect on mechanical stresses then performing numerical fatigue life analysis, ensuring that the results are not affected by the mesh size is primordial. For this purpose, a mesh convergence study has been conducted. Figure 7 shows the curve representing the stress concentration factor, representing the maximal stress at stress concentration point (named A, shown in Figure 7) divided by the stress in the gross section against mesh density. The stress concentration coefficient K is determined at point A as a function of the size of the elements (ratio S/R). The results obtained lead us to choose a ratio between the hole edge mesh size S and the hole radius R equal to 0.1. Local mesh refinement has been applied to the area close to the hole, whilst larger elements has been applied to the remaining plate material.

3.4-Model global validation under static loading

First level numerical validation was dedicated to the global response of the joint under static loading. Figure 8 shows the numerical / experimental comparison of the behavior of the reference joint under static loading (for reasons of readability, only 2 of the 3 tested preload levels are displayed). The numerical model makes it possible to accurately predict the behavior of the joint, whether in the elastic domain or the plastic domain. Stiffness of the joint is correctly predicted either for low and high preload levels, the characteristic data of the static resistance of the joint, including elastic and plastic stiffness's, macro-slipping load and displacement and yield load shows also good correlation.

It is therefore visible that for relatively low external loads, increasing the preload and thus the macro-slipping load has the noticeable effect of making the joint work mainly in the adhesion phase, characterized by the greatest stiffness, this effect consequently help to increase the durability of the joint given that cyclic loads, leading to high cycle fatigue lives are relatively low. Analysis of friction formulation law to represent the tangential behavior has also been performed to choose the most reliable friction law. It was found that the use of the Lagrange type formulation acts mainly on the adhesion phase: the stiffness of this phase is 10 to 15% higher than that produced by the classical penalty formulation. In terms of computation time, calculations using a Lagrange multiplier law tend to diverge for high preload, probably due to the nonlinearity generated by the change of status between slip and adherence of the nodes in contact. For lower preload, the Lagrange multiplier convergence rate is significantly lower than the standard penalty method. On the other hand, considering the exponential type law, including static and kinetic friction coefficients, the main effect was linked to the transition between adhesion and bearing phases and shows a gradual decrease in carried load, according to the coefficients of the equation under consideration. Nevertheless, for radial clearance values less than 30μm, it has been observed that the exponential law has a behavior that is substantially identical to the standard penalty laws. For these reasons, the isotropic penalty formulation will be considered for the rest of the study.

In addition, the static tests allow through the identification of the macro-slipping load and displacement, to identify and to confirm the values of global friction coefficient used between the plates.

3.4-Effect of preload

In order to assess the stress distributions in the joint, following the introduction of several preload values and under an external stress of 100MPa, calculated on the net section of the specimen (see Figure 9), seven paths were selected as described in Figure 9. This aim of this section is to analyze the mechanical stresses along the most critical paths of the joined plates in order to better understand the influence of the preload on the mechanical loading transferred through the joint. Figure 10 shows the evolution of the stresses σ 11 through 3 selected critical paths for different levels of preload. Looking at the evolution of the stresses σ 11 along the path PA, it is observed that the longitudinal stress remains substantially high for low preloads. Increasing the preload magnitudes decreases in a significant way the magnitudes of longitudinal stress, and this decrease occurs through the hole thickness of the external plate. Thus, the stresses obtained at the edge of the hole at the interface between the plates depend strongly on the level of preload with a significant decrease between a low preload and a high preload.

Analysis of load transfer modes, realized through monitoring contact forces through the different contact surfaces, confirms the observed tendency: through the application of high preload values, friction mode is used to transfer a high amount of external load, allowing to reduce the contribution of bearing mode, and thus reducing the stress concentration factor.

Analysis of failed specimens shows that failure mostly occurs at the interface between external and internal plates, and this was independent of the preload level and joint architecture. Analysis of stresses evolution through the paths PTi and PCi, 2 paths that are located on the interface, is therefore considered and shown in Figure 10. The evolution of stresses σ 11 along the path PTi and PCi confirm the tendency highlighted on PA path: stress concentration at point A, at the edge of a hole (for L = 0 or for θ = 90°), is very high for null or low preloads.

Radially, the longitudinal normal stress tends towards a horizontal asymptote when moving away from the hole edge, whatever the preload. At the edge of a hole, the evolution of the stress σ 11 as a function of the preload is significant: at zero preload, a high value of tensile stress is measured which illustrates the phenomenon of stress concentration. When the preload increases, the value at the edge of the hole evolves very substantially and becomes much lower at high preload. Indeed, the axial compression stresses introduced in the preload phase make it possible to reduce the value of σ 11 when the external force is introduced for a sufficiently high preload. Finally, the stresses analysis has also been conducted for the remaining paths and indicates the same tendencies observed for PTi and PCi paths.

3.5-Extension to fatigue loading

The numerical model has been extended to assess the fatigue behavior: subsequent to preload application (from t=0s to t=1s) and preload maintain (t=1s to t=2s), cyclic loading has been applied with the aim of evaluating the main critical values leading to fatigue crack initiation (t=2s to t=13s).

For most of the existing fatigue estimation criteria dedicated to metallic materials and structures, it is possible to state three observations (Qilafku et al. [START_REF] Qilafku | Fatigue of specimens subjected to combined loading. Role of hydrostatic pressure[END_REF]):

-The fatigue life depends strongly on the shear stress variation amplitude. It decreases when the maximum amplitude of shear variation increases. -The fatigue life is not very sensitive to the mean value of the shear stress.

-The fatigue life depends strongly on the value of the maximum hydrostatic pressure, which is equal to the average of the three normal stress components. A compression state (corresponding to a negative hydrostatic pressure) substantially increases the duration of fatigue crack initiation.

These different qualitative findings seem to be accepted by specialists and have led to many criteria (Susmel [27]). Indeed, the aim of this section is to evaluate the effect of preload on the two following variables: hydrostatic pressure and amplitude of shear stress variation.

The evolution of the hydrostatic pressure at stress concentration point (point A, Figure 9) has been plotted in Figure 11 for an external loading of 100MPa. Compressive axial stresses, introduced through preload, considerably modify the magnitude of the hydrostatic pressure throughout the loading cycle, leading to negative mean and maximal hydrostatic pressure through the loading cycle, this reduction in hydrostatic pressure substantially increases the duration to fatigue crack initiation. Loading paths corresponding to the shear stress variation during the fatigue cycle are plotted in Figure 12 for two preload levels: 5.9 and 17.6kN and for an external loading of 100MPa. The calculation of the shear stress amplitude was performed on each increment of loading, by determining the critical plane (P1 to P11, corresponding to 11 increments of loading) and plotting the shear stress variation through it. The same axis limits were selected to allow the visualization of the changes in terms of amplitude but also in terms of shape. The loading paths thus have a more refined and less extended shape under the effect of a high preload value, and the maximal amplitude of the shear stress variation 𝜏𝜏 𝑎𝑎,𝑚𝑚𝑎𝑎𝑚𝑚 decreases, leading to a higher fatigue life.

3.6-Application of SWT and Crossland criteria

In double lap fastened joints, the stress and strain distribution near the bolt hole is multiaxial as a result of preload and external loading. Therefore, in this study, fatigue life prediction was performed using Crossland and SWT method. The choice of these two methods was motivated by previous research studies, stating that these criteria provides better calculation accuracy than other methods for a multi-axial stress state problem (Duprat et al. [START_REF] Duprat | Fatigue life prediction of interference fit fastener and cold worked holes[END_REF]).

The SWT criterion, first proposed by Smith et al. [START_REF] Smith | A stress-strain function for the fatigue of metals (stressstrain function for metal fatigue including mean stress effect)[END_REF] incorporates both cyclic strain range and maximum stress and was selected because it had the capability to predict multi-axial fatigue life (fatigue life due to complex stress states where the three principal stresses are nonproportional or whose directions change during the loading cycle). The equation ( 1) below presents the SWT formula that has been used:

∆𝜀𝜀 2 𝜎𝜎 𝑚𝑚𝑎𝑎𝑚𝑚 = (𝜎𝜎 𝑓𝑓 ′ )² 𝐸𝐸 (2𝑁𝑁 𝑓𝑓 ) 2𝑏𝑏 + 𝜎𝜎 𝑓𝑓 ′ 𝜀𝜀 𝑓𝑓 ′ (2𝑁𝑁 𝑓𝑓 ) 𝑏𝑏+𝑐𝑐 (1)
Where ∆𝜀𝜀 2

is the maximum principal strain amplitude, 𝜎𝜎 𝑚𝑚𝑎𝑎𝑚𝑚 is the maximum stress on the maximum principal strain plane, E is the elastic modulus, N f is the number of cycles to crack nucleation under a certain stress and strain level, 𝜎𝜎 𝑓𝑓 ′ (=475MPa) is the fatigue strength coefficient, 𝜀𝜀 𝑓𝑓 ′ (=0.27) is the fatigue ductility coefficient, b (=-0.125) is the fatigue strength exponent and c (=-0.595) is the fatigue ductility exponent.

The Crossland criterion (Crossland [START_REF] Crossland | Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel[END_REF]) is a stress based multiaxial fatigue criterion which uses the second invariant of deviatoric stress tensor and maximum hydrostatic pressure in its equation [START_REF] Benhaddou | Pre-tensioning effect on fatigue life of bolted shear joints[END_REF].

�𝐽𝐽 2,𝑎𝑎 + 𝜆𝜆𝜎𝜎 𝐻𝐻,𝑚𝑚𝑎𝑎𝑚𝑚 = 𝜎𝜎 𝑓𝑓 ′ (2𝑁𝑁 𝑓𝑓 ) 𝑏𝑏 (2)
Where J 2,a is the amplitude of second invariant of deviatoric stress tensor, σ H,max is the maximum value of the hydrostatic pressure and 𝜆𝜆 is a material dependent constant. Given that components of the stress tensor are not proportional to loading and the principal directions vary during the loading, the amplitude of second invariant of deviatoric stress tensor has been replaced by the maximum amplitude of equivalent shear stress variation, thus allowing for more adapted calculations. Specific details on this calculation technique has been developed in previous research (Benhaddou et al., [START_REF] Benhaddou | Pre-tensioning effect on fatigue life of bolted shear joints[END_REF]), leading to the main equation used to calculate the fatigue life which is reported in the equation [START_REF] Bickford | Introduction to the Design and Behavior of Bolted Joints, Fourth Edition: Non-Gasketed Joints[END_REF].

𝑁𝑁 𝑓𝑓 = 10 5 ( 𝐴𝐴 𝑁𝑁 𝑇𝑇 𝑒𝑒𝑒𝑒,𝑎𝑎,𝑚𝑚𝑎𝑎𝑚𝑚 + 𝐵𝐵 𝑁𝑁 . 𝜎𝜎 𝐻𝐻,𝑚𝑚𝑚𝑚𝑚𝑚 ) 𝑝𝑝 (3)
𝐵𝐵 𝑁𝑁 (=0.875), 𝐴𝐴 𝑁𝑁 (=179.4) and 𝑝𝑝(=5) were given by two fatigue tests (at R=0.1 and R=-1) of the considered material and considering that for different loading ratios R on a logarithmic diagram (Log (Stress), Log (Fatigue life) forms a network of parallel straight lines of slope -p.

The estimated and the experimental fatigue lives are plotted in Figure 13 for tested specimens with preload values equal to 5.9, 11. The fatigue crack initiation site, shown as red spots, are reported in Figure 14. A unified legend has been applied and show the modification of crack initiation site for the two criteria: at low preload level, the crack initiation site is located at the hole edge of the net section. The crack initiation site migrates outside of the compressed area for high preload levels, leading to a gross section failure. This observation tends to confirm experimental observations reported in Figure 5.

4-Conclusion

The application of controlled axial preload, through the use of innovative tightening techniques, shows that high preload levels, enhances in a very significant way the fatigue life of double lap bolted joints. The failure area was also affected by the preload levels. Numerical model, allowing to analyze the effect of preload on longitudinal tensile stresses, shows that higher preload levels allows to reduce significantly the magnitude of longitudinal tensile stresses at stress concentration areas, allowing to reduce the severity of stress concentration zones and redistribute the stresses in a more effective manner. High preload allows for the reduction of hydrostatic pressure and shear stress amplitude, leading to higher fatigue lives for most of tested configurations. Fatigue life analysis, through the application of multiaxial criteria, provides a better understanding of the effects of preloading process as a fatigue life enhancement technique of fastened joints. Thus, the successful application of Crossland criterion, either in terms of fatigue life estimation and crack position determination, can be extended to the determination of optimal preload value, but also the corresponding scatter leading to the choice of the appropriate tightening mean. Finally, the choice of an average equivalent friction coefficient and a constant preload seems in good accordance with experimental results but the detailed effect of friction and preload evolutions during fatigue testing will be discussed more thoroughly in further studies.

Figure 1 :

 1 Figure 1 : Iterative tightening mean: Spindle overview (left) -example of iterative tightening sequence (right).

Figure 2 .Figure 2 )

 22 It includes the following elements: EN6114-K4 Ta6V titanium fasteners with a nominal diameter of 6.35mm (Ref 1, Figure 2), ASNA2531-4 Cadmium steel self-locking nuts (Ref 2, Figure 2) and 3 2024-T351 Aluminum joined plates: 2 external plates of thickness 2.5mm (Ref 3, and an internal plate of thickness 7.5mm (Ref 4, Figure 2). The distances to the edge have been chosen in accordance with Airbus Structure Design Manual and the joining operation was made without sealant.

Figure 2 :

 2 Figure 2: Joint overview: Mounting (a) -close-up view of the joint and dimensions in mm (b).

Figure 3 :

 3 Figure 3: Experimental static tests results of the joint of Figure 2.

Figure 4 :

 4 Figure 4: S-N data and Wöhler curves with regard to preload.

Figure 5 :

 5 Figure 5: Failure modes in relation to preload of the joint of Figure 2.

Figure 6 :

 6 Figure 6: Finite element model -boundary conditions (a), interactions (b) and mesh (c).

Figure 7 :

 7 Figure 7: Mesh convergence analysis.

Figure 8 :

 8 Figure 8: Simulated and experimental load displacement curves under a static loading.

Figure 9 :

 9 Figure 9: Definition of stress distribution paths (a), definition of applied load on gross section (b).

Figure 10 :

 10 Figure 10: Evolution of the stresses σ 11 along the paths, PA, PCi and PTi as a function of the preload, external loading of 100MPa.

Figure 11 :

 11 Figure 11: Evolution of the hydrostatic pressure during the loading cycle for two preload levels: 5.9 and 17.6kN.

Figure 12 :

 12 Figure 12: Plot of loading paths corresponding to shear stress variation for two preload levels: 5.9 and 17.6kN.

  7 and 17.6kN respectively. The horizontal axis represents the experimental life and the vertical axis represents the estimated life. Both axes are in log scale. The solid line indicates that the estimated results are equal with experimental tests results. The inner bound dashed line show an estimation in accordance with the life factor of 2. The outer bound dashed line show an estimation is in accordance with the life factor of 3. It is seen that the Crossland criterion agree with the experimental tests results very well and that SWT criterion tends to slightly overestimate fatigue lives, especially at high preload level. Consequently, Crossland criterion seems more adapted to the analysis of the studied configuration of double lap bolted joints.

Figure 13 :

 13 Figure 13: Correlation between predicted and experimental fatigue life.

Figure 14 :

 14 Figure 14: Location of minimal fatigue life for the different configurations (nut side is up).

Table 1 : Evolution of AFI with regard to preload.

 1

Table 2 : Elastic properties of considered materials.

 2

		Fastener	Nut	Plates
	Material	Titanium	Steel	Aluminium
	Young Modulus (GPa)	116	210	70
	Poisson ratio ()	0.30	0.30	0.33

Acknowledgements

The authors would like to acknowledge CALMIP platform for the generous grant of computing time (project P1121).

Highlights

-Effect of axial preload on double lap fastened joints durability has been studied.

-Innovative iterative tightening technique has been used to achieve an accurate preload level.

-High preload has a negligible effect on static life and beneficial effect on fatigue life.

-Numerical model allows to explain experimental findings through analysis of fatigue-linked parameters. Fatigue life analysis shows that Crossland criteria is well adapted for double lap fastened joints durability prediction.


F Golfier

L Van De Steene

S Salvador

F Mermoud

C Oltean

M A Bues

Impact of peripheral fragmentation on the steam gasification of an isolated wood charcoal particle in a diffusion-controlled regime

Keywords:

This study focuses on the effect of peripheral fragmentation during gasification of an isolated wood charcoal particle in the regime of diffusional limitations. In fact, all the models available in the literature fail to reproduce the specific changes in biomass charcoals for which conversion increases quasi linearly versus time, whatever the particle size. We assumed this discrepancy is partly due to the fragmentation phenomenon, i.e. the detachment of fragments from the surface of the particle, which is not usually taken into account in models. The classical assumption in percolation models is a critical value of porosity beyond which fragmentation occurs. We compared our model results with experimental data. We demonstrated that in the diffusion-limited regime, the direct extrapolation of such a method to a continuum model is not satisfying. A new criterion of fragmentation based on a critical porosity and a critical porosity gradient is proposed which considerably improves modelling of gasification charcoal particles.

Introduction

Gasification has emerged as a clean and effective way of producing gas from biomass, particularly from wood. The main transformation in such processes is the reaction between the charcoal -or char-(carbonaceous solid residue after wood pyrolysis) and steam. This heterogeneous C-H 2 O reaction produces the majority of H 2 and CO that can be valorised either for electricity or for the production of biofuels. In fixed bed gasifiers, this endothermic reaction occurs at temperatures in the range of 800-1200 °C and requires H 2 O and a heat supply inside the bed. Regarding the size of the particles (around 1-2 cm), limitations due to mass and heat transfers cannot be ignored. Basically, the reaction can be described as follows:

C þ H 2 O ! CO þ H 2 ð1Þ
Given the lack of knowledge about the charcoal gasification process, a single particle gasified under steam has frequently been studied both experimentally and numerically.

As a result, several numerical models have been developed at the single particle scale [START_REF] Dasappa | The gasification of wood-char spheres in CO 2 -N 2 mixtures-analysis and experiments[END_REF][START_REF] Dasappa | Wood-char gasification: experiments and analysis on single particles and packed beds[END_REF][START_REF] Gobel | Progress in thermochemical biomass conversion[END_REF][START_REF] Peters | A flexible and stable numerical method for simulating the thermal decomposition of wood particles[END_REF][START_REF] Wang | A generalised dynamic model for char particle gasification with structure evolution and peripheral fragmentation[END_REF][START_REF] Mermoud | Experimental and numerical study of steam gasification of a single charcoal particle[END_REF]. Although these models enable a better understanding of all the highly coupled phenomena involved at the particle scale, in the case of biomass charcoals, none has succeeded in satisfactorily matching experimental results beyond a critical value of conversion. The term ''conversion" is used in this study to refer to the macroscopic average conversion, i.e. the ratio of the total mass lost -by gasification or fragmentation -to the initial mass of the particle. Indeed, this kind of char presents a conversion level X which increases quasi-linearly versus time, in spite of diffusional effects. As emphasized by Struis et al. [START_REF] Struis | Gasification reactivity of charcoal with CO 2 . Part I: conversion and structural phenomena[END_REF] and Mermoud et al. [START_REF] Mermoud | Experimental and numerical study of steam gasification of a single charcoal particle[END_REF], models tend to overestimate the influence of internal transfers for large particles and hence fail to reproduce this linear evolution satisfactorily. In Dasappa et al. [START_REF] Dasappa | The gasification of wood-char spheres in CO 2 -N 2 mixtures-analysis and experiments[END_REF], for example, the comparison between predictions and experiments was valid only up to a conversion level of 60%. A similar comment can be made about the simulations conducted by Mermoud et al. [START_REF] Mermoud | Experimental and numerical study of steam gasification of a single charcoal particle[END_REF].

There are several possible explanations for this discrepancy. Firstly, it should be recalled that all the models available in the literature are spherically symmetric one-dimensional models, based on the assumption of an isotropic material. But, as the wood is initially anisotropic, the charcoal loses its initial shape during gasification and shrinks, preferentially in a transversal direction. This shorter dimension facilitates mass and heat transfers inside the particle in comparison with the predictions of the model. As a consequence, although this assumption is relevant at low conversion levels, i.e. before the particle size decreases, at the end of gasification process it leads to overestimation of the conversion rate dX/dt.

Secondly, several authors [START_REF] Wang | A generalised dynamic model for char particle gasification with structure evolution and peripheral fragmentation[END_REF][START_REF] Mermoud | Experimental and numerical study of steam gasification of a single charcoal particle[END_REF][START_REF] Struis | Gasification reactivity of charcoal with CO 2 . Part I: conversion and structural phenomena[END_REF][START_REF] Standish | Gasification of single wood charcoal particles in CO 2[END_REF][START_REF] Reyes | Percolation concepts in modelling of gas-solid reactions -II. Application to char gasification in the diffusion regime[END_REF][START_REF] Marban | Influence of percolation on the modification of overall particle properties during gasification of porous solids[END_REF] have suggested that such behaviour might result from peripheral fragmentation (also called secondary fragmentation), i.e. the disintegration of the porous charcoal into small fragments at the particle surface. Note that a similar process can be observed with coal chars [START_REF] Kerstein | Fragmentation during carbon conversion: predictions and measurements[END_REF][START_REF] Sundback | Fragmentation behavior of single coal particle in a fluidized bed[END_REF][START_REF] Mitchell | The impact of fragmentation on char conversion during pulverized coal combustion[END_REF][START_REF] Zhang | The fragmentation of coal particles during the coal combustion in a fluidized bed[END_REF][START_REF] Feng | Percolative fragmentation of char particles during gasification[END_REF][START_REF] Baxter | Char fragmentation and fly ash formation during pulverized-coal combustion[END_REF], even if the fragmentation mode may differ between both chars due to the difference of structure (e.g., variation of volatile-matter content or of the initial porosity). The above-mentioned phenomenon can be considered as a percolative fragmentation process [START_REF] Bar-Ziv | Mutual effects of porosity and reactivity in char oxidation[END_REF][START_REF] Daccord | evaporations, etchings[END_REF][START_REF] Sahimi | Statistical and continuum models of fluid solid reactions in porous media[END_REF]. Indeed, under the control of external diffusion, fragmentation can enhance the conversion rate. In these conditions, the gasification process involves the disappearance of the solid phase and the formation of a porosity gradient due to the mass transfer effects, until the local porosity of the superficial slice of the particle reaches a critical value. At this point, the charcoal structure disintegrates at the surface and the radius of the particles decreases. The physical result is that when too much carbon has been removed from the charcoal structure, the remaining matrix can no longer maintain its structure and a fragment detaches from the particle. The existence of this process was first confirmed experimentally by Kerstein and Niksa [START_REF] Kerstein | Fragmentation during carbon conversion: predictions and measurements[END_REF] and Sundback et al. [START_REF] Sundback | Fragmentation behavior of single coal particle in a fluidized bed[END_REF] for coal chars. Mitchell and Akanetuk [START_REF] Mitchell | The impact of fragmentation on char conversion during pulverized coal combustion[END_REF] suggested that fragmentation can reduce the initial particle mass by around 20-30%. Zhang et al. [START_REF] Zhang | The fragmentation of coal particles during the coal combustion in a fluidized bed[END_REF] studied the influence of operating conditions on the fragmentation of coal particles in a fluidized bed. Feng and Bhatia [START_REF] Feng | Percolative fragmentation of char particles during gasification[END_REF] contributed to the understanding of the phenomenon by measuring the variation in electrical resistivity with conversion and by direct observation, of the fragments. Their results imply, in particular, that peripheral fragmentation occurs at early stages of gasification with low conversion levels and is conditioned by diffusive effects. Baxter [START_REF] Baxter | Char fragmentation and fly ash formation during pulverized-coal combustion[END_REF] also reported a strong influence of particle size on the extent of fragmentation. It should be noted that, even in the kinetically controlled regime, fragmentation can induce a variation in the conversion rate, due to the exposure of a fresh surface area, and hence an increase in reactive area, accompanying the particle gasification process [START_REF] Struis | Gasification reactivity of charcoal with CO 2 . Part I: conversion and structural phenomena[END_REF][START_REF] Fuertes | Modelling gasification reactions including the percolation phenomenon[END_REF].

Many researchers have studied the fragmentation phenomenon in both particle models [START_REF] Wang | A generalised dynamic model for char particle gasification with structure evolution and peripheral fragmentation[END_REF][START_REF] Reyes | Percolation concepts in modelling of gas-solid reactions -II. Application to char gasification in the diffusion regime[END_REF][START_REF] Fuertes | Modelling gasification reactions including the percolation phenomenon[END_REF][START_REF] Reyes | Percolation concepts in modelling of gas-solid reactions -I. Application to char gasification in the kinetic regime[END_REF][START_REF] Biggs | The CO/CO2 product ratio for a porous char particle within an incipiently fluidized bed: a numerical study[END_REF][START_REF] Yamashita | Modeling of gasification and fragmentation behavior of char particles having complicated structures[END_REF][START_REF] Liu | Modeling the fragmentation of non-uniform porous char particles during pulverized coal combustion[END_REF] and in reactors models, laminar flow reactors [START_REF] Zolin | Coupling thermal deactivation with oxidation for predicting the combustion of a solid fuel[END_REF] or fluidized beds [START_REF] Marban | Simulation of secondary fragmentation during fluidized bed combustion of char particles[END_REF][START_REF] Pinho | Fragmentation on batches of coke or char particles during fluidized bed combustion[END_REF]. Such models can be classified in three distinct approaches:

-A discrete or network approach [START_REF] Reyes | Percolation concepts in modelling of gas-solid reactions -II. Application to char gasification in the diffusion regime[END_REF][START_REF] Reyes | Percolation concepts in modelling of gas-solid reactions -I. Application to char gasification in the kinetic regime[END_REF][START_REF] Yamashita | Modeling of gasification and fragmentation behavior of char particles having complicated structures[END_REF][START_REF] Liu | Modeling the fragmentation of non-uniform porous char particles during pulverized coal combustion[END_REF][START_REF] Kerstein | Percolation model for simulation of char oxidation and fragmentation time-histories[END_REF][START_REF] Sahimi | Dynamic scaling for the fragmentation of reactive porous media[END_REF] where the single char particle is seen as a two or three-dimensional lattice composed of a large number of small clusters (cf. Sahimi et al. [START_REF] Sahimi | Statistical and continuum models of fluid solid reactions in porous media[END_REF] for a review of these models). Results are interesting from a qualitative point of view but require previous knowledge of the char microstructure. -A stochastic or statistical approach [START_REF] Kerstein | Fragmentation during carbon conversion: predictions and measurements[END_REF][START_REF] Chirone | Secondary fragmentation of char particles during combustion in a fluidized bed[END_REF][START_REF] Cui | In-bed char combustion of Australian coals in PFBC. 3. Secondary fragmentation[END_REF][START_REF] Miccio | Monte-Carlo Simulation of combustion-induced percolative fragmentation of carbons[END_REF][START_REF] Salatino | Combustion and percolative fragmentation of carbons[END_REF] based on the Monte Carlo method. The fragmentation process is described by different statistical functions such as the number of fragments generated, which need to be fitted. -A continuum approach [START_REF] Wang | A generalised dynamic model for char particle gasification with structure evolution and peripheral fragmentation[END_REF][START_REF] Mitchell | The impact of fragmentation on char conversion during pulverized coal combustion[END_REF][START_REF] Zolin | Coupling thermal deactivation with oxidation for predicting the combustion of a solid fuel[END_REF][START_REF] Gavalas | Analysis of char combustion including the effect of pore enlargement[END_REF] based on percolation theory, where particle fragmentation occurs when porosity exceeds a critical value e à . The relative simplicity of this method means it can easily be incorporated in gasification continuum models. However, few studies included quantitative comparison with experimental results, and still fewer for particles with a radius of more than a few millimetres.

This study focuses on the latter formulation and attempts to explain the discrepancy observed at high conversion levels between experimental and numerical results by taking the fragmentation phenomenon into account in a particle gasification model. In order to validate our assumptions, we used the Mermoud et al. [START_REF] Mermoud | Experimental and numerical study of steam gasification of a single charcoal particle[END_REF] model. First, we show that the classical percolative approach is not really suitable to predict char gasification when there are significant limitations due to mass and heat transfer within the particle. Second, we propose a new formulation to characterize the peripheral percolation phenomenon and compare it with experimental results obtained by Mermoud et al. [START_REF] Mermoud | Experimental and numerical study of steam gasification of a single charcoal particle[END_REF].

One-dimensional gasification modelling at the scale of one particle

In this section, we first briefly recall the numerical model developed by Mermoud et al. [START_REF] Mermoud | Experimental and numerical study of steam gasification of a single charcoal particle[END_REF] for steam gasification of a single charcoal particle and the main results obtained. This model, based on balance equations for heat and gas species and including heterogeneous reaction kinetics, is derived under a spherically symmetric one-dimensional form. A Langmuir-Hinshelwood formulation that accounts for hydrogen inhibition is used for the steam gasification kinetics.

As for the equation of state, the ideal gas law is applied. No simplifying assumption is adopted concerning the evolution of the charcoal particle during gasification. Thus, the description of the transport of the products in conjunction with the increase in porosity covers the entire range from the limiting case of a reacting-core model (reaction-rate limited regime) to a shrinking core model (mass-transfer limited regime).

The boundary conditions are obtained by considering the heat and mass fluxes (including radiative transfer) at the charcoal/ atmosphere interface.

At each time step, once the pressure and velocity fields are determined, temperature and concentration fields can be calculated for the different species. At this point, the new properties of the medium (particle diameter, porosity, permeability, effective diffusion and conductivity) can be updated. In particular, local porosity varies during gasification as a function of the reaction term, i.e. the quantity of solid carbon consumed by the chemical reaction. In the current version of the model -in which fragmentation is not included -when the porosity of the cell considered reaches 1 (0.999 for numerical purposes), this cell is removed and the particle shrinks.

Comparison with experimental data indicated that the model predictions are both qualitatively and quantitatively satisfactory, despite the fact that the phenomena of reactive surface variation and particle fracturing are not well understood.

Indeed, during experiments under all conditions, conversion follows an almost linear slope with time. Fig. 1 shows the typical evolution of the conversion as a function of time for different particle sizes (7, 14 and 21 mm diameter). As can be seen in the figure, the model can predict the slope in the case of small particles where the reaction kinetics is dominant, but cannot show the conversion of larger particles beyond roughly 60% conversion. As mentioned above, this relative lack of accuracy beyond a critical value of conversion is not specific to this model but its origin still remains problematic. We suspect that this discrepancy is due to the fact that the model does not take into account the particle fragmentation phenomenon that occurs during gasification.

In the following section, we examine the peripheral fragmentation process and determine which criterion of fragmentation is the most convenient.

Peripheral fragmentation: which is the most convenient criterion?

Peripheral fragmentation is usually related to the percolation phenomenon [START_REF] Mohanty | Reaction and transport in disordered composite media: introduction of percolation concepts[END_REF][START_REF] Shah | Transport and reaction in evolving;disordered composites. I: gasification of porous solids[END_REF][START_REF] Sahimi | Transport, reaction and fragmentation in evolving porous media[END_REF]. Basically, a porous medium can be seen as a square lattice, i.e. an infinite array of squares. All of them correspond to a site which can be either occupied or vacant, depending on whether it represents solid material or a pore. The sites are assumed to be occupied randomly with a probability p, which depends on porosity. When two adjacent sites are occupied, they are said to be connected. Several sites that are interconnected form clusters. Thus, the whole porous matrix constitutes an infinite cluster.

Carbon is consumed throughout the gasification reaction, and as a result, some sites are removed. If a particle remains at constant size -no compaction or shrinking -the disappearance of sites leads to an increase in porosity, and hence a decrease in probability p. As we will see later, such a transition usually suggests a major transformation in the infinite cluster. For instance, one could ask what is the smallest probability at which the porous material can maintain its structure.

In the kinetic regime, percolation theory predicts that a low enough probability, or high enough porosity, will cause the particle to disintegrate into fragments. This critical probability p à or critical porosity e à at which the porous matrix can no longer maintain its structure is called percolation threshold. In the kinetic regime, this condition is reached simultaneously throughout the particle, whereas in the diffusion-controlled regime, a porosity gradient appears leading to the formation of a fragmentation front. Indeed, under the action of the chemical reaction, connections that link a group of neighbouring occupied sites to the general cluster can be broken and cause part of the porous material to detach: this is the peripheral fragmentation phenomenon.

Several authors [START_REF] Wang | A generalised dynamic model for char particle gasification with structure evolution and peripheral fragmentation[END_REF][START_REF] Mitchell | The impact of fragmentation on char conversion during pulverized coal combustion[END_REF][START_REF] Zolin | Coupling thermal deactivation with oxidation for predicting the combustion of a solid fuel[END_REF][START_REF] Gavalas | Analysis of char combustion including the effect of pore enlargement[END_REF] have used this approach to include the fragmentation process in their gasification continuum model. At each node, porosity increases over time proportionally to the amount of carbon consumed until a certain critical value e à is reached when the total solid mass suddenly disintegrates by disaggregation. The above value of e à takes into account only topological characteristics of the char microstructure leading to particle breakup [START_REF] Reyes | Percolation concepts in modelling of gas-solid reactions -II. Application to char gasification in the diffusion regime[END_REF]. Hence, it is assumed to be constant, regardless of particle size. Although this approach is convenient in the kinetic regime, it is problematic in the diffusion-limited regime.

Fig. 2 illustrates the coupling of this method to the numerical model developed by Mermoud et al. [START_REF] Mermoud | Experimental and numerical study of steam gasification of a single charcoal particle[END_REF]. Two different values of e à are used here, 0.9 and 0.999 (i.e. no fragmentation occurs). Simulations are performed with both criteria and two different particle sizes (7 and 21 mm diameter), and compared with experimental data. Temperature and steam partial pressure are constant and fixed respectively, at 930 °C and 0.2 atm. Fig. 2 shows that, as explained in the previous section, without fragmentation, the model satisfactorily reproduces the behaviour of a small gasifying particle for which the reaction kinetics is dominant, but internal diffusive limitations are overestimated when the particle size increases. Conversely, for a critical porosity value equal to 0.9, the model successfully predicts the gasification of the 21 mm diameter particle but fails to reproduce the slope for the smallest one.

Indeed, several experimental studies [START_REF] Feng | Percolative fragmentation of char particles during gasification[END_REF][START_REF] Baxter | Char fragmentation and fly ash formation during pulverized-coal combustion[END_REF][START_REF] Salatino | Combustion and percolative fragmentation of carbons[END_REF] confirmed that the extent of fragmentation depends to a great extent on mass and heat transfer limitations, which can vary as a function of the operating conditions such as particle size and temperature. In particular, it has been observed that peripheral fragmentation may start at a value of p higher than the percolation threshold. As the reaction progresses, porosity increases locally -especially at the surfaceunder the effect of diffusive gradients. Thus, some fragments are isolated from the core of the particle, i.e. the infinite cluster, and detach from the surface, removing the carbon at this location. Nevertheless, the remaining matrix maintains its structure and does not collapse until the percolation threshold is reached. If the above-mentioned approach can capture such a process, it predicts a mass of detached fragments directly proportional to the mass of the initial char particle. But the percolation theory [START_REF] Sahimi | Transport, reaction and fragmentation in evolving porous media[END_REF] predicts that the distribution of fragments varies linearly with particle mass in a log-log plot. Experimental measurements by Kerstein and Niksa [START_REF] Kerstein | Fragmentation during carbon conversion: predictions and measurements[END_REF] and Feng and Bhatia [START_REF] Feng | Percolative fragmentation of char particles during gasification[END_REF] support this observation.

Simulations by Sahimi and Tsotsis [START_REF] Sahimi | Dynamic scaling for the fragmentation of reactive porous media[END_REF] also provided an interesting insight into this problem. In their model, based on a network approach, each occupied site can react with a probability r representing the reactivity of the solid matrix. If r ' 1 (limit of the diffusion regime or shrinking core model), only the surface of the initial matrix is consumed and no major fragmentation of the particle takes place. This is the phenomenon of peripheral fragmentation. Conversely, if r ' 0 (kinetic regime or uniform particle model), the reactant penetrates deeply inside the network and matrix fragmentation can occur when the percolation threshold is reached. Between these limits, a porosity gradient appears within the particle (from a macroscopic point of view) and many fragments with a wide variety of shapes and sizes are produced as a function of the relative importance of kinetic and diffusional limitations. These are the operating conditions on which we focus.

From this discussion, it is obvious that the classical continuum approach based on a single critical value of porosity does not allow the peripheral fragmentation phenomenon to be satisfactorily taken into account. The underlying idea is that for the same value of probability p (and hence porosity e), several more or less stable configurations (with a different number of connections) are possible. As a consequence, a criterion of the percolation threshold, based only on porosity which is a macroscopic property, cannot describe such a phenomenon, and an additional criterion is required. Fig. 3 shows two different configurations of percolation sites. Both clusters lead to the same value of porosity, calculated on the entire domain represented (same number of occupied sites) but do not present the same stability from the point of view of the number of connections. For instance, let us consider that site A of configuration 1 is consumed. This does not modify the global structure of the percolation cluster; porosity increases but fragmentation does not occur. Conversely, in configuration 2, if site B is consumed, the connection between the smaller cluster composed of three adjacent cells and the base, i.e. the charcoal particle, is broken, and part of the porous material detaches. Feng and Bhatia [START_REF] Feng | Percolative fragmentation of char particles during gasification[END_REF] suggest considering isolated clusters in the formation of fragments at the surface in addition to the percolation threshold. Based on this observation, we propose a new conceptual model where the particle is seen as an ensemble of interconnected isolated clusters, themselves made up of several occupied sites. From a macroscopic point of view, the occurrence of breakage of the remaining connections between isolated clusters and the matrix depends on stresses undergone by the particle, such as those induced by the weight of the char itself. The greater the diffusional effects exerted within the char, the more fragile the particle structure at the surface. Therefore, we suggest using the porosity gradient re c as the criterion of particle structure weakness. Under this assumption, the fragmentation phenomenon occurs if e P e à (uniform percolation threshold) or if e P e c and re P re c (peripheral percolation threshold) with e c 6 e à . If particle gasification corresponds to a shrinking core or a uniform particle model, no porosity gradient appears and the classical criterion of critical porosity controls the fragmentation process. In intermediate conditions, a porosity gradient occurs, and an additional criterion of fragmentation is required.

Below, by comparing our model with experimental results we demonstrate that such a criterion is valid and depends only on the intrinsic properties of the porous material.

Results and discussion

As explained above, the fragmentation process introduced in the model of Mermoud et al. [START_REF] Mermoud | Experimental and numerical study of steam gasification of a single charcoal particle[END_REF] is now modified to start as soon as a critical value of the porosity and of the porosity gradient is reached. Simulation results are presented in Fig. 4 for three particle sizes, of which the operating conditions are similar to those detailed in Fig. 1. The results of the simulation are compared both to experiments and to the model without fragmentation. e c and re c is fixed here at 0.92 and 18 m À1 , respectively, from the observation of simulations. The percolation threshold e à is kept at 0.999.

In this way, the same values of critical porosity and critical porosity gradient are used for all the simulations. Both parameters depend only on the intrinsic properties of the char under consideration and can consequently be modified if a different charcoal particle (with a different initial porosity for instance) is studied.

When reaction kinetics is the predominant mechanism (for the 7 mm diameter particle), the agreement obtained with the experimental curve remains excellent with or without fragmentation. On the other hand, when the particle is larger and the diffusional effects increase, the model including fragmentation is much more accurate.

Actually, a small particle is gasified practically uniformly without core shrinking. Porosity increases in a quasi homogeneous way and the porosity gradient consequently always remains lower than the fixed critical value. The char is completely gasified and no fragmentation occurs. Conversely, in the case of larger particles, the structure does not evolve uniformly and a porosity gradient quickly appears. Because of diffusional effects, porosity at the surface increases faster than inside the core. As soon as the two criteria (critical porosity and critical porosity gradient) are fulfilled, fragmentation occurs. Thus, as early as 30-40% of average conversion, the model predicts fragments will detach from the surface of the charcoal.

The ratio of mass lost by fragmentation and truly gasified mass for particles of 14 and 21 mm diameter is shown in Figs. 5 and6, respectively. The fragmentation effect is negligible for the smallest particle of 7 mm diameter. As expected, the fragmentation process starts earlier with the largest particle and has a greater impact (30% instead of 26% of mass lost by fragmentation).

Another interesting comparison is the changes in the radius of the charcoal particle during gasification predicted by the model depending on whether fragmentation is taken into account or not. For this purpose, we used the experimental results obtained by Mermoud et al. [START_REF] Mermoud | Experimental and numerical study of steam gasification of a single charcoal particle[END_REF] for a charcoal particle with an initial diameter of 30 mm which was partially gasified to 20%, 50% and 70% at 930 °C and 0.2 atm H 2 O and recovered for macro-and microscopic observations. The size of the particle was measured at each conversion value. Given the wood anisotropy, the diameter of the particle is represented with an error interval representing the ovoid shape of the particle. Results are presented in Fig. 7 and compared with the predictions made by the two models. There was much better agreement between experimental measurements and model predictions when the fragmentation phenomenon was taken into account. The discrepancy between the two numerical models corresponds to shrinking of the radius due to the peripheral fragmentation process predicted by our theory.

Conclusion

In this study, we focused on the problem of modelling gasification at high conversion levels for a single large particle of wood charcoal. In fact, none of the current models succeeds in reproducing the quasi-constant conversion rate observed experimentally. In this study, the classically observed discrepancy between model predictions and experiments was assumed to be due to peripheral fragmentation. From the point of view of the percolation theory, this phenomenon can be considered as the breakage of the connections between isolated clusters and the particle surface. The detachment of additional mass could explain the conservation of the conversion rate in spite of the increase in diffusional effects with an increase in particle size. Different approaches were tested to take into account such a phenomenon and model results were compared to experimental results obtained by Mermoud et al. [START_REF] Mermoud | Experimental and numerical study of steam gasification of a single charcoal particle[END_REF]. Two main conclusions can be drawn from this study:

-Peripheral fragmentation is sensitive to mass and heat transfer limitations within the particle and hence to particle size. As a consequence, a fragmentation criterion based only on a critical porosity value cannot explain the behaviour observed experimentally. -We have demonstrated that adding a criterion based on the porosity gradient at the surface allowed us to significantly improve modelling of charcoal particle gasification. The introduction of such a parameter enables the impact of diffusional effects on fragmentation to be taken into account.

The upscaling of fragmentation processes to the reactor-scale, which represents the ultimate goal of such a work, is a tremendous task and is beyond the scope of this paper. Moreover, additional mechanisms known to induce fragmentation in industrial gasifiers such as particle attrition [START_REF] Salatino | Combustion and percolative fragmentation of carbons[END_REF], bed compaction [START_REF] Fernandes | Steady-state simulation of a continuous moving bed reactor in the pulp and paper industry[END_REF] or the pressure increase during devolatization [START_REF] Mitchell | The impact of fragmentation on char conversion during pulverized coal combustion[END_REF] should be also taken into account. The reader interested by the way of implementing fragmentation in reactor modelling and by a state-of-art on the subject may refer to a recent study of Syred et al. [START_REF] Syred | Development of fragmentation models for solid fuel combustion and gasification as subroutines for inclusion in CFD codes[END_REF]. Nevertheless, in order to give a brief insight into possible routes for an extent of this study to the upper scale, two approaches should be considered as a function of the size (and the type) of char particles under consideration, and hence of the kind of reactor:

-For large particles as the ones under study here, the fragments size produced during gasification can be considered as negligible. For reactors involving such particles (e.g., in fixed bed reactors), the implementation could be quite straightforward. The fragments can be assumed to be instantaneously consumed and consequently, the fragmentation process can be directly coupled to gasification [START_REF] Zolin | Coupling thermal deactivation with oxidation for predicting the combustion of a solid fuel[END_REF]. A new criterion -including the above-mentioned fragmentation mechanisms -should be defined in that scope. -If char particles involved in the gasification process are smaller (few millimetres or less), the fragments size may be significant and their presence cannot be neglected anymore. For such reactors (e.g., in fluidized bed reactors), if our criterion allows to predict the char mass lost by fragmentation, it is not sufficient to determine the evolution of the particle size distribution during the char gasification phase. For certain coals or charcoals, indeed, fragmentation can lead to the formation of several hundreds of new particles -of size almost equal to the fragmenting particles [START_REF] Fernandes | Steady-state simulation of a continuous moving bed reactor in the pulp and paper industry[END_REF]. Hence, a statistical model is required in parallel for predicting the number and the size of fragments generated [START_REF] Chirone | Secondary fragmentation of char particles during combustion in a fluidized bed[END_REF][START_REF] Cui | In-bed char combustion of Australian coals in PFBC. 3. Secondary fragmentation[END_REF][START_REF] Miccio | Monte-Carlo Simulation of combustion-induced percolative fragmentation of carbons[END_REF][START_REF] Salatino | Combustion and percolative fragmentation of carbons[END_REF].

  Nomenclature d charcoal particle diameter (mm) p probability of site occupation (-) p * percolation threshold (-) t time (s) X conversion level (-) e porosity (v%) e * critical porosity associated to uniform percolation (-) e c critical porosity associated to peripheral percolation (-) re c critical porosity gradient (m À1 )

Fig. 1 .

 1 Fig. 1. Influence of particle diameter on changes in the conversion rate (T = 930 °C, P H2O = 0.2 atm) (from Ref. [6]).

Fig. 2 .

 2 Fig.2. Impact of the critical porosity criterion on simulations.

Fig. 3 .

 3 Fig. 3. Two different configurations of clusters with the same probability p.

Fig. 4 .

 4 Fig. 4. Influence of particle diameter on changes in the conversion rate (T = 930 °C, P H2O = 0.2 atm) -comparison between numerical simulations with or without fragmentation and experiments.

Fig. 5 .

 5 Fig. 5. Ratio of mass lost by fragmentation and truly gasified for a 14 mm diameter particle (T = 930 °C, P H2O = 0.2 atm).

Fig. 6 .

 6 Fig. 6. Ratio of mass lost by fragmentation and truly gasified for a 21 mm diameter particle (T = 930 °C, P H2O = 0.2 atm).

Fig. 7 .

 7 Fig. 7. Macroscopic evolution of a gasifying particle versus conversion (0%, 20%, 50% and 70% conversion) and comparison with model predictions with or without fragmentation.