A De Pannemaecker
email: adepannemaecker@gmail.com

S Fouvry

J Y Buffiere

M Brochu

Modelling the fretting fatigue crack growth: from short crack correction strategies to microstructural approaches

Keywords: Fretting fatigue, Crack growth propagation, In situ tests, Short cracks, Fatigue life 1

Predicting the fretting fatigue crack propagation is complex and uncertain. It implies to consider stress gradient and short crack behavior. To address this challenge former in situ synchrotron Xray imaging of micro-fretting fatigue experiments are simulated. First a Pugno et al. / Fleury et al. short crack correction (SCC) approach is considered. Then a microstructural Navarro-Rios (NR) model originally developed to analyze fatigue crack growth for short crack is transposed. Both strategies are compared. Very good agreement with experiments are observed. However the NR model provides a more physical description of the short crack behavior allowing the quantification of the non-monotonous crack growth rate when this later shifts from fretting dominant to fatigue dominant stress domain.

Introduction

First studied by Eden [START_REF] Eden | The endurance of metals[END_REF], fretting is defined as a small oscillatory movement between two surfaces in contact. When combined with fatigue loadings, the crack nucleated by fretting stresses can propagate until failure of the structure [START_REF] Ferre | Prediction of the Fretting Fatigue Crack Nucleation Endurance of a Ti-6V-4Al/Ti-6V-4Al Interface: Influence of Plasticity and Tensile/Shear Fatigue Properties[END_REF][START_REF] Ferré | Stress gradient effect on the crack nucleation process of a Ti-6Al-4V titanium alloy under fretting loading: Comparison between non-local fatigue approaches[END_REF][START_REF] Lindley | Fretting fatigue in engineering alloys[END_REF]. When cracking mechanisms are involved in fretting configurations, the fretting life can be split into three different phases: a crack nucleation period which corresponds to the formation of microcracks mainly controlled by the cyclic fretting contact stresses [START_REF] Araújo | On the prediction of highcycle fretting fatigue strength: Theory of critical distances vs. hot-spot approach[END_REF], a crack growth period corresponding to the transition from micro-to macrocracks. First controlled by the contact stress the crack extension is progressively driven by the bulk fatigue loading. Note that a crack arrest condition can be achieved in the short or the long crack regime depending on the combined fretting and fatigue stressing conditions [START_REF] Araújo | Analysis of pad size effects in fretting fatigue using short crack arrest methodologies[END_REF]. From this analysis, it was shown that fretting, by favoring the crack nucleation and promoting an initial crack propagation can drastically reduce the fatigue endurance of assemblies [START_REF] Zhou | Fissuration Induite en Petits Débattements: Application au Cas d'Alliages d'Aluminium Aéronautiques[END_REF]. When the crack growth rate is within the Paris region, for fretting fatigue tests, a competition between fretting and fatigue will occur leading to higher crack growth rates than with simple fatigue tests. On the contrary, if the crack tip is far enough from the contact, fretting loads have no influence any more on the crack tip, meaning that crack growth rates will be identical for fretting fatigue and fatigue tests within the Paris region.

Fretting fatigue crack prediction requires non-local multiaxial fatigue criteria to capture the stress gradient and complex state imposed by the contact loading [START_REF] Amargier | Stress gradient effect on crack initiation in fretting using a multiaxial fatigue framework[END_REF][START_REF] Araújo | Early cracking orientation under high stress gradients: The fretting case[END_REF][START_REF] Fouvry | From uni-to multi-axial fretting-fatigue crack nucleation: Development of a stress-gradient-dependent critical distance approach[END_REF][START_REF] Antunes | Stress intensity factor solutions for fretting fatigue using stress gradient factor[END_REF]. In order to model the crack growth propagation, LEFM approach under mixed mode loading conditions is usually considered. A modified Paris relation is adopted defining an equivalent stress intensity factor taking into account the mode I and mode II components. Combining crack nucleation and propagation, the global fretting fatigue life can be evaluated [START_REF] Gandiolle | Lifetime prediction methodology for variable fretting fatigue loading: Plasticity effect[END_REF][START_REF] Hojjati-Talemi | Prediction of fretting fatigue crack initiation and propagation lifetime for cylindrical contact configuration[END_REF][START_REF] Fontes Do Rêgo | A methodology for fretting fatigue life estimation using strain-based fracture mechanics[END_REF]. Rather good correlations are usually observed. One limitation of this model concerns the crack path prediction. The crack growth prediction deals with two issues: the crack path determination and the crack growth rate. Recently Baietto et al. propose a global-local strategy based on the XFEM (extended finite element method) which takes into account scale effects, structure, components, the contact interface, created cracks and interfacial non-linearities occurring within the crack lips [START_REF] Baietto | Fretting fatigue crack growth simulation based on a combined experimental and XFEM strategy[END_REF]. A mixed mode Paris crack growth relation is introduced for plain fretting and preloaded fretting cracking configurations. Then the Hourlier [START_REF] Hourlier | Propagation of fatigue cracks under polymodal loading[END_REF] criterion is applied to determine the propagation direction of the crack and a very good correlation is observed. Another limitation of the LEFM approach concerns the short crack propagation description. Recently a short crack propagation strategy extracted from the El Haddad et al. approach was introduced by Fleury et al. to capture the stress gradient effect in notched specimen [START_REF] Fleury | Evaluating the influence of residual stresses and surface damage on fatigue life of nickel superalloys[END_REF]. This approach will be examined in the present fretting fatigue analysis. However such approach is questionable from a physical point of view.

To overcome those limitations, the present work proposes another approach based on the Navarro-Rios formulations [START_REF] Navarro | Short and long fatigue crack growth: A unified model[END_REF][START_REF] Navarro | An alternative model for the blocking of dislocations at grain boudaries[END_REF][START_REF] Navarro | Fatigue crack growth modelling by successive blocking of dislocations[END_REF] and some extensions proposed by Vallellano et al [START_REF] Vallellano | Fatigue crack growth threshold coniditons at notches. Part I: Theory[END_REF][START_REF] Vallellano | Fatigue crack growth threshold coniditons at notches. Part II: Generalization and application to experimental results[END_REF][START_REF] Vallellano | Two-parameter fatigue crack growth driving force: Successive blocking of the monotonic and cyclic plastic zones at microstructural barriers[END_REF][START_REF] Vallellano | A micromechanical model for small fatigue crack growth: an approach based on two threshold conditions[END_REF]. A micromechanical model based on the Crack Tip Opening Displacement (CTOD) calculations, originally formulated for fatigue configurations, is extended to implement fretting effects. The model is considering both short and long crack regimes, takes into account high stress gradient effects, a variable stress ratio and the geometry of the specimen. The model is compared to 7050-T7451 aluminum alloy experimental data obtained using in situ small-scale fretting fatigue setup especially developed to fit within the synchrotron facility (ESRF, Grenoble, France).

Nomenclature

Small-scale fretting fatigue experiments

Material

The present analysis was performed on the 7050-T7451 aluminum alloy widely used in aeronautical structures. Its chemical composition and mechanical properties are exposed in 2. This alloy displays a mixed structure of non-recrystallized and recrystallized grains fully detailed in [START_REF] Shahzad | Surface characterization and influence of anodizing process on fatigue life of Al 7050 alloy[END_REF]. Large grains display an average grain size about whereas small ones are about . Note that the average grain size along the transverse direction (T) for our configuration is about and will be used in the model later on.

Fretting fatigue specimens were extracted from a rolled aluminum block with 530mm along the longitudinal direction (L), 910mm along the transverse direction (T) and 110mm along the short direction (S). All samples were machined using electrical discharge machining and were manually polished to reach a low surface roughness (

). Machining was chosen in order to apply the fretting fatigue load along the L-direction making the crack propagate along the T-direction. The cylindrical fretting pad, displaying a low surface roughness of , is a heat treated 52100 steel ensuring that the crack only initiates and propagates within the aluminum samples. The fretting fatigue experiment was done using a fatigue stress ratio around . Unfortunately we do not display the fatigue limit at the same stress ratio for our aluminum alloy. This value is however known for the same orientation L-T at with and at with [START_REF]Metallic materials and elements for aerospace and vehicle sutructures[END_REF]. Therefore we will assume an intermediate value, so to say . We previously performed fatigue crack growth tests on C(T) specimens at [START_REF] De Pannemaecker | Identification of the fatigue stress intensity factor threshold for different load ratios R: From fretting fatigue to C(T) fatigue experiments[END_REF] leading to a long crack stress intensity factor range threshold of and a maximum stress intensity factor threshold of . Applying the Kitagawa Takahashi approximation, the short crack transition is evaluated around:

(1)

Hence the material properties related to studied fretting fatigue experiments are compiled in Table 3.

Experimental setup

If both crack nucleation and crack arrest conditions are now well formalized through close correlations between experiments and simulations, the prediction of the crack propagation rate under complex fretting fatigue stressing remains a very challenging problem. In contrast to crack nucleation and crack arrests which can be investigated using post mortem destructive observations measurement of the crack propagation rate requires in situ characterization of the crack extension. This is a complex experimental issue as in fretting fatigue the presence of the contact prevents a direct observation of the specimen surface. Fretting fatigue crack propagation rates can be measured using post mortem examination of several specimens obtained for several contact loading conditions and test durations [START_REF] Baietto | Fretting fatigue crack growth simulation based on a combined experimental and XFEM strategy[END_REF]. In addition of being very tedious this experimental approach induces a relatively large scatter on the measured crack lengths which critically influences the crack rate calculation and cannot allow optimal modelling analysis. The potential drop method extensively applied in standard fatigue propagation experiments can also be used for fretting and fretting fatigue experiments [START_REF] Meriaux | Characterization of crack nucleation in TA6V under fretting-fatigue loading using the potential drop technique[END_REF] with the help of a calibration curve, crack propagation rates can be measured. However, in contrast to fatigue configurations, the presence of the contact pad induces a large scatter of the potential drop measurements so that only semi-quantitative estimation of the crack propagation rate can be extracted [START_REF] Meriaux | Characterization of crack nucleation in TA6V under fretting-fatigue loading using the potential drop technique[END_REF].

To overcome those limitations, one promising technique consists in performing X-ray tomography or radiography during a fretting fatigue test. An original device was specially developed by the MATEIS and LTDS laboratories in order to perform in situ fretting fatigue tests for the first time using synchrotron radiographic images [START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF].

Figure 1 shows the fretting fatigue setup principle (Figure 1.a) and its implementation within the synchrotron facility (ESRF, ID19 beamline, see Figure 1.c and Figure 1.d). The aluminum sample (Figure 1.b) is clamped at its top and bottom where the upper grip is fixed and the lower one mobile, in order to apply the cyclic fatigue loading at positive stress ratios . A hydraulic actuator is used to generate the fatigue with the average load, and the amplitude. To apply alternated fretting tangential force, a static normal force is imposed by the fretting pad on the specimen side. Bimetallic flexure elements are controlling the stiffness pad holder. Therefore the tangential load is created by both the sample elongation due to fatigue and the controlled stiffness of the fretting pad holder. The axial fatigue loads are recorded using and shown on Figure 1.a while the tangential load is calculated with . The crack nucleation and propagation were studied using 2D radiographs. Although this type of imaging provides an integrated view of damage though the specimen thickness, the geometry of the mechanical tests makes very difficult 3D tomographic imaging for which the sample should be exposed to the X-ray beam during a rotation of 180° (parallel beam). In order to fit within the synchrotron facility and because of the small maximum voxel available (pixel size of for a 2D view of an area about ), the fretting fatigue device had to be considerably reduced compared to usual macroscopic tests. Therefore in order keep the plane strain hypothesis, the sample cross section was chosen to be of with the following loading parameters: the linear normal force generating a maximum hertzian pressure of , and the cylinder pad radius is leading to a hertzian half contact width (distance from contact border to center). Finally the friction coefficient at the fretted interface was obtained thanks to variable displacement tests [START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF].

For the following analysis, we are focusing on the partial slip condition corresponding to the regime generating cracks in fretting. This condition is fulfilled as long as the maximum tangential load amplitude is relatively small compared to the normal load so that . Figure 2 summarizes the procedure to extract 2D-radiographic images taken at the synchrotron facility. The beam was pointed on the aluminum sample along the S-direction with its window covering the contact area, enabling images acquisition of the fretting fatigue crack propagation. A full analysis of the first results was previously completed, so that you can refer to [START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF] for additional information on the setup and test procedure. Note that radios obtained from Test A were previously analyzed in [START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF] as well so that only major results will be reminded in this paper for the sake of the discussion.

Experimental results

Many tests were performed to adjust the procedure and optimize the cylinder/plane alignment which is a key aspect to generate reliable fretting fatigue cracking data. The present investigation focuses on two tests conditions A and B for which a perfect contact alignment was obtained (Table 4). Test A was performed with radio observations allowing the crack propagation rate analysis. This online analysis is compared to post mortem SEM and 3D surface profiles observations of the cracked surface. Test B was performed without radio observations so only post mortem expertises were achieved.

Both X-ray radio and post mortem crack extension analyses of test condition A are illustrated in Figure 3 and Figure 4 where the projected crack length along the vertical axis is plotted versus the fretting cycles.

By analyzing the radios taken from the micro fretting fatigue tests, the crack nucleation is quite hard to evaluate. After about 20000 fretting fatigue cycles, it is assumed that a through crack is obtained as the crack becomes clearly visible on the radios. The length of the first through crack is extended to be of the order of 100µm.

FEM modelling

The loading conditions imposed by the micro fretting fatigue setup are complex and cannot be addressed using analytical formulations [START_REF] Hills | Mechanics of fretting fatigue[END_REF]. The fretting fatigue experiment is therefore described using a 2D plane strain elastic finite element model (Abaqus software) described in Figure 5 simulating the experimental conditions (more details on the model are available in [START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF]). The model is composed of a moving steel cylinder (

), the aluminium sample, and a fixed roller bearing on its back. The mesh uses triangular (CPE3) linear elements outside the contact. The mesh size is then refined down to at the contact interface with quadrilateral (CPE4R) linear elements (Figure 5.a). The friction at the interface is described using Lagrange multipliers with the friction coefficient previously experimentally determined.

The applied loads are presented in Figure 5.b. The mean fatigue load is imposed, followed by the normal load . As we are dealing with a mono-actuator where fretting and fatigue are in phase, fretting fatigue loads are then applied at the same time during one and a half cycle by adding alternative amplitudes, namely for the fatigue, and for the fretting. As described in Figure 5.c, due to the complex fretting fatigue loadings, the shear and pressure distributions are very disturbed and require FEM computations. The stress intensity factors of the fretting fatigue crack are computed using a dedicate 2D-plane strain FE model coupled with an automatic remeshing procedure using Matlab and previously described in [START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF]. A kinked crack shape defined from the synchrotron radiographies is implemented in the previously exposed FEM contact modelling at where propagation occurs according to x (mm) q max =328MPa p max =302MPa p(x) q(x) c) used as described in [START_REF] De Pannemaecker | Introduction of a reverse simulation approach to identify the fatigue stress intensity factor crack arrest threshold from fretting cracking experiments[END_REF]. A single crack is computed at the trailing contact border at the two stages corresponding to the most severe stress conditions: the stress intensity factor is extracted at and is extracted at using the Rice's integrals method (Figure 6, [START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF][START_REF] De Pannemaecker | Introduction of a reverse simulation approach to identify the fatigue stress intensity factor crack arrest threshold from fretting cracking experiments[END_REF]). Note that due to the bending phenomena induced by the fixed roller bearing, the mode I maximum and minimum values will occur either at depending on the crack length so that the stress ratio characterizing at the equivalent fretting fatigue crack tip can be expressed as follows:

(

where and are numerical stress intensity factor values calculated by the Rice's integrals method.

The mode I stress intensity factor range (i.e.

) is computed using equation (3). Only the positive part of is considered (a negative value corresponding to a closed crack is not contributing to the propagation).

(

The mode II contribution is expressed by: (4) An equivalent mixed mode stress intensity factor is therefore considered: [START_REF] Araújo | On the prediction of highcycle fretting fatigue strength: Theory of critical distances vs. hot-spot approach[END_REF] The crack growth rate between cycles and where the crack length changes from and is defined by equation ( 6) in agreement with the ASTM E647 [START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF][START_REF] Astm | Standard test method for measurement of fatigue crack growth rates[END_REF]:

(6) and ( 7)

Crack growth analysis

The X-ray radio conditions of Test A are simulated and the corresponding crack growth rate is reported in the diagram (Figure 7). These micro fretting fatigue crack growth rates are compared to the conventional C(T) experiments of the studied 7050-T7451 aluminum alloy.

ZOOM ZOOM=PLASTIC ZONE

Fretting pad a : Projected crack length Computation for each radiography (MPa) Figure 7: Correlation between micro fretting fatigue experiment and conventional C(T) tests for the 7050-T7451 aluminum alloy (Micro fretting fatigue at [START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF], C(T) at [START_REF] De Pannemaecker | Identification of the fatigue stress intensity factor threshold for different load ratios R: From fretting fatigue to C(T) fatigue experiments[END_REF], C(T) at [START_REF] Handbook | Properties and selection: nonferrous alloys and special purpose materials[END_REF])

Focusing on the long crack regime ( ) it is interesting to note that "micro fretting fatigue" crack growth rates are very close to those obtained with macro C(T) experiments following a linear trend on a log-log scale so that a Paris equation can be fitted: [START_REF] Amargier | Stress gradient effect on crack initiation in fretting using a multiaxial fatigue framework[END_REF] with and .

As developed in [START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF], it confirms the stability of the micro fretting fatigue experiment to characterize long crack behaviours. However, for cracks smaller than , a short crack behavior is observed. A nearly constant crack growth rate is obtained so that:

(9) This transition from low to high domain is limited in many aspects. First it requires a discontinuous formulation which is not suitable for an optimal numerical model. Secondly it does not explicit the transition from short to long crack propagation regimes which is in fact the controlling factor of such crack growth rate formulation.

Short Crack Correction (SCC) model

Former investigations focusing on the crack arrest conditions conclude that the stress intensity factor range threshold value is not constant and may be expressed as a function of the crack length [START_REF] Araújo | The effect of rapidly varying contact stress fields on fretting fatigue[END_REF]. Hence, considering the Kitagawa-Takahashi (K-T) diagram, Araújo and Nowell suggested that the stress intensity factor range threshold may be expressed through a discontinuous formulation so that: [START_REF] Fouvry | From uni-to multi-axial fretting-fatigue crack nucleation: Development of a stress-gradient-dependent critical distance approach[END_REF] With the long crack threshold and the transition from short to long crack regime previously approximated by equation [START_REF] Eden | The endurance of metals[END_REF].

Using the El Haddad et al. approach, they also derived a more continuous formulation so that whatever the crack length : [START_REF] Antunes | Stress intensity factor solutions for fretting fatigue using stress gradient factor[END_REF] A similar modified short crack growth expression was successively suggested by Pugno et al. [START_REF] Pugno | A generalized Paris' law for fatigue crack growth[END_REF] and by Fleury et al. [START_REF] Fleury | Evaluating the influence of residual stresses and surface damage on fatigue life of nickel superalloys[END_REF] to provide a generalized Paris equation for fatigue crack growth: [START_REF] Gandiolle | Lifetime prediction methodology for variable fretting fatigue loading: Plasticity effect[END_REF] Where is the geometrical parameter, and for surface crack .

This equivalent short crack stress intensity factor range value can be related to the nominal one according that:

(13) So that: [START_REF] Fontes Do Rêgo | A methodology for fretting fatigue life estimation using strain-based fracture mechanics[END_REF] This short crack modified stress intensity factor range can easily be integrated within the Paris equation: [START_REF] Baietto | Fretting fatigue crack growth simulation based on a combined experimental and XFEM strategy[END_REF] A more general formulation can be considered: [START_REF] Hourlier | Propagation of fatigue cracks under polymodal loading[END_REF] With an exponent conditioning the short crack correlation. When , no short crack correction is considered whereas corresponds to the El Haddad description. Figure 8 compares the proposal versus the studied micro fretting fatigue analysis. Like previously underlined, the plain (ie. for ) formulation is unable to capture the short crack response. The El Haddad approach ( ) improves the correlation but the constant crack growth rate in the small domain is not described. The model extrapolates a crack growth rate around the order of magnitude of for against for the experiment. Hence the El Haddad approximation of can lead to dangerous underestimation of the short crack growth rate. A parametric analysis is done and shows that the short crack rate is better approximated when which implies: [START_REF] Fleury | Evaluating the influence of residual stresses and surface damage on fatigue life of nickel superalloys[END_REF] This rather high value is physically questionable but provides a better description of the global short to long crack growth rate response. This short crack correction of the crack growth rate formulation is simply very practical to predict the crack propagation evolution but is limited in many aspects. First the rather high value found for is physically questionable. Secondly the approach does not consider the microstructural effect influencing the discontinuous evolution of the short crack propagation. Hence an alternative microstructural strategy needs to be considered to better illustrate this aspect.

1.E-12

1.E-11

Microstructural crack propagation model

Navarro-Rios (NR) model: principles

The proposed model is using the Navarro-Rios formulation detailed in [START_REF] Navarro | Short and long fatigue crack growth: A unified model[END_REF][START_REF] Navarro | An alternative model for the blocking of dislocations at grain boudaries[END_REF][START_REF] Navarro | Fatigue crack growth modelling by successive blocking of dislocations[END_REF] which was extended by Vallellano et al. [START_REF] Vallellano | Two-parameter fatigue crack growth driving force: Successive blocking of the monotonic and cyclic plastic zones at microstructural barriers[END_REF][START_REF] Vallellano | A micromechanical model for small fatigue crack growth: an approach based on two threshold conditions[END_REF] later on. The main idea is that a crack and its plastic zones need to overcome microstructural barriers of the material in order to propagate. The present work uses Navarro-Rios hypothesis assuming that only the monotonic plastic zone can be blocked at a grain boundary acting as a barrier, and is responsible for the crack arrest or propagation. By making the sum of stresses applied on the crack, one can calculate the Crack Tip Opening Displacement (CTOD) which is related to the crack growth rate.

Figure 9 illustrates the model principles with the loads involved. A fatigue crack of a length initiates within a grain displaying an average size and propagates in mode I. The crack is subjected to cyclic fatigue loads with a maximal external stress tending to open the crack. The latter also deals with which is the stress opposed to opening and closing, is the stress opposed to deformation and is the stress needed to propagate. usually represents residual stresses and will not be considered in the present work. The present model uses an odd parameter representing the half grain interval while the crack propagates (

). The crack will grow until its monotonic plastic zone reaches a grain boundary at acting like a microstructural barrier where it is blocked. The monotonic plastic zone needs to overcome the first dislocation at (with ), where is the associated stress, right after the grain boundary so that the crack can propagate.

Figure 9: Schematic description of the Navarro-Rios model where is the average grain size, the half grain interval, the crack, the maximal external stress, the stress opposed to opening and closing, the stress opposed to deformation, the stress needed to propagate, the distance between grain boundary and the first dislocation [START_REF] Navarro | Short and long fatigue crack growth: A unified model[END_REF][START_REF] Navarro | An alternative model for the blocking of dislocations at grain boudaries[END_REF][START_REF] Navarro | Fatigue crack growth modelling by successive blocking of dislocations[END_REF][START_REF] Curtis | Analysis of the effects of controlled shot peening on fatigue damage of high strength aluminium alloys[END_REF] The model uses a non-dimensional crack length and is given by (see Figure 9): [START_REF] Navarro | Short and long fatigue crack growth: A unified model[END_REF] The monotonic plastic zone occurring at the crack tip grows within the grain, is blocked by a barrier, and jumps after each grain boundary. Therefore the crack needs to grow within the monoplastic zone blocked at a barrier, before expanding within the following grain. The growth is then determined by the CTOD value described by [START_REF] Navarro | Fatigue crack growth modelling by successive blocking of dislocations[END_REF][START_REF] Vallellano | A micromechanical model for small fatigue crack growth: an approach based on two threshold conditions[END_REF]: [START_REF] Navarro | An alternative model for the blocking of dislocations at grain boudaries[END_REF] where, for plane strain conditions: [START_REF] Navarro | Fatigue crack growth modelling by successive blocking of dislocations[END_REF] According to Curtis et al. can be approximated by the cyclic yield ( ) [START_REF] Curtis | Analysis of the effects of controlled shot peening on fatigue damage of high strength aluminium alloys[END_REF].

The crack growth rate is a function of the CTOD and is given by: ( 21)

i 3  i 2  i 1  0 0 a n 1 n 2 1 2 iD 0 2 r iD  x ) 2 / /(iD x  max grain boundary crack D 0 r
where and are parameters experimentally determined as we will see in the following section.

A threshold condition is reached at a grain boundary and is given by the non-dimensional critical crack length [START_REF] Vallellano | Two-parameter fatigue crack growth driving force: Successive blocking of the monotonic and cyclic plastic zones at microstructural barriers[END_REF][START_REF] Vallellano | A micromechanical model for small fatigue crack growth: an approach based on two threshold conditions[END_REF][START_REF] Vallellano | On the estimation of fatigue failure under fretting conditions using notch methodologies[END_REF]: [START_REF] Vallellano | Fatigue crack growth threshold coniditons at notches. Part II: Generalization and application to experimental results[END_REF] where is the maximum stress threshold. The latter should be obtained from the experiment. Nevertheless, as proposed by Vallellano et al. [START_REF] Vallellano | Fatigue crack growth threshold coniditons at notches. Part I: Theory[END_REF][START_REF] Vallellano | Fatigue crack growth threshold coniditons at notches. Part II: Generalization and application to experimental results[END_REF], it can be approximated by: [START_REF] Vallellano | Two-parameter fatigue crack growth driving force: Successive blocking of the monotonic and cyclic plastic zones at microstructural barriers[END_REF] with the crack length is the short-long crack transition obtained in equation ( 1). where is the fatigue limit of the material at the corresponding stress ratio, and is a parameter equal to 2.5 for aluminium alloys as specified by Vallellano et al. [START_REF] Vallellano | Fatigue crack growth threshold coniditons at notches. Part I: Theory[END_REF][START_REF] Vallellano | Fatigue crack growth threshold coniditons at notches. Part II: Generalization and application to experimental results[END_REF].

Finally, if the monotonic plastic zone is able to overcome the grain boundary, meaning that the crack can propagate within the following grain, then the following step is given by: (

The number of cycle is then extracted by integrating Equation ( 21), namely: [START_REF] Shahzad | Surface characterization and influence of anodizing process on fatigue life of Al 7050 alloy[END_REF] where corresponds to the "starting" crack length within grain and is the "critical" crack length corresponding to the crack reaching the next grain boundary.

Application to fretting fatigue stressing using the FE modelling of the micro fretting fatigue experiments

Stresses are extracted at the exact contact border , where maximum stress gradients occur. As a rather straight crack path is experimentally observed we will focus on the mode I propagation, so that only the first component of the stress tensor will be considered.

Figure 10.b plots the obtained maximum and range stresses as functions of the distance from the surface related to Test F. The corresponded stress ratio is also added. It is clear that high stress gradients specially appear near the surface (

), leading to a variable stress ratio. Such variations have to be taken into account into the presented micromechanical fretting fatigue life model as they have a significant influence on the crack propagation.

As reminded by Aguirre et al. [START_REF] Aguirre | On the application of a micromechanical small fatigue crack growth model to predict fretting fatigue life in AA7075-T6 under spherical contact[END_REF], an equivalent stress has to be considered in order to take into account stress gradients, variable stress ratio and the crack nucleation at an edge. Vallellano et al. previously determined such an equivalent stress [START_REF] Vallellano | On the estimation of fatigue failure under fretting conditions using notch methodologies[END_REF] using the following expression: [START_REF] Newman | Fatigue and crack growth in 7050-T7451 aluminium alloy under constant-and variable-amplitude loading[END_REF] where is the correction factor and is a polynomial expression determined from the FEM [START_REF] Hartranft | Alternating method applied to edge and surface crack problems[END_REF]: [START_REF] Moustafa | Experimental study of the stress gradient effect under fretting loading by full field measurement techniques[END_REF] Figure 10 shows the calculation procedure to obtain equivalent stresses. The FE model exposed in Figure 5 is first computed. At the studied fretting fatigue load stage, (maximum, minimum, or both) is extracted at the exact contact border where the stress is maximum (

). Equation ( 26) is applied on the extracted stress to finally be able to plot the equivalent stress as a function of the depth. The maximum equivalent stress and the equivalent stress range are introduced in the micostructural model by replacing with and with .

Parameters calibration

The proposed model uses an initial defect to start the computation of the crack extension. This value is material dependent and has a significant impact on the life prediction. Even if it is known that is a representative inclusion size [START_REF] Lankford | The growth of small fatigue cracks in 7075-T6 aluminium[END_REF], Aguirre et al. [START_REF] Aguirre | On the application of a micromechanical small fatigue crack growth model to predict fretting fatigue life in AA7075-T6 under spherical contact[END_REF] reminded that it is essential to perform a deeper investigation on this value. Thus a parametric study on the initial 0 200 400 600 0,E+00

1,E-03  eq,max (MPa)

a (m) z d z a z a z f Y a a eq     0 2 2 max , 11 max , ) ( )) / ( 1 ( 2 ) (     11,max (z)
D max  min  max b) R min / max c) Depth z(m) Depth z(m)  11 (MPa)
defect was preliminary performed in order to check the model stability. It appeared that for a defect larger than , there is a drastic drop of the life prediction, whereas for defects between and , small variations are observed. For defects smaller than , no variation is observed. Assuming that fretting crack nucleation is led by severe stress gradients more than microstructural effects within the very early stage, an initial defect of will be used for the computations.

The variable was related to so that . The crack shape correction used in Equation ( 26) was set to corresponding to the theoretical value for semielliptical cracks [START_REF] Aguirre | On the application of a micromechanical small fatigue crack growth model to predict fretting fatigue life in AA7075-T6 under spherical contact[END_REF][START_REF] Irwin | Crack-extension force for a part-through crack in plate[END_REF]. The so called "Navarro-Rios coefficients" and in equation ( 21) are determined with the experiment by fitting the LEFM long crack regime [START_REF] Vallellano | Two-parameter fatigue crack growth driving force: Successive blocking of the monotonic and cyclic plastic zones at microstructural barriers[END_REF][START_REF] Vallellano | A micromechanical model for small fatigue crack growth: an approach based on two threshold conditions[END_REF]. Therefore the following equivalence can be made for long cracks:

The presented model was first computed using only fatigue loads in order to determine the "Navarro-Rios coefficients" and at the stress ratio . Figure 11 plots the crack growth rate as a function of the stress intensity factor range with . Figure 11 compares experimental data extracted from fatigue crack growth tests [START_REF]Metallic materials and elements for aerospace and vehicle sutructures[END_REF][START_REF] De Pannemaecker | Identification of the fatigue stress intensity factor threshold for different load ratios R: From fretting fatigue to C(T) fatigue experiments[END_REF] for long cracks performed at and the corresponding micromechanical small fatigue crack growth model. The long crack regime is fitting the experiment by taking (1/cycles) and (-).

Figure 11: Fatigue crack growth: model prediction versus experimental data at the stress ratio [START_REF] De Pannemaecker | Identification of the fatigue stress intensity factor threshold for different load ratios R: From fretting fatigue to C(T) fatigue experiments[END_REF][START_REF] Handbook | Properties and selection: nonferrous alloys and special purpose materials[END_REF] 1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-01 1.E+00 1.E+01 1.E+02 da/dN (m/cycle) DK (MPa.m 0.5 ) C(T) at R fa =0.3

Model Prediction

Long crack regime fitting with the experiment

Comparison between fretting fatigue experiments and micromechanical modelling

representation

Figure 12.a displays the in situ fretting fatigue data obtained at the synchrotron facility related to Test A. It appears that the experimental points seen on the ESRF radios are very close to the prediction line, which even enables us to identify the five zones observed on the SEM (Figure 4). During the first zone, the crack nucleation is driven by fretting loadings which generate acceleration. Arriving on the second zone, deeper within the sample, the crack decelerates as the crack tip is less influenced by fretting, creating an inflection point between the first and second zones. The third zone is crucial as this is where fatigue becomes the main driving parameter making the crack grow. Thereby the crack will drastically accelerate until failure of the specimen. By comparison, Figure 12.b shows the same trends on Test B. The latter was not in situ performed but the SEM observations are making the five zones identifiable on the crack growth predicted curve. The fretting and fatigue contributions can be analyzed thanks to Figure 13 by plotting the crack growth for both tests B and A as a function of the crack length . The continuous line represents Test A whereas dashes are related to Test B. For both tests, a first acceleration occurs, validating the observations made on Figure 12. The crack then decelerates around which exactly corresponds to the transition from the first to the second zone. The crack is obviously lead by fretting from its initiation up to . In this region, the curve corresponding to Test B is below the one obtained for Test A (Figure 13) which is , fatigue is monitoring the crack growth with a rather straight slope until reaching failure. Again, the curve related to Test B is below the one for Test A as the maximum fatigue stress is lower for Test B with versus for Test A. If those two differences on the fretting and fatigue loadings do not seem to be that significant on a mechanical point of view, they have a dramatic influence on the fretting fatigue life as Test B is longer than Test A with almost 50000 additional fretting fatigue cycles. Thus the short-long crack transition can easily be identified by combining Figure 12 and Figure 13. It appears that the transition is exactly at the one passing from the first to the second zone (Figure 12). The number of cycles related to short and long crack can directly be identified with the proposed model at cycles for Test B, and for Test A. Thereby the short and long crack domains represent 30% and 70% respectively for the fatigue life regarding Test B, versus 15% and 85% for Test A.

representation

The micromechanical model is based on a crack growth description allowing a more representative analysis of material microstructural effects on short crack propagation. It can also be transposed to the engineering representation. Indeed, assuming semi-circular cracks, Akiniwa et al. [START_REF] Akiniwa | Statistical characteristics of propagation of small fatigue cracks in smooth specimens of aluminium alloy 2024-T3[END_REF] proposed the following equation: [START_REF] De Pannemaecker | Identification of the fatigue stress intensity factor threshold for different load ratios R: From fretting fatigue to C(T) fatigue experiments[END_REF] In their research work, Akiniwa et al. focused on the 2024-T3 aluminum alloy where samples were machined using thin sheets about thick. Therefore the same correction factor can be used in the present work as the studied samples displayed a thickness which is in the same order of specimen thickness. The equivalent stress intensity factor range can be calculated with:

The corresponding were computed and compared to the experimental results defined from the FEM-CI stress intensity factor past processing (Figure 14). All curves are converging for the long crack regime. For short cracks, both FE model (red squares) and the micromechanical fretting fatigue life model (black line) are superimposed confirming that the proposed model is relevant for the studied case. By zooming on the short-long crack transition, points related to the FE model based on the LEFM display same trends as the predicted model with a deceleration followed by a drastic acceleration corresponding to the end of the first zone (Figure 12).

Comparison between the Short Crack Correction (SCC) and the Navarro-Rios (NR) approaches

As previously discussed, the Navarro-Rios approach is the only one able to describe the nonmonotonous crack growth evolution when the crack moves from the fretting dominant domain to the deeper fatigue dominant stress domain. To better compare the different modellings, the The analysis clearly illustrates the very poor correlation given by the plain Paris approximation (

). The El Haddad approach allows a better prediction and the optimal gives a closer estimation of the crack length's evolution, particularly in the very short crack length domain. The best prediction is given by the Navarro-Rios microstructural approach which is able to describe the relative influence of fretting and fatigue stresses.

Another way to analyze the models consists in comparing both predicted and experimental fretting fatigue endurances and the related relative errors (see equation 31, Table 5 and Figure 16): [START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF] This analysis clearly underlines the stability of the proposed transposition of the Navarro-Rios (NR) model to predict the fretting fatigue crack propagation endurance. It also suggests that despite its simplicity and rather poor physical support of its formulation, the optimal Short Crack Correction (SCC, ) model allows conservative predictions.

Conclusions

Using in situ the X-ray synchrotron facility, the fretting fatigue crack growth rate of a 7050-T7451 aluminum alloy was quantified for two specific fretting fatigue loadings. The following points can be highlighted:

-The calculated crack growth rates are consistent with the Paris formulation obtained from conventional C(T) fatigue experiments but differs in the short crack regime (for low ) where a rather constant crack growth rate occurs ( when ). -The experimental fretting fatigue analysis shows a discontinuous evolution of the crack propagation in the short crack domain and an unexpected non-monotonous evolution characterized by a deceleration and acceleration when the crack extends deeper below the contact stress influence. -To capture such peculiar behavior, the Pugno et al. and Fleury et al. short crack correction strategy, inspired by the El Haddad approach, is applied and rewritten using a basic power expression: where is the short-long crack transition.

-The comparison with X-ray micro fretting fatigue results shows that the plain Paris expression ( ) gives dangerous overestimated predictions. The correlation is improved using the original El Haddad approximation (

). The best prediction is however achieved with . This basic approach provides more realistic and conservative predictions, but still requires deeper investigations to explain the high value of the exponent. -To provide a more physical description of the complex short crack propagation behavior enduring severe stress gradients imposed by fretting stressing, the Navarro-Rios model is transposed: and curves are obtained and show short and long crack behaviours. The curves are well fitting the long crack regime obtained with conventional C(T) fatigue crack growth tests. Additionally the short crack region is perfectly superimposed with data previously obtained thanks to in situ fretting fatigue test analysis.

-The short crack contribution for the total life is smaller than for usual fatigue experiments (from 15% to 30%) and significantly reduces the endurance due to fretting stress gradient effects. -Despite the fact that the given Navarro-Rios model only considers a mode I description, very good predictions were obtained leading to relative errors inferior to compared to the experiment.

  Crack length (m) Short crack/long crack transition (m) Coefficient describing the Paris region ( ) Coefficient describing the Navarro-Rios model ( ) Half width of the contact area (m) Average grain size (m) Crack growth rate (m/cycle) Young's modulus (Pa) Young's modulus corresponding to plane strain conditions (Pa) Relative error (%) Material dependent parameter (-) Polynomial function determined by the FEM (-) Fatigue load (N) Fatigue amplitude (N) Maximum fatigue amplitude (N) Average fatigue load (N) Parameter corresponding to a certain state (-) Stress intensity factor ( ) Equivalent stress intensity factor ( ) Long crack stress intensity factor threshold ( ) Exponent describing the Paris region (-) Exponent describing the Navarro-Rios model (-) Exponent describing the Short Crack Correction model (-) Number of cycles (-) Number of cycles at failure (-) Related to the Navarro-Rios model Non-dimensional crack length (-) Non-dimensional crack length located at a grain boundary (-) Critical non-dimensional crack length (-) Critical non-dimensional crack length located at a grain boundary (-) Linear normal load (N/m) Surface pressure (Pa) Maximum surface pressure (Pa) Fretting linear tangential load (N/m) Fretting linear tangential load amplitude (N/m) Surface shear (Pa) Maximum surface shear (Pa) Stress ratio (-) Surface roughness (m) Equivalent stress ratio (-) Stress ratio related to the fatigue load (-) Stress ratio related to the fretting load (-) Radius of the fretting pad (m) Distance of the first dislocation located after a grain boundary (m) Related to the Short Crack Correction model Correction factor (-) Depth (m) Greek letters: Stress intensity factor range ( ) Equivalent stress intensity factor range ( ) Long crack stress intensity factor range threshold ( ) Crack tip opening displacement (m) Friction coefficient within the crack lips (-) Friction coefficient at the sliding transition (-) Poisson coefficient (-) Stress (Pa) Fatigue stress amplitude (Pa) Equivalent stress (Pa) Maximum stress threshold (Pa) Ultimate stress (Pa) Yield stress (Pa) Conventional yield stress at 0.2% (Pa) Stress opposed to the crack tip opening and closing (Pa) Stress opposed to deformation (Pa) Stress needed for a crack to propagate (Pa)

Figure 1 :Figure 2 :

 12 Figure 1: Micro fretting fatigue device (a) schematic diagram of the setup, (b) specimen geometry, (c) global view at the ESRF on ID19 beamline, (d) zoom on the contact[START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF]

  on the contact: ESRF Radiography: specimen

Figure 3 :Figure 4 :

 34 Figure 3: SEM observations of the fracture surfaces of the sample broken during Test A (a) secondary electron mode (b) back scattered electron mode (c) zoom (arrows indicate different crack initiation sites) and schematic description of the nucleation zone due to fretting ()[START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF]

Figure 5 :

 5 Figure 5: FE modeling of the micro fretting fatigue experiments (a) FE contact description, (b) loading sequence on an elastic configuration (TestA). The stress within the fretting fatigue specimen is maximum when as and minimum when as . (c) Micro fretting fatigue experiments: surface shear and pressure distributions at the maximum stress level from the FE simulation[START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF] (with the half-width contact, the half-width sticking zone, the eccentricity due to fatigue)

  Figure 4.a. A friction coefficient between the crack lips is a)

Figure 6 :

 6 Figure 6: Micro fretting fatigue FEM cracked modelling (Test A after 132521 cycles, at and ): schematic configuration and zoom at the crack tip illustrating the monotonic plastic zone for a crack length measured just before failure

" 3 (

 3 Micro fretting fatigue" at R fa =0.Test A) C(T) at R fa =0.3 C(T) at R fa =0.33 (literature) Trend line (for short crack)

Figure 8 :

 8 Figure 8: Comparison between the proposed Short Crack Correction (SCC[START_REF] Fleury | Evaluating the influence of residual stresses and surface damage on fatigue life of nickel superalloys[END_REF][START_REF] Pugno | A generalized Paris' law for fatigue crack growth[END_REF]) and the studied micro fretting fatigue analysis (Test A[START_REF] De Pannemaecker | In situ fretting fatigue crack propagation analysis using synchrotron X-ray radiography[END_REF])

Figure 10 :

 10 Figure 10: Equivalent stress extraction procedure (a) FE modeling with the stress extraction at the contact border along the depth (Test A) and at the maximum fretting fatigue load stage, (b) Stress range variation and the corresponding stress ratio as a function of the depth (c) Maximum equivalent stress calculation (Test A) [39, 40] (Test A: cycles)

Figure 12 :

 12 Figure 12: Fretting fatigue crack length evolution as a function of the fretting fatigue cycles and comparison versus SEM fracture surfaces (correlation versus fractographic features 1 to 5)) (a) In situ fretting fatigue crack propagation observation (Test A) versus the model prediction, (b) Model prediction versus experimental failure (Test B)

  experiment as Test B displays a lower fretting contribution with a maximum shear stress about compared to for Test A. When cracks are above

Figure 13 :

 13 Figure 13: Predicted fretting fatigue crack growth as a function of the crack length for Tests B ( ) and Test A ( )

Figure 14 :

 14 Figure 14: Predicted and experimental small crack propagation curves related to Test A ( )

  evolution of the crack length as a function of the fretting fatigue number of cycles are compared (Figure15).

Figure 15 :

 15 Figure 15: Comparison between experimental (Test A) and simulated evolution of the crack length as a function of the fretting fatigue cycles (NR: Navarro-Rios model; SCC: Short Crack Correction with : no correction, : El Haddad model, : optimal value obtained from the chart)

Figure 16 :

 16 Figure 16: Comparison of the relative errors related to each prediction model

Table

  1 and Table

Table 1 :

 1 Chemical composition (wt.%) of the 7050-T7451 aluminum alloy[START_REF] Shahzad | Surface characterization and influence of anodizing process on fatigue life of Al 7050 alloy[END_REF]

	Si	Fe	Cu	Mn	Mg	Ti	Zn	Zr	Al
	0.28	0.19	1.76	0.07	2.42	0.06	6.15	0.14	Bal

Table 2 :

 2 Mechanical properties of the two studied materials

	Material
	7050-T7151 [26]	73.4	0.33	460	525
	52100 [27]	200	0.3	813	1500

Table 3 : Studied fretting fatigue experiments

 3

	Material
	7050-
	T7151	73.4 0.33	460	80	162	1.8	2.57
	[28, 29]
	52100 [27]	200	0.3	813	N/A	N/A	N/A	N/A

Table 4 :

 4 Micro fretting fatigue test configurations ( )

	Test

Table 5 :

 5 Experimental and predicted fretting fatigue lives -Relative errors (%)

			Experiment	Short Crack Correction (SCC)	Navarro-Rios
	Test		(reference)			(NR)
	A	Number of cycles Relative error E(%)	1.33E+05 -	1.08E+07 8025.73	1.27E+06 1.93E+05 857.04 45.83	1.38E+05 4.32
	B	Number of cycles Relative error E(%)	1.79E+05 -	1.21E+07 6664.37	1.24E+06 2.58E+05 592.90 43.83	1.71E+05 -4.88



  In situ X-ray imaging of micro-fretting fatigue experiments are simulated.  A Pugno et al./Fleury et al. short crack correction (SCC) approach is considered.  A microstructural Navarro-Rios (NR) model is transposed.  Strategies are compared and very good agreement with experiments are observed.  The NR model provides a more physical description of the short crack behaviour.

Acknowledgments

The authors thank the Ecole Polytechnique de Montréal, département Génie Mécanique, for the new established partnership with the LTDS and MATEIS, and the Carnot Institute (I@L, project Tomo-Fretting) for financial support.


A B Lemullier
email: aude.lemullier@live.fr

F T J M Fortuin

S W Omta

L Sauvée

Impact of R&D & Marketing integration on innovation and business performance in French food processing companies

Keywords: R&D, Marketing, Integration, Innovation, Performance, French food processing companies 2009;

The globalization and the rapid changes of consumers' needs make the food industry turbulent. Companies have to become more market-oriented and to implement an adequate innovation strategy to respond to these changing needs. In management literature, cross functional integration, especially between R&D & Marketing, is thought to be crucial to achieve better market-oriented innovation and ultimately better business performance and success in the market place. It is the aim of this study to show the impact of integration barriers and mechanisms on R&D & Marketing integration and consequently on innovation and business performance. A questionnaire survey in 30 French food processing companies with more than 120 employees and 3 in-depth interviews with two R&D directors and one Marketing director were conducted in order to get specific insights in R&D & Marketing integration in French food processing companies.

R&D & Marketing integration positively impacts innovation and business performance in the participating French food processing companies. The quality of R&D & Marketing integration is impacted by integration barriers and mechanisms such as communication and cooperation, organizational structure (i.e. cross functional teams), the sharing of goals, formal integrative management processes, language, ICT (Information Communication Technology) tools, informal social systems (e.g. trips and group events) and well balanced incentives and rewards.

Introduction

According to the CIAA (Confédération des Industries Agro-Alimentaires de l'Union Européenne) review of key competitiveness indicators, the objective of the food-industry is to grant safe, nutritious, sustainable, healthy and affordable food and drinks to society (CIAA report, 2008). The French food processing industry is the first industry of the country with a turnover of €147billion in 2008 (Ministère de l 'Agriculture, 2010). With an export value of €44 billion (against €38.5 billion of imports) France is the fourth exporter of food products in the world after the USA, The Netherlands and Germany (Ania, 2010;[START_REF] France | Enjeux des industries agroalimentaires[END_REF]. In terms of processed food products, France, first exporter for a long time, is now the fourth in the world after The Netherlands, Germany and USA (Ministère de l'Agriculture, 2010). In France,10.568 French companies conduct their activities in the food processing industry, including 97% of firms with less than 250 employees and 73% of firms with less than 20 employees (Ania, 2010).

Since the 1970s, a lot of innovations in process, product, organization of production and distribution in the French food industry have occurred [START_REF] Nicolas | Apprentissage technologique et innovation en agro-alimentaire[END_REF]. According to a study of the Dutch Institute LEI (Landbouw Economisch Institut), the competitiveness of European food industry is weak compared to the American or the Canadian one [START_REF] Wijnands | Competitiveness of the European food industry -An economic and legal assessment[END_REF]. The growth and competitiveness of food processing companies depend on their capacity to differ themselves from competitors. This differentiation is link to the capacity to innovate. By constructing "pôles de compétitivité" (competitiveness clusters) since 2005 in all its industries, France has a good advantage for competitiveness. The "pôles de compétitivité" of the food sector have as principal objective to create a local R&D's dynamic (Ministère de l 'Agriculture, 2008).

Concerning agricultural and food markets, recent changes are observed. The most important one is the shift from production to market orientation [START_REF] Omta | Financiarisation et globalisation des stratégies d'entreprises : le cas des Multinationales agroalimentaire en Europe[END_REF]. Consumers' needs and wants concerning food are changing and the companies have to quickly adapt themselves to keep competitive advantages. One consequence is that innovation, as a primordial factor for business competition, is becoming crucial in agri-food marketing strategy. According to [START_REF] Matthyssens | Value innovation in the functional foods industry[END_REF], value innovation, which is "a strategic approach to business growth", "can be achieved by implementing a focus on innovation and creation of new market possibilities" (www.qfinance.com, 2010) and is one of the fastest growing domains in the food industry. The implementation of an internal culture is one of the important management practices useful to achieve it. One of the most important factors of innovation success is R&D & Marketing integration (Fortuin and Omta,

Theoretical background and study domain

Interdepartmental integration

Integration is defined as "the degree to which there is communication, collaboration and a cooperative relationship between R&D and marketing" (Leenders and Wierenga, 2002, p.306). Interdepartmental integration is described as a single dimensional concept or a multidimensional concept. First, the single dimensional view can be interaction-based (structured part of interdepartmental integration) or collaboration-based (unstructured nature of interdepartmental integration). Second, the multidimensional concept is mentioned when both interaction and collaboration are taken into account [START_REF] Kahn | Interdepartmental Integration: A Definition with Implications for Product Development Performance[END_REF]. The composite view of integration reflects the distinct nature of interaction and collaboration [START_REF] Kahn | Marketing's Integration with Other Departments[END_REF].

According to [START_REF] Kahn | Marketing's Integration with Other Departments[END_REF], collaboration is the strongest factor and the most appreciate one to well managed R&D & Marketing integration and ultimately to achieve better performance. Interaction can also improve companies' performance but too much interaction may be negative for companies. [START_REF] Kahn | Marketing's Integration with Other Departments[END_REF] recommended to R&D & Marketing managers to focalize on collaboration to integrate the two functional areas. Figure 1 shows the impact of interaction and collaboration on companies' performance.

Figure 1 : Influence of interaction and collaboration on performance (adapted from [START_REF] Kahn | Interdepartmental Integration: A Definition with Implications for Product Development Performance[END_REF]Kahn and[START_REF] Kahn | Marketing's Integration with Other Departments[END_REF].

R&D & Marketing integration barriers

Different R&D & Marketing integration barriers exist in companies and often lead to communication, cooperation and collaboration problems. The main barriers presented in the management literature are stated below. [START_REF] Grifffin | Integrating R&D and Marketing: a review and analysis of the literature[END_REF]. Training and background are often different between R&D & Marketing employees. The differences of culture inside firm's departments increase these barriers. Even if employees work for common goals, their interpretations of these goals can be different (Griffin and Hauser, 1996).

Organizational responsibilities: This barrier includes the difference of priorities and responsibilities of R&D & Marketing employees. When functional success measures are unsupportive of integration, difficulties appear between R&D & Marketing employees to communicate and cooperate (Griffin and Hauser, 1996).

Goal incongruity: This barrier is defined as "the extent to which members of marketing and R&D subscribe to different goals, use different decision criteria, and think differently about time" (Song and Song, 2010, p. 384).

Physical barriers: When R&D & Marketing departments are isolated it is harder for employees to communicate. The number of meetings, the quality of information transfer as well as the good problems' clarification decrease in the same way as the physical distance (Griffin and Hauser, 1996).

Communication barriers and cooperation: Communication barriers are created by the differences of properties between the two departments, the time pressure and the poor timing free for communication [START_REF] Gupta | The R&D-Marketing interface in High-Technology Firms|[END_REF].

Insensitivity to each other's capabilities and perspectives: Marketing managers point out the fact that R&D managers are not sensitive to marketing's perspectives and to the characteristics of the market environment [START_REF] Gupta | The R&D-Marketing interface in High-Technology Firms|[END_REF].

Lack of senior management support: In the study of [START_REF] Gupta | The R&D-Marketing interface in High-Technology Firms|[END_REF], R&D & Marketing managers mentioned that senior management miss long run orientation and push too much in order to get near term profits. Another primordial problem is the promotion of marketing performance measures that obstruct successful R&D & Marketing integration [START_REF] Gupta | The R&D-Marketing interface in High-Technology Firms|[END_REF].

Lack of market knowledge: The lack of market knowledge such as knowledge of competition, customers and product appliance forms barriers to R&D & Marketing integration. [START_REF] Gupta | The Credibility-Cooperation Connection at the R&D-Marketing Interface[END_REF] affirmed that a problem of credibilitycooperation connection exist in high tech firms to integrate R&D & Marketing departments. R&D managers think that Marketing managers do not provide enough information (e.g. about customers requirements, test marketing results and competitors' strategy) and lack credibility [START_REF] Gupta | The R&D-Marketing interface in High-Technology Firms|[END_REF].

R&D & Marketing integration mechanisms

Some mechanisms can be implemented to facilitate R&D & Marketing integration. Those mechanisms are:

Relocation and physical facilities: According to Griffin and Hauser (1996), it is primordial to relocate the departments because R&D & Marketing integration is improved when the two departments are close to each other. It allows the improvement of communication and cooperation between employees as well as the achievement of a better market place success (Griffin and Hauser, 1996).

Personnel movement: It means that employees can move between different functions to reduce the probability of isolated thought worlds [START_REF] Leenders | The effectiveness of different mechanisms for integrating marketing and R&D[END_REF]. It allows the development of information flows between departments and enables a better understanding by bringing common language. [START_REF] Monaert | R&D-Marketing Integration Mechanisms, Communication Flows, and Innovation Success[END_REF] mentioned that job rotation increases interaction between employees and Griffin and Hauser (1996) sated that personnel movement increases marketplace success and decreases time to market. As this mechanism is difficult to achieve, firms sometimes ask the employees to have both R&D & Marketing competencies before to work in one of those departments. Temporary transfer is also a good solution for employees to obtain training in both areas.

Informal social systems and culture: The implementation of informal social systems enables open communication and provides contacts between employees. This mechanism helps to reduce language and cultural through worlds barriers as well as physical barriers by improving the communication, the utilization of information and the coordination and by reducing uncertainties (Griffin and Hauser, 1996). Informal networks can be implemented by organizing informal group events such as trips including employees of different functional areas [START_REF] Leenders | The effectiveness of different mechanisms for integrating marketing and R&D[END_REF].

Organizational structure: The delegation of an "integrator" to manage a project is an important factor to improve communication and cooperation between employees (Griffin and Hauser, 1996). Another solution concerning organizational structure is the promotion of specific relationships between R&D & Marketing employees for each innovation project. Via coordinating groups, matrix organizations and projects teams, both coordination and communication are increased (Griffin and Hauser, 1996). According to [START_REF] Leenders | The effectiveness of different mechanisms for integrating marketing and R&D[END_REF], the structure the most appropriate for R&D & Marketing integration is the cross-functional project teams. Those teams support cooperation and information exchange and ultimately have a positive effect on integration [START_REF] Leenders | The effectiveness of different mechanisms for integrating marketing and R&D[END_REF].

Incentives and rewards: According to Griffin and Hauser (1996), R&D & Marketing employees can get bonuses base on technology improvement for R&D employees and base on the increase of market share for Marketing employees for instance.

This kind of rewards augments differentiation between R&D & Marketing employees. Interrelated rewards for R&D & Marketing employees base on product-development profits decrease organizational responsibilities barriers and increase profits. Moreover well balanced incentives and rewards lead to cross-functional decision-making and help to resolve conflicts between the two departments. [START_REF] Leenders | The effectiveness of different mechanisms for integrating marketing and R&D[END_REF] mentioned that the different opportunities of remuneration and career have to be harmonized and identical for R&D & Marketing employees to improve integration. In opposite, [START_REF] Igbaria | Career orientations and career success among research, and development and engineering professionals[END_REF] stated that R&D employees need specific mechanisms to drive motivation. Individual needs and personal competences should be taken into account when evaluating researchers [START_REF] Debackere | Strategic career development for R&D professionals: lessons from field research[END_REF][START_REF] Igbaria | Career orientations and career success among research, and development and engineering professionals[END_REF]. Then, incentives and rewards should be harmonized but also adapted to the employees' needs.

Formal integrative management processes: Griffin and Hauser (1996) announced that the objective of this mechanism is to

Conceptual model, questionnaire construction and data collection

Conceptual model and questionnaire construction

Figure 2 shows the way in which we expect integration barriers and mechanisms impact the quality of R&D & Marketing integration and ultimately innovation and business performance.

Integration barriers and mechanisms: They refer to the different factors previously described. In this study, the majority of the factors have been statistically analyzed. Cultural through worlds, language and insensitivity to each other's capabilities perspectives' factors have been analyzed only with interviews.

R&D & Marketing integration: the quality of R&D & Marketing integration has been analyzed by looking at the items below:

-The employees of different functions are working together to identify and select new product/technology opportunities.

-Current market information is passed on by marketing to R&D on a regular basis.

-There is an excellent communication between R&D & Marketing employees.

-There is an excellent cooperation between R&D & Marketing employees.

-R&D & Marketing employees share common ideas, collective goals, common vision of the company, "esprit de corps".

Performance: Innovation performance is operationalized by looking at the effectiveness of the R&D process, the protections of the products and processes (e.g. patents) compared to the main competitors, the new products' capacity to enter the market, the returns from R&D investments and the percentage of current sales generated by new or strongly improved products.

Business performance is operationalized by creating questionnaires' items about financial position, reputation in the market, flexibility of market responses compared to the main competitors, the yearly growth rate and the operating profit margin. The questionnaire was adapted from WIAT (Wageningen Innovation Assessment Tool) and was composed by 38 questions.

To obtain in depth information about the subject, three semi structured interviews have been conducted.

Data collection

The data have been collected during the summer 2010. A total of 30 questionnaires (out of 181) have been sent back. The sample is composed of 63% R&D oriented and 37% Marketing oriented employees. 33% of the companies practice their activity in the dairy sector, 24% in the meat sector and 19% in the vegetables and fruits sector. Concerning the size of the companies, 50% are medium sized (less than 600 employees) and 50% are large companies (more than 600 employees). The small companies (less than 120 employees) have been excluded because those companies do not have separate R&D & Marketing departments and R&D & Marketing integration is not a big issue for them. The three in-depth interviews have been performing with two R&D directors and one Marketing director.

Results

Impact of the quality of R&D & Marketing integration on innovation and business performance

R&D & Marketing integration positively impacts the effectiveness of the R&D process and ultimately the innovative performance of the participating French food processing companies. R&D & Marketing integration, by impacting the yearly growth rate and the operating profit margin of the companies as well as the good reputation in the market and the position in the market compared to their main competitors, has an impact on business performance of the companies.

In this study, interaction has been measured by asking respondents to mention if their company use or not ICT tools (e.g. emails and videoconferencing) and to precise the frequency of the use of interaction tools such as cross functional meetings.

Collaboration has been measured by asking to the respondents how far R&D & Marketing employees share ideas and goals.

These concepts have also been discussed during the interviews. It appears that both interaction and collaboration are used in the participating French food processing companies but collaboration is the most implemented one. Interaction and collaboration are both positively linked to the quality of R&D & Marketing integration. Interaction and collaboration, by impacting the effectiveness of the R&D process, positively impact the innovative performance of the participating French food processing companies. Moreover collaboration, by positively impacting the yearly growth rate compared to the main competitors, impacts the business performance of the companies. The findings of the study of [START_REF] Kahn | Marketing's Integration with Other Departments[END_REF], mentioning that collaboration is the most important factor to well managed R&D & Marketing integration and ultimately to achieve better performance, is confirmed. Concerning interaction, it appears that the frequency of the use of cross functional meetings is much more important to achieve good quality of R&D & Marketing integration than the use of ICT tools such as emails, videoconferencing and phone calls. Personnel movement: Griffin and Hauser (1996) stated that personnel change is an essential tool to reduce language barrier.

Impact of integration barriers and mechanisms on the quality of R&D & Marketing integration

In this study, the results of the questionnaires survey do not prove the positive impact of personnel change on R&D & Marketing integration. Moreover, two interviewees said that it is not a good solution. They think that individual specificities are needed to successfully achieved innovation projects and that, by changing of function, the employees could lose their specificities.

Organizational structure: Griffin and Hauser (1996) mentioned that it is very important to select integrators and the interviewees agreed with that but no link have been statistically found between the quality of R&D & Marketing integration and the selection of integrators. [START_REF] Fortuin | Innovation barriers and drivers in food processing[END_REF] and [START_REF] Fortuin | Key success factors of innovation in multinational agrifood prospector companies[END_REF] Lack of market knowledge: Management literature states that companies should take customer needs into account to gain innovation performance and that marketing should become more and more future oriented [START_REF] Fortuin | Innovation barriers and drivers in food processing[END_REF][START_REF] Fortuin | Key success factors of innovation in multinational agrifood prospector companies[END_REF]. The participating French food processing companies align their products and processes to customers' needs and the market information is well transferred from marketing to R&D department. It has a positive impact on the quality of R&D & Marketing integration. The frequent use of interactions tools such as meetings help to overcome the lack of market knowledge. [START_REF] Costa | New insights into consumer-led food product development[END_REF] expect the existence of a positive link between market-orientation and business performance. The empirical part of this study found out a positive and direct link between the transfer of market information and innovation (i.e. effectiveness of the R&D process compared to the main competitors) and business (i.e. position of the company in the market and operating profit margin compared to the main competitors) performance. Therefore, more R&D employees become market-oriented (thanks to the Marketing employees who transfer the market information) and more the companies achieve better performance.

  Language / Cultural through worlds: R&D & Marketing employees use different technical terms. Marketing employees talk about products benefits and positioning while R&D professionals talk about specifications and performance (Griffin and PERFORMANCE -

  specify to the employees what tasks have to be achieved, in what order and by whom. The implementation of interactive sessions between R&D & Marketing employees allows the exchange of information, the modification of personal value etc. By formally structuring decision-making processes between R&D & Marketing, harmony appears between them.ICT (Information Communication Technology) tools: It helps to transfer information inside the companies.[START_REF] Leenders | The effectiveness of different mechanisms for integrating marketing and R&D[END_REF] argued that ICT tools lead to a better R&D & Marketing integration and ultimately to better new products developments. ICT tools include different communication technologies such as e-mail, electronic bulletin board, video conferencing and also decision-aiding technologies such as expert systems and executive information systems(Song and Song, 2010). According to Song and Song (2010), ICT tools help to reduce the negative impact of physical distance, goal incongruity and cultural barriers on R&D & Marketing integration.

Figure 2 :

 2 Figure 2 : Conceptual model

Figure 3 ,Figure 3 :Figure 4 :Figure 5 :Figure 6 :

 33456 Figure 3, Figure 4, Figure 5 and Figure 6 present the significant correlations between questionnaires' items to demonstrate the links between R&D & Marketing integration and performance and between integration barriers and mechanisms and the quality of R&D & Marketing integration.

  Communication and cooperation: Internal and external communication and cooperation are essential for innovation. A bad communication is described as one of the most important barrier to achieve good R&D & Marketing integration. Good communication and cooperation exist in the participating French food processing companies and positively impact the quality of R&D & Marketing integration. Moreover, communication positively impact both innovative (i.e. effectiveness of R&D process) and business (i.e. yearly growth rate) performance. Language barrier: Griffin and Hauser (1996) stated that the use of specific terms (e.g. benefits, positioning, specifications and performance) by R&D & Marketing employees leads to misunderstanding. The interviewees mentioned this barrier as one of the most important to achieve good R&D & Marketing integration and common efforts are required from both R&D & Marketing employees. By overcoming language barrier, R&D & Marketing become more sensitive to each other's capabilities and perspectives.

  stated that teams including employees from different areas are required for successful innovation projects. The participating French food processing companies implement cross functional teams including R&D & Marketing employees. This study shows that the small physical distance between R&D & Marketing department is positively linked to the implementation of cross functional teams. The positive impact of the implementation of cross functional teams on R&D & Marketing integration has been proved and the managers are aware of the importance of the creation of those teams during innovation projects.Physical distance between R&D & Marketing and ICT tools:The empirical outcomes of this study provide evidence that the respondents do not think that physical distance is an obstacle for communication. According to them, when efficient management processes such as meetings and committees and ICT tools such as emails and videoconferencing are used, the physical distance is not an obstacle anymore. As ICT tools are rather well used in the participating French food processing companies, especially in the largest ones, the physical distance is not perceived as an obstacle for communication and cooperation between R&D & Marketing employees. This study shows that the use of ICT tools positively impact the quality of R&D & Marketing integration. Formal integrative management processes: The implementation of interactive sessions including R&D & Marketing employees allows the exchange of market information. Moreover, integrative management processes increase communication and cooperation between employees and ultimately increase the quality of R&D & Marketing integration. This integration mechanism is well used in the participating French food processing companies and positively impacts the quality of R&D & Marketing.Informal social systems: Informal social systems as parties and trips including R&D & Marketing employees enable communication and help to decrease the language barrier. This mechanism is moderately used in the participating French food processing companies whereas it has a positive impact on the quality of R&D & Marketing integration. Innovative companies organize more social events than the other companies.Incentives and rewards: The implementation of well balanced incentives and rewards is positively linked to the quality of R&D & Marketing integration but is moderately achieved in the participating French food processing companies. Incentives and rewards should be better balanced by taking into account individual needs of R&D & Marketing employees; well balanced do not signify identical. According toGriffin and Hauser (1996), interrelated rewards to R&D & Marketing increase profit. The interviewees said that motivation drivers to stimulate innovation are not implemented in their companies whereas it has a positive impact on the fact that employees work together to select opportunities and ultimately on the quality of R&D & Marketing integration. According to the interviewees, motivation drivers for innovation are already included in their gross salary. According to them, one good solution to stimulate innovation could be to organize campaign for innovation inside the company by asking employees for new ideas and reward the best ideas with a non financial gift (e.g. tickets for sport activities, trips etc.).Lack of senior management support: The lack of senior management support to define responsibilities is described as an important barrier to implement good communication and cooperation between R&D & Marketing employees. Even if top managers actively participate in the selection of R&D projects and in the definition of goals and responsibilities, especially in the participating innovative companies, this study do not statistically prove the impact of senior management support on R&D & Marketing integration and ultimately on innovation and business performance. In opposite, the interviewees mentioned the active participation of top managers in the definition of goals, priorities and responsibilities as essential to successfully achieve innovation projects.Goal incongruity: According to the interviewees, top management commitment to innovation decreases the goal incongruity barrier. In the participating French food processing companies, R&D & Marketing employees share ideas and goals and it is positively linked to the quality of R&D & Marketing integration.

Discussions and conclusions

As the response rate is low (16.57%), the representativeness and the generalization of the outcomes is limited to the participating French Food processing companies. Moreover, more than three interviews would increase the quality of the study and the relevancy of the information gathered. Following conclusions are drawn.

Management literature sates that R&D & Marketing integration is positively linked to innovation and business performance (Griffin and Hauser, 1996;[START_REF] Kahn | An Empirical Study of the Relationships among Co-location, Integration, Performance, and Satisfaction[END_REF]Leenders and Wierenge, 20002;Viaene and Januszeska, 1999;[START_REF] Monaert | R&D-Marketing Integration Mechanisms, Communication Flows, and Innovation Success[END_REF][START_REF] Kahn | Interdepartmental Integration: A Definition with Implications for Product Development Performance[END_REF][START_REF] Kahn | Marketing's Integration with Other Departments[END_REF][START_REF] Fortuin | Innovation barriers and drivers in food processing[END_REF]. Griffin and Hauser (1996) and Song and Song (2010) mentioned that R&D & Marketing integration is essential to firms' success in the market place and, according to [START_REF] Kahn | Marketing's Integration with Other Departments[END_REF], R&D & Marketing integration positively impacts department performance, company performance, product development performance and product management performance. As mention in Section 4, the impact of R&D & Marketing integration on company performance has been verified in this study and especially the impact on the effectiveness of the R&D process (i.e. innovation performance and R&D department performance), the yearly growth rate, the operating profit margin, the reputation in the market and the position in the market (i.e. business performance). The first objective, which was to show the impact of R&D & Marketing integration on innovation and business performance in the French food processing industry is achieved.

Concerning the second research objective, this study shows that several integration barriers and mechanisms impact the quality of R&D & Marketing integration. The positive impact of a good communication and cooperation, the transfer of market knowledge, the implementation of informal social systems, common goals, well balanced incentives and rewards, formal integrative management processes and the use of ICT tools on the quality of R&D & Marketing integration have been confirmed. Furthermore, as mention in the literature, interviewees stated the differences of language and culture and the insensitivity to each other's capabilities and perspectives as important barriers for R&D & Marketing integration. Physical distance is not perceived as an integration barrier insofar as ICT tools and cross-functional meetings are well implemented.

As the participating French food processing companies implement those mechanisms, departments' relocation is not necessary (as suggested by Griffin and Hauser, 1996). Concerning organizational structure, [START_REF] Leenders | The effectiveness of different mechanisms for integrating marketing and R&D[END_REF] stated that cross functional teams have a positive impact on departments' integration; it has been confirmed in this study. In opposite, the positive impact of the delegation of an integrator to manage innovation projects on the quality of R&D & Marketing integration has not been confirmed. The literature mentioned that personnel movement can be a mechanism to achieve a good R&D & Marketing integration, but no relationship has been demonstrated in this study.

According to the outcomes of this study, the main managerial recommendations are: -Create cross functional teams for each innovation project with regular meetings (at least once a month).

-Implement group events (e.g. trips and parties) including R&D & Marketing employees to develop affinities.

-Create a well balanced incentive system and motivation drivers and rewards for R&D & Marketing employees.

-Stimulate the intensive use of ICT tools (e.g. emails, videoconferencing), especially when the physical distance between R&D & Marketing is large to increase the communication between R&D & Marketing employees.