Lucas Fourel

Jean-Philippe Noyel

Etienne Bossy

Xavier Kleber

Philippe Sainsot

Fabrice Ville
email: fabrice.ville@insa-lyon.fr

Towards a Grain-scale Modeling of Crack Initiation in Rolling Contact Fatigue -Part 2: Persistent Slip Band Modeling

Keywords:

Rolling Contact Fatigue (RCF) is the result of crack initiation and propagation leading to surface damages. This study proposes a mesoscopic model for RCF crack initiation simulation. Finite Element Method (FEM) is used to obtain stresses using cubic elasticity. Persistent Slip Bands (PSBs) are modeled using polycrystalline geometry and grain orientations. PSBs can pass through Grain Boundaries (GBs) if the misorientation of the adjacent grains is lower than a critical angle. The Tanaka-Mura micromechanical model is then used to calculate the number of loading cycles required to initiate cracks. The results are compared to previous models and experiments. Initiation depths appear to be consistent.

Introduction

Rolling Contact Fatigue (RCF) concerns most of the components that endure repeated contact pressure such as rolling bearings, gears or railway tracks. Historically, the service life of these components was determined by empirical models such as Wöhler curves [START_REF] Wöhler | Über die festigkeitsversuche mit eisen und stahl[END_REF] or Coffin-Manson law [START_REF] Manson | Behavior of materials under conditions of thermal stress[END_REF][START_REF] Coffin | A study of the effects of cyclic thermal stresses on a ductile metal[END_REF]. ISO 281 is a current standard which is used to predict the service life of rolling bearings, it is based on Lundberg and Palmgren's works [START_REF] Lundberg | Dynamic capacity of rolling bearings[END_REF][START_REF] Pålmgren | Die lebensdauer von kugellagern[END_REF] which also arise from experimental results. In the early 1980's, researchers have started to adopt deterministic approaches in order to predict service life based on the actual physical mechanisms that cause fatigue [START_REF] Sadeghi | A Review of Rolling Contact Fatigue[END_REF]. RCF modeling was then separated into two categories of models: models treating crack initiation and models treating crack propagation although some studies combined both [START_REF] Zhou | Micropitting in Rolling and Sliding Contact Under Mixed Lubrication[END_REF]. Concerning crack initiation, the role of the microstructure of the materials is all the more important, as in this case the deterioration processes are very often associated with microstructural changes [START_REF] Sadeghi | A Review of Rolling Contact Fatigue[END_REF]. Physical phenomena operate at the atomic scale which makes a complete nanoscale model extremely hard to develop and compute.

Some researchers have nonetheless developed micromechanical model based on fatigue physics in order to evaluate number of cycles to crack initiation (or fatigue crack threshold) as a function of stresses based on simplifying assumptions. This is the case of Tanaka and Mura who developed a fatigue crack initiation law based on the assumption that crack initiation is a result of dislocation slip and accumulation [START_REF] Tanaka | A Dislocation Model for Fatigue Crack Initiation[END_REF]. As first pointed out by Ewing and Humphrey [START_REF] Ewing | The fracture of metals under repeated alternations of stress[END_REF], dislocations form Persistent Slip Bands (PSBs) which are preferred locations for plastic strain. Dörr and Blochwitz [START_REF] Dörr | Microcracks in fatigued fcc polycrystals by interaction between persistent slip bands and grain boundaries[END_REF][START_REF] Blochwitz | Misorientation measurements near grain boundary cracks after fatigue tests[END_REF][START_REF] Blochwitz | Analysis of activated slip systems in fatigue nickel polycrystals using the ebsd-technique in the scanning electron microscope[END_REF] and then Zhang and Wang [START_REF] Zhang | Comparison of fatigue cracking possibility along large-and low-angle grain boundaries[END_REF][START_REF] Zhang | Relationship between the fatigue cracking probability and the grain-boundary category[END_REF][START_REF] Zhang | Dependence of intergranular fatigue cracking on the interactions of persistent slip bands with grain boundaries[END_REF] studied this theory and extended the knowledge concerning fatigue crack initiation at PSBs. The Tanaka-Mura model is particularly interesting because it is physically based and it can be easily implemented into a numerical model in combination with stress field computation. Several studies used Tanaka-Mura to predict fatigue crack threshold for structural fatigue. Bruckner-Foit et al. [START_REF] Brückner-Foit | Numerical simulation of microcrack initiation of martensitic steel under fatigue loading[END_REF][START_REF] Bruckner-Foit | On the determination of material parameters in crack initiation laws[END_REF] studied the fatigue crack threshold on martensitic steel and Toyoda et al. [START_REF] Toyoda | Numerical simulation of fatigue crack initiation in thinwalled high strength steel as modeled by voronoi-polygons[END_REF] on thin-walled high strength steel. Hilgendorff et al. [START_REF] Hilgendorff | Simulation of irreversible damage accumulation in the very high cycle fatigue (vhcf) regime using the boundary element method[END_REF] used a boundary element method to compute stresses and Briffod et al. [START_REF] Briffod | Microstructure modeling and crystal plasticity simulations for the evaluation of fatigue crack initiation in -iron specimen including an elliptic defect[END_REF] used a crystal plasticity model in combination with the Tanaka-Mura model in pure -iron.

In this study, Tanaka-Mura's law is used in order to simulate crack initiation in RCF which differs from structural fatigue on multiple aspects [START_REF] Sadeghi | A Review of Rolling Contact Fatigue[END_REF][START_REF] Olver | The mechanism of rolling contact fatigue: An update[END_REF]. In RCF, the stress state is multi-axial and non-proportional, moreover, stresses are highly localized at the contact region. These specific conditions are one of the main reasons to model crack initiation at a mesoscopic length scale to account for the most fundamental principles of fatigue. Raje et al. developed a mesoscopic statistical model to estimate life scatter in RCF [START_REF] Raje | A discrete damage mechanics model for high cycle fatigue in polycrystalline materials subject to rolling contact[END_REF]. This model use the random nature of the material microstructure as a cause of the Weibull distribution of fatigue life. They found very good agreement with experimental data. Starting from the same idea that mesoscopic modeling is the key to explain the complexity of RCF, this paper proposes a grainscale numerical model. This model focuses on the elementary principle of fatigue that large scale damages result from smaller scale defect accumulation and propagation.

In a previous study [START_REF] Fourel | Towards a grain-scale modeling of crack initiation in rolling contact fatiguepart 1: Shear stress considerations[END_REF], an isotropic model was developed using Tanaka-Mura's approach to predict crack initiation in rolling contact fatigue. This article goes further in the modeling of the physics of fatigue crack initiation.

In the next section, physics of fatigue crack initiation and the Tanaka-Mura micromechanical model are described. Then, the numerical model is presented along with the polycrystalline geometry and the anisotropic properties. Finally, shear stress range calculation is detailed and a method to model PSBs in polycrystals is proposed.

Physics of Fatigue Crack Initiation

Sangid's review article [START_REF] Sangid | The physics of fatigue crack initiation[END_REF] provides extensive explanation on fatigue crack initiation. The main physical aspects on which the following numerical model is based are described in this section. Fatigue is characterised by series of forward and reverse loading. This repeated loading induces stresses in the material which may cause micro-plasticity at certain favorable locations [START_REF] Sangid | The physics of fatigue crack initiation[END_REF], and shear stress is widely recognized to be the primary cause [START_REF] Olver | The mechanism of rolling contact fatigue: An update[END_REF]. In order to minimize the energy of the material structure exposed to shear stress, dislocations form, move via slip and accumulate forming more stable configurations [START_REF] Basinski | Fundamental aspects of low amplitude cyclic deformation in face-centred cubic crystals[END_REF]. These phenomena are the core principles of plastic deformation in metals. In crystalline materials, the dislocation slip can be concentrated into Persistent Slip Bands (PSBs) [START_REF] Ewing | The fracture of metals under repeated alternations of stress[END_REF] because they constitute preferred locations for slip due to softening [START_REF] Grosskreutz | The mechanisms of metal fatigue (i)[END_REF]. Even though PSBs appear only under certain conditions, they have been observed in most face-centered cubic (fcc) and bodycentered cubic (bcc) materials [START_REF] Lukáš | Role of persistent slip bands in fatigue[END_REF]. PSBs seem to not occur at very high temperature (above one-half of the melting temperature [START_REF] Boehme | Psb observation in copper fatigued at one half the melting temperature[END_REF]) and only for a nearly symmetrical loading [START_REF] Lukáš | Role of persistent slip bands in fatigue[END_REF]. However, they still constitute the most common location for fatigue crack initiation in polycrystalline materials [START_REF] Sangid | The physics of fatigue crack initiation[END_REF][START_REF] Grosskreutz | The mechanisms of metal fatigue (i)[END_REF][START_REF] Lukáš | Role of persistent slip bands in fatigue[END_REF]. PSBs under contact fatigue conditions have been observed by Forster et al. [START_REF] Forster | Rolling contact fatigue life and spall propagation characteristics of aisi m50, m50 nil, and aisi 52100, part iii: Metallurgical examination[END_REF] in steel and by Medina et al. [START_REF] Medina | Development of rolling contact damage in two bronze alloys[END_REF] in bronze.

In single crystals and surface grains of polycrystals, a PSB ends on the specimen surface in intrusions and extrusions (see Figure 1). This is due to close-enough positive and negative dislocations annihilating via cross-slip (for the screw components) or climb (for the edge components), which produces vacancies [START_REF] Sangid | The physics of fatigue crack initiation[END_REF]. PSBs are composed of multiple parallel layers which are induced by inter-dislocation attractive and repulsive forces.

In polycrystalline materials, dislocation arrangements are similar to those in single crystals, except that the dislocations are impeded by the Grain Boundaries (GBs) resulting in pile-up [START_REF] Sangid | The physics of fatigue crack initiation[END_REF].

Crack initiation can also occur within a grain (not at the GB), although this is generally the case for materials with defects, inhomogeneities or large grain size. In superalloys, improvements in processing have led to a significant reduction in material defects and therefore reduced intra-granular initiation [START_REF] Sangid | The role of grain boundaries on fatigue crack initiation -an energy approach[END_REF]. It has been experimentally shown that during initiation, micro-cracks can develop along the GB, along the PSB or along both depending on the angles formed by the directions of the GB, of the PSB and the loading direction [START_REF] Li | Amorçage de fissures en fatigue dans un acier 304L: influence de la microstructure et d'un chargement d'amplitude variable[END_REF].

Tanaka and Mura developed a micromechanical model for fatigue crack initiation [START_REF] Tanaka | A Dislocation Model for Fatigue Crack Initiation[END_REF]. Their theory is based on dislocation pile-up at the PSB-GB interface. Forward and reverse plastic flow is modeled by dislocations with different signs moving on two closely located layers, and it is assumed that their movement is irreversible.

Each cycle induces an increase of the plastic strain energy per unit width in each layer of the PSB:

Δ = (1 -) 8 (Δ -2 ) 2 2 (1)
Where Δ is the shear stress range, is the frictional stress, is the PSB length, is the Poisson's ratio and is the shear modulus. Frictional stress is the critical stress that needs to be overcame in order to let the dislocations slip.

The definition of crack initiation depends on the length scale being considered [START_REF] Krupp | Fatigue crack propagation in metals and alloys: microstructural aspects and modelling concepts[END_REF]. From a microscopic point of view, a crack corresponds to the local overcoming of interatomic cohesive forces and thus, the creation of new surfaces. From a mesoscopic scale, an initiation can be considered when a crack reaches a significant size compared to the grain size order. At this length scale, crack propagation depends on microstructural features such as grain orientations and grain boundaries [START_REF] Krupp | Fatigue crack propagation in metals and alloys: microstructural aspects and modelling concepts[END_REF]. This is no longer the case at a macroscopic length scale where the material can be considered as homogeneous and thus crack propagation rate can be modeled by Linear Elastic Fracture Mechanics (LEFM) for example.

Tanaka and Mura originally considered that a crack initiation occurs when the accumulated energy at layer I and layer II reaches specific fracture energy of a crack of the same length as the PSB. This leads to:

+ = 2 Δ = 2 (2)
With the fatigue crack threshold (number of cycles) and the specific fracture energy for a unit area. Substituting Equation 1 into Equation 2 and rearranging gives:

= 8 (1 -) (Δ -2 ) 2 (3)
Improvements of the model have been proposed by Mura and Nakasone [START_REF] Mura | A theory of fatigue crack initiation in solids[END_REF][START_REF] Mura | A theory of fatigue crack initiation[END_REF] who used Gibb's free energy. Other authors proposed variants of this model with the use of additional parameters such as Burgers vector [START_REF] Wu | A fatigue crack nucleation model for anisotropic materials[END_REF] or the PSB width [START_REF] Chan | A microstructure-based fatigue-crack-initiation model[END_REF]. The original model has nevertheless received great popularity within the scientific community because it captures the essence of crack initiation via slip and accumulation of dislocations. Several studies have been able to correlate the Tanaka-Mura fatigue crack initiation model with experimental results [START_REF] Bruckner-Foit | On the determination of material parameters in crack initiation laws[END_REF][START_REF] Tryon | A reliability-based model to predict scatter in fatigue crack nucleation life[END_REF][START_REF] Jezernik | Numerical modelling of fatigue crack initiation and growth of martensitic steels[END_REF][START_REF] Kramberger | Extension of the tanaka-mura model for fatigue crack initiation in thermally cut martensitic steels[END_REF][START_REF] Hoshide | Life prediction by simulation of crack growth in notched components with different microstructures and under multiaxial fatigue[END_REF]. Cheng and Cheng compared the same type of model with experimental results in RCF conditions [START_REF] Cheng | Micromechanics Modeling of Crack Initiation Under Contact Fatigue[END_REF][START_REF] Cheng | Semi-Analytical Modeling of Crack Initiation Dominant Contact Fatigue Life for Roller Bearings[END_REF].

At first, GBs were considered as impenetrable [START_REF] Dörr | Microcracks in fatigued fcc polycrystals by interaction between persistent slip bands and grain boundaries[END_REF] but later, Blochwitz et al. showed that misorientation of adjacent grains have an influence on PSB-GB interaction [START_REF] Blochwitz | Misorientation measurements near grain boundary cracks after fatigue tests[END_REF]. Zhang and Wang [START_REF] Zhang | Comparison of fatigue cracking possibility along large-and low-angle grain boundaries[END_REF][START_REF] Zhang | Relationship between the fatigue cracking probability and the grain-boundary category[END_REF][START_REF] Zhang | Dependence of intergranular fatigue cracking on the interactions of persistent slip bands with grain boundaries[END_REF] showed that PSBs can pass through Low Angle Grain Boundaries (LAGBs). Therefore, cracking does not occur at these GBs. LAGBs are defined as a misorientation between adjacent grains of less than = 15° [START_REF] Sangid | The physics of fatigue crack initiation[END_REF]. Concerning High Angle Grain Boundaries (HAGBs), dislocations pile-up at the PSB-GB interface, building up stress concentration over the repeated loading cycles, eventually leading to crack initiation (see Figure 2).

As discussed, fatigue crack initiation results from the accumulation of plastic strain due to dislocation slip along PSBs and interacting with the microstructure. In the next section a numerical model for crack initiation in RCF is presented.

Model

Two variables need to be calculated in order to use Tanaka-Mura's law for crack initiation. The first one is shear stress range Δ which has been previously calculated according to different methods in an isotropic model [START_REF] Fourel | Towards a grain-scale modeling of crack initiation in rolling contact fatiguepart 1: Shear stress considerations[END_REF]. In this present paper, the use of cubic elasticity in a 2D polycrystalline model opens new perspectives. Secondly, PSB length which has been previously approximated by the grain size can now be calculated in the relevant directions and eventually allowing PSBs to pass through LAGBs.

The numerical model used in this study is similar to the previous one [START_REF] Fourel | Towards a grain-scale modeling of crack initiation in rolling contact fatiguepart 1: Shear stress considerations[END_REF] on multiple aspects. It uses a moving contact pressure to simulate the rolling contact onto the surface of the material. A loading cycle is represented by the displacement of the contact pressure from = -4 to = 4 , with contact half-width (see Figure 3). The simulation is performed for dry contact, in order to analyze the most se-vere case of damage [START_REF] Coulon | Effect of a dent on the pressure distribution in dry point contacts[END_REF]. In order to initiate surface and sub-surface cracks, two surface conditions are simulated:

-Smooth contact surface ( = ℎ ) -Dented contact surface ( = ℎ + Δ ) ℎ is calculated using Hertz contact theory [START_REF] Johnson | Contact mechanics and the wear of metals[END_REF] and Δ is calculated using Coulon et al. [START_REF] Coulon | Effect of a dent on the pressure distribution in dry point contacts[END_REF][START_REF] Coulon | An abacus for predicting the rolling contact fatigue life reduction due to debris dents[END_REF]. The contact characteristics are given Table 1. Stresses are calculated by FEM computation with quasistatic conditions for each incremental position of the contact pressure (274 increments). Stress components , , and are obtained with plane strain assumptions. An area of analysis with a width of 2.5 is embedded into a wider body, large enough to behave as a semi-infinite body (see Figure 3). The polycrystalline geometry of the area of analysis has been generated using the open access software package Neper [START_REF] Quey | Large-scale 3d random polycrystals for the finite element method: Generation, meshing and remeshing[END_REF][START_REF] Quey | Optimal polyhedral description of 3d polycrystals: Method and application to statistical and synchrotron xray diffraction data[END_REF][START_REF] Quey | Nearly uniform sampling of crystal orientations[END_REF] which uses Voronoi tessellation algorithms. The mean grain size is equal to [START_REF] Basinski | Fundamental aspects of low amplitude cyclic deformation in face-centred cubic crystals[END_REF] and the size distribution follow a log-normal law with a standard deviation of 11 . Each grain is meshed into quadratic quadrangles.

In this study, the anisotropic character of each crystal of the model is based on Noyel [START_REF] Noyel | Development of a granular cohesive model for rolling contact fatigue analysis: Crystal anisotropy modeling[END_REF]. Applying crystal elasticity to each grain with random crystal orientations allows to preserve the material macroscopic isotropic behavior. With a large enough number of randomly oriented grains, the macroscopic behavior of the locally-anisotropic mesoscopic model converges to the behavior of an isotropic material. It has been previously verified that the values used in this paper are relevant for both local and global considerations [START_REF] Noyel | Analyse de l'initiation de fissures en fatigue de contact : Approche mésoscopique[END_REF]. Even though RCF models could be applied to rolling element bearings, gears or railway tracks that can be composed of several material constituents, Noyel considered the main material constituent: -iron. The crystal coordinate system ( , ) is aligned with the atoms of the crystal lattice of the grain (see Figure 4).

Cubic elasticity stiffness matrix [ ] (Equation 4) is described by three independent constants 11 , 12 , and 44 . The values defined by Courtney [START_REF] Courtney | Mechanical Behavior of Materials: Second Edition[END_REF] are used in this study (see Table 2). Table 2 Stiffness matrix components of -iron [START_REF] Noyel | Development of a granular cohesive model for rolling contact fatigue analysis: Crystal anisotropy modeling[END_REF]. 237 GPa 141 GPa 116 GPa

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ × ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 2 2 2 ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (4)
The same type of crystal cubic elasticity has been used in RCF mesoscopic models in previous works [START_REF] Paulson | Effects of crystal elasticity on rolling contact fatigue[END_REF][START_REF] Vijay | A continuum damage mechanics framework for modeling the effect of crystalline anisotropy on rolling contact fatigue[END_REF][START_REF] Zhang | Effect of material anisotropy on rolling contact fatigue life under dry and lubricated point contact conditions: A numerical study[END_REF][START_REF] Beyer | A method to model crystalline anisotropy in contact using semi-analytical method[END_REF]].

An in-plane orientation is assigned to each grain. These orientations are defined by random angles varying between 0°and 90°which represents the rotation angle around axis between the global coordinate system ( , ) and the grain local coordinate system ( , ) (see Figure 5).

FEM computation provides stress distribution at each point of the meshed geometry and for each position of the moving contact pressure. In order to illustrate the effect of cubic elasticity on the stress field, Figure 6 shows the orthogonal shear stress for a centered contact pressure under smooth surface condition for (a) isotropic properties and (b) cubic elasticity properties.

Next section develops fatigue crack initiation simulation in RCF using the previously addressed model and singlegrain PSBs.

Crack Initiation with Single-Grain PSBs

The Tanaka-Mura micromechanical model considers shear stress range Δ as a variable influencing crack initiation. As discussed in section 2, shear stress produces dislocations, induces slip and is therefore the cause of strain energy accumulation. Multiple options exist concerning the calculation of the shear stress: maximum shear stress, orthogonal shear stress, octahedral shear stress, etc. These different possibilities are compared and discussed in a previous study [START_REF] Fourel | Towards a grain-scale modeling of crack initiation in rolling contact fatiguepart 1: Shear stress considerations[END_REF]. Since grain orientations are considered in this paper, a more physical approach is possible: calculate shear stress range directly in the slip directions. This is called resolved shear stress.

In this study, only in-plane slip systems are considered. Since stress components are calculated for a bcc material, the possible slip systems are positioned at +45°and -45°f rom the grain local coordinate system (see Figure 4). After changing coordinate system, it can be shown that resolved shear stress along is given by:

= sin 2 + 1 2 ( -) cos 2 (5)
is the opposite of for each contact pressure incremental position. Therefore, resolved shear stress range Δ is identical for every PSBs inside a grain whether the PSB is oriented according to or . Δ is calculated using the maximum and minimum of during the displacement of the moving contact pressure:

Δ = ( ) - ( ) (6)
, , and correspond to averaged values of each element of a grain and is the grain orientation angle which has already been used is the FEM anisotropic computation.

Figure 7 shows the resolved shear stress range Δ for the smooth contact and the dented contact.

Material parameters can take different values depending on bearing steel grades. Material property values that are used in the present study [START_REF] Fourel | Towards a grain-scale modeling of crack initiation in rolling contact fatiguepart 1: Shear stress considerations[END_REF] are in Table 3. The value of was determined by Tryon et al. [START_REF] Tryon | A reliability-based model to predict scatter in fatigue crack nucleation life[END_REF] and the value of can be found in Kamimura et al.'s review [START_REF] Kamimura | Experimental evaluation of the peierls stresses in a variety of crystals and their relation to the crystal structure[END_REF]. [-] 0.3 Fracture energy [START_REF] Tryon | A reliability-based model to predict scatter in fatigue crack nucleation life[END_REF] [ ∕ 2 ] 440 Frictional stress [START_REF] Kamimura | Experimental evaluation of the peierls stresses in a variety of crystals and their relation to the crystal structure[END_REF] [ ] 390 Several crack initiation criteria can be considered depending on the length scale and the physics considered in a model. Tanaka and Mura choose to consider a grain-scale (i.e. mesoscopic) criterion for crack initiation. This results in initiation of cracks of the same length as the PSBs that contributed to strain energy accumulation. In other words, they predicted that a PSB converts into a crack when sufficient energy is piled-up. Other models could arguably consider other hypotheses concerning the length scale of the initiated cracks, however, this study proceeds with the initial mesoscopic hypothesis.

The model which is developed in this paper is varied into three different combinations of hypotheses. These model variants are defined in Table 4. Model A is obtained with the simplest hypotheses, grain properties are isotropic, Δ is calculated in the direction that maximizes it and PSB length is approximated by grain size. This first model has already been used in the previous study [START_REF] Fourel | Towards a grain-scale modeling of crack initiation in rolling contact fatiguepart 1: Shear stress considerations[END_REF]. Model B differs from previous one because grains are considered as anisotropic and crystal orientations are randomly distributed. This allows computation of Δ in the slip directions as described in section 3. At this point, there is no multi-grain PSB. PSB lengths are approximated by grain size by finding the longest segment in the direction of Δ of each grain. Model C, which considered more advanced hypotheses, is discussed in the next section.

Figure 8 shows a comparison between models A and B on the number of cycles required for crack initiation. Model A is more critical than model B. This is consistent since in model A, shear stress range Δ is calculated in the direction that maximizes it [START_REF] Fourel | Towards a grain-scale modeling of crack initiation in rolling contact fatiguepart 1: Shear stress considerations[END_REF]. In model A, distribution basically depends on the grain position. For instance, in the case of the smooth surface, the value of is highly influenced by the depth of the grain. Grain size differentiates adjacent grains that are at the same depth.

distribution has more of a random aspect in model B since crack initiation depends not only on grain size, but also on grain orientation which can be very favorable or very unfavorable towards crack initiation. This leads to a more scattered distribution both in the case of the smooth surface and in the case of the dented surface.

Thus far, PSB length was approximated by grain size. In the next section, it is proposed to model multi-grain PSBs in order to get a more physically based approach.

Crack Initiation with Multi-Grain PSBs

It is possible to estimate the length of real PSBs by taking into account the misorientations between grains. As Tanaka-Mura's equation implies (see Equation 3), PSB length as an influence on fatigue crack initiation. Longest PSBs are more critical for material due to the larger amount of accumulated dislocations. This statement is consistent with the Hall-Petch relationship. As previously described in section 2, if the misorientation angle of two adjacent grains is small enough, dislocations can cross the GB which is therefore called a LAGB (see Figure 2). This phenomenon leads to the formation of multi-grain PSBs. In this section, an algorithm to model multi-grain PSBs in polycrystals is proposed.

A grid is generated inside each grain, following the two possible slip directions and of the slip local coordinate system (see Figure 4). In the algorithm, the number of PSBs per grain and per direction can be control by parameter . When a line of the grid crosses a GB, the intersection point is found, both adjacent grains are identified and if their misorientation is lower than the critical misorientation angle = 15° [START_REF] Sangid | The physics of fatigue crack initiation[END_REF], the line passes through the GB following the new slip direction. A PSB which has been generated following the direction of the first grain can be extended following either or of a second grain as long as the angular difference between the first and the second direction is lower than . Figure 9 shows the algorithm applied to one grain only: in case that = 10 (a), in case that = 100 (b). In the second case, more PSB paths are considered as an additional grain can be reached, leading to longer PSBs.

When the algorithm is applied to each grain of the analysis area, a PSB field is generated (see Figure 10). Each PSB is colored according to its length. In the present case, the longest PSB equals 220 which corresponds to around 9 grains.

The number of cycles required for crack initiation can be calculated from the Tanaka-Mura model (Equation 3) using resolved shear stress range Δ , PSB length and the material properties. The multiple segments of the PSBs are used in this calculation. Each segment contributes to the energy that is accumulated in a PSB. Building from Tanaka-Mura's principle, the total accumulated energy per cycle in a PSB Considering the initiation of a crack of the same length as the PSB, the energy criterion is:

2 , Δ = 2 (8)
With , the number of cycles required to initiate a crack along the PSB, the multi-grain PSB length and the specific fracture energy for a unit area. This leads to:

, = 8 (1 -) ∑ =1 ∑ =1 2 (Δ -2 ) 2 (9)
The assumption that a crack is initiated when the energy level reaches the requirement for cracking the entire PSB is still considered here. Even though an initiated crack is generally smaller than a PSB and then grows into a full PSB length [START_REF] Krupp | Fatigue crack propagation in metals and alloys: microstructural aspects and modelling concepts[END_REF], it is more convenient to start with Tanaka-Mura's original hypothesis as this model operates at a mesoscopic length scale. = 100 has been found to be the convergence criterion to not have results depending on the number of PSBs per grain and per direction.

As expected, contact surface conditions have a predominant influence on fatigue crack threshold with either singlegrain PSBs or multi-grain PSBs. A damaged surface highly decreases , .

There is a slight difference between the predictions of model B' (single-grain PSBs) and model C (multi-grain PSBs). Model B' is more critical (lower , ) than model C. This can be explained by the fact that energy accumulation needs to reach a level which is proportional to the PSB length in order to initiate a crack. Even though longer PSBs accumulate more energy than smaller PSBs (at the same stress level), they also need more energy to be entirely cracked.

Average depth of early crack initiation ( < 10× , ) in the case of a smooth surface (Hertzian contact pressure) is equal to: ]. This experimental value was obtained with only 13 measurements of crack depth but averages are still similar. This agreement between experimental and numerical results for Hertzian type contacts allows a reasonable confidence in the method developed in the present study and in its application to other surface conditions.

Conclusions

In this study, a 2D polycrystalline model has been developed using crystal anisotropy in order to compute realistic stress distribution in the surface of a material exposed to rolling contact pressure. Two independent cases have been considered: smooth and dented surface conditions. These cases could be combined by summing the different accumulated energies. Benefits of these assumptions over a previous isotropic model on crack initiation are shown using Tanaka-Mura's approach in order to account for the fundamental mechanics of fatigue, i.e. strain energy accumulation through dislocation formation and slip.

The main interest of this paper lies in the modeling of multi-grain PSBs, allowing a more realistic simulation in agreement with experimental observations showing PSBs passing through low angle grain boundaries.

As material parameters are still an important source of uncertainty, validation of the model could be made by comparing the results with experiments. These correlations would provide more confidence, in particular concerning fracture energy and frictional stress . Further improvements could be made concerning the material behavior by modeling crystal plasticity or material heterogeneities. Grain size variation depending on depth could be considered as well as macroscopic anisotropy induced by crystal orientation texture. Concerning the surface, friction and roughness could be modeled. In-plane grain orientations are also a questionable assumption. A 3D model would help to solve this issue along with modeling the PSBs in a 3D space.

Figure 1 :

 1 Figure 1: Persistent Slip Band (PSB) inducing intrusions and extrusions at the free surface of the material.

Figure 2 :

 2 Figure 2: Dislocations moving along Persistent Slip Bands, piling-up at High-Angle Grain Boundaries (HAGBs) and passing trough Low-Angle Grain Boundaries (LAGBs).

Figure 3 :

 3 Figure 3: Model dimensions and polycrystalline geometry.

Figure 4 :

 4 Figure 4: Global coordinate system ( , ), grain local coordinate system ( , ), slip local coordinate system ( , ).

Figure 5 :

 5 Figure 5: Distribution of grains orientations .

Figure 6 :

 6 Figure 6: Orthogonal shear stress for a centered contact pressure in smooth surface condition for (a) isotropic properties, (b) cubic elasticity properties.

Figure 7 :

 7 Figure 7: Resolved shear stress range for (a) smooth surface, (b) dented surface.

Figure 8 :

 8 Figure 8: Fatigue crack threshold for (a,b) model A, (c,d) model B, in the case of (a,c) smooth surface, (b,d) dented surface.

Figure 9 :

 9 Figure 9: PSB passing though GBs for (a) = 10, (b) = 100.

Figure 10 :

 10 Figure 10: PSB field in the polycristalline geometry of the area of analysis using = 100.

Figure 11 :

 11 Figure 11: Accumulated strain energy per unit width per cycle for (a) smooth surface, (b) dented surface.

Figure 12 :

 12 Figure 12: Fatigue crack threshold for (a,b) model B', (c,d) model C in the case of (a,c) smooth surface, (b,d) dented surface.

Figure 12

 12 Figure 12 compares two models: models B' and C. Model B' is equivalent to model B but the representations are different. Contrary to model B in Figure 8 where only the value of the longest PSB is displayed, in model B' each PSB is displayed in order to compare it to model C. Hence model B' simulates the presence of PSBs without allowing for passing through GBs. In model C, passing through LAGBs is allowed, so that multi-grain PSBs are simulated.= 100 has been found to be the convergence criterion to not have results depending on the number of PSBs per grain and per direction.As expected, contact surface conditions have a predominant influence on fatigue crack threshold with either singlegrain PSBs or multi-grain PSBs. A damaged surface highly decreases , .There is a slight difference between the predictions of model B' (single-grain PSBs) and model C (multi-grain PSBs). Model B' is more critical (lower , ) than model C. This can be explained by the fact that energy accumulation needs to reach a level which is proportional to the PSB length in order to initiate a crack. Even though longer PSBs accumulate more energy than smaller PSBs (at the same stress level), they also need more energy to be entirely cracked.Average depth of early crack initiation ( < 10× , ) in the case of a smooth surface (Hertzian contact pressure) is equal to:

-

  = -0.62 for model A -= -0.56 for model B -= -0.57 for model C Chen et al. found experimentally an average value of = -0.522 [58, 59

Table 1

 1 Contact and dent characteristics.

	Characteristic	Notation	Unit		Value
	Half-width		[ ]		615
	Max contact pressure	0	[	]	2100
	Dent diameter		[ ]		250
	Dent depth	ℎ	[ ]		37

Table 3

 3 Tanaka-Mura model parameters for bearing steel.

	Parameter	Notation	Unit		Value
	Shear modulus		[	]	80
	Poisson's ratio

Table 4

 4 Hypotheses of the different models.

	Model	A		B		C
	Grain properties	Isotropy		Cubic elasticity		Cubic elasticity
	Δ direction	Maximizes (Δ	)	Slip direction (Δ	)	Slip direction (Δ	)
	PSB length	Grain size		Grain size		Multi-grain PSB


F Gambiez

C Kolski

X Tang

P Millot

MAN-MACHINE INTERFACE FOR PREDEX, AN EXPERT SYSTEM SHELL FOR PROCESS CONTROL AND SUPERVISION

This paper aims at describing our research work related to Man-Machine interface for an expert system shell for process control and supervision, called PREDEX. The first part of the paper presents the PREDEX framework. This expert system shell integrates a qualitative model of the process. The second part of the paper describes the Man-Machine cooperation model and the main displays constituting the interface.

1.

INTRODUCTION

When computerizing the control rooms of continuous industrial processes, Man-Machine interfaces were first based on classic representation modes such as synopsis displays, curves, bargraphs, etc. Lind's work [12; 13] has lead to implementing a new mode of representation: flow models, which give a comprehensive functional description of the process. This process is described in terms of "mass" and "energy" and in terms of "transfer" and "storage". This symbolism has already been used in industrial applications such as those by Siebert, Sicart, Thebault [START_REF] Siebert | Comparaison d'interfaces de conduite. Essais incidentels sur pupitre imageries graphiques[END_REF] or Praetorius, Duncan [START_REF] Praetorius | Flow Modelling of Plant Process For Fault Diagnosis[END_REF].

Progress made in the field of qualitative reasoning has lead to creating decision aid tools based on a qualitative model of the industrial process which needs to be controled. Using such as model has, in the short run, helped take into focus and even improve potential "aid-devices" for the operator; for instance, predicting the future state of the process; focussing on a sub-system in case of problem; making the diagnosis process much easier; simulating the evolution of a variable in response to a change in another. It has thus become necessary to develop a new kind of interface meeting these new standards. These interfaces do also have to fit into the operator's decision-making process.

Our research work on Man-Machine Communication is based on these concepts, and this, mainly for decision aid systems which integrate a qualitative model of the process. This paper presents our work on man-machine interface design for PREDEX, an expert system shell for process control and supervision.

This paper is composed of two main parts: (i) the first one describes the different modules constituting the expert system shell PREDEX, which integrates a qualitative model of the process; (ii) the second part first describes the Man-Machine cooperation model. This model is used as a conceptual basis for our works related to PREDEX Man-Machine interface design. Then the different types of graphical displays constituting the interface are specified.

2.

PRESENTATION OF PREDEX

PREDEX is an expert system shell for process control and supervision of industrial processes [START_REF] Tang | PREDEX: un générateur de systèmes experts de seconde génération pour la supervision des procédés continus[END_REF]. It is composed of a set of modules which interchange control information, and data. There are four modules: "Perception", "Prediction", "Diagnosis", and "User interface management", Figure 1. These four modules may run on different calculators connected together by an Ethernet network. Their information interchanges are monitored by a communication controller. These modules are autonomous, each of them realizing specific functions and operating on their own data and knowledge. In this context, the controller role consists in (i) initializing each module, then in (ii) assigning the data communication channels and finally in (iii) synchronizing in real time the different processes. The cooperation between the different modules allows to insure the complex task of assistance to the operator. These modules are briefly presented below. The perception module constitutes the physical interface between the aid system and the underlying process. Its purpose consists in acquiring data from the real world, in processing them and in providing the different modules with these data. This module is also able (i) to compute unmeasurable data, (ii) to realize specific treatments with the object of filtering noise, (iii) to validate data, (iv) to generate alarms which are stored in order to be used by the operator. This set of acquired or generated data is transmitted to the diagnosis process, which is described below.

The diagnosis is realized according to two different approaches which are implemented in two different modules: (i) the shallow diagnosis, using effective knowledge issued from a human expertise, (ii) the deep diagnosis, using descriptive knowledge on the process to be controled. So, first, the shallow diagnosis module -developed according to the Kramer approach [START_REF] Kramer | A rule based approach to fault diagnosis using the directed graph[END_REF] -detects and identifies the failing system which is at the origin of the dysfunctioning. This module has been implemented in PREDEX with NEMO, which is a first order inference engine. NEMO uses production rules and temporal primitives for knowledge representation. This first diagnosis phase provides a list of failures which are likely to be at the origin of this dysfunctioning. This list is transmitted, through the supervisor to the deep diagnosis module.

So the deep diagnosis module is activated when the shallow diagnosis module has identified the failing system(s). The shallow diagnosis module provides the deep diagnosis module with a list of suspected failures which could have affected the process. So the deep diagnosis module must discriminate between these hypothesis in a way to make the diagnosis better, and to propose to the Human operator a reduced classified list of elementary failures. The implemented diagnosis method combines three different methods: (i) the structural reasoning method issued from Davis [START_REF] Davis | Diagnosis based on description of structure and function[END_REF], (ii) the causal reasoning method issued from Console [START_REF] Console | Integration of causal & temporal reasoning in diagnostic poblem solving[END_REF], and (iii) the qualitative reasoning method issued from Kramer [START_REF] Kramer | A rule based approach to fault diagnosis using the directed graph[END_REF]. The deep knowledge is based on several types of deep knowledge: a network of functional causality, a structural model, a fault model, a description of failure modes. When a failing system is identified, an action plan is then defined. This plan has to guide the operator in the creation of a procedure related to the abnormal state of the process.

In order to predict the behavior of the process to be controled, the system PREDEX also uses a qualitative modelling of it. The modelling corresponds to the one proposed by Tang [START_REF] Tang | Contribution à la conception des systèmes à base de connaissances "Temps réel" pour l'aide au contrôle de procédés continus[END_REF]. This approach is based on "event propagation". Concepts of automation are introduced in this modelling [START_REF] Feray-Beaumont | Modèle qualitatif de comportement pour un système d'aide à la supervision des procédés[END_REF][START_REF] Gentil | Qualitative modelling for process supervision systems[END_REF]: that is to say that interaction links between variables are defined by information such as 'gain', 'delay', and 'response time'. In this case, a dynamic process is modelled by an interaction network: the nodes represent process variables and links correspond to the relations between these variables. Temporal interactions between the variables are done by events exchange. They are the only sources of changes in the nodes and variables. The technique used in PREDEX consists in modelling a dynamical system by a propagation network of temporal events, and in optimizing the response time by using a qualitative hierarchical model of the process. So, a rule-based system -built on this technique -can optimize its treatment according to the real needs of the current task, for instance a diagnosis or prediction task, by identifying the smaller sub-model satisfying the task´s objectives. For that purpose, it is able to focus its reasoning by using a minimal set of necessary data, and by avoiding useless treatments and reasonings.

The activation of the prediction module comes into effect after a failure has been identified. The prediction of the process behavior serves several purposes: fault hypothesis verification, fault anticipation and detection, proposed advice justification, possibility for the operator to evaluate the future state of the process with applying or not a proposed action plan. This module may provide, with an operator request, three types of predictions: (i) value and tendency to a fixed horizon, (ii) threshold over or underflow with maximum and minimum values, (iii) evolution curves in a fixed time-period. This information and also the data gathered from the interface-dedicated prediction module are at the disposition of the user interface management system. It aims at supplying interface displays through data files. Man-Machine interface is described in the following part.

MAN-MACHINE INTERFACE FOR PREDEX

A decision support system realized with the expert system shell PREDEX provides the Human operator with diversified helps: diagnosis, supervision, prediction for instance. In a way to optimize Human-Machine Interaction, it is therefore necessary to base the design works on a model of cooperation between the operator and the decision support system. This model is now described.

"Man-decision support system" cooperation model [10]

The theorical basis of the Man-Machine cooperation is the General Problem Solving Model proposed by Rasmussen [15; 16]. This model is henceforth commonly used in Human Engineering. In this context, we suppose that during the normal functioning of the process, the human task consists in supervising it and in improving the production parameters. During the abnormal functioning of the process, the operator aims at (i) detecting dysfunctionings, (ii) making diagnosis, (iii) deciding action plans, and finally (iv) performing the adequate actions on the process.

So, it is necessary to specify the organization of the displays which ensure the presentation of the set of useful information to the operator. This phase is crucial because the specified organization sets the future cooperation between the decision aid system and the human operator. The solving of the possible conflicts, which can happen if the operator does not agree with the advices provided by the computer, depends on this cooperation. Indeed the Human-Machine interface must lead the operator to check the results corresponding to the different steps of his reasoning process, in order to find the origin of the conflict. For that purpose a solution consists in analysing in parallel the human's and the system's problem solving process, in order to provide the operator with relevant information according to different justification levels, with a view to finding a consensus as quickly as possible (see the works by Taborin and Millot [9;[START_REF] Taborin | Cooperation entre Opérateur et Système d'Aide à la Décision pour la Conduite de Procédés Continus: Application à l'[END_REF][START_REF] Taborin | Cooperation Between Man and Decision Aid System in Supervisory Loop of Continuous Processes[END_REF][START_REF] Millot | Two approaches for man-computer cooperation in supervision tasks[END_REF]). In order to specify the chaining between the different displays of the interface, a man-machine cooperation model is necessary.

Our model is summarized in figure 2. It gives a global overview on the communication between the operator and the interface, when a dysfunctioning appears. It is decomposed into three interconnected phases: (i) first, the decision support tool considers the situation and generates a set of hierarchical alarms; (ii) secondly, there is a phase of communication between the decision support system and the operator. So, facing the situation gravity, some complementary information may be displayed on the interface, such as advice action plans, justifications, and so on; (iii) finally, the third phase consists in problem solving, in which the operator and the system have to cooperate. In this phase, the three different behaviour levels quoted by Rasmussen [START_REF] Rasmussen | Skills, rules and knowledge; signals, signs and symbols, and other distinctions in human performance models[END_REF] may happen: Skill-based behaviour, Rule-based behaviour and Knowledge-based behaviour (see figure 2).

Figure 2 : "Man-decision support system" cooperation model [START_REF] Kolski | Methodology for designing Man-Machine "Intelligent" interface: The "Decisional Module of Imagery" as a case study[END_REF] Thus, for each of these three levels, the interface has to propose different graphical helps. The final aim is to obtain a consensus between the operator and the decision support tool, when a dysfunctioning appears and when the operator has to make a decision. Thus, if the operator does not agree with the action advice issued from the decision tool, the interface must be able to explain them under many justification levels using supplementary graphical displays [18; 19]. The three different behaviour levels are presented below:

• The operator can agree with the decision support system and then, acts on the process. This behaviour can be identified to a Skill-based behaviour. • However, he can be undecided and try to evaluate the situation in a better way, in order to make a diagnosis and to take a decision. He can base his reasoning on complementary information. If "shallow" justifications allow him to identify the process situation, he can admit them and adopt the (proposed or not) intervention rules. This behaviour can be identified to a Rule-based behaviour. • To a third behaviour level (Knowledge-based behaviour), the operator uses "deep" justifications, helping him to make his own diagnosis. This one leads him to make a decision and to planify actions to be performed on the process.

After having globally presented the cooperation model, we describe in the following part ergonomical specifications for the different displays which have to ensure this cooperation.

Interface Description

Several types of displays have been or are being realized [START_REF] Gambiez | Etude et mise en oeuvre d'un module d'interface pour l'outil PREDEX de développement de système expert de filtrage et de gestion d'alarmes pour les procédés continus[END_REF]. They are described below.

"Monitoring" display

The first type of display aims at synthesizing the process current state. Indeed, during a normal operation, the operator supervizes the variables in order to predict a failure. An alert state is generally not activated by an alarm generated on the process or by the aid system but by prediction activities which are proper to the operator. In accordance with (i) the type of sub-process to be controledconstituted with independant or interconnected variables -, and (ii) the training level of the operator (it is more or less important), the operator is induced to scan either the set of variables or a subset of characteristic variables in order to compare them with the objectives of operations which are assigned to him. In this case, his cognitive approach deals essentially with the situation assessment. If the observed state or the prediction made by the operator are different from the fixed objectives, then the process is considered as being in an abnormal state. The "Star display", presented at figure 3, synthesizes in a particular way the more important variables of the process. Accessible through the command "Supervision" (see figure 3), the star display include the more significant variables of the process. This display consists in eight radii (in the presented case study) of a same circle. The characteristics of each variable are displayed on each radius: the set value, and five thresholds (normal, high, very high, low and very low alarm). These characteristics are normalised for each variable so that they are linked together by concentric circles. The temporal evolution of each variable is represented by a point on its radius. The eight points are linked with straight lines to form a regular polygon when the process operation mode is normal . When a defect appears, the polygon gets out of shape: the operator is alerted according to the expert system's prediction. Moreover, the star display pattern changes according to the prediction made on the variables presented on it. This prediction polygon is presented to the operator in order to help him in his anticipation of a defect. This representation, which principle is not new (see [START_REF] Coekin | A versatile presentation of parameters for rapid recognition of total state International Symposium on Man-Machine Systems[END_REF]), has been studied by many authors. So Jacob et al. [START_REF] Jacob | The face as a data display[END_REF] show that the interpretation time of a situation is lower than the one which is necessary when using a numerical value display. Other authors like Goodstein [START_REF] Goodstein | An integrated display set for process operators[END_REF] advise to use the star display for the supervision of a process. Furthermore, an experimentation has shown the superiority of the star display over displays containing bar-graphs and dials, according to the criterion of the time which is necessary to detect a disturbance [START_REF] Taborin | Cooperation entre Opérateur et Système d'Aide à la Décision pour la Conduite de Procédés Continus: Application à l'[END_REF]. In order to help the operator in his failure predictive detection task, essentially during the transient phases of state changes, it is recommanded to display only the more significant information according to the considered functionning mode. Nevertheless let us note that in order to anticipate an abnormal event and to facilitate the decision making, il is necessary to display a prediction of these "supervision" variables (If these data can be provided by the decision support system). In order to represent the predicted tendency of these variables, the interface has to present their speeded-up evolution -for instance a second corresponding to a minute in the reality -. Concerning this point, it is interesting to note that a study by Van der veldt and Van den boomgaard [START_REF] Van Der Veldt | Predictive information in the control room[END_REF] explains that the operators' performances are improved when using a predictive display, even if this one is not perfect. The star display is first and foremost a view of the smooth running of the process: if the variables presented in this view are all in their interval of normality, the process is in a normal functioning mode. In this context, each abnormal operation must affect these variables of control at a more or less long term. The prediction module provides the future values of these variables, on a time interval which has to be fixed a priori. It is therefore necessary to know when the disturbance will be propagated towards these variables of control. We can notice that the variables research or validation can be automated from the knowledge of the qualitative model of the process. It is therefore possible at this level to advise the interface designer about the choice of the variables of control. This option is under consideration for the next interface version.

The "action advice" display

When a failure has been detected, PREDEX deduces a recovery procedure and suggests a task to the operator, in the form of an action plan which contains the following information (see figure 4 in the upper zone, on the left): (i) The main objective of the plan: this objective is defined when the origin of the disturbance has been identified by the diagnosis module. It aims at offsetting this one. (ii) A decomposition of the main task in sub-tasks consisting in a set of organised and dated orders able to reach the main objective. Action plan are organised into a hierarchy according to the delay of intervention.

At present time, only the two more important action plans are continuously presented to the operator, both for problems of lisibility and for the limitation of the operator's workload. Indeed two action plans can correspond to eight action advice. As seen on figure 4, action plans are displayed in the more scanned area of the screen (upper left area). These action plans include: (i) the disturbance to be corrected, corresponding to the action plan aim; (ii) the action maximum delay to correct the disturbance; (iii) one or more action means to succeed the correction; (iv) the amplitude of the correction to be realized; (v) the correction mode -automatic or manual -, according to the variable to be regulated or not. It is very important to know the level of description of the commands which have to be presented to the operator according to his knowledge level: indeed, for a skilled operator, the main objective should probably be sufficient, whereas for an unskilled operator, action to be executed should have to be more detailled. Moreover, if the number of action plans is greater than two, the operator must have the possibility to visualize them in a secondary drawport. This opportunity is accessible with the "Suite" button, figure 4. Parallel to the expert system diagnosis module which process reasoning will lead to give advise to the operator, this latter may have processed his own reasoning and may have found his own diagnosis and recovery procedure. In accordance with the cooperation model described on figure 2, many situations may appear: (i) the operator agrees with the advised action plan proposed by the expert system, then he just has to act in accordance with the proposed action. In this case, the expert system reinforces his decision; (ii) the operator does not agree with the expert system deductions, or he may have not yet succeeded in an action plan definition: as a consequence, he wants some explanations. For this purpose, different levels of justification are at the disposal of the operator: the first one consists in presenting to the operator a flow model of the action plan propagation; the second one proposes variables implied by the action plan and presents them in form of curves. These two levels are more precisely described below.

The first justification level: flow model

If the operator wants explanation on the advised action, he may access to a display called "Fluence". An example may be seen on figure 5. In this case, the star display is automatically shown in the left lower area of the screen. The operator is shown by this 'Fluence' display the failure diagnosis. On a simplified process synopsis are displayed the important process variables implied with the action plan linked together. These links represent the propagated correction due to the proposed action. To be created, this display uses the qualitative model of the process: the variables to be presented to the operator are deduced according to the knowledge of the action plan entry and issue point. This explanation level is accessible through the 'Fluence' button. If the operator wants more detailed explanations on the predicted propagation or if he wants to dispose of more information on the amplitude of the propagation, he may then access to a second level of justification: the "Prediction" display. The "Prediction" display presents the tendency prediction of a disturbed variable, figure 6. This prediction is given by a module integrating a qualitative model of the process. The number of variables to be presented must be determined according to the operator informational needs. This display has to be generated according to the specific knowledge of the characteristics of the variables to be presented, that is: number of variables, references, change interval, threshold, historics and prediction time. This second level is accessible through the "Prediction" button. On figure 6, the automatical move of the "propagation" display in the lower left hand area can also be set into evidence.

Failure identification and explanation

The diagnosis module purpose is to recognize the failure which is at the origin of the disturbance. A failure may be either an equipment failure or a disturbance on a system entry variable. If a disturbance may be explained by a curve, this is not the case for an equipment failure. So, this disturbance has to be localised on the process synopsis with variables and their tendency which proof the failure. The same case appears when the system has to prove the disturbance on an unmeasurable variable. An example concerning Failure identification and explanation for an equipment failure is given on figure 7. The purpose of this "failure" display (on figure 7, the french term "panne" is employed) is to justify the detected failure. So, on this display, the failure is located. Furthermore, the variables and their tendency, which have permitted the detection of this one, are presented. As several failures may be identified at one time, the operator has to select the failure he wants to be explained. In this case, the operator selects "failure" display by clicking the "Panne" button. Then, he selects the action plan on which he wants a justification about the diagnosis. The operator may need to visualize the tendency of a variable which has not been planned by the decision support system. In that case, such a variable must be easily accessible by the operator. For this purpose, a display called "Evolution" has been developed.

As the process to be controlled may have many variables, these one have been grouped by type. For instance, in the case study presented on figure 8, four types have been distinguished: temperature, flow, pressure and analyzer. So the operator has to select a type. Then the name of each variable of this type is displayed on a panel. So the operator has to select the variable he wants to know the tendency. This functionality is accessible through the "Evolution" button.

CONCLUSION

This paper has mainly detailed the interface of PREDEX, an expert system shell for process control and supervision. The integration in PREDEX of a qualitative model of the process offers interesting prospects concerning the Man-Machine Interaction. Indeed it is possible to assist the Human operator by (i) predicting him the future state of the process, (ii) providing him different types of diagnosis: deep and shallow, (iii) displaying him different justification scales of the presented information, and so on.

The interface which is being realized has now to be dynamically validated on industrial sites, with the operators. Subjective and Objective evaluation methods will be used in order to evaluate the workload and the performances of the operators using a decision support system realized with PREDEX.

According to the real needs of the operators, the displays will be ergonomically improved. In the same way, other types of displays could be developed.

Figure 1 :

 1 Figure 1: The modules constituting a PREDEX application

Figure 3 :

 3 Figure 3: "Monitoring" display

Figure 4 : 3 . 2 . 3 .

 4323 Figure 4: Action plan displayed in the upper zone, on the left 3.2.3. Accessibilty to the different justification levels of the diagnosis module.

Figure 5 : 3 . 2 . 3 . 2 .Figure 6 :

 532326 Figure 5: first justification level by displaying the action plan propagation

Figure 7 :

 7 Figure 7: Example of Failure identification and explanation for an equipment failure 3.2.5. "Evolution" display

Figure 8 :

 8 Figure 8: "Evolution" display

ACKNOWLEDGEMENTS -

The PREDEX realization was supported by the French Industry Ministry in the Program: "Automatisation des procédés continus APC'88"