Ghiath Monnet
email: ghiath.monnet@edf.fr

Lionel Gélébart

Prediction of the mechanical behavior of irradiated Reactor Pressure Vessel steels: multiscale modeling of crystal plasticity

In this paper, we present a set of equations capitalizing the multiscale modeling of the mechanical behavior of Reactor Pressure Vessel (RPV) steels before and after irradiation. The equations capture the temperature and strain rate sensitivity in addition to the contributions of microstructure features peculiar to RPV steels, such as carbides, initial dislocation network and the Hall-Petch grain size sensitivity. Dislocations are assumed to move on the {110} and {112} crystallographic planes and a simplified interaction matrix is proposed. The predicted yield stress is found in close agreement with a large number of experimental results over a large temperature range. Finally, contribution of radiation defects is accounted for using atomistic and dislocation dynamics results, revealing the effect of the solute cluster size and density on the mechanical behavior. Results are discussed and compared with an experimental database on neutron-irradiated RPV steels.

Introduction

Recently, a complete set of constitutive equations has been proposed to predict crystal plasticity in pure iron [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF]. These equations, forming a so-called Crystalline Law (CL), are well-adapted for the prediction of the local behavior of ferrite in macroscopic modeling through homogenization techniques (full-field or self-consistent methods). The reported CL [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF] accounted for the activation of 12 slip systems on the {110} planes, the Burgers vectors being ½ <111>. The significant breakthrough of this CL was the proposal of a thermally activated rate equation, based on the analysis of Dislocation Dynamics (DD) results [START_REF] Naamane | Low temperature deformation in iron studied with dislocation dynamics simulations[END_REF][START_REF] Monnet | Orowan strengthening at low temperatures in bcc materials studied by dislocation dynamics simulations[END_REF]. Hence, by accounting for the curvature of non-screw dislocations, it was shown that hardening induced by local obstacles (such as junctions and precipitates) decreases substantially with decreasing temperature. The line tension contribution to strain hardening was also found to increase with temperature and with density of local obstacles (dislocation network, carbides, etc) and to reach a plateau beyond a given temperature close to 400 K defining a so-called athermal regime. The line tension contribution appears only when the average length of screw dislocation segments decreases to a minimum critical value. In the athermal regime, where dislocation mobility is no longer controlled by lattice friction but by jog drag, a simpler CL can be used [START_REF] Monnet | Multiscale Modeling of Radiation Hardening[END_REF]. Another important contribution of the CL proposed in [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF] was the proposition for a smooth transition between the two regimes (thermally activated vs athermal) through a harmonic superposition of the two rate equations, which assumes that the applied stress must provide at the same time enough nucleation rate of kink-pairs (controlling the low temperature flow) and sufficient driving force for the jog-dragged dislocations to pass through the local obstacle field. In other terms, the harmonic superposition rule is conservative. Nevertheless, this CL cannot be applied directly to Reactor Pressure Vessel (RPV) steels, because other ingredients must be implemented, owing to the specific features of the bainitic microstructure. For more metallurgical details on the bainitic microstructure, readers are invited to consult the excellent textbook in ref [START_REF] Pickering | Constitution and Properties of Steels[END_REF]. The design and fabrication of RPV steels have recently been reviewed [START_REF] Soneda | Irradiation Embrittlement of Reactor Pressure Vessels (RPVs) in Nuclear Power Plants[END_REF]. These materials experience significant radiation embrittlement due to the accumulation of radiation-induced defects of nanometric size. Although many phenomenological approaches for predicting radiation hardening were reported (see for ex. [START_REF] Eason | A physically-based correlation of irradiation-induced transition temperature shifts for RPV steels[END_REF] and [START_REF] Soneda | Embrittlement Correlation Method for the Japanese Reactor Pressure Vessel Materials[END_REF]), these approaches did not provide a global modeling of plastic deformation of irradiated RPV steels. Constitutive equations of the mechanical behavior are still missing. This is the main objective of the present paper. The next chapter explains how the CL reported in [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF] can be simplified and adapted to RPV steels, including the introduction of slip activity on the {112} planes and of the grain size (Hall-Petch) effect on the flow stress. Chapter 3 presents a multiscale modeling of the radiation-induced hardening at the grain scale, accounting for the recent results of Molecular Dynamics (MD) and Dislocation Dynamics (DD) simulations. Chapter 4 is dedicated to the implementation of the CL in larger scale simulations (self-consistent and full-field methods) and to the presentation of simulation results. Finally the results are discussed and compared with experimental observations.

Simplified crystalline law for irradiated RPV steels

The objective of the construction of the CL reported in [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF] was to reproduce plastic deformation in pure iron single crystal in details. The physical basis of the constitutive equations was recalled in details in [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF]. It is not recalled here. To adapt this CL to RPV steels, different steps must be considered. First, when focusing on the behavior of polycrystals, not all details are relevant. Second, specific features of bainitic steels must be accounted for, such as solid solution and grain size effects. Third, when deforming polycrystals, grains of all orientations are involved. In this context, slip activity on the {112} planes cannot be ignored. In the following, we present simple methods to take into account these modifications. For the sake of simplicity, the set of constitutive equations is introduced with the small strain approximation but can be extended straightforwardly to the finite strains framework [START_REF] Teodosiu | Large Plastic Deformation of Crystalline Aggregates[END_REF].

Rate equation

The plastic slip rate on a reference slip system s is a harmonic sum of the slip rates controlled by the jog-drag (noted by the lower index "drag") and lattice friction (lower index "friction"):

s friction s drag s       1 1 1   . ( 1
)
By convention in this paper, upper indexes refer to the slip system number, while lower indexes denote the nature of the variable. For the sake of simplicity, the upper indexes may be omitted in the text, but not in the equations. The rate equation for the regime controlled by jog drag is given by:

) ( s n s c s app o s drag sign                 , ( 2
)
where o  is a constant in the order of 10 -5 s -1 , app is the resolved shear stress on the reference slip system, c the critical stress (defined below). The shear rate in the lattice friction regime can be expressed as:

  s s o s eff o s sc s m s friction sign kT G l H b                          1 exp  , ( 3
)
where m is the mobile dislocation density (constant in our model), b the norm of Burgers vector, H a frequency factor, s sc l is the average length of screw dislocation segments, k the Boltzmann constant, T the absolute temperature, Go and o two parameters fitted on experimental results. eff is the effective stress, that is, the net driving force for dislocation motion. It is given by:

         s c app s c s app s c s app s eff        0 (4)
The absolute value of the applied stress is considered in order to assign the same sign for the applied stress and the shear stress.

Average obstacle spacing

As it will be seen in the next section, the computation of the critical stress c requires the knowledge of the average length of screw segments lsc, which is a function of the densities of local obstacles and their strengths. In the lower bainitic structure, local obstacles are lattice dislocations and intragranular carbides [START_REF] Pickering | Constitution and Properties of Steels[END_REF]. When irradiated, other defects appears in the matrix [START_REF] Fukuya | Current understanding of radiation-induced degradation in light water reactor structural materials[END_REF], basically Dislocation Loops (DLs) and Solute Clusters (SCs) considered as spherical precipitates in our model. According to the dispersed barrier hardening model [START_REF] Seeger | Proc. 2nd UN Int. Conf. on Peaceful Uses of Atomic Energy[END_REF] (cited by [START_REF] Singh | Radiation hardening revisited: role of intracascade clustering[END_REF]), interaction statistics between moving dislocations and radiation-induced defects are controlled by the planner density of defects. For solute clusters we have SC = DSCCSC, and DL = DDLCDL for dislocation loops, where D and C are respectively the diameter and concentration of the defect family. Consequently, the total density of local obstacles in irradiated RPV steels can be given as:

DL SC carbide s j j dis s obs            , (5)
where j dis  denotes dislocation density on slip system "j". The sum in eq 14 is to be done over all slip systems (see next section). carbides, SC and DL are respectively the planner densities of carbides, solute clusters and dislocation loops. To this total obstacle density, one may associate an average obstacle strength on system s:

      s j DL DL SC SC carbide carbide j dis sj s obs s a a a a       1 , (6)
with sj a the interaction coefficient between slip systems s and j, determined from DD simulations [START_REF] Queyreau | Slip systems interactions in α-iron determined by dislocation dynamics simulations[END_REF][START_REF] Madec | Dislocation strengthening in FCC metals and in BCC metals at high temperatures[END_REF]. It is important to note here that the interaction coefficients with forest dislocations are note constant. The logarithmic correction [START_REF] Madec | From Dislocation Junctions to Forest Hardening[END_REF] is necessary to account for the decrease in the dislocation line tension with the dislocation density. The following expression is used to update the values of the coefficients [START_REF] Devincre | Physical analyses of crystal plasticity by DD simulations[END_REF]: with the reference coefficients aref computed in DD simulations at the reference density ref = 10 12 m -2 (see Figure 1).

    sj ref ref
The other interaction coefficients appearing under the square root in eq. 6 are determined using multiscale modeling, as will be shown in the next sections. Eq. 6 implies an implicit assumption about the superposition rule of strengthening sources. As discussed by Was [START_REF] Was | Fundamentals of Radiation Materials Science: Metals and Alloys[END_REF], adding hardening contributions is not obvious. It depends on the nature and strength of the defects families. However, since all obstacles in eq. 6 are of local type, and since their planner density and stress contribution are comparable, the quadratic superposition rule seems to be the best choice. This has been demonstrated by DD simulations in the case of superposition of forest hardening and carbide strengthening [START_REF] Queyreau | Orowan strengthening and forest hardening superposition examined by dislocation dynamics simulations[END_REF]. For more details about this issue, readers are invited to consult the discussion reported in [START_REF] Monnet | Multiscale modeling of irradiation hardening: Application to important nuclear materials[END_REF].

In the drag regime, dislocation mobility is isotropic and interaction statistic obeys the twodimensional Poisson's distribution leading to the well-known spacing of 1/√𝜌 𝑜𝑏𝑠 . However, at low temperature, it has been shown [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF] that interaction statistics follow the onedimensional Poisson's distribution, parallel to the Burgers vector. DD simulations [START_REF] Monnet | Orowan strengthening at low temperatures in bcc materials studied by dislocation dynamics simulations[END_REF] have shown that the low mobility of screw dislocations induces a curvature of non-screw dislocations in the order of

2𝜏 𝑒𝑓𝑓 µ𝑏
, which leads to a curvature diameter:

s eff s µb D   . ( 8
)
The one-dimensional distribution provides an average spacing close to (Dobs) -1 . With increasing temperature, the effective stress decreases and D increases, such that (Dobs) -1 exceeds the isotropic spacing 1/√𝜌 𝑜𝑏𝑠 . To make a smooth transition between the two statistical distributions, one may consider the average spacing to be given by:

𝜆 𝑠 -1 = 𝑚𝑖𝑛{√𝜌 𝑜𝑏𝑠 𝑠 ; 𝐷 𝑠 𝜌 𝑜𝑏𝑠 𝑠 } (9)
Once the spacing is obtained, the average length of screw dislocation involved in eq. 3 can be given by [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF]:

  c s s s s sc l D l ; max     , ( 10
)
where lc is the minimum length of a screw dislocation segment, taken equal to 10nm.

Critical stress

The critical stress represents the athermal part of the flow stress. It accounts for different athermal contributions: the dipolar interaction self (between dislocations belonging to the same slip system), the grain size or Hall-Petch (HP) effect HP and the line tension LT. The athermal character of the Hall-Petch effect has been evidenced in many experimental observations [START_REF] Petch | The Ductile-Brittle Transition in the Fracture of \ensuremath\Alpha-Iron: I[END_REF][START_REF] Conrad | Cottrell locking and the flow stress in iron[END_REF]. Recently, Tsuchida et a. [START_REF] Tsuchida | Effect of ferrite grain size on tensile deformation behavior of a ferrite-cementite low carbon steel[END_REF], have shown that grain size effect is also independent of strain rate. These observations indicate that this component must be added linearly to the other components of the critical stress. Solid solution hardening must also be added linearly as attested in other experimental results [START_REF] Ebeling | Dispersion hardening of copper single crystals[END_REF][START_REF] Hirsch | The Deformation of Single Crystals of Copper and Copper-Zinc Alloys Containing Alumina Particles. I. Macroscopic Properties and Workhardening Theory[END_REF], but it is expected to be much lower than the HP component. For the sake of simplicity we keep only one linear component accounting for the Hall-Petch effect and the solid solution, called HP. Consequently, the critical stress can be expressed as:

HP s LT s self s c        2 2 (11)
The self-interaction is given by the classical equation:

s self s self a µb    , ( 12
)
where µ is the shear modulus. The HPcomponent component is obtained by :

grain HP d K K µ µ ) 300 (   , ( 13
)
where dgrain is the grain size. In most models of the grain size effects, the HP constant K is found proportional to the shear modulus. It is thus necessary to account for this dependency in our simulations. Since the experimentally determined values of K were mostly determined at room temperature, eq. 13 allows using the so-obtained value at any simulation temperature. It is known that the HP effect depends on the crystallographic structure and the alloying elements in industrial materials [START_REF] Cordero | Six decades of the Hall-Petch effecta survey of grain-size strengthening studies on pure metals[END_REF]26]. For RPV steels, we believe that the appropriate value of K is that reported by Tsuchida et al. [22 in the ferritic grains of the Fe-C system. Concerning the line tension contribution, the assessment is not trivial. The reason is that straight screw dislocations are not always affected by the curvature of non-screw dislocation, as revealed in DD simulation [START_REF] Monnet | Orowan strengthening at low temperatures in bcc materials studied by dislocation dynamics simulations[END_REF]. Consequently, as long as screw dislocation segments are of finite size, that is lsc > lc, one can consider that there is no line tension contribution. But at high obstacle density and/or high temperature, lsc tends to lc, and if the current curvature of non-screw dislocations at the vicinity of obstacles is not high enough to release dislocation segments from these obstacles, an increase of flow stress is mandatory to increase this curvature, inducing a line tension contribution [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF]. After simplification, this contribution can be given as:

         0 , max s eff c s s s TL l µb     . ( 14
)
At high temperature where the effective stress becomes negligible and the average obstacle spacing 𝜆 equals 1/√𝜌 𝑜𝑏𝑠 (see eq. 9), LT recovers its well-known expression:

s obs s s TL µb     , ( 15
) known as the Taylor equation.

Introducing radiation defects

Assessment of radiation hardening at the grain scale has received a large attention from the scientific community, see for example [START_REF] Was | Fundamentals of Radiation Materials Science: Metals and Alloys[END_REF]. Different approaches have been proposed ranging from detailed interaction description (see for ex. [27]) to more or less phenomenological treatments [START_REF] Shi | Interaction of 〈100〉 dislocation loops with dislocations studied by dislocation dynamics in α-iron[END_REF][START_REF] Soneda | Development of neutron irradiation embrittlement correlation of reactor pressure vessel materials of light water reactors[END_REF]. The common feature of these models is the assumption that radiation hardening at the macroscopic scale is proportional to the increase in the critical stress due to interaction with radiation. Given the order of magnitude of radiation hardening, this assumption may be realistic in irradiated annealed austenitic steels, but can hardly be justified in the case of RPV steels [START_REF] Monnet | Multiscale modeling of irradiation hardening: Application to important nuclear materials[END_REF]. Typical sizes of radiation defects are within few nanometers, which is quite comparable to the size of dislocation cores [START_REF] Monnet | Structure and mobility of the edge dislocation in BCC iron studied by molecular dynamics[END_REF][START_REF] Khater | Dislocation glide in Fe-carbon solid solution: From atomistic to continuum level description[END_REF]. This confirms that classical elasticity theory is not always appropriate to predict the interaction strength or outcome. MD simulations are thus needed to track the interaction details and quantitatively characterize the defects resistance at the atomic level. In principle, these results can be used to implement appropriate interaction rules at larger scale simulations, such as DD simulations. Unfortunately, the majority of MD simulations aimed at the rationalization of the obtained results by tracking the details of the interaction mechanisms. In other terms, very rare attempts were made to analyze MD results at the continuum level, see for example the treatments in [START_REF] Rodney | Activation enthalpy for kink-pair nucleation on dislocations: Comparison between static and dynamic atomic-scale simulations[END_REF][START_REF] Monnet | Determination of the activation energy by stochastic analyses of molecular dynamics simulations of dislocation processes[END_REF]. In some of these attempts, the classical cusp angle concept [START_REF] Friedel | Dislocations[END_REF] is used to quantify directly the defects strength (ex. [START_REF] Patinet | Depinning transition for a screw dislocation in a model solid solution[END_REF]) following the Russell-Brown model [START_REF] Russell | A dispersion strengthening model based on differing elastic moduli applied to the iron-copper system[END_REF] (ex. [START_REF] Osetsky | Atomic modelling of strengthening mechanisms due to voids and copper precipitates in α-iron[END_REF]). However, as discussed in [START_REF] Monnet | Transition methods in the multiscale simulation framework: from the atomic to the continuum scale[END_REF], this method is of little help in describing interactions with radiation defects. One may even question the relevance of this concept in the case of any defect of finite size. In our model, we use alternatively the concept of obstacle shear resistance, that has been used to describe interactions with voids [START_REF] Monnet | Mesoscale thermodynamic analysis of atomicscale dislocation-obstacle interactions simulated by molecular dynamics[END_REF][START_REF] Byggmästar | Molecular dynamics simulations of thermally activated edge dislocation unpinning from voids in $\ensuremath{\alpha}$-Fe[END_REF] and other precipitates [START_REF] Monnet | Transition methods in the multiscale simulation framework: from the atomic to the continuum scale[END_REF]. It has been shown that implementing this resistance as a local rule in DD simulations is straightforward. Applying this concept in massive DD simulations of dislocation interactions with spherical precipitates allows thus for a multiscale treatment. Monnet [START_REF] Monnet | Multiscale modeling of irradiation hardening: Application to important nuclear materials[END_REF] presented a general framework for the treatment of precipitation hardening and proposed a constitutive equation giving precipitation hardening as a function of the precipitate size Dprc, density Cprc and resistance prc. If SCs are considered as coherent precipitates (which has been recently confirmed in experiment by Shu et al. by [START_REF] Pascuet | Solute effects on edge dislocation pinning in complex alpha-Fe alloys[END_REF] and in numerical simulations by Pascuet et al. [42]), application of the constitutive equation in the case of SCs gives:

  SC SC SC SC SC SC SC D C µb b L b L b D   2 ) / ln( ) / ln( / 2 ln 2 / 3             , (16)
with SC, DSC, LSC (not to be confused with the average length of screw dislocation segments lsc in eq 10) are respectively the shear resistance, size and average free planner spacing of SCs. DSC is the harmonic mean of DSC and LSC, which equals (DSC LSC,)/(DSC + LSC).  is the maximum resistance of 4.5 GPa beyond which hardening saturates because dislocations cannot penetrate obstacles (Orowan mechanism). For pure Cu or Cr coherent precipitate in iron, the resistance is close to 2.3 GPa [START_REF] Monnet | Transition methods in the multiscale simulation framework: from the atomic to the continuum scale[END_REF]. But recent MD simulations [42] have shown that SC is rather close to 2.8 GPa. In the rigorous treatment, LSC is equal to [(DSCCSC) -0.5 -DSC],

but since the average spacing is significantly larger than the size, LSC reduces to (DSCCSC) -0.5 , which explains the presence of the square root in eq. 16. Comparing eq. 16 with the classical form of hardening,

SC SC SC a µb   
, it is easy to identify an appropriate value of the interaction coefficient aSC, appearing in eq. 6. It is also easy to check that for low to intermediate radiation damage CSC is in the order of few 10 23 m -3 , with a volume fraction below 0.005, aSC does not vary much and remains approximately close to 0.04. Concerning hardening induced by dislocation loops, two tendencies can be isolated depending on loop size DDL [START_REF] Bacon | Dislocation-Obstacle Interactions at the Atomic Level[END_REF]. Interaction with large loops is similar to dislocationdislocation interactions (shearing, junction annihilation, etc.). This population's contribution is thus similar to the forest contribution. However, loops of size less than 4 nm are absorbed by the edge character, leaving a super-jog, while a helical turn is formed on screw dislocations inducing a high pinning force. Massive DD simulations were performed to compute hardening due to the absorption mechanism [START_REF] Monnet | New insights into radiation hardening in face-centered cubic alloys[END_REF]. It was shown that for dislocation loops of 2 -4 nm size and of density up to 10 23 m -3 , a simple Taylor-type equation reproduces all DD simulation results within a good accuracy:

DL DL s DL C D µb 5 . 0   (17)
Again, comparing eq. 17 with

DL DL s DL a µb   
, it is possible to identify a value of aDL = 0.25 that can be used in eq.6. The contribution of DLs to hardening was recently reviewed in [START_REF] Monnet | Multiscale modeling of irradiation hardening: Application to important nuclear materials[END_REF] and it was shown that the induced hardening as revealed by DD simulations is quite similar to that induced by network dislocations, using the same scaling parameters (average distance between dislocation lines) and interaction strength.

Implementation of slip activity on {112} planes

In the initial CL [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF], slip was constrained on {110} planes, generating 12 slip systems. However, many experimental results (see for ex. [START_REF] Franciosi | Investigation of slip system activity in iron at room temperature by SEM and AFM in-situ tensile and compression tests of iron single crystals[END_REF]) evidenced the activity of other slip planes, at least the {112} planes. This missing feature did not affect results in [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF] because the simulated behavior concerned only orientations in the center of the standard stereographic triangle, implying a primary and a secondary slip plane of {110} type [START_REF] Jaoul | Deformation plastique de monocristaux de fer[END_REF]. But activity on the {112} planes may play an important role in the deformation of polycrystals such as RPV steels, because increasing the number of slip systems induces a decrease in the Taylor factor [START_REF] Rosenberg | Calculation of the taylor factor and lattice rotations for bcc metals deforming by pencil glide[END_REF]. In the following we present the method used to implement this feature in the CL. The activation energy in eq. 3 has been fitted on experimental data of slip activity on the {110} planes. This is the easy part because {110} planes are the principal slip planes in BCC metals [START_REF] Keh | Work hardening and deformation sub-structure in iron single crystals deformed in tension at 298[END_REF][START_REF] Takeuchi | Orientation Dependence of Work-Hardening in Iron Single Crystals[END_REF]. Much less experimental data are available on the activation parameters for slip on the {112} planes. Spitzig and Keh [START_REF] Spitzig | The effect of orientation and temperature on the plastic flow properties of iron single crystals[END_REF][START_REF] Spitzig | Orientation dependence of the strain-rate sensitivity and thermally activated flow in iron single crystals[END_REF] measured the flow stress as a function of the tensile axis orientation. At all test temperatures, the CRSS of Fe single crystals can be sorted as 110 < 112-TW < 110-AT, where 110, 112-TW and 110-AT are the CRSS on the {110}, {112} in the twining direction and {112} (in the anti-twining direction) respectively. They have also shown that the CRSSs on the different planes scale with different values of the parameter o in eq. 3. In this paper, we adopt the same strategy in accounting for the activity on the {112} planes. We thus consider the same value of Go and different values for o: 360 MPa (for the {110} planes), 410 MPa (for the {112} planes in the twinning direction) and 480 MPa (for the {112} planes in the anti-twining direction). Another important feature must be highlighted. The interaction coefficients between the {110} and {112} planes must account for the relaxation induced by the cross slip of screw dislocations between the two plane families. This relaxation is not considered in the classical representation of interaction coefficients. In fact, since during cross slip the two activated slip systems share the same Burgers vector, the resulting interaction is considered to be of collinear type, that is, the strongest interaction. Consequently, instead of considering the collinear interaction coefficient (which amounts to acol = 0.7), we propose to use a new coefficient, called the Cross-slip coefficient (acs) with a strength equal to that of the selfinteraction (asc = aself = 0.1). The total interaction matrix with the reference values of the different interaction coefficients (in the inset table) are given in Figure 1. For the sake of simplicity, only two values of the coefficients are used: 0.7 for the collinear interaction and 0.1 for all the others. In our model we do not distinguish the cross-slip between the {110} planes and the {112} in the twinning direction from that in the antitwinning direction.

Evolution of defects densities

A classical Kocks-Mecking formulation [START_REF] Kocks | Physics and phenomenology of strain hardening: the FCC case[END_REF] is used to predict the evolution of dislocation densities as a function of plastic glide:

        s s s s s y Λ b    1 1   , ( 18
)
where  represents the so-called dislocation mean free path, namely the average displacement of mobile dislocations before immobilization, and y a distance below which dislocation annihilation is systematic. As explained in [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF], Temperature dependent interaction statistics affect  and y, which can be accounted for by the following expressions:

                      obs s obs s s self s self o s eff grain s K K a d Λ       1 1 1 , (19)
Burgers -1 1 1 -1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 1 1 -1 1 -1 1 1 -1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 1 1 -1 1 plan 0 -1 1 1 0 1 1 1 0 0 -1 1 -1 0 1 -1 1 0 0 1 1 1 0 1 -1 1 0 0 1 1 -1 0 1 1 1 0 1 -1 2 -2 -1 -1 1 2 -1 -1 2 -1 2 -1 -1 -1 -1 2 1 -2 -1 1 1 2 -2 1 -1 -1 1 2 2 1 -1 -1 -2 -1 A2 A3 A6 B2 B4 B5 C1 C3 C5 D1 D4 D6 A6' A2" A3' B4" B2' B5" C3" C5' C1 "
Where Kself (Kobs) are constants representing the number of intersections with primary dislocations (local obstacles) before immobilization. They correspond to the permeability of primary dislocations (obstacles) to dislocation crossing and ydrag is the annihilation distance that prevails at high temperature in the drag regime. Since radiation defects are usually sheared, one must also account for the evolution of the local density of these defects. Consider interactions with DLs of size DDL. Let V be a small volume subjected to a shear increment dγ. The latter must equal bdS/V, where dS is the area swept by dislocations. If we consider that a moving dislocation can drag or absorbe all loops contained in a layer of thickness (DLDDL), centered on the dislocation slip plane, then the decrease in the loop density on system s within the volume V is simply dCDL = -DDLdSCDL/V. Combining dγ and dCDL and dividing by the time increment, one gets :

s s DL s DL DL s DL C b D C       . ( 21
)
The parameter  accounts for the efficiency of the dislocation in dragging the loops. By definition, when  , only loops cutting the slip plane are absorbed.

A similar reasoning can be applied to SCs. The softening here results from shearing rather than dragging or absorption. We may write:

s s SC s SC SC s SC C b D C       . ( 22
)
Of course, in the case of SCs, there is no reason that SC can be larger than one. On the contrary, we expect the SCs to be completely sheared after the passage of n = DSC/b dislocations. In other terms, a plausible value for this parameter would be SC = 1/n = b /DSC, which allows to simplify eq. 22 to :

s s SC s SC C C     . ( 23
)
However, this equation questions the relevance of the cross graining (scale transition) procedure used here, that is, the transition from the dislocation discrete space to the continuum space of larger scale simulations such as finite-element method.

Simulation of the macroscopic behavior

Simulation methods

In this work, three simulation techniques for the integration of the crystalline law have been used: the classical self-consistent homogenization scheme with the Berveiller-Zaoui localization rule [START_REF] Berveiller | An extension of the self-consistent scheme to plasticallyflowing polycrystals[END_REF], called in the following the BZ method, and two full field simulation techniques: the Finite Element (FE) method using the code CAST3M [54] and the FFT based solver AMITEX-FFTP [START_REF] Gélébart | Non-linear extension of FFT-based methods accelerated by conjugate gradients to evaluate the mechanical behavior of composite materials[END_REF], called FFT method. The BZ method was used in all the applications below, while the FE method was used for the tensile simulations of single crystals and the FFT method for the simulations of tensile tests of RPV steels.

FFT-based simulations have been performed on Voronoï cubic grain aggregates, with various spatial resolutions and various number of grain orientations (taken from a random isotropic crystallographic texture). Finally, the converged average behavior of the Voronoï polycrystal obtained with 1000 orientations and a spatial resolution of 7x7x7 voxels per grain was found to be satisfactorily reproduced with a much simpler cubic grain aggregate, with 125 grains and one voxel per grain, leading to drastically reduced computation times, comparable to the BZ model. Note also that, in the following, the BZ and FE simulations use the small strain approximation whereas FFT simulations use the complete finite strains framework. In all simulations, the values of the constants and the material parameters are listed in

Validation of the implementation of {112} planes

FEM simulations of tensile test of single crystal beam of [1 ̅ 49] has been reported in [START_REF] Monnet | Dislocation-dynamics based crystal plasticity law for the low-and high-temperature deformation regimes of bcc crystal[END_REF]. This orientation is close to the center of the standard stereographic triangle (orientation D in the inset of Figure 2a), and activates slip on the (1 ̅ 01) plane. Using exactly the same simulation conditions and sample geometry, we simulate using the FE method tensile tests of beams orientations [1 ̅ 18] (close to orientation A) and [2 ̅ 88] (close to orientation B), activating, respectively, slip on the (11 ̅ 2) planes in the twinning direction and on the (2 ̅ 11) plane in the antitwining direction. In these simulations of pure Fe single crystals, we set SC = DL = carbide = 0, with a high enough value of grain size dgrain to suppress the Hall-Petch effect. The comparison between the measured and computed values of the CRSS as a function of temperature and single crystal orientation is shown in Figure 2a andb. At each simulation temperature, the CRSS on the (1 ̅ 01) plane is lower than that on the (11 ̅ 2) plane, which is also lower than that on the (2 ̅ 11) plane. The obtained sorting and ratios between the different CRSS is in agreement with experimental observations [START_REF] Spitzig | The effect of orientation and temperature on the plastic flow properties of iron single crystals[END_REF][START_REF] Spitzig | Orientation dependence of the strain-rate sensitivity and thermally activated flow in iron single crystals[END_REF]. This indicates that the implementation of slip activity on the {112} planes meets the rare requirements reported in the literature.

Behavior of pure iron polycrystals

The second set of simulations was obtained using the BZ method on pure Fe polycrystals. Materials parameters are the same as above, apart from dislocation densities on the different slip systems. They were taken such that the total dislocation density is close to 10 13 m -2 . Since the conventional offset of 0.2% strain is usually affected by static aging in iron, we use stress at 2% offset of deformation to compare with experiment [START_REF] Cottu | Etude en traction de l'adoucissement du fer de haute pureté par le carbone à basse temperature[END_REF]. The evolution of the yield stress predicted by the BZ method is depicted in Figure 3. As can be noticed from the figure, there is a good agreement with experiment in the whole temperature range. The surprising feature is that the implementation of the {112} planes with different activation scheme does not alter strongly the predicted yield stress. Passing from 12 to 24 slip systems does not seem to soften the mechanical behavior.

Simulated behavior of the RPV steel

In this Section we target RPV steels with metallurgical characteristics close to those of the SA508 class specifications. For these steels, it is important to account for the intra-lath carbide density and the high dislocation density in both the un-irradiated and irradiated states.

To do so, it is important to select a reference material that has been characterized using experimental tests. This material is called Euromaterial A and has been characterized in the PERFECT project [START_REF] Massoud | PERFECT: Prediction of Irradiation Damage Effects on Reactor Components[END_REF] by Vincent et al in [START_REF] Vincent | Towards a modelling of RPV steel brittle fracture using crystal plasticity computations on polycrystalline aggregates[END_REF] and others [START_REF] Meslin | Characterization of neutron-irradiated ferritic model alloys and a RPV steel from combined APT, SANS, TEM and PAS analyses[END_REF][START_REF] Lambrecht | On the correlation between irradiation-induced microstructural features and the hardening of reactor pressure vessel steels[END_REF]. The values of the constants found for this material can be consulted in Table 1.

The comparison between the experimental and simulated tensile curves can be seen in Figure 4. The comparison is made at three different values of temperature. The materials parameters, that is, the dislocation density, the mean free paths and the grain size are the same. It can be seen that without adjusting the crystalline law at every temperature, the predicted tensile curves are close to the experimental ones, which highlights the predictive power of the CL. However, at the onset of plastic deformation, the predicted curves are smooth and do not fit correctly the experimental curves. This feature is due to the moderate static aging, known to prevail in most of Fe-C alloys and steels [START_REF] Krauss | Steels: Processing, Structure, And Performance[END_REF]. The constitutive description of static aging is still a difficult task and, to the knowledge of the authors, has not been addressed deeply in the literature.

The self-consistent scheme adopted here [START_REF] Berveiller | An extension of the self-consistent scheme to plasticallyflowing polycrystals[END_REF] is known to be well adapted for elasto-plastic behavior. However, the BZ homogenization model is also well known to provide estimates that are too stiff (even violating bounds in some specific cases) [START_REF] Gilormini | A Critical Evaluation for Various Nonlinear Extensions of the Self-Consistent Model[END_REF][START_REF] Masson | An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals[END_REF]. Hence, it is interesting to compare the BZ self-consistent results with the ones of a full field FFT method. The two sets of simulated tensile curves are depicted in Figure 5, considering for both implementations a small strain formulation framework. It can be seen that, at the 0.2% plastic strain offset, the difference between predictions is negligible for the all tested temperatures.

Uniaxial Strain

Uniaxial Stress (MPa) At larger strains, the BZ method is too stiff, as expected, and predicts significantly higher tensile stresses. The deviation seems to increase monotonously with the flow stress and plastic deformation. However the shift remains in the order of 5% at the offset of 2% of plastic strain. At low strain, The FFT and BZ methods provide thus consistent predictions of the flow stress, and the BZ method is used in the following for that purpose. This result encourages us to check for the effect of solute cluster on the flow stress and internal variables of the model. We consider the typical values of density and size of solute clusters (using APT characterization) of moderately irradiated low copper RPV steels recalled in Table 1. For the as-received (AR) materials (non-irradiated), the simulation predictions are compared with results obtained by Renevey [START_REF] Renevey | Approches globale et locale de la rupture dans le domaine de la transition fragile-ductile d'un acier faiblement allié[END_REF], Libert [START_REF] Libert | Temperature dependent polycrystal model application to bainitic steel behavior under tri-axial loading in the ductile-brittle transition[END_REF] and Euromateriala A [START_REF] Vincent | Towards a modelling of RPV steel brittle fracture using crystal plasticity computations on polycrystalline aggregates[END_REF]. Radiation hardening has been measured on the Euromateriala A [START_REF] Vincent | Towards a modelling of RPV steel brittle fracture using crystal plasticity computations on polycrystalline aggregates[END_REF] materials. The comparison is shown in Figure 7 for all the reported results. Since RPV experience static aging, especially at low temperature, we consider the flow stress at the offset strain of 2%. Two important remarks can be made: (i) there is a fare agreement between experimental and simulation results in the AR and irradiated cases, obtained with no adjustable parameters and (ii) radiation hardening seems to vanish at low temperatures. Contrary to the popular belief, radiation hardening does depend on temperature. At the first glance, this appears to be in contradiction with experimental results (see for ex. [START_REF] Williams | Irradiation Strengthening and Fracture embrittlement of A533-B Pressure Vessel Steels plate and submerged-arc weld[END_REF][START_REF] Hunter | Fracture and tensile behavior of neutron-irradiated A533-B pressure vessel steel[END_REF]). Our interpretation of this discrepancy is that the offset of 0.2% for the measurement of the yield stress is not the most appropriate. This is due to static aging strongly accentuated by irradiation. The Lüders plateau increases with decreasing temperature and increasing radiation dose, as can be seen on Figure 4. However, according to our model, at very low temperature, the flow stress of RPV at strains larger than, say 2%, must not be strongly affected by irradiation. We could not confirm this important result because we could not find available tensile curves on irradiated RPV in the literature. On the other hand, Figure 6 shows that radiation hardening is little sensitive to temperature above 300 K. The small decrease in irradiation hardening is due to the classical elastic softening. This athermal high-temperature irradiation hardening has been discussed in details in recent papers [START_REF] Monnet | Multiscale Modeling of Radiation Hardening[END_REF][START_REF] Monnet | Multiscale modeling of irradiation hardening: Application to important nuclear materials[END_REF], in which radiation hardening is predicted and compared with experimental results in RPV, Fe-Cr ferriticmartensitic and austenitic stainless steels. One of the most important parameter in our model is the evolution of the average length of screw dislocation segments lsc. This variable delineates the domain of appearance of the line tension contribution. When, lsc decreases to its minimal value lc = 10 nm, the line tension contribution is no longer zero, because the fraction term in the min expression involving  in eq. 14 becomes larger than the effective stress.

Temperature (K) RP 2%

(MPa) In order to track the evolution of lsc in the simulations of RPV (AR and irradiated) we choose to record the value of lsc associated with the most active slip system in the best oriented grain, that is, with the highest Schmid factor. This evolution is given in Figure 7 as a function of strain for different simulation temperatures in the irradiated (with solute clusters) and unirradiated case. For all profiles at larger simulation temperatures than the one plotted in the figure, lsc reaches its minimal value at the beginning of deformation. lsc is found to decease as expected with deformation because of the increase in the obstacle density, which in this case corresponds to the stored dislocations. At low temperature, lsc is large because the effective stress is large, which increases the curvature and the average obstacle spacing. When SCs are added as a new component of the microstructure, the same behavior is found but lsc is lower in irradiated materials because of the increase in the obstacle density. Consequently, the line tension contribution increases with the density of radiation defects. This is the origin of radiation hardening, which explains the insensitivity of hardening to temperature for T > 300 K. Another interesting feature is that lsc in the irradiated material becomes larger than that in the unirradiated materials for uniaxial strain larger than 8 %, as can be noticed in Figure 7. This is a tricky property and can be explained as follows. The increase in the critical stress due to the presence of solute clusters delays the activation of secondary slip. Since Kcopla is substantially higher than Kforst, (see Table 1), the mean free path of primary dislocations ( s in eq. 19) decreases less with strain, which slows down dislocation storage (eq. 18) and, consequently, slows down the decrease in lsc. Although in the irradiated material, the starting value of lsc is lower than in the unirradiated material, at enough strain, lsc in the irradiated material exceeds that in the unirradiated one. In order to give an insight into the different contributions of the microstructure components to the flow stress, we show in Figure 8 the evolution of every contribution as a function of temperature in the unirradiated (a) and irradiated (b) RPV steels.

Conclusions

In this paper, we present physically based constitutive equations (crystalline law) designed for polycrystalline iron and RPV steels that accounts for the thermally activated (below 250 K) and athermal (above 350 K) regimes, with a smooth transition in the intermediate temperature range. The flow stress is expressed in terms of the microstructural components: dislocation network, carbides, grain size and, in the case of irradiated RPV, solute clusters. This law has been integrated using the self-consistent method (using the Berveiller-Zaoui localization rule) and the full field methods: the FFT (AMITEX) and Finite-Elements (CASTEM) solvers. From the results reported in this paper, we can draw the following conclusions:

-The BZ homogenization model provides satisfying results at the onset of plastic deformation but when the applied strain increases it overestimates significantly the macroscopic response with respect to the FFT-based full field predictions, -For the behavior of polycrystals, it is not necessary to account for the intragranular behavior in full details. The proposed simplified law is found to reproduce appropriately the macroscopic behavior of polycrystals, -The implementation of the {112} slip planes does not seem to strongly affect the yield stress. No softening is obtained when passing from 12 to 24 slip systems. -The large initial dislocation and carbide densities in RPV steels decreases strongly the average length of screw dislocation segments, which in turn increases the effective stress with respect to a fully annealed polycrystalline iron. -Radiation defects accentuate the decrease in the screw segment length and generate a line tension contribution at lower temperatures. -The Hall-Petch effect (grain size effect) contributes strongly to the yield stress in RPV steels. It amounts to more than 35%. This component must not be neglected in modeling plastic deformation of microstructures such as bainite and martensite. -Radiation hardening computed at 2% offset of plastic strain seems to decrease with decreasing temperature in the thermally activated regime and vanishes below 100 K. For temperatures higher than 250 K, hardening becomes athermal. This important feature deserves further investigation for validation at very low temperature..

Figure 1 :

 1 Figure 1: Interaction matrix of slip systems on the {110} and {112} planes with the reference values.

  a_col a_copla a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_cs a_col a_cs a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc A3 a_self a_col a_junc a_junc a_junc a_junc a_copla a_junc a_junc a_junc a_junc a_cs a_cs a_col a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc A6 a_self a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_copla a_col a_cs a_cs a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc B2 a_self a_col a_col a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_cs a_col a_cs a_junc a_junc a_junc a_junc a_junc a_junc B4 a_self a_col a_junc a_junc a_junc a_junc a_copla a_junc a_junc a_junc a_junc a_col a_cs a_cs a_junc a_junc a_junc a_junc a_junc a_junc B5 a_self a_junc a_junc a_copla a_junc a_junc a_junc a_junc a_junc a_junc a_cs a_cs a_col a_junc a_junc a_junc a_junc a_junc a_junc C1 a_self a_col a_col a_copla a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_cs a_cs a_col a_junc a_junc a_junc C3 a_self a_col a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_col a_cs a_cs a_junc a_junc a_junc C5 a_self a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_cs a_col a_cs a_junc a_junc a_junc D1 a_self a_col a_col a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_cs a_col a_cs D4 a_self a_col a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_cs a_cs a_col D6 a_self a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_col a_cs a_cs A6' a_self a_col a_col a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc A2" a_self a_col a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc A3' a_self a_col a_junc a_cs a_copla a_self a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc a_junc

Figure 2 :

 2 Figure 2: evolution of the critical resolved shear stress with temperature for three orientations, (a) experimental results from [50] and (b) predicted values in FEM simulations.

Figure 3 :

 3 Figure 3: comparison of the yield stress at 2% strain measured in experiment and predicted by the BZ homogenization technique, using all slip planes ({110} and {112}) and only {110} planes.

Figure 4 :

 4 Figure 4: simulated (BZ method) vs experimental tensile curves of the RPV material Euromateriala A at three different temperatures.

Figure 5 :

 5 Figure 5: comparison between the tensile curves predicted by the BZ (self-consistent) and FFT (full field) methods at different temperature.

Figure 6 :

 6 Figure 6: comparison between the BZ predictions (full symbols) and experimental values (open symbols) of the flow stress at.2% offset strain in RPV steels before (in bleu) and after (in red) irradiation as a function of the simulation/experimental temperature of the tensile tests.

Figure 7 :

 7 Figure 7: evolution of the average length of screw segments as a function of strain and BZ simulation temperature with (Ir for irradtaed) and without (AR for as-received unirradiated) solute clusters defects.

Figure 8 :

 8 Figure 8: contributions of the microstructure components in un-irradiated (a) and irradiated (b) RPV steels as a function of temperature.

Table 1 .

 1 Of course, the parameter values depend on the tested materials.

	Parameter	Value	Parameter	value
	o  (rate constant) Go H b (Burgers vector) E (Young modulus) in GPa a ij (non collinear) ydrag (dist. annihi.)  s ini= s m (primary system) singlecrystal simulation  s ini= s mFe polycrystal Kself Ccarbide (RPV) dgrain (RPV) Dcarbide (RPV)	10 -5 s -1 0.84 eV 2 10 11 s -1 0.248 nm 236 -0.0459 T k (Boltzmann constant) o ({110}) n (rate power) o ({112} TW) o ({110} AT) 0.1  (Poisson's ratio) 2 nm 10 12 m -2  s ini= s m (other systems) singlecrystal simulations 10 11 m -2  s ini= s m RPV a ij (collinear) 17 Kforest 7.6×10 17 m -3 lc (minimal length) 6.9 µm SC 80 nm DL	100 360 MPa 410 MPa 480 MPa 8.6 10 -5 eVK -1 0.35 0.7 10 112 m -2 10 13 m -2 Kself /3 10 nm 0 0

Table 1 :

 1 Values of the model parameters and material constants used for the simulations

Acknowledgment

The authors acknowledge the financial support of Euratom research and training program 2014-2018 SOTERIA (http://www.soteria-project.eu/)


S Jaillet
email: jaillet@lirmm.fr

A Laurent
email: laurent@lirmm.fr

M Teisseire
email: teisseire@lirmm.fr

J Chauch
email: chauche@lirmm.fr

Order and Mess in Text Categorization: Why Using Sequential Patterns to Classify

Keywords: text mining, categorization, sequential patterns

Text categorization is a well-known task essentially based on statistical approaches using neural networks, Support Vector Machines and other machine learning algorithms. Texts are generally considered as bags of words without any order. Although these approaches have proven to be efficient, they do not provide users with comprehensive and reusable rules about their data. These rules are however very important for users in order to describe the trends from the data they have to analyze. In this framework, an associationrule based approach has been proposed by Bing Liu (CBA). In this paper, we propose to extend this approach by using sequential patterns in the SPaC method (Sequential Patterns for Classification). Taking order into account allows us to represent the succession of words through a document without complex and time-consuming representations and treatments such as those performed in natural language and grammatical methods. The original method we propose here consists in mining sequential patterns in order to build a classifier. We show on experiments that our proposition is relevant, and that it is very interesting compared to other methods. In particular, our method has better results than SVM when SVM do not perform well. Moreover, our approach is very interesting for huge volumes of data.

INTRODUCTION

Automatic text classification goes back at least to the 1960's [START_REF] Maron | Automatic indexing: An experimental inquiry[END_REF]. But with the growing volume of available digital documents, researches on automatic classification have been extensively addressed in the past few years in order to define efficient and scalable methods [START_REF] Sebastiani | Machine learning in automated categorisation[END_REF][START_REF] Yang | An evaluation of statistical approaches to text categorization[END_REF].

There are two distinct types of approaches in automatic clas-sification: the supervised one and the unsupervised one. In the supervised classification or categorization, categories are defined by an expert while they are automatically learned in the other (also called clustering) [START_REF] Salton | Introduction to modern information retrieval[END_REF][START_REF] Iwayama | Cluster-based text categorization: a comparison of category search strategies[END_REF]. In this article we focus on text categorization or, in other words, on supervised learning algorithms.

Currently, existing classifiers are mostly based on statistical methods and Support Vector Machines. These methods are based on word frequencies such as T F -IDF (Term Frequency, Inverse Document Frequency) [START_REF] Salton | Introduction to modern information retrieval[END_REF]. However, these methods do not provide understandable descriptions of the extracted knowledge. In order to cope with this problem, an association-rule based approach has first been proposed by Bing Liu (CBA) [START_REF] Liu | Integrating classification and association rule mining[END_REF], which has been enhanced by [START_REF] Wang | Growing decision trees on support-less association rules[END_REF], [START_REF] Li | CMAR: Accurate and Efficient Classification Based on Multiple Class-Association Rules[END_REF], [START_REF] Janssens | Adapting the cba-algorithm by means of intensity of implication[END_REF] and [START_REF] Baralis | Majority classification by means of association rules[END_REF], and by ARC-BC in [START_REF] Antonie | Text document categorization by term association[END_REF] and ARC-PAN in [START_REF] Antonie | An associative classifier based on positive and negative rules[END_REF]. Association rules have indeed been extensively studied and are very efficient and scalable.

All these methods consider each text as a so-called bag of words where no order between words is taken into account for categorization. This textual representation has proven to be useful and almost as efficient as complex representations which require time-consuming methods like syntactic analysis. It is thus interesting to investigate methods that take order into account and remain scalable.

In this paper, we propose thus to extend the CBA method by taking order into account. In our approach, the order management is performed by considering the extraction of sequential patterns instead of association rules. A sequential pattern is a set of items appearing consecutively in the database.

The originality of our approach SPaC (Sequential PAtterns for Categorization) is that we use sequential patterns for text categorization. For this purpose, two steps are considered. The first step consists in mining sequential patterns from texts. The granularity we consider is the sentence, meaning that each sentence is considered as an unordered set of items (words) whereas texts are considered as ordered sets of sentences. The second step consists in classifying documents using these sequential patterns.

In this paper, we show that sequential pattern-based classification is very efficient when SVM do not perform well. In this framework, sequential patterns have three main advantages: first they provide understandable rules (contrary to SVM, Rocchio, naïve Bayes,...). Second they allow trend analysis, as shown in [START_REF] Lent | Discovering trends in text databases[END_REF]. Third, they extract patterns that are more precise and informative than association rules. Note that our approach is evaluated using precision-recall merged in F β -measure [START_REF] Rijsbergen | Information Retrieval[END_REF] and not only accuracy. These measures have indeed proven to be more relevant to compare text classification methods [START_REF] Sebastiani | Machine learning in automated text categorization[END_REF].

The paper is organized as follows. Section 2 presents the background of the problem being addressed by introducing sequential patterns and textual representations. Section 3 details existing methods dealing with text mining with "frequent patterns" and "sequential patterns". Section 4 details our method based on Sequential Patterns (SPaC). Section 5 shows that our method performs well on datasets in French and English. Finally, Section 6 summaries the paper and presents future work.

PROBLEM STATEMENT

First, we formulate the concept of sequence mining by summarizing the formal description of the problem introduced in [START_REF] Agrawal | Mining Sequential Patterns[END_REF] and extended in [START_REF] Srikant | Mining Sequential Patterns: Generalizations and Performance Improvements[END_REF]. Second we look at the categorization problem.

Mining of sequential patterns

Let DB be a set of customers' transactions where each transaction T consists of customer-id, transaction time and a set of items involved in the transaction.

Let I = {i1, i2, ..., im} be a set of literals called items. An itemset is a non empty set of items. A sequence s is a set of itemsets ordered according to their time stamp. It is denoted by < s1s2...sp > where sj, j ∈ 1..n, is an itemset. A n-sequence is a sequence of n items (or of length n). For example, let us consider a given customer who purchased items 1, 2, 3, 4, 5, according to the following sequence: s =< (1) (2, 3) (4) (5)>. This means that apart from 2 and 3 that were purchased together, i.e. during the same transaction, items in the sequence were bought separately. s is a 5sequence.

A sequence < s1s2...sp > is a sub-sequence of another sequence < s 1 s 2 ...s m > if there exist integers i1 < i2 < ...ij... < in such that s1 ⊆ s i 1 , s2 ⊆ s i 2 , ..., sp ⊆ s in . For example, the sequence s = < (2) (5) > is a sub-sequence of s because (2) ⊆ (2, 3) and ( 5) ⊆ (5). However < (2) (3) > is not a sub-sequence of s since items were not bought during the same transaction.

All transactions from the same customer are grouped together and sorted in increasing order. They are called a data sequence. A support value (supp(s)) for a sequence gives its number of actual occurrences in DB. Nevertheless, a sequence in a data sequence is taken into account only once to compute the support even if several occurrences are discovered. In other words, the support of a sequence is defined as the fraction of total distinct data sequences that contain s. A data sequence contains a sequence s if s is a sub-sequence of the data sequence. In order to decide whether a sequence is frequent or not, a minimum support value (minSupp) is specified by the user. The sequence is said to be frequent if the condition supp(s) ≥ minSupp holds.

Given a database of customer transactions the problem of sequential pattern mining is to find all sequences whose support is greater than a specified threshold (minimum support). Each of which represents a sequential pattern, also called a frequent sequence.

Sequential patterns are usually extracted from a database built on the following schema: date, client, items. For instance, we consider the database of client purchases in a supermarket, as shown in Table 1. Each line (transaction, tuple) from this table corresponds to the set of items bought by the client at the corresponding date.

In this example, Peter has bought the items 1, 2, 3, 4, 5 as the sequence <(1)(2,3)(4)(5)>.

Client

Textual Representation and Categorization

Text categorization is the task of assigning a boolean value to each pair (document, category). A categorization process has to define: (1) a formalization of texts and category (2) a (text, category) measurement (3) a categorization policy.

The textual database is partitioned into two databases TT rain and TT est where TT rain stands for a training set and TT est stands for a test set.

In usual methods, texts are represented as bags of words [START_REF] Sebastiani | Machine learning in automated text categorization[END_REF]. The order is not considered. Each document is represented by a vector where each component is a word weighted by a numerical value. The most used weighting is T F -IDF (Term Frequency -Inverse Document Frequency) [START_REF] Salton | Introduction to modern information retrieval[END_REF]. For a word w, we have:

tf idf (w) = tf (w).log N df (w)
where tf (w) is the number of occurrences of w in the document, df (w) if the number of documents containing w and N is the total number of documents. The weight tf idf (w) represents thus the relative importance of the word in the document.

These vectors describing documents are then used in order to extract knowledge using common algorithms such as knearest neighbors, SVM, naive bayes, etc.

RELATED WORK

Text mining has been widely investigated [START_REF] Lent | Discovering trends in text databases[END_REF][START_REF] Ali | Partial classification using association rules[END_REF][START_REF] Agrawal | Athena: Mining-based interactive management of text databases[END_REF][START_REF] Sebastiani | Machine learning in automated text categorization[END_REF]. In this section, we focus on text classification and frequent patterns.

Classification Based on Associations: the CBA Method

In [START_REF] Liu | Integrating classification and association rule mining[END_REF] the authors propose CBA: a text categorization method based on association rules. CBA consists of two parts, a rule generator (CBA-RG) which is based on the well-known Apriori algorithm [START_REF] Agrawal | Fast Algorithms for Mining Generalized Association Rules[END_REF] and a classifier builder (CBA-CB) which is based on generated rules.

CBA-RG

In this first step, the key is to find all ruleitems whose support is greater than a specified threshold (minimum support). A ruleitem is defined by: < conset, Ci > where condset is a set of items and Ci is a class label. Each ruleitem ρ represents a rule condset → Ci and the support and the confidence of such a rule is defined by:

sup(ρ) = #texts from Ci matching condset |D| conf (ρ) = #texts in Ci matching condset #texts in D matching condset
CARs (classification rules) thus consists of all the ruleitems that satisfy the minimum support and a minimum confidence.

In these approaches, frequent patterns are extracted using a single minimum support threshold. However, categories are not always equi-distributed. It is thus not relevant to consider such a single value. Choosing a relevant minimum support threshold is crucial so that frequent patterns are relevant for the categorization task. A high support will indeed prevent the system from finding frequent patterns for a small category, while a low support will lead to the generation of a huge number of rules, which is not interesting because it will lead to overfitting.

Works have been proposed in order to define a multiple minimum support application (msCBA) [START_REF] Janssens | Adapting the cba-algorithm by means of intensity of implication[END_REF][START_REF] Liu | Improving an association rule based classifier[END_REF]. In these approaches, ruleitems are extracted using a multiple minimum support strategy. The minimum support of each category is defined according to the distribution frequency of each category and the user minimum support threshold:

minSupC i = minSupuser * f reqDistr(Ci)
where, f reqDistr(Ci) = #texts from Ci #texts .

CBA-CB

Let R be the set of CARs and D the training data. The basic idea of the algorithm is to choose a set of high precedence rules in R to cover D. The categorizer is thus represented by a list of rules ri ∈ R ordered according a total order based on the confidence. We have thus, < (r1, r2, ..., r k ), Ci > (where Ci is the target category and rj one of the associated rules).

Each rule is then tested over D. If a rule does not improve the accuracy of the classifier, this rule and the following ones are discarded from the list.

Once the categorizer has been built, classification rules are tested until a condition is matched. The text is then associated with the target class of the classification rule.

Enhancements and Other Approaches

In [START_REF] Janssens | Adapting the cba-algorithm by means of intensity of implication[END_REF], the authors propose to replace the confidence by the intensity of implication when sorting the rules to build the classifier. In [START_REF] Kumar | Classification Using Association Rules: Weaknesses and Enhancements[END_REF], the authors integrate the CBA method with other methods such as decision trees.

In [START_REF] Baralis | Majority classification by means of association rules[END_REF], the authors investigate a method for association rule classification. Contrary to CBA-CB which takes only one rule into account, they propose to determine the category of a text by considering several rules that are mixed using a majority voting. In order to cope with the huge number of rules, the authors propose a pruning method during the extraction of the classification rules using χ 2 , as done in [START_REF] Li | CMAR: Accurate and Efficient Classification Based on Multiple Class-Association Rules[END_REF]. maxrules stands for the maximum number of rules used to classify new cases. Moreover, rules are used at different levels. This approach is enhanced in [START_REF] Baralis | On support thresholds in associative classification[END_REF] by considering several minimum support thresholds.

In [START_REF] Ali | Partial classification using association rules[END_REF], association rules are used for partial classification. Partial classification means that the classifier does not cover all cases. In particular, this work is interesting when dealing with missing values. [START_REF] Lent | Discovering trends in text databases[END_REF][START_REF] Wong | Visualizing sequential patterns for text mining[END_REF] propose to use sequential patterns in the text mining framework. In [START_REF] Wong | Visualizing sequential patterns for text mining[END_REF], the proposition is based on two methods. The first method is based on the visualization of word occurrences in order to detect sequential patterns. The second method is based on classical methods to extract sequential patterns. However, the authors do not propose a method to classify texts from the sequential patterns. Moreover, the texts considered are associated with a date. The corpus consists of 1,170 articles collected over 6 years. This point makes it very different from our proposal and more difficult to apply since texts are rarely associated with a date.

In [START_REF] Lent | Discovering trends in text databases[END_REF], authors demonstrate how sequential patterns are useful for text mining. Sequential patterns are used in order to extract trends from textual databases.

As we show in this section, text mining with frequent patterns is either performed as a classification task using association rules, or as a non classification task using sequential patterns. Note that [START_REF] Antonie | Text document categorization by term association[END_REF][START_REF] Antonie | An associative classifier based on positive and negative rules[END_REF] propose a classification method based on association rules. However this method is neither based on sequential patterns nor devoted to text classification. We propose thus an original method based on sequential patterns for classification. We argue that this method is able to deal with order in texts without being time-consuming. Next section details our approach. Section 5 shows that our method obtains good results compared to other ones.

SEQUENTIAL PATTERNS FOR CLASSI-FICATION: THE SPAC METHOD

In this paper, we propose an original method (SPaC) for text classification based on sequential patterns. This methods consists in two steps. In the first step, we build sequential patterns from texts. In the second step, sequential patterns are used to classify texts.

From Texts to Sequential Patterns

Each text is a set of words. Our method is based on sequential pattern mining. Texts are represented as ordered sets of words using the T F -IDF representation. Each text is thus considered as being the equivalent of a client. The text is constituted by a set of sentences which is associated with a date (its position in the text). Finally the set of words contained in each sentence corresponds to the set of items purchased by the client in the market basket analysis framework. Table 2 summaries the two terminologies.

Note that if order is considered in the text, a sentence remains treated as a bag of words. This adds flexibility to our approach since we argue that the order of the words within a sentence is not as important as the order of ideas in the sentences themselves.

This representation is coupled with a stemming step and a stop-list. The stemming step consists in replacing each word by its root word. The stop-list prevents the system from learning from noisy words such as "the, a". Some words are discarded by considering the entropy of each stem over the corpus. This method eliminates words that could skew the classifier since they are not discriminant enough. Moreover, this method allows us to apply low supports in the sequential pattern discovery without deteriorating results. For this purpose, a user-defined threshold is considered. For each word w, we consider its entropy H(w) over all classes Ci defined as: corresponding to the sequence spj, the category Ci and the confidence confi,j of the rule spj → Ci.

H(w) = -C i p (
Table 3: Notations automatically by considering 3 texts per category. This assumption leads to consider for each category Ci the following multisupport:

minsupC i = 3 * 100 #texts in training set Ci
In our approach the training set is divided into n training sets, one for each category. Texts are thus grouped depending on their category. Sequential pattern mining algorithms are applied separately on these n databases using the corresponding minimum supports.

For each category, frequent sequential patterns are computed and their supports are stored. The support of a frequent pattern is the number of texts containing the sequence of words. Definition 1. Let < s1 . . . sp > be a sequence. The support of < s1 . . . sp > is defined as:

supp(< s1 . . . sp >) = #texts matching < s1 . . . sp > #texts
Algorithm 1 describes the SPaC sequential pattern generation. The SPAM algorithm is used through the SP M ining() function in order to find all frequent sequences in the transactional databases (DB) [START_REF] Ayres | Sequential pattern mining using bitmaps[END_REF]. end For instance, the following frequent patterns have been extracted from the category "Purchasing-Logistics" of our French database:

< (cacao) (ivoir) (abidjan)> < (blé soja) (maï)> < (soj)(blé lespin victor)(maï soj )(maï )(grain soj)(soj)>
The first sequential pattern means that texts contain words cacao then ivoire then Abidjan in three different sentences.

The second sequential pattern means that texts contain the words blé and soja in the same sentence and then maïs 1 (mai).

The third sequential pattern means the word maï occurs in two successive sentences before the word grain.

Note that the use of sequential patterns allows the apparition of a word several times in the text, contrary to association rules. Moreover, some frequent co-occurrences can be identified with sequential patterns mining.

From Sequential Patterns to Categories

Once sequential patterns have been extracted, the goal is to derive a categorizer from the obtained rules. This is done by computing for each rule in a category the corresponding confidence. For each sequential pattern < s1 . . . sp > found for the category Ci, a rule γ is computed:

γ :< s1 . . . sp >→ Ci.
This rule means that if a text contains s1 then s2 . . . then sp then it will belong to the category Ci. Each rule is associated with its confidence, indicating to which extent the sequential pattern is characteristic to this category: end Rules are sorted depending on their confidence and the size of the associated sequence. When considering a new text to be classified, a simple categorization policy is applied: the K rules having the best confidence and being supported are applied. The text is then assigned to the class mainly obtained within the K rules. This method is the same as the majority voting in [START_REF] Baralis | Majority classification by means of association rules[END_REF]. If two categories obtain the same score, a random choice is carried out. This prevents the system from always choosing the same category. The SPaC classifier step (SPaC-C) is described in Algorithm 2.

EXPERIMENTS

Experiments are led on three databases. The two first ones are the well-known english databases 20 Newsgroups and Reuters [START_REF] Hettich | The UCI KDD Archive[END_REF]. The third database describes French texts (news) with 8239 texts divided into 28 categories.

Experiments compare our approach to the results obtained with CBA and SVM. Table 4, 5 and 6 detail these results.

Comparisons are based on the F β measure [START_REF] Sebastiani | Machine learning in automated text categorization[END_REF]. This measure allows us to combine recall and precision for a global evaluation. The F β measure is thus more relevant than accuracy since accuracy does not take into account the case when a text is not classified. This leads to consider that classifying no text (thus never making errors) would have an accuracy near 100%! Accuracy has been taken as the reference measure in [START_REF] Baralis | Majority classification by means of association rules[END_REF][START_REF] Liu | Integrating classification and association rule mining[END_REF]. However, for the reasons presented above, we argue that this is not as relevant as relying on recall and precision, as mentioned in [START_REF] Sebastiani | Machine learning in automated text categorization[END_REF] Definition 2. The F β measure is defined as follows:

F β = (β 2 + 1)πiρi β 2 πi + ρi
where ρ stands for the recall, and π stands for the precision. This measure is computed for each class Ci.

Definition 3. Precision and recall are defined as follows:

πi = T Pi T Pi + F Pi , ρi = T Pi T Pi + F Ni
where T Pi, F Pi, F Ni stand respectively for the number of texts in class well classified i (True Positive), the number of texts put in class Ci by error (False Positive), the number of texts put in a different class from i by error (False Negative).

Accuracy is defined as follows: Definition 4. Accuracy:

Accuracyi = T Pi + T Ni T Pi + T Ni + F Pi + F Ni
In order to evaluate the classifier for all classes, we consider Micro-averaging (µ) and Macro-averaging (M ) [START_REF] Sebastiani | Machine learning in automated text categorization[END_REF][START_REF] Yang | An evaluation of statistical approaches to text categorization[END_REF] defined as follows:

Definition 5. Macro-averaging and Micro-averaging

πM = |C| i=1 πi |C| , ρM = |C| i=1 ρi |C| πµ = |C| i=1 V P i |C| i=1 (V P i + F P i ) , ρµ = |C| i=1 V P i |C| i=1 (V P i + F N i )
Micro-averaging gives the same importance to each document, contrary to macro-averaging which computes the average class by class (puting thus more importance to small categories).

In this paper, we consider that recall and precision have the same importance. We have thus: β = 1 for the F β measure. Results for SVM are obtained using a linear kernel (no better result is provided by a more complex kernel) with SVMLight [START_REF] Joachims | Text categorization with support vector machines: learning with many relevant features[END_REF]. The supports chosen here are the values providing the best results while remaining computable (too low supports lead to a very time-consuming application). SPaC is applied with K = 10 rules taken into account when classifying new documents.

SPaC

In this work, we argue that getting understandable knowledge is as important, and even more important than getting the most accurate classifier. For this reason, comparisons are essentially studied between CBA and SPaC. CBA is tested with the msCBA version of the algorithm, using the best results we have obtained when testing different supports. The results show that SPaC is always better than CBA when considering the macro-average. This is due to the fact that SPaC obtains good results for every category while CBA obtains very good results for the categories having many texts and low results for little categories. SPaC is thus more efficient when dealing with a difficult learning task. For this reason, SPaC is better than CBA and similar to SVM for French texts, and is better than SVM when dealing with the 20 Newsgroups database. In Fig. 1 we compare the results obtained for the F 1 measure regarding the number of rules considered for the choice of the class. Experiments have shown that K = 10 provides good results (similar to [START_REF] Baralis | Majority classification by means of association rules[END_REF] where best results correspond to maxrules = 9). Experiments are done (1) with order: the sequential pattern (sequence) is supported by the text (2) without order: the sub-sequences are supported by the text in an unspecified order. The corresponding sentences are in the document but they are not ordered as in the sequential pattern. We can notice that results are always better when order is taken into account.

SPaC

CONCLUSION AND FURTHER WORK

In this paper, we address the problem of text categorization using sequential patterns. In our framework, texts are represented by T F -IDF vectors, and each category is associated with a set of sequential patterns. When classifying new data, a text is matched to a category depending on the number of sequential patterns holding. The corresponding category is determined using a majority voting. Even if SVM have proven to be efficient in such a task, we argue that it is very important to provide users with understandable knowledge about their data. In this framework, sequential patterns are well-adapted. They provide rules that are used for the classification. We show that this approach is efficient and relevant, in particular when SVM do not perform well.

Moreover, the method we propose is simple and adaptable to drifting concepts since it is possible to update sequential patterns without performing the whole process using incremental sequential patterns mining [START_REF] Masseglia | Incremental mining of sequential patterns in large databases[END_REF]. This possibility is of high importance for text categorization, in particular for the automatic analysis of news which is a very fast variable area. This is thus a first step towards On-Line Classification Process (OLCP) using sequential patterns.

Future works include the integration of our approach on different foreign languages, in order to determine how order is important for each language. We are working on the automatic definition of the best number number of rules to take into account (the K parameter of our method). Our approach may also be enhanced by mining generalized sequential patterns [START_REF] Agrawal | Mining sequential patterns[END_REF]. This framework allows to integrate time constraints as shown in [START_REF] Masseglia | Pre-processing Time Constraints for Efficiently Mining Generalized Sequential Patterns[END_REF]. Finally, we aim at integrating muti-level sequential patterns, as proposed in [START_REF] Baralis | On support thresholds in associative classification[END_REF]. This allows indeed to keep very specific rules without damaging the classifier performances. In this framework, we argue that it is interesting to build a very compact set of rules (rules where the left-hand part is as short as possible). For this purpose, we aim at extending works on δ-free sets [START_REF] Cremilleux | Simplest rules characterizing classes generated by delta-free sets[END_REF] to sequential patterns.

Figure 1 :

 1 Figure 1: SPaC: F 1 in function of the number of rules considered

Table 1 :

 1 Database of Purchases

		Date	Items
	Peter	04/01/12	TV (1)
	Martin 04/02/28	Chocolate(5)
	Peter	04/03/02 DVD Player (2) , Camera (3)
	Peter	04/03/12	Printer (4)
	Peter	04/04/26	Chocolate (5)

Table 2 :

 2 Application of the Sequential Pattern Terminology to Textual Data

	Usual Databases		Textual Databases
	client	↔	text
	item	↔	word
	items/transaction	↔	sentence (set of words)
	date	↔	position of the sentence
	In the sequel of the paper, we use the notations introduced
	in Table 3. Experiments have led us to consider a threshold
	that eliminates about 5 to 10% (according to the Zipf law
	[27]) of the words.
	In SPaC, the sequential patterns are extracted using a mul-
	tiple minimum support strategy as done in msCBA. This
	means that a different support is applied for each category
	Ci. Contrary to msCBA, the minimum supports are defined

w).p(Ci|w).log(p(Ci|w))

+((1 -p(w)).p(Ci| w).log(p(Ci| w)))

Algorithm 1 :

 1 SPaC rules generation Data : TT rain : the training Set {minSupC i }: the set of minimum supports for each category Ci Result: SP: the set of sequential patterns begin SEQ ← ∅; cust ← 0; date ← 0; foreach Category Ci ∈ C do foreach Text Tj ∈ T C i do foreach Sentence S k ∈ Tj do

	Vs=TFIDF( Stem(S k )); // Compute the
	T F -IDF vector of the sentence
	for (s = 0; s <| Vs |; s + +) do
	if Vs[s] > 0 then
	SEQ[Ci][cust][date].additem(s);
	date++;
	date ← 0; cust + +;
	cust ← 0;
	foreach Category Ci ∈ C do
	SP[Ci]=SPMining(SEQ[Ci],minSupC i );

  Ci matching<s 1 ...sp> #texts matching<s 1 ...sp> Algorithm 2: SPaC-C Method Data : TT est: the test Set KFS: the K First Satisfied Parameters SP: the sequential Pattern Table from SPaC-RG begin nb ← 1 ; foreach Category Ci ∈ C do foreach spj ∈ SP[Ci] do RuleSP [nb] ← (spj, Ci, conf (spj → Ci) ; nb + + ; Sort RuleSP by rule confidence and size of sequence ; nf s ← 0; classable ← 0; foreach Text T k ∈ TT est do foreach rule (spj → Ci)∈ RuleSP do if T k supports SPj then T k .score[Ci]++; classable ← 1; nf s++ ; if nf s ≥ KF S then break if classable then Set T k to the Most Valued Category;

1

In French, blé means wheat and maïs stands for corn conf (γ) = #texts from classable ← 0; nfs ← 0 ;

Table 4 :

 4 Comparison of results by SPaC, msCBA and SVM on French news. Training Set = 33%

			msCBA	SVM
	F 1 M	0.444	0.367	0.485
	F 1 µ	0.487	0.401	0.486
	Accuracy	0.962	0.956	0.969
	Parameters multisupp. multisupp. = 0.7% C = 1

Table 5 :

 5 Comparison of results by SPaC, msCBA and SVM on English texts (REUTERS)

	msCBA	SVM

Table 6 :

 6 Comparison of results by SPaC, msCBA and SVM on English texts (20 Newsgroups). Training Set = 33%