Thilo F Morgeneyer
email: thilo.morgeneyer@ensmp.fr

Thibault Taillandier-Thomas

Ante Buljac

Lukas Helfen

François Hild

On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3)

Keywords: A. Fracture mechanisms, A. Voids and inclusions, B. Metallic material, C. Finite elements, C. Mechanical testing

Strain and damage interactions during tearing of a ductile Al-alloy with high work hardening are assessed in situ and in 3D combining two recently developed experimental techniques, namely, synchrotron laminography and digital volume correlation. Digital volume correlation consists of registering 3D laminography images. Via simultaneous assessments of 3D strain and damage at a distance of 1-mm ahead of a notch root of a thin Compact Tension-like specimen, it is found that parallel crossing slant strained bands are active from the beginning of loading in a region where the crack will be slanted. These bands have an intermittent activity but are stable in space. Even at late stages of deformation strained bands can stop their activity highlighting the importance of plasticity on the failure process rather than damage softening. One void is followed over the loading history and seen to grow and orient along the slant strained band at very late stages of deformation. Void growth and strain are quantified. Gurson-Tvergaard-Needleman-type simulations using damage nucleation for shear, which is based on the Lode parameter, are performed and capture

Introduction

The understanding and prediction of strain and damage interactions in ductile alloys remain a challenging topic due to the development of novel alloys and joining techniques, e.g., for transport applications [START_REF] Jolu | Microstructural characterization of internal welding defects and their effect on the plastic behavior of FSW joints of AA2198 Al-Cu-Li alloy[END_REF][START_REF] Jolu | Fatigue lifetime and tearing resistance of AA2198 alloy friction stir welds: Effect of defects[END_REF]. This understanding is important for micromechanics-based development of new materials and the assessment of structural integrity.

The prediction of failure at various stress states still remains a challenge [START_REF] Papasidero | Determination of the Effect of Stress State on the Onset of Ductile Fracture Through Tension-Torsion Experiments[END_REF].

In particular, strain localization phenomena are important issues as they may reduce the toughness of structures [START_REF] Rice | The Localization of Plastic Deformations[END_REF]. Their prediction is difficult as the roles of plasticity and damage softening on localization need to be clarified [START_REF] Tekoglu | On localization and void coalescence as a precursor to ductile fracture[END_REF]. Such strain localization phenomena in metals have been studied theoretically by Needleman and Tvergaard [START_REF] Needleman | Analyses of Plastic Flow Localization in Metals[END_REF] who found that they are favored by material behaviors that form a sharp vertex on the yield surface, by material damage and by heating due to plastic deformation leading to thermal softening at high strain rates. The effect of softening in bands due to strain rate sensitivity has also been investigated [START_REF] Needleman | The Effect of Rate Dependence on Localization of Deformation and Failure in Softening Solids[END_REF]. Huang and Hutchinson assessed numerically the effect of voids and their distribution on strain localization in a band. Fleck et al. [START_REF] Fleck | A phenomelogical theory for strain gradient effects in plasticity[END_REF] analyzed theoretically the effect of nucleation and growth of voids in a shear band and showed that damage softening favors localization.

Another poorly understood phenomenon linked to localization is slant fracture during ductile tearing [START_REF] Mahgoub | Three dimensional stress and deformation fields around flat and slant cracks under remote mode I loading conditions[END_REF]. Slant fracture is hard to reproduce numerically [START_REF] Besson | Continuum models of ductile fracture: A review[END_REF]. This could be linked to numerical problems due to mesh dependence of local models [START_REF] Besson | Modeling of crack growth in round bars and plane strain specimens[END_REF]. Regularized models may overcome this issue but the reg-ularization length then needs to be identified in an appropriate way. Slant fracture has successfully been simulated using shear void nucleation based on the Lode parameter [START_REF] Morgeneyer | Flat to slant ductile fracture transition: Tomography examination and simulations using shear-controlled void nucleation[END_REF] that actually resembles the shear modification of the Gurson model [START_REF] Nahshon | Modification of the gurson model for shear failure[END_REF]. In Ref. [START_REF] Besson | Modeling flat to slant fracture transition using the computational cell methodology[END_REF] it has been shown via a computational cell simulation that the dissipated energy reaches a minimum when the crack is slanted.

Experimentally, slant fracture is widely observed in thin walled structures [START_REF] Pardoen | Mode I fracture of sheet metal[END_REF] and can also lead to flip-flopping phenomena [START_REF] El-Naaman | Observations on mode I ductile tearing in sheet metals[END_REF][START_REF] Simonsen | Experimental and numerical modelling of ductile crack propagation in large-scale shell structures[END_REF] that are neither understood nor reproduced numerically. Shear banding has also been attributed to Portevin-Le-Chatelier effect that leads to equally spaced rough bands on the surface of a Kahn tear test specimen made of Al-Li alloy [START_REF] Delafosse | Dynamic strain-aging and crack propagation in the 2091 Al-Li alloy[END_REF]. In Ref. [START_REF] Rousselier | Combining porous plasticity with Coulomb and Portevin-Le Chatelier models for ductile fracture analyses[END_REF] the Coulomb fracture model and the Portevin-Le-Chatelier (PLC) model or dynamic strain aging (DSA) are formulated at the slip system scale and slant fracture could be reproduced.

In ductile damage modeling, where nucleation, growth and coalescence of voids is generally taken into account, efforts have been made to account for void shape effects. In particular, the changes of ellipsoidal voids have been accounted for in homogenization models [START_REF] Cao | A model for ductile damage prediction at low stress triaxialities incorporating void shape change and void rotation[END_REF]. It has been shown numerically that void reorientation can increase the susceptibility of the material to shear localization for triaxialities ranging between approximately 0.3 and 0.8 [START_REF] Song | The evolution of pore shape and orientation in plastically deforming metals: Implications for macroscopic response and shear localization[END_REF]. Void coalescence in combined tension and shear fields has been studied numerically in Ref. [START_REF] Torki | On Void Coalescence Under Combined Tension and Shear[END_REF]. However, these models need to be validated via in situ experimental observations. With the recent developments in 3D imaging techniques such as synchrotron tomography it has become possible to follow and quantify damage processes in situ at micrometer resolutions [START_REF] Maire | Quantitative X-ray tomography[END_REF]. Even nondestructive observations at resolutions of tens of nanometers are now possible [START_REF] Morgeneyer | Nanovoid morphology and distribution in deformed HDPE studied by magnified synchrotron radiation holotomography[END_REF][START_REF] Nizery | Three dimensional characterization of intermetallic particles in highstrength aluminium alloys using synchrotron X-ray tomography[END_REF]. In situ tomography is limited to axisymmetric (or stick-like) specimens with diameters of the order of 1 mm, which, in practice, may be relaxed by a factor of 3 to 5 if local tomography can be used [START_REF] Laiarinandrasana | Effect of multiaxial stress state on morphology and spatial distribution of voids in deformed semi-crystalline polymer assessed by X-ray tomography[END_REF]. In contrast, regions of interest in flat specimens can be imaged via synchrotron laminography [START_REF] Helfen | High-resolution three-dimensional imaging of flat objects by synchrotron-radiation computed laminography[END_REF][START_REF] Helfen | On the implementation of computed laminography using synchrotron radiation[END_REF]. This technique has been applied to assess the underlying physical mechanisms of ductile crack initiation and propagation [START_REF] Shen | Three-dimensional quantitative in-situ study of crack initiation and propagation in AA6061 Al-alloy sheets via synchrotron laminography and finiteelement simulations[END_REF][START_REF] Morgeneyer | Ductile crack initiation and propagation assessed via in situ synchrotron radiation computed laminography[END_REF].

To evaluate 3D displacement fields in the bulk of imaged samples, two techniques are used, namely, feature tracking and volume correlation. The tracking technique consists of following the motion of particles or other features. It is performed either manually [START_REF] Toda | Assessment of the fatigue crack closure phenomenon in damage-tolerant aluminium alloy by in-situ high-resolution synchrotron X-ray microtomography[END_REF] or automatically [START_REF] Toda | A 3D measurement procedure for internal local crack driving forces via synchrotron X-ray microtomography[END_REF][START_REF] Kobayashi | High-density three-dimensional mapping of internal strain by tracking microstructural features[END_REF]. Digital volume correlation (DVC) provides either a cloud of 3D displacements for local-based registrations [START_REF] Bay | Digital volume correlation: threedimensional strain mapping using X-ray tomography[END_REF][START_REF] Bornert | Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l'analyse des matériaux et des structures[END_REF][START_REF] Verhulp | A three-dimensional digital image correlation technique for strain measurements in microstructures[END_REF] or spatially continuous and dense displacement fields for global registrations [START_REF] Roux | Three dimensional image correlation from X-Ray computed tomography of solid foam[END_REF][START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF][START_REF] Réthoré | Digital volume correlation analyses of synchrotron tomographic images[END_REF]. For both techniques, additional markers have sometimes to be added to enable the tracking or registration procedure to be performed [START_REF] Nielsen | Measurements of plastic displacement gradient components in three dimensions using marker particles and synchrotron X-ray absorption microtomography[END_REF][START_REF] Bornert | Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l'analyse des matériaux et des structures[END_REF][START_REF] Haldrup | A General Methodology for Full-Field Plastic Strain Measurements Using X-ray Absorption Tomography and Internal Markers[END_REF]. However, these additional markers may alter the deformation and degradation mechanisms. Even though the volume fraction of secondary features is very low in the alloy investigated in this study (i.e., less than 1 % volume fraction) it was decided not to add any additional marker since they may become damage nucleation sites. In a feasibility analysis, it has been proven that global DVC could be used to measure 3D displacement fields even under such extreme conditions [START_REF] Morgeneyer | 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: An initial feasibility study[END_REF].

For an Al-Cu-Li alloy with low work hardening it has been shown with these techniques that slant strained bands precede the onset of damage and the slant crack [START_REF] Morgeneyer | In situ 3D observation of early strain localisation during failure of thin Al alloy (2198) sheet[END_REF]. This region of the sample has been shown to be in a plane strain state with respect to the crack propagation direction [START_REF] Buljac | Slant strained band development during flat to slant crack transition in AA 2198 T8 sheet: in situ 3D measurements[END_REF]. It has also been shown that the ratio between the strain in the band and outside of the band was of the order of 2 [START_REF] Morgeneyer | In situ 3D observation of early strain localisation during failure of thin Al alloy (2198) sheet[END_REF]. In other words the material outside the bands also deforms during every load step, except for the very last ones, but to a lesser extend than the material within the band. For the last load step deformation only occurs within the band. If the term "localization" refers to a state where plastic yielding takes place within the band and the outer region is plastically inactive, localization has only been measured at the very latest step. However, strain concentrations in a slant strained band were observed very early on and the scene for localization and fracture was already set. It is of interest for the understanding of strain and damage interactions if this way to deform is a general mechanism for these types of materials or only found in some particular cases.

In this paper the interactions between strain, porosity and the final crack are assessed via DVC for laminography data already reported in Ref. [START_REF] Morgeneyer | Ductile crack initiation and propagation assessed via in situ synchrotron radiation computed laminography[END_REF], namely, for a ductile aluminum alloy containing pre-existing voids (AA2139 T3). After introducing the material and the experimental approach, the strain fields and their corresponding microstructure are investigated in a region 1-mm ahead of the notch root. Incremental and cumulated strain fields will be given. The response of one void in such a strain field is then assessed. The findings are compared with a Gurson-Tvergaard-Needleman-type model simulation using shear-based void nucleation in terms of strain fields. The strain distribution will be further discussed using 2D plane strain simulations that account for soft bands or randomly distributed soft zones.

Experimental configuration

Studied Material

The investigated material is an Al-Cu-Mg alloy (AA2139) in a naturally aged (i.e., T3) condition. The material processing directions are the rolling direction (L), the transverse direction (T) and the short-transverse direction (S) in the through thickness direction. Figure 1(a) shows the grain size (mean value: 60 µm) in the L direction, in the T (mean value: 52 µm ) and S (mean value: 24 µm) directions. These data have been obtained from optical microscopy on etched sections using a mean linear intercept method [START_REF] Morgeneyer | Evolution of voids during ductile crack propagation in an Al-alloy sheet toughness test studied by microcomputed tomography[END_REF]. Figure 1 The spacing is in the order of 25 µm [START_REF] Morgeneyer | Experimental and numerical analysis of toughness anisotropy in AA2139 Al-alloy sheet[END_REF], which is an average upper bound.

The stress/strain curves in the L and T directions are shown in Figure 2 up to the ultimate tensile strength showing strong work hardening and high (b) 2D sections of reconstructed 3D synchrotron tomography data (after Ref. [START_REF] Morgeneyer | Evolution of voids during ductile crack propagation in an Al-alloy sheet toughness test studied by microcomputed tomography[END_REF]) ductility (see Ref. [START_REF] Morgeneyer | Experimental and numerical analysis of toughness anisotropy in AA2139 Al-alloy sheet[END_REF] for mechanical properties and chemical composition).

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The material has a moderate texture and very limited anisotropy measured in stress-strain curves.

Laminography

Laminography imaging was performed on the instrument installed by the Karlsruhe Institute of Technology at the imaging beamline ID19 [START_REF] Weitkamp | Parallel-beam imaging at the ESRF beamline ID19: current status and plans for the future[END_REF] of the Euro-120 pean Synchrotron Radiation Facility (ESRF, Grenoble, France) in the conditions described by Morgeneyer et al. [START_REF] Morgeneyer | Ductile crack initiation and propagation assessed via in situ synchrotron radiation computed laminography[END_REF][START_REF] Morgeneyer | 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: An initial feasibility study[END_REF]. The main advantage of laminography compared to tomography is that large flat specimens can be used. Synchrotron laminography benefits from various contrast modes including propagation-based phase contrast used herein in addition to the conventional absorption employed in laboratory X-ray imaging. For large source-specimen distances, large sample environments (e.g., mechanical testing devices) can be employed for highresolution imaging without risk of collision during the scanning motion.

Similar to tomography, a series of digital radiographs is acquired while rotating the sample about the laminographic axis [START_REF] Weitkamp | Parallel-beam imaging at the ESRF beamline ID19: current status and plans for the future[END_REF]. The latter is inclined with respect to the transmitted beam at an angle 0 • < θ < 90 • , see Figure 3. With the flat specimen oriented approximately perpendicular to the rotation axis, the average beam transmission can be optimized during the entire laminographic scan. Using filtered back-projection algorithms [START_REF] Myagotin | Efficient volume reconstruction for parallel-beam computed laminography by filtered backprojection on multi-core clusters[END_REF] different planes or complete 3D images with down to sub-micrometer resolution zoomed onto the large specimen can be reconstructed by a single scanning motion. An inclination angle of the rotation axis of about 25 • with respect to the beam normal was chosen as well as a monochromatic beam of 25 keV X-ray energy. Volumes were reconstructed from 1,500 angularly equidistant radiographs using a microscope-based indirect detector system with ×20 magnification [START_REF] Douissard | A versatile indirect detector design for hard X-ray microimaging[END_REF] and an ESRF Frelon camera. The exposure time of each projection was 250 ms.

The scanned region is ≈ 1 mm 3 in volume with a voxel size of 0.7 µm. This is where the flat-to-slant transition is typically observed [START_REF] Ueda | In-situ laminography study of threedimensional individual void shape evolution at crack initiation and comparison with GTN-type simulations[END_REF]. The minimum specimen to detector distance is 70 mm leading to relatively strong edge enhancement due to phase contrast. The resulting strongly contrasted edges facilitate the detection of voids and damage. For easier data handling, all 3D images consisting of 32-bit floating point values were converted into 8-bit gray level 3D images using the same linear dependence. As the same affine transformation is performed after reconstruction, the gray level distributions between the images are comparable. The final reconstructed volumes have a size of 2040 × 2040 × 2040 voxels.

However, only the voxels where all projections contribute to the reconstruction exhibit all the directional information available, show similar noise statistics and therefore have been considered herein.

In situ mechanical experiment

To investigate strain and damage interactions in situ, a loading device was adapted to the environment given by the laminography scanning geometry (Figure 4(a)) [START_REF] Morgeneyer | Ductile crack initiation and propagation assessed via in situ synchrotron radiation computed laminography[END_REF]. In particular, the device was designed to present a low profile perpendicular to the sample plane, thereby avoiding obstruction of the beam during rotation. The screws (see Figure 4(b)) allow a stepwise crack mouth opening displacement (CMOD) to be prescribed [START_REF] Morgeneyer | Ductile crack initiation and propagation assessed via in situ synchrotron radiation computed laminography[END_REF]. Before each "load step," the region ahead of the notch was scanned by laminography. A load step consists of turning the 2 screws of the loading device leading to incremental openings of the notch, i.e., a stepwise monotonic and quasi-static loading was applied.

The specimen (with dimensions 60 × 70 × 1 mm) contains a round notch with a radius of R = 0.17 mm and a 24-mm ligament (see Figure 4(b)). The loading configuration is T-L. Being close to standard tearing test geometries, this specimen provides relevant engineering conditions and allows for the development of plastic zones sizes as found in standard tests. The region ahead of the notch is loaded in tension and bending. A simple anti-buckling device (i.e., flat plates) is used to limit out-of-plane displacements.

All the mechanical parameters are characterized as functions of the nominal CMOD obtained by the opening of the loading device in this study. The definitions of CMOD, crack length a are shown in Figure 4(b,d). The crack length is measured in the center of the 3D image/specimen and may, due to

3D kinematic measurements

The acquired scans can be used to measure 3D displacements fields by registering one gray level volume in the reference configuration f and a second volume g in the deformed configuration via DVC. Two different types of approaches can be followed. First local approaches consist of subdividing the region of interest (ROI) into a set of independent sub-volumes that are registered [START_REF] Bay | Digital volume correlation: threedimensional strain mapping using X-ray tomography[END_REF][START_REF] Smith | Digital volume correlation including rotational degrees of freedom during minimization[END_REF][START_REF] Bornert | Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l'analyse des matériaux et des structures[END_REF][START_REF] Verhulp | A three-dimensional digital image correlation technique for strain measurements in microstructures[END_REF].

An alternative approach is global so that the warping of the reference volume is analyzed globally and not locally [START_REF] Roux | Three dimensional image correlation from X-Ray computed tomography of solid foam[END_REF]. One of the advantages of the latter approaches is that finite element kinematics can be considered thereby allowing for regularized [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF][START_REF] Taillandier-Thomas | Localized strain field measurement on laminography data with mechanical regularization[END_REF] or integrated [START_REF] Bouterf | Digital volume correlation applied to X-ray tomography images from spherical indentation tests on lightweight gypsum[END_REF][START_REF] Hild | Toward 4d mechanical correlation[END_REF] approaches.

In the following C8-DVC [START_REF] Roux | Three dimensional image correlation from X-Ray computed tomography of solid foam[END_REF] will be considered. It consists of discretizing the volume of interest with 8-noded cubes (i.e., C8 elements) whose edge length is denoted by . Trilinear displacement interpolations are used within each considered element. The DVC procedure consists of minimizing the global correlation residuals

Φ c = ROI ϕ 2 (x) (1)
where

ϕ(x) = |f (x) -g(x + u(x))| (2)
with respect to the degrees of freedom associated with the parameterization of the displacement field u(x), where x denotes the location of all the voxels belonging to the ROI. The overall quality of the registration is measured by the global dimensionless residual

ρ c = 1 ∆f Φ c |ROI| (3)
where ∆f = max ROI f -min ROI f denotes the dynamic range of the gray levels of the ROI in the reference configuration.

When a series of volumes in the deformed configuration is analyzed, two different strategies can be followed. First, when the strain levels remain lower 190 than 10 %, the multiscale implementation [START_REF] Roux | Three dimensional image correlation from X-Ray computed tomography of solid foam[END_REF] usually allows the whole series to be analyzed with f being the volume of the first scan [START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF]. Second, when larger strains occur, a first step of the analysis consists of evaluating the incremental displacement fields for which the deformed volume of the previous calculation becomes the reference volume of the next one. Once this first series of calcula-

195
tions is performed, it is possible to cumulate the displacements in a Lagrangian way.

For example, let us denote u 01 (x 0 ) the displacement field of scan 1 with respect to scan 0 expressed in the reference frame of scan 0. An incremental calculation between scan 1 and 2 will yield the displacement field u 12 (x 1 ) expressed in the frame of scan 1. To be cumulated, the incremental displacement field u 12 has to be evaluated for the location of x 0 in the deformed configuration x1 (x 0 ) = x 0 + u 01 (x 0 ). Knowing the measured displacement field u 12 (x 1 ), it can be interpolated to be evaluated at x1 by using the shape functions of the C8 discretization. Let û12 denote this interpolated displacement field. The cumulated displacement field then reads

u 02 (x 0 ) = u 01 (x 0 ) + û12 (x 0 ) (4)
with

û12 (x 0 ) = u 12 (x 0 + u 01 (x 0 )) (5)
In general, some of the nodal positions may require an extrapolation, namely, they lie outside the ROI in the reference configuration. In that case, among many options, two are implemented. If the aim is to stop the analysis at this stage, no extrapolation is required, they are considered as NaN (i.e., not a number) and they are not visualized. Direct calculations may also be performed in which this first evaluation of the cumulated displacement field corresponds to the initial guess of a direct calculation from scan 0 to 2 [START_REF] Morgeneyer | In situ 3D observation of early strain localisation during failure of thin Al alloy (2198) sheet[END_REF]. In that case, the previous option cannot be used and extrapolated values should be assessed.

The extrapolated value corresponds to the displacement of the closest node for which a measured estimate exists.

In the present DVC analyses, a volume of interest has been extracted from the scanned volume. Within this volume an initial ROI of 864×608×992 voxels (i.e., ≈ 600 µm ×425 µm ×700 µm) has been considered for correlation purposes and has been discretized with 32-voxel C8 elements. With such a discretization, ≈ 54, 000 kinematic degrees of freedom are measured. The choice of the element size in any DVC calculation results from a compromise between measurement uncertainty and spatial resolution [START_REF] Leclerc | Digital volume correlation: What are the limits to the spatial resolution?[END_REF][START_REF] Morgeneyer | 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: An initial feasibility study[END_REF] (here twice the element size, i.e., ≈ 45 µm). To evaluate the resolution of the measurement technique, two scans of the undeformed configuration were reconstructed when a small motion was prescribed between the two acquisitions. For 32-voxel (≈ 22 µm) elements, the standard displacement resolution was found to be of the order of 0.2 voxel, and the corresponding standard equivalent strain resolution equal to 0.5 %.

These values are the largest levels below which the measurement cannot be distinguished from noise. From the loading state CMOD = 2.313 mm onward, the ROI was shifted away from the notch by 280 µm to avoid the damaged region and reach correlation convergence. In addition, a bigger ROI has been used (i.e., 1248 × 512 × 992 voxels) to be able to follow two voids in the microstructure in spite of large strain levels.

Based upon these results, the convergence criterion of the correlation calculations, which corresponds to the mean displacement increment between two consecutive iterations, is chosen to be equal to 10 -4 voxel. It was checked that the mean correlation residual has reached a steady state for this criterion. The results are also checked by computing the correlation residuals voxel-wise, and their average value (i.e., Φ c or equivalently ρ c ). The mean and local residual levels are similar to those found by considering two subsequent acquisitions in which no deformation of the sample has occurred (i.e., for the resolution analysis). Consequently, the reported results are deemed trustworthy.

The correlation analyses reported herein consist of estimating the total displacement field for each analyzed step with respect to the reference configuration up to loading step for which the CMOD is equal to 1.625 mm. From these Lagrangian measurements, the incremental displacement fields are directly given by the total displacement difference corresponding to two consecutive steps. The Green-Lagrange strain fields are estimated either with the total displacement fields, or with the incremental displacement fields. In both cases, the mean values per C8 element are reported. For the displacement field between the initial state and any load step for which the CMOD is greater than 1.5 mm the direct correlation could not yield a converged solution as strains are very high and damage nucleation started to set in. For this reason incremental correlations taking load state CMOD = 1.5 mm as a reference have been carried out. This procedure was also performed for the subsequent load steps. This representation has been used for different load steps and incremental correlations in Figure 6. The underlying reference microstructure (at step n) is

Strain and damage interaction

shown with an overlay of the incremental von Mises equivalent strain field for the correlation between steps n and n + 1.

Figure 6(a) shows the strain field for a load increment starting from the unloaded state to load step CMOD = 1.5 mm. This is the largest direct correlation step that can be performed. Several intermediate steps were performed during the in situ experiment that will be shown in the sequel. When CMOD = 1.5 mm, there is no crack that has initiated from the initial notch. The von Mises equivalent strain field shows a heterogeneous distribution with an average level ≈ 10 %.

Several parallel slant bands are observed at this relatively early stage of loading.

Three bands oriented from the lower left part of the section to the upper right in the bands is about twice as high (i.e., 13 %) as in the material between the bands. This ratio has already been found for an aluminum alloy with less work hardening and initial porosity [START_REF] Morgeneyer | In situ 3D observation of early strain localisation during failure of thin Al alloy (2198) sheet[END_REF]. However, the latter material mainly showed one localized band and not several parallel bands as seen here. In addition, the slant strained band was less diffuse than in the present case. On this 2D section two particular voids of the underlying microstructure are encircled. They are also highlighted for the following load increments. This representation allows the spatial location of the strained bands to be assessed with respect to their underlying microstructure and if these bands "stick" to the microstructure.

In Figure 6(b) an overlay of the incremental strain field between CMOD = 1.5 mm and CMOD = 2.0 mm is shown over the underlying microstructure of the load It is still located above the two encircled voids. Normal to this band two other parallel strained bands are active. The strain field is now substantially more localized than in Figure 6(c). A very similar strain pattern is seen for the next load step from CMOD = 2.375 mm to CMOD = 2.438 mm (Figure 6(e)). There is still very little damage to be seen in the microstructure at this location and, if any, it is rather diffuse and also located in regions that do not belong to any strained band.

In Figure 6(f) the same strained band from the lower left to the upper right corner of the image is mainly active. Only one out of the two bands normal to this one, i.e., the upper band, is now active. The latter is particularly active in the lower part of the investigated region. It is noteworthy that at these late stages of loading there is still not only a single strained band active but a competition between several bands that have undergone tens of percents of strain.

The next load step from CMOD = 2.438 mm to CMOD = 2.5 mm is the last one for which a trustworthy correlation could be achieved (Figure 6(g)), namely, the residuals were still very close to those observed when two undeformed volumes are registered. It can be seen for the underlying microstructure that a notable damage evolution now sets in. In the region where the strained bands cross, voids grow. The upper one of the two encircled voids shows a reorientation that has already been reported [START_REF] Ueda | In-situ laminography study of threedimensional individual void shape evolution at crack initiation and comparison with GTN-type simulations[END_REF] for the same experiment. With the present work the corresponding strain fields for the void shape changes are known.

Figure 6(h) shows the microstructure of load step CMOD = 2.5 mm without the corresponding strain field. Voids in the region where the two strained bands cross grow further. This highlights that in this area the main assumption for trustworthy DVC, which is the conservation of gray levels is no longer satisfied as damage sets in. However, the measured strain concentration where the two strained bands cross is plausible, as, in addition, void growth is seen here that is surely a result of highly localized strain activity. The lower half of the strained bands is more active than the upper half. In Figure 6(i) the final crack path is observed (i.e., CMOD = 2.625 mm). A roof top crack is seen, which has not fractured the full section yet. The crack has followed the second slant strained band that is present from the very early loading onward. The two voids that had been encircled in the previous load increments now contribute to the final fracture surface.

In summary, several slant, crossing and parallel strained bands with inter-mittent activity were found. Failure occurred along one band that was already active since the very beginning of the loading history.

Cumulated fields

In this section some cumulated strain fields are assessed for very large load intervals from the unloaded state. This is only possible via numerically cumulating the incremental displacement fields as explained above. Due to the large strains direct correlations are no longer possible. Accumulation of the fields is only possible up to CMOD = 2.313 mm as after this step the 3D ROI has been shifted away from the notch to avoid damaged areas for correlation purposes.

For the cumulated displacement fields the equivalent strain fields are then derived and shown in Figure 7. In the lower parts of Figures 7(b-c) the information on the displacement is no longer available. This is due to the fact that the ROI size was kept constant for all correlations and that the material elongated along the loading direction.

The striking finding in Figure 7(b) compared to Figure 7(a) is that the strained bands do not move in space. The band in the upper left corner is the most active, which is consistent with the fact that the strain field shown in Figure 6(b) was "added." The maximum equivalent strain in this slice is now increased to ≈ 0.24. In Figure 7(c) a cumulated field is shown corresponding to strains between load step CMOD = 0.0 mm and CMOD = 2.313 mm. The equivalent strained band pattern is still similar to the previous ones but the maximum magnitude has now increased to ≈ 0.33. The bands do not move in space and, as a consequence, do not even out the strain distribution but keep a concentrated pattern made up of several bands. This is surprising as the regions between the strained bands could be expected to be softer than the bands and easier to deform if a uniform material behavior is assumed. Local variability in stress/strain behavior could be at the origin of such behavior. Damage evolution is not striking until this loading state and thus should not be of prior importance to softening.

The strain profile along a line perpendicular to the strained bands is shown in Figure 8. This graph confirms the spatial stability of the strained bands.

The ratio of the equivalent strain in the main bands over the strain outside the bands lies between 1.5 and 2. For the band that will lead to the final failure 385 this ratio is somewhat lower. This result is comparable to the strain ratio that was reported for AA 2198 T8 [START_REF] Morgeneyer | In situ 3D observation of early strain localisation during failure of thin Al alloy (2198) sheet[END_REF]. The main difference is that several bands are active in the present case.

Small load increments from the initial state

In this section strain fields for small steps are reported when DVC is per-390 formed from the unloaded step (CMOD = 0.0 mm) to the load step before crack initiation (i.e., CMOD = 1.5 mm). The incremental strain fields in this load interval from the unloaded state are also given. The latter investigation is made in order to interpret the incremental strain activity (∆p eq ) as could also be done in an FE simulation that accounts for instabilities.

Figure 9(a) shows the strain field obtained for the first load step carried out during the in situ test (i.e., CMOD = 0.5 mm). In this step a maximum equivalent strain of 0.03 is obtained. However, the three strained bands from the lower left corner of the image to the upper right corner can already be seen. These bands remain visible for a substantially larger load increment (Figure 6(a)).

Damage nucleation or growth is negligible at this step of loading. A similar result was also found for AA2198 T8 [START_REF] Morgeneyer | In situ 3D observation of early strain localisation during failure of thin Al alloy (2198) sheet[END_REF]. The strained bands found herein are less localized.

Figure 9(b) shows the strain field for a correlation from the initial unloaded state to the second load step (i.e., CMOD = 0.75 mm). The strain field has a similar pattern as the first one, but the strain magnitude increases up to 0.05.

For the subsequent load steps (Figures 9 (c-f)) the same observation holds. It should be noted that the last strain field (see Figure 9(f)) is the same as that shown in Figure 6(a). This result indicates that the characteristic strain pattern hardly evolves over the loading history, which is also consistent with the very large strain measurements shown in Figure 7. reported. The noisy nature of the field is due to a lower signal to noise ratio of the measurements (i.e., the maximum strain is about 0.02). The three strained bands can still be guessed. In the subsequent load step the upper strained bands seem more active than the lower ones. The next three incremental fields appear However, the strain activity is higher in the lower part of the field.

In Figure 11, incremental strain fields in the same loading interval (CMOD = 0.0 mm to CMOD = 1.5 mm) are also given. However, bigger load increments are assessed in the correlation (n to n + 2) to obtain a higher signal to noise ratio. In Figure 11(c) the largest strain activity is seen in the lower part of the 2D section. This activity may be consistent with Portevin Le Chatelier (PLC)-like behavior spreading the strain in an even manner. However, the characteristic strain pattern always remains, i.e., most of the strain activity takes place in the strained bands and not in the surrounding strain "valleys."

Void shape evolution in strained bands

In previous sections the existence of intermittent strained bands has been shown. In this section the interest lies on the evolution of a pre-existing void (i.e., hydrogen micropore [START_REF] Toda | Roles of Pre-Existing Hydrogen Micropores on Ductile Fracture[END_REF]) up to fracture in such a strained band. The void is located 45 µm farther away from the notch than the 2D section shown in the previous figures. It is situated at about mid-thickness of the specimen (see Figures 13(a-b) [START_REF] Ueda | In-situ laminography study of threedimensional individual void shape evolution at crack initiation and comparison with GTN-type simulations[END_REF]). This void was located in the final strained band that led to fracture as illustrated by the overlay of the strain field over the underlying microstructure (Figure 13(b)). It undergoes substantial growth only at late stages of deformation. From Figure 7(c) it is known that the corresponding equivalent strain is of the order of 0.30 in this region, but growth is moderate as seen in Figure 13(e).

In the subsequent load steps the void grows in the loading direction and turns toward the orientation of the strained band. During the next load steps, void growth mainly occurs along the strained band direction. This is consistent with a local mode I and III loading state [START_REF] Mahgoub | Three dimensional stress and deformation fields around flat and slant cracks under remote mode I loading conditions[END_REF] that lets the void grow in a mixture of tensile and simple shear fields [START_REF] Song | The evolution of pore shape and orientation in plastically deforming metals: Implications for macroscopic response and shear localization[END_REF]. This void reorientation has been shown numerically to facilitate localization phenomena. In the final stages smaller damage features become visible that are consistent with void nucleation and growth on a second population of particles at high levels of strain. However, very narrow crack regions are also seen in Figure 13(i), which are consistent with the void sheeting phenomenon. This void fully contributes to the final crack.

The void growth versus local equivalent von Mises strain has been further quantified in Figure 14. This figure confirms that despite the fact that the strain profiles are already heterogeneous from early loading stages onward no substantial void growth occurs until elevated levels of strain (more than 0.30 in the present case). This is also in agreement with results obtained by unit cell calculations [START_REF] Morgeneyer | Experimental and numerical analysis of toughness anisotropy in AA2139 Al-alloy sheet[END_REF]. They predict for a stress triaxiality of 1.25 a void volume (vv) of 3 times the initial volume at about 0.15-0.35 strain. However, void coalescence is predicted in unit cell calculations at higher void volume fraction level than in the experiment (about 12 times the initial vv) and also at higher strain (0.3-0.5). In the experimental data void growth acceleration occurs at 0.3 strain and for a vv that is 3 times higher than the initial one, i.e., substantially earlier than in the unit cell calculations. The complex strain field encountered herein, which may lead to mixed tensile and shear states, may be at the origin of this behavior [START_REF] Song | The evolution of pore shape and orientation in plastically deforming metals: Implications for macroscopic response and shear localization[END_REF]. It is worth noting that only one void has been studied. An investigation considering more voids would be useful to gain a better statistical representation of these processes.

Simulations

To interpret these experimental results, and in particular the slant strained band formation at early loading stages, simulations are carried out. In Ref. [START_REF] Ueda | In-situ laminography study of threedimensional individual void shape evolution at crack initiation and comparison with GTN-type simulations[END_REF] Gurson-Tvergaard-Needleman-type simulations consistent with the experimentally observed crack growth have been carried out for the present material and sample geometry. These simulations did not show inclined bands in the strain fields nor slant fracture. A shear-based void nucleation term is used [START_REF] Morgeneyer | Flat to slant ductile fracture transition: Tomography examination and simulations using shear-controlled void nucleation[END_REF], which had yielded slant fracture, to compare the initial measured and predicted strain fields and final fracture.

3D Gurson-Tvergaard-Needleman-type simulations

The modification for void nucleation under shear [START_REF] Morgeneyer | Flat to slant ductile fracture transition: Tomography examination and simulations using shear-controlled void nucleation[END_REF] is briefly recalled. The idea is to use the Lode parameter of the strain rate tensor to detect if the material deforms under a shear strain state and nucleate voids in that case.

The Lode parameter for the strain rate tensor assuming incompressibility reads

π = 3 ṗ2 ṗ1 -ṗ3 with ṗ1 ≥ ṗ2 ≥ ṗ3 (6)
where ṗi are the eigen values of the plastic strain rate tensor. Using the Lode parameter of the strain rate tensor, the shear-controlled nucleation rate is then expressed as

ḟn = A n ṗ (7)
with

A n = A shear 2 exp - π π0 2 when p ≥ p shear crit ( 8
)
where π0 is an adjustable parameter of the Gaussian function. For the rest of the Gurson-Tvergaard-Needleman-type model the reader is referred to Refs. [START_REF] Morgeneyer | Experimental and numerical analysis of toughness anisotropy in AA2139 Al-alloy sheet[END_REF][START_REF] Morgeneyer | Flat to slant ductile fracture transition: Tomography examination and simulations using shear-controlled void nucleation[END_REF].

The model parameters are given in Table 1. The results of the simulations with the new term for void nucleation are

shown in Figure 15. The strain fields are given in the same ROI as the measured fields. For some increments the strain magnitude is different from the experiment. It can be seen that the initial simulated strain fields do not resemble the measured ones. The simulated strain fields resemble those of von Mises-type plasticity simulations or classical Gurson-Tvergaard-Needleman-type simulation [START_REF] Morgeneyer | In situ 3D observation of early strain localisation during failure of thin Al alloy (2198) sheet[END_REF] with a characteristic circular strain pattern. This is expected as the shear nucleation terms only sets in after a critical strain of 0.10. Only at late stages do the strain fields begin to be slanted. In contrast to the experimental findings only one band is active and not several parallel strain bands with intermittent activity. The slant fracture and flat-to-slant crack transition are successfully reproduced in this simulation. The fact that the initial strain fields are not found by the simulation indicates that the plasticity model needs to be modified to enhance the predictions.

Effect of material heterogeneity 4.2.1. Soft slant bands

The measurements of the strain fields ahead of the notch show parallel strained bands that are stable in space but have intermittent activity. The ratio of the strains within the bands and outside is of the order of 1.5-2.0 (see ). To interpret further the existence of these bands and in particular the strain distribution 2D simulations are performed to account for material heterogeneity [START_REF] Hutchinson | Shear band formation in plane-strain[END_REF]. The simulations are carried out in plane strain conditions as it has been shown that crack propagation occurs under such conditions. Isotropic hardening is described by

R(p) = R 0 1 + K 0 p + K 1 1 -e -k1p + K 2 1 -e -k2p (9)
where the material parameters R 0 , K 0 , K 1 , k 1 , K 2 , k 2 are given in Table 1. The weakening is described by multiplying the hardening law with a perturbation factor 0 < x < 1

R(p) = (1 -x)R 0 1 + K 0 p + K 1 1 -e -k1p + K 2 1 -e -k2p (10)
The mesh with the soft zone and the line along which the strain profile is analyzed are given in Figure 16(a). Loading of the upper and lower ends of The strain profiles normal to the meshed soft band are reported in Figure 17.

The equivalent plastic strain increases in both regions (i.e., within and outside the band). The strain ratio is of the order of 2 for a perturbation x = 10 %. shows the change of the strain ratio for different perturbation levels. The stronger the perturbation the higher the initial strain ratio and the earlier the final increase of the strain ratio. In the experimental case the strain ratio is still of the order of 2 for a strain outside the band of ≈ 0.22. A soft perturbation of the order of 10 % reproduces rather well these experimental findings.

Randomly distributed soft zones

There is no evident reason for soft zones in the material to be distributed in slant bands. Consequently soft zones with a size of 50 µm×50 µm are randomly distributed in an additional simulation. These dimensions are close to the mean grain size of the studied aluminum alloy. The soft zones were randomly distributed in the central part of the meshed region. The soft phase fraction has been chosen to be the same as in the previously reported calculation with a soft slant band. The soft zone distribution is depicted in Figure 19(a). The resulting equivalent strain field is shown in Figure 19(b). Some development of slant bands occurs at 0.14 average equivalent strain, which is relatively early in comparison with the experimental evidence reported herein. The ratio between the bands and the surrounding material is less (i.e., ≈ 1.15) than for the simulation where all soft zones are located in a single slant band.

Discussion

Combining three dimensional in situ imaging of a flat specimen with DVC it is shown that slant strained bands precede the development of damage in the slant fracture region. These parallel slant bands have an intermittent activity.

The crossing bands initially form a "waffle"-like pattern due to strain ridges and valleys. The ratio of the strain within the bands and the surrounding material is of the order of 1.5 to 2. The strain partitioning is further investigated via plane strain simulations where a slant soft zone is modeled. A plane strain condition had been identified

for the crack propagation direction in the region where the crack is slanted [START_REF] Buljac | Slant strained band development during flat to slant crack transition in AA 2198 T8 sheet: in situ 3D measurements[END_REF].

For a perturbation of 10%, strain partitioning between the band and the surrounding material, and a strain level at localization are in agreement with the experimental findings. A softening effect is thus identified rather than shear localization where all strains would localize within the band.

The question that arises is what could be at the origin of these bands. The heterogeneous strain fields could be linked to heterogeneous microstructures and resulting plasticity [START_REF] Hutchinson | Shear band formation in plane-strain[END_REF]. Local material heterogeneity caused by crystallographic texture, which is referred to as roping, could also be a possibility [START_REF] Qin | Analysis of roping of aluminum sheet materials based on the meso-scale moving window approach[END_REF]. In that case these local textures are the result of ex-super grains that existed at some point during material processing. However, the characteristic lengths identified in Ref. [START_REF] Qin | Analysis of roping of aluminum sheet materials based on the meso-scale moving window approach[END_REF] are substantially larger than what is found herein, namely, more Using a mesh with randomly distributed soft zones to model soft grain distribution, slant bands are obtained at early loading stages under plane strain conditions. These soft zones may be a possible origin of the phenomena reported herein. However, the strain ratio within and outside the bands is not similar to the experimental findings. Using different surface fractions or softness, results might be obtained that could be closer to the experimental findings. However, the intermittent activity of the bands is not captured through these effects.

In the present models no ingredients to reproduce the intermittent strained band activity are used. Origins of the intermittent nature of the strained bands could be DSA (i.e., Portevin-Le-Chatelier effect [START_REF] Wang | Le-835 Delliou, Portevin-Le Chatelier (PLC) instabilities and slant fracture in C-Mn steel round tensile specimens[END_REF][START_REF] Rousselier | Combining porous plasticity with Coulomb and Portevin-Le Chatelier models for ductile fracture analyses[END_REF]). This observation is also supported by the fact that even at very late stages of deformation strained bands may become inactive and damage softening does not seem to influence these processes on the first order. It is, however, surprising that the experimentally measured strained bands are stable in space with respect to microstructural features. This is generally not found for moving PLC bands on smooth specimens as in these cases the strained bands move in time and even out the strain heterogeneities in the different regions. Here, however, the notch naturally concentrates the strain fields and may prevent the bands from moving. Similar strained bands have been found on Kahn tear test sample surfaces made of Al-Li alloy [START_REF] Delafosse | Dynamic strain-aging and crack propagation in the 2091 Al-Li alloy[END_REF] after tearing. Deformation patterns in slant crossing bands ahead of notches have also been suggested in early works on these phenomena [START_REF] Hahn | Local yielding and extension of a crack under plane stress[END_REF].

Further, it is experimentally shown that inside the strained bands voids change shape and rotate to align along the direction of the band. The question arises whether the shape evolution of pre-existing voids can lead to strain localization and damage softening. It is noteworthy that this evolution takes place only at the latest stages of deformation. It is thus more likely that this void growth is the result of previous, and thus governing, deformation and concentration in strained bands. It is certainly of interest to investigate the effect of void shape changes in such strained bands on the fracture resistance of the material. It is likely that there will be an additional localization effect due to void rotations [START_REF] Song | The evolution of pore shape and orientation in plastically deforming metals: Implications for macroscopic response and shear localization[END_REF].

The problem in modeling slant fracture remains the mesh dependence of the simulations [START_REF] Besson | Modeling of crack growth in round bars and plane strain specimens[END_REF]. The slant fracture found herein was reproduced by a shear void nucleation term based on the Lode parameter [START_REF] Morgeneyer | Flat to slant ductile fracture transition: Tomography examination and simulations using shear-controlled void nucleation[END_REF]. However, the mesh dependence is still an issue. In addition, several slant bands could not be reproduced. To create bands with intermittent activity a PLC-type model may be used [START_REF] Rousselier | Combining porous plasticity with Coulomb and Portevin-Le Chatelier models for ductile fracture analyses[END_REF] but mesh dependence and computation times are also problematic for such types of models.

Experiments on other types of Al alloys and specimen geometries should also be performed to confirm the presence of slant bands and PLC effects. Nonlocal models are needed to get more reliable predictions of localized phenomena but the regularization length is still to be physically understood and identified. The present finding could also add another length to the commonly used particle or void spacing, which is the strained band spacing.

Conclusions

Damage and strain interactions have been assessed in situ and within the material bulk during ductile tearing. The region of interest is located 1080 µm ahead of a notched 1-mm thick plate made of ductile aluminum alloy containing initial porosities (≈ 0.33 vol%) and particles (≈ 0.45 vol%). This analysis is made possible by the combination of in situ synchrotron laminography and digital volume correlation using the material microstructure contrast. The overlay of strain and microstructure maps has yielded the following results:

• Several parallel, crossing and slant strained bands are active from the very early stages of loading (i.e., max. 0.03 equivalent von Mises strain).

• The analysis of incremental strain fields shows that the activity of the bands is intermittent over the whole loading history.

• The spatial location of these bands compared to the microstructure is stable, i.e., the strain activity is not evening out the strain fields.

• Even very concentrated (strained) bands with high von Mises equivalent strain levels (i.e., 0.10) may become inactive and other bands become active at very late stages of loading.

• The typical strained band spacing is 150 -250 µm.

• The ratio of the equivalent strain within and outside the bands is about 1.5 to 2.

• In the local area of observation substantial void growth could not be detected at micrometer resolution before the occurrence of strained bands.

• The fact that strained bands may switch off at the end of the test indicates that the mechanisms at play for strain concentration may differ from damage softening.

• The observed slant strained bands at early loading stages cannot be reproduced using a Gurson-Tvergaard-Needleman-type model with Lode parameter-based shear void nucleation. The flat to slant crack transition is however successfully reproduced.

• 2D plane strain simulations accounting for a soft zone yield strain ratios within the band and the surrounding material similar to the measurements for a 10-% weakening. A softening effect is thus identified rather than shear localization where all strains would localize within the band.

Using randomly distributed soft zones, slant strained bands can be obtained. The ratio of the strain within and outside the bands is, however, substantially lower than in the experimental findings.

Last, investigations on other Al-alloys and specimen geometries would be useful to confirm the present results, which have also be found on a low work hardening alloy (i.e., AA2198 T8 [START_REF] Morgeneyer | In situ 3D observation of early strain localisation during failure of thin Al alloy (2198) sheet[END_REF]).

  (b) shows typical L-S and T-S sections of the material obtained by tomography. Intermetallic particles are seen in white and the aluminum matrix in gray. The intermetallic content has been determined to be 0.45 vol% and the initial void volume fraction 0.34 %[START_REF] Morgeneyer | Experimental and numerical analysis of toughness anisotropy in AA2139 Al-alloy sheet[END_REF]. The spacing of the particles and voids has been estimated using Feret dimensions of Voronoi cells around particles and pores.

Figure 1 :

 1 Figure 1: (a) Pseudo 3D cube showing the grain shapes (optical microscopy after etching).

Figure 2 :

 2 Figure 2: Engineering stress-strain curve for AA2139 T3 when loading in the L and T directions. The thickness variation is also given (after Ref. [46])

Figure 3 :

 3 Figure 3: Schematic drawing of the laminography setup (after Ref. [42])

Figure 4 :

 4 Figure 4: (a) Photograph of the specimen, loading device and anti-buckling frame mounted onto the laminography specimen holder; (b) drawing of the flat (1 mm) CT-like specimen and loading device; (c) measured crack length versus CMOD curve (every dot on the curve corresponds to one load increment and corresponding scan); (d) 2D section of reconstructed laminography data at mid thickness for load step CMOD = 2.125 mm

3. 1 .

 1 Incremental fieldsThe objective of this work is to investigate the interactions between damage and strain during ductile tearing and, in particular, during the flat-to-slant crack transition. The challenge is to show both values simultaneously. Figure 5 highlights the way in which this simultaneous assessment of strain and damage is performed. A volume of interest has been extracted (Figure 5(a)).Its outer face is located 1080 µm ahead of the notch root and normal to the crack propagation direction. Figure5(b)shows the 3D von Mises equivalent strain field in the ROI for a DVC analysis between the undeformed state (i.e., CMOD = 0.0 mm) and when CMOD = 1.5 mm. In Figure5(c) a new way to simultaneously show microstructure and strain fields is given. The strain field is made half transparent and it is laid over the underlying material microstructure as observed by laminography (i.e., particles and porosity). The 2D slices (strain and microstructure) have been obtained at the same location in the investigated 3D volume. The underlying microstructure section shows the microstructure before loading as this is the reference state for the performed correlation.

Figure 5 :

 5 Figure 5: (a) 3D rendering of the initial microstructure and the ROI. (b) 3D rendering of the equivalent strain field in the extracted volume when CMOD = 1.5 mm. (c) 2D overlay of strain field over the initial microstructure

Figure 6 :

 6 Figure 6: Overlay of incremental equivalent strain fields (half transparent) over the microstructure 1080 µm ahead of the notch root. (a)-(g) 2D sections of laminography data and equivalent strain fields. The microstructure is shown in state n for incremental correlations between n and n+i. Two reference voids are encircled in white on the microstructure for every load step; (h),(i) microstructure showing damage and the final crack path; (j) load steps corresponding to the shown sections

Figure 7 :

 7 Figure 7: 2D sections of cumulated equivalent von Mises strain fields. (a) Direct correlation from CMOD = 0.0 mm to CMOD = 1.5 mm. (b) One cumulation from CMOD = 0.0 mm to CMOD = 2.0 mm. (c) Two cumulations from CMOD = 0.0 mm to CMOD = 2.313 mm (the coordinate system is shown in its initial state)

Figure 8 :

 8 Figure 8: (a) 2D section of cumulated fields (equivalent von Mises strain) with indication of the location for the strain profile. (b) Strain profile for the direct and cumulated strain fields from CMOD = 0.0 mm to CMOD = 2.313 mm

Figure 9 :

 9 Figure 9: 2D section of 3D laminography and DVC data. Overlay of the smallest possible direct equivalent strain fields (half transparent) over the microstructure in its initial state

Figure 10 :

 10 Figure 10: 2D section of 3D laminography and DVC data. Overlay of the smallest possible incremental equivalent strain fields (half transparent) over the microstructure in its initial state

  420

Figure 11 (Figure 9

 119 Figure 11(a) corresponds to Figure 9(b). The three strained bands are visible.425

Figure 11 :

 11 Figure 11: 2D section of 3D laminography and DVC data. Overlay of incremental equivalent strain fields n to n + 2 (half transparent) over microstructure (the microstructure in its initial state)

Figure 12 :

 12 Figure 12: 2D section of 3D laminography and DVC data. Overlay of incremental equivalent strain fields n to n + 3 (half transparent) over the microstructure in the initial state

Figure 13 :

 13 Figure 13: 3D void visualization of reconstructed laminography data. (a) Through-thickness section with location of the void 1125 µm from the notch. (b) Overlay of strain field for step CMOD = 2.5 mm to CMOD = 2.563 mm over microstructure at CMOD = 2.5 mm. (c)-(j) Void growth. (k) Indication of the corresponding load step

Figure 14 :

 14 Figure 14: Void volume fraction for one void (see Figure 13) versus measured von Mises equivalent strain.

Figure 15 :

 15 Figure 15: Results of a 3D simulation of the CT-like specimen using a Gurson-Tvergaard-Needleman model combined with a shear void nucleation term. (a) Stress field in the loading direction. (b) Equivalent strain field in the plastic zone. (c)-(i) incremental strain fields in the ROI for load steps corresponding to Figure 6 1050 µm ahead from the notch root. (j) Final crack position. (k) Indication of the corresponding load step

Figure 8

 8 Figure 8). To interpret further the existence of these bands and in particular the

Figure 16 :

 16 Figure 16: 2D plane strain simulation using a slant soft band. (a) 2D mesh. (b) Equivalent plastic strain field when x = 10 %

  510

Figure 17 :Figure 18

 1718 Figure 17: Equivalent strain profiles normal to the band (see Figure 16(a)) for a 10-% perturbation

  Three-dimensional FE simulations using von Mises plasticity or a Gurson-Tvergaard-Needleman-type model do not capture the slant bands. With a standard Gurson-Tvergaard-Needleman-model the crack remains flat. The flat-toslant crack transition can be reproduced successfully using a Lode parameterbased void nucleation term in a Gurson-Tvergaard-Needleman-model. The early slant bands are, however, not reproduced.

Figure 18 :Figure 19 :

 1819 Figure 18: Strain ratio versus strain outside the band for different perturbation levels (2 % ≤ x ≤ 14 %)

Table 1 :

 1 Material parameters for the laws used herein

	Behavior
	Elastoplastic E (GPa) ν	R 0 (M P a)	K 0	K 1	k 1	K 2	k 2
		70	0.3	250	0.054 1.12 7.15	0.273	187
	Damage	q 1	q 2	f 0	f c	A 1 n	A shear n	p shear c	π0
		1.97	0.91 0.33 % avg. 4.5 % 0.05 2.0	0.1	0.033

Acknowledgements

The financial support by Fédération Francilienne de Mécanique and Agence Nationale de la Recherche (ANR-14-CE07-0034-02 grant for COMINSIDE project) are gratefully acknowledged. Constellium C-Tec is acknowledged for material supply. The authors would also like to acknowledge ESRF for providing beamtime (experiment MA1006), Dr. M. Mavrogordato and Prof. I. Sinclair for help with the experiment. They particularly thank Prof. J.W. Hutchinson for discussing strain partitioning.


Yosra Ben Slimen
email: yosra.benslimen@orange.com

Sylvain Allio
email: sylvain.allio@orange.com

Julien Jacques
email: julien.jacques@univ-lyon2.fr

Julien Jacques Model

Model-based co-clustering for functional data

Keywords: coclustering, données fonctionnelles, algorithme SEM-Gibbs model-based coclustering, functional data, SEM-Gibbs algorithm

come

Introduction

With the introduction of new technologies and services in mobile networks, the complexity of these latter have increasingly grown creating an heterogeneous environment where different architectures (micro-, macro-, pico-, femto-cells) and different radio access technologies (GSM, UMTS, LTE, . . .) coexist. In this context, mobile operators need to deal with the new challenges in order to provide a top quality of services without increasing costs.

The quality of services is measured by data captured from the network. These data are generated from different sources such as probes, robots and key performance indicators (KPI). They are used for different needs such as network maintenance, optimization, troubleshooting and management. They are also used for self-organizing networks e.g for self-healing and self-optimization tasks.

KPIs are measurements used to monitor quality of service perceived by the user and the network performance. The KPIs concern different network elements such as transceivers, cells, sites, . . . and they are defined by mathematical formulas derived from different counters, computed periodically from the network. Therefore, KPIs are continuous data that are computed with different temporal granularities (hourly or less, daily, weekly). Figure 1 provides an illustration of 20 daily evolutions of 30 KPIs, with a temporal granularity of 15 minutes. Some KPIs are common across the different radio access technologies while others are specific to each one of them. Even under the same technology, the counters/KPIs may differ from one constructor to another. Therefore, as the number of technologies, services, cell types, and constructors grows, the number of KPIs observed by the support team becomes enormous and network planning and operating becomes more complex. However, nowadays, the number of procedures, e.g related to optimization and fault management, that are manually carried out is still considerable. This has triggered a significant research effort, gathered under the term "automatic networking".

In this context, with this work, we are interested in KPIs and the study of their behaviors. We define a clustering of not only the different KPIs, but also of observations. The proposed model can help the technical support team of mobile operators and it can afford additional information to self-organizing networks by discovering new relationships between the different KPIs and new similarities between their behaviors in different days. In other terms, we define a model for co-clustering of both KPIs and days.

The rest of this paper is organized as follows. Section 2 presents the data representation and its pre-processing. Section 3 presents the latent block model for functional data that we propose and Section 4 details its inference. Finally, Section 5 describes briefly the ongoing work.

The data

The data under study are a sample of n observations (days), each observation being described by a set of p curves (functional features, KPIs). The statistical model underlying data represented by (multivariate) curves is a stochastic process with continuous time:

X = {X(t)} t∈[0,T ] with X(t) = (X 1 (t), . . . , X p (t)) ∈ R p , p ≥ 2.
A sample path of X is represented by a set of p curves.

Transformation of the observed discretized curves

In practice, data are generally observed at discrete time points and with some noise. In order to reflect the functional nature of data, smoothing methods consider that the true curve belongs to a finite dimensional space spanned by some basis of functions. Let us assume that each observed curve x ij (1 ≤ i ≤ n, 1 ≤ j ≤ p) can be expressed as a linear combination of basis functions {φ j } =1,...,m j :

x ij (t) = m j =1 a ij φ j (t), t ∈ [0, T ]. (1)
The basis expansion coefficients a ij = {a ij } =1,...,m j can be estimated by least square smoothing (see [START_REF] Ramsay | Functional data analysis[END_REF] for instance). In this work, the same basis {φ } =1,...,m is used for all the functional features (KPIs).

Principal components analysis for functional data

Principal components analysis is often used to give a simple representation of the functional data. Principal components analysis for functional data (FPCA, [START_REF] Ramsay | Functional data analysis[END_REF]) consists in computing the principal components C h and principal factors f h of the Karhunen-Loeve expansion:

X(t) = µ(t) + h≥1 C h f h (t), t ∈ [0, T ]. (2)
When curves are assumed to be decomposed into a finite basis of function [START_REF] Govaert | Co-Clustering[END_REF], FPCA consists in a classical PCA of the basis expansion coefficients using a metric defined by the inner product between the basis functions. FPCA has been extended to multivariate functional data in [START_REF] Jacques | Model-based clustering of multivariate functional data[END_REF].

In theory, the number of principal components are infinite. However, in practice, due to the fact that the curves are observed at discrete time points and then approximated on a finite basis of functions, the maximum number of components one can compute is equal to the number m of basis functions used for approximation.

Latent block model for functional data

In a co-clustering study, the goal is to gather observations as well as (functional) features into clusters. For this, we use the latent block model which assumes that data into a block (defined by a cluster of observations and a cluster of features) are independent and identically distributed. Let K r be the number of clusters in row and let K c be the number of clusters in column.

Let x = (x ij ) 1≤i≤n,1≤j≤p be the matrix of curves (x ij : x ij (t), t ∈ [0, T ]) whose rows are observations (days) and whose columns are the functional features. Let c = (c h ij ) 1≤i≤n,1≤j≤p,1≤h≤m be the principal components resulting from a (univariate) FPCA of all the curves x, without distinction between curves from different observations or different features. In the following, the straightforward ranges for i, j, h, k r and k c will be omitted for simplicity of notations.

The latent block model for functional data we propose is defined by: p(x; θ) = v∈V w∈W p(v; θ)p(w; θ)f (c|v, w; θ)

where,

• V is the set of all possible partitions of the rows into K groups, W is the set of all possible partitions of the columns into M groups,

• p(v; θ) = ikr α v ikr
kr , α kr being the row-mixing proportion and v = (v ikr ) ikr with v ikr = 1 if observation i belongs to the row cluster k r , 0 otherwise, • p(w; θ) = jkc β w jkc kc , β kc being the column-mixing proportion, and w = (w jkc ) jkc with w jkc = 1 if the functional covariate j belongs to the column cluster k c , 0 otherwise,

• f (c|v, w; θ) = ijkrkc p(c ij ; µ krkc , Σ krkc ) v ikr w jkc with, -c ij = (c h ij ) 1≤h≤m are the functional principal components of x ij (t),
p(•; µ krkc , Σ krkc ) is the m-variate Gaussian density with mean µ krkc = (µ hkrkc ) 1≤h≤m and diagonal variance matrix Σ krkc with diagonal (σ 2 hkrkc ) 1≤h≤m , • θ = (α kr , β kc , µ krkc , Σ krkc ) 1≤kc≤Kc,1≤kr≤Kr .

Inference via a SEM-Gibbs algorithm

Let us assume that a FPCA has been carried out on the whole data set of curves, and then each curve x ij is represented by its principal components c h ij for h = 1, . . . , m. Inference of the latent block model is computationally infeasible with an EM algorithm [START_REF] Govaert | Co-Clustering[END_REF] and we choose to use its stochastic version SEM coupled with a Gibbs sampling. Starting from an initial value of the parameter θ (0) and of the missing data w (0) , the q th iteration of the partial SEM-Gibbs alternates the following SE and M steps.

SE step Execute a small number (at least 1) of successive iterations of the two following steps:

1. generate the row partition v

(q+1) ikr |c, w (q) for all 1 ≤ i ≤ n, 1 ≤ k r ≤ K r : p(v ikr = 1|c, w (q) ; θ (q) ) = α (q) kr f kr (c i |w (q) ; θ (q) ) k r α (q) k r f k r (c i |w (q) ; θ (q) )
where c i = (c h ij ) j,h and f kr (c i |w (q) ; θ (q) ) = jkc p(c ij ; µ

(q) krkc , Σ (q)
krkc ) w (q) jkc 2. generate the column partition w

(q+1) jkc |c, v (q+1) for all 1 ≤ j ≤ p, 1 ≤ k c ≤ K c : p(w jkc = 1|c, v (q+1) ; θ (q) ) = β (q) kc f kc (c j |v (q+1) ; θ (q) ) k c β (q) k c f k c (c j |v (q+1) ; θ (q) )
where c j = (c h ij ) i,h and f kc (c j |v (q+1) ; θ (q) ) = ikr p(c ij ; µ

(q) krkc , Σ (q) krkc ) v (q+1) ikr M step Estimate θ (q+1) conditionally on v (q+1) , w (q+1) : α (q+1) kr = 1 n i v (q+1) ikr β (q+1) kc = 1 p j w (q+1) jkc µ (q+1) krkc = 1 n (q+1) krkc i j c v (q+1) ikr w (q+1) jkc ij Σ krkc (q+1) = 1 n (q+1) krkc -1 i j (c ij -µ (q+1) krkc ) t (c ij -µ (q+1) krkc ) v (q+1) ikr w (q+1) jkc
where n (q+1)

krkc = i j v (q+1) ikr w (q+1) jkc .
Choosing the parameter estimation and the final partition After a brun in period, the final estimation θ of the parameter θ is defined by the mean of sample distribution. The final partition is estimated by maximum a posteriori according to θ.

Ongoing work

In this paper, we defined a co-clustering model for functional data. Our ongoing work includes the establishment of an experimental study in order to test our approach. The experimentation is composed of tests with simulated data as well as tests with real data. The results will be presented during the conference. Our model is characterized to be generic so it can be applied to any application that uses multivariate functional data and that aims to discover the relationship between them and the different observations. In particular, in the mobile network domain, our model can extract new knowledge about KPIs which will help the technical support team of network operators in optimization and troubleshooting tasks. Moreover, it can be applied as a pre-processing step of self-organizing networks procedures in order to explore the enormous number of the used KPIs.

The main advantage of this model is that it offers a way to get labeled data, which is very hard to obtain in the area of mobile network. Actually, in some use cases such as self-healing, where there is no dataset available (neither issued from live networks nor artificial), designing and evaluating data mining approaches is still a challenge that should be tackled. Therefore, starting from unlabeled data, we proposed a technique of clustering which result i.e the different clusters can be addressed to mobile network experts who can identify them. The resulted labeled data will allow the use of supervised techniques and prediction of new observations.

Figure 1 :

 1 Figure 1: Sample of daily evolutions of 30 KPIs.


Haofan Yang

Tharam S Dillon

Yi-Ping Phoebe Chen

Evaluation of Recent Computational Approaches in Short-term Traffic Forecasting

Keywords:

Computational technologies under the domain of intelligent systems are expected to help the rapidly increasing traffic congestion problem in recent traffic management. Traffic management requires efficient and accurate forecasting models to assist real time traffic control systems. Researchers have proposed various computational approaches, especially in short-term traffic flow forecasting, in order to establish reliable traffic patterns models and generate timely prediction results. Forecasting models should have high accuracy and low computational time to be applied in intelligent traffic management. Therefore, this paper aims to evaluate recent computational modeling approaches utilized in short-term traffic flow forecasting. These approaches are evaluated by real-world data collected on the British freeway (M6) from 1 st to 30 th November in 2014. The results indicate that neural network model outperforms generalized additive model and autoregressive integrated moving average model on the accuracy of freeway traffic forecasting.

Introduction

Traffic during peak hours in developed and developing countries is usually congested and such problem is apace rising since the past three decades. Computational technologies particularly intelligent systems are expected to help the rapidly increasing traffic congestion problem in recent traffic management. The use of intelligent systems is an important trend in traffic management, and its objective is to provide innovative services to transportation development and enable various users to better use of transport networks. Advances in computing and communications technologies promote using intelligent systems to manage many problems in transportation, especially the traffic congestion [START_REF] Gu | Study on the Intelligent Transport System and Social Economic Development[END_REF][START_REF] Ben-Akiva | A dynamic traffic assignment model for highly congested urban networks[END_REF][START_REF] Hamilton | The evolution of urban traffic control: changing policy and technology[END_REF]. Congestion can be reduced by redistributing traffic spatially and temporally. To achieve traffic flow redistribution, it is necessary to get future traffic conditions [START_REF] Daganzo | A behavioral theory of multi-lane traffic flow. Part II: Merges and the onset of congestion[END_REF][START_REF] Kamal | Smart Driving of a Vehicle Using Model Predictive Control for Improving Traffic Flow[END_REF]. The authors in [START_REF] Daraghmi | Negative Binomial Additive Models for Short-Term Traffic Flow Forecasting in Urban Areas[END_REF] also emphasized that it is necessary to continuously forecast the traffic conditions for short time ahead to enable dynamic traffic control. These cause establishing efficient and accurate traffic flow forecasting models to become an important issue in traffic management.

Lieu, et al. [START_REF] Lieu | Traffic flow theory[END_REF][START_REF] Lieu | Traffic estimation and prediction system[END_REF] stated that the way to establish a reliable and efficient intelligent traffic system relies on providing continuous information of the traffic conditions over time. Such traffic information need to be updated in a timely fashion and should generate projections on the expected traffic networks [START_REF] Vlahogianni | Short-term traffic forecasting: Overview of objectives and methods[END_REF]. In the new era of intelligent traffic systems, research has focus on establishing forecasting models to manage the traffic networks [START_REF] Zhang | A hybrid short-term traffic flow forecasting method based on spectral analysis and statistical volatility model[END_REF][START_REF] Wu | Improved k-nn for Short-Term Traffic Forecasting Using Temporal and Spatial Information[END_REF][START_REF] Zhang | Hierarchical fuzzy rule-based system optimized with genetic algorithms for short term traffic congestion prediction[END_REF]. Many computational approaches have been commonly applied to build forecasting models, such as neural network (NN), generalized additive model (GAM), and autoregressive integrated moving average (ARIMA) [START_REF] Daraghmi | Negative Binomial Additive Models for Short-Term Traffic Flow Forecasting in Urban Areas[END_REF][START_REF] Chan | Prediction of short-term traffic variables using intelligent swarm-based neural networks[END_REF][START_REF] Chen | Short-time traffic flow prediction with ARIMA-GARCH model[END_REF][START_REF] Tjondronegoro | Knowledge-discounted event detection in sports video[END_REF][START_REF] Nahar | Kernel-based naive bayes classifier for breast cancer prediction[END_REF]. In this paper, above approaches are applied to the traffic data collected on the British freeway (M6) from 1 st to 30 th November in 2014 for evaluation. We evaluate three different approaches and report on their performance. The rest of this paper is organized as follows. Section 2 describes the computational modeling approaches used in this study. Section 3 explains the experimental design. Section 4 discusses the experiment results. Finally, the conclusion is given in Section 5.

2

Description of Used Computational Modeling Approaches

Neural Network (NN)

NN is a kind of information processing technique, and it can be trained to learn relationships in a dataset. The NN model has been applied in short-term traffic forecasting for many years, and it has been proven to be effective in solving problems that existing complex relationships, such as traffic flow forecasting [START_REF] Chan | Prediction of short-term traffic variables using intelligent swarm-based neural networks[END_REF][START_REF] Chan | Selection of significant on-road sensor data for short-term traffic flow forecasting using the Taguchi method[END_REF][START_REF] Vlahogianni | Testing and comparing neural network and statistical approaches for predicting transportation time series[END_REF][START_REF] Vanajakshi | A comparison of the performance of artificial neural networks and support vector machines for the prediction of traffic speed[END_REF]. Since the traffic data with lumpiness may reduce the generalization capability on the short-term traffic flow forecasting on unseen data [START_REF] Chan | Neural-network-based models for short-term traffic flow forecasting using a hybrid exponential smoothing and Levenberg-Marquardt algorithm[END_REF], we applied the exponential smoothing method [START_REF] Delurgio | Forecasting principles and applications[END_REF][START_REF] Wanas | On the optimal number of hidden nodes in a neural network[END_REF] to remove lumpiness in the collected traffic data. After removing the lumpiness, a three-layer feed-forward NN with levenberg-Marquardt algorithm was trained to establish a short-term traffic forecasting model. The traffic data is collected from n sensing stations (S 1 , S 2 , S 3 ,…,S n ), which are located on the freeway (M6). S i captures two traffic condition measures, i.e., the average vehicle speed 𝑉 � 𝑖 (𝑡) and the average headway ℎ � 𝑖 (𝑡) between time t and time t+T S , where T S is the sampling time. Future traffic conditions can be forecasted by the NN model, according to the current and past traffic conditions. The current traffic condition is denoted by the current average speed 𝑉 � 𝑖 (𝑡) and current average headway ℎ � 𝑖 (𝑡). The past traffic condition is indicated by the past average speed 𝑉 � 𝑖 (𝑡 -𝑘𝑇 𝑠 ) and past average headway ℎ � 𝑖 (𝑡 -𝑘𝑇 𝑠 ), which was collected by Si at time (𝑡 -𝑘𝑇 𝑠 ) with i=1,2,3,…,n and k=1,2,3,…,p, whereas the past traffic data within p sampling time period are collected. The future traffic condition can then be generated by the NN model, which is indicated by ℎ � 𝐿 (𝑡 + 𝑚𝑇 𝑠 ) passing through the L th sensing station S L at time (𝑡 + 𝑚𝑇 𝑠 ), where future traffic condition with m sampling time ahead is forecasted. The NN model is formulated as:

ℎ � 𝐿 (𝑡 + 𝑚𝑇 𝑠 ) = � � �𝛽 𝑗,𝑖 𝑉 Ѱ �𝛾 0,𝑗,𝑖 𝑉 + � 𝛾 𝑘,𝑗,𝑖 𝑉 𝑉 𝑖 (𝑡 -𝑘𝑇 𝑠 ) 𝑝 𝑘=1 � 𝑀 𝑗=1 𝑛 𝑖=1 + 𝛽 𝑗,𝑖 ℎ Ѱ �𝛾 0,𝑗,𝑖 ℎ + � 𝛾 𝑘,𝑗,𝑖 ℎ ℎ 𝑖 (𝑡 -𝑘𝑇 𝑠 ) 𝑝 𝑘=1 �� + 𝛼 0 ( 1
)
where M is the number of nodes in the hidden layer; Ѱ() is the activation function of the hidden set (sigmoid function is used in this study); 𝛽 𝑗,𝑖 𝑉 , 𝛽 𝑗,𝑖 ℎ , 𝛾 0,𝑗,𝑖 𝑉 , 𝛾 0,𝑗,𝑖 ℎ , 𝛾 𝑘,𝑗,𝑖 𝑉 , 𝛾 𝑘,𝑗,𝑖 ℎ , and 𝛼 0 are the parameters of the NN model.

Generalized Additive Model (GAM)

The GAM has the ability to allow non-parametric fits with relaxed assumptions between predicted and actual values. With this ability, the GAM can provide better fits to data than purely parametric models. The GAM is based on the additive model (AM), which is a nonparametric regression method that assumes the mean of a response variable depends on predictors by a nonlinear function. The AM relates a univariate response variable to a set of other response variables. The GAM extends the generalized AM which assumes linear dependence, by allowing the dependence of the response variable to be nonlinear [START_REF] Hastie | Generalized additive models[END_REF]. The GAM is generalizing the AM to allow the response variable to follow an exponential family distribution [START_REF] Wood | Generalized additive models: an introduction with R[END_REF][START_REF] Wood | Generalized additive models for large data sets[END_REF] along with a link function. Given a data set �𝑦 𝑖 , 𝑥 𝑖1 , 𝑥 𝑖2 , … , 𝑥 𝑖𝑝 � 𝑖=1 𝑛 of data size n, where 𝑦 𝑖 and 𝑥 𝑖 are the i th observations of the traffic flow, the AM is formulated as:

𝑦 𝑖 = � 𝑓 𝑗 (𝑥 𝑖𝑗 ) 𝑝 𝑗=1 + 𝜀 𝑖 ( 2
)
where 𝑓 𝑗 (𝑥 𝑖𝑗 ) is a smooth function of the i th observation of covariate j, and 𝜀 𝑖 is the residual. With a link function G, the GAM can be structured as:

G �𝐸�𝑦 𝑖 |𝑥 𝑖1 , 𝑥 𝑖2 , … , 𝑥 𝑖𝑝 �� = 𝑓 0 + � 𝑓 𝑗 (𝑥 𝑖𝑗 ) 𝑝 𝑗=1 + 𝜀 𝑖 ( 3
)
where 𝑓 0 is the intercept that is equal to the overall mean of the 𝑦 𝑖 [START_REF] Hastie | Generalized additive models[END_REF][START_REF] Wood | Generalized additive models for large data sets[END_REF]. In this study, a local scoring algorithm which maximizes a likelihood function is used to estimate the used smooth functions in GAM. Also, the generalized cross-sectional validation approach is used to avoid over-fitting in the short-term traffic flow forecasting.

2.3

Auto-Regressive Integrated Moving Average (ARIMA)

The ARIMA model, one of the most popular time series approach applied in traffic flow forecasting, is generally referred as ARIMA (p, d, q), where p, d, and q are non-negative integers [START_REF] Williams | Modeling and forecasting vehicular traffic flow as a seasonal ARIMA process: Theoretical basis and empirical results[END_REF]. p, d, and q refer to the order of the autoregressive, integrated, and moving average parts of the model, respectively. To empirically create a proper model, we followed the Box-Jenkins ARIMA modeling procedure which consists of the following steps: identification, estimation, diagnostic checking, and forecasting [START_REF] Pankratz | Forecasting with univariate Box-Jenkins models: Concepts and cases[END_REF]. Through continuous modification, the most proper forecasting model can be generated.

In the identification step, the collected traffic data is assessed to determine whether it is stationary. Autocorrelation function (ACF) and Partial Autocorrelation Function (PACF) are used to detect it [START_REF] Liu | Identification of seasonal ARIMA models using a filtering method[END_REF]. If the collected traffic data is not stationary, the differencing approach is applied. The value d is the lowest order of differencing applied to achieve stationary. The second step is to decide whether AR(p) or MA(q) should be used in the model. According to the ACF and PACF plots of the differenced series, it is possible to tentatively identify the value p and/or q. The error residuals can be calculated when the candidate model ARIMA (p, d, q) is estimated. The candidate model is further tested by Akaike Information Criterion (AIC) and Schwartz's Bayesian Criterion (SBC) in diagnostic checking step [START_REF] Claeskens | Model selection and model averaging[END_REF]. The lower value of AIC and SBC, the more suitable ARIMA (p, d, q) should be. Q-test is also used to determine whether the estimated model is suitable. If the candidate model cannot pass the test, the process should go back to the identification step to develop a better model. The candidate model which passed diagnostic checking is set as the used ARIMA model in short-term traffic flow forecasting. In the ARIMA model, we predict ℎ � 𝐿 (𝑡 + 𝑚𝑇 𝑠 ), the average traffic flow at the L th sensing station S L at time (𝑡 + 𝑚𝑇 𝑠 ), where future traffic condition with m sampling time ahead is forecasted based on the past traffic flow condition. The past traffic flow condition 𝑃 𝑡𝑡 is defined as

𝑃 𝑡𝑡 = (1 -𝐿) 𝑑 ℎ � 𝐿 (𝑡 + 𝑚𝑇 𝑠 ) (4)
L is the lag operation and the ARIMA model is structured as:

(1 -∑ 𝛷 𝑖 𝐿 𝑖 )𝑃 𝑡𝑡 𝑝 𝑖=1 = (1 + ∑ 𝜃 𝑖 𝐿 𝑖 )𝜀 𝑇 𝑞 𝑖=1 (5)
where p is the order of autoregressive and q is the order of moving average. 𝛷 𝑘 and 𝜃 𝑘 are the parameters of the autoregressive and moving average parts, respectively. 𝜀 𝑇 are error terms for the stationary distribution.

Experimental Design

The traffic flow data was collected from the British traffic warehouse, and it was captured by ANPR cameras, in-vehicle GPS and inductive loops built into the road surface located between junctions, J40 and J41, of the freeway M6 in the United Kingdom from 1 st to 30 th November in 2014. The main aim of traffic flow forecasting is to help for reducing traffic congestion, therefore the peak traffic period on the business days should be the main target. The traffic time periods are set to 15-minute intervals in the day which also refers to 0 to 95 where 0 indicates 00:00 to 00:15 and 95 refers to 23:45 to 24:00. The collected traffic data was analyzed and it was found all business days had similar traffic headway distribution as shown in Fig. 1. The figure illustrates how the data have daily seasonal patterns, which are periodically repeated. The peak traffic flow condition was occurred in mornings between 7:00 and 9:30 (time period 28 to 37) and evenings between 17:00 and 19:30 (time period 68 to 77) on the business days. To train the models (NN, GAM, and ARIMA) of the selected road section, we excluded weekend traffic data since it has different daily patterns. The total number of days used for evaluation is 20 (business days between 3 rd to 28 th November). The significance level is set to 5% and the confidence interval is 95% in all analyses used in this study. The remained traffic data was divided into two subsets. The data of first subset, namely the training set, was used for establishing the NN, GAM, and ARIMA models. The data of second subset, namely the test set, was applied to evaluate the generalization capability of the trained models. The business days of the first three weeks in November 2014 were set in the training set and the business days of the last week in November 2014 were concluded in the test set. For the time period with 15minute intervals, training data set and testing data set contain 1440 (3*5*96) data and 480 (1*5*96) data, respectively. Since the generalization capability on traffic flow forecasting would be improved by removing the lumpiness from the raw traffic data, we applied the exponential smoothing method to remove lumpiness in the collected data before applying the data to develop proposed models.

In order to evaluate the accuracy and efficiency of the NN, GAM, and ARIMA models, the performance of these models was evaluated by Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE). MAPE and RMSE indicate the mean and the sample standard deviation of the differences between observed values and predicted values, respectively.

Experiment Results and Discussion

Since the main target in this study is the peak traffic period on the business days, we evaluate the forecast an hour ahead (in the peak traffic period 34 to 37 or 74 to 77) using the models developed in the training phase. The traffic flow conditions on the road section M6-J40 to M6-J41 from the previous one and half hours (in the peak traffic period 28 to 33 or 68 to 73) of the same peak traffic period (morning or evening peak) are used, i.e., the previous 6 records of a peak traffic period are used to predict 4 records ahead during the same peak traffic period. The forecasting models (NN, GAM, and ARIMA) are evaluated in different business days using the testing data set. The forecasted results are compared with real observations in order to know the performance of each model which is evaluated by the MAPE and RMSE. The ARIMA model considers the number of lags to forecast a future value, therefore it is used to forecast the time (𝑡 + 2 * 𝑚𝑇 𝑠 ) value. The forecasting performance of NN, GAM, and ARIMA models applied to peak traffic periods of the testing data set are presented in Table 1 andTable 2. For the comparison results displayed in Table 1 and Table 2, we notice that the MAPEs and RMSEs of the NN model are smaller than those of the GAM and ARIMA models during the morning peak and evening peak. Fig. 2. shows the comparison between the observed traffic headway values and the predicted headway values which are generated by NN, GAM or ARIMA models on Monday 24 th November 2014. The comparison between the observed traffic headway values and the predicted headway values on 25 th , 26 th , 27 th , and 28 th November (Tuesday to Friday) presents similar results as Monday 24 th . Both Tables and Fig. 2 show that the NN model performs much better than the other models as its predicted headway values are close to the observed headway values. Moreover, its MAPE and RMSE values are less than GAM and ARIMA models.

Conclusion

Due to the fact that traffic congestion problem is rapidly increasing, it is urgent to establish intelligent traffic control systems that can redistribute the traffic flow spatially and temporally. In order to reach it, an efficient and accurate forecasting model is necessary to assist real time traffic control systems. Research has applied many computational approaches on short-term traffic forecasting to build forecasting models. While the traffic flow data of a road link can be collected much easier than before, selecting appropriate computational approaches to deal with the demand of forecasting accuracy and efficiency is getting significant. In this paper, we proposed to evaluate some computational approaches in short-term traffic flow forecasting. From

Fig. 1 .

 1 Fig. 1. Traffic headway distribution on the business day between 1 st to 30 th November in 2014. The time period index (0 to 1920) starts on Monday 00:15 at 3 rd November and ends on Friday 24:00 at 28 th November in 2014 with 96 intervals (15-minutes intervals) per day.

  experiment results, we found that on forecasting accuracy, NN model outperforms GAM and ARIMA models on forecasting accuracy of the short-term freeway traffic forecasting.

Table 1 .

 1 Comparison of forecasting models based on MAPE in peak traffic periods. Morning peak 34 to 37 indicates 08:30 to 09:30 and evening peak 74 to 77 denotes 18:30 to 19:30.

	Traffic				MAPE%
		Model
	Peak period		Mon Tue	Wed Thu	Fri
		NN	2.19	1.85	1.99	2.11	2.61
	Morning peak 34 to 37	GAM	2.76	4.02	4.28	3.36	3.63
		ARIMA 13.32 15.17 16.05 14.54 14.91
		NN	1.73	2.08	2.24	2.83	1.90
	Evening peak 74 to 77	GAM	3.04	3.88	3.61	2.98	4.11
		ARIMA 17.43 17.58 15.68 16.38 17.09
		NN	1.15	1.20	1.72	1.41	1.53
	Evening peak 74 to 77	GAM	2.65	1.69	2.85	1.88	2.03
		ARIMA 7.43	7.26	4.91	8.66	7.59

Table 2 .

 2 Comparison of forecasting models based on RMSE in peak traffic periods. Morning peak 34 to 37 indicates 08:30 to 09:30 and evening peak 74 to 77 denotes 18:30 to 19:30.

	Traffic		RMSE
		Model
	Peak period		Mon Tue Wed Thu Fri
		NN	1.20 1.01 1.36 1.05 1.14
	Morning peak 34 to 37	GAM	2.38 1.54 3.05 1.56 1.68
		ARIMA 5.60 6.18 4.92 7.60 6.57