Yin Cheng
email: yin.cheng@gmx.com

Lucien Laiarinandrasana

Lukas Helfen

Henry Proudhon

Olga Klinkova

Tilo Baumbach

Thilo F Morgeneyer
email: thilo.morgeneyer@mines-paristech.fr

¶ H Proudhon

3D damage micromechanisms in polyamide 6 ahead of a severe notch studied by in situ synchrotron laminography

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

3D damage micromechanisms in polyamide 6

ahead of a severe notch studied by in situ synchrotron laminography

Introduction

Semicrystalline polymers are frequently employed in engineering components that are subjected to thermomechanical loadings, therefore considerable efforts have been made in the past few decades to understand their mechanical behaviour in terms of deformation and damage mechanisms in connection with their microstructural organization. Under tensile drawing, cavitation or crazing [1][2][3][4][5][6][START_REF] Kramer | Crazing in Polymers[END_REF][START_REF] Kausch | Crazing in Polymers[END_REF][START_REF] Kausch | Crazing in Polymers[END_REF][10][11][12] have been identified as major damage processes in many semicrystalline polymers 13 including polyamide 6 (PA6).

Various techniques have been exploited to ob-serve such phenomena over a wide range of length scales. Many mechanisms of deformation can be revealed by electron microscopes including SEM and TEM, which reveal damage features with sizes from nanometers to a few micrometer. In the literature, Galeski and coauthors presented direct TEM evidence for cavitation, chemical damage and/or density reduction in localized regions of the microstructure of bulk spherulitic nylon 6 under plastic deformation. They proposed a physical picture suggesting possible complex deformation processes in a spherulite due to uniaxial tension. 14 Aboulfaraj et al. differentiated the α-and β-spherulites in polypropylene (PP) by SEM after appropriate etching 15 and further revealed specific responses with both types of spherulites with respect to uniaxial tensile and simple shear deformation. 16 Dijkstra et al. 17 proposed a failure transition model from yielding to crazing based on their SEM observations of PP specimens subjected to varying strain rates. Henning et al. 18 compared the micromechanical behaviours between the α-and β-modified isotactic PP. At room temperature (23 • C) the microvoids were visible in the polar region of the α-iPP, whilst in the β-iPP intensive microvoid formation took place in the whole volume. At -40 • C only a few but long and coarse crazes were found for the α-form, however much smaller, thin and numerous crazes were typical for the β-form. The measured enhanced toughness of the β-iPP was then linked to these micromechanisms by the authors and attributed to the lamellar nanostructure arrangement of β-spherulite. However, the required sample preparations (i.e. chemical etching) constrained both methods to either ultra-thin (100-500 nm) films or interrupted and unloaded (usually with deformed samples) investigations. By applying atomic force microscopy (AFM), Ferreiro et al. 19,20 showed the competitive nanoscopic observation (down to the lamellar scale) of shear band deformation of a relatively thick (80 µm) PA6 film under uniaxial tension at yield. The authors demonstrated that below 160 • C shear bands which intersect at an angle of 104 • within the amorphous interlamellar phase of the spherulites govern the plastic deformation in this plane stress state.

Thomas et al. 21,22 further applied in situ AFM, they captured craze initiation and growth in several 100 µm thick polybutene (PB) films with different spherulite sizes and concluded the structural size dependence of plastic processes in PB. At a similar length scale, small-angle X-ray scattering (SAXS) and wide-angle Xray scattering (WAXS) have also been broadly utilized. They allow for an averaged in situ investigation of even thicker (mm size) samples at the 1-100 nm scale taking advantage of high energy and short wavelengths of X-rays. Castagnet et al. 23 studied α polyvinylidene fluoride (PVDF) by volume strain measurements, WAXS and SAXS experiments. They showed that cavitation predominated in equatorial intraspherulitic amorphous layers before yielding, and futher nucleated in other areas for higher strains. This process dominated the deformation in PVDF deformation over crystallite fragmentation. Pawlak and co-authors [24][25][26] characterized cavity evolution during the tensile deformation of PP. Amongst their findings was that the cavities initiated in the amorphous phase in the equatorial part of spherulites. Nanovoids as well as microvoids changed their orientation from elongated perpendicular to being parallel to the deformation axis, i.e. the damage in the polar fans of spherulites. They are responsible for the whitening of samples. Assisted by SEM, the "polar fan" damage in deformed sections has also been observed. Since SAXS is an indirect method to assess damage features, synchrotron X-ray microtomography (SRµCT) has been recently explored by several researchers in order to better understand the deformation induced whitening at the mesoscale level (from hundreds of nanometers to a few micrometers) as well as in a nondestructive way as compared to SEM. Laiarinandrasana et al. 27 made the first attempt on smooth round bar semicrystalline PA6 specimens after interrupted tensile tests. With a voxel size of 0.7 µm, they imaged 3D bulk volumes of the necked region and revealed pennyshaped crazes and void columnar-structures that aligned along the tensile axis. Further studies were conducted on notched round bar specimens under interrupted tests accompanied by image analysis 28 and reinforced the conclusion that the multiaxial stress state produced by the notch increased the size of damage features and defined the distribution of voids. Similar SRµCT measurements have been described by N. Brusselle-Dupend et al. 29 and Rosenberg et al. [START_REF] Rosenberg | Epsztein[END_REF] on PVDF.

Blaise et al 31 who performed postmortem SRµCT examinations coupled with 2D FFT analyses on a predeformed high density polyethylene (HDPE), suggesting that the commonly admitted crazing mechanism could not account for the whitening of such "noncavitating" materials upon necking, 32 but could result from the crystalline transformation. Soon after, a comparable technique at the same spatial range called incoherent polarized steady-light transport (IPSLT) was used by those authors and confirmed their SRµCT results. 33 Further work followed by Farge et al., 34 who combined both IPSLT and postmortem SRµCT on highly cavitating PP, enriched Blaise's result that the origin of whitening may differ between different semicrystalline polymers. In their study, "zebra patterns" and "cigar shape" array of holes were observed that were similar to the observation in PA6 by Laiarinandrasana et al. They speculated that the whitening phenomenon in PP was due to confined groups of nanovoids in the equatorial disks of spherulites, which however could not be detected by the IPSLT and SRµCT. Very recently, Morgeneyer et al. 35 applied the synchrotron nano-holotomography 36,37 to a polymer showing promising results of the nanovoid morphology and distribution in a deformed HDPE with a voxel size of 59.7 nm. Nanovoids were found that were elongated in the tensile direction. Their attempt may answer the question of Blaise and Farge, and fill the gap between SAXS and IPSLT/SRµCT. It should also be noted that IPSLT is an averaged volume measurement. Apparently all these tools assessing damage at different length scales can effectively complement each other. In a recent paper, Farge et al. 38 combined SAXS, in situ IPSLT, postmortem SRµCT and 3D digital image correlation covering a multiscale study of the deformation mechanisms in a semicrys-talline HDPE, and emphasized the impact of microscale organizations on the macroscale behaviours.

Despite tremendous efforts paid to understand the cavitation, 13 little information can be found concerning the crack initiation and propagation mechanisms in polymers, [39][START_REF] Friedrich | In Physik der Duroplaste und anderer Polymerer[END_REF][START_REF] Friedrich | In Anwendungsbezogene physikalische Charakterisierung von Polymeren, insbesondere im festen Zustand[END_REF][42][START_REF] Zafeiropoulos | Macromolecular Rapid Communications[END_REF] which is highly important for mechanical modeling. Narisawa and Ishikawa [START_REF] Narisawa | [END_REF][START_REF] Narisawa | Crazing in Polymers[END_REF] carried out numerous polarized optical micrograph inspections on various deformed semicrystalline thermoplastics. However, the inner microstructure evolution was inaccessible and little evidence was provided on the proposed crack propagation criteria. In the present study, thanks to the synchrotron computed laminography (SRµCL), [46][47][48][49] it has been possible to trace the volumetric micro-crack initiation and propagation in a 2 mm thick notched PA6 laminate under severe triaxial stresses. SRµCL is a nondestructive 3D X-ray imaging technique that allows for the assessment of the inner microstructure of a plate-like object. It overcomes the general limitation in µCT measurement for which small cylindrical or dog-bone-shaped specimens are used because these small specimens are sometimes still very different from standard mechanical tests and their plastic zones at the notch tips can be possibly larger than the specimen thickness: SRµCL fulfills the most realistic engineering boundary conditions and particularly when coupled with in situ tests. The observed fracture mechanisms will be described and a comparison made between these findings and damage features in less deformed materials from previous SRµCT observations.

In this study, two in situ SRµCL experiments were conducted on two PA6 specimens. The first experiment was conducted at the European Synchrotron Radiation Facility (ESRF), Grenoble, France. Local laminography [START_REF] Ramm | The Radon transform and local tomography[END_REF] was applied in which the 2 mm specimen thickness exceeded the effective field of view (FOV ∼ 1.4 mm) of the detector. There was a smooth defect-free notch surface (Figure 2a), however a limited number of loading steps were recorded during the measurement. For this reason, the same experiment was repeated at the ANKA light source, Karlsruhe, Germany, in order to follow DOI: 10.1002/macp.201500316 more load increments as well as to examine the entire specimen thickness in the FOV. However, a notch roughness defect was introduced, due to a poor machining, in the latter experiment (Figure 2b) which turned out to be an advantage because it brought new information and could be seen as having an impact on the failure mechanisms.

Experimental Section Material

The PolyAmide 6 (PA6) under study was a semicrystalline polymer provided by Angst & Pfister as 610 mm × 1230 mm flow moulded plates 10 mm in thickness. Its physicochemical properties have been published elsewhere 27,28 (crystallinity was 43%, Supporting Information Table S1). To reveal its spherulitic microstructure, examinations by SEM and optical microscope of the samples can be found in Refs. 27,51,52. The average spherulite diameter was ∼ 5 µm from SEM image analysis.

The plate was cut into 40 × 50 mm 2 specimens and further machined symmetrically in the center to a thickness of 2 mm in order to guarantee sufficient X-ray transmission in the beam. 48,53 The notch was machined close to a radius of 0.25 mm by wire saw cutting in the case of a defective notch, and by a milling tool in the case of a defect-free notch. The loading direction, the crack propagation direction (notch direction) and the short transverse direction (through-thickness direction) are referenced to the letters L, P and S respectively as indicated in Figure 1. The loading test was performed in the L-P configuration.

Mechanical Test

The geometry of the flat notched specimen is shown in Figure 1. To ensure the reproducibility, several mechanical pre-tests have been performed before conducting the synchrotron experiment, and also one in situ x-ray test was carried out on a 1 mm thick specimen. Both cases produced the same trend as the results reported in this study. Mechanical tests were carried out at room temperature (20 • C) and 50% relative humidity (RH). The loading was achieved via a 2-screw displacement controlled wedging rig that imposed the specimen crack mouth opening displacement (CMOD) without measuring the load level. Stepwise loading was applied between different laminography scans. Each scan began after waiting 5 min, ensuring that the short-term stress relaxation and the material's viscoplasticity did not perturb the image quality. The applied CMOD was 0.7 mm and 1 mm per turn on the defective notch and defect-free notch, respectively. An anti-buckling device (Supporting Information Figure S1) was mounted around the specimen, leaving a window for the synchrotron X-ray laminography close to the notch so as to prevent the thin sheet from buckling in the compressive zone as well as out-of-plane motion. The scanned region of interest (ROI) was close to the notch and moved along with the propagating crack tip to image the damaged material ahead of the notch tip.

Micrometer In Situ SRCL Imaging

SRµCT is a well established tool for 3D imaging of one-dimensionally extended objects (e.g. cylindrical samples) that stay in the field of view of the detector system during rotation. In contrast, SRµCL is representing a scheme which allows 3D imaging without destruction of the specimen the lateral dimensions of which largely exceed the effective FOV (laterally extended i.e. plate-like specimens). For laminography, the rotation axis of the specimen is inclined at an angle of θ < 90 • with respect to the incident X-ray beam direction, while in the case of CT θ = 90 • (Supporting Information Figure S2).

Imaging of the defect-free notch was performed on KIT's laminography instrument installed at beamline ID19 54 of the European Synchrotron Radiation Facility (ESRF, Grenoble, France). An inclination angle of the specimen rotation axis of 25 • with respect to the beam normal (θ ≈ 65 • ) was chosen as well as a monochromatic beam of 25 keV X-ray energy. Volumes were reconstructed from 1500 angularly equidistant radiographs, the exposure time of each projection was 50 ms. The scanned region was ∼ (1.5 mm) 3 in volume with a voxel size of 0.75 µm. The minimum specimen to detector distance was 70 mm leading to relatively strong edge enhancement 55 due to phase contrast. 3D cross-sectional slices of the imaged specimen normal to the rotation axis have been reconstructed from the 2D projections using a filtered-backprojection algorithm. 56 The final reconstructed volumes have a size of 2040 × 2040 × 2040 voxels.

Imaging of the defective notch was performed at the TopoTomo beamline 57 of the ANKA light source (Karlsruhe, Germany). The inclined angle of laminography was also chosen to be 25 • . A white X-ray beam in a photon energy window of 9.6-24 keV with the maximum flux density located at 14.5 keV was used. A high speed camera pco.dimax (PCO AG, Germany) combined with continuous sample rotation was employed. The pco.dimax camera had a 2016 × 2016 pixels CMOS sensor with 11 µm × 11 µm pixel size. It was coupled to an optical microscope (Optique Peter, France) consisting of a 10× objective (NA=0.28) without eyepiece, which resulted in an effective pixel size of 1.1 µm, and a corresponding FOV of 2.2 × 2.2 mm 2 . A 10 µm thin LSO crystal (Tb-doped Lu 2 SiO 5 ) scintillator was chosen to be combined with the optical microscope according to optimization of the depth of focus and scintillator efficiency. Volumes were reconstructed from 2000 angularly equidistant radio-graphs, the exposure time of each projection was ∼20 ms, guaranteeing a total scanning time of around 50 s for a single laminogram. The final reconstructed volumes had a size of 2016 × 2016 × 2016 voxels.

The main experimental parameters of the two measurements are summarized in Table 1.

Results

The polymer matrix is shown in grey, the machined notch is in dark grey, and the voids/cracks are in black, surrounded by white X-ray phase contrast fringes. [START_REF] Cloetens | [END_REF] Figure 3a shows the as-received material, it seems that there is no pre-existing cavity in the bulk material. However, it should be mentioned that this conclusion is only valid for voids that can be detected by the pixel resolution limit of 0.75 µm. In addition, polymers develop crazes that can be considered as cracks the opening of which are bridged with fine fibrils. A craze transforms into a real crack when the fibrils are broken down. Since those nanometric fibrils could not be resolved at the given resolution, it did not allow differentiation between crazes and cracks, therefore all the craze-like features found in the current study will be referred to as cracks. The following discussions will be based on these consideration. DOI: 10.1002/macp.201500316

Defect-Free Notch

2D sections of reconstructed laminography data in the sheet (L-P) plane at the mid-thickness of the sample are presented in Figure 3, showing the damage evolution ahead of the defect-free notch up to a CMOD of 6 mm. Further than that, the notch was missed in the FOV during the X-ray measurement so as we failed to follow the crack propagation any longer (Supporting Information Figure S4). Nevertheless, the observed damage features in this early deformation stage (crack initiation) showed good agreements with the levels of stress states ahead of a notch tip. [START_REF] Anderson | Fracture Mechanics: Fundamentals and Applications[END_REF] That is, a stress gradient is known to exist with higher stresses close to the notch tip and lower stresses at distances further away from the notch. The well distributed damage zones along the notch root axis can be clearly distinguished, as illustrated in Fig- ure 3c-d. It should be noted that the morphology and the "arrangement" (i.e. in polar fans or tunnel cracks composed of radial coalesced disks) were attributed to different levels of the stress triaxiality ratio (τ σ ) by Laiarinandrasana et al : [START_REF] Laiarinandrasana | Fracture mechanics concepts applied to PVDF polymeric material exhibiting porosity, time and temperature dependency[END_REF][START_REF] Laiarinandrasana | Cavitation in semi-crystalline polymers shown by x-ray tomography: application to constitutive modelling and durability[END_REF] Zone 0) A non-damage layer ∼ 33 µm in the vicinity of the notch surface, where the stress state is uniaxial (τ σ ∼ 1/3). It is consistent with the SRµCT observations on smooth round bar specimens where a surface layer of ∼ 30 µm is also free from Zone i) Polar fan composed of pile ups of penny shaped cracks arranged as void column like features; the crack surfaces being perpendicular to the load direction. This domain corresponds to moderate τ σ although τ σ > 1/3. The maximum principal stress is parallel to the load direction so that it is normal to the crack surfaces;

Zone ii) Longer cracks (tunnel crack) with typical length of 30-45 µm were piling up parallelly in the loading direction: interor intra-spherulitic cracks prevailed here crossing several spherulites. 18,[START_REF] Schultz | [END_REF] It is con-sidered that the maximum τ σ as well as the maximum largest principal stress were reached in this area;

Zone iii) Further away from the notch and beyond zone ii, tiny round cavities were observed. In this region, the level of τ σ and the largest principal stress have decreased compared with those of zone ii.

More pronounced damage can be differentiated in different zones in (d) -cracks increase in height during deformation and it is expected that the main crack would take place in the center of the notch root by breaking zone 0 and zone i.

DOI: 10.1002/macp.201500316 An attempt was made to measure the characteristic lengths of individual voids within the volume corresponding to CMOD = 6 mm. To this end, the voids were selected at some specific locations related to each zone. The cavity dimensions are denoted as length, width and height corresponding to the directions of P, S and L. Figure 4 illustrates the collected data. Around the mid-thickness in the L-P plane, the obtained cavity sizes were plotted along the P-direction starting from the notch tip (zero position, Figure 4a). The damage zones were clearly discerned: the tunnel cracks in zone ii had maximum dimensions in length and width, whilst the polar fan cracks in zone i possessed the maximum height dimension, which were in good agreement with our observation in Figure 3c-d.

Furthermore, looking at different zones, three distances of about 54, 225 and 840 µm ahead of the notch tip were respectively chosen to analyse the cavity size distribution along the thickness (S) direction. In Figure 4b, it can be seen that, within zone i, the diameter and height of the polar fan penny shaped cracks decreased from the center to the surface of the specimen. Inside zone ii (Figure 4c), the length and width of the tunnel cracks decreased whilst their heights relatively stayed the same from the core to the surface. Further away from the notch tip in zone iii (Figure 4d), the cavities maintained a similar scale and tendency in length and width dimensions, and their height also did not vary much through the specimen's thickness analogous to the observation in its neighbourhood zone ii. (Note: In Figure 4bd, the undeformed thickness of 2 mm was used although lateral neckings appeared around the notch roots. Actually, whereas the lateral surface is visible in the volumes, the plane at midthickness could not be precisely located.) DOI: 10.1002/macp.201500316 To go further, a concept of void shape factor noted as χ is introduced as the ratio between the length/width and the height: χ P = P/L and χ S = S/L. Figure 5 reports the χ P , χ S relationship based on the Figure 4 plots. Obviously, for axisymmetrical voids (penny shape, cylinder) χ P = χ S , which is the case corresponds to zone i (χ P = χ S = 2) and zone iii (χ P χ S = 2.8). Whereareas zone ii is the only case where χ P > χ S owing to the coalescence of voids in the P direction in this zone. Therefore the onset of χ P /χ S > 1 can be considered as the criterion for crack initiation. It is also found that the shape factor reaches its maximum value at the mid-thickness and gradually decreases towards the lateral surfaces.

Defective Notch

The damage evolution in the defective notch specimen in the L-P midplane is shown in Figure 6 corresponding to twelve loading steps. At the initial state, the machined notch with de-fects is seen, with initial voids appearing in the vicinity of the notch surface.

At the first two loading steps (b-c), there was no obvious effect except a slight lateral opening of the notch along the tensile drawing direction. At CMOD = 0.7 mm (d), it is seen that damage started to set in: small cracks initiated, grew and became visible either in the defect area (red arrow) or ahead of the notch somewhere within the material bulk (red ellipse). The latter damage is consistent with the fact that the highest stress triaxiality was located at some distances ahead of the notch surface rather than at the site of the notch. [START_REF] Zafeiropoulos | Macromolecular Rapid Communications[END_REF]48,63 At CMOD = 1.05 mm (e) these cracks have a typical size of 14-19 µm and leave a zone free from damage (zone 0 ) with a thickness ∼ 33 µm in front of the notch (indicator in Figure 6e). The defect-free zone is consistent with plane stress conditions found. This is also in good agreement with the finding for the defect-free notch specimen reported in Figure 3.

It is worth noting that fairly large cracks (Fig- ure 6f, zone ii ) are oriented radially with regard to the notch circumference: this can be correlated to the characteristics of craze/crack in semicrystalline polymers: crazes are known to propagate perpendicularly to the maximum principal stress. [START_REF] Zafeiropoulos | Macromolecular Rapid Communications[END_REF][START_REF] Strobl | The Physics of Polymers: Concepts for Understanding Their Structures and Behavior[END_REF][START_REF] Sternstein | Properties of Solid Polymeric Materials[END_REF] In our case, the principal stress is the tangential component of the tensile force along the notch girth circumference and reaches a maximum value at the mid-thickness, as indicated by the red component in the inset image of Figure 6f (CMOD = 1.4 mm). Further away from the notch round voids appear (zone iii ). The cracks that started from the notch imperfection (zone ii ) are caused by the very locally generated stress concentration. This concentration leads to amplified damage growth from CMOD = 2.1 mm (g) to CMOD = 2.8 mm (h). The upper part of the notch (zone ii ) became the favorable site for damage nucleation, due both to the defect and also attributed to the residual material after notch machining seen previously in 3D in Figure 2, which acts as a stress relief in the central part of the notch and as a stress raiser in the corners of the notch.

As a consequence, apart from this stress intensification zone: at the center of the notch, void columns that aligned parallel to the tensile direction were found (zone i, Figure 6h) consistent with lower stress triaxiality levels. This will be seen below from a 3D rendering in Figure 9 which finely shows their morphologycylindrical in the middle and conical at both ends. The void column can be interpreted as "polar fan" damage in the polar regions of two adjacent spherulites. 17,24,25 They are typically found for deformations under less severe (mod-DOI: 10.1002/macp.201500316 erate) states of stress. 27,28 Beside, at the bottom corner of the notch, crack growth got retarded.

At CMOD = 3.5 mm (Figure 6i), the material ligament in the upper corner of the notch was fractured and coalesced with the micro-cracks in front. At this stage, the main crack initiated. At the same time, several new cracks with typical lengths ranging from 26-43 µm originated in front of the main crack. They are much larger than the size of the ones in zone i and much larger than the average spherulite diameter. It appears that the elevated level of stress triaxiality allowed the damage features to overcome spherulite borders such as seen in PP for deformations at low temperatures. 18 From this moment on, the crack entered the propagation regime, and to the authors' knowledge, it is the first time that the crack propagation mechanisms have been revealed in PA6 in situ.

In the propagation regime, whilst the main crack was progressing by coalescence with the forward micro-cracks, at CMOD = 4.2 mm (Figure 6j), another big secondary crack (red circle) can be clearly seen to have formed in front of the main crack tip with a length of ∼ 150 µm and there is a less damaged ligament (length about 124 µm) between the main crack tip and this big secondary crack. It will be shown below that this big crack is a "tunnel" structured crack but not a penny-shaped crack. Indeed, this crack has its largest dimension in its crack propagation direction and three times less in the perpendicular (L-S) section. The observed crack propagation scenario here is to some extent similar to that of slow crack growth (SCG) mechanisms which have been ob-DOI: 10.1002/macp.201500316 served in an MDPE under creep condition. 66,67 However, in creep damage, the zone in front of the secondary craze tip stays undamaged and the crack evolves always into the "virgin" materials. Whereas in our case, damage zones (i.e. crazes, polar fan, round voids) were instantaneously formed ahead of the secondary crack tip causing the main crack always to advance into the damaged materials.

In the next step, at CMOD = 4.9 mm (Figure 6k), the main crack coalesced with the big tunnel crack and to continue propagating. The crack tip progressed by 282 µm compared to the last loading stage (j). This value is in agreement with the length of the big crack plus the length of the ligament in (j), which implies that the main crack overcame the remaining ligaments and merged the big secondary crack leading to the crack propagation. New long cracks initiated and grew ahead of the new main crack front, and new round cavities formed further away. These are consistent with the stress states that are similar to the observation in the early stages as well as to the crack initiation in the smooth notch specimen (Figure 3c zone i, ii, iii): a stress gradient redistributed ahead of the propagating crack tip. [START_REF] Anderson | Fracture Mechanics: Fundamentals and Applications[END_REF] Moreover, the regions which are not in the crack path, i.e.; above or below the main crack, are unloaded by crack shielding causing a slight closure of the damage features but their overall shape remains similar to the loaded state (Fig. 6k). This means that the elastic relaxation does not dramatically change the void shape. DOI: 10.1002/macp.201500316 From (k) to CMOD = 5.25 mm (Figure 6l), since the current crack progression rate is very high, no other intermediate big secondary crack formation between these two steps was observed. The crack runs through the full specimen thickness at (l) (Supporting Information Figure S9g).

SRCL makes it possible to visualize the damage evolution in three dimensions, as shown in Figure 7, the crack initiation as well as crack propagation are clearly depicted. During deformation, the residual material of the defective notch induced a local secondary notch effect which pronely influenced the stress distribution in front of the notch root. As can be seen in Figure 7b, the central grey part (zone 0 ) in the vicinity of the remnant was free from damage. However, in the upper corner the residual material caused a local notch radius of ∼ 0.08 mm highly intensified stress and strain (zone ii ), while the lower corner (zone i ) was only moderately deformed. As the crack propagated (Figure 7c-d), new stressdamage sites corresponding to new zone i, ii, iii were subsequently reproduced ahead of the main crack tip. The crack tip opening displacement (CTOD) [START_REF] Anderson | Fracture Mechanics: Fundamentals and Applications[END_REF]68 was measured to incubate with the applied CMOD (Supporting Information Table S2). In addition, the local CTOD profile was descending from the interior mid-DOI: 10.1002/macp.201500316 To further highlight the microstructure evolution in three dimensions, only voids/cracks in a (100 µm) 3 cubic volume around the midthickness were picked out at different CMODs (location indicated as the red box in Figure 6i). The crack growth and propagation mechanism in terms of void to crazes/cracks transition (i.e., zone iii →ii ), crack coalescence, and progression are displayed in detail (Supporting Information Figure S11).

In the L-S planes (through-thickness plane), a location ∼ 350 µm distant from the initial notch root (indicated by the magenta dashed line in Figure 6i) was chosen to illustrate the damage evolution and mechanism. This plane was chosen with respect to the features indicated in Figure 6i that are tracked for all sections.

At CMOD = 3.5 mm (Figure 8a), cracks aligned perpendicular to the loading direction. In addition, most cracks congregated at the middle of the thickness, consistent with the maximum triaxial stress in the center of bulk material. The contrasting densities of cracks in zone i and zone ii arises from the notch defect which is consistent with the L-P observation plane. Necking of the specimen was observed due to deformation.

At CMOD = 4.2 mm (b), layers of cracks opened in height and joined together in the loading direction, forming a tunnel-like structure. At this stage, the biggest crack (labelled in green) was still found to be located in the center of the thickness, which was about 40 µm in width and 30 µm in height. It is just the same tunnel crack that has already been followed in the L-P plane in Figure 6j. In regions other than the upper part, void columns are the main damage features and may be due to much lower stresses. At CMOD = 4.9 mm (c), necking was pronounced due to large plastic deformation, the main crack propagated and reached 350 µm distance. The cleavage of ligaments and merger of layers of cracks are clearly DOI: 10.1002/macp.201500316 visible from Figure 8b-c.

Two 3D segmented volumes in Figure 9 represent some typical damage features distributed in zone i and zone ii in the bulk material of both specimens. For Figure 9(a) the different 3D shapes of the damage features can be seen: close to the notch root cracks are penny-shaped, and with increasing stress they become tunnelshaped.

Discussion

Influence of Sample Geometry and Generated Stress State on the Damage Features. Compared to the axisymmetrical specimens observed in tomography, 27,28 complex damage mechanisms can be seen: The sizes of the cracks that initiated were substantially bigger in diameter. This is indeed an effect of the stresses that: with higher triaxiality and associated normal stress, agglomerated polymer chains, the so-called fibrils may fracture more easily and the large crack may form. These cracks overcame spherulite borders. 18,[START_REF] Schultz | [END_REF] In the smooth (notch root radius r 0 = ∞ and at necking r necking ≈ 5 mm) or notched round bar (r 0 = 4 mm) specimens, only spherulitic damage that are penny-shaped crazes and polar fan void columns can be observed. 27 In the smooth round bar specimen, the mean diameter of the central part of the void column is 7 µm, and 8 µm in height (Figure 10a). In the notched round bar (r 0 = 4 mm) specimen, the maximum damage occurs at the center of the specimen. 28 The mean diameter of the central part of the void column is 11 µm, and 8 µm in height (Figure 10b). In the notched round bar (r 0 = 0.45 mm) specimen, the maximum voiding is located in between the center and the notch root. The increased multiaxial stress state leads to the formation of larger crazes consisting of a cluster of coalesced penny-shaped cracks (Figure 10c) in the net sections of the specimen. [START_REF] Cayzac | Laiarinandrasana[END_REF] The mean diameter of the cluster is 17.5-35 µm. The enlargement of those crazes and their transformation to the next morphology are not captured because the stress state is relatively severe and the material fails abruptly after this stage. In the compact-tension like specimens, even higher maximal triaxial stresses are created, in particular a stress gradient is allocated in front of the notch tip. A mixture of specific damage features corresponding to varying levels of stress was observed. The void columns only occurred in the regions with moderate stress triaxiality, and their average size was 9 µm (length) by 5 µm (height). In these specimens, during crack initiation, long cracks (cluster of coalesced penny-shaped cracks) that were initiated Tunnel Crack Formation. From previous microstructure assessment of the material (the damage in the round bar specimen was shown to be isotropic in the plane normal to the loading direction 28 ), the average spherulitic dimensions of materials should be isotropic. The question is why the tunnel crack is anisotropic and not a round penny-shaped crack. Concerning the knowledge of the stress-state ahead of a propagating crack in an elastoplastic material, it is established that there is no strain in crack propagation direction, this implies that there can be stresses in the crack propagation direction. 70,71 These stresses may lead to the coalescence of small cracks into the tunnel crack. In the through-thickness direction the material necked, which means there was plastic deformation and stresses can be relieved. This is a pos-sible explanation for the fact that the tunnellike crack does hardly propagate laterally.

Slanted Crack Coalescence in Semicrystalline PA6. A hypothetical shear band damage mechanism has been observed in the L-S plane at a distance of ∼ 74 µm from the notch root (Figure 11). The slanted cracks were formed between multiple initiated parallelaligned cracks (perpendicular to the loading direction, of an average length ∼ 11 µm (equivalent to 2-3 spherulites), they also have a similar spacing around 11 µm), causing a zig-zag cracking path that is likely to undergo a coexistence of inter-and intra-spherulitic fracture course. The slanted cracks intersect at an angle between 50 -55 • with respect to the loading direction. As has been documented by various authors, shear banding can exist alongside crazing in many semicrystalline polymers such as HDPE, atactic PS or PP under compression. [START_REF] Friedrich | Crazing in Polymers[END_REF][73][74][75][76][77][78] Although the tests reported in this study were performed under tension, it is noted that the observation of the zone occurs very close to the notch surface. As has been introduced at the beginning, Ferreiro et al. have reported finding fine parallel shear/slip bands in PA6 which cross the spherulites and run uniformly through the whole sample surface and intersect at 52 • to the direction of the tensile axis. 19 However the authors also pointed out that the angle deviation from 45 • in their quenched specimen before unloading should not be attributed to a pressure effect due to elastic recovery 73,79 which actually decreases the angle. The origin for such angles is still unknown. A long straight crack of length ∼ 70 µm is also found in this cross-section. A comprehensive depiction of the in situ formation and evolution of these possible shear bands have been supplied in the Supporting Information, Figure S8.

The Impact of Notch Quality on Fracture Toughness. Two qualities of notch have been studied, the defective notch caused early crack progression, and the cracks started both from the notch surface and inside the material. By contrast for the defect-free one, the damage set in later and was not activated directly from the surface. The notch defect prompted very early stress concentration, causing the intermediate damage feature (i.e. void column of zone i ) being suppressed, for instance, after the main crack initiated in the defective notch specimen the damage in zone i was almost arrested and remained in height about 2.2 µm in the center of the void column (Figure 6i-l, Figure 8b), compared to 4 µm height (Figure 3c) while still increasing in the case of the defect-free notch specimen.

Conclusion

We have achieved the first time in situ observation of micro-damage evolution in a semicrystalline PolyAmide 6 with a severe notch root radius (0.25 mm) generating high levels of triaxial stresses and also maximum principal stress. Thanks to synchrotron radiation computed laminography (SRµCL) -examination of the crack initiation and propagation in PA6, in a flat 2 mm-thick specimen with a notch, at the micrometer scale in three-dimension was possible. The large flat specimens lead to the formation of plastic zones with large sizes.

In this configuration, the cracks/crazes have larger characteristic diameters than for drawing of initially smooth specimens or specimens with gentler notch root radii. 28 These crack dimensions are of the order of several average spherulite diameters (average diameter 5 µm here). This indicates that higher levels of stress triaxiality and maximum principal stress promote the trans-spherulitic failure mechanisms. Damage is to a lesser extent affected by microstructural length scales.

Consistent with the presumed role of maximum stress three main damage zones are found: 0) A damage-free zone ∼ 33 µm immediately in the vicinity of the notch surface due to the plain strain state.

i) Short cracks about one spherulite diameter ahead of zone 0. These cracks grow radially, align perpendicular to the maximum principal stress and increase in height along the loading direction. These cracks are identified as the "polar fan" damage within two adjacent spherulites and display as the void column morphology.

ii) Even longer cracks are initiated ahead of these polar fan damages -these cracks are generally larger than one spherulite diameter (30-45 µm in length), and piling up parallel to the loading direction. This zone sustains the maximum triaxial stress in the bulk material.

iii) Further away, small round cavities were seen to nucleate and be associated with decreasing stress levels compared to zone ii.

Tunnel shaped cracks are formed ahead of the main crack during crack propagation. They are separated from the main crack by a ligament of material that is less damaged. The main crack subsequently coalesces with these tunnel cracks overcoming the ligament of material. The tunnel-like shape of these secondary cracks is consistent with the presumed plane strain conditions, in the propagation direction, of the crack.

During crack propagation, stresses are concentrated in front of the main crack tip; the stress level elsewhere is then partially relieved. New stress zones corresponding to zone i, ii, iii are formed and distributed subsequently ahead of the main crack tip. The crack progression is constantly accompanied by stress zone transitions from zone iii (mild stress) to zone i (moderate stress) to zone ii (severe stress) in front of the main crack tip.

In the immediate vicinity of the notch, shear like features were found that have been reported for PA6 thin film in uniaxial tensile loading in the past. 19 This can be linked to the plane stress state close to the notch surface. This observation highlights the rich information that can be gained in a 3D experiment that screens regions subjected to different stress states.

The impact of notch preparation should be highlighted: The notch prepared by a milling tool showed a smooth surface and damage nucleated away from the notch. In contrast, the specimen with the defective notch made by wire cutting showed a rough surface with remaining material and also initial damage features close to the notch root. For the latter sample crack initiation and propagation occurred at lower strain levels and the location of the damage initiation was shifted to the notch surface. This is consistent with the fact that the defect decreased the effective notch root radius substantially and shifted the maximum principal stress location closer to the notch root. This underlines the importance of notch cutting quality for comparable polymer toughness values testing. Testing using a defective notch will provide conservative lower bound values.

In the present study, the individual spherulite can not be resolved. It would be interesting to synthesize polymers (such as PP) which can possess larger spherulitic sizes in order to map their microstructual 3D damage at the spherulitic level.

Finally, a 3D FE modelling of the experiment would be interesting in order to quantify further the link between maximum principal stresses and the dimensions of the damage features. the use of ANKA GPU computing servers. Dr. Laurent Corté is acknowledged for helpful discussions. Dr. Anthony Bunsell is greatly thanked for proofreading.

Supporting

Information Available: Physicochemical properties of the PA6 under study; the anti-buckling device and the entire in situ loading machine; concise comparison of the principles of CT and CL as well as their respective experimental setups at ANKA and ESRF; results of L-S plane at distances of ∼ 13 µm and 200 µm ahead of the notch tip; video 1 shows the crack initiation and propagation in 3D; video 2 shows distinct damage features and the crack shape in 3D. This material is available free of charge via the Internet at http://pubs.acs.org/.

Figure 1 :

 1 Figure 1: (a) Compact-tension like specimen geometry. (b) In situ loading device and the 2 mm thick notched specimen. The notch radius is ∼ 0.25 mm (the anti-buckling device is not shown here). L: loading direction, P: crack propagation direction, S: short transverse direction. (c) Schematic of 3D synchrotron computed laminography for imaging of regions of interest inside plate-like specimens.

Figure 2 :

 2 Figure 2: Comparison of the two initial notch surfaces in 3D. (a) defect-free notch, and (b) defective notch. The notch (black), notch curvature (golden), and polymer matrix (grey) are rendered independently.

Figure 3 :

 3 Figure 3: Reconstructed 2D cross-sections (L-P) at the mid-thickness of the PA6 (defect-free notch) sheet plane, showing homogeneously initiating and growing damage features lying in distinct zones which are well associated with stress levels. (a) As-received material; (b) CMOD = 2.5 mm; (c) CMOD = 4.5 mm; (d) CMOD = 6 mm. The inset is an enlarged view of the red dashed box, zone 0, i, ii and iii are clearly discernible.

Figure 4 :

 4 Figure 4: 3D Cavity distribution at CMOD = 6 mm of the defect-free notch specimen. (a) cavity dimensions at the mid-thickness along the crack propagation (P) direction; cavity dimensions along the through-thickness (S) direction, from the center to the surface of the specimen at (b) a distance ∼ 54 µm ahead of the notch tip (inside zone i ), (c) a distance ∼ 225 µm ahead of the notch tip (inside zone ii ), and (d) a distance ∼ 840 µm ahead of the notch tip (inside zone iii ).

Figure 5 :

 5 Figure 5: (a) χ P χ S except for zone ii. (b) and (d): eventually χ P = χ S but the value slightly evolves through the thickness. Increasing or decreasing towards the surface depending on the relative decrease of P/S and L. (c) corresponding to zone ii shows the only case where χ P > χ S .

Figure 6 :

 6 Figure 6: 2D cross-sections (L-P) at the mid-thickness of the PA6 (defective notch) sheet plane of reconstructed laminography data showing crack initiation at incremental CMODs. (a) As-received material; (b) CMOD = 0.35 mm; (c) CMOD = 0.525 mm; (d) CMOD = 0.7 mm.

Figure 6 :

 6 Figure 6: (Continued) 2D cross-sections (L-P) at the mid-thickness of the PA6 (defective notch) sheet plane of reconstructed laminography data showing crack growth at CMODs. (e) CMOD = 1.05 mm; (f) CMOD = 1.4 mm; (g) CMOD = 2.1 mm; (h) CMOD = 2.8 mm.

Figure 6 :

 6 Figure 6: (Continued) 2D cross-sections (L-P) at the mid-thickness of the PA6 (defective notch) sheet plane of reconstructed laminography data showing crack initiation and propagation at CMODs. (i) CMOD = 3.5 mm; (j) CMOD = 4.2 mm; (k) CMOD = 4.9 mm; (l) CMOD = 5.25 mm.

Figure 7 :

 7 Figure 7: 3D renderings (330 µm thick volume) around the (defective notch) specimen's midplane at different CMODs. (a) CMOD = 0 mm; (b) CMOD = 2.8 mm; (c) CMOD = 4.2 mm; (d) CMOD = 5.25 mm. The notch is in black, the polymer matrix is shown in grey and the damages are shown varying from light to deep golden depending on the cavitation density of a specific area. Information on the supplementary loading steps is given in video 1.

Figure 8 :

 8 Figure 8: 2D cross-sections at the through-thickness (L-S) plane (through the magenta line indicated in Figure 6i) at a distance of ∼ 350 µm from the initial notch shown for different CMODs. (a) CMOD = 3.5 mm; (b) CMOD = 4.2 mm, the insets are magnified images of the red dashed boxes. The width-height cross-section of the tunnel crack is labeled in green; (c) CMOD = 4.9 mm.

Figure 9 :

 9 Figure 9: 3D representations of typical damage features in terms of void column in zone i and elongated cracks in zone ii of both (a) defect-free notch at CMOD = 6 mm, and (b) defective notch at CMOD = 4.2 mm deformed specimens. The scale bars are 100 µm. More information in video 2.

Figure 10 :

 10 Figure 10: Different specimen geometries that generate different triaxial stress states and corresponding typical associated damage features: (a) Smooth round bar specimen; 27 (b) Notched round bar specimen with radius 4 mm; 28 (c) Notched round bar specimen with radius 0.45 mm; 69 (d) Compact-tension like specimen with notch radius 0.25 mm.

Figure 11 :

 11 Figure 11: (a) 2D section of reconstructed SRCL data in the through-thickness (L-S) plane at a distance of ∼ 74 µm from the initial notch at CMOD = 1.4 mm. (b) Zoomed area of the red dashed box in (a); (c) the highlighted image of (b).

Table 1 :

 1 The parameters of two X-ray laminographic imaging experiments

		Defect-free	Defective
		notch		notch
	Synchrotron	ESRF		ANKA
	Notch machining milling		rotating
		tool cut	wire saw
				cut
	Notch root radius 0.25 mm	0.28 mm
	Loading steps	10		12
	X-ray energy	25	keV	White-
		monochro-	beam with
		matic		15	keV
				peak
	Projections	1500		2000
	Voxel size	0.75 µm	1.1 µm
	Detector	2048×2048	2016×2016
		pixels		pixels
		Frelon 58	CMOS
				pco.dimax
	FOV	∼1.5×1.5	∼2.2×2.2
		mm 2		mm 2
	Scanning type	local		full-field
		laminogra-	laminogra-
		phy		phy
	Exposure time	50 ms		20 ms
	Scintillator	20	µm	10 µm thin
		GGG:Eu	LSO:Tb
		crystal		crystal

DOI: 10.1002/macp.201500316

Acknowledgement The authors thank ANKA and ESRF for the provision of beamtime MAT-50 and MA-1276, respectively. We thank Harald Schade, Stefan Uhlemann, Tomy dos Santos Rolo, David Haas, Thorsten Müller for their excellent beamline support, and Dr. Martin Köhl for his excellent assistance for DOI: 10.1002/macp.201500316

Graphical TOC Entry

The in-situ micro-damage mechanisms in semicrystalline PolyAmide 6 are examined in 3D by X-ray imaging for crack initiation and propagation in a large flat specimen. The characteristic sizes of the damage features are quantified. Distinct damage zones are found well distributed ahead of the initial notch, and characterized as a function of stress states. The formation of tunnel cracks during crack propagation and the effect of notch defect are discussed.


Aekta Aggarwal

Paola Goatin

Crowd Dynamics through Non-Local Conservation Laws

Keywords: Nonlocal Conservation Laws, Lax-Friedrichs Scheme, Lane Formation AMS MSC: 35L65, 65M12

We present a Lax-Friedrichs type scheme to compute the solutions of a class of non-local and non-linear systems of conservation laws in several space dimensions. The convergence of the approximate solutions is proved by providing suitable L 1 , L ∞ and BV uniform bounds. To illustrate the performances of the scheme, we consider an application to crowd dynamics. Numerical integrations show the formation of lanes in groups moving in opposite directions.

Introduction

We consider the following class of non-local systems of N conservation laws in 2 space dimensions:

∂ t U k + ∂ x f k (t, x, y, U k , η * U ) + ∂ y g k (t, x, y, U k , ϑ * U ) = 0, k ∈ {1, . . . , N }, (1.1)
where Here, (η * U )(t, x, y) and (ϑ * U )(t, x, y) are vectors in R m whose ℓ-component, for ℓ = 1, . . . , m, is

U (t,
(η * U ) ℓ (t, x, y) = R 2 N k=1 η ℓ,k (x -x ′ , y -y ′ ) U k (t, x ′ , y ′ ) dx ′ dy ′ ∈ R , (ϑ * U ) ℓ (t, x, y) = R 2 N k=1 ϑ ℓ,k (x -x ′ , y -y ′ ) U k (t, x ′ , y ′ ) dx ′ dy ′ ∈ R .
In (1.1), as usual, t is time, x, y the space variables, U the vector of the unknown densities, (f k , g k ) k=1,...,N the matrix valued flow and η and ϑ are smooth convolution kernels. The coupling among the equations is only due to the non-local terms η * U and ϑ * U .

The above class of systems can model many physical phenomena such as sedimentation [START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF], one-dimensional vehicular traffic [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF][START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF], biological applications, e.g., in structured population dynamics [START_REF] Perthame | Transport equations in biology[END_REF], supply chains [START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF], conveyor belts [START_REF] Göttlich | Modeling, simulation and validation of material flow on conveyor belts[END_REF], etc. The nonlocal nature of (1.1) is particularly suitable to describe the behavior of crowds, where each individual moves according to the average crowd density/density variation in a neighborhood of his position. The size of this neighborhood is related to the individual visual horizon. Other crowd dynamics models fitting into (1.1) were recently considered in the literature, see for example [START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF][START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF].

Here, we use the Lax-Friedrichs type algorithm that was recently proposed in [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF], generating a sequence of approximate solutions to (1.1) which, up to a subsequence, converges to a weak entropy solution of (1.1). The key point in the related convergence proof is to obtain a uniform bound on the total variation of the approximate solutions. However, a bound on the L ∞ norm and the L 1 Lipschitz continuous dependence of the solution on time are also obtained. These estimates allow to extend to a wider class of equations the existence results in [START_REF] Amorim | On the numerical integration of scalar nonlocal conservation laws[END_REF][START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF][START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF][START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF][START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF]. In particular, the actual number of space dimensions is not relevant, no divergence free assumption is required, see [START_REF] Chainais-Hillairet | Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate[END_REF], and the convolution kernels η and ϑ are not required to have compact support.

We are interested in crowd dynamics models that capture the phenomenon of so-called lanes or paths or trails formation, which is a behavior commonly observed in the engineering literature, when two groups of people move in opposite directions. Some models were recently introduced in the mathematical literature in this regard, see [START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF][START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF][START_REF] Tory | An adaptive finite-volume method for a model of two-phase pedestrian flow[END_REF]. A new nonlocal model close to [8, eq. (3.5)] is proposed in Section 3, which is shown to fit the settings of Theorem 2.2, see also [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF]Theorem 2.3], and hence possesses a weak entropy solution. The numerical scheme developed in [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF] is then used to compute solutions of the proposed model, showing the formation of various patterns, in particular of lanes.

The next section revisits the main results of [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF], while Section 3 is devoted to sample numerical integrations of (1.1).

Main Results

Throughout, we denote

R + = [0, +∞[, R N + = [0, +∞[ N ,
|ξ| for a the modulus of a scalar ξ ∈ R and, for ξ ∈ R 2 , ξ stands for the usual Euclidean norm. The following assumptions on the functions f k = f k (t, x, y, u, A), g k = g k (t, x, y, u, B) and η, ϑ in (1.1), for k = 1, . . . , N are required:

(H0) f k , g k ∈ C 2 (R + × R 2 × R × R m ; R); ∂ u f k , ∂ u g k ∈ L ∞ (R + × R 2 × R × R m ; R); for all t ∈ R + , (x, y) ∈ R 2 and A, B ∈ R m , f k (t,
x, y, 0, A) = g k (t, x, y, 0, B) = 0.

(H1) There exists a constant M > 0 such that max t,x,y,A

∂ x f k , ∂ 2 xx f k , ∂ 2 xy f k , ∇ A f k , ∂ x ∇ A f k , ∂ y ∇ A f k , ∇ 2 AA f k ≤ M |u|, max t,x,y,B ∂ y g k , ∂ 2 yy g k , ∂ 2 xy g k , ∇ B g k , ∂ x ∇ B g k , ∂ y ∇ B g k , ∇ 2 BB g k ≤ M |u|. (H2) ∂ u f k , ∂ u g k ∈ W 1,∞ (R + × R 2 × R × R m ; R). (H3) η, ϑ ∈ (C 2 ∩ W 2,∞ )(R 2 ; R m×N ).
We recall the definition of solution to the Cauchy problem for (1.1), see also [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF][START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF][START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF][START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF][START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF][START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF].

Definition 2.1. Let U o ∈ L ∞ (R 2 ; R N ). A map U : [0, T ] → L ∞ (R 2 ; R N ) is a solution to (1.1) with initial datum U o if, for k = 1, . . . , N , setting for all w ∈ R f k (t, x, y, w) = f k t, x, y, w, (η * U )(t, x, y) , gk (t, x, y, w) = g k t, x, y, w, (ϑ * U )(t, x, y) ,
the map U k is a Kružkov solution to the scalar Cauchy problem

∂ t U k + ∂ x f k (t, x, y, U k ) + ∂ y gk (t, x, y, U k ) = 0 U k (0, x, y) = U k o (x, y) . (2.1)
Above, by Kružkov solution we refer to [START_REF] Kružhkov | First order quasilinear equations with several independent variables[END_REF]Definition 1]. It is immediate to observe that (H0) ensures that U (t, x, y) ≡ 0 solves (1.1) in the sense of Definition 2.1. Fix a rectangular grid with mesh sizes ∆x and ∆y in R 2 and a time step ∆t. Also, define

t n = n ∆t, n ∈ Z ; x i = i ∆x, x i+1/2 = i + 1 2 ∆x, i ∈ Z ; y j = j ∆y, y j+1/2 = j + 1 2 ∆y, j ∈ Z ; λ x = ∆t/∆x , λ y = ∆t/∆y . Throughout, an initial datum U o ∈ (L ∞ ∩ BV)(R 2 ; R N
) is fixed and we denote

u k,0 ij = 1 ∆x ∆y x i+1/2 x i-1/2 y j+1/2 y j-1/2 U k o (x, y) dx dy for i, j ∈ Z , k ∈ {1, . . . , N }.
Iteratively, we define a piecewise constant approximate solution u ∆ (t, x, y)

≡ u 1 ∆ , . . . , u N ∆ (t, x, y) to (1.1) such that u k ∆ (t, x, y) = u k,n ij for      t ∈ [t n , t n+1 [ , x ∈ [x i-1/2 , x i+1/2 [ , y ∈ [y j-1/2 , y j+1/2 [ , where n ∈ N i, j ∈ Z k ∈ {1, . . . , N }
through the 5-points algorithm based on dimensional splitting, see [9, Section 3]:

u k,n+1/2 ij = u k,n ij -λ x F k,n i+1/2,j (u k,n ij , u k,n i+1,j ) - F k,n i-1/2,j (u k,n i-1,j , u k,n ij ) u k,n+1 ij = u k,n+1/2 ij -λ y G k,n i,j+1/2 (u k,n+1/2 ij , u k,n+1/2 i,j+1 ) -G k,n i,j-1/2 (u k,n+1/2 i,j-1 , u k,n+1/2 ij ) (2.2)
where for fixed α, β in 0, 2/3 , we define

F k,n i+1/2,j (u, v) = f k,n i+1/2,j (u) + f k,n i+1/2,j (v) 2 - α(v -u) 2 λ x , f k,n i+1/2,j (u) = f k (t n , x i+1/2 , y j , u, A n i+1/2,j ) , G k,n i,j+1/2 (u, v) = g k,n+1/2 i,j+1/2 (u) + g k,n+1/2 i,j+1/2 (v) 2 - β(v -u) 2 λ y , g k,n+1/2 i,j+1/2 (u) = g k (t n+1/2 , x i , y j+1/2 , u, B n+1/2 i,j+1/2 ) .
The convolution terms are computed through quadrature formulae, so that they are given by

A n i+ 1 2 ,j =∆x∆y l,p∈Z N k=1 η 1,k i+1/2-l,j-p u k,n l+1/2,p , . . . , l,p∈Z N k=1 η m,k i+1/2-l,j-p u k,n l+1/2,p B n+1/2 i,j+ 1 2 =∆x∆y l,p∈Z N k=1 ϑ 1,k i-l,j+1/2-p u k,n+1/2 l,p+1/2 , . . . , l,p∈Z N k=1 ϑ m,k i-l,j+1/2-p u k,n+1/2 l,p+1/2 (2.3)
where, for instance, u k,n l+1/2,p is a convex combination of u k,n l,p and u k,n l+1,p , moreover

η ℓ,k i+1/2,j = η ℓ,k (x i+1/2 , y j ) and ϑ ℓ,k i,j+1/2 = ϑ ℓ,k (x i , y j+1/2 ) . (2.4)
Throughout, ∆t must satisfy the CFL condition

λ x ≤ min{6α, 4 -6α, 1} 1 + 6 max k ∂ u f k L ∞ , λ y ≤ min{6β, 4 -6β, 1} 1 + 6 max k ∂ u g k L ∞ , (2.5)
and we assume that the mesh sizes are sufficiently small, in the sense that ∆x, ∆y ≤ 1/(3M ), where M is as in (H1). The main result in [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF] is as follows:

Theorem 2.2. Let assumptions (H0), (H1), (H2), (H3) and conditions (2.5) hold. Fix an initial datum

U o ∈ (L 1 ∩ L ∞ ∩ BV)(R 2 ; R N + ).
Then, the algorithm (2.2) defines a sequence of approximate solutions which converges, up to a subsequence, to a solution U ∈ C 0 R + ; L 1 (R 2 ; R N + ) of (1.1) in the sense of Definition 2.1. Moreover, U is locally Lipschitz continuous with respect to the L 1 norm and for all k ∈ {1, . . . , N } and for all t ∈ R + , the following bounds hold:

U (t) L ∞ (R 2 ;R N ) ≤ e C t(1+ Uo L 1 ) U o L ∞ (R 2 ;R N ) , U k (t) L 1 (R 2 ;R) = U k o L 1 (R 2 ;R) , TV(U k (t)) ≤ e K1 t TV(U k o ) + K 2 e K1 t -1 , U (t + τ ) -U (t) L 1 (R 2 ;R N ) ≤ C(t) τ,
where C, K

∈ (L 1 ∩ L ∞ ∩ BV)(R 2 ; R N + ).
Then, the approximate solution u ∆ defined by the algorithm

(2.2) satisfies (a) (Positivity) u k ∆ (t, x, y) ≥ 0 for all k ∈ {1, . . . , N }, t ∈ R + and (x, y) ∈ R 2 . (b) (L 1 bound) u k ∆ (t) L 1 = u k ∆ (0) L 1 for all k ∈ {1, . . . , N } and t ∈ R + . (c) (L ∞ bound) For all t ∈ R + , u ∆ (t) L ∞ ≤ e C t(1+ Uo L 1 ) U o L ∞ , (2.6)
where C depends only on η, ϑ, f 1 , . . . , f N , g 1 , . . . , g N .

(d) (BV bound) For all n, for all t ∈ t n , t n+1 and for all k = 1, . . . ,

N ij u k,n i+1,j -u k,n ij ∆y + u k,n i,j+1 -u k,n ij ∆x ≤ e K1 t ij u k,0 i+1,j -u k,0 ij ∆y + u k,0 i,j+1 -u k,0 ij ∆x + K 2 e K1 t -1 .
where K 1 and K 2 are constants depending on U o L 1 and on the functions η, ϑ, f k and g k .

(e) (Discrete entropy condition) For all i, j ∈ Z, for all k = 1, . . . , N and for all κ ∈ R,

u k,n+1 ij -κ -u k,n ij -κ + λ x Φ k,n,κ i+1/2,j (u k,n ij , u k,n i+1,j ) -Φ k,n,κ i-1/2,j (u k,n i-1,j , u k,n ij ) + λ x sgn(u k,n+1/2 ij -κ) f k,n i+1/2,j (κ) -f k,n i-1/2,j (κ) + λ y Γ k,n,κ i,j+1/2 (u k,n+1/2 ij , u k,n+1/2 i,j+1 ) -Γ k,n,κ i,j-1/2 (u k,n+1/2 i,j-1 , u k,n+1/2 ij ) + λ y sgn(u k,n+1 ij -κ) g k,n+1/2 i,j+1/2 (κ) -g k,n+1/2 i,
Φ k,n,κ i+1/2,j (u 1 , u 2 ) = f k,n i+1/2,j (u 1 ∨ κ, u 2 ∨ κ) -f k,n i+1/2,j (u 1 ∧ κ, u 2 ∧ κ) , Γ k,n,κ i,j+1/2 (u 1 , u 2 ) = g k,n+1/2 i,j+1/2 (u 1 ∨ κ, u 2 ∨ κ) -g k,n+1/2 i,j+1/2 (u 1 ∧ κ, u 2 ∧ κ) . . (f ) (L 1 Lipschitz Continuity in time) For all n = 1, . . . , n, n ∈ N fixed, u ∆ (t n+1 ) -u ∆ (t n ) L 1 ≤ C ∆t .
where C is a constant depending on U o L 1 , TV(U o ), t n , λ x , λ y α, β and on the functions η, ϑ, f k and g k , for k ∈ {1, . . . , N }.

For the proofs of Theorem 2.2 and Lemma 2.3, refer to [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF].

Numerical Integrations

We refer to the class of nonlocal crowd dynamics models introduced in [START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF], for two populations trying to avoid each other and described by their densities U 1 , U 2 . In particular, we consider y) defines the preferred path for i-th population where each pedestrian moves in the preferred direction v i (x, y) but scales his speed according to (1

                   ∂ t U 1 + div   c 1 U 1 (1 -U 1 ) 1 -ε 1 U 1 * µ 1+ U 1 * µ 2 v 1 (x, y) -ε 2 ∇U 2 * µ 1+ ∇U 2 * µ 2   = 0, ∂ t U 2 + div   c 2 U 2 (1 -U 2 ) 1 -ε 1 U 2 * µ 1+ U 2 * µ 2 v 2 (x, y) -ε 2 ∇U 1 * µ 1+ ∇U 1 * µ 2   = 0. (3.1) For i, j = 1, 2, j = i, c i U i (1 -U i ), c i > 0 is the crowding factor and v i (x, y) = [v i1 , v i2 ](x,
-ε i (U i * µ)/ 1 + U i * µ
2 ), i.e., according to the density of its own population, which he sees in his horizon. The i-th population also deviates from its preferred trajectory due to the interaction with the individuals of the j-th population according to -ε j (∇U j * µ)/ 1 + ∇U j * µ 2 .

Lemma 3.1. System (3.1) fits into (1.1) setting, N = 2, m = 6 and

f (t, x, y, U, A) =      c 1 U 1 (1 -U 1 ) 1 -ε 1 A1 √ 1+A1 2 v 11 (x, y) -ε 2 A5 √ 1+A5 2 +A6 2 c 2 U 2 (1 -U 2 ) 1 -ε 1 A4 √ 1+A4 2 v 21 (x, y) -ε 2 A2 √ 1+A2 2 +A3 2      , g(t, x, y, U, B) =      c 1 U 1 (1 -U 1 ) 1 -ε 1 B1 √ 1+B1 2 v 12 (x, y) -ε 2 B6 √ 1+B5 2 +B6 2 c 2 U 2 (1 -U 2 ) 1 -ε 1 B4 √ 1+B4 2 v 22 (x, y) -ε 2 B3 √ 1+B2 2 +B3 2      , and
η = ϑ = µ ∂ x µ ∂ y µ 0 0 0 0 0 0 µ ∂ x µ ∂ y µ T , with A = η * U , B = ϑ * U , A, B ∈ R 6 . Moreover, if v i ∈ (C 2 ∩ W 2,∞ )(R 2 ; R 2 ) and µ ∈ (C 3 ∩ W 3,∞ )(R 2 ; R), for any compactly supported initial datum (U 1 o , U 2 o ) ∈ (L 1 ∩ L ∞ ∩ BV)(R 2 ; [0, 1]
2 ) problem (3.1) admits a solution defined for all t ∈ R + and satisfying the estimates in Theorem 2.2.

The proof is immediate and follows the same lines of [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF]Lemma 3.1]. Two different situations are considered and numerical integrations obtained with the algorithm described in Section 2 are exhibited. By Definition 2.1, [12, Theorem 3], the choices (3.5) and (3.9) of the initial data, the maximal density of solutions can not exceed 1, which is consistent with the results of the integrations.

Two Crowds Moving Radially in Opposite Directions

In the domain [-4, 4] × [-4, 4], two groups of people, one concentrated in a circle of radius √ 4.3 at the origin and other concentrated in an annulus between concentric circles (centered at origin) of radii √ 4.5 and √ 8.8, move radially outward and inward respectively. Thus, in (3.1) we choose: As in [START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF], the preferred path of each pedestrian is given by the sum g+δ. The vector g is tangent to the geodesic towards the pedestrian's target or 0 when pedestrians would not move. The vector δ describes the discomfort of pedestrians when walking close to a wall. It is perpendicular to the walls and pointing inward. Its maximum modulus is δ max = 0.8 at walls and decreases to zero at a distance δ r = 0.6 from walls. As indicated earlier, both crowds scale their speed due to their own density and deviate from their preferred trajectory due the presence of the other group present in their horizon. In (3.1) we choose: (3.9)

c 1 = c 2 = 4, ε 1 = 0.3, ε 2 = 0.7 (3.2) v 1 (x, y) = -v 2 (x, y) = [x, y] 1 + x 2 + y 2 , ( 3
c 1 = 2, c 2 = 1, ε 1 = 0.3, ε 2 = 0.7 (3.6)
The numerical integration of (3.1)-(3.9) is shown in Figure 3. The first group moves to the right. The second crowd, though prefers not to move, moves and forms lanes to pave the path for the first crowd to move. After the first crowd has completely vacated the room, the density of the second crowd keeps moving for a while.

Figure 4 shows the L 1 norm of the solution (U 1 , U 2 ) as a function of time t. It can be seen that the mass of the first crowd decreases to zero, while the mass of second crowd, remains constant.

  x, y) = (U 1 , . . . , U N )(t, x, y) ∈ R N , U k (t, x, y) ∈ R , k = 1, . . . y) ∈ R m×N , η ℓ,k (x, y) ∈ R , ℓ = 1, . . . , m , k = 1, . . . , N, ϑ(x, y) = , y) ∈ R m×N , ϑ ℓ,k (x, y) ∈ R , ℓ=1, . . . , m , k = 1, . . . , N.

. 3 ) 1 R 2 μFigure 1 Figure 1 :

 31211 Figure1shows the L 1 norm of the solution (U 1 , U 2 ) as a function of time t. It can be seen that the

3. 2 A

 2 Moving Crowd Interacting with a Standing Crowd Consider two groups of people in the domain [-1.7, 4.2] × [-1.7, 1.7] with a single exit at x = 4.2. At time 0, one group represented by U 1 starts moving towards the exit while the other group, U 2 , wants to stay and moves only as a reaction to the exiting population. The space available to the pedestrians is the rectangle [-1.4, 4.2] × [-1.4, 1.4] due to the presence of the walls of 0.3 width along {-1.7} × [-1.7, 1.7], and [-1.7, 4.2] × {-1.7} and [-1.7, 4.2] × {1.7}.

Figure 2 : 1 R 2 μ

 212 Figure 2: Solution to (3.1)-(3.5) with space mesh with ∆x = ∆y = .0125, λ x = λ y = .05, ∆t = 6.25E -4 and α = β = .3333. A symmetric(radial) formation of lanes is noticed for both the crowds. Each crowd forms lanes to pave paths for the other crowd to move in/out of the domain. The lanes formed by the two crowds are complimentary to each other. The formation of the lanes is noticed till the first crowd U 1 completely exits the domain.

Figure 3 :

 3 Figure 3: Solution to (3.1)-(3.9) computed with ∆x = ∆y = .01, λ x = λ y = .05, ∆t = .005 and α = β = .3333. Note that though the second crowd U 2 does not prefer to move, the model makes U 2 to adjust and develop lanes to allow the first crowd U 1 to move along its preferred path to right. The lane formation ceases to exist after the first crowd U 1 has completely vacated the domain.

Figure 4 :

 4 Figure 4: Time evolution of the L 1 norm of (U 1 , U 2 ) as a function of t.

  [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF] , K 2 and C(t) are constants defined in Lemma 2.3 below.The proof is based on Lemma 2.3 stated below. In view of [1, Remark 2.1], (H0) and (H3), Theorem 2.2 ensures the existence of a solution and the validity of the bounds above on any finite time interval, whenever the initial datum is compactly supported.

Lemma 2.3. Let assumptions (H0), (H1), (H2), (H3) and condition (2.5) hold. Fix U o

Acknowledgements. This work was partially supported by the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013) ERC Grant Agreement n. 257661. The authors would like to thank the third co-author R. M. Colombo (INDAM Unit, University of Brescia) who contributed significantly to this work.


Laurent Goffart

Parallel processing for the generation of saccades to simultaneously moving centrifugal targets

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Results

Laurent Goffart

Institut de Neurosciences de la Timone, CNRS & Aix-Marseille Université, Marseille, France

BB5

Methods

The simultaneous presentation of two static targets in the visual field leads to a saccade which lands on a location situated along a line that connects the two targets. The landing position depends upon the relative salience of the targets. The more salient the target, the stronger its influence on the landing position. A similar process happens during saccades made in response to two sequentially presented targets. Saccades with the shortest latencies land on the location of the first target whereas saccades with longer latencies land closer to the second target location. Obviously, the visuomotor brain does not switch abruptly between two targets; the transition is gradual and made continuously. The spatiotemporal properties of this averaging process were further explored by testing in the monkey the saccades made in response to two centrifugal targets moving rapidly and simultaneously in different directions.

In response to two centrifugal targets, the monkeys do not select one of the two targets prior to its saccadic foveation. A first saccade can indeed be made toward an intermediate location, a location which seems to correspond to a variable average of two commands weighted by their speed (or the duration of retinal exposure). The scattering of saccade endpoints between lines of target isochrony and the observation of some curved saccades indicate parallel and continuous visuomotor flows until the saccade is completed. Following the "averaging"-like saccade, the eye moves in the same direction with a slow velocity for duration which can last more than 200ms. Then, for "reasons" which would be interesting to further investigate, a "choice" is made between the two targets and a second saccade is made toward its changing location, with an accuracy which is relatively good.

Conclusion

Three rhesus monkeys (A, B, C) were used in this study. The movements of one eye were recorded with the scleral coil in magnetic field technique. After fixating a central target for a variable interval, two conditions were possible. During the single-target condition, one target moved toward the periphery along the cardinal (horizontal or vertical) or oblique axes. During the double-target condition, the central target was replaced by two identical targets moving away from the center toward the periphery. One target moved along an oblique axis while the other target moved along one of the cardinal axes. The speed of the target moving along the oblique axis was fixed (30 deg/s constant or accelerating from 0 to 60°/s or decelerating from 60 to 0°/s) whereas the target moving along the cardinal axis had a speed that was varied (from 15 to 60 deg/s, average speed, constant, accelerating or decelerating). Different pairs of speed were tested during separate block of trials. The monkeys were free to track anyone of the targets and were rewarded on every trial.

  cardinal target Paths of the oblique targetThe examination of saccades made in response to targets moving in the same visual quadrant show different scatterings of landing positions between the testing conditions. During the double-target condition, the landing positions were scattered in the region of the visual field which is situated between the paths of the targets. Although their distributions were biased toward the slower target, the orientation of the scatters depended upon the speed of the two targets: the landing positions were scattered between lines of isochronous target locations.


Aremu Idowu Raheem
email: raheemidowu50@yahoo.com

ANALYSIS OF THE EFFECTS OF OIL AND NON-OIL EXPORT ON ECONOMIC GROWTH IN NIGERIA

Keywords: Economic growth, oil exports, non-oil exports and Granger causality 1.0

This study investigated the role of oil and non-oil exports on the Nigerian economy over the period of 1981 to 2015. The ADF and PP unit root test, Johansen cointegration test, Granger causality test, impulse response functions (IRF) and variance decomposition (VD) were used in the analysis of the study. The cointegration test indicates that GDP, Oil and Non-oil exports were cointegrated. The Granger causality test indicates short run unidirectional causality running from oil export to GDP. There are also bidirectional long run causality relationship between oil export and GDP, and unidirectional long run causality running from non-oil export to GDP. The study result indicates that oil exports have inverse relationship with economic growth while non-oil exports have positive relationship with economic growth.

Introduction

Exports of goods and services represent one of the most important sources of foreign exchange income that ease the pressure on the balance of payments and create employments opportunities, [START_REF] Ruba | The Causal Relationship between Exports and Economic Growth in Jordan[END_REF]. Generally export activities are said to stimulate economic growth in a number of ways such as: through production and demand linkages, and economies of scale due to larger international markets.

Export led Growth is said to be an economic development strategy in which export expansion play a central role in a country's economic growth. Although practical evidence in support of export led growth may not be universal, it is widely acknowledge that carefully managed openness to trade through an export led growth can be a mechanism for achieving rapid growth, [START_REF] Giles | Export-Led Growth: A Survey of the Empirical Literature and Some Causality Results, Part 1[END_REF].

Nigeria been a developing country, has been grappling with the realities of developmental process not only politically and socially but also economically. In the 1960s, agriculture contributed 80% of the total export making agriculture the main stay of the Nigeria economy and the greatest foreign exchange earner. By the middle of 1970, the situation changed in favor of oil which then contributes 94% of total export making oil the main stay and the greatest foreign exchange earner of the Nigeria economy. Since 70's, till the present moment, oil has been playing the leading role in the Nigeria economy being the major source of foreign exchange. This mono cultural nature of the economy makes Nigeria susceptible to the effects of oil price shock. The over reliance of the country on oil was manifested in the inability of the country to manage her economy as a result of fall in the price of oil globally towards the end of 2015 which now push Nigeria economy into recession according to report. Therefore, not only that export is important for the survival of an economy but also the composition of export is of paramount important.

Several researchers which include, Javad et.al (2014); Kilavuz and Topcu (2012); [START_REF] Udude | Exports and Nigerian's Economic Growth: a Cointegration Analysis[END_REF]; Safdari and Zaroki (2012); Oyatoye et.al (2011); among others studied the relationship between export and economic growth within the neo-classical framework. These studies concluded in support of the export-led growth. Syed (2015) and Noula et.al (2013) found negative relationship and mixed effect of export on economic growth. Researchers such as Abayomi et.al (2015), Adedokun (2012), Baghebo and Atima (2013) among others had also tended to focus attention on the relationship between oil export and economic growth. While some studies had focused attention on the relationship between non-oil export and economic growth these include Ali Shah et.al (2015), Abogan et.al (2014), [START_REF] Kalu | Investigation of the Impact of Non-oil Revenue on Economic Growth in Nigeria[END_REF], Mehrara (2013). But few research works which include Muhrabadi et.al (2012[START_REF] Mehrara | The Relationship between non-oil Trade and GDP in Petroleum Exporting Countries[END_REF], Hosseini and Tang (2014) had sought to examine the effect of oil and non-oil export on economic growth. It is important therefore to contribute to this area by investigating the extent of the contribution of oil and non oil export on the growth of Nigeria economy and to offer appropriate suggestions based on the findings of the study.

Literature Review

Several studies have examined the export-led growth hypothesis, findings from the empirical literature point to the possibility of several types of relationships between exports and economic growth. Depending on the econometric model, data frequency, and the country or region studied, export is causing growth, growth is causing export, there is bidirectional causality, and there is no causality, [START_REF] Konya | Export-Led Growth, Growth-Driven Export, Both or None? Granger Causality Analysis on OECD Countries[END_REF]. Javad et.al (2014), examined the relationship between exports and economic growth in the industrial sector in Iran. Based on the research results, the hypothesis of a positive impact of increased exports on the growth of the industrial sector in Iran is to be accepted. [START_REF] Ruba | The Causal Relationship between Exports and Economic Growth in Jordan[END_REF], examined the causal relationship between economic growth and exports in Jordan using the Granger methodology in order to determine the direction of the relationship between the two variables during the period 2000-2012. The study found that there is a causal relationship going from the economic growth to Export, and not vice versa. [START_REF] Turan | An Empirical Study on Import, Export and Economic Growth in Albania[END_REF], observed the relationship between export, import and Gross Domestic Product (GDP) in Albania by using annual data for the period between 1984 and 2012. Different empirical researches and macro econometric models indicates that there is an equilibrium relationship between exports, imports and GDP in the long term. Based on the study done, the imports have negative relationship with GDP while exports have a significant positive relationship with GDP. [START_REF] Istaiteyeh | A causal Relationship Between Foreign Direct Investment, Economic Growth and Export: Empirical Case For Jordan[END_REF], analyzed the relationship between foreign direct investment, economic growth and exports in Jordan. The co-integration method and vector error correction model were applied. The results confirm the existence of long-term causal links between variables studied. The results show a positive impact of export on GDP, rather foreign direct investment has no effect on GDP.

Data and Methodology

For the purpose of this study, annual time series data for Nigerian for the period 1981 to 2015 will be collected from the Central Bank of Nigeria Statstical Bulletin of 2015. The model will consists of three variables: the gross domestic product (GDP), oil exports (OX), and non-oil exports (NOX). This study employed Vector Autoregressive Models (VAR)

technique for the analysis. α is the intercept, 1 α and 2 α are the slope coefficients to be estimated, lnGDP is the natural log of the real gross domestic product (GDP), lnOX is the natural log of real oil exports, lnNOX is the natural log of real non-oil exports and t ε is the error term. From table 2 above, the result of ADF and PP tests shows that all the three variables are integrated of order one and that none of the variables is integrated of order two.

Empirical Results and Discussion

Co-integration Test

From the unit root, all the variables are integrated of order one which justified the use of Johansen Co-integration test to examine whether a long-run association between variables exists and to examine the presence of co-integrating relationships in the data. The null hypothesis in the Johansen test is that there is no co-integration among variables and the alternative hypothesis states that there is co-integration. The estimated long-run equation above reveals that both variables are statistically significant. It also reveals that export of non-oil products (lnNOX) has positive relationship with economic growth in Nigeria while export of oil and gas products (lnOX) has negative effects on the economic growth of Nigeria. That is, for every one percent increase in oil export, the GDP will reduce by 14% while for every one percent increase in non-oil export, the GDP will be increased by 29%. Even though the contribution of non oil export to GDP during the period under consideration was small, it has positive impact on the economic growth. The negative effect of oil and gas export on the economic growth can be attributed to the resource curse hypothesis introduced by [START_REF] Auty | Sustaining development in mineral economies: The resource curse thesis[END_REF]. There has been a belief that the export of mineral resources is a blessing, especially for developing countries Nigeria inclusive to promote their economic growth. Nevertheless, [START_REF] Auty | Sustaining development in mineral economies: The resource curse thesis[END_REF] argued that developing countries usually lack the skill and technology for processing their mineral resources before such resources can be exported to other countries. [START_REF] Stokes | Another look at the export of raw materials and economic growth[END_REF] also found that the exports of raw materials or goods with low levels of processing are less likely to promote economic growth. the export of mineral resources does not necessarily contribute to economic growth and sometimes worsens the process of development, particularly for countries and regions with an abundance of mineral resources, Hosseini and Tang (2014). [START_REF] Palley | Lifting the natural resource curse[END_REF] from his own view added that the resource curse hypothesis happens because the revenues from these resources are usually mismanaged and are not channeled into productive sectors that can effectively stimulate economic growth, due to corruption and other institutional problems. Therefore, it is not surprising finding the negative impact of oil export on economic growth of Nigeria.

Having established the existence of long run co-integration between the variables within the period under study, the study will proceed to estimate VECM.

The VECM adopted in this paper is specified as follows: The table above shows the direction of causality between the variables with the VECM framework. The F-test results show the significance of the short-run causal effects, while the long-run causal effect based on the significance of the one period lagged errorcorrection term is presented with [ECT(-1)]. It is clear from the Table above that in the shortrun, there is only a unidirectional causality running from lnOX to lnGDP which means that lnNOX does not cause either lnGDP or lnNOX in the short-run. But in the long-run, there are bidirectional causality relationship between lnOX and lnGDP, and unidirectional causality running from lnNOX to lnGDP. Response of LOG(NOIL) to LOG(NOIL)

1 1 1 1 1 1 1 1 1 1 ln ln ln ln .....................(2) k k k t i t i i t i i t i t t i i i GDP GDP OX NOX EC α β δ φ λ ε - - - - = = = ∆ = + ∆ + ∆ + ∆ + + ∑ ∑ ∑ 2 2 2 2 2 1 2 1 1 1 ln ln ln ln ......................(3) k k k t i t i i t i i t i t t i i i OX GDP OX NOX EC α β δ φ λ ε - - - - = = = ∆ = + ∆ + ∆ + ∆ + + ∑ ∑ ∑ 3 3 3 3 3 1 3 1 1 1 ln ln ln ln ....................(4) k k k t i t i i t i i t i t t i i i NOX GDP OX NOX EC α β δ φ λ ε - - - - = = = ∆ = + ∆ + ∆ + ∆ + + ∑ ∑ ∑ Table5:

Granger Causality Test Results

Independent Variables ln GDP

∆ ∑ ln OX ∆ ∑ ln NOX ∆ ∑ ECT(-1) ln GDP ∆ _ 3.
Response to Cholesky One S.D. Innovations

Fig: 1

Impulse response function will allow us to study the effects of a shock in a particular variable on the other variables that are included in the model. According to the diagram above, the real GDP respond negatively to shocks in oil export and positively to shocks in non oil export. This figure complement the result presented earlier that oil export has negative relationship with GDP. The positive response of real GDP to shocks in non oil export approached zero by period nine, this can be linked to decrease in the contribution of non oil to export over the period of time.

Also, the figure shows the response of oil to shocks in real GDP and non oil export.

Oil export responds negatively to shocks in real GDP and positively to shocks in non oil export. Lastly, the response of non oil export to GDP was positive until fourth period when it turns negative while it responds positively to shocks in oil export.

Source: Computed by the Author

The results of Variance Decomposition (VDC) are presented in the table above. The results were reported for a 10-month horizon. The ordering of the variables is based on the Cholesky decomposition method which suggests the following order of the variables: LOG(GDP), LOG(OIL) and LOG(NOIL).

It can be seen from the table 7 above that in the short run, both oil and non-oil export caused variation in GDP with non-oil export responsible for 14% out of the total 15%. But in the long run, oil export was responsible for higher variation in GDP with 21% coming from oil export and 3% is coming from non-oil export. This signifies that both oil and non-oil exports are responsible for variation in GDP.

Conclusions

This study investigated the effect of oil and non-oil export on the economic growth of Nigeria using annual time series data from 1981 to 2015. The ADF and PP unit root test, Johansen cointegration test, Granger causality tests, Impulse response functions (IRF) and variance decomposition (VD) analysis were used in this study. The Johansen cointegration test showed all the variables are cointegrated. The cointegrating equation showed that oil export has negative effect on economic growth while non-oil export has positive effect on economic growth. The result of Granger causality tests showed that there is unidirectional causality running from lnOX to lnGDP in the short-run. But in the long-run, there are bidirectional causality relationship between lnOX and lnGDP, and unidirectional causality running from lnNOX to lnGDP. The impulse response functions indicated that when there is a shock to oil and non-oil export, GDP will respond positively to non-oil and negatively to oil export.

Based on the results of this study, Nigeria government should diversify her export from oil to non-oil because despite the huge revenue from oil, it impacted negatively on the economic growth of the country. Also, government should ensure that all the refineries are working up to date so that the country can depend less on the importation of finished product.

  Abogan et.al (2014), observed the impact of non-oil export on economic growth in Nigeria between 1980 and 2010. It examined the significant role of non-oil export on economic growth which the previous studies might have ignored and the aggregate non-oil exports data used by them might bias their conclusions. This study revealed that the impact of non-oil export on the economic growth was moderate and not all heartening as a unit increase in non-oil export impacted positively by 29% on the productive capacity of goods and services in Nigeria during the period.[START_REF] Mehrara | The Relationship between non-oil Trade and GDP in Petroleum Exporting Countries[END_REF], investigated the role of oil and non-oil exports in the Syrian economic over the period of 1975-2010. The cointegration test indicates that GDP is positively and significantly related to oil and non-oil exports. The Granger causality test indicates bidirectional short-run causality relationships between GDP, oil exports and non-oil exports. There are also bidirectional long-run causality relationship between non-oil exports to GDP, and unidirectional long-run causality relationship running from oil exports to GDP.The study result indicates that oil exports have the biggest effect on the GDP.Syed et.al (2015), estimated the relationship between Gross domestic product (GDP) and agricultural and non-agricultural exports for Pakistan employing Johansen co-integration technique by using secondary data for the period 1972-2008. It was found that agricultural exports have a negative relationship with economic growth of Pakistan while non-agricultural exports have positive relation with economic growth.

  The findings showed that the agricultural exports have mixed effect on economic growth in Cameroon. Coffee export and banana export has a positive and significant relationship with economic growth. On the other hand, cocoa export was found to have a negative and insignificant effect on economic growth.

	correlated with GDP while Exports (EXC) was negatively related with GDP. The short-run
	dynamics adjusts to the long-run equilibrium at the rate of 0.866% per annum.
	Noula et.al (2013), explored and quantified the contribution of agricultural exports to
	economic growth in Cameroon. It employs an extended generalized Cobb Douglas
	production function model, using food and agricultural organization data and World Bank
	Data from 1975 to 2009.
	The result of Co-
	integration reveals existence of one positive co-integrating equation. The result of Causality
	test show short run and long run causality run from GDP to exports. The result concludes that
	both in short and long run only growth in production cause exports growth.
	Safdari and Zaroki (2012), observed the effect of exports on economic growth
	(industry & mining sector, services and agriculture). The data were collected from 1961-2006
	and were analyzed using Ordinary Least Squares (OLS) model. The results of this study show
	that each section export growth has a positive effect on the growth of value added in the same
	section. But the effect of export growth on the value added in industry and mining sector is
	more than other sectors.
	Mehrabadi et.al (2012), examined the effects of oil and non-oil export on economic
	growth. Time series data and the method of VAR (Vector Auto Regressive) were used in the
	analysis. It was found that both oil and non-oil export had positive effect on the economic
	growth of Iran.
	Udude and Okulegu (2012), examined whether there is bi-directional relationship
	between exports and economic growth in Nigeria. It also tries to evaluate significant impact
	of exports on the economic growth in Nigeria. It was found that there exist a long-run
	relationship with economic growth and export in Nigeria. Having integrated the short run
	dynamics and long run equilibrium, Imports (IMP) and Exchange Rate were positively

[START_REF] Abbas | Causality between Exports and Economic Growth: Investigating Suitable Trade Policy for Pakistan[END_REF]

, examined the causal relationship between GDP and exports for the period of 1975 to 2010. The aim of the study is to check affectivity of export promotion policy adopted by Pakistan during 1990s. Johansen test of Co-integration and Granger Causality employed to determine short run and long run causality.

  The basic model employed in this paper can be expressed as

	follows:
	ln	0 .t 1 2 ln ln t t t GDP OX NOX α α α ε = + + +
	Where 0

...............................................(1)

Table 1 : Descriptive Statistics

 1

		lnGDP	lnOX	lnNOX
	Mean	10.19444	6.407228	3.090832
	Median	10.01381	7.100439	3.211767
	Maximum	11.14221	9.569633	7.030124
	Minimum	9.530920	1.974248	-1.593565
	Std. Dev.	0.519951	2.656751	2.717442
	Skewness	0.492623	-0.428095	-0.160318
	Kurtosis	1.862179	1.757264	1.871064
	Jarque-Bera	3.303632	3.321289	2.008568
	Probability	0.191701	0.190016	0.366307
	Sum	356.8054	224.2530	108.1791
	Sum Sq. Dev.	9.191874	239.9832	251.0726
	Observations	35	35	35

According to the table 1 above, all the series display a high level of consistency because their mean and median values are within maximum and minimum values of the series. Also, all the series are normally distributed which manifested in the probability value of the Jarque-Bera which accepts the null hypothesis at 10% significant level.

Unit Root Test Table2A: Unit Root Result LEVEL

		Augmented Dickey-Fuller		Phillip Perron
		Constant	Constant and	None	Constant	Constant and	None
			Trend			Trend
	lnGDP	0.724410	-2.219401	2.658256	1.799207	-2.311338	4.571776
	lnOX	-1.223317	-0.971545	1.909118	-1.274879	-0.971545	1.893532
	lnNOX -0.904175	-2.917969	1.381813	-0.998075	-3.057758	1.389652
			FIRST DIFFERENCE
	lnGDP -3.378729** -3.602994**	-1.985629	-3.220256**	-3.461891	-1.793479
	lnOX	-5.959428

*** -4.910210*** -4.869608*** -5.964082*** -6.640349*** -4.953879 lnNOX -6.792092*** -6.833623*** -5.236281*** -7.651386*** -10.75930*** -5.306522*** Table2B:

Summary of Unit Root Test

		Augmented Dickey-Fuller (ADF)		Phillip-Perron (PP)
		Level	First Difference I(d)	Level	First Difference	I(d)
	lnGDP	-2.219401 b	-3.602994 b**	I(1)	-2.311338 b	-3.220256 a**	I(1)
	lnOX	-1.223317 a	-5.959428 a***	I(1)	-1.274879 a	-6.640349 b***	I(1)
	lnNOX	-2.917969 b	-6.833623 b***	I(1)	-3.057758 b	-1075930 b***	I(1)
	Note: ***,** imply statistical significance at 1% and 5% respectively. Also, 'a' denotes
	model with constant, 'b' is for model with constant and trend.

Table 3 : Johansen-Juselius Cointegration Test

 3

	Null	Alternative r Max-Eigen	Critical Value Trace	Critical Value
	r ≤ 0	1	41.59585*	21.13162	72.89005*	29.79707
	r ≤ 1	2	25.26180*	14.26460	31.29420*	15.49471
	r ≤ 2	3	6.032397*	3.841466	6.032397*	3.841466
	Note: * denotes significance at 5% level
	Based on the table 3 above, the result of the Johansen cointegration test shows that
	there are three cointegrating equations based on the Trace and Max-Eigen value tests. That is,
	the results indicate that there is a long-run relationship between lnGDP, lnOX and lnNOX,
	the cointegrating equation was normalized by using the real GDP variable as thus:
	Table4: Cointegrating equation normalized with respect to GDP
	lnGDP		lnOX	lnNOX	C
	1.000000		0.140978	-0.294127	-10.16777
			(5.79359)	(-11.0478)
	From the Table 4 above, the long run lnGDP equation can be written as:

Table 6 : Results of the statistical diagnostic tests on the VECM

 6 Table 6 above presented the result of diagnostic tests which the VECM was subjected to. The results of normality tests, serial correlation tests and heteroskedasticity tests accepted the null hypothesis for all the series at 5%.

	The Dependent Variables	lnGDP	lnOX	lnNOX
	Normality tests	0.842300	1.334707	0.948559
		[0.656292]	[0.513065]	[0.622333]
	Serial correlation LM tests	1.732435	0.398559	0.173408
		[0.2454]	[0.8358]	[0.9643]
	Heteroskedasticity(ARCH) test 1.171977	1.181272	1.473184
		[0.3609]	[0.3568]	[0.2473]