Aldo Cocchi
email: cocchi@lma.cnrs-mrs.fr

O Montagnier
email: olivier.montagnier@ecole-air.fr

C Hochard
email: hochard@lma.cnrs-mrs.fr

Study of hourglass-shaped specimens for the analysis of compression behaviour in fibre direction of FRP composites using compression and four-point bending tests

Keywords: Glass fibres, Failure, Mechanical testing, Finite element analysis (FEA)

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

Introduction

Determination of compression properties of composite materials is a difficult task. As reported by multiple authors [START_REF] Woolstencroft | A comparison of test techniques used for the evaluation of the unidirectional compressive strength of carbon fibre-reinforced plastic[END_REF][START_REF] Harper | Problems associated with the compression testing of fibre reinforced plastic composites[END_REF][START_REF] Schneider | Determination of Compressive Properties of Fibre Composites in the In-plane Direction According to ISO 14126. Part 1: A Round Robin Test[END_REF][START_REF] Bogetti | Evaluation of the IITRI Compression Test Method for Stiffness and Strength Determination[END_REF] standard tests tend to underestimate compression strength of composites materials. These premature failures are mainly due to stress concentrations at the load introduction zones or, sometimes, uncontrolled buckling of the specimen. With modern day necessity to reduce weight in composites applications, an underestimation of compression properties can lead to overly conservative designs.

Many testing methods to characterize the compressive properties of composite materials can be found in literature. In the case of pure compression tests on constant cross-section specimens, standards like ASTM D3410 prescribe shear loading. In this case failure generally takes place near the tabs where the concentration of compressive and shear stress are maximum [START_REF] Tan | Stress analysis and the testing of celanese and iitri compression specimens[END_REF]. In end compression tests (standard or modified ASTM D-695), specimens have rectangular or dog-bone shape and anti-buckling fixtures are used.

In these tests failure generally occurs at the ends of the rectangular specimens or just after the connection radius of the dog-bone specimen, a zone of sharp cross-section variation introducing a stress concentration.

In bending,, failure usually occurs at the load point rather than supports due to the high stress concentration and high compressive stress in the material under the loading nose [START_REF] Cui | Contact finite element analysis of three-and four-point short-beam bending of unidirectional composites[END_REF][START_REF] Wisnom | The effect of specimen size on the bending strength of unidirectional carbon fibre-epoxy[END_REF]. A solution to avoid this kind of premature failure is to perform hinged buckling tests [START_REF] Wisnom | Constrained buckling tests show increasing compressive strain to failure with increasing strain gradient[END_REF][START_REF] Wisnom | Reduction in compressive strain to failure with increasing specimen size in pin-ended buckling tests[END_REF][START_REF] Arhant | Effect of sea water and humidity on the tensile and compressive properties of carbon-polyamide 6 laminates[END_REF][START_REF] Fukuda | Simplified compression bending test method for advanced composites[END_REF], which ensures failure in the middle of the specimen where the bending moment is at its maximum. However, the imposed moment and the stress state generate in the coupon depend on its deformed configuration.

Another solution to ensure failure away from the load introduction zones is to use waisted coupons with a smaller cross-section in the central zone whilst limiting the stress concentration generated by the crosssection variation. This technique has been used in tension on flat specimens with a large radius of curvature in the gauge zone to ensure a gradual cross-section variation [START_REF] De Baere | On the tension-tension fatigue behaviour of a carbon reinforced thermoplastic part II: Evaluation of a dumbbell-shaped specimen[END_REF][START_REF] Payan | Damage modelling of laminated carbon/epoxy composites under static and fatigue loadings[END_REF]. The radius of curvature is of the order of one meter and ensures failure at the centre of the specimen while guaranteeing an extremely low stress concentration. This geometry is described by Payan [START_REF] Payan | Damage modelling of laminated carbon/epoxy composites under static and fatigue loadings[END_REF] and also adopted by De Baere et al. [START_REF] De Baere | On the tension-tension fatigue behaviour of a carbon reinforced thermoplastic part II: Evaluation of a dumbbell-shaped specimen[END_REF] and more recently by Öztürk et al. [START_REF] Öztürk | Finite element based design of a new dogbone specimen for low cycle fatigue testing of highly filled epoxy-based adhesives for automotive applications[END_REF]. The authors refer to this shape as 'dumbbell' and 'dogbone' but this nomenclature is also used for common specimens with a central constant cross-section. To put emphasis on the constant curvature radius in the central part of the specimen, we will refer to the shape as 'hourglass'.

In this paper, specimens with this type of geometry are proposed for an ASTM D695 compression test and a four-point bending test. It should be noted that in the literature there are also specimens that present a smooth waisted section but in thickness to ensure failure away from the load introduction zones. For example, Wisnom [START_REF] Wisnom | Reduction in tensile and flexural strength of unidirectional glass fibre-epoxy with increasing specimen size[END_REF] has proposed waisted specimens with in-thickness plies drop-off for tensile purposes. Similarly, the authors have proposed tubular coupons with in-thickness plies-drop-off for tension and compression tests [START_REF] Eyer | Design of a composite tube to analyze the compressive behavior of CFRP[END_REF][START_REF] Eyer | Effet de l'endommagement transverse sur la résistance en compression sens fibre d'un composite carbone/époxy[END_REF][START_REF] Eyer | Rupture des matériaux composites en compression sens fibre: analyse de l'effet de l'endommagement[END_REF].

In this work, first of all, results obtained in tension tests are presented. Afterwards, we explain the design of a flat hourglass shaped specimen well-adapted for compression and four-point bending tests. Next, bending tests are presented and analysed using a geometrical protocol to estimate bending moment taking into account large rotations and displacements. An unbalanced satin woven E-Glass/Epoxy and a balanced satin woven Carbon/PEEK prepreg composites are tested to validate the protocol. Then, compression test are performed on a fixture conform to ASTM D695 using standard specimens and an adapted shorter version of the hourglass specimens. Only the E-Glass/Epoxy composite is tested in compression. Finally, results are presented and analysed.

Coupons preparation and tensile test

In this paper, two different materials are considered. The first one is an E-glass fibre reinforced epoxy produced by Hexcel (M42ST/1055) 5-harness satin weave. The main characteristic of this woven is to be unbalanced with 83% of the fibres in warp direction and only 17% in weft. Press moulding at 150 • C for 5h at a pressure of 5 bar is the recommended cure cycle. Due to impossibility to perform press moulding process, we opted for autoclave consolidation and increased pressure to 7 bar. Coupons were water-jet cut from plates with a ply thickness of 0.33 mm for an average fibre volume fraction of 48%. The second one is a Carbon/PEEK balanced woven composite (T300J, PiPreg 3106-1250P0378). Specimens were cut via water-jet from plates consolidated by press moulding at 400 • C with a cured ply thickness of 0.28 mm.

To begin, tensile tests were carried-out on

[0 • ] 8 , [90 • ] 8 , [45 • , -45 • ] 8 E-
glass/Epoxy specimens to assess the material tensile properties in longitudinal, transverse and shear directions. Specimens were shaped according to the geometry depicted in Figure 1 [START_REF] De Baere | On the tension-tension fatigue behaviour of a carbon reinforced thermoplastic part II: Evaluation of a dumbbell-shaped specimen[END_REF][START_REF] Payan | Damage modelling of laminated carbon/epoxy composites under static and fatigue loadings[END_REF], the dimensions of this specimen come from the adaptation of a 300 × 30 mm 2 rectangular tabbed specimen. To position the specimen in the machine grips, a 50 × 30 mm 2 rectangular zone was left at each extremity of the coupon. Consequently the length of the waisted central part was fixed to 200 mm, a ratio of 2/3 between the width at the extremities (30 mm) and at the centre (20 mm) was chosen, this resulted in a constant radius of 1002 mm in the waisted part.

The described shape ensures failure in the central zone of the coupon, avoiding premature ends failure. On the other hand, the radius is sufficiently large so that only a very limited in-plane stress concentration is generated in this central zone as depicted in Fig. 2a. The stress concentration generated in tension by the geometry depicted in Fig. 1 was estimated simulating the specimen on FEM software ABAQUS. The coupon was simulated as a 3D orthotropic solid of thickness 1 mm meshed with C3D6 elements and engineering constants reported in Tab. 1. A load of 2000 N, corresponding to a nominal stress (σ nom ) of 100 MPa for the 20 × 1 mm 2 central cross-section, was imposed on one end of the coupon while the other was fixed.

Stress distribution in the section of minimal width is depicted in Fig. 2b. In this article we define the in plane stress concentration coefficient of a coupon (S coupon ) as the ratio between the maximum stress in fibre direction (σ M AX ) and the nominal stress:

S coupon = σ M AX σ nom (1)
In the present case of an hourglass coupon with a curvature radius of 1002 mm in tension, the coefficient S hourglass T is equal to 1.029 (Fig. 2b).

Tests were carried out on an MTS1000 universal testing machine at a constant displacement rate of 1 mm/min. Two specimens were tested for each layup for a total of six specimens. Figure 3a shows tensile behaviour in fibre's direction. Failure in warp direction was brittle and occurred, as expected, at the centre of the specimens, [0 • ] 8 coupons failed at 750 ± 8 MPa for a strain of 2.59 ± 0.03%. Observed behaviour is stress distribution generated by the hourglass shape at the centre of the specimen slightly non-linear due to damage generated in the 17% of fibres in weft direction as described in [START_REF] Hochard | A generalized damage model for woven ply laminates under static and fatigue loading conditions[END_REF]. For [90 • ] 8 samples (Fig. 3b), failure also occurred at the centre of the coupon at 146 ± 5 MPa for a strain of 2.15 ± 0.07%. Transverse and shear direction behaviours (Figs. 3b and3c), are strongly non-linear due to damage and plasticity in matrix. For simulation purpose, only the linear part of the material constitutive equation in transverse and shear direction was taken into account (see red lines on the figures). The three orientations made possible to identify engineering constants as reported in Table 1.

A single tensile test on a [0] 8 balanced woven carbon/PEEK specimen was carried out as we are not interested in the full determination of the material's engineering constants. The hourglass geometry of Fig. 1 was utilized and the specimen failed once again at its centre. The result of the test is depicted in Fig. 4.

Behaviour in 0°direction is linear until failure that occurred at 920 MPa for a strain of 1.55 %. Young's

Modulus is measured at a value of 55 GPa.

Definition of hourglass geometry for compression and bending tests

It was chosen to adopt the hourglass shape for the specimens to test in bending and compression to ensure failure at the centre of the coupons. An adapted hourglass coupon was designed starting from the one utilized in tension [START_REF] De Baere | On the tension-tension fatigue behaviour of a carbon reinforced thermoplastic part II: Evaluation of a dumbbell-shaped specimen[END_REF][START_REF] Payan | Damage modelling of laminated carbon/epoxy composites under static and fatigue loadings[END_REF] (Fig. 1). Its dimensions were adjusted for usage in four-point bending and end-loading compression. The total length of the specimen was fixed to 80 mm as per the standardized dog-bone coupon for the ASTM D695 end-loading compression test (Fig. 5a). The length of the waisted part was fixed at 75 mm to leave a small straight part between the ends and the waisted part. It was chosen to keep the same ratio of 2/3 between the maximum and minimum width as in the tension hourglass specimens. Maximum and minimum width were chosen with the aim to maximize the radius of the waisted part as the greater the radius the smaller the stress concentration. First the two widths were taken equal to 19.0 mm and 12.7 mm as in the standardized bog-bone. The ratio between these widths is close enough to the desired one of 2/3 and the curvature radius of the waisted part in a sample with these widths would be

Four-point bending tests

Numerical simulation of the hourglass specimen geometry in bending

A parametric study was conducted on FEM software ABAQUS with the aim to find the right dimensions for the loading and supporting spans and the thickness of the coupon. The simulation model is depicted in Fig. 7. The fixture was simulated as four half cylinders (support and loading rollers) to reduce computational time. The four entities were defined as solids meshed with R3D4 elements. The support rollers were completely fixed. A downward displacement was imposed on the loading rollers and rotations were blocked.

Taking advantage of symmetry, only half coupon was simulated as a 2D shell entities meshed with S3

elements. The thickness of the specimen has been taken into account in the definition of the contact points.

Contacts between specimen and rollers were defined as frictionless to ease the simulation. Not having yet completed compression characterization, the material was simulated as linear elastic with the mechanical constants reported in Table 1. FEM convergence has been evaluated. S3 elements with a size of 0.6 mm and R3D4 elements with a size of 0.5 mm gave satisfactory results.

Due to limitations imposed by the fixture, the simulations were carried out until one of two conditions occurred:

1. load exceed 800 N (maximum applicable load of the fixture);

2. maximum specimen's deflection exceed 20 mm.

The two conditions were necessary as once one of the two was met during testing, the fixture's support rollers slipped out of their grooves, invalidating the final part of the test. Finally, the result of the simulation was considered valid if the average longitudinal stress in the elements at the centre of the specimen exceeded 800 MPa, meaning approximatively specimen failure. To limit deflection and applied load, the loading span was fixed at 50 mm and the support span at 80 mm.

Simulations were performed to find the optimal thickness of the coupon respecting the constraints dictated by the fixture. Finally a [0] 14 layup was choose as it failed for a load of 600 N (Fig. 9a), thicker laminates broke for loads greater than the limit of the fixture while, thinner laminate yielded an excessive deflection.

Fig. 8a depicts the longitudinal stress field in the out-most ply in tension. Even if the radius is smaller with respect to the 1002 mm of the tension specimen, simulation shows how the chosen hourglass shape helps at maintaining a homogeneous stress field in the central area of the specimen. The in-plane stress concentration coefficient of this coupon in bending, S hourglass B , is estimated to a value of 1.035 as depicted in Fig. 8b. It should be noted that the stress gradient near the edge of the coupon (Fig. 8b) should not be confused with the through-thickness gradient studied by Wisnom et al. in [START_REF] Wisnom | The effect of specimen size on the bending strength of unidirectional carbon fibre-epoxy[END_REF][START_REF] Wisnom | On the high compressive strains achieved in bending tests on unidirectional carbon-fibre/epoxy[END_REF][START_REF] Wisnom | Constrained buckling tests show increasing compressive strain to failure with increasing strain gradient[END_REF][START_REF] Wisnom | Reduction in compressive strain to failure with increasing specimen size in pin-ended buckling tests[END_REF] necessarily present in bending and much higher.

Simulations were also carried out to verify that parasite mid-span beam axial force were not generated by the fixture due to large displacements and rotations. If present, this force would subject the specimen to a mix of compression or tension and bending making more difficult behaviour identification. By comparing the strains on the two surfaces, no significant difference was found, meaning that no parasite longitudinal loading is generated which signify that, under ideal conditions, the specimen is loaded in pure bending.

Evaluation of hourglass specimen and methodology on an E-Glass/epoxy composite

Four-point bending tests were conducted on a MTS1000 universal testing machine using a fixture conforming to ASTM-D6272 on the specimen defined above (Fig. 6). Strain measurement was performed via 2D digital image correlation (DIC) using ARAMIS software and strain gauges were used to validate DIC measures.

The moment applied to the specimen was estimated with the geometrical procedure described in appendix to this article starting from the machine load and displacement without the need for a complex FEM analysis. The result of this geometrical procedure, using only the displacement and the force recorded from the machine, leads to a non-linear law linking applied load and imposed moment (solid black line in Fig. 9a).

This curve can be compared to the results of the FEM simulation (dash-dotted black line). They are in very good agreement except for the final part where damage starts to develop in the specimen. The figure shows also a large difference between a linear computation of the moment (grey dotted line) and a more realistic one. This discrepancy is also noticeable in the load versus displacement plot (Fig. 9b) where simulation and test result are in very good agreement. In conclusion, this comparison confirms previous assumptions and the approach proposed to compute bending moment.

Strains were measured via 2D DIC on one side of the specimens, on a 10×4.6 mm 2 surface located in the central part (Fig. 10). The camera used was equipped with a CCD chip with a resolution of 2448×2050 pixel 2 and an objective with a focal length of 50 mm. The camera was positioned 0.60 m away from the specimen. The acquisition was set to 20 pictures per millimetre of displacement with an exposure time between 40 ms and 65 ms depending on the speckles pattern on the coupons and an aperture of f /8. DIC generates a 2D mesh on the specimen surface (visible in Fig. 10) and computes the strains in its nodes. Strain values were averaged lengthwise on the rows of the mesh to obtain their thickness-wise distribution:

ε avg (y) = 1 N nodes N nodes i=1 ε(x i , y) (2)
Fig. 11 shows strains estimated after this averaging method for two example of specimens just before collapse (white points). In accordance to Kirchhoff hypothesis, strains are linear along the thickness. These measures show the absence of inter-ply delamination due to the continuity of the deformation and lack of 3D effects on the free edge. This permits to estimate strains on top and bottom surfaces via linear extrapolation that compression and tensile modulus are similar up to failure but not necessarily linear. For the case of the E-glass/Epoxy studied here, material behaviour in tension was demonstrated as slightly non-linear (Fig. 3a) due to the presence of irreversible transverse damage in the 17% of weft fibres. To justify the symmetric strain distribution, compressive behaviour must also be slightly non-linear, in this case a fibre non-linearity.

Here we will neglect the slight non-linearity and approximate the material constitutive law as linear with compression and tensile modulus equivalent and constant up to failure:

σ(y) = Eε avg (y) (3)
The assumption of linear behaviour leads to a great simplification in the estimation of maximum stresses in the out-most plies applying Classical Beam Theory:

σ max = M f I t 2 = 6M f wt 2 ( 4
)
where I is the area moment of inertia of the specimen at the minimum width. This classical results gives also directly the Young's modulus with the relation

E = 6M f /(ε max wt 2 ).
Skin stress-strain plot of the three tests are depicted in Fig. 12 using Eqs. 2 and 4 under the assumption of linear behaviour, this assumption appears to be valid up to a deformation of 2%. Stress failure values are reported in Table 2. All specimens have broken in the central part of the face in tension without any visible delamination phenomena. The only compression linked phenomenon that we were able to notice was localized buckling of woven tows in correspondence of the interlocks where tow curvature is maximum (see Fig. 10). This phenomenon was limited to the tows, the rest of the surface did not present any sign of As we can see on Fig. 12, strain values from DIC and gauges are very close except for the final part of the compression curves. Unsticking of the strain gauges explains these large discrepancies. This comparison validates the strain estimation protocol via DIC and shows the advantage of this method here.

Finally, the measured values for the average ultimate tensile stress, the ultimate tensile strain and the Young's modulus are equal to 881 ± 5.51 MPa, 3.28 ± 0.09% and 29 ± 0.7 MPa, respectively. Higher values of tensile strain at failure are observed in bending tests in comparison with tensile tests (Tab. 2). This result on the ultimate tensile strain has also been found by Wisnom and Atkinson in four-point bending tests in [START_REF] Wisnom | Reduction in tensile and flexural strength of unidirectional glass fibre-epoxy with increasing specimen size[END_REF]. These authors linked this phenomena to the size effect studying specimens of different thicknesses. This effect can be also observed in the presence of in-plane stress gradients generated by a stress concentration (plate with a hole, a notch, etc.) where strains much higher than tensile failure strain are recorded before failure [START_REF] Hochard | A ply scale non-local fibre rupture criterion for CFRP woven ply laminated structures[END_REF]. It should be noted that, for the thickness tested here, E-glass/epoxy coupons always failed in tension. In conclusion, the hourglass shape assures to obtain failure at the centre of the specimen in four-point bending.

Evaluation of hourglass specimen and methodology on a carbon/PEEK composite

To complete the analysis of the optimized flat hourglass geometry, the same testing protocol was applied to a carbon/PEEK 5-harness satin weave 3K woven composite (T300J, PiPreg 3106-1250P0378). Only one hourglass specimen with the same dimensions depicted in Fig. 6 was tested. The specimen failed again in the centre but, in this case, on the compression side. Test was stopped as soon as kink-bands appeared on the surface of the specimen. Fig. 13 depicts the strain field and local out-of-plane fibre buckling on the surface similar to that observed by Soutis and Turkmen [START_REF] Soutis | Moisture and Temperature Effects of the Compressive Failure of CFRP Unidirectional Laminates[END_REF].

Fig. 14 shows strain distribution for the Carbon/PEEK specimen. Material behaviour is non-linear as extremal strains sensibly differ (approximately, 1.45 % and -1.6 % on the figure) [START_REF] Jones | Apparent Flexural Modulus and Strength of Multimodulus Materials[END_REF][START_REF] Mujika | Determination of tensile and compressive moduli by flexural tests[END_REF], the bending neutral axis is shifted towards the tensile part implying a tangential compressive modulus inferior to Young's Modulus, consequentially the estimation of the material behaviour is complicated by this non-linearity and a different procedure has to be used to analyse the results of the test. To estimate the stress-strain curve, the power-law model already proposed by Allix [START_REF] Allix | Modelling and identification of the mechanical behaviour of composite laminates in compression[END_REF] is used, as in [START_REF] Montagnier | Compression Characterization of High-modulus Carbon Fibers[END_REF], for the compressive constitutive law:

σ = (1 + β ε γ ) E ε ( 5
)
where E is the Young's modulus identified in tension (see. 2).

This permits to identify only the non-linearity parameters γ and β for the compressive constitutive law (Eq. 5). Using the error minimization procedure proposed in [START_REF] Montagnier | Compression Characterization of High-modulus Carbon Fibers[END_REF], γ was found near a value of 1 because the In bending, coupons are subjected to a through thickness strain gradient in the failure zone. As found in literature, the effect of this gradient is to generate higher compressive failure strains compared to the ones measured in its absence [START_REF] Wisnom | Constrained buckling tests show increasing compressive strain to failure with increasing strain gradient[END_REF][START_REF] Wisnom | Reduction in compressive strain to failure with increasing specimen size in pin-ended buckling tests[END_REF][START_REF] Wisnom | On the high compressive strains achieved in bending tests on unidirectional carbon-fibre/epoxy[END_REF].

In conclusion, for the carbon/PEEK composite, the hourglass-shaped specimen still permits to avoid failure far from the load introduction points. However, the effect of the strain gradient in thickness does not make possible to directly estimate the material strength. Moreover, the behaviour determination, when non-linear, demands a more complex calculation procedure when compared to a pure compression test as the one studied in the second part of the article.

Compressive tests

Definition of hourglass specimen and testing procedure

Compressive tests were performed on a 3R Syntax 100 with the help of an anti-buckling fixture conform to ASTM D695 [START_REF] Astm | Astm d695 standard, standard test method for compressive properties of rigid plastics[END_REF]. Both dog-bone shaped ASTM D695 and new flat hourglass shaped specimens were tested using this fixture thanks to the same height and thickness, and close width (Fig. 5). The dog-bone and hourglass shapes utilized for the specimens are depicted in Fig. 5. In the case of the adapted hourglass shape the stress concentration factor in compression is calculated from a FEM simulation as the ratio between the nominal stress imposed to the specimen and the highest stress recorded along its centre. This yielded a value of S hourglass C = 1.045. Before testing, particular attention was put in the preparation of the coupons, parallelism of the upper and lower faces is mandatory to avoid coupon misalignment and consequent premature ends failure. Specimens were positioned between the anti-buckling surfaces of the fixture, afterwards the retaining screws were finger tightened. The screws were not over-tightened to avoid specimen jamming under load and introduction of a compression force in thickness direction. Longitudinal strains were measured for both specimen types using a video-extensometer from Limess on the side of the coupons and video-extensometer images were also treated with 2D DIC (Fig. 16). Video-extensometer requires two line markers with good contrast.

Black-white stickers were used with a calibrated distance of 20 mm for this reason.

Evaluation of hourglass specimen and methodology on an E-Glass/epoxy composite

First of all, the measurement obtained from the video-extensometer was compared to the measure from DIC in the same region of interest (ROI). For the dog-bone shaped specimens, the ROI is flat and strain measured via DIC varied in a range between -2.2% and -2.6% (Fig. 17) possibly due to heterogeneities in the material and errors in the measurement technique. However, the averaged DIC and the video-extensometer resulted in the same values. For the hourglass specimens, the ROI is curved (highlighted in red in Fig. 18). DIC measurements in the ROI varied between -2.5% and -2.9% with the same variation as the values obtained for the dog-bone coupon. Consequentially we can consider that the strain field is homogeneous in the waisted part. As with the dog-bone coupon the averaged DIC strain and the video-extensometer yielded the same values ( black and red lines in Fig. 18).

Seven standard dog-bone shaped and three hourglass shaped E-Glass/epoxy specimens were tested.

Only three hourglass specimens were tested because, due to manufacturing problems (poor quality of the end surfaces), other specimens were deemed not suitable for testing. Dog-bone specimens failed for a stress level of 688 ± 18 MPa with an average longitudinal strain at failure of -2.24 ± 0.08% (Fig. 19). Full results are reported in Table 2. As usual, it can be noted in the table that the coefficient of variation (COV) is larger in compression (around 3%) than in bending or in tension (around 1%). Experimental curves confirmed behaviour linearity in compression up to a strain of 2%. Failure occurred systematically in the upper tapered part, in proximity of the loaded end. This failure position can be explained by the presence of friction between specimen and fixture and will be discussed in the next paragraph. The three hourglass shaped specimens failed in the central part of the coupon but not in the middle where highest compression stresses should be present. The average ultimate stress at failure was equal to 765 ± 23 MPa and average longitudinal strain at failure was equal to -2.72 ± 0.04 % (Fig. 19). This ultimate strain at failure is close to the values obtained by Kim and Crasto [START_REF] Kim | Longitudinal Compression Strength of Glass Fiber-Reinforced Composites[END_REF] on E-Glass/Epoxy fabric using original miniature sandwich specimens, more complex to manufacture. Looking only at failure stresses, the use of the hourglass geometry lead to a gain of 11% compared to the dog-bone, avoiding failure in a zone characterized by a sharp width variation and large stress concentration. The adoption of the hourglass shape lead to an even greater increase of failure strain with a gain of 24% due to the non-linearity of the material.

To understand the influence of specimen geometry and why failure occurred systematically in proximity of the loaded side, qualitative FEM simulations taking into account friction, were conducted on ABAQUS software (Fig. 20). The specimens were simulated as 3D deformable solids with C3D6 and C3D4 elements to take into account out of plane effects (e.g. Poisson's effect and out-of-plane compression), minimum element size in contact zones was 0.5 mm to limit simulation time. Engineering constants of the material are reported in Table 1. Constants in out-of plane direction were supposed equal to constants in transverse direction due to the quasi-unidirectional nature of the Glass/Epoxy woven. For the fixture, only the two surfaces in contact with the specimen (anti-buckling grooves) were simulated as fixed rigid surfaces. Therefore, deformation of the anti-buckling fixture (guides, screws, etc.) is not considered. Contact with friction is considered with a friction coefficient of 0.3. Specimens were fixed on one end and on the other a compression load was imposed. Simulations were stopped as the mechanical test failure load was reached, no failure criteria were implemented. As the compression load increases, the friction effect is more and more important due to the Poisson's effect in the thickness. Simulated strain fields were superimposed over images of broken specimens in Fig. 21. Results of the simulations are very different for both specimens showing the interest of the new specimen geometry. From the images of the dog-bone specimen (Fig. 21a), it appears clearly that failure initiates from a zone where a sharp longitudinal stress gradient is present independently of the friction coefficient, in absence of friction the stress concentration coefficient (S dogbone C ) in this zone was estimated equal to 1.20 by FEM simulation. In the case of the hourglass shape specimen (Fig. 21b) the stress concentration coefficient (S hourglass C ) of the sample yielded a value of 1.045. The 3D simulations showed how the zone with the maximum longitudinal stress shifts towards the loaded end if friction is considered. For the hourglass coupon, failure position does not correspond with the maximum simulated stress due to the complexity of the problem and the qualitative nature of our FEM simulation (unknown friction coefficient).

In conclusion, the hourglass-shaped specimen geometry gives satisfactory results in compression for this kind of material even if the failure happen in the central part of the specimen but not in the middle. This last point can be explained by the combined effect of material Poisson ratio and friction in the anti-buckling system.

Conclusion

In this paper we have proposed the use of an hourglass shaped specimen to study compression behaviour of FRP in four-point bending and compression tests. As previously shown in tensile testing [START_REF] Payan | Damage modelling of laminated carbon/epoxy composites under static and fatigue loadings[END_REF][START_REF] De Baere | On the tension-tension fatigue behaviour of a carbon reinforced thermoplastic part II: Evaluation of a dumbbell-shaped specimen[END_REF], this particular shape, ensures failure at the centre of the coupon in a zone where the stress field is subjected to a negligible in-plane gradient, avoiding failure due to stress concentrations. Dimensions of the specimens were adapted for four-point bending test and end compression test. This geometry generated only a slight in-plane stress gradient in the central part of the specimen in both bending and compression, resulting in stress concentration coefficients of 1.035 and 1.045 respectively. In four-point bending, failure always occurred at the centre of the hourglass coupons.

The short hourglass shape specimen was compared to a standardized dog-bone specimen in pure compression tests on an end-compression fixture conform to ASTM D-695. Hourglass specimens performed better than the standardized dog-bone as stress and strain at failure were higher. Failure position was also different, dog-bone specimens failed in a zone characterized by a sharp width variation far from the centre.

Hourglass specimens failed close to the centre. FEM simulations of the ASTM D695 test demonstrated how the dog-bone shape generates the strongest stress concentration causing premature failure of the specimens.

The use of the hourglass coupons permitted to identify the constitutive equation of the material.

Both four-point bending and ASTM D-695 compression testing techniques present advantages and drawbacks. The ASTM D-695 compression test is simple to analyse, it gives direct access to the constitutive equation of the material and permits to get closer to the pure compressive resistance when using the antibuckling fixture together with the hourglass shaped coupons. Four point bending permits to avoid buckling in compression. The determination of behaviour laws is complicated by the necessity to estimate the bending moment in presence of large displacements and rotations. To accomplish this task a geometrical protocol is proposed and validated against FEM simulations. Other factors that complicate the procedure are nonlinearities of the material, while ultimate strength determination is not directly possible due to the presence of the strain gradient through the thickness as reported by Wisnom et al. [START_REF] Wisnom | The effect of specimen size on the bending strength of unidirectional carbon fibre-epoxy[END_REF][START_REF] Wisnom | Constrained buckling tests show increasing compressive strain to failure with increasing strain gradient[END_REF][START_REF] Wisnom | Reduction in compressive strain to failure with increasing specimen size in pin-ended buckling tests[END_REF][START_REF] Wisnom | On the high compressive strains achieved in bending tests on unidirectional carbon-fibre/epoxy[END_REF]. contact points constantly changes, consequentially magnitude of imposed moment is not directly proportional to applied load. This means that a correction must be applied to estimate the real magnitude of the moment.

A procedure is proposed here to compute this correction (Fig. 22a). Focusing on one side of the fixture, vertical position of loading roller is always known from machine's displacement and can be utilized to estimate the position of the contact points. We can assume that the coupon remains always straight between the supporting roller and the loading roller (this assumption is confirmed during testing as shown in Fig. 22b) and always touches both. Under these assumptions, the specimen can be represented by a straight line (representing his mid plane) tangent to two circumferences of radius equal to that of the rollers plus half of the specimen's thickness (dashed circles in Fig. 22a). The known parameters of the geometry are given in Table 3. Geometrically, two incident lines are tangent to two circles and they intersect in a point P int of coordinates (x P int ,y P int ) (black cross in Fig. 22a) given by the following equation:

   x P int = 1 2 (x cL + x cS ) y P int = 1 2 (y cL + y cS ) (6)
One of these lines represents the mid-plane of the specimen (black dash-dotted line in Fig. 22a). Its slope (m tg ) can be computed via the coordinates of its two tangent points (blue crosses in Fig. 22a):

m tg = y P tgL -y P tgS x P tgL -x P tgS (7) where coordinates of these tangent points are given using the following equations:

x P tg k = x c k + r * 2 (x P int -x c k ) (x P int -x c k ) 2 + (y P int -y c k ) 2

+

-r * (y P int -y c k ) (x P int -x c k ) 2 + (y P int -y c k ) 2 -r * 2 (x P int -x c k ) 2 + (y P int -y c k ) 2 for k ∈ {L, S}

with r * = r + t/2. This way, the specimen's angle respect to the horizontal is linked to its slope by: α = tan -1 (m tg ).

Under the hypothesis that specimen is always straight between the two rollers we can assume that the slope of its edges (green and red dash-dotted lines in Fig. 22a) is the same as its mid-plane (m tg ). Intercepts of the two lines representing the edges of the specimen are found by solving the two systems given by the equation of the lines and the equations of the two circles representing the rollers (solid lines in Fig. 22a) and imposing tangency between the two entities. The searched intercepts quantities q L and q S , for the lower and upper edge respectively, are given by:    q L = -m tg x c L + r m 2 tg + 1 + y c L q S = -m tg x c S -r m 2 tg + 1 + y c S [START_REF] Arhant | Effect of sea water and humidity on the tensile and compressive properties of carbon-polyamide 6 laminates[END_REF] Once all parameters describing the lines are known, the same systems can be solved for x and y to find the coordinates of the tangent points representing the contact points between the specimen and the rollers (green and red crosses in Fig. 22a):

x contact k = - m tg q k -x c k -m tg y c k m 2 tg + 1 for k ∈ {L, S} (11)
y contact k = m tg x contact k + q k for k ∈ {L, S}

Their horizontal span (δ x ) and vertical span (δ y ) are respectively equal to:

δ x = x contact S -x contact L (13)
δ y = y contact S -y contact L [START_REF] Öztürk | Finite element based design of a new dogbone specimen for low cycle fatigue testing of highly filled epoxy-based adhesives for automotive applications[END_REF] Neglecting friction, the effort normal to the neutral axis of the specimen F n , can be decomposed in a vertical force equal to half the machine applied load (F ), and a horizontal force F c :

F 2 = F n cos(α) (15)
F c = F n sin(α) (16)
The span (δ) between the two F n is equal to:

δ = δ x cos(α) -δ sin(α) (17)
The second member of Eq. 17 is negligible when compared to the first as the vertical span δ y is negligible compared to δ x .

Finally, pure bending moment is calculated:

M f = -F n δ = - δ x F 2 + F c δ y = F 2 (δ y tan(α) -δ x ) (18)
This procedure permits to estimate the real bending moment during a four-point bending test only knowing the position of the loading rollers without the need to constantly track the position of the specimen.

Figure 1 :Figure 2 :

 12 Figure1: Flat hourglass shaped specimen used for traction and fatigue testing (dimensions in mm)[START_REF] De Baere | On the tension-tension fatigue behaviour of a carbon reinforced thermoplastic part II: Evaluation of a dumbbell-shaped specimen[END_REF][START_REF] Payan | Damage modelling of laminated carbon/epoxy composites under static and fatigue loadings[END_REF]

Figure 3 :Figure 4 :

 34 Figure 3: Tensile test on E-glass/Epoxy specimens (V f = 0.48): a) [0] 8 layup; b) [90] 8 layup, c) [45, -45] 4S layup

Figure 5 :Figure 6 :

 56 Figure 5: Flat specimen geometries used for pure compression test: a) dog-bone shaped ASTM-D695 specimen [20] ; b) new flat hourglass shaped specimen (dimensions are in mm)[START_REF] Cocchi | Influence de l'endommagement matriciel sur la résistance en compression sens fibre pour des composites stratifiés[END_REF]

Figure 7 :

 7 Figure 7: Four-point bending simulation model scheme

Figure 8 :

 8 Figure 8: Four-point bending test simulation: a) stress field on the face in tension of the flat hourglass shaped specimen for a nominal stress of 881 MPa [21]; b) stress distribution generated by the hourglass shape at the centre of the specimen on the face in tension

Figure 9 :Figure 10 :

 910 Figure 9: Four-point bending test: comparison between an E-glass/epoxy specimen experimental test and FEM simulation: a) bending moment versus machine applied load ; b) machine applied load versus loading rollers displacement

Figure 11 :

 11 Figure 11: Four-point bending test: interpolated strain fields for E-Glass/epoxy specimens 0 and 2 just before collapse

Figure 12 :

 12 Figure 12: Four-point bending test: skin stress/strain results for three different E-glass/Epoxy specimens (cross = failure) and picture of the specimens just after collapse

Figure 13 :

 13 Figure 13: Four-point bending test: strain field measured with DIC on Carbon/PEEK specimen side just before collapse and out-of-plane kink-band failure (at right).

Figure 14 :

 14 Figure 14: Four-point bending test: interpolated strain fields for Carbon/PEEK specimen just before collapse

Figure 15 :

 15 Figure 15: Four-point bending test: skin strain/stress results for a Carbon/PEEK specimen (cross = failure)

Figure 16 :

 16 Figure 16: Pure compression test: specimen DIC measurement area (after failure, only one Black-white sticker is visible)

Figure 17 :Figure 18 :Figure 19 :

 171819 Figure 17: Pure compression test: longitudinal strain fields of E-Glass/Epoxy dog-bone specimen before failure measured via DIC

Figure 20 :

 20 Figure 20: Pure compression test: FEM simulation model scheme for ASTM D695 fixture in ABAQUS

Figure 21 :

 21 Figure 21: Pure compression test: FEM simulations superimposed over E-Glass/Epoxy failed specimens: a) ASTM D695 dog-bone shaped specimens , b) flat hourglass shaped specimens

Figure 22 :

 22 Figure 22: Four-point bending test: a) span estimation procedure ; b) forces decomposition for moment estimation

y

  P tg k = y c k + r * 2 (y P int -y c k ) (x P int -x c k ) 2 + (y P int -y c k ) 2 + r * (x P int -x c k ) (x P int -x c k ) 2 + (y P int -y c k ) 2 -r * 2 (x P int -x c k ) 2 + (y P int -y c k ) 2 for k ∈ {L, S}

Table 2

 2

	: Testing results on various configurations of E-Glass/epoxy (M42ST/1055) (Scoupon = stress concentration coefficient;
	C = coupon; Av.= average; StD = standard deviation; COV = coefficient of variation)

Table 3 :

 3 Four-point bending test: geometrical notations (n/a = not applicable)

	Parameter	Notation	Unit Value
	Minimum width of the specimen	w	mm	10
	Thickness of the specimen	t	mm	2.64
	Rollers radius	r	mm	5
	Loading roller (L) centre	(x cL , y cL )	mm	n/a
	Supporting roller (S) centre	(x cS , y cS )	mm	n/a
	Loading roller contact point coordinates	(x contact L , y contact L )	mm	n/a
	Supporting roller contact point coordinates	(x contact S , y contact S )	mm	n/a
	specimen's slope	mtg	-	n/a

Acknowledgements

This research is financially supported by the French Ministry of the Armed Forces through the Directorate General of Armaments.

Appendix: Bending moment calculation method

As demonstrated by Mujika [START_REF] Mujika | On the difference between flexural moduli obtained by three-point and four-point bending tests[END_REF][START_REF] Mujika | New correction terms concerning three-point and four-point bending tests[END_REF], the main sources of error in estimation of mechanical properties from four-point bending tests are the contact points between specimen and fixture. During a test, position of


domain: chim sdv spi

text:
DDS Marion Dehurtevent
email: marion.dehurtevent@univ-lille.fr

DDS Lieven Robberecht

PhD Anthony Thuault

DDS Etienne Deveaux

PhD Anne Leriche

PhD Fabrice G Petit

DDS Corentin Denis

PhD Jean-Christophe Hornez

DDS, PhD Pascal Béhin

Effect of build orientation on the manufacturing process and the properties of stereolithographic dental ceramics for crown frameworks

INTRODUCTION

Additive manufacturing is a technology that builds 3D objects by adding material in layers. This method is being used to manufacture metal and polymer objects for dental use but is still in the early stages of development for shaping dental ceramics. [START_REF] Antanasova | Influence of surface airborne-particle abrasion and bonding agent application on porcelain bonding to titanium dental alloys fabricated by milling and by selective laser melting[END_REF][START_REF] Liu | Internal adaptation of cobalt-chromium posts fabricated by selective laser melting technology[END_REF][START_REF] Van Noort | The future of dental devices is digital[END_REF][START_REF] Denry | Emerging ceramic-based materials for dentistry[END_REF][START_REF] Eckel | Additive manufacturing of polymer-derived ceramics[END_REF][START_REF] Dehurtevent | Stereolithography: a new method for processing dental ceramics by additive CAD/CAM[END_REF] Additive manufacturing is an effective way to avoid the major drawbacks associated with subtractive manufacturing, which involves shaping a solid block of material, including waste of raw material, wear of milling tools, shape limitations, and microscopic cracks that can weaken restorations. [START_REF] Wang | Strength influencing variables on CAD/CAM zirconia frameworks[END_REF] Stereolithography (SLA) is an additive manufacturing process that produces objects in a layer-by-layer fashion using a photosensitive resin. The monomers are polymerized to form the body of a 3D solid. The monomers can also be mixed with ceramic particles to form a photosensitive slurry for producing ceramic objects. SLA shows more promise than other additive manufacturing methods because it can be used to build crown frameworks that meet dental standards (ISO 6872:2015). [START_REF]Dentistry -Ceramic materials[END_REF] SLA involves designing 3D objects supported by pillars, slicing, transferring to SLA software and equipment, layer-by-layer polymerization of a ceramic slurry containing a photosensitive resin, hand sectioning of the pillars, thermal debinding, and sintering. A complex object can be manufactured with different build orientations, which has an impact on the size of the different layers. [START_REF] Dehurtevent | Stereolithography: a new method for processing dental ceramics by additive CAD/CAM[END_REF] To avoid deformations of the object during the manufacturing process, a minimum slurry resting period must be adopted immediately after the polymerization of a layer.

SLA alumina ceramic crown frameworks with high density, flexural strength, and

Weibull modulus values can be built and are suitable for clinical use. [START_REF] Dehurtevent | Stereolithography: a new method for processing dental ceramics by additive CAD/CAM[END_REF] A smaller slurry organic phase also results in fewer defects in SLA alumina ceramics, and an 80% (w/w) slurry dry matter content is the best compromise for the manufacturing process. [START_REF] Dehurtevent | Stereolithography: a new method for processing dental ceramics by additive CAD/CAM[END_REF] Despite the high mechanical strength, feasibility, and structural reliability of SLA objects, this approach remains controversial because of the anisotropic shrinkage of alumina ceramics that occurs during the densification step, [START_REF] Cawley | Solid freeform fabrication of ceramics[END_REF] which could have a negative impact on the mechanical behavior of the ceramic. The orientation of ceramic particles has an effect on the mechanical behavior of ceramics as does the general build orientation on the mechanical properties of SLA resins. [START_REF] Cesar | ADM guidance-ceramics: fracture toughness testing and method selection[END_REF][START_REF] Puebla | Effects of environmental conditions, aging, and build orientations on the mechanical properties of ASTM type 1 specimens manufactured via stereolithography[END_REF] This results in an anisotropic behavior of additive-manufactured materials. [START_REF] Unkoskiy | Objects build orientation, positioning, and curing influence dimensional accuracy and flexural properties of stereolithography printed resin[END_REF] This highlights the need to investigate the effect of this parameter on ceramics. Pfaffinger et al 13 investigated 2 build orientations for densified tricalcium phosphate ceramics and reported that slurry composition affected flexural strength in the build orientation. Dental restorations require high flexural strength, density, hardness, fracture toughness, and structural reliability to ensure durability. Fracture toughness can be measured on the surface of a ceramic specimen by using the indentation technique but is influenced by the microstructure of the specimen. [START_REF] Anstis | A critical evaluation of indentation techniques for measuring fracture toughness: I, direct crack measurements[END_REF][START_REF] Tricoteaux | Influence of porosity on the mechanical properties of microporous β-TCP bioceramics by usual and instrumented Vickers microindentation[END_REF][START_REF] Ghorbal | Comparison of conventional Knoop and Vickers hardness of ceramic materials[END_REF] The single-edge-V-notched-beam (SEVNB) method is a precise technique for measuring fracture toughness, but the microstructure of the ceramic may affect notch preparation, and precracks may underestimate the results. [START_REF] Domingues | Comparison of the indentation strength and single-edge-v-notched beam methods for dental ceramic fracture toughness testing[END_REF] The 2 methods are thus complementary.

Manufacturing time should be minimized, so the minimal resting period of the slurry between the polymerization of 2 layers must be determined with respect to the build orientation.

These physical and mechanical properties must be studied in 3 orthogonal planes (ZX, ZY, and XY) of 3 build orientations in order to meet the ISO6872:2015 8 standard and to optimize the orientation of the SLA crown frameworks and obtain the best mechanical behavior for clinical use. This will make it possible to determine the optimal build orientation with respect to the main masticatory stresses of prostheses and their position in the dental arch, which will provide the best mechanical strength and reproducibility and the shortest manufacturing time (buccolingual, mesiodistal, or occlusocervical directions).

The purpose of this in vitro study was to evaluate the impact of 3 build orientations on the physical properties (minimum resting period between the polymerization of 2 layers, density), mechanical properties (3-point flexural strength, Weibull modulus, hardness, fracture toughness by indentation and by SEVNB), and microstructure of SLA alumina dental ceramics. The null hypotheses were that the build orientation would not affect the physical properties, the mechanical properties, or the microstructure of SLA alumina dental ceramics.

MATERIAL AND METHODS

Slurries were prepared using alumina powders (CT1200SG; Almatis). The powders were dried for 1 hour at 90 °C in an oven (Beschickung loading model 100-800; Memmert). The powders were incorporated into a photosensitive (λ=350-400 nm) acrylic resin (C1-alumina; CryoBeryl Software) to obtain 80% (w/w) dry matter content slurries that were homogenized at 150 rpm by planetary milling for 30 minutes (PM100; Retsch). The rectangular parallelepipedal specimens were designed using CAD software (Catia; Dassault Systèmes). Their dimensions were 1.3×4×22 mm for the evaluation of density, 3-point flexural strength, Weibull modulus, hardness, fracture toughness by indentation, and microstructure assays and 3×4×40 mm for the evaluation of fracture toughness by the SEVNB -3-point bending test (Fig. 1A).

All the specimens were oriented in 3 different planes to ensure that their widths and lengths were situated within the ZX, ZY, and XY axes (Fig. 1B), making it possible to produce ZX-, ZY-, and XY-oriented specimens. The specimens were supported on pillars (end diameter: 0.2 mm, base diameter: 0.7 mm, height: 2 mm) during the design step. The models were oversized to compensate for the anisotropic shrinkage that occurs during the thermal treatment of SLA alumina. [START_REF] Dehurtevent | Stereolithography: a new method for processing dental ceramics by additive CAD/CAM[END_REF] The specimens were then sliced (layer thickness=50 µm) (Creation Workshop software; Datatree3D), and the data were transferred to SLA software (CryoCeram; CryoBeryl Software) and equipment (CryoCeram Printer; CryoBeryl Software) for additive manufacturing (polymerization energy=10 mW.cm -2 , resolution=50 µm).

The entire surface of a layer was polymerized at once using a dynamic mask of digital micromirror devices. The support platform was then moved down into the slurry. The pillars were hand sectioned after the manufacturing phase. The specimens were debinded (0.1 °C/minute, 450 °C, 2-hour dwell time) in an oven (LHT; Carbolite) and were sintered (15 °C/minute, 1690 °C, 3-hour dwell time) (HT16/17; Nabertherm). Specimens (n=3/group) were prepared with different slurry resting periods (6, 7, 8, 9, 10, 11, and 12 seconds) before polymerizing a layer. The minimum slurry resting period between the polymerization of 2 layers was set as the time when no visual macroscopic deformations of the manufactured object were observed. The density of the specimens was determined by hydrostatic weighing using the Archimedes principle (3 vacuum cycles, n=10/group).

The 3-point flexural strength was determined using a universal testing machine (DY30; Adamel Lhomargy) (n=30/group) with a maximum force of 1 kN. Based on the ISO 6872:2015 standard, 8 a progressive central vertical load was applied at a speed of 0.5 mm/minute on the dark grey face of the specimens (Fig. 1B). The distance between the lower supports was 20 mm.

The 3-point flexural strength in MPa (σ) was (equation 1):

,

where F is the maximum force to failure (N), L is the distance between the lower supports (mm), W is the specimen width (mm), and H is the specimen height (mm). A Weibull distribution analysis using 3-point flexural strength data was used to assess the structural reliability of the ceramics (n=30/group). The Weibull modulus (m) was , where Pf is the probability of failure between 0 and 1, σ is the 3-point flexural strength in MPa, and σ0 is the characteristic strength in MPa.

A Vickers indentation hardness test (ZHU 0.2, Zwick) was performed (ISO 14705:2016) (n=15/group). 18 A 2-N load was applied to force the indenter into the test piece and hold it in place for 13 seconds, and the diagonals (d) of the impression were measured. The Vickers hardness in Hv was , where P is the load (N) and d is the impression diagonal (mm). The microhardness was evaluated on 3 faces: thickness-length, length-width, and widththickness (n=15/face). Fracture toughness (KIC) was determined by SEVNB -3-point bending using a universal testing machine (Zwick Z100). A V-notch of uniform depth was created in all the specimens by using a 0.3-mm cutoff wheel mounted on a cutting machine (Struers Discotom 100). The specimens were loaded using a 3-point bend fixture (30-mm outer span and 15-mm inner span) with a crosshead speed of 0.5 mm/minute (n = 4/group). Fracture toughness was , where α is the initial precrack relative length (m), W is the specimen width (m), B is the specimen thickness (m), F is the fracture load (MN), and S is the support span (m).

The indentation fracture toughness was determined using the indentation cracking method (ASTM STP 678). [START_REF]ASTM STP678-EB. Fracture mechanics applied to brittle materials (Freiman 1979)[END_REF] Vickers indentations were impressed (F=20 N) (n=4/group). The crack lengths were measured on 3 faces using a scanning electron microscope. The fracture toughness by indentation in MPa.m 1/2 (K1C) was , where F is the indentation load (N), c is the average crack length (mm), E is the Young modulus (GPa), and Hv is the hardness (GPa).

The microstructures of the mirror-polished surfaces were observed by scanning electron microscopy (S-3500N; Hitachi) (magnification ×2000) on 3 faces (Fig. 1). Additionally, the fracture surfaces of the 3-point flexural strength specimens were observed by scanning electron microscopy (magnification ×25 and ×1500).

Specimen sizes were determined in order to reach a minimal statistical power of 95%.

The normality of data distribution was verified using the Kolmogorov-Smirnov test. The

RESULTS

The results of the physical and mechanical measurements are summarized in Table 1. Figure 2 shows box plots of these results. The minimum slurry resting period between the polymerization of 2 layers varied from 7 seconds to 11 seconds depending on the build orientation (Table 1).

The minimum resting period was 11 seconds for the XY-oriented specimen, 8 seconds for the ZY-oriented specimen, and 7 seconds for the ZX-oriented specimen. The general manufacturing time was 35 minutes for the XY-oriented specimen, 47 minutes for the ZY-oriented specimen, and 3 hours 6 minutes for the ZX-oriented specimen. The densities of the SLA specimens were similar for all groups (P>.05) (Fig. 2, Table 1).

The 3-point flexural strength of the ZY-oriented specimens (409.7 ±29.6 MPa) was significantly higher than that of the ZX-oriented specimens (313.8 ±27.9 MPa and the XYoriented specimens (334.3 ± 62.1 MPa) (P<.05) (Fig. 2, Table 1). The Weibull moduli of the ZX-oriented specimens (13.7 [12.4-15.1]) and the ZY-oriented specimens (16.1 [14.6-17.5])

were higher than that of the XY-oriented specimens (6.2 [5.6-6.8]) (Fig. 3, Table 1). No difference in hardness was found regardless of the face considered for all the groups (P>.05) (Fig. 2, Table 1). The fracture toughness of the ZY-oriented specimens (4.6 ±0.2 MPa.m ½ ) as determined by SEVNB was significantly higher than that of the ZX-oriented specimens (3.2 ±0.2 MPa.m ½ ) (P<.05). In contrast, the fracture toughness of all the faces as determined by indentation was similar for all the specimens (P>.05) (Fig. 2, Table 1).

All the specimens were microporous and displayed no signs of delamination. In addition, the major axis of the large grains of alumina was parallel to the orientation of the layers (Fig. 4).

In terms of the fracture surface analysis, more transgranular failures were observed inside the alumina particles with the ZY-and XY-oriented specimens than with the ZX-oriented specimens (Fig. 5), while more intergranular failures were observed with the ZX-oriented specimens.

DISCUSSION

The null hypotheses that the build orientation would have no impact on the physical properties, the mechanical properties, or the microstructure of SLA alumina dental ceramics were rejected.

In general, the physical and mechanical properties and microstructure of SLA alumina ceramics were influenced by the orientation of the printed layers.

The ZX-oriented specimens had the smallest layer surface and required the shortest minimum resting period between the polymerization of 2 layers. The present study showed that the smaller the layer surface, the shorter the minimum resting period. However, the ZX orientation increased the number of layers and had a negative impact on the manufacturing time.

The build orientation had an impact on the minimum resting period and the manufacturing time.

Given this, a compromise that took the shortest minimum resting period of the slurry and the lowest number of layers into account had to be found. The XY-oriented specimens best met this requirement, with an 11-second minimum resting period for the slurry, 34 layers, and a general manufacturing time of 35 minutes. The ZY-oriented specimens (general manufacturing time: 50 minutes) were also a good compromise with respect to the ZX-oriented specimens (general manufacturing time: 3 hours). Although the XY-oriented specimens had the shortest general manufacturing time, their mechanical strength was lower.

In terms of the physical properties of densified SLA alumina ceramics, the density of the alumina ceramics was close to 98% after the thermal treatment and was similar for all the build orientations. It has been previously reported that high density results in fewer cracks and reduces the delamination of SLA ceramics. [START_REF] Pfaffinger | Stabilization of tricalcium phosphate slurries against sedimentation for stereolithographic additive manufacturing and influence on the final mechanical properties[END_REF][START_REF] Braga | Factors involved in the development of polymerization shrinkage stress in resin-composites: a systematic review[END_REF] The densities of the ceramics used in the present study were thus acceptable for clinical use and were consistent with previous work on alumina and tricalcium phosphate. [START_REF] Dehurtevent | Stereolithography: a new method for processing dental ceramics by additive CAD/CAM[END_REF][START_REF] Pfaffinger | Stabilization of tricalcium phosphate slurries against sedimentation for stereolithographic additive manufacturing and influence on the final mechanical properties[END_REF] In terms of mechanical properties, the SLA specimens displayed anisotropic behavior.

The 3-point flexural strengths of all the specimens were above 300 MPa and corresponded to the dental ISO6872:2015 standard 8 regardless of the build orientation. The ZY-oriented specimens had the highest 3-point flexural strength (409.7 ±29.6 MPa). More transgranular failures were observed in the ZY-oriented (perpendicular to the load surface) and XY-oriented specimens (perpendicular to the load direction) and more intergranular failures in the ZX-oriented specimens. Furthermore, the microstructural analysis revealed heterogeneous texturing, with the main axis of the large grains and the orientation of the layers parallel for all specimens.

The flexural strength results can be explained by previous studies, which showed that ceramic specimens with the load perpendicular to the grain axis display more transgranular failures, which results in a higher mechanical strength than when the load is parallel to the grain axis. [START_REF] Suzuki | Control of texture in alumina by colloidal processing in a strong magnetic field[END_REF][START_REF] Quinn | A practical and systematic review of Weibull statistics for reporting strengths of dental materials[END_REF] Further studies on the microstructural modifications that occur during the densification of ceramics will be required to better understand this texturing. However, this does not explain why the ZY-oriented specimens (perpendicular to the test load surface) displayed a higher 3point flexural strength than the XY-oriented specimens (perpendicular to the direction of the test load), which also have a main axis of large grains perpendicular to the direction of the load.

These results might, however, be explained by the morphological texturing and spatial orientation of the large grains, which should be investigated in future studies. The evaluation of the structural reliability of dental ceramics by the Weibull analysis provided additional answers to this issue. The Weibull moduli of the ZX-and ZY-oriented specimens were similar to those reported in the literature. [START_REF] Guazzato | Strength, fracture toughness and microstructure of a selection of all-ceramic materials. Part II. Zirconia-based dental ceramics[END_REF] The XY-oriented specimens had the lowest Weibull modulus, indicating that they have less structural reliability than the ZX-and ZY-oriented specimens. [START_REF] Guazzato | Strength, fracture toughness and microstructure of a selection of all-ceramic materials. Part II. Zirconia-based dental ceramics[END_REF] Pfaffinger et al [START_REF] Pfaffinger | Stabilization of tricalcium phosphate slurries against sedimentation for stereolithographic additive manufacturing and influence on the final mechanical properties[END_REF] suggested that low Weibull moduli can result from defects in SLA ceramics, which leads to anisotropic mechanical behavior. In the present study, the XY-oriented specimens had the greatest layer surface with the longest resting periods. This can increase the risk of defects within the layers and may explain the lower Weibull moduli observed in this group. To test this hypothesis, the maximum layer surface that does not reduce the Weibull moduli should be determined.

The hardness values followed the same trend as the density values. The hardness of the ceramic specimens was similar, regardless of the SLA orientation, suggesting that this parameter is isotropic in nature and only depends on grain size and relative density. [START_REF] Tricoteaux | Influence of porosity on the mechanical properties of microporous β-TCP bioceramics by usual and instrumented Vickers microindentation[END_REF] However, consistent with Uçar et al, [START_REF] Uçar | Layered manufacturing of dental ceramics: fracture mechanics, microstructure, and elemental composition of lithography-sintered ceramic[END_REF] these values were higher than those of other alumina ceramics manufactured by subtractive methods for dental applications. [START_REF] Sakar-Deliormanli | 2.5 ± 1.9 k σ, 3-point flexural strength; m, Weibull modulus; K1C, fracture toughness[END_REF] This could be explained by the textured aspect of SLA ceramics. [START_REF] Tricoteaux | Influence of porosity on the mechanical properties of microporous β-TCP bioceramics by usual and instrumented Vickers microindentation[END_REF][START_REF] Ghorbal | Comparison of conventional Knoop and Vickers hardness of ceramic materials[END_REF] The fracture toughness of the SLA ceramics depended on the evaluation method used.

However, these results should be considered preliminary given the small sample size. Fracture toughness evaluated by indentation has been used for a first analysis of a new ceramic. [START_REF] Anstis | A critical evaluation of indentation techniques for measuring fracture toughness: I, direct crack measurements[END_REF] In the present study, fracture toughness evaluated by indentation followed the same trend as hardness.

SEVNB is a more precise method for evaluating fracture toughness than indentation, a local measurement method. [START_REF] Tricoteaux | Influence of porosity on the mechanical properties of microporous β-TCP bioceramics by usual and instrumented Vickers microindentation[END_REF] This technique gave higher fracture toughness values for the ZYoriented specimens than for the ZX-oriented specimens. Once again, this shows that the microstructure of SLA ceramics had an impact on their mechanical behavior, suggesting that the heterogeneous texturing increased the fracture toughness as determined by the SEVNB method when the layer is perpendicular to the load surface. [START_REF] Cesar | ADM guidance-ceramics: fracture toughness testing and method selection[END_REF] Transposed to a clinical context, these results showed that ZY-and ZX-oriented frameworks generate an orientation of the printed layers perpendicular to the occlusal surface and thus parallel to the direction of the main masticatory stresses applied to the occlusal surface (Fig. 6). In this situation, and with these experimental conditions, both oriented ceramics would have the best reproducibility. Furthermore, preference should be given to a short general manufacturing time (compromise between a small number of layers and a small layer surface).

The ZY-orientation should thus be preferred because it has the shortest general manufacturing time and has layers that are parallel to the main direction of the masticatory stresses. For a complex crown framework shape, and if the Z axis is considered as the occlusocervical direction, the orientation of the layers that results in both the shortest manufacturing time and layers parallel to the direction of the masticatory stresses are the buccolingual direction for incisors and premolars and the mesiodistal or buccolingual directions for molars in terms of the anatomy of the crown (Fig. 7). However, a clinical extrapolation of these experimental results must be tempered, as the complexity of a crown shape represents both the ZY-and ZX-oriented experimental conditions. These results must be confirmed with future investigations that simulate the more complex masticatory stresses that ceramics and other materials such as zirconia are subjected to. The maximum layer surface of a complex crown framework must be determined in this way and other layer orientations should be investigated. 11

CONCLUSIONS

Based on the findings of this in vitro study, the following conclusions were drawn:

1. All the SLA alumina ceramics studied met dental standards.

2. The orientations of the SLA ceramics in 3 orthogonal planes affected the minimum resting period of the slurry between the polymerization of 2 layers, as well as the physical and mechanical properties of the sintered objects (P<.05).

3. The minimum resting period of the slurry needed between the polymerization of 2 layers increased with the layer surface.

4. The ZY-oriented specimens with the smallest number of layers and a layer orientation perpendicular to the occlusal surface provided the best compromise between mechanical strength and general manufacturing time (P<.05).

TABLES

  density, microhardness, and fracture toughness values were analyzed by using the 1-way Kruskal-Wallis test and Dunn post hoc test. Three-point flexural strength was analyzed using a 1-way ANOVA test and the Tukey multiple comparison post hoc test (GraphPad 5; Prism) (α=.05).

Figure 2 .

 2 Figure 2. Boxplots (box: median, Q1, Q3; whiskers: 10-90 percentile) of 3 orientations (ZX, ZY,

Figure 3 :

 3 Figure 3: Weibull plot comparing strength distribution of SLA alumina ceramics in 4

Figure 4 :

 4 Figure 4: Micrographs of 3 faces of ZX-, ZY-, and XY-oriented SLA alumina specimens

Figure 5 .

 5 Figure 5. Micrographs of fracture surface after 3-point flexural testing of SLA alumina ceramic

Figure 6 :

 6 Figure 6: Transposition of orientations of printed layers from specimens studied to complex

Figure 7 .P

 7 Figure 7. A, Recommended SLA orientations to manufacture complex frameworks (white bars:

Table 1 .

 1 Physical properties of 3 orientations (ZX, ZY, and XY) of SLA-manufactured ceramics

	Evaluated parameter

Dr Marion Dehurtevent

Department of Prosthodontics

Faculty of Dentistry

University of Lille FRANCE Email: marion.dehurtevent@univ-lille.fr


B Scherrer

M Dojat

F Forbes

C Garbay

LOCUS: LOcal Cooperative Unified Segmentation of MRI Brain Scans

We propose to carry out cooperatively both tissue and structure segmentations by distributing a set of local and cooperative models in a unified MRF framework. Tissue segmentation is performed by partitionning the volume into subvolumes where local MRFs are estimated in cooperation with their neighbors to ensure consistency. Local estimation fits precisely to the local intensity distribution and thus handles nonuniformity of intensity without any bias field modelization. Structure segmentation is performed via local MRFs that integrate localization constraints provided by a priori general fuzzy description of brain anatomy. Structure segmentation is not reduced to a postprocessing step but cooperates with tissue segmentation to gradually and conjointly improve models accuracy. The evaluation was performed using phantoms and real 3T brain scans. It shows good results and in particular robustness to nonuniformity and noise with a low computational cost.

Introduction

MRI brain scan segmentation is a challenging task and has been widely addressed in the last 15 years. Difficulties in automatic segmentation arise from various sources including the size of the data, the low contrast between tissues, the limitations of available a priori knowledge, local perturbations such as noise or global perturbations such as intensity nonuniformity. Current approaches share three main characteristics: first, tissue and structure segmentations are considered as two separate tasks whereas they are clearly linked. Second, for a robust to noise segmentation, the Markov Random Field (MRF) probabilistic framework is classically used to introduce spatial dependencies between voxels [START_REF] Van Leemput | Automated model-based tissue classification of MR images of the brain[END_REF][START_REF] Zhang | Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximisation algorithm[END_REF]. Third, tissue models are generally estimated globally through the entire volume and do not reflect spatial intensity variations within each tissue, due mainly to biological tissue properties and to MRI hardware imperfections. Only the latter is generally addressed, modeled by the introduction of an explicit so called "bias field" model to estimate. Local segmentation is an attractive alternative. The principle is to compute models in various subvolumes to fit better to local image properties. However, the few local approaches proposed to date are clearly limited: they use local estimation as a preprocessing step only to estimate a bias field model [START_REF] Shattuck | Magnetic resonance image tissue classification using a partial volume model[END_REF], a training set for statistical local shape modelling [START_REF] Pohl | Logarithm odds maps for shape representation[END_REF], redondant information to ensure consistency and smoothnesss between local estimated models [START_REF] Rajapakse | Statistical approach to segmentation of single-channel cerebral MR images[END_REF][START_REF] Zhu | Multicontextual fuzzy clustering for separation of brain tissues in magnetic resonance images[END_REF], or an atlas providing a priori local spatial information [START_REF] Fischl | Whole brain segmentation: automated labeling of neuroanatomical structures in the human brain[END_REF] greedily increasing computational cost. We present in this paper an original LOcal Cooperative Unified Segmentation (LOCUS) approach which 1) performs tissue and structure segmentation by distributing a set of cooperating local MRF models through the volume, 2) segments structures by introducing prior localization constraints in a MRF framework and 3) ensures local models consistency and tractable computational time via specific cooperation and coordination mechanisms.

Method

MRF Segmentation

We consider a finite set of N voxels V = {1, ...N } on a regular 3-D grid. Our aim is to assign each voxel i to one of K classes considering the observed greylevel intensity y i at voxel i. Both observed intensities and unknown classes are considered to be random fields denoted respectively by Y = {Y 1 , ..., Y N } and Z = {Z 1 , ..., Z N }. Each random variable Z i takes its value in {e 1 , ..., e K } where e k is a K-dimensional binary vector corresponding to class k. Only the k th component of this vector is non zero and is set to 1. In a traditionnal Markov model based segmentation framework, it is assumed that the conditional field

Z given Y = y is a Markov random field, ie. p (z |Y = y, Φ ) = W -1 y,Φ exp (-H (z |y, Φ ))
, where H (z |y, Φ ) is an energy function depending on some parameters Φ = (Φ y , Φ z ) and given by:

H (z |y, Φ ) = H (z |Φ z ) - i∈V log p (y i |z i , Φ y ). ( 1
)
This energy is a combination of two terms: the first term is a regularization term that accounts for spatial dependencies between voxels. Denoting by N (i) the neighbors of voxel i and by t z i the transpose of vector z i , we will consider a Potts model with external field:

H(z |Φ z ) = i∈V ( t z i v i - β 2 j∈N (i) t z i z j ). (2)
The second summation in (2) tends to favor neighbors that are in the same class when β is positive. This β parameter accounts for the strengh of spatial interaction. Other parameters are the v i ´s that are K-dimensional vectors defining the so-called external field. In this case Φ z = {v 1 , ..., v N , β}. The v i 's can be related to a priori weights accounting for the relative importance of the K classes at site i. The introduction of these extra parameters in the standard Potts model enables us to integrate a priori knowledge on classes. The second term in (1) is a data-driven term based on intensities. For MRI we generally consider Gaussian probability density functions for each k, p (y

i |z i = e k , Φ y ) = g µ k ,σ k (y i ), with Φ y = {µ k , σ k , k = 1...K}.
Segmentation is then performed according to the Maximum A Posteriori principle (MAP) by maximizing over z the probability p (z |y, Φ ). This requires the evaluation of an intractable normalizing constant W y,Φ and the estimation of the unknown parameters Φ. A standard approach is to use EM-based algorithms to globally estimate the parameters through the entire volume. We propose in the next subsection a LOcal and Cooperative version of EM (LOC-EM) for local segmentation approaches.

Local Cooperative Tissue Segmentation (LOCUS-T).

We

H c (t |y, Φ c ) = i∈Vc   t t i λ c i - β c 2 j∈N (i) t t i t j -log p y i t i , Φ c y   , (3)
where the parameters Φ c = Φ c t , Φ c y have to be estimated. However, the external field denoted by {λ c 1 , ...λ c N } is not estimated but used to incorporate information coming from structure segmentation to perform cooperation. Φ c t reduces then to {β c }, while Φ c y are the estimated parameters of the local Gaussian tissue intensity models. The MRF model M c introduces spatial dependencies between voxels in its subvolume V c , providing consistent neighboring labels. Because the estimation is local, some tissue classes are likely to be under-represented in some subvolumes, leading to poor model estimations with a classical EM scheme. k is larger than a given threshold, we compute the corrected mean and variance of class k from a linear combination of intensity models in M c and M c using D c k to determine the linear coefficients. -Model Interpolation: from local estimations in neighbouring subvolumes we get then one intensity model per voxel by using cubic splines interpolation between corrected models of M c and of N (M c ). This results in a non-stationary fieldlike approach and has the advantage to ensure smooth model variation between neighboring subvolumes and to intrinsically handle nonuniformity of intensity inside each subvolume. Coordination between MRF models.

-System starting: each local EM enters in idle mode after its local initialization. A global intensity model is computed using the Fuzzy C-Mean algorithm and then only the MRF models closest to the global model are activated.

-Knowledge spreading: when the EM algorithm for M c is stabilized, its neighbors are activated to perform estimation in turn. For already stabilized EM, model checking is performed. If it results in model correction and model interpolation, the corresponding EM are restarted to take into account the updated models modifications.

Cooperative Tissue and Structure Segmentation (LOCUS-TS).

We extend the approach above to segment both tissues and structures. We currently consider L = 9 subcortical structures: the ventricular system, the Frontal Horns, the Caudate Nuclei, the Thalamus, and the Putamens. For each target structure l we define a local Markov model M l that labels voxels of its subvolume V l in K = 2 classes referred to as structure and background. Denoting by s = {s i , i ∈ V l } the hidden classes, the energy function of M l is given by:

H l s y, Ψ l = i∈V l   t s i α l i - β l 2 j∈N (i) t s i s j -log p y i s i , Ψ l y,i   , (4)
with Ψ l = β l , α l i , Ψ l y,i , i ∈ V l and s i ∈ {e 1 , e 2 } = {e B , e S } for a voxel of the background or a voxel belonging to structure l. Integration of prior localization constraints in the MRF. Automatic structure segmentation cannot rely only on radiometry information because intensity distributions of grey nuclei are largely overlapping. A priori knowledge should be introduced. A recent way to provide it is to describe brain anatomy with generic fuzzy spatial relations [START_REF] Barra | Automatic segmentation of subcortical brain structures in MR images using information fusion[END_REF][START_REF] Colliot | Integration of fuzzy spatial relations in deformable models -Application to brain MRI segmentation[END_REF]. Three kind of relations are generally considered: distance, symmetry and orientation relations. They are expressed as 3D fuzzy maps to take into account the generic nature of the provided knowledge. Each subcortical structure is described by a set of such generic fuzzy spatial relations provided by a brain anatomist. Fusion operators between fuzzy sets are then used to combine the knowledge provided by each spatial relation and provide a generic Fuzzy Localization Map (FLM) of the structure in the volume. The FLM f l of structure l is used in two ways: first it dynamically provides the structure subvolume V l containing the structure l by a simple thresholding. Second, it can be integrated as an a priori anatomical knowledge in the MRF framework via the external field {α i , i ∈ V l }. We denote by f l i the value of f l at voxel i and propose to introduce the prior fuzzy knowledge of the FLM as relative prior weights for each voxel i, by setting

α l i = t [α l i (e B ), α l i (e S )] to α l i = γ t [ -log 1 -f l i , -log f l i ],
where γ > 0 adjusts the influence of the external field. When f l i ≈ 0, voxel i is unlikely to belong to the structure. It follows α l i (e B ) < α l i (e S ) which favors in (4) the background class. When f l i ≈ 1, voxel i is likely to belong to the structure. In that case α l i (e B ) > α l i (e S ) and the class structure is favored. Cooperation and coordination mechanisms between MRF models. Let C T →S (l) (resp. C S→T (c)) denotes the tissue (resp. structure) subvolumes that overlap with the structure subvolume V l (resp. tissue subvolume V c ). C S→S (l) denotes structures using l as a reference in a spatial relation. MRF models cooperate to make the segmentations gradually more accurate as described below.

-Structure segmentation starting: Structure models wait for their corresponding tissue models convergence to start their segmentation with sufficient reliable tissue knowledge.

-Updating structure models via tissue models: each structure l being composed of a single tissue T l ∈ {e CSF , e GM , e W M }, we do not estimate intensity models of class structure and class background. We rather compute them from tissue intensity models by setting for i ∈ V l :

p y i s i = e S , Ψ l y,i = p y i t i = T l , Ψ y,i p y i s i = e B , Ψ l y,i = max t∈{e CSF ,e GM ,e W M } p (y i |t i = t, Ψ y,i ) ,
so that improvements in tissue intensity models estimation are dynamically taken into account by structure models.

-Feedback of Structure Segmentation on Tissue Segmentation: conversely, results from structure models are integrated in the tissue model via the external field λ c (see Eq. 3). We express it as the disjunctive fusion over l of posteriori probabilities p s y, Ψ l coming from structures l of C S→T (c). It follows that structure segmentation is not reduced to a second step but is combined to tissue segmentation to improve their performances.

-Updating Fuzzy Maps: when the segmentation of structure l is updated the structure models of C S→S (l) take it into account by re-computing their spatial relations with respect to l, making the knowledge gradually more accurate.

A synthetic view of our approach is given in Fig 1.

Results

The evaluation was performed using both phantoms and real 3T brain scans. We choose not to estimate the β parameter but considered it as β = 1/T with T a decreasing temperature. Experimentally T : 10 → 5 provided good results. We first quantitatively compared LOCUS-T to two well known approaches, FAST [START_REF] Zhang | Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximisation algorithm[END_REF] of FSL and SPM5 [START_REF] Ashburner | Unified segmentation[END_REF], with the Dice similarity metric on the BrainWeb phantoms with 40% of nonuniformity and different noise values (see Fig. 2). Fig. 3 shows a visual evaluation on a very high bias field real 3T brain scan 4 . Fig. 3 shows that SPM5 failed, probably due to the use of a priori information hard to match with a surface coil brain acquisition. Next, we evaluated the cooperative tissue and structure segmentations. Three experts have manually segmented on BrainWeb the left caudate nucleus, the left putamen and the left thalamus, from which we computed a ground truth segmentation using STAPLE [START_REF] Warfield | Simultaneous truth and performance level estimation (STAPLE): An algorithm for the validation of image segmentation[END_REF]. Fig. 4 illustrates the gradual improvements of tissue and structure segmentations provided by LOCUS-TS. At the first convergence (time t 0 ) of EM, the Dice index is respectively 0.76, 0.77 and 0.72 for the caudate nucleus, the putamen and the thalamus. At the end, it reaches 0.76, 0.79 and 0.80. Fig. 5 shows qualitative evaluation of LOCUS-TS on a real 3T brain scan.

Discussion

Classical global approaches require to estimate an explicit bias field model to take into account the tissue intensity inhomogeneities [START_REF] Ashburner | Unified segmentation[END_REF][START_REF] Wells | Adaptative segmentation of MRI data[END_REF] due to MRI hardware imperfections. This model relies on the non realistic assumption of a single multiplicative bias field affecting all tissue classes equally. In contrast, the local estimation of MRF parameters in different subvolumes intrinsically handles the different sources of tissue intensity inhomogeneities. Our approach, with specific cooperation mechanisms between local models, appears to be an elegant and time efficient way to ensure global consistency of local models for tissue segmentation. It shows a significantly higher robustness to noise when compared to SPM5 (see Fig. 2), and more generally comparable results for a reduced computing time, namely, approximately 4min for LOCUS-T and respectively 8min and 14min for FSL and SPM5 on a 4Ghz Pentium, 1Go RAM. It illustrates that easy-to-segment subvolumes converge quickly, allowing the system to focus on other areas. LOCUS-T appears to be robust to very high intensity inhomogeneities as well (see Fig. 3), while SPM5, which uses an a priori atlas, fails in the segmentation and FSL does not estimate a correct bias field. In addition, instead of considering structure segmentation as a postprocessing, we propose to combine tissue and structure segmentations in a cooperative way: tissue and structure models are mutually updated, making both models gradually more accurate and providing optimal results (see Fig. 4 and5). Improvements are particularly significant for structures such as thalamus or putamen for which contrast to noise ratio is low (see Dice index improvement in Section 3). As regards the additional use of a priori anatomical knowledge, standard structure segmentation approaches rely on a global atlas. Atlas warping methods are classicaly time consuming and more or less limited due to inter-subject variability.

[7] introduced an atlas knowledge via the interaction energy term in a MRF. To be tractable, this solution considers that only the first order conditional dependence is important. This simplifying assumption, which reduces the Markov property, is not required with our approach. We consider instead an a priori description of brain anatomy based on fuzzy spatial relations. This was introduced in [START_REF] Barra | Automatic segmentation of subcortical brain structures in MR images using information fusion[END_REF] with a region-based approach, while it is used in [START_REF] Colliot | Integration of fuzzy spatial relations in deformable models -Application to brain MRI segmentation[END_REF] in a deformable model framework. However, the image preprocessing steps required make this approach difficult to apply on high field images, with high intensity nonuniformity, or on non homogeneous structures such as putamen. Our solution consists in introducing fuzzy localization constraints as relative prior weights for each voxel in the MRF framework via an external field term. It does not suffer from such difficulties as can be illustrated on structures such as putamen, and is still time efficient (10 to 15min). Note that currently LOCUS segments only a subset of the 37 structures segmented by the method proposed in [START_REF] Fischl | Whole brain segmentation: automated labeling of neuroanatomical structures in the human brain[END_REF]. To conclude, the robustness and modularity of our LOCUS approach appear as interesting features when handling complex segmentation tasks. Efficiency is improved both in term of results quality and computing time.

  partition the volume into a set of C non-overlapping local subvolumes V c , c ∈ C and distribute one local MRF model M c per subvolume. We consider K = 3 tissue classes: CSF (Cephalo-Spinal Fluid), GM (Grey Matter) and W M (White Matter). The hidden tissue classes t i 's take their values in {e 1 , e 2 , e 3 } respectively for classes {e CSF , e GM , e W M }. Each local MRF model M c is defined by the Gibbs distribution of energy (see Section 2.1):

  We propose a LOcal and Cooperative version of EM (LOC-EM) for spatially organized subvolumes to ensure a global consistency of local models. We denote by N (M c ) the set of MRF models neighbouring M c and introduce in EM a set of cooperation and coordination mechanisms as follows: Cooperation between M c and N (M c ). -Model Checking: we compute for each M c a model M c averaging the models of N (M c ). Then, for each class k, we compute the KullBack-Leibler distance D c k between intensity models of M c and M c . -Model Correction: if D c

Fig. 1 .

 1 Fig.1. Cooperative LOCUS-TS approach: for tissues (left), a LOC-EM cycle is distributed to each subvolume. For structures (right), each structure subvolume is associated to an EM cycle which cooperates with tissues.

Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .

 2345 Fig. 2. Comparison of LOCUS-T, FSL and SPM5 on the BrainWeb phantoms.

This image was acquired with a surface coil which provides a high sensitivity in a small region (here the occipital lobe) for functional imaging applications.