Clémence Petit

Éric Maire

Sylvain Meille

Jérôme Adrien

Shingo Kurosu

Akihiko Chiba

Eric Maire
email: eric.maire@insa-lyon.fr

CoCrMo cellular structures made by Electron Beam Melting studied by local tomography and finite element modelling

Keywords: Porous samples, X-ray Tomography, Mechanical behaviour, Finite Element

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Introduction

For many years, cobalt-chromium-molybdenum CoCrMo alloys have been used as biomedical devices, e.g. for orthopaedic prostheses or dental implants, due to an interesting combination of properties: biocompatibility, high strength and hardness, and good corrosion resistance. Various studies have been performed on the relationship between microstructure and mechanical properties of the biomedical Co29Cr6Mo alloy [START_REF] Ramírez | Precipitation path of secondary phases during solidification of the Co-25.5%Cr-5.5%Mo-0.26%C alloy[END_REF][START_REF] Herrera | Effect of C content on the mechanical properties of solution treated as-cast ASTM F-75 alloys[END_REF][START_REF] Lee | Effect of Fe addition on microstructures and mechanical properties of Ni-and C-Free Co-Cr-Mo alloys[END_REF][START_REF] Giacchi | Microstructural evolution during solution treatment of Co-Cr-Mo-C biocompatible alloys[END_REF][START_REF] Podrez-Radziszewska | Characteristic of intermetallic phases in cast dental CoCrMo alloy[END_REF]. Dense metallic implants have been extensively used but they have some drawbacks: a high density and mechanical properties that do not match with that of bone. For example, Co-Cr-Mo alloys have a Young's modulus of 230 GPa while the value for bone is 15 GPa. This difference of stiffness creates a phenomenon called stress shielding, which causes bone resorption around the implant and reduces the life span of the device. Development of porous implants have been proved to be a solution to overcome these problems [START_REF] Murr | Next generation orthopaedic implants by additive manufacturing using electron beam melting[END_REF]. The porosity enables to lower the mechanical properties towards those of bone. The bone in-growth around the implants is also improved by colonization of pores by living cells. Conventional fabrication processes failed in providing porous samples with complex shape, desired porosity and mechanical properties.

Producing such samples from casting of liquid alloy would lead different defects such as residual pores or other heterogeneities. For these reasons, Additive Manufacturing (AM) techniques have been developed to create innovative samples with desired porosity and shape [START_REF] Murr | Next generation orthopaedic implants by additive manufacturing using electron beam melting[END_REF]. For metals, Selective Laser Melting (SLM) and Electron Beam Melting (EBM) are currently the most popular techniques.

Different authors fabricated biomedical devices with these processes with titanium alloys [START_REF] Hollander | Structural, mechanical and in vitro characterization of individually structured Ti-6Al-4V produced by direct laser forming[END_REF][START_REF] Ryan | Porous titanium scaffolds fabricated using a rapid prototyping and powder metallurgy technique[END_REF][START_REF] Lopez-Heredia | Bone growth in rapid prototyped porous titanium implants[END_REF][START_REF] Gorny | In situ characterization of the deformation and failure behaviour of non-stochastic porous structures processed by selective laser melting[END_REF][START_REF] Wieding | The effect of structural design on mechanical properties and cellular response of additive manufactured titanium scaffolds[END_REF][START_REF] Ponader | In vivo performance of selective electron beam-melted Ti-6Al-4V structures[END_REF][START_REF] Yan | Evaluations of cellular lattice structures manufactured using selective laser melting[END_REF] and also with the above mentioned biomedical cobalt-based alloy [START_REF] Gaytan | Comparison of microstructures and mechanical properties for solid and mesh cobalt-base alloy prototypes fabricated by Electron Beam Melting[END_REF][START_REF] Gaytan | A TEM study of cobalt-based alloy prototypes fabricated by EBM[END_REF][START_REF] Murr | Microstructure and mechanical properties of open-cellular biomaterials prototypes for total knee replacement implants fabricated by electron beam melting[END_REF][START_REF] Rivera | Development of dense and cellular solids CrCoMo alloy for orthopaedic applications[END_REF].

However, this also creates new specific defects that can have detrimental effects on mechanical properties. The aim of this paper is to characterize the structural and mechanical properties of CoCrMo samples produced by EBM. For this purpose, samples were characterized thanks to X-Ray Computed Tomography (XRCT). This technique has become the most popular means of characterization of highly porous materials because it is non-destructive and enables to provide three dimensional information about the cellular samples with no special sample preparation [START_REF] Maire | X-Ray tomography applied to the characterization of highly porous materials[END_REF]. Typical resolutions of few microns are used to observe the mesoscopic architecture of the samples. But, it is also possible to obtain information about the fine scale microstructural features of the solid phase thanks to the so-called local tomography mode [START_REF] Zhang | Local tomography study of the fracture of an ERG metal foam[END_REF]. In this mode, the sample is placed near the Xray source and a high resolution image of the irradiated part of the specimen is obtained. The samples were compressed in situ in XRCT to obtain images at different levels of strain. Finally, a Finite Element (FE) model was built from the tomographic images at low resolution. The tomographic images at two resolutions, the in situ test and the FE calculations were combined to explain the mechanical behaviour of the material.

Material and methods

Fabrication of the materials

We prepared the samples according to the process described in the paper published by Sun et al. [START_REF] Sun | Build direction dependence of microstructure and high-temperature tensile property of Co-Cr-Mo alloy fabricated by electron beam melting[END_REF].

The cobalt powder used in the experiment had the composition given in table 1 and was made of spherical particles with an average particle size of 64 µm. It has to be noted that the powder composition is in the range of the ASTM F75 standard. The samples were fabricated with a special device (Arcam AB, Mölndal, Sweden). An electron beam was focused thanks to electromagnetic lenses and sent to a layer of powder. The powder was raked on the substrate from a powder stock. A first beam scan pre-heated the powder at a temperature of 840°C and the powder was selectively melted at a temperature of 1430°C (i.e. the melting point of the cobalt-based alloy). A new layer was then subsequently created onto the previous layers already built which formed the sample under construction. The powder is selectively melted according to a model previously generated by a Computer Aided Design (CAD) software. The samples had a cubic shape with a dimension of 20 mm. The cells were targeted to have a size of 1.2 mm and wall size of 400 µm. The direction 3 is referred as the building direction. After fabrication, the samples underwent an ageing heat treatment at 800°C during 24 hours to transform the γ (fcp) phase into ε (hcp) phase. In a previous paper, microstructural analysis evidenced that the grains were fully transformed into the single ε phase by the ageing treatment [START_REF] Sun | Build direction dependence of microstructure and high-temperature tensile property of Co-Cr-Mo alloy fabricated by electron beam melting[END_REF]. Precipitates such as M2N nitrides and M23C6 carbides were also identified with M being Cr, Mo and Si.

A small sample with dimensions of 4.6 x 4.7 x 3.9 mm 3 (respectively directions 1, 2 and 3) was cut from one of the larger samples with a micro-saw (IsoMet 1000, Buehler, Germany). These dimensions were chosen to have a sample small enough to be inserted into the in situ mechanical test device.

The geometric relative density was calculated from the weight and the dimensions of the large samples, considering a density of 8.4 g.cm -3 for the fully dense CoCrMo alloy.

The sample was observed thanks to Scanning Electron Microscope (SEM, SUPRA VP55, Zeiss, Germany) at a voltage of 10 kV.

Nanoindentation was used to determine the mechanical properties of the solid phase of the sample.

For this purpose, a sample was embedded in a plastic resin and carefully polished down to 1 µm with diamond paste. The nanoindentation system (Agilent G200, Agilent Technologies, USA) was equipped with a Berkovich tip. Due to the geometry of the Berkovich tip, the indent length is equal to seven times the indent depth. The applied load was chosen so that the indent was fully localized in the solid wall. A series of 10 indents was made with a speed of 10 nm.s -1 and dwell time of 10 s. The calculation of Young's modulus from nanoindentation experiments was done through the use of the continuous stiffness measurement mode (CSM).

X-ray tomography

A laboratory X-ray tomograph (Phoenix vTomeX / X ray, Germany) described in [START_REF] Buffière | In situ experiments with X ray tomography: an attractive tool for experimental mechanics[END_REF] was used to scan the sample. Due to the high absorption coefficient of the cobalt-based alloys, the tomograph was operated at 100 kV and 140 µA with a 0.3 mm copper filter on the incident X-rays. The whole sample was scanned at a resolution of 15 µm (i.e. voxel size of 15 µm). The source to dectector is fixed to 80 cm in our device. To obtain low resolution, the distance between the X-ray source and the sample was 10 cm. In this configuration, the whole sample was irradiated by the X-ray conical beam. In order to observe the finer details of the microstructure, local tomography was subsequently used [START_REF] Buffière | In situ experiments with X ray tomography: an attractive tool for experimental mechanics[END_REF]. In this mode, only a part of the sample was irradiated by X-rays under all the viewing angles as the field of view of the detector was smaller than the size of the sample. A part of the sample is not seen by the detector for some angles of rotation. But, some authors demonstrated that it is possible to reconstruct a 3D image without noise [START_REF] Youssef | Finite element modelling of the actual structure of cellular materials determined by X-ray tomography[END_REF][START_REF] Toda | Quantitative assessment of microstructure and its effects on compression behavior of aluminum foams via high-resolution synchrotron X-ray tomography[END_REF]. For this purpose, the sample was placed very close to the X-ray source: the distance between the source and the sample was then only 1 cm. Using this mode, a high resolution reconstruction of the irradiated part of the sample could be obtained with a voxel size of 3 µm. To scan another part of the sample, it was then sufficient to physically displace the specimen on the rotating stage. Reconstructing the whole volume of the sample at high resolution required that we acquired 8 images, each representing a part of the sample.

A home-made compression rig was inserted in the tomograph to perform the compression test as described in [START_REF] Zhang | Local tomography study of the fracture of an ERG metal foam[END_REF]. The sample was placed between two grips acting as compression platens. The upper platen was screwed to a force sensor. The set (force sensor, platens and sample) was placed into an aluminium tube with an inner diameter of 16 mm and a thickness of 4 mm. The latter was then screwed to the motor. Aluminium has a mass attenuation coefficient of 0.37 cm 2 /g for a beam with a voltage of 100 kV [START_REF] Hubbell | Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients (version 1.4)[END_REF]. This value is lower than that of cobalt (2.14 cm 2 /g). Therefore, the presence of the aluminium tube does not impeach scanning of the sample during the in situ test. The displacement was applied by the motor via a home-made software with a speed of 0.01 mm.s -1 . The reaction force was measured by the force sensor and recorded by a second home-made software. The test was interrupted to scan the sample at different strains with the voxel size of 20 µm. The difference of length between images of initial and deformed states was used to calculate average strain.

Image processing and analysis

Image processing of the 3D images was performed using the ImageJ software. The low resolution image of the initial sample was processed to improve the image quality: adjustment of contrast and brightness, application of median filtering over neighbours of two voxels. Then, it was thresholded to separate the pores from the solid phase.

The thresholded image was used to perform different morphological characterizations. A more detailed description of these measurements is explained in [START_REF] Maire | Characterization of the morphology of cellular ceramics by 3D image processing of X-ray tomography[END_REF].

The relative density was determined as the ratio of total voxels belonging to the solid phase compared to the total number of voxels.

The distributions of the pore size and of the cell wall thickness were measured by a technique of granulometry comparable to sieving of a powder. It consists in applying a series of erosion and dilation operations to the studied phase of the image. A histogram giving the pore size or the cell wall thickness distribution is obtained.

FE modelling

The low resolution 3D image of the initial sample was meshed with 1200000 quadratic tetrahedra having an edge size of 54 µm by Avizo 8 software. A negative displacement corresponding to the strain reached during the mechanical test was applied to the top nodes in the direction 3 whereas the nodes of the opposite faces were displacement constrained. An elastoplastic behaviour was assumed for the CoCrMo material. A Young's modulus of 230 GPa (determined by nanoindentation) and a Poisson's ratio of 0.3 were considered for elastic properties. This value of Young's modulus is close to one determined in literature [START_REF] Balagna | Tantalum-based multilayer coating on cobalt alloys in total hip and knee replacement[END_REF][START_REF] Ortega-Saenza | Tribological and corrosion testing of surface engineered surgical grade CoCrMo alloy[END_REF]. Plastic stresses and strains were defined following the compression stress/strain curve from Montero-Ocampo et al. [START_REF] Montero-Ocampo | Effect of alloy preheating on the mechanical properties of as-cast Co-Cr-Mo-C alloys[END_REF]. This curve was obtained for a CoCrMo alloy (having a chemical composition in accordance with the ASTM F75 standard) cast at a temperature of 1430°C (melting temperature of the alloy) and heat treated at 815°C during 24 hours.

The melting of powder during EBM is certainly a different process from the casting of liquid alloy.

Therefore, the compression stress/strain of a dense CoCrMo alloy made by EBM would certainly not have exactly this profile. But, without existing data about a uniaxial compression test of dense samples made by EBM, it was necessary to choose data from alloys made by casting.

Results

Qualitative and quantitative description of the sample

The periodic square cell structure is visible thanks to 3D visualization (fig. 1a) and a SEM image of the sample (fig. 1b). The 3D visualization was built from low resolution tomographic image with a voxel size of 15 µm. The sample has an apparent density of 0.24. The relative density calculated by ImageJ is 0.25 in good agreement with the previous value. The relative density distribution (fig. 2)

shows alternation of dense slices and more porous ones. The direction 3 corresponds to the building axis during the EBM process. Each curve is periodic which is consistent with the cubic pore structure (fig. 2).

The low resolution image (fig. 3a) enables to notice the important roughness at the surface of the sample in all the struts. Some pores are also visible in the solid phase. This is confirmed by images obtained thanks to local tomography. The high resolution images revealed the presence of many morphological defects in the solid struts. We present hereafter tomographic images (3D visualization and slices) of some zones in the sample. First, different pores with various shapes are noticed in the struts (see fig. 3a showing a horizontal strut). In fig. 4a, sets of small solid spheres are visible in the solid phase (see the red ellipse). They give a rough aspect to the surface (visible in 3D image in fig. 4a). The spheres are partially melted particles which attached to the sample during the process. Fig. 3b shows a strut where no attached particles are present. Its surface has an important roughness which is the cause of irregular thickness along the struts (see the vertical struts in 3D in fig. 3b and in 2D in fig. 4b). These defects are not visible in lower resolution images and are revealed by local tomography. The interest of local tomography lies in the possibility to scan the whole sample at high resolution and to characterize completely the presence of defects.

The strut thickness and cell size distributions obtained from 3D granulometry on tomographic volume are shown in fig. 5a and5b. The structural element used to measure the granulometric distribution was a sphere. The strut thickness distribution from tomographic image is rather wide around the main peak. This is consistent with the presence of defects seen on fig. 3 and4. The pore size distribution possesses a main peak corresponding to the macro-pores with a square cross-section and a smaller peak around 120 µm, indicating the presence of smaller pores. The wide size distributions of solid and porous phases illustrate the presence of heterogeneities in the structure.

In situ compression test

Fig. 6 presents the compression engineering stress versus engineering strain curve. The stress/strain curve presents a first elastic domain followed by a plastic part. The low ductility of the CoCrMo alloy is an explanation of the brittle failure of the struts and of the absence of stress plateau and densification regions.

The mechanical properties obtained from the compression test for this sample and for a second one are summarized in table 2, as well as some data extracted from literature on compression of CoCrMo cellular samples. The quoted authors of the table studied CoCrMo cubic lattice structures made by AM (EBM or SLM) in compression. Our results are in relatively good agreement with these previous experiments. The ratio of the cell over the sample size has already been shown to have an influence on the mechanical properties, with a smaller strength for samples with fewer cells [START_REF] Andrews | Size effects in ductile cellular solids. Part II: experimental results[END_REF]. The quoted authors tested samples with 6 cells in each direction whereas our sample has only 2 cells in each direction. This can explain their higher relative yield strength and relative compression strength.

Moreover, the difference obtained by the two quoted authors can be explained by the different processes used (i.e. EBM) and the different relative densities achieved. Fig. 7 shows a 3D visualization of some struts of the sample during the in situ compression test.

Deformation localizes in the upper struts and especially in the left one. Buckling is visible in this strut at a strain of 2.5% (see fig. 7b). This strain corresponds to the macroscopic linear elastic part of the stress/strain curve (see fig. 6). The deformation visible in the images could be caused by local microplasticity. With increasing strain, stress exceeded yield point. At this stage (i.e. strain of 4.7% shown in fig. 7c), the deformation in the upper struts increases. This localization of deformation in some struts created a deformation band perpendicular to the loading direction typical of a cellular material tested in compression. This corresponds in the stress/strain curve to the beginning of the stress plateau. But, the vertical struts progressively broke leading to a decrease of stress until separation of the sample into two parts. This is not shown in the figure. Fig. 7c also shows the deformation localizes in a part of the strut (see the red frame). The corresponding high resolution image of this strut (fig. 7d) evidences the presence of an agglomerate of partially melted particles at the exact location of the deformation.

FE modelling

Fig. 8 presents the experimental and the modelled stress/strain curve. The calculated Young's modulus was ten times higher than the experimental one. This has already been observed for finite element modelling of cellular samples [START_REF] Caty | Modelling the properties of closed-cell cellular materials from tomography images using finite shell elements[END_REF]. No clear explanation exists about this overestimation [START_REF] Petit | Cellular solids studied by X-ray tomography and finite element modeling -a review[END_REF]. On the contrary, there was a good agreement between experiment and modelling for the plastic part of the curve (between the yield point and the compression strength).

The normal σ33 stress distributions for the struts (at a strain of 4.7 %) are shown in fig. 9. It is seen that the finite element model predicts higher stress in the upper struts. The defects due to fabrication such as reduced thickness in the struts concentrate the stresses as it is visible in fig. 9. This confirms the experimental findings about the detrimental influence of these specific defects.

Discussion

CoCrMo cellular samples exhibit a mechanical behaviour which has already been observed by other authors for cubic unit cell structures [START_REF] Hazlehurst | Evaluation of the stiffness characteristics of square pore CoCrMo cellular structures manufactured by laser melting technology for potential orthopaedic applications[END_REF][START_REF] Kadkhodapour | Failure mechanisms of additively manufactured porous biomaterials: Effects of porosity and type of unit cell[END_REF]. The deformation occurs by buckling of vertical struts followed by a layer by layer collapse of the sample [START_REF] Ahmadi | Additively manufactured open-cell porous biomaterials made from six different space-filling unit cells: the mechanical and morphological properties[END_REF].

The plastic deformation initiates in weak zones where some defects are present. These defects are difficult to observe with low resolution image of the whole sample and are revealed by local tomography. These defects are caused by the EBM process: small pores in the solid phase, attached particles in the surface, irregular thickness along the strut. Other authors have already noticed the presence of defects in samples made by EBM or SLM. Murr et al. [START_REF] Murr | Microstructure and mechanical properties of open-cellular biomaterials prototypes for total knee replacement implants fabricated by electron beam melting[END_REF] or Pupo et al. [START_REF] Pupo | Surface quality analysis in Selective Laser Melting with CoCrMo powders[END_REF] found the important role of roughness in CoCrMo structures made by SLM. The presence of partially melted particles on the surface has already been observed by SEM by Yan et al. with stainless steel samples made by SLM [START_REF] Yan | Evaluations of cellular lattice structures manufactured using selective laser melting[END_REF]. But, up to now, local tomography has not been used to scan a whole sample at high resolution. The scans performed in this study enable to observe the presence of the defects in all the sample and to link them to the zones of deformation during the compression. This is confirmed by the FE calculation which predicts that the defects constitute weak points in the structure. This new information should encourage the use of local tomography to better characterize the structure of the EBM-or SLM-built samples. They also prove that improvements of these AM processes are necessary to remove these defects which are caused by a non-optimized process. The presence of spherical residual pores can be linked to entrapment of gas in the powder in the EBM build chamber. The gas bubbles could not escape during melting of powder and generated pores. More irregularly shaped pores are linked to an incomplete melting of powder during the process [START_REF] Tammas-Williams | XCT analysis of the influence of melt strategies on defect population in Ti-6Al-4V components manufactured by Selective Electron Beam Melting[END_REF]. The presence of attached particles is caused by a partial melting of some powder particles during the EBM.

Different parameters of the process can be optimized: powder characteristics, scanning parameters, electron beam or laser beam parameters. Post-processing operations can also be done: Yan et al. [START_REF] Yan | Evaluations of cellular lattice structures manufactured using selective laser melting[END_REF] propose to sand-blast the samples after EBM to remove the particles, Pattanayak et al. [START_REF] Pattanayak | Bioactive Ti metal analogous to human cancellous bone: fabrication by selective laser melting[END_REF] heat treat their samples to fuse the particles and to bond them to the surface.

Conclusion

The EBM manufacturing technique was successfully used to produce porous samples with cubic unit cells. X-ray computed tomography is useful to characterize quantitatively and qualitatively the architecture of the samples. Local tomography revealed the presence of different defects due to the fabrication process, especially at the strut surface: pores in the solid phase, partially melted particles attached to the surface giving a rough aspect to the surface, irregular thickness along the struts. In situ compression test performed on this sample showed a typical cellular behaviour with deformation followed by fracture of the upper struts by buckling. Careful examination of the tomographic volumes showed an influence of the fabrication defects on the initiation of the deformation. The presence of thinner zones surrounded by bump of partially melted particles constituted weak points of the sample which initiated the buckling. Consequently, the FE calculation predicted the highest stresses in the zones of reduced thickness and of partially melted particles in the vertical struts.

Figures and tables

Table 1: Chemical composition of the CoCrMo powder used for fabrication (wt. %) [START_REF] Sun | Build direction dependence of microstructure and high-temperature tensile property of Co-Cr-Mo alloy fabricated by electron beam melting[END_REF] Table 2: Young's modulus, yield strength and compression strength obtained from the compression test in this study and comparison with values from the literature for other CoCrMo cellular samples

Fig. 1 :

 1 Fig.1: Compression stress/strain curve of a biomedical CoCrMo alloy used in the model[START_REF] Hubbell | Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients (version 1.4)[END_REF]

Fig. 2 :Fig. 3 :Fig. 4 :

 234 Fig. 2: (a) 3D image of the whole sample and (b) relative density profile along the three directions

Fig. 5 : 21 Fig. 6 :Fig. 7 :Fig. 8 :Fig. 9 :

 5216789 Fig. 5: (a) Strut size and (b) pore size distribution of the sample measured by image processing

Fig. 1

 1 Fig. 1

Fig 2 Fig 3 24

 23 Fig 2

Fig 4 Fig 5 25

 45 Fig 4

Fig 6 Fig 7 Fig 8

 678 Fig 6

Table 2

 2

Acknowledgements

The authors acknowledge José Ferreira (MATEIS laboratory, INSA Lyon) who developed the softwares used for the in situ mechanical test.


Clémence Petit

Eric Maire
email: eric.maire@insa-lyon.fr

Sylvain Meille

Jérôme Adrien

Two-scale study of the fracture of an aluminum foam by X-ray tomography and finite element modeling

Keywords: Aluminum foam, X-ray tomography, mechanical behavior, finite element modeling Highlights

An aluminum foam can be characterized by its architecture and by the solid phase' microstructure. Our aim is to link the foam's morphological and microstructural features with its mechanical properties thanks to X-ray tomography and finite element (FE). An approach combining X-ray tomography at different resolutions, image processing, and FE modeling was developed to take into account the influence of the intermetallics on the foam's fracture. First, the samples were scanned with "local" tomography, where the specimen is placed close to the X-ray source. These images allowed for observing intermetallics. Then an in situ tensile test was performed in the tomograph to follow the sample's deformation at low resolution. The images obtained from local tomography were processed to create one low-resolution image of the initial sample including details from high resolution. This was done by a series of thresholding and scaling of the high-resolution images. This image was used to generate a FE mesh. A FE input file was obtained thanks to Java programs associating the elements to the phases. At the local scale, the calculated stress distribution and the images of the struts were analysed. Our work confirms that the presence of inclusions can explain the fracture of struts.

Introduction

Cellular metals have received more and more attention due to their unique combination of properties (i.e., lightness, shock and sound absorption) [START_REF] Banhart | Light-metal foams -history of innovation and technological challenges[END_REF]. Aluminum cellular samples are particularly of interest due to the low weight and low melting point of aluminum alloys. Among them, Duocel ® foam is an open-cell aluminum sample made of 6101 alloy. The relation between the architecture of this foam and its mechanical behavior was extensively studied in the literature [START_REF] Andrews | Compressive and tensile behavior of aluminum foams[END_REF][START_REF] Amsterdam | The influence of strain induced damage on the mechanical response of open-cell aluminum foam[END_REF][START_REF] Gioux | Failure of aluminum foams under multiaxial loads[END_REF][START_REF] Nieh | Effect of cell morphology on the compressive properties of open-cell aluminum foams[END_REF]. However only few studies [START_REF] Amsterdam | Fracture and microstructure of open cell aluminum foam[END_REF][START_REF] Wang | Solidification and precipitation behavior of Al-Si-Mg casting alloys[END_REF][START_REF] Tanihata | Effect of casting and homogenizing treatment conditions on the formation of Al-Fe-Si intermetallic compounds in 6063 Al-Mg-Si alloys[END_REF] focused on the influence of the microstructural features of the solid phase on the mechanical properties of such foams. In the case of 6xxx Al alloys, the presence of Fe-rich intermetallic inclusions [START_REF] Wang | Solidification and precipitation behavior of Al-Si-Mg casting alloys[END_REF][START_REF] Tanihata | Effect of casting and homogenizing treatment conditions on the formation of Al-Fe-Si intermetallic compounds in 6063 Al-Mg-Si alloys[END_REF] is known to possibly have an influence on the macroscopic behavior of porous aluminum foams. As an example, in situ tensile tests with scanning electron microscope (SEM) showed a brittle intergranular rupture in the struts containing Fe-rich precipitates [START_REF] Amsterdam | Fracture and microstructure of open cell aluminum foam[END_REF].

Cellular solids have been often studied thanks to X-ray computed tomography [START_REF] Maire | X-Ray tomography applied to the characterization of highly porous materials[END_REF][START_REF] Burteau | Impact of material processing and deformation on cell morphology and mechanical behavior of polyurethane and nickel foams[END_REF][START_REF] Guillen | Compressive behavior of bovine cancellous bone and bone analogous materials, microCT characterization and FE analysis[END_REF], a non-destructive technique which provides 3D visualization and enables a 3D characterization of a sample. The architectural parameters (e.g., relative density, distribution of pores size) can be determined with 3D images having resolutions of few microns. The deformation of the samples can also be followed with this technique through in situ mechanical tests [START_REF] Buffière | In situ experiments with X ray tomography: an attractive tool for experimental mechanics[END_REF].

Moreover, the 3D images can be meshed to build finite element (FE) models [START_REF] Guillen | Compressive behavior of bovine cancellous bone and bone analogous materials, microCT characterization and FE analysis[END_REF][START_REF] Youssef | Finite element modeling of the actual structure of cellular materials determined by X-ray tomography[END_REF][START_REF] Miedzinska | Numerical and experimental aluminum foam microstructure testing with the use of computed tomography[END_REF][START_REF] Jeon | Finite element simulation of the plastic collapse of closed-cell aluminum foams with X-ray computed tomography[END_REF].

Recently, 3D microstructural characterization at a higher resolution was made possible using the so-called "local tomography" mode [START_REF] Toda | Quantitative assessment of microstructure and its effect on compressive behavior of aluminum foams via high-resolution synchrotron X-ray tomography[END_REF][START_REF] Zhang | Local tomography study of the fracture of an ERG metal foam[END_REF]. In this mode, the sample is placed near the Xray source and a high-resolution image of the irradiated part of the sample is obtained. These high-resolution images can be analysed to characterize the mechanical behavior of the samples, especially the initiation and the propagation of cracks [START_REF] Toda | Quantitative assessment of microstructure and its effect on compressive behavior of aluminum foams via high-resolution synchrotron X-ray tomography[END_REF]. This approach has been developed initially for the study of the tensile fracture of Duocel foams. The authors noticed the link between the presence of inclusions and the fracture of the slightly loaded struts [START_REF] Zhang | Local tomography study of the fracture of an ERG metal foam[END_REF].

The aim of this paper is to refine this first approach by building a FE model including the microstructural features of the Duocel foams obtained from local tomography. Our work is based on the use of X-ray tomography at two scales, i.e., at two resolutions. A Duocel foam was characterized thanks to local tomography, enabling to observe the Fe-rich inclusions. Then an in situ tensile test was performed in the tomograph. Compression test is often chosen for metallic foam to study the capacity of these samples to absorb energy [START_REF] Banhart | Light-metal foams -history of innovation and technological challenges[END_REF][START_REF] Andrews | Compressive and tensile behavior of aluminum foams[END_REF][START_REF] Nieh | Effect of cell morphology on the compressive properties of open-cell aluminum foams[END_REF]. But, the fracture behavior of these materials is also of interest. In this work, a tensile test was chosen to follow the progressive fracture of the foam and to make the link with its microstructural features.

During the test, the sample was scanned at a low resolution. Finally, the high-resolution images obtained from local tomography were processed to create a FE model at low resolution, including the presence of the inclusions in the calculation. The comparison between high and low-resolution 3D images finally shows that the struts' fracture mainly occurs in the inclusionrich regions. The stress fields calculated by FE taking into account the inclusions enable us to explain the fracture of some struts in which no architectural feature was found.

Material and methods

Material

The Duocel foam considered in this study is an open-cell aluminum foam kindly provided by ERG Aerospace (Oakland, CA, USA). Its composition, as determined by inductively coupled plasma atomic emission spectroscopy by Zhou et al. [START_REF] Zhou | Mechanisms and mechanics of compressive deformation in open-cell Al foams[END_REF], is presented in Table 1. The Duocel foams are fabricated by investment casting as described in detail in [START_REF] Banhart | Manufacture, characterization and application of cellular metals and metal foams[END_REF]. An open-cell polyurethane foam is first infiltrated by a ceramic slurry. After heat treatment, a porous ceramic having the negative shape of the polyurethane foam is obtained. Then the pores of the ceramic are infiltrated by liquid aluminum. After removal of the ceramic by pressurized water, a porous metallic sample is obtained and is a replication of the initial polyurethane foam. After fabrication, the Duocel foams were heat treated by ERG with a T6 strengthening heat treatment: the samples were first heat treated at a temperature of 527°C for 8 hours and then quenched in water at room temperature. Finally, they were aged at 177°C during 5 hours. 1: Chemical composition of the Duocel foam (wt %) [START_REF] Zhou | Mechanisms and mechanics of compressive deformation in open-cell Al foams[END_REF] A square prism shaped sample (thickness: 4.1 mm, width: 9.6 mm, and height: 22 mm, dimensions referred hereafter as in directions 1, 2 and 3 respectively) was cut from a foam block. The thickness was the thickness of the initial foam block. The width and the length were imposed by the tensile loading device, which will be described in the next section. The entire approach developed in this paper, from the in situ testing to the FE modeling was applied to this sample, used for illustration in the paper. This is due to the difficulty to reproduce this timeconsuming approach for a large number of samples. Therefore, our goal was to develop a qualitative approach to determine the influence of artificial defects on the macroscopic behavior of scaffolds.

Al

Characterization by X-ray tomography

A laboratory X-ray tomograph (Phoenix vTomeX / X ray, Germany) located in the MATEIS lab, described in detail in [START_REF] Buffière | In situ experiments with X ray tomography: an attractive tool for experimental mechanics[END_REF] was used. It operated at a voltage of 80 kV and an intensity of 280 µA with no filtering on incident X-rays. The spot size of the source was approximately 1 µm in these conditions. The detector (Varian Paxscan) has an active surface of 200 x 250 mm with a matrix of 1920 x 1536 pixels (pixel size 127*127 µm 2 ). Each scan was performed with the following conditions: 900 projections, scan timing of 333 ms for each angular position. The distance from X-ray tube to the detector is 577 mm. The 3D final images were reconstructed thanks to a standard filtered back projection algorithm implemented in the software coupled to the tomograph [START_REF] Kak | Principles of computerized tomographic imaging[END_REF]. Firstly, the whole sample was scanned at a low resolution (20 µm per voxel). In this configuration, the X-ray tube/object distance was 90.8 mm. Then high-resolution 3D images (voxel size of 3 µm) were obtained thanks to local tomography. In that experiment, the sample was placed near the X-ray source and only a part of the sample was irradiated by Xrays under all the viewing angles. The X-ray tube/object distance was then 13.6 mm. Some authors proved that high quality images can be obtained in this mode when the sample is homogeneous outside of the field of view of the detector [START_REF] Youssef | Finite element modeling of the actual structure of cellular materials determined by X-ray tomography[END_REF][START_REF] Toda | Quantitative assessment of microstructure and its effect on compressive behavior of aluminum foams via high-resolution synchrotron X-ray tomography[END_REF]. It was observed that the images reconstructed by local tomography have a lower signal to noise ratio than in conventional tomography. However, due to their low density, porous samples are particularly adapted to local tomography [START_REF] Zhang | Local tomography study of the fracture of an ERG metal foam[END_REF]. After the high-resolution scan of the irradiated part of the sample, it was necessary to physically displace the specimen on the rotating stage to scan another part of the sample. Finally, 18 3D images, each representing a different region of the sample were acquired to have a full description of the sample at high resolution.

After the initial characterization, the sample underwent an in situ tensile test in the tomograph.

For this purpose, a custom-made device described in [START_REF] Zhang | Local tomography study of the fracture of an ERG metal foam[END_REF] was inserted on the rotating stage. This device is shown in Fig. 1 as unmounted. First, the sample was glued to two screws, the alignment of the screws was maintained thanks to a device designed for this purpose. Then, the sample was screwed to two grips (Fig. 1, arrow 1). The upper grip was attached to a force sensor. The set {force sensor, grips and sample} was placed into an aluminum tube (Fig. 1, arrow 2). This tube has an inner diameter of 15 mm and a thickness of 1 mm. It does not affect the reconstruction because it is almost transparent to the X-rays. The inner diameter of the tube imposed to have a sample's width lower than 10 mm. Then, the aluminum tube containing the set {force sensor, grips, and sample} was attached to the driving axis of the motor (Fig. 1,arrow 3). Finally, the assembly was placed onto the rotating stage in the tomograph. The length of the sample was imposed by the entire length of the device to scan the entire sample. A uniaxial tensile test was applied: the vertical displacement of the lower grip was applied by the motor via a custom-made software with a speed of 0.01 mm.s -1 . The displacement of the upper grip was totally constrained. The reaction force was measured as a function of time by the force sensor and recorded by the software. The test was interrupted fifteen times to scan the sample at different applied strains with the same voltage and intensity as for the initial characterization and using a voxel size of 20 µm (referred to hereafter as "low-resolution"). The mean applied strain at different loading steps was determined from the displacement obtained from the comparison of the images of initial and deformed states. It was calculated relatively to the total height of the initial sample. Between two interruptions to scan the sample, the linear applied displacement varied between 0.13 and 0.4 mm (i.e., between 7 and 20 voxels with a voxel size of 20 µm). This corresponds to incremental strains between 0.6% and 1.8 %. To calculate the mean engineering stress, the applied force was simply divided by the initial section of the sample. The logarithmic differential method was used to calculate the uncertainties on the stress, taking into account an uncertainty of 2 N for the force and 0.05 mm for the sample's dimensions. To complete the characterization of the microstructure by X-ray tomography, some struts of aluminum foam were scanned with another tomograph operating at higher resolution (Easy Tom Nano, RX Solutions, France, equipped with a Hamamatsu X-ray tube and a Hamamatsu detector). In this case, different struts were cut from a foam block and scanned. Higher resolution images of these particular struts with a voxel size of 0.4 μm were obtained without using local tomography. The scans were made using a voltage of 60 kV and an intensity of 150 µA.

Image processing steps

As previously mentioned, the 18 high-resolution 3D images of the initial sample were acquired and then processed with the ImageJ software [21]. ImageJ is a public domain, open-source Javabased software for image processing and analysis and is extensible via plugins and macros. Image processing steps described below were performed on reconstructed volumes. First, the quality of each volume was improved by the adjustment of contrast and brightness and by the application of median filtering over neighbours of two voxels. Fig. 2a shows a slice of a highresolution 3D image after treatment. Next, two thresholding operations separated the three different phases: voids, aluminum matrix and intermetallic inclusions (see Fig. 2b). At this stage, it could have been possible to stitch the 18 volumes together to obtain a high-resolution 3D image of the whole sample. However, the result would have been a large volume which could not be easily used for analysis and FE modeling. Instead, each thresholded image was down-scaled to change its voxel size from 3 to 20 µm. The segmentation made at high resolution was preserved after scaling as shown in Fig. 2c. However, scaling blurred the images by spreading out the grey levels around the initial value especially at the interface between the phases. Thus, a second thresholding was made using a conservative value of the grey level to make sure that no inclusion was lost during this step (Fig. 2d). The 18 initial high-resolution 3D images (Fig. 2a) were transformed into 18 low-resolution thresholded 3D images. Finally, these tomograms were stitched together until the image of the whole sample was created. It must be clearly stated that the second thresholding step results in a non-physical increase of the size, thus of the volume fraction of intermetallics (see Fig. 2b and2d). For this reason, the final 3D image should be considered as "containing the information" of the presence of the inclusions in different locations inside the sample, rather than actually describing their morphology in detail.

Image analysis

The final 3D image of the initial sample (Fig. 2d) was used to quantify the sample's architecture using Java programs implemented in ImageJ software. Three parameters were calculated to characterize the scale of the architecture: the relative density and its variation in the sample, the pore size distribution and the strut size distribution. For highly porous materials, 3D data are needed to obtain a complete characterization. The principle of these quantifications is described in [START_REF] Maire | Characterization of the morphology of cellular ceramics by 3D image processing of X-ray tomography[END_REF]. We give here below a brief summary of this procedure.

The relative density was determined as the ratio of the number of voxels belonging to the solid phase (i.e., white voxels in the thresholded 3D images) as compared to the number of voxels in the entire volume. Moreover, the local variations of density in the sample are given by plots of the relative density in each slice as a function of the slice position in a given direction. Three plots are obtained, each corresponding to one direction. For comparison, the geometric relative density of the sample was also determined from the weight and the dimensions of the samples, considering a density of 2.7 g.cm -3 for the fully dense aluminum alloy. The uncertainty on this measurement was calculated considering the uncertainties in the measurements of the scale and of the caliper.

The reconstructed volume was also used to measure the characteristic size distribution of each phase (i.e., solid and voids). The method used in this work can be considered as a virtual granulometry technique and is comparable to sieving of a powder [START_REF] Maire | Characterization of the morphology of cellular ceramics by 3D image processing of X-ray tomography[END_REF]. It consists in the application of a sequence of three-dimensional erosion and dilation operations to the phase of interest. The sequence is repeated with increasing size of the structural element, spherical in the present study. A histogram was obtained giving the amount of the phase as a function of the structural element size. This operation was applied to the voids and the solid phase to obtain the pore size and the strut size distributions. Because the sample is an open-cell foam, the characteristic size of the solid phase is described in terms of strut size. We also applied this technique to the 3D images of single struts (voxel size of 0.4 μm) to measure the inclusion size distribution.

Building of the FE model

First, the solid phase (i.e., intermetallics and aluminum) was meshed using Avizo 8 software with 1 228 000 tetrahedral elements having an average edge size of 76 µm. Because the goal of this study was to take into account the inclusions on the fracture properties, the calculation was performed with the finest mesh which could be used in relation with the computer capacity (Dell, Intel Xeon, RAM 100 Go). That is why no specific convergence study was performed in this work, dedicated to the comparison of calculations including or not the presence of inclusions. To mesh the volume, the first step consisted of the generation of a surface meshing.

A triangulated closed surface separating the volume from the voids was obtained thanks to a marching cube algorithm [START_REF] Lorensen | Marching cubes: high resolution 3-D surface reconstruction algorithm[END_REF] implemented in Avizo [START_REF] Zilske | Adaptive remeshing of non-manifold surface[END_REF]. After different steps of optimization of the triangulated surface, the three-dimensional mesh was created by the advancing front method implemented in Avizo [START_REF] Frey | 3D Delaunay mesh generation coupled with an advancing-front approach[END_REF]. The 3D mesh generated by Avizo was made of tetrahedra with four nodes (at each corner).

Then, the Avizo generated mesh had to be transformed into an input file readable by Abaqus.

The presence of the intermetallics in the solid phase had to be taken into account in the model.

For this purpose, a special Java program was developed to create an Abaqus input file. This procedure is similar to what was done in [START_REF] Caty | Modeling the properties of closed-cell cellular materials from tomography images using finite shell elements[END_REF] to model a fatigue test of a stainless steel hollow sphere structure by associating each shell element of a mesh to the local thickness of the hollow spheres. The Java program first calculated the barycentre coordinates of each tetrahedral element of the mesh. Then it associated each element to the grey level located at its barycentre as indicated in the tomographic image. As the grey level represented a specific phase (i.e., 128 for inclusions and 255 for aluminum), each element was associated to a phase. Therefore in the Abaqus input file, two element sets were defined: one for elements corresponding to the aluminum phase and the other for elements corresponding to the inclusions or to a zone containing inclusions. Some inclusions could be larger than 76 µm, the size of the elements. Some elements could be then associated to an inclusion whereas they correspond to a zone containing inclusions and aluminum matrix. Each element set was subsequently associated to one of the two material behaviors described in the following section. A second calculation was performed as a reference neglecting the presence of the inclusions: for this, all the elements were considered as having the constitutive law of the aluminum matrix.

The two calculations were made from these two input files readable by Abaqus and containing all the information required. The element type used was first-order tetrahedral element with linear interpolation. For this type of element, the degrees of freedom are the translations. They are fully integrated elements (i.e., there is one point of integration). The FE analysis, performed with Abaqus/Explicit, was a dynamic and nonlinear analysis due to the nonlinearity of the material constitutive law (see section 2.6). The total volume of the sample tested experimentally was meshed and used for the calculation. To reproduce the experimental conditions in the FE model, a positive displacement equivalent to the vertical displacement reached during the in situ test was applied to the nodes on the top surface of the sample. On the bottom face, the nodes were totally constrained. The stress was calculated making the sum of the reaction forces at each node of the top face divided by the section of the sample. The strain was calculated dividing the displacement of the top face by the total height of the initial sample. The presence of the glue in the experimental device was not taken into account in the modeling because the zone of interest (i.e., where the fracture occurred) is far from the sample's borders.

Material constitutive laws defined in the FE model

For the aluminum matrix, an elasto-plastic behavior was considered and was identified from the work of Zhou et al. [START_REF] Zhou | Microscale testing of the strut in open cell aluminum foams[END_REF] (Fig. 3). The main mechanical properties were a Young's modulus of 70 GPa, a Poisson's ratio of 0.3, a yield strength of 192.5 MPa and an ultimate tensile strength of 205.3 MPa. The damage behavior of the sample was modelled using the "Metal porous plasticity" model from Abaqus material library. This behaviour is based on Gurson-Tvegaard-Needleman (GTN) model [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: part I -Yield criteria and flow rules for porous ductile media[END_REF][START_REF] Tvergaard | Analysis of the cup-cone fracture in round tensile bar[END_REF]. For ductile materials, the damage mechanism is linked to the nucleation, growth and coalescence of voids. The Von Mises yield condition Φ is thus modified by the introduction of a variable defining the volume fraction of pores. Gurson [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: part I -Yield criteria and flow rules for porous ductile media[END_REF] proposed a yield condition which was later modified by Tvegaard and Needleman [START_REF] Tvergaard | Analysis of the cup-cone fracture in round tensile bar[END_REF] (equation ( 1)):

𝛷 = ( 𝜎 𝑒𝑞 𝜎 𝑦 ) 2 + 2 𝑞 1 𝑓𝑐𝑜𝑠ℎ (- 3 2 𝑞 2 2 𝑝 𝜎 𝑦 ) -1 -𝑞 3 𝑓 2 = 0 ( 1
)
where σeq is the Von Mises equivalent stress, p is the hydrostatic stress, σy is the yield stress of the fully dense material, f is the volume fraction of pores in the materials and q1, q2 and q3 are fitting parameters. f is generally considered as lower than 0.1 in Abaqus [START_REF] Simulia | Abaqus Analysis User's Manual[END_REF]. The values generally reported in the literature for metallic alloys for q1, q2 and q3 are 1.5, 1 and 2.25 respectively (q3 = q1 2 ) [START_REF] Tvergaard | Localization in ductile materials containing spherical voids[END_REF].

The model implemented in Abaqus assumes that nucleation rate increases with plastic strain rate following equation (2) proposed by Chu et al. [START_REF] Chu | Void nucleation effects in biaxially stretched sheets[END_REF]:

𝑓 ̇= 𝐴𝜀̇p l eq ( 2
)
where ḟ is the nucleation rate, 𝜀̇p l eq is the plastic strain rate, and A is a proportionality coefficient which follows the normal distribution given by the equation ( 3):

𝐴 = 𝑓 𝑁 √2𝜋 𝑠 𝑁 exp [- 1 2 ( 𝜀 𝑝𝑙 -𝑠 𝑁 𝑠 𝑁 )] (3)
where εN is the mean value of the normal distribution, sN is the standard deviation of the normal distribution, and fN is the volume fraction of nucleated voids. The following values are generally reported in the literature for metallic alloys: εN = 0.3, sN = 0.1 and fN = 0.04 [START_REF] Simulia | Abaqus Analysis User's Manual[END_REF][START_REF] Li | Ductile fracture: experiments and computations[END_REF].

No data were found in the literature to describe the mechanical behavior of the elements containing inclusions. Nanoindentation was therefore performed in several inclusions to estimate the Young's modulus of the inclusions, using a G200 apparatus (Agilent Technologies, USA) equipped with a Berkovich tip. A series of 10 indents was made with a speed of 10 nm.s -1 , a dwell time of 10 s and a maximum penetration depth of 300 nm. The Young's modulus was determined thanks to the Oliver-Phar method [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF]. Before the measurements, the precise location of the tip was calibrated by indenting a fused silica reference sample. This depth was chosen to characterize only the inclusions (having a size between 20 and 120 μm). A Young's modulus of 160 GPa was determined and used in the FE model. An elasto-plastic constitutive law obtained from the one of the aluminum phase (Fig. 3) was assumed. The values of stresses were arbitrarly multiplied by 1.5 and the values of strains were divided by 1.5 to obtain a slightly stronger and less ductile behavior than the one of aluminum. For the damage behavior, the same values of the criterion were used for the aluminum phase and the inclusions.

In this simulation, two materials models were defined: one for the aluminium matrix and one for the inclusions. The interfaces between the two phases were assumed to be perfect. The sample is characterized by a geometric density of 0.090 ± 0.004. Fig. 5 presents the variation of relative density along the three directions obtained by 3D image analysis. The relative density in each direction is 0.07 ± 0.02, which is a bit lower than the measured apparent density. The difference can be related to the modifications of the image caused by the resolution of the images. During thresholding, some voxels at the interface between the phases can be assigned to the air whereas they belong to the solid phase or inversely. An error on few voxels leads to a larger error on the measurement of the volume fraction of a phase, especially for the thinnest struts [START_REF] Maire | X-ray tomography and threedimensional image analysis of epoxy-glass syntactic foams[END_REF]. The variations of relative density are similar along the three directions, highlighting a random distribution of pores in the sample.

Results

Architecture of the sample

Characterization of the solid phase

Local tomography images with a voxel size of 3 µm enable to reveal the presence of Fe-rich rounded inclusions in the matrix as found in the literature (see Fig. 7a). The strong contrast between the inclusions and the aluminum matrix indicates that the inclusions contain at least one element heavier than aluminum. Fig. 7b presents a slice of a higher resolution 3D image (voxel size of 0.4 µm) obtained on a single strut. Elongated particles are also present in the sample in the same quantity as the rounded ones. These latter ones have been already observed along the grain boundaries in ERG aluminum foams [START_REF] Amsterdam | Fracture and microstructure of open cell aluminum foam[END_REF][START_REF] Zhou | Microscale testing of the strut in open cell aluminum foams[END_REF]. The inclusion size distribution (Fig. 8) determined with the highest resolution images of single struts (Fig. 7b) is between 10 and 160 μm. This distribution is explained by the different shapes of the inclusions. Observation of the tomographic slices (Fig. 7) proves that the rounded inclusions have a size of around tens of microns and the elongated ones have a length of hundreds of microns and a thickness of less than 10 μm. The volume fraction of inclusions in a single strut is 0.06 ± 0.02 %. The value was calculated from four high-resolution 3D images of single struts. The standard deviation is high because the number, shape and size of inclusions are variable from one strut to another.

In situ tensile test

The sample was tested in uniaxial tension along the direction 3, parallel to the cell elongation.

The mechanical properties of the sample as determined from the stress/strain curves are the following ones: Young's modulus of 135 MPa, and ultimate tensile strength of 1.54 MPa. The experimental values of Young's modulus and tensile strength are in the range of those obtained in the literature. The experimental tensile stress/strain curve of the sample is given in Fig. 9: each point of the curve represents a strain at which the test was stopped to perform a scan. Error bars represent the uncertainties on the engineering stress calculated from the uncertainties on the load measurement. It is typical of a porous metallic sample with an elastic part followed by a plastic domain. Once the ultimate tensile strength is reached, the stress gradually decreases, due to the progressive rupture of the struts which reduces the effective number of struts carrying the load. The images show the progressive alignment of the struts in the tensile direction (direction 3, see the black arrows in Fig. 10) and the progressive fracture of struts with increasing applied strain. The comparison of the images of the deformed sample and of the initial sample at high resolution enables to check the presence of inclusion in the broken struts. The broken struts were found by manual inspection of the 3D images of the deformed sample. It was found that 20 out of the 25 broken struts contained an inclusion located in the fracture path. It is not possible to treat in detail the 25 broken struts. Therefore, it was decided to focus on four broken struts, which are pointed out by red rectangles in Fig. 10. These four broken struts are presented in Fig. 11. Fig. 11a and11b show one of the first struts to break (Fig. 11a) in which no inclusion can be detected (Fig. 11b). In this case, the favourable orientation of the strut is responsible for its early fracture. The three struts shown in Fig. 11c (one strut) and 11e (two struts) also broke at the beginning of the tests. However, they were all inclined relatively to the loading direction. These fractures are therefore difficult to link with architectural features. In the three cases, the corresponding high-resolution 3D images (Fig. 11d and 11f) show the presence of inclusions at the location of the fracture of the struts. In Fig. 11c and 11d, the strut broke at one of its extremity and not in its thinnest section. At the exact fracture location, an inclusion is present along the strut thickness (see the white circles in the images). In Fig. 11e and 11f, a big inclusion runs along the upper part of the strut on the left side and is present in the whole strut thickness. The strut rupture occurred where the inclusion was slightly less thick (see the white circles in the images). The strut on the right side broke at the location of a small inclusion which is not centred relatively to the strut thickness (see the white dotted circles in the images). In the three latter cases, these inclusions then appear to be clearly responsible for the fracture of the struts.

(

(e) (f) Fig. 11: (a,c,e) Magnified views of different broken struts localized by red rectangles in Fig. 10b and10c (voxel size of the images: 20 µm): these struts fractured at an applied strain of 4.4% (a, e) and 5. 7 % (b),(b,d,f) same struts from the image of the initial sample (voxel size of 20 µm, obtained thanks to the image processing from local tomography described in section 2.4) showing the inclusions in red: (b) no inclusion is present in the broken strut and (d, f) inclusions are present in the struts, at the fracture location. In Fig. 11c, 11d, 11e and 11f, circles represent the zones where the struts broke.

FE modeling

Fig. 12 reports the experimental and the two numerical stress/strain curves considering or not the presence of the inclusions. The calculated stress/strain curve with aluminum only reproduces well the shape of the experimental curve. This means that the elasto-plastic behavior and the Gurson-Needleman-Tvegaard model seem to be adapted to the Duocel foam behavior.

The calculated Young's modulus (266 MPa) is overestimated in comparison with the experimental one (134 MPa). This overestimation has already been observed with FE modeling of cellular materials [START_REF] Caty | Modeling the properties of closed-cell cellular materials from tomography images using finite shell elements[END_REF][START_REF] Burteau | Impact of material processing and deformation on cell morphology and mechanical behavior of polyurethane and nickel foams[END_REF][START_REF] Marcadon | Mechanical behavior of hollow-tube stackings: Experimental characterization and modeling of the role of their constitutive material behavior[END_REF] and will be discussed in the next section. An experimental point around a strain of 1% would have been interesting as a supplementary information to compare the experimental and calculated curves. The calculated curve determined from the FE model with the inclusions has the same general shape as for the calculation with pure aluminum alloy but it is shifted to higher stresses. The Young's modulus of this curve (305 MPa) is slightly higher than the one obtained with the previous calculation. 11a, the FE calculation without inclusions (Fig. 13a) confirms that this strut broke mainly due to its alignment in the loading direction. The calculation taking into account the inclusions (Fig. 13b) gives similar stress fields. Maximum stresses appear where the strut section is reduced. The strut presented in Fig. 11c is not favourably aligned in the tensile direction and no local reduction in the section is detected. The FE calculation without consideration of the inclusions (white circle in Fig. 13c) does not show a stress concentration at the location of the fracture of the strut. Thus, it confirms that the architectural features of the foam are not sufficient to explain the rupture of this strut. The stress distribution determined using a specific constitutive law for the inclusions (white circle in Fig. 13d) shows a stress concentration around the inclusion at the location of the fracture of the strut. Fig. 13e and13f show the case of the two struts shown in Fig. 11e. In these two cases, the calculations with inclusions result in important modifications of the stress fields (see white circles). High stress concentrations are found in the vicinity of the inclusions, which can explain the struts' fracture. without and (f) with inclusions. In Fig. 13c, 13d, 13e and 13f, circles are placed at the same location as in Fig. 11. For all figures, the loading is applied along direction 3.

Discussion

A previous study proved the possibility to observe microstructural features thanks to local tomography in aluminum foams [START_REF] Zhang | Local tomography study of the fracture of an ERG metal foam[END_REF]. This work confims the relation between the location of these features and the fracture of some struts. In addition, tomography images have been processed to build a FE model. For this purpose, different image processing steps were developed to create a low sized image from the high-resolution images. This allows for using this low sized image as an input for FE meshing. However, the image scaling step overestimates the inclusions size and volume fraction. Consequently, the morphology and size of the inclusions are not precisely described by the low sized image. Moreover, the voxel size of 3 μm reached with the local tomography does not enable to distinguish the different inclusions: all of them have the same grey levels.

The calculated Young's moduli are higher than the experimental one (fig. 12). Caty et al. [START_REF] Caty | Modeling the properties of closed-cell cellular materials from tomography images using finite shell elements[END_REF] also noticed this overestimation and pointed out the presence of different defects in the solid phase such as some pores as a possible reason. Otherwise, the choice of the material constitutive law can explain the phenomenon. An elasto-plastic law of a bulk 6101 aluminum alloy was used to model this sample. But, Zhou et al. [START_REF] Zhou | Microscale testing of the strut in open cell aluminum foams[END_REF] performed in situ tensile tests of Duocel foam struts in a SEM. In particular, they obtained a lower Young's modulus than the one of the aluminum bulk alloy. These tests tend to indicate that the mechanical behavior of an aluminum strut is different from that of an aluminum bulk alloy. The authors related it to some differences of microstructure (i.e., chemical composition and type of inclusions) in the aluminum struts and in the bulk alloy. It can also be noted that the small size of the tested struts can explain their results. The measurement of the cross sectional area of a strut is also delicate easy. For these reasons, we preferred using the data of a bulk alloy. Moreover, the measured weight fractions of the alloying elements (and especially of Mg and Si) in the foam are low compared to the nominal composition of the 6xxx alloys [START_REF] Zhou | Mechanisms and mechanics of compressive deformation in open-cell Al foams[END_REF]. It is possible that the elasto-plastic law of a bulk 6101 alloy is not the most adequate behavior for this sample. Performing different calculations with different elasto-plastic constitutive laws for aluminum alloys would be necessary.

The experimental plastic domain is well reproduced by the FE model which does not take into account the inclusions (Fig. 12). It indicates that the choice of the material behaviors (i.e., elasto-plastic law combined with a GTN model) enables to model the tensile test of this sample.

However, an optimization of the parameters (especially for the GTN model) is still needed to understand better their respective influence on the calculations. The calculated macroscopic stress is increased when the inclusions are present in the model. It is due to the constitutive law chosen for this phase. In the absence of data from the literature, we "built" a constitutive stress/strain law from the one of aluminum. For simplicity, we used the same Gurson parameters for the elements containing inclusions. A better choice would have been to promote damage in these elements. Gurson's model being mostly deformation-driven, damage is obviously underpredicted in these elements in our model compared to reality. A deeper study would be necessary to test different damage behaviors for this phase or to characterize better the mechanical properties of the inclusions.

The analysis of the local stress distributions given by the FE calculation brings important information to explain the local struts fracture. The local stress distributions obtained by the two calculations are different. It allows for explaining the fracture of some struts for which no architectural feature was found (Fig. 13b and13d). The analysis of the 3D images and the stress distributions in the struts evidence the importance of taking into account the two characteristic scales of the cellular samples (i.e., mesoscale and microscale).

It is important to note that the approach presented here is more qualitative than quantitative. To perform calculations with a reasonable time, the image was meshed with tetrahedra having an average size of 76 μm. This size is slightly high compared to the inclusions size (between 10 and 160 μm, Fig 8). As already mentioned, the image processing steps artificially increase the inclusions size and volume fraction in the final image. As explained in section 2.6, when associating an element to a grey level of the image, each element is represented by its barycentre. For these reasons, some elements may be associated to the inclusions although they physically correspond to zones containing both inclusions and aluminum matrix. 0.3% of the elements were associated to the inclusions whereas the volume fraction of inclusions in the struts was evaluated to 0.06%. This large difference could be partly corrected by a refinement of the mesh, but it would be detrimental to the duration of the calculation. Scanning the initial samples at higher resolution would also bring a finer characterization of the inclusions shape and size. The high-resolution scans of the struts (Fig. 7b) show that a scan with a voxel size smaller than 1 μm can provide a precise characterization of the inclusions size and morphology.

In the model presented here, the tetrahedra are separated in two groups, i.e., the ones associated to the aluminum phase and the ones associated to the inclusions. This means that all the inclusions are modelled by a unique material behavior. However, a careful examination of the high-resolution 3D images shows that slight variations of grey levels appear in some inclusions (Fig. 14a), which are related to a local difference of volume fraction or of composition of inclusions. This can be preserved in the thresholded 3D image by a series of different thresholding. This resulted that thresholded 3D image would contain a gradient of grey levels to describe the inclusions (Fig. 14b). By assigning a different material constitutive law to each grey level, it would be possible to obtain a finer description of the sample behavior. Higher resolution images and characterization of the local properties of the inclusions could also result in a classification of the inclusions according to their chemical composition. Different types of Fe-rich inclusions have already been identified in the AlMgSi alloys [START_REF] Amsterdam | Fracture and microstructure of open cell aluminum foam[END_REF][START_REF] Wang | Solidification and precipitation behavior of Al-Si-Mg casting alloys[END_REF][START_REF] Tanihata | Effect of casting and homogenizing treatment conditions on the formation of Al-Fe-Si intermetallic compounds in 6063 Al-Mg-Si alloys[END_REF], that could be taken into account in the model.

Our approach needs to be validated with experimental data at local scale to better characterize the local constitutive laws of the different phases considered. Some authors have already performed local mechanical tests on aluminum foams [START_REF] Amsterdam | Fracture and microstructure of open cell aluminum foam[END_REF][START_REF] Zhou | Microscale testing of the strut in open cell aluminum foams[END_REF] or on FIB-machined samples in fibers of 10 µm diameter [START_REF] Mueller | Fracture toughness testing of nanocrystalline alumina and fused quartz using chevron-notched microbeams[END_REF], smaller than the inclusions found in the present study. To go further, it could be possible to perform in situ tensile tests of a single strut using our approach.

The use of Digital Image Correlation could also improve our approach to evaluate the strains in the struts and to compare it with simulated ones [START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF][START_REF] Madi | Computation of full-field displacements in a scaffold implant using digital volume correlation and finite element analysis[END_REF].

Conclusion

This paper reports an original approach based on X-ray tomography to characterize and model the mechanical behavior of an aluminum foam taking into account its microstructural features at two length scales. The architecture and the microstructure of the solid phase were characterized thanks to scanning of the samples at two resolutions. The higher resolution 3D images (voxel size of 3 µm) were obtained from local tomography and reveal the presence of intermetallic inclusions typical of 6xxx aluminum alloys. Then the progressive deformation of the sample was followed thanks to an in situ tensile test. The comparison between the 3D images of the deformed sample and the high-resolution 3D images of the initial sample showed the presence of inclusions along the fracture path for most broken struts. For this reason, FE models including the presence of these inclusions were built. The model was based on the meshing of a low sized 3D image containing the inclusions and processed from the high-resolution images.

The model used specific constitutive laws for the aluminum matrix and the inclusions. A classical elasto-plastic law combined with a GTN model was chosen to model the damageable behavior of the aluminum matrix. The Young's modulus of the inclusion was determined using nano-indentation. Inclusions were considered as stiffer and less ductile than the matrix. The presence of the inclusions in the model resulted in local stress concentration explaining the early fracture of some struts where no architectural specific features were noticed. It showed the importance of the microstructural features to explain the macroscopic properties of the cellular samples. The approach we propose in this paper could complete previous works on cellular samples where the intermetallic inclusions were not considered. However, it is important to note that our results are mainly qualitative at this stage. A better fitting between the experimental and simulated stress/strain curves could be obtained. Different improvements were proposed in the paper to improve the quantification of properties (use of higher resolution 3D images, consider gradients in properties).

Fig. 1 :

 1 Fig. 1: Loading device used to load the sample (the arrows indicate the different steps to mount

  Fig. 2: (a) Slice of a local tomography 3D image of a part of the foam (voxel size: 3 µm): pores in dark grey, aluminum in medium grey and inclusions in light grey, (b) the same slice after thresholding into three different phases (voxel size: 3 µm): pores in black (grey level of 0), aluminum in white (grey level of 255) and inclusions in grey (grey level of 128), (c) the slice after scaling (voxel size: 20 µm): pores in black, aluminum in white and inclusions with a gradient of grey and (d) the down-scaled slice after the second thresholding (voxel size: 20 µm)

Fig. 3 :

 3 Fig. 3: Engineering stress/engineering strain curve of the 6101 aluminum alloy used to model

Fig. 4 Fig. 4 :

 44 Fig.4presents a 3D image of a typical cell of a Duocel foam showing its polyhedral shape. The

Fig. 5 :Fig. 6 :

 56 Fig. 5: Relative density in slice profiles along the three main directions of the sample obtained

Fig. 7 :

 7 (a) Slice of a high-resolution image of a strut with local tomography showing intermetallic inclusions (voxel size of 3 µm) and (b) higher resolution image (voxel size of 0.4 µm) showing an elongated inclusion

Fig. 8 :

 8 Fig. 8: Inclusion size distribution measured by the granulometry technique on the 3D image of

Fig. 9 :

 9 Fig. 9: Tensile engineering stress/engineering strain curve of the sample (error bars represent

Fig. 10 :

 10 3D image of the sample during the in situ test (voxel size of 20 µm): (a) at initial state, with no applied strain, (b) at an applied strain of 4.4 % and (c) at an applied strain of 5.7 %. The black arrows point out one strut which undergoes a progressive alignment in the tensile direction.

Fig. 12 :

 12 Fig. 12: Experimental and calculated stress/strain curves: the curve "FE -Only aluminum

Fig. 13 :

 13 (a) and (b) Normal stress σ33 contours predicted for the strut shown in Fig. 11a by the FE models (a) without and (b) with inclusions, (c) and (d) normal stress σ33 contours predicted for the strut shown in Fig. 11c by the FE models (c) without and (d) with inclusions, (e) and (f) normal stress σ33 contours predicted for the struts shown in Fig. 11e by the FE models (e)

Fig. 14 :

 14 Illustration of a possible improvement of the FE models including a gradient in inclusions properties: (a) slice of a high-resolution image (voxel size of 3 μm) showing differences in grey levels in the inclusions and (b) slice of a thresholded high-resolution image where the difference of grey levels in the initial image can be taken into account by a change of grey levels thanks to different thresholding steps.

Acknowledgements

The authors acknowledge José Ferreira (MATEIS laboratory, INSA Lyon) who developed the softwares used for the in situ mechanical test and ERG Aerospace ® which kindly provided the samples.

Funding sources

This work was supported by the French Ministry of Education and Research (grant for C. Petit PhD, 2012-2015).