Walid Fekih Ahmed

Mohamed Ben

Hadj Sghaier

Samir Allaoui
email: samir.allaoui@univ-reims.fr

Investigation of the effect of three-point-bending testing parameters on the behaviour of 3D printed sand

Keywords: 3D printed sand, furan binder, three-point bending test, digital image correlation

3D sand mould printing allows rapid and direct fabrication of complex-shaped moulds and cores that helps to overcome the limitations of conventional casting. The manufactured moulds and cores, having complex geometries, are subjected to severe thermomechanical loads during casting and thus must have sufficient rigidity to meet the technical requirements. Therefore, a mastery of their behaviour and intrinsic sand properties becomes essential for a better dimensioning of moulds and cores but also the simulation of casting processes. The bending is the most critical load type, for sand cores under casting conditions, and therefore needs to be characterized. This study proposes to investigate the effect of testing parameters, that are spanto-thickness and strain rate, on the three-point bending behaviour of printed sand. The results showed that the elastic modulus increases with the rise span-to-thickness parameter and is not affected by the strain rate. The bending strength, for its part, is only affected by strain rate. The tests results and DIC measurement made it possible to give recommendation for the optimal testing parameters.

Introduction

The 3D printing of moulds and cores using binder jet technology make possible many improvements in casting process. Compared to conventional casting it provides the capability to fabricate any complex moulds and cores geometries, which are impossible to obtain without 3D printing, using layer-by-layer deposition without the use of patterns and core-box [START_REF] Bassoli | 3D printing technique applied to rapid casting[END_REF][START_REF] Gibson | Binder Jetting, Additive Manufacturing Technologies[END_REF]. Indeed, 3D sand mould printing helps to overcome the limitations of conventional casting such as limitations on geometry, providing undercuts and higher lead times due to the pattern making. This technology allows rapid and direct fabrication of complex-shaped moulds and cores of custom-made and production of unit pieces.

The design freedom offered by additive manufacturing (AM) combined with the good dimensional accuracies of the obtained parts allows to increase the capability of casting process by: The decrease of the required metal by using of hemispherical riser instead of cylindrical riser [START_REF] Sama | Non-conventional mold design for metal casting using 3D sand-printing[END_REF]; improving the casted part quality and dimensional accuracy following the fast and uniform cooling of the casting allowed by the use of shell-truss sand mould [START_REF] Shangguan | 3D-printed shell-truss sand mold for aluminum castings[END_REF][START_REF] Shangguan | The Design of 3D-Printed Lattice-Reinforced Thickness-Varying Shell Molds for Castings[END_REF]; improving the melt flow control that leads to reducing the presence of inclusion by using optimized parabolic and conical-helix sprue [START_REF] Sama | Novel sprue designs in metal casting via 3D sand-printing[END_REF]; increasing the freedom in design of riser and gating system [START_REF] Walker | 3D printed smart molds for sand casting[END_REF], …. Additive manufacturing of sand mold has revolutionized the molding industry by allowing weight reductions of complex parts of up to 33% [START_REF] Upadhyaya | 3D printing for rapid sand casting-A review[END_REF], without losing the technical requirements of the component, while reducing inclusions [START_REF] Low | Perspective on 3D printing of separation membranes and comparison to related unconventional fabrication techniques[END_REF] and offering energy savings [START_REF] Sachs | Three-Dimensional Printing: Rapid Tooling and Prototypes Directly from a CAD Model[END_REF].

However, to achieve these performances, the mechanical behavior of the printed sand must have sufficient rigidity to meet the technical requirements and not too high to avoid generating excessive residual stress that may give casting defects [START_REF] Motoyama | A verification of the thermal stress analysis, including the furan sand mold, used to predict the thermal stress in castings[END_REF][START_REF] Marumoto | Dynamic measurements of the load on gray cast iron castings and contraction of castings during cooling in furan sand molds[END_REF]. Several works in the literature have therefore focused on the characterization of the mechanical behavior of printed sands, through classical and non-classical mechanical tests, to identify the intrinsic characteristics of the material and to dimension the molds and cores for foundry applications as accurately as possible. Due to the difficulty in clamping the 3D printed sand specimen on the tensile machine, which is due to the porous nature and brittle behavior, the most used tests to identify the mechanical parameters are bending test using single prismatic specimen [START_REF] Bernhard | Foundry sand core property assessment by 3-point bending test evaluation[END_REF] [START_REF] Sivarupan | A review on the progress and challenges of binder jet 3D printing of sand moulds for advanced casting[END_REF], tensile test using dog bone specimens [START_REF] Shangguan | 3D-printed shell-truss sand mold for aluminum castings[END_REF], compression using unconfined compressive strength specimen [START_REF] Khandelwal | Effect of binder composition on the shrinkage of chemically bonded sand cores[END_REF][START_REF] Khandelwal | Effect of molding parameters on chemically bonded sand mold properties[END_REF] and indirect tensile strength test using Brazilian test [START_REF] Perras | Compressive and Tensile Behavior of 3D-Printed and Natural[END_REF][START_REF] Voglera | A comparison of tensile failure in 3D-printed and natural sandstone[END_REF].

Three-point bending is the most widely used method for testing the quality of cores and molds prepared with printed sand in the foundry industry. This is due to the technical simplicity of conducting these tests coupled with the need for mechanical performance of the cores in terms of sufficient strength to allow handling and manipulation operations and high bending strength. Indeed, bending is the most critical type of loading, for sand cores under casting conditions, which can be imposed by clamping forces, thermal load, flow drag and buoyancy loads.

The different results of the literature showed that the final properties and quality of the 3D sand printed mould depend on the printing parameters. A high recoater speed results in high porosity, which when followed by curing at lower temperatures improve polymerization of resin and increase the evaporation of solvent and hence the sand particles get tightly packed [START_REF] Mitra | Experimental and numerical characterization of functional properties of sand molds produced by additive manufacturing (3D printing by jet binding) in a fast foundry[END_REF][START_REF] Sivarupan | A review on the progress and challenges of binder jet 3D printing of sand moulds for advanced casting[END_REF]. Sand compaction, which affect the strength and permeability of the mould and thus the casting quality, is also affected by the orientation and the position of the specimens in the job box [START_REF] Coniglio | Investigation of process parameter effect on anisotropic properties of 3D printed sand molds[END_REF]. This is due to the layering which strongly affect the behaviour of 3D-printed sands [START_REF] Perras | Compressive and Tensile Behavior of 3D-Printed and Natural[END_REF][START_REF] Voglera | A comparison of tensile failure in 3D-printed and natural sandstone[END_REF]. This anisotropic behaviour of printed specimens is due to more oriented porosity along direction of printhead movement which impact the trajectory of the droplet [START_REF] Perras | Compressive and Tensile Behavior of 3D-Printed and Natural[END_REF].

Therefore, it is necessary to position the moulds judiciously before printing to take this effect into account this anisotropy.

Another important parameter that has been studied is the layer thickness. Indeed, increasing the layer thickness under the same binder saturation conditions leads to an increase in flexural strength due to improved powder pulverization and binder penetration in the vertical and lateral directions [START_REF] Vaezi | Effects of layer thickness and binder saturation level parameters on 3D printing process[END_REF]. Finally, concerning the effect of the constituents properties of the printed sand, it has been shown in the literature that increasing the grain fineness index leads to a decrease in mechanical properties while increasing the binder ratio and curing time improves the mechanical properties [START_REF] Khandelwal | Effect of binder composition on the shrinkage of chemically bonded sand cores[END_REF][START_REF] Khandelwal | Effect of molding parameters on chemically bonded sand mold properties[END_REF].

In addition to the effect of processes, test conditions and environment also play a nonnegligible role on the measured behaviour. Indeed, the final properties and quality of the 3D sand printed mould depend on the time and storage conditions prior to testing due to the interaction of the material with the environment and specially with the humidity uptake. [START_REF] Bernhard | Foundry sand core property assessment by 3-point bending test evaluation[END_REF].

The analysis of the results of the different studies shows a difficulty in reproducing the mechanical properties which can be attributed to the porous nature of the printed specimens but also to the testing conditions [START_REF] Dana | Mechanical characterisation of anisotropic silica sand/furan resin compound induced by binder jet 3D additive manufacturing technology[END_REF][START_REF] Del Giudice | Mechanical properties of 3D printed material with binder jet technology and potential applications of additive manufacturing in seismic testing of structures[END_REF][START_REF] Mitra | On the rapid manufacturing process of functional 3D printed sand molds[END_REF] or to test conditions. Some strategies were proposed to avoid dispersed mechanical properties obtained by three-point bending strength like the use of test with anti-buckling device [START_REF] Dana | Mechanical characterisation of anisotropic silica sand/furan resin compound induced by binder jet 3D additive manufacturing technology[END_REF].

Despite the numerous studies published in this field, the behavior of printed molds remains not fully mastered because of the several parameters influencing the behavior of printed sand.

The studies of the literature highlighted the important effect of the processes and the storage conditions on the behavior of the printed sand but no studies, to our knowledge, addressing the effect of test conditions. Moreover, the studies use different experimental protocols to carry out the three-point bending tests, which show a significant dispersion of results making it difficult to compare and cross-reference their results. However, the dimensioning of foundry moulds and cores requires the behaviour laws and the real intrinsic elastic parameters of the material allowing to have robust numerical simulations.

It is within this framework that this study aims to contribute to the mastery of the printed sands behavior through the evaluation of the effect of test conditions on the stiffness measured by three-point bending test. Digital Image Correlation (DIC) technique, that is a contactless method, was used to measure the strain field of sand specimens during tests to analyse the obtained behaviours according to the different conditions.

Experimental procedure a) Material and method

The three-point bending test specimens were printed, layer by layer, by the binder jetting 3D printer ExOne S-Max Machine, which is equipped with 1800 × 1000 × 700 mm 3 job box. The re-coater spread a 280 μm layer of sand pre-mixed with the acid activator on the job box and the inkjet printer head apply a furfuryl-alcohol based binder to the appropriate zone of the layer. A slice of specimen is formed by particles sand bonding within the zone where the binder was applied. The other areas with unbound sand, which will be removed at the end of printing, are used as support for next slices during printing. The job box moves downwards, and a same process continues until the last slice is printed. The recoating speed used for printing the specimens is 150 mm/s with a sand that has an average grain size of Silica particles is 140 μm and its standard deviation is 25 μm. The drop frequency of the inkjet printer head is 10 kHz with print resolution of 0.11 mm and print head voltage, witch control drop volume, is 79.8 V. All the tested specimens were printed in the "Z" direction to overcome the effect of the printing direction on the characterized mechanical behavior which has already been characterized in the literature (Fig. 1). The specimens were printed with several lengths but with a 22 mm square cross-section. This cross-section has been chosen because being the most used in the literature for the bending specimens of printed sand as well as the sand obtained using core box [START_REF] Bernhard | Foundry sand core property assessment by 3-point bending test evaluation[END_REF][START_REF] Mitra | On the rapid manufacturing process of functional 3D printed sand molds[END_REF] but also the most widely applied for the quality control of sand cores as described by American Foundrymen's Society [START_REF] Afs | Molding Methods and Materials[END_REF] and similarly by the German standard VDG [START_REF] Vdg-Merkblatt | Biegefestigkeit von warmhärtenden kunstharzgebundenen feuchten Formstoffen[END_REF]. The specimens have an apparent density of 1315 ±7 kg/m 3 . Two testing parameters were investigated in this study to highlight their effect on the mechanical behaviour of the printed sand: testing speed and the span-to-thickness (Length / height) ration. Three-point bending tests were performed at two displacement rate levels (0.05 and 5 mm/min) to investigate the effect of strain-rate on the 3D sand printed behaviour.

To assess the effect of the span-to-thickness ration (L/h), several configurations were tested.

The three-point bending test can give rise to a combined effect of bending and shear behaviors. This combination is governed by the value of span-to-thickness ratio. When it is important one tends to have a pure bending, whereas when it is weak the shear behavior is predominant.

Between the two, one obtains a behavior combining pure bending and shearing. Bending tests with different span-to-thickness ratio (L/h) were therefore carried out to define a ratio minimizing the shearing effect. Seven distances between the two supporting pins (L from 60 mm to 200 mm) allowed by the used bending device (table 1). This range of variation integrates the most used distance in the literature which is 150 mm [START_REF] Bernhard | Foundry sand core property assessment by 3-point bending test evaluation[END_REF][START_REF] Coniglio | Investigation of process parameter effect on anisotropic properties of 3D printed sand molds[END_REF][START_REF] Mitra | On the rapid manufacturing process of functional 3D printed sand molds[END_REF]. The specimens section remained fixed (22x22 mm); this gives a span-to-thickness ration going from 2.73 to 9.09 (Table 1). The different batches of specimens were printed with length equal to the distance between the two supporting pins (L) to which it was added 10 mm overhang on each side (specimen length = L +20 mm). Therefore, the specimens are not of the same length.

This avoids the parts of the specimen, which are located beyond the supports, generating forces opposite to the applied load and therefore disturbing the result, given the forces reached and the density of the material In a general case of three-points bending tests, the deflection is given by the Timoshenko beam theory below:

∆ = 𝑃 .𝐿 3 48 .𝑏 .𝐷 + 𝑃 . 𝐿 4 . 𝐴 .𝐺 (1)
Where  is the deflexion, P the applied force, b the width of the specimen, L the distance between the two supporting pins, D the bending stiffness, G the shear modulus and A the section of the specimen.

The resulting behavior is a coupling of pure bending and shear behavior. To measure the bending stiffness, it is necessary to know span-to-thickness threshold value beyond which there is a pure bending configuration, without a coupling with shear, and for which the theory of Euler-Bernoulli beams can be applied. This is one of the objectives of this study which aims to determine this threshold for the case of printed sand. For the sake of simplicity, the elastic modulus (E), will be evaluated by the theory of Bernoulli beams in this study because the use of coupling requires a characterization of the shear beforehand whose testing is complex. The bending strength (𝜎 𝑀𝑎𝑥 ) and the fracture energy (𝑊 𝑡𝑜𝑡 ) are calculated according to continuum mechanics.

In addition, tests were also performed using Digital Image Correlation (DIC) measurements to obtain the strain fields and compare them between different configurations of span-to-thickness ratio (L/h). The configurations used for these measurements are those with distance between the two supporting pins of 100, 150 and 200 mm. The principle of DIC is to assess the displacements fields, and thus strains fields, over the surface of a deforming specimen by comparing two images acquired at different steps of deformation. The first image, acquired at the beginning of the test, is used as a reference image and the second one, acquired after some increment of deformation is the deformed state of the specimen. Prior to apply the measuring process, a region of interest is defined on the reference image and divided into small subsets.

The software performs an image correlation by identifying the most similar subset in the deformed image and then calculate the local strain tensor from the spatial distribution of the displacement field for each image.

The stereovision system used is LAVISION Strain Master composed of two CCD (charge coupled device) cameras, that has 5-million-pixel resolution, connected to a computer which allows the acquisition and processing of images via software. 30 frames per second were saved for each test. Prior the tests, a random pattern is spread on the sample surface with a painting to enhance the image contrast.

All the specimens have been stored after printing and tested at room temperature with a 30-50% relative humidity. The mechanical behavior of the sand varies during the first hours of printing and stabilizes after a few hours [START_REF] Bernhard | Foundry sand core property assessment by 3-point bending test evaluation[END_REF]. To limit this variability and their effect on the results of this study, a sufficient time to stabilize the behavior and overcome the environment effect has been set at 24 hours after production. This also represent the typical time between printing the moulds and core and their end usage in foundry practice.

At least three specimens were used for each testing configuration to verify repeatability.

b) Quality control of the specimens

All specimens were printed in the same direction (Direction "Z") to avoid the anisotropy of the printed sand due to non-uniform resolution of printing. The cross-section dimensions of each specimen were controlled after cleaning to assess accurately the mechanical parameters such as the flexural strength and modulus. The measured dimension tolerances of width and thickness are ± 0,15 mm and 0,11 mm respectively.

The loss on ignition was used to measure the amount of binder in the printed specimens. To do, A quantity of printed sand with a mass of about 14 g was extracted from 3D printed specimen, then put in ceramic crucible and finally heated to calcinate the binder. The procedure followed and the associated measures are described below. The first step of loss on ignition procedure is weighting empty crucible to get it is mass M1. Then, about of 14 g of sand samples introduced into the crucible and are weighed again to get total exact mass M2 of the crucible with sand.

The crucible is put in a furnace at 900°C. After 2 hours, the crucible is removed from the furnace and placed in a desiccator for cooling. After cooling, the total mass of the burned-out sand and the crucible is measured again to get mass M3. All of weighing were carried out by a precise balance with an accuracy of ±0.001 g. The loss on ignition was determined by the following equation:

𝐿𝑜𝑠𝑠 𝑜𝑛 𝑖𝑔𝑛𝑖𝑡𝑖𝑜𝑛 (%) = 100. 𝑀 2 -𝑀 3 𝑀 2 -𝑀 1 (2)
Three measurements for each set of specimens with the same length were performed, and an average of the results has been done. The of polymeric binder measured is equal to 2.033±0.025 % of the mass of the printed sand.

Results and discussion

An example of the force displacement curve is given on Fig. 2 for the case of a span length of 150 mm. A change in the slope shown in the figure was observed on all the curves with three distinct phases of the behaviour (Fig. 2). The beginning of the bending test (phase 1) is characterized by the crushing and sliding of sand particles located on the specimen surface at the contact zones with the loading and supporting pins. This leads to generates low forces and therefore very low strains at the specimen/device contact areas. This can be observed on the displacement measured by DIC in the loading direction where the most important values, but still weak (less than 0.01mm), remain localized at the contact area with the loading pin (Fig. 3a). In this phase, variations and instability of the load can occur (Fig. 2) due to the instability of the phenomena and the sliding which occur. The tested specimen does not undergo bending in this phase considering the very small displacement, which is of the order of the precision of the DIC measurements, measured on the rest of the specimen as well as the lower part of the specimen.

Once local stability between sample and pins is achieved, phase 2 of the behaviour begins and the load increases non-linearly (Fig. 2). In this phase, two concurrent phenomena occur simultaneously: bending of the specimen and a local crushing of the material due to Hertz pressure. Indeed, with the increase in the force, we observe a beginning of the specimen bending in this phase characterized by an increase in the deflection. In the state of loading in phase 2 illustrated by Fig. 3-b, a deflection of the specimen of the order of 0.024 mm is measured. Part of the machine displacement was not transmitted to the specimen as a whole and was absorbed by the local compression, due to the hertz pressure, at the contact area with the loading pin where the displacement is 50% greater (displacement of 0.038 mm). Finally, once this phase is over, phase 3 is entered, characterized by a linear behaviour of the material (Fig. 2) and where its deflection increases significantly but always with the presence of compression on the upper part of the specimen in contact with the device pins (Fig. 3-c). To overcome or limit the local crushing due to the Hertz pressure, it would be necessary to favour the use of loading and supporting pins having a large diameter and not use small value as in the literature [START_REF] Bernhard | Foundry sand core property assessment by 3-point bending test evaluation[END_REF][START_REF] Mitra | Experimental and numerical characterization of functional properties of sand molds produced by additive manufacturing (3D printing by jet binding) in a fast foundry[END_REF].

The behaviour exhibits a brittle fracture that occurs spontaneously at the middle of specimen when the maximum force is reached (Fig. 4). The fracture initiation is localised at the lower face of specimen where tensile stress value is maximum, which generates decohesion of sand grains. The comparison of the force-deflection curves of three points bending tests performed for different span lengths with a fixed testing speed of 5 mm/min are shown in Fig. 5. For better readability, only one curve per configuration is plotted in the figure . A change in the slope with the presence of the three phases described before is observed for all tests of each configuration. However, while increasing the span length, the extent of phase 1 and 2 decreases and becomes weaker for the configuration like with the spans 175 mm and 200 mm. The curves obtained for these two configurations are also smoother than the others and contain less irregularity in the evolution trend of the force. All tested configurations exhibit a brittle fracture and a good repeatability of the curves slope illustrated by a maximum dispersion of around 7% of the calculated parameters (Table 2). The configuration with a span length of 75 mm is the one that has the maximum dispersions (7% for the maximal strength and 12% for the fracture energy). The increase of supporting span value leads to a significant decrease in the maximum force level and an increase in the deflection value at failure point (Fig. 5). The average values of the bending strength have ranged from 6 MPa to 7 MPa. We observe a low standard deviation for each value (Table 2), with a deviation less than 7%, in contrast to the results reported by a previous studies done on three-point bending tests [START_REF] Dana | Mechanical characterisation of anisotropic silica sand/furan resin compound induced by binder jet 3D additive manufacturing technology[END_REF] or even the four-bending test [START_REF] Del Giudice | Mechanical properties of 3D printed material with binder jet technology and potential applications of additive manufacturing in seismic testing of structures[END_REF],

where the test conditions are cleaner because they are free of shear, with adapted protocols and setups. The evolution of the bending strength as function of supporting span seems to be not significant (Fig. 6-a), which is not the case for the fracture energy value that was assessed by the integration of the area under the curve. Indeed, the increase of distance between the supporting pins leads to a decrease of fracture energy value of about 24 MJ between the span lengths of 60 mm and 125 mm (Table 2). From the value of 125 mm, corresponding to a spanto-thickness ration of 5.68, the fracture energy reaches a plateau with a value of around 30 MJ and no longer evolves significant (Fig. 6-b).

Concerning the stiffness of the material, it was evaluated by calculating the Young's modulus on phase 3 of the bending curves according to Euler-Bernoulli beam theory instead the Timoshenko beam theory as discussed before. The results show good repeatability with a maximum variation of 10% obtained for the span length of 60 mm (Table 2). The elastic parameter seems to increase continuously, without stabilizing, with the span length evolution.

To confirm this trend, loading-unloading tests were performed with the different configurations to have more reliable measures. The first remark which can be made is that the values of the Young modulus evaluated during the loading-unloading tests are greater and with less dispersion (maximum variation of 1.3%) than those evaluated with the conventional bending tests (Fig. 8). We reach a ratio of 3.2 and 1.3 times between the two configurations respectively for the span lengths of 60 mm and 200 mm. This deviation between the values measured by the two methods decreases with the increase of span length. This difference can be explained by the fact that the evaluation of the elastic moduli during the loading-unloading tests, that are more realistic, makes it possible to overcome the effect of local compression, due to Hertz pressure, at the contact zones between specimen and the pins. Elastic modulus increases with increasing span length. However, for those measured during loading-unloading tests, we reach a plateau at a span length of 175 mm, corresponding to a span-to-thickness ratio of 7.95, where the value stabilizes around 2400 MPa (Fig. 8) that indicates that the pure bending is reached after this. Remember that the evolution of fracture energy reached a plateau starting from a span-to-thickness ratio of 5.68 corresponding to a span length of 125 mm. Consequently, for the characterisation of the bending behaviour of sand using three-point bending tests, it will be appropriate to choose a span-to-thickness ratio greater than 8 to overcome to overcome the coupling with the shear behavior and the parasitic phenomena, related to the specific behaviour of sand or the experimental conditions, which occur during the three-bending test. Indeed, given its composition, sand can be considered as a composite material and therefore a heterogeneous material. Its mechanical behavior is therefore governed by the combination of those of its constituents. Given the modest values of the stresses reached during the tests, the sand grains will tend to be animated by a rigid body movement and not to deform, because of their great rigidity, while the binder which is a polymer will deform. These mechanisms will occur at the local (microscopic) scale of the material.

On a macroscopic scale, these behaviors will result in a disparity of the deformations on a specimen or structure which will depend on the level of the stresses which will be reached locally, the test conditions, the homogeneity of the composition, the printing direction, .. Distance between supports (mm)

Loading-unloading tests

Loading tests these measurements made on the elastic phase of the curves (phase 3), the scale of the deformation iso-values was kept fixed.

As expected, we observe the presence of tensile strain on the upper part of the test specimens and of compression on the lower part delimited by horizontal green lines on the images which is positioned at the middle of the specimens (Fig. 9). The location of the neutral plane which characterised by zero bending stress is function of modular ratio ET/EC (where ET and EC are tensile modulus and compression modulus respectively) [START_REF] Chamis | Analysis of the three-point-bend test for materials with unequal tension and compression properties[END_REF]. The equal distances from neutral plane to top and bottom of specimen indicate hence that the tensile modulus is equal to compression modulus. This symmetry of behaviour can be explained by the porous nature of structure that allows the sliding of sand in compressed section without compaction. For all the tested configurations, the strain distribution is consistent and where the high strains are measured at the center of the specimens where the stresses are higher. The maximum values of the strains, both in compression and in tensile, are measured in the case of weak span length, i.e. at 100 mm, and are in sections located on the central part of the tested specimen. This is consistent because the forces, and therefore the stresses given that the section is constant, are greater for low span lengths tests (Fig. 5).

If we look to the symmetry of the measurements, we observe that the strains located on the same section at identical distances from the central axis of the specimen are not identical. This is illustrated by the values reported on Fig. 9 which were measured on the ends of the central sections, where the pin applies the bending force, represented by the vertical green line on the images. We observed in the case of specimens having with a span length of 100 mm, where the difference is significant, that the tensile strain (0.133%) has an absolute value greater than that of the compression (-0.1%). This can be attributed to the local disparity in material composition and stresses but also to the effect of the onset of fracture initiation since these measurements were made just before failure. The symmetry of the strain's measurements remains correct in both directions with respect to the axes of symmetry of the specimen. The difference between the deformations measured, at the ends of the sections considered in the two directions (Fig. 10), remain small and acceptable.

The evolution trend of the strains between the ends of the sections, both vertical and horizontal, also remains correct. However, these evolutions are not linear according to the vertical sections and are very disturbed and not smooth for the horizontal sections. This effect is attributed to the local variability local of the printed sand leading to a variability of local stresses and specifically that of shear. That is confirmed by the measurements of shear strains (xy) representative of the state of the shearing stresses, where one notes this variability which is more pronounced in the case of small span length where the level of stresses is more important like for normal stresses (Fig. 11). The results of shear strain distribution obtained with span length equal to 100 mm indicate that the bending specimen undergo high level of shear stress that requires considering the shear rigidity which is neglected by the used Euler-Bernoulli Beam equations. The Timoshenko beam theory must be used to get exact solution when bending is combined shear [28]. The axial strain and the shear strain are function of deflection, horizontal displacement, and rotation of cross-section [START_REF] Dao-Kuili | Large deflection and rotation of Timoshenko beams with frictional end supports under three-point bending[END_REF]. Whereas in our study the rotation of cross-section is not considered. For this reason, the Elastic modulus values calculated for span length less or equal to 150 mm are different than values determined for span lengths equal to 175 mm and 200 mm. The comparison of the behaviors is illustrated on Fig. 12 where only one curve per configuration is plotted for better readability. Results of the mechanical parameters function of the testing speed are given on the table 3. Thus, we observe that tests performed at 5 mm/min (table 3) exhibit the same results than those obtained previously for the tests with different span length on another batch of sample (table 2). This therefore confirms the repeatability of the results, the robustness of the characterization protocol as well as the reproducibility of the 3D printing process used in this study. The decrease of displacement rate from 5 mm/min to 0.05 mm/min leads to a decrease of maximal bending strength value of about 26% whereas the elastic modulus only drops by 3.2%, which remains largely within the dispersion of the results, which is between 2% and 6%

for the two configurations (table 3). The ultimate stress, and therefore the dissipated energy, evolves as a function of the strain rate, elastic modulus is not affected.

Moreover, the force-displacement curves for the tests performed at high displacement speeds are characterized by several load drops while those at low speed are quite smooth (Fig. 12).

This effect may be due to the potential effect of sliding between the sample and the device pins or even progressive decohesion of sand particles. we can therefore conclude that the effect of the strain rate is not significant on the elastic parameters of the printed sand while the breaking stress increases according to the testing speed.

Conclusion

This study proposes to evaluate the effect of span-to-thickness (specimen length / specimen height) and strains rates on the printed sand mechanical behavior during three-point bending tests instrumented with optical measurements. The DIC measurements highlighted the presence of a local crushing at the specimen/pin contacts, due to the Hertz pressure, that affects the onset of the behaviour. To overcome or limit this phenomenon, it would be necessary to favour the use of loading and supporting pins having a large diameter.

The assessed flexural strength of sand printed specimens under three-point bending is not affected by span length value contrary to the elastic modulus, which increases with span length. This evolution was observed both during the simple bending test, but without values stabilization, and with the loading-unloading, more reliable method, that gave rise to higher results with a stabilization of the elastic modulus values after a span-to-thickness-value of 8.

The fracture energy, for its part, decreased and reached a plateau starting after a span-tothickness ratio of 5.7. Moreover, when the span length increase, the DIC measurements showed that the distribution of the strains fields becomes more homogeneous both locally and globally. Therefore, for the characterization of the intrinsic elastic properties of printed sand using three-point bending tests, it will be appropriate to use a loading-unloading tests and to choose a span-to-thickness greater than 8 to overcome parasitic phenomena and coupling with shear behaviour. The increase of the strain rate, in the values range used, leads to an increase of the breaking stress. Hence, for the determination of the intrinsic stress of printed sands, it would be necessary to choose the strain rate close to conditions during casting for the intended application.

Future research will focus on study of mechanical behavior at high temperatures for relevant sizing of the foundry molds and cores and to make simulations of the molding process more robust.

Fig. 1 .

 1 Fig. 1. Printing configuration of the specimens

Fig. 2 .(c) Phase 3 Fig. 3 .

 233 Fig. 2. Example of load-displacement curve of a specimen with a span length of L=150mm.

Fig. 4 .

 4 Fig. 4. Failure of the specimens

Fig. 5 .

 5 Fig. 5. Load-displacement curves of the printed specimen at different distance between

Fig. 6 .

 6 Fig. 6. Bending strength (a) and fracture energy (b) evolution as function of span length.

Fig. 7 .

 7 Fig. 7. Load-displacement curve of cyclic bending test performed at span length of 200 mm.

Fig. 8 .

 8 Fig. 8. Elastic modulus evolution as function of span length.

  . The strain fields εxx obtained by the DIC for a loading state before fracture at different span lengths (100, 150 et 200 mm), presented on Fig. 9, illustrate this well. To facilitate the comparison of

Fig. 9 .Fig. 10 .

 910 Fig. 9. stain field (εxx) measurements before fracture at different span lengths

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Shear stain field (εxy) measurements before fracture point at different span lengths

Table 1 :

 1 Span-to-thickness ratio used for the three-point bending tests

	Span length (mm)	60	75	100	125	150	175	200
	Span length/Thickness L/h (mm) 2.73	3.41	4.55	5.68	6.82	7.95	9.09

Table 2 :

 2 Results of the mechanical parameters

		Maximal Strength	Elastic Modulus	Fracture Energy (MJ)
	Span length (mm)	Average (MPa)	Standard deviation (%)	Average (MPa)	Standard deviation (%)	Average (MJ)	Standard deviation (%)
	60	6.73 ±0.12	1.76	308.64 ±30.23	9.80	54.11 ±1.96	3.63
	75	6.78 ±0.47	6.95	504.54 ±17.82	3.53	43.88 ±5.31	12.11
	100	6.76 ±0.25	3.63	832.02 ±56.46	6.79	37.88 ±1.34	3.54
	125	6.39 ±0.32	5.01	1063.70 ±56.81	5.34	31.30 ±2.03	6.49
	150	6.22 ±0.13	2.09	1372.41 ±6.91	0.50	28.39 ±1.26	4.45
	175	6.68 ±0.25	3.75	1631.10 ±45.78	2.81	33.96 ±1.30	3.82
	200	6.37 ±0.36	5.67	1821.02 ±5.17	0.28	30.74 ±3.56	11.57

Table 3 :

 3 Results of the mechanical parameters function of the testing speed

	3	Parameter	Test 1	Test 2	Test 3	Average	Standard deviation (%)
		Maximal Stress (MPa)	6.24	6.09	6.21	6.18 ±0.08	1.29%
	5mm/min
		Young Modulus (MPa)	1348.19	1363.90	1401.70	1371.27 ±27.5	2.01%
		Maximal Stress (MPa)	4.44	4.53	4.67	4.55 ±0.11	2.51%
	0.5mm/min
		Young Modulus (MPa)	1422.05	1271.36	1289.25	1327.56 ±82.33	6.20%

Acknowledgments

The authors are thankful to the company 3D Metal industrie for providing sand printed samples.

Funding

This work was supported by French Région Grand Est (Grant number D201507798), the European Union (grant number D201507799) and UIMM (Grant number within the Chair MATUR.

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Conflict of interest

The authors declare no competing interests.

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