45 When the particles are larger and the second phase occupies a considerable volume, the mean edge-to-edge distance or the mean free distance between particles, λ, can be used to assess the inter-particle spacing. The conventional method involves a counting of the phase interceptions with random straight lines per unit line length [115]. In the absence of this function in the image analysis software, the dilation and counting technique seems to be the most appropriate as it also provides the particle edge-to-edge spacing (in contrary to center-to-center spacing regarded in some other methods) and takes into account morphological features of the second phase [122].
The dilation and counting technique used in this study implies a successive dilation of the second phase particles with counting the particle number after each dilation step until all particles are merged together into a single particle [122]. The resulting data is transferred into the graph where the inter-particle spacing is plotted versus the cumulative percolation. Here, the inter-particle spacing is defined as two times the number of dilation steps, i, multiplied by the pixel size. The cumulative percolation is computed as 1-(Ni/N0), where Ni is the number of Si particles after the dilation step i and N0 is the original number of Si particles. Then, the inter-particle spacing, λ, is defined as a value which corresponds to the median value of the cumulative percolation, i.e. to the percolation of 0.5 [127].
3.5 Characterization of mechanical behavior
3.5.1 FEM simulations
FEM simulations have been carried out on virtually designed and experimentally obtained samples of Al-Si eutectic phase. While virtually generated structures have been produced with an isotropic voxel size of 46 × 46 × 46 nm3, 3D images obtained by the tomographic reconstruction have featured an anisotropic voxel size due to the experimental setup (see Section 3.3). Therefore, the experimental images have first been rescaled to the isotropic voxel size in accordance with the virtual samples.
The series of 2D binary images issued from the tomographic reconstruction have been merged into a 3D volume mesh and the material properties have been mapped on the barycenter of each voxel. The resulting amount of mesh cells have been very large requiring a high computational power. To speed up the computations and reduce memory requirements, the meshes have been coarsened with the help of an algorithm that combines several voxels into a larger voxel with respect to the volume fraction and properties of the materials therein. The influence of the coarsening level on the number of mesh cells and computational time has
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been studied in [128,129]. Finally, the coarsening level 8 has been used for all the structures, which implies that every 8 × 8 × 8 voxels have been combined into 1 voxel.
The following properties have been assigned to the materials: A Young’s modulus of E = 70 GPa, a Poisson’s ratio of ν = 0.34, and a yield strength of σy = 40 MPa have been applied to the aluminium, whereas a Young’s modulus of E = 107 GPa, a Poisson’s ratio of ν = 0.27, and a yield strength of σy = 7 GPa have been assigned to the silicon.
The FEM simulations have been carried out with the help of the structural mechanics module of COMSOL Multiphysics, whereby an elasto-plastic material model with an isotropic hardening has been used. In particular, the aluminium has been modeled as an elasto-plastic material and the silicon has been modeled as a linear elastic material. The damage of the material and residual stresses have not been taken into account.
The following boundary conditions have been applied to the 3D samples for the FEM simulations of uniaxial compression: one side of the meshed volume has been fixed while the opposite side has been subjected to the compressive loading. Then, the stress-strain curves have been computed by using volume averages of stress, , and strain, [13]:
3.4
3.5
where V is the volume of the 3D sample, m is the cell number and N is the total number of mesh cells. Here, the von Mises stresses in the material have been calculated as they are usually applied to the mechanical characterization of ductile materials like Al-Si alloys [9].
More details on the FEM simulations procedure can be found in [129].
3.5.2 Compression tests
The compression tests have been carried out with the help of the E10000 Linear-Torsion Floor Instrument of Instron® Company which allows a maximum load capacity of 10 kN. All test specimens have been cut from similar parts of the castings into cubes with the side length of 5 mm. The tests have been performed at a constant cross-head speed of 0.001 mm/s. After the compression loading, the specimens have been sectioned in the center along the loading axis for further metallographic investigations.
3.5 Characterization of mechanical behavior
47 3.5.3 Tensile tests
The tensile tests have been carried out with the same machine as the compression tests. The specimens were machined according to the European Standard EN ISO 527-2 [130], where the gage has a square cross-section of 2 × 2 mm2 and is 10 mm long. The specimen is fixed in the machine, so that the one end of the specimen remains static and the tensile load is applied to the other end of the specimen. The applied force and the corresponding elongation (amount of stretch) are recorded during the test. Then, the engineering stress-strain curve is computed by dividing the force by the initial cross-section area of the specimen and the amount of stretch of the gage by the initial gage length of the specimen, respectively. The results deduced from the stress-strain curve include the following mechanical characteristics: the Young’s modulus, the yield strength, the ultimate tensile strength and the total elongation [131].
Since Al-Si alloys do not exhibit a pronounced yield point, an offset deformation of 0.2% has been used to define the yield strength. To do so, a line intersecting the strain axis at the pre-defined offset has been drawn parallel to the elastic part of the stress-strain curve. The stress corresponding to the intersection of this line with the stress-strain curve is the offset yield strength. The modulus of elasticity is equal to the slope of the elastic part of the stress-strain curve. The ultimate tensile strength has been defined as a maximum stress carried by the specimen. The elongation-at-fracture has been computed by dividing the amount of stretch to rupture by the initial gage length of the specimen. It can also be read from the stress-strain curve as a strain corresponding to the stress decreased by 10% from its maximum value [131,132].
In relation to the strain rate, the general rule is the following: the lower strain rate, the lower the strength and the larger the elongation obtained. For the aluminium, the maximal recommended strain rate is 15 × 10-5 s-1 [131]; for the gage length of 10 mm it results in a cross-head speed of 0.09 mm/min. In [39], the tensile tests have been performed on cylindrical specimens of modified and unmodified AlSi8 alloys with a gage length of 30 mm and at the cross-head speed of 0.5 mm/min, i.e. the strain rate of 2.8 × 10-4 s-1. In [75,86], the tensile tests of Al-Si-Mg alloys have been carried out at the strain rate of 1 × 10-4 s-1. In [20,53], the strain rate of 1 × 10-3 s-1 has been applied for the tensile testing of Al-Si-Mg alloys (gage length of 15 mm). Thus, to define an optimal strain rate for the measurements, a complementary set of experiments has been carried out at different strain rates. The stress-strain curves obtained during the tests are shown in Figure 3.8.
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Figure 3.8: Influence of the strain rate (the cross-head speed) on the mechanical behavior of Al-Si specimens (alloy 7X_AC).
As can be seen from Figure 3.8, no explicit influence of the strain rate on the mechanical behavior of the specimens can be observed. Therefore, the cross-head speed of 0.01 mm/s which corresponds to the strain rate of 1 × 10-3 s-1 has been chosen for the tensile testing of Al-Si alloys. After the tension loading, the specimens have been sectioned in the center along the loading axis for further metallographic investigations.