Correlation between dimples and Si particles geometry

The microstructure of a material has a deterministic role for the fracture surface properties. In particular, the morphology of Si particles, which the fracture meets on its way, defines the dimple morphology, so that the dimple size increases with the particle size and inter-particle spacing, on one hand, while the number of dimples increases with the number of particles in a bulk microstructure, on the other hand [47]. A nearly one-to-one relation between the average dimple size and the average inter-particle spacing has been found for aluminium alloys [69].

However, there are no specific quantitative relations with respect to the influence of Si particles’ size-shape distributions in a bulk structure on the fracture surface geometry.

Therefore, the following section aims to determine which geometrical features of Si particles exert an influence on the dimple geometry and to what extent they do so. Thus, average properties and their distributions for both particles and dimples from corresponding alloys have been analysed. At the end, the number of investigated features has been limited to the area describing the size of objects, the aspect ratio and the shape factor describing their shape.

Other size parameters, in particular the equivalent circle diameter and MaxFeret diameter, have shown equivalent distributions to the area distribution and will therefore not be presented here. The same holds for the shape parameters such as roundness and circularity, which describe different aspects of similarity between an object and a circle.

Figure 4.33 shows histograms of the area distribution of particles on metallographic sections that represent a bulk microstructure and the area distribution of dimples on the fracture surfaces of the corresponding alloys. Si particles and dimples exhibit very similar distributions with a low percentage of outliers consisting of particles/dimples with the area exceeding the average area by the factor of 3 or more. For the sake of comparison, the area and the frequency of particles/dimples have been normalised to the average value of objects area and the number of objects in every class, respectively. Matching distributions indicate a proportional relationship between the size features of particles and dimples. Thus, the area of

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117 the Si particles’ cross sections is a characteristic feature that directly influences the fracture surface geometry.

Figure 4.33: Distributions of particles’ vs. dimples’ area in the unmodified (2X_AC), Sr-modified (3X_AC and 7X_AC) alloys in as-cast state and Sr-Sr-modified alloy after heat treatment (3X_ST). Other ST alloys are not presented here because they have similar distributions to the one of the alloy 3X_ST (extended from [133]).

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Though the nature of the relationship between the size of particles and dimples can theoretically be determined, it is practically hardly possible. The difficulty to compute this relationship is due to the fact that analyzing the size of a Si particle and the size of a dimple corresponding to this particle is in principle not possible since:

 The size of particles is measured on the metallographic sections and the size of dimples is computed on the fractographs.

 Even if the Si particles and the corresponding dimples are both manually segmented on the fractographs, available image processing tools do not allow a tracking of their corresponding properties in the output data.

Attempts to filter particles and dimples by size, assuming that the smaller dimples correspond to the smaller particles, and fit an equation to the data scatter have provided different ratios for different samples. A comparison of the average size values for corresponding micrographs and fractographs has resulted in the ratio of the dimple equivalent area circle diameter to the particle equivalent area circle diameter, which varies between 1.3 and 3. Thus, there is no unique coefficient of proportionality that would fit all structures. The size of dimples depends not only on the size of nucleus-particles but also on the particles arrangement, i.e. the particles density and the inter-particle spacing [47]. Therefore, to predict dimple characteristics on the fracture surface, the geometry and arrangement of microstructural constituents have to be taken into account simultaneously.

Analogously to results reported by Broek [69], Figure 4.34 compares the average dimple equivalent diameter versus the average inter-particle spacing (more precisely, a sum of the average particle equivalent diameter and the interparticle spacing computed in Section 4.4).

Few data points fall onto the one-to-one relationship, but in general, the dimple diameter is slightly below the one-to-one trendline which indicates a smaller particle density on the fracture surfaces as compared to the random sections though the material. Here, every particle on the fracture surface is assumed to cause a void initiation. Therefore, the particle density on the fracture surface is identical to the dimple density therein. On the other hand, the particle density on the metallographic sections has been compared with the dimple density on the fractographs and the corresponding data have been summarised in Table 4.8. A good correlation between the two densities can be found for most of the samples. The comparatively higher dimple density observed for the modified alloys 3X_AC and 7X_AC could be either due to the slight overestimation of the number of dimples that resulted from

4.6 Fractography of Al-Si eutectic

119 the sophisticated dimple segmentation procedure or the low statistical representativeness of high resolution fractographs. In fact, the particle density and the average inter-particle spacing have been computed on the metallographic images with a high population of Si particles. The dimple density and the average dimple equivalent diameter have been determined for every alloy as the mean value of several fractographs showing different fields of the sample at high magnification (i.e. low statistics). Hence, a high standard deviation of the measurements can be seen in Table 4.8. Nevertheless, the values of both densities still have the same order of magnitude. Taking into account conclusions drawn from Figure 4.34 and Table 4.8, one can assume a certain degree of randomness of the fracture propagation path. This means that the distribution of Si particles in random metallographic sections through the structure can give a good approximation to the distribution of particles on the fracture surface.

Figure 4.34: Dimple equivalent diameter vs. interparticle spacing in a bulk microstructure in the unmodified (2X), Sr-modified (3X) and inhomogeneously modified (7X) alloys in the as-cast (AC) and solution treated (ST) states. A straight line defines a one-to-one relationship between two data sets.

Table 4.8: Particle vs. dimple densities in different Al-Si alloys. The standard deviation of the dimple density is placed in parentheses. There is a good correlation between the two densities: corresponding densities for the samples have either similar values or at least, the values of the same order of magnitude (extended from [133]).

2X_AC 3X_AC 7X_AC 2X_ST 3X_ST 7X_ST

Particle density (µm^-2) 0.31 0.37 0.16 0.04 0.05 0.05 Dimple density (µm^-2) 0.31 0.72 0.28 0.05 0.05 0.07 (0.07) (0.18) (0.10) (0.02) (0.01) (0.03)

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A different situation is observed for the shape factor. As can be seen in Figure 4.35, the distributions of the shape factors of dimples and particles exhibit different shapes: particles show a higher standard deviation than the dimples for the shape factor, except from the solution treated alloy. Particles also have a higher fraction of objects with lower shape factors than dimples, whereas the dimple system is more homogeneous, i.e. it has a lower variability of the shape factor. According to the definition of the shape factor [124], which compares a particle area to the area of a circle with the same perimeter as the particle’s perimeter, decreasing the shape irregularities is associated with a reduction of the object’s perimeter, which, as a consequence, increases its shape factor. In this case it means that the shape irregularities present in cross sections of Si particles are transmitted to dimples in a reduced or smoothed manner. Thus, the complexity of the nucleus-particles’ shape is not directly reflected by the shape of dimples.

Figure 4.35: Distributions of particles’ vs. dimples’ shape factor in different Al-Si alloys.

The shape factors of dimples and particles exhibit different frequency distribution shapes.

No direct relation can be observed between the two data sets of the shape factors (extended from [133]).

The aspect ratio, however, shows better distribution correspondence between particles and dimples than the shape factor (see Figure 4.36). The relationship between the average aspect ratios of dimples and particles for corresponding samples is close to one-to-one (see Table

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121 4.9). Therefore, the elongation of particles is broadly transmitted to the dimples. Thus, the aspect ratio can be considered to be another characteristic feature of Si particles with a direct influence on the shape of dimples.

Figure 4.36: Distributions of particles’ vs. dimples’ aspect ratio in different Al-Si alloys.

High conformity between the distributions can be observed (extended from [133]).

Table 4.9: Average particles’ vs. dimples’ aspect ratio for distributions presented in Figure 4.36. Particles and dimples from corresponding samples have similar average aspect ratios (extended from [133]). structural features on micrographs and fractographs is that there is a correlation between the geometry of dimples and Si particles. In particular, such features as the area, or more general, the size and the elongation of Si particles cross sections together with the arrangement of Si particles relative to each other define for the most part the fracture surface geometry.

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