## 4 Theoretical Modelling of Mechanical Behavior
\begin{table}
\begin{tabular}{c c c c} \hline  & \multicolumn{2}{c}{**Al5 Cr with a Volume**} & \multicolumn{2}{c}{**Al5 Cr with a Volume**} \\  & **Fraction (\(f_{y}\)) of Porosity** & **m** & **Fraction of Porosity (\(f_{y}\))** \\  & **of 0.003** & & **of 0.017** \\ \hline \multicolumn{3}{c}{**Yield Strength (Y\({}_{s}\)),**} & \multicolumn{2}{c}{**Yield Strength (Y\({}_{s}\)), Al5 Cr**} & \multicolumn{2}{c}{**Yield Strength (Y\({}_{s}\)), Al7.5 Cr**} \\ \hline  & & 2 & 39.3 \\
40 & 39.78 & 3 & 38.9 \\  & & 4 & 38.6 \\ \hline  & & 2 & 35.4 \\
36 & 35.8 & 3 & 35.1 \\  & & 4 & 34.7 \\ \hline  & & 2 & 29.4 \\
30 & 29.86 & 3 & 29.2 \\  & & 4 & 29 \\ \hline \end{tabular}
\end{table} 
Table 2: Yield strength values as a function of volume fraction of porosity (\(f_{y}\)) for Al5 Cr and Al7.Cr specimens. Figure 18: (**a**) Hardness and (**b**) Yield strength (\(\sigma_{y}\)) dependence with volume fraction of porosity (\(f_{y}\)). On the other hand, Al7.5 Cr yield strength values (with a volume fraction of porosity = 0.017) were sensitive to the \(m\) exponent in Equation (3). When \(\sigma_{y}\) of pure Al was established to 40 MPa, modeling yielded values of 39.7, 35.8, and 29.8 MPa for m = 2, 3, and 4, respectively. Precipitation of Cr\({}_{2}\)Al on Al5 Cr \"matrix\" and Cr(Al) on Al7.5 Cr \"matrix\" occurred during sintering treatment; for example, Al and Cr\({}_{2}\)Al phases together can be considered a composite material. Thus, to compute hardness or yield strength values, the rule of mixtures can be applied and can be represented in Equation (4). \[p_{c}=\sum f_{i}p_{i}=f_{1}p_{1}+f_{2}p_{2}+\ldots+f_{n}p_{n} \tag{4}\]  
where \(p_{c}\) is the property of compound; \(p_{1}\), \(p_{2}\),... \(p_{n}\) are the properties of each of the constituents of composite material, and \(f_{1}\),\(f_{2}\),... \(f_{n}\) are the volume fractions of each of the components [50]. For the modeling process, the following values of hardness and yield strength for Cr\({}_{2}\)Al and Cr(Al) phases acting as \"reinforcement\" phases were employed: For Cr\({}_{2}\)Al, a hardness of 600 H\({}_{\rm v}\) and \(\sigma_{\rm y}\) of 600 MPa, and for Cr(Al), a hardness of 250 H\({}_{\rm v}\) and \(\sigma_{\rm y}\) of 200 MPa [51; 52; 53; 54; 55]. Hardness and yield strength oh Al5 Cr composite were computed from Equation (4). In this case, starting hardness of Al5 Cr \"matrix\" was 17.96 H\({}_{\rm v}\) (see Figure 19a), and considering a volume fraction of Cr\({}_{2}\)Al of 0.068, the resulting hardness was equal to 58.6 H\({}_{\rm v}\). Also, initial yield strength of Al5 Cr \"matrix\" was 35.8 MPa (see Figure 19), and considering a volume fraction of Cr\({}_{2}\)Al of 0.068, the modeled \(\sigma_{\rm y}\) was 75.39 MPa. Hardness and yield strength of Al7.5 Cr composites were calculated from Equation (4). In this case, the starting hardness of the Al7.5 Cr \"matrix\" was 17.7 H\({}_{\rm v}\) (see Figure 20a), and considering a volume fraction of Cr(Al) precipitates of 0.1, the resulting hardness was equal to 42.2 H\({}_{\rm v}\). Also, the initial yield strength of the Al7.5 Cr \"matrix\" was 35.8 H\({}_{\rm v}\) (see Figure 20b), and considering a volume fraction of Cr(Al) of 0.1, the modeled \(\sigma_{\rm y}\) was 52.4 MPa. Figure 19: (a) Hardness and (b) yield strength dependence with volume fraction of Al matrix in Al5 Cr Matrix Composite.