## 4 Theoretical Modelling of Mechanical Behavior  
This section presents a theoretical model of the behavior of the most important mechanical characteristics. Properties like yield strength and hardness, for instance were expressed as functions of volume fraction of porosity. The hardness values for pure Al in this model were gathered from literature and ranged from 15 to 20 H\({}_{\text{v}}\), and they were utilized when performing the modeling procedure [40; 41; 42]. Similarly, pure aluminum yield strength values obtained from literature research ranged from 30 to 40 MPa [43; 44; 45]. And these values were used for the modeling process. In this model, it was assumed that the yield strength and hardness values of pure aluminum were assigned to the Al5 Cr and Al7.5 Cr alloys. Various yield strength and hardness values were taken for pure Al (which were assigned to the Al-Cr alloys studied here), and from those data, intermediate values were selected. The following equation was used to describe hardness as a function of porosity volume fraction:  
\[H=H_{0}*(1-f_{v}) \tag{2}\]  
where \(H\) is the hardness of the sintered material, H\({}_{0}\) is the hardness of the material without porosity (i.e., the theoretical or maximum hardness), f\({}_{\text{v}}\) is the volume fraction of porosity in the material, and (1 - f\({}_{\text{v}}\)) is the relative density factor of the sintered material [46; 47; 48]. The volume fraction porosity values of sintered Al5 Cr and Al7.5 Cr in this work were 0.003 and 0.017, respectively. The hardness of Al5 Cr, determined from Equation (2), ranged from 14.9 to 19.9 H\({}_{\text{v}}\) (as mentioned above, with hardness values of pure Al ranging from 15 to 20 Hv). And the hardness values of the Al7.5 Cr specimens (computed from Equation (2)) fluctuated from 14.7 to 19.66 H\({}_{\text{v}}\). (see Figure 18a). Hardness as a function of volume \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Alloy** & **Area Fraction** & \begin{tabular}{c} **Area Fraction** \\ **(Matrix)** \\ \end{tabular} & \begin{tabular}{c} **Area Fraction** \\ **(Precipitates Dispersed** \\ **in Al Matrix)** \\ \end{tabular} & \begin{tabular}{c} **Area Fraction** \\ **of Pores** \\ \end{tabular} & \
\begin{tabular}{c} **Precipitate** \\ **Sizes (\(\mu\)m)** \\ \end{tabular} \\ \hline Al5 Cr & 0.932 & 0.068 & 0.003 &
\begin{tabular}{c} Width (1.7) \\ Length (6.5) \\ \end{tabular} \\ Al7.5 Cr & 0.897 & 0.103 & 0.0178 & 4.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Microstructural features of Al based alloys. Figure 17: (a) Microstructure of Al-7.5 Cr consolidated sample, chemical mappings of (**b**) Al and (c) Cr. fraction of porosity was calculated for both Al5 Cr and Al7.5 Cr samples using pure Al's starting hardness. To predict the yield strength values, the Gibson and Ashby model for porous materials was applied. The mathematical expression for this model is displayed in Equation (3). \[\sigma_{y}=\sigma_{0}(1-f_{y})^{m/2} \tag{3}\]  
where \(\sigma_{y}\) is the yield strength of the material, \(\sigma_{o}\) is the strength of the material without pores, _fv_ is the volumetric porosity fraction, and \(m\) is the strength exponent, which depends on the geometry and distribution of pores in the material. In general, typical values of \(m\) for sintered materials are in the range of 2 to 4 [46; 47; 48; 49]. Results derived from modeling \(\sigma_{y}\) values from Equation (3) considering the starting yield strength values of 30, 36 and 40 MPa are presented in Figure 18b and Table 2. In this sense, \(\sigma_{y}\) values were estimated using Equation (3) (with m = 2, 3, and 4). For an Al5 Cr specimen with a porosity of 0.003, the m exponent of Equation (3) had no significant effect on \(\sigma_{y}\) values. In this regard, the values of \(\sigma_{y}\) for this specimen were 39.7, 35.8, and 29.8 MPa, compared to pure Al starting values of 40, 36, and 30 MPa, respectively.