## 3 Results  
### Elastic Moduli Dependence on CF Content  
In continuous fiber composites, the contribution of the reinforcing phase is initially reflected on the change in the elastic modulus, which varies linearly with respect to the corresponding moduli of the alloy and the reinforcement, according to the rule of mixtures. This rule also serves as the theoretical foundation according to which the drawn results are evaluated. In composites with unidirectional fibers, the elastic modulus varies with the stress direction. Thus, for a stress applied in the direction of the fibers, the modulus of elasticity of the composite (\(E_{ct}\)) is expressed as:  
\[E_{ct}=E_{m}(1-f)+E_{f}f \tag{4}\]  
where \(f\) stands for the fiber volume fraction and \(E_{m}\), \(E_{f}\) stand for the elastic moduli of the alloy and the fibers, respectively. For stress applied normal to the direction of the fibers, \(E_{ct}\) is expressed as:  
\[\frac{1}{E_{ct}}=\frac{1-f}{E_{m}}+\frac{f}{E_{f}} \tag{5}\]  
Figure 6a depicts the tensile stress application along and perpendicularly to the direction of the fibers. Figure 6b shows a fibrous composite with discontinuous fibers and a random directional distribution. In this configuration, a linear relationship holds again, taking into account a fiber efficiency parameter, \(K\), according to the relation:  
\[E_{ct}=E_{m}(1-f)+KE_{f}f \tag{6}\]  
Figure 6: (**a**) Elastic moduli in the longitudinal and transverse fiber direction of composites with continuous fibers; (**b**) Elastic modulus of fibrous composite with discontinuous fibers and random directional distribution. Parameter \(K\) (Krenchel parameter) is less than 1, usually between 0.1 and 0.6. Its value strongly depends on the geometry and orientation of the amplifying phase [52]. A typical 160,000-element ensemble with features representing the precipitates and the CF is shown in Figure 7 for an MMC with \(f=12\%\) as an example. The tensile stress simulation of the same sample results in the deformed shape shown in Figure 8, with regard to the total strain (Figure 8a) and von Mises stress averaging at 75% (Figure 8b). Figure 8: Deformed MMC sample after 1.4% elongation and distribution of (**a**) total strain and (**b**) von Mises stress in MPa. Figure 7: Abaqus model of a 12% AA/705/CF MMC, with the corresponding elements representing (**a**) Al\({}_{7}\)Cu\({}_{2}\)Fe, (**b**) Mg\({}_{2}\)Si, and (**c**) CF phases. Figure 9 shows the stress-strain curves of the four types of MMCs tested, for elongations of up to 1.4%. In total, 16 simulations (four for each volume fraction \(f\)) were carried out. It is noted that with increasing CF volume fraction, the elastic modulus also increases. Comparing the results of the simulation with those of [38], it is concluded that there is no close match for room temperature; a higher elastic modulus value for pure AA7075 is found and almost the same value is found for the three MMCs with \(f\) = 4%, 8%, and 12%. Nevertheless, the present results match more closely with those of [38] for 100 \({}^{\circ}\)C, as well as those by [PERSON] et al. [53], who found that with increasing CF content, the elastic modulus increases almost linearly. The elastic moduli that correspond to the curves of Figure 9 are shown in Figure 10. On the same plot, the values expected from uniaxial tensile stress in the longitudinal and transverse direction for the continuous-fiber-reinforced composite are presented, as described by Equations (4) and (5). In these equations, \(E_{m}\) was considered to be equal to 45.8 GPa, which is the slope of the elastic part of the pure alloy curve in Figure 9. Theoretically, the conducted results should be between the curves corresponding to these two equations and, more specifically, match Equation (6), as the MMCs are considered discontinuous, with their length-to-diameter ratio being less than 100 [37; 54]. Indeed, this trend is followed and, using Equation (6), the values of \(K\) are estimated as 0.59, 0.56, and 0.59 for 4%, 8%, and 12% MMCs, respectively.