## 4 Discussion
### The Mechanism by Which Composite Nanostructures Influence Strength  
The mechanical properties of the materials were determined by the microstructure. The contributions of composite nanostructures to strength were determined by the quantitative characterization of the dislocation density, grain size and characteristic parameters of precipitates in the long-aged structure. Generally, the YS of age-hardenable Al alloys could be calculated by the following equation [27]:  
\[\sigma_{y}=\Delta_{gb}+M_{A}\Big{(}\tau_{0}+\tau_{s}+\Big{(}\Delta\tau_{d}^{2}+ \Delta\tau_{p}^{2}\Big{)}^{1/2}\Big{)} \tag{1}\]  
where \(\Delta_{gb}\) is the grain boundary strengthening determined by the grain size, \(M_{A}\) represents the Taylor factor that depends on the types and percentage of textures, \(\tau_{0}\) is the intrinsic critical resolved shear stress of pure Al that has a value of approximately 10 MPa, \(\tau_{s}\) is the solid solution strengthening that depends on the types and concentrations of solutes, \(\Delta\tau_{d}\) is the dislocation strengthening from the dislocation density of the samples and \(\Delta\tau_{p}\) is the precipitation strengthening attributed to the type, size and density of precipitates. For the CR-373K (1200 min) samples containing composite nanostructures, the precipitation of a large number of nanoprecipitates introduced precipitation strengthening. Studies have shown [30] that deformed samples continuously consume solute atoms in supersaturated solid solutions during the subsequent aging to produce precipitates. When the peak-aged state is reached, the solid solution strengthening is almost removed. Figure 1a shows that the hardness of the deformed samples was the largest when they were aged at 373 K for 1200 min, which was when they could be considered to have reached the peak-aged state. Therefore, the contribution of solid solution strengthening to the strength of the CR-373K (1200 min) sample was approximately 0 MPa. On the other hand, Figure 6c shows that the average grain size of the CR-373K (1200 min) samples was 184 nm. At the same time, the dislocations in the CR-373K (1200 min) samples could still provide dislocation strengthening. Therefore, the YS of the CR-373K (1200 min) samples could be calculated with the following equation [16; 25]:  
\[\sigma_{y/\text{CR-373K (1200 min)}}=\Delta_{gb}+M_{A}\Big{(}\tau_{0}+\Big{(} \Delta\tau_{d}^{2}+\Delta\tau_{p}^{2}\Big{)}^{1/2}\Big{)}. \tag{2}\]  
Generally, grain boundary strengthening is described by the Hall-Petch relationship [31]:  
\[\Delta_{gb}=\sigma_{0}+k_{1}D^{-1/2} \tag{3}\]  
where \(\sigma_{0}\) is the friction stress and \(k_{1}D^{-1/2}\) is the grain boundary strengthening (\(k_{1}\) is the Hall-Petch slope and \(D\) is the effective grain size). In this paper, \(\sigma_{0}\) and \(k_{1}\) were approximately equal to 20 MPa and 0.04 MPa m\({}^{-1/2}\), respectively [31]. Therefore, the calculated value for the CR-373K (1200 min) samples was 113.26 MPa. In this paper, the classic Bailey-Hirsch relationship was used to determine how dislocation strengthening affects the YS [25]:  
\[\Delta\tau_{d}=\alpha\text{Gb}\rho^{1/2} \tag{4}\]  
where \(\alpha\) is a constant with a value of 0.27, G is the shear modulus of the aluminum alloy that has a value of 26.2 GPa and b is the Burgers vector, whose magnitude is equal to 0.286 [17]. According to Equation (4), \(\Delta\sigma_{d}\) for the CR-373K (1200 min) samples was 39.95 MPa. Precipitation strengthening is the most important strengthening mechanism for age-hardenable Al alloys and depends on the size, distribution and spacing of the precipitates. \(\Delta\sigma_{p}\) can be simply expressed by the following equation [27; 28]:  
\[\Delta\tau_{p}=0.85\text{Gbln}(x/b)/2\pi(l-x) \tag{5}\]  
where the average diameter (\(x\)) and spacing (\(l\)) of the precipitated phases in the CR-373K (1200 min) samples were estimated from the HRTEM and TEM images and were equal to 16 nm and 44 nm,respectively.