# Advanced Metal Matrix Nanocomposites
## 5 Strengthening Mechanisms
### Hall-Petch  
The addition of ceramic particles or the primary precipitation of phases during solidification can lead to a grain refinement of the microstructure. In magnesium alloys, for example, zirconium is used for grain refinement, which can reduce the grain size in aluminum-free casting alloys by one order of magnitude. The addition of nanoparticles can therefore also lead to an increase in yield strength, which can be attributed to grain refinement. This increase in yield strength is described in Equation (10), called the Hall-Petch equation [148; 149] where \(\sigma_{\rm y}\) is the yield strength, \(\sigma_{0}\) is the friction stress that allows dislocations to move on slip planes in a single crystal in the absence of any strengthening mechanisms, \(k_{\rm y}\) is the stress concentration factor, and D is the average grain size. \[\sigma_{\rm y}{=\sigma_{0}+k_{\rm y}D^{-1/2}} \tag{10}\]  
The increase in yield strength can also be expressed by Equation (11), premised that grain refinement was achieved by nanoparticles and exactly the same casting process was used as for the unreinforced alloy [150]. D\({}_{\text{MMNC}}\) and D\({}_{0}\) are the grain sizes of the nanocomposite and the unreinforced alloy. \[\Delta\sigma_{\text{GR}}=\text{k}_{y}\bigg{(}\frac{1}{\sqrt{\text{D}_{\text{ MMNC}}}}-\frac{1}{\sqrt{\text{D}_{0}}}\bigg{)} \tag{11}\]  
The grain refining effect of the introduced nanoparticles is very clearly illustrated when, as a part of the EU project EXOMET, a die-casting alloy AM60 was reinforced with 1 wt% AlN nanoparticles [127]. The AM60 is a commercial alloy used in the automotive industry from which, for example, steering wheels, seat frames, or housings are die-cast. The components have very good room temperature strength and ductility. The AM60-based nanocomposite was produced using an ultrasound-assisted casting process. After adding the nanoparticles, the melt was stirred and an ultrasonic sonotrode was immersed in the melt, which emits sound at a frequency of 20 kHz. As can be seen in Figure 17, an immense grain refinement of the microstructure could be achieved by adding the AlN nanoparticles. Since both materials were produced identically, grain refinement is therefore only due to the presence of nanoparticles; a change from 1277.0 \(\pm\) 301.3 \(\upmu\)m to 84.9 \(\pm\) 6.2 \(\upmu\)m doubles the yield strength (+103%) and tensile strength (+115%), see Table 4, while the ductility at the same time was increased by 140% from 6.4 to 15.4% [127]. Calculations have shown that the contribution of grain refinement to the yield strength increase is the largest with 42.7 MPa, followed by the contribution of Orowan strengthening with 8.3 MPa; other contributions to the increase in strength are negligible. All properties are summarized in Table 4.  
As already mentioned in Table 1, AlN has a hexagonal structure with lattice constants of a = 0.311 nm and c = 0.498 nm. These are very similar to those of magnesium (a = 0.321 nm \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **AM60** & **AM60 + AlN** \\ \hline Grain size [\(\upmu\)m] & 1277.0 \(\pm\) 301.3 & 84.9 \(\pm\) 6.2 \\ Hardness [HV5] & 48.0 \(\pm\) 4.0 & 46.4 \(\pm\) 6.0 \\ Density [g/cm\({}^{3}\)] & 1.7848 \(\pm\) 0.0004 & 1.783 \(\pm\) 0 \\ Porosity [\%] & - & 0.919 \\ Yield strength [MPa] & 44.9 \(\pm\) 6.9 & 91.2 \(\pm\) 3.8 \\ UTS [MPa] & 109.3 \(\pm\) 19.2 & 235.1 \(\pm\) 6.4 \\ Elongation [\%] & 6.4 \(\pm\) 3.4 & 15.4 \(\pm\) 4.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Grain size, hardness, density, porosity proportion, and mechanical properties of AM60 reinforced by AlN reinforcing agent with \(\sim\)1500% reduction in grain size and 200% improvement in yield strength [127]. Figure 17: Microstructure of (**a**) AM60; and (**b**) AM60 + AlN [127]. Microstructure is significantly refined using the addition of AlN nanoparticle as well as by irradiating the melt by an ultrasonic transducer. and \(c=0.512\) nm), which leads to an easy growth of magnesium on an AlN nucleus during solidification [86].