## 5 Mechanical Properties
### Grain Refinement Strengthening  
Grain refinement is a highly effective strengthening mechanism to improve both strength and ductility for Mg alloys, which are a low fault energy metal. This strengthening strategy states that the smaller the grain size, the stronger the mechanical properties were. The relationship between grain size and yield strength can be described by the Hall-Petch equation [87]:  
\[\sigma_{y}=\sigma_{0}+k_{y}D^{-\frac{1}{2}} \tag{2}\]  
where \(\sigma_{y}\) is the yield strength, \(\sigma_{0}\) is the intragranular friction stress, \(k_{y}\) is a constant, which is related to multiple metallurgical factors, such as twins, texture, grain size microstructure and compositions, and \(D\) is the average grain size.  
Figure 8 shows the tensile test results of AZ31 alloys, and the Hall-Petch relation of yield strength and flow stress is observed by [PERSON] et al. [88]. It can be seen that two distinguished Hall-Petch slopes were derived with varying grain sizes, and the critical grain size is 17 \(\upmu\)m (\(<\)17 \(\upmu\)m was defined as fine grains, \(>\)17 \(\upmu\)m was defined as coarse grains) for the transition from twining to slip dominated flow. Thus, it can be calculated that \(\sigma_{fine\ grains}\) = 131 MPa and \(\sigma_{coarse\ grains}\) = 787 MPa.  
Figure 8: The results of tensile tests of AZ31 alloys: (**a**) engineering stress–strain curves, (**b**) true stress–strain curves, (**c**) Hall–Petch relation between yield strength and average grain size and (**d**) Hall–Petch relation bet flow stress [88] (The dot line represents the value of AZ31 alloys with average grains size of 17 \(\upmu\)m).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & **Processing History** & **UTS (MPa)** & **YS (MPa)** & **El (\%)** & **Ref.** \\ \hline Mg-6.85 Gd-4.52Y-1.15 Nd-0.55 Zr & MDF with accumulative strain of 1.50 & 320 & 253 & 7.5 & [80] \\ \hline Mg-6.4 Li-3.6 Zn-0.37 Al-0.36Y & MDF with accumulative strain of 3.60 & 286 & - & 31.8 & [81] \\ \hline Mg-8.0 Gd-3.7Y-0.3 Ag-0.4 Zr & MDF with accumulative strain of 4.50 & 366 & 301 & 10.0 & [82] \\ \hline Mg-8.0 Gd-3.7Y-0.3 Ag-0.4 Zr & MDF with accumulative strain of 5.40 & 364 & 309 & 10.2 & [82] \\ \hline Mg-10.2 Li-2.1 Al-2.23 Zn-0.2 Sr & MDF with accumulative strain of 3.0 & 242 & - & 23.59 & [83] \\ \hline Mg-8.9 Gd-1.8Y-0.5 Zr-0.2 Ag & MDF with accumulative strain of 3.0 & 320 & 220 & 26.1 & [84] \\ \hline Mg-6.2 Gd-3.7Y-0.9 Zn-0.3 Zr-0.3 Ag & MDF with accumulative strain of 4.05 & 384 & 292 & 9.0 & [70] \\ \hline Mg-6 Gd-4Y-0.7 Zn-0.4 Zr-0.3 Ag & MDF with accumulative strain of 4.05 & 386 & 290 & 9.6 & [85] \\ \hline Mg–8.2 Gd–3.8Y–1.0 Zn–0.4 Zr & MDF with equivalent strain of 4.8 & 419 & 399 & 14.8 & [86] \\ \hline \hline \end{tabular}
\end{table}
Table 3: _Cont._[PERSON] et al. [89] prepared the pure Mg with average grain sizes ranging from 31 nm to 9.8 \(\upmu\)m and observed that there existed three Hall-Petch relationships: a positive correlation with a large stress concentration factor and a main deformation mechanism of twinning of 145 MPa \(\upmu\)m\({}^{1/2}\) for YS and 221 MPa \(\upmu\)m\({}^{1/2}\) for UTS, a positive correlation with a small stress concentration factor and a main mechanism of dislocation of 9 MPa \(\upmu\)m\({}^{1/2}\) for YS and 14 MPa \(\upmu\)m\({}^{1/2}\) for UTS and an inverse correlation with a negative stress concentration factor with a main mechanism of grain boundary motion of \(-\)10 MPa \(\upmu\)m\({}^{1/2}\) for YS and \(-\)21 MPa \(\upmu\)m\({}^{1/2}\) for UTS.