# Anisotropic Yield Criterion of Rolled AZ31 Magnesium Alloy via Nanoindentation
## 3 Results and Discussion
The relationship between the square of hardness (\(H^{2}\)) and the reciprocal of indentation depth (\(1/h\)) should be linear, as shown in Equation (6). We performed the linear fitting of \(H^{2}\) vs. \(1/h\). Based on the indentation data collected between 500 and 1500 nm, the slope and intercept of the fitted straight line are \(H_{0}^{2}\cdot h*\) and \(H_{0}^{2}\), respectively. Thus, \(H_{0}\) independent of the indentation size effect can be obtained (Figure 6). \(H_{0}\) may be determined from the results of linear fitting, as shown in Figure 1:5 \begin{table}
\begin{tabular}{c c c c c} \hline Angle (\({}^{\circ}\)) & 0\({}^{\circ}\) & 45\({}^{\circ}\) & 90\({}^{\circ}\) & ND \\ \hline \(H_{0}\) (GPa) & 0.35 & 0.36 & 0.42 & 0.28 \\ \hline \end{tabular}
\end{table} 
Table 2: Fitting values of \(H_{0}\) under different angles. \begin{table}
\begin{tabular}{c c c c} \hline Angle (\({}^{\circ}\)) & 0\({}^{\circ}\) & 45\({}^{\circ}\) & 90\({}^{\circ}\) & ND \\ \hline \(H_{0}\) (GPa) & 0.35 & 0.36 & 0.42 & 0.28 \\ \hline \end{tabular}
\end{table} 
Table 1: Fitting values of hardness under different angles. Figure 6: H\({}^{2}\) versus \(1/h\) curves of the rolled AZ31 alloys in four angles. A Tabor's factor of 3 can be used to convert the hardness values into yield stress values [14]:  
\[H=3\sigma \tag{7}\]  
The strength of magnesium alloys can be obtained using Equation (7). The yield stresses in different directions are presented in Table 3  
The anisotropy yield criterion of rolled AZ31 magnesium alloys is established according to the [PERSON] 1948 yield function. The three-dimensional yield function proposed by [PERSON] [15] is defined as follows:  
\[f_{y}=F_{y}(\sigma_{22}-\sigma_{33})^{2}+G_{y}(\sigma_{33}-\sigma_{11})^{2}+H_ {y}(\sigma_{11}-\sigma_{22})^{2}+2L_{y}\sigma_{23}^{2}+2M_{y}\sigma_{31}^{2}+ 2N_{y}\sigma_{12}^{2}-\overline{\sigma}_{y}^{2} \tag{8}\]  
where \(F_{y},G_{y},H_{y},L_{y},M_{y},N_{y}\) are used to describe direction-dependent yield stresses. These coefficients can be obtained by considering the uniaxial compression at some angle relative to the rolling direction and denoting the uniaxial compressive yield stress \(\sigma_{\theta}\). The stress components in the Cartesian axis system are:  
\[\sigma_{11}=\sigma_{\theta}\cos^{2}\theta\;\sigma_{12}=\sigma_{\theta}\sin \theta\cos\theta\;\sigma_{22}=\sigma_{\theta}\sin^{2}\theta \tag{9}\]  
Substituting (9) for \(\sigma_{11},\sigma_{12},\sigma_{22}\) in Equation (8) yields:  
\[\sigma_{\theta}=((1)((F_{y}+H_{y})\sin^{4}\theta)+(G_{y}+H_{y})\cos^{4}\theta -2H_{y}\sin^{2}\theta\cos^{2}\theta+2N_{y}\sin^{2}\theta\cos^{2}\theta)^{-1} )^{1/2}\overline{\sigma}_{y} \tag{10}\]  
Based on the primitive function, the uniaxial compressive yield stresses for the rolling direction (\(0^{\circ}\)), diagonal direction (\(45^{\circ}\)), and transverse direction (\(90^{\circ}\)) are formulated as follows:  
\[ \begin{array}{l}\sigma_{0}=(\frac{1}{G_{y}+H_{y}})^{1/2}\overline{\sigma}_{ y},\\ \sigma_{45}=(\frac{4}{F_{y}+G_{y}+2N_{y}})^{1/2}\overline{\sigma}_{y},\\ \sigma_{90}=(\frac{1}{F_{y}+H_{y}})^{1/2}\overline{\sigma}_{y}.\end{array}  \tag{11}\]  
The yield stress in the rolled-plane direction, which is denoted \(\sigma_{Z}\), can be concisely expressed as follows (12):  
\[\sigma_{z}=(\frac{1}{F_{y}+G_{y}})^{1/2}\overline{\sigma}_{y} \tag{12}\]  
The following expression can be acquired by finding the primitive function, and four anisotropic parameters are formulated by solving Equations (11) and (12) as follows:  
\[F_{y}=\frac{1}{2}(\frac{1}{\sigma_{90}^{2}}-\frac{1}{\sigma_{0}^{2}}+\frac{ 1}{\sigma_{z}^{2}})\overline{\sigma}_{y}^{2} \tag{13}\]  
\[G_{y}=\frac{1}{2}(\frac{1}{\sigma_{0}^{2}}-\frac{1}{\sigma_{90}^{2}}+\frac{ 1}{\sigma_{z}^{2}})\overline{\sigma}_{y}^{2} \tag{14}\]  
\[H_{y}=\frac{1}{2}(\frac{1}{\sigma_{0}^{2}}+\frac{1}{\sigma_{90}^{2}}-\frac{1} {\sigma_{z}^{2}})\overline{\sigma}_{y}^{2} \tag{15}\]