# Anisotropic Yield Criterion of Rolled AZ31 Magnesium Alloy via Nanoindentation
## 3 Results and Discussion  
Figure 2 shows the optical microscopic images of the rolled AZ31 samples in the 0\({}^{\circ}\), 45\({}^{\circ}\), 90\({}^{\circ}\), and ND directions (these directions are marked by the arrows in Figure 1); coarse grains of 5-20 \(\upmu\)m could be observed in the samples. [PERSON] et al. found that grain size affects the mechanical responses of wrought magnesium [8]. The grain sizes of the rolled samples in the four directions were significantly different, which could explain the observed differences in their strength. The nanoindentation experiments were carried out in four directions, as shown in Figure 1.  
Figure 1: Schematic of the loading of rolled AZ31 alloys in four directions. Figure 2: Optical microscopic images of the rolled AZ31 samples in the 0\({}^{\circ}\), 45\({}^{\circ}\), 90\({}^{\circ}\), and vertical (ND) directions. In CSM mode, the function of contact stiffness can be expressed as follows [6]:  
\[S=\left[\frac{1}{\left(F_{amp}/h_{amp}\right)\cos\phi-\left(K_{s}-m_{\omega}{}^{2} \right)}-\frac{1}{K_{f}}\right]^{-1} \tag{1}\]  
where \(S\), \(F_{amp}\), and \(h_{amp}\) are the contact stiffness, amplitude of the harmonic excitation force, and response displacement amplitude, respectively. In addition, \(\phi\) is the phase shift, \(\omega=2\pi f\) is the angular frequency (\(f=45\)Hz), and \(K_{s}\), \(K_{f}\), and m are the spring constant in the vertical direction, frame stiffness, and mass of the indenter, respectively. The contact stiffness increases nearly linearly with indentation depth in the three directions, as shown in Figure 3.  
The projected contact area \(A_{c}\) of a perfect Berkovich diamond indenter can be calculated as follows [9; 10]:  
\[A_{c}=24.56h_{c}^{2} \tag{2}\]  
where \(h_{c}\) is the contact depth and calculated by the equation:  
\[h_{c}=h-\varepsilon\frac{p}{s} \tag{3}\]  
where \(P\) and \(\varepsilon=0.75\) are the contact load and a contact for the Berkovich indenter, respectively. Hardness (\(H\)) is defined by the follow equation:  
\[H=\frac{P}{A_{c}} \tag{4}\]  
where \(P\) is the contact load. The hardness-depth curves of the samples in four directions can then be obtained. As shown in Figure 4, the experimental data of 45\({}^{\circ}\) have better repetition. Figure 3: Nanoindentation contact stiffness-depth curves obtained during the loading of rolled AZ31 alloys in four directions. The hardness values of rolled AZ31 alloys during loading in the \(0^{\circ}\), \(45^{\circ}\), \(90^{\circ}\), and ND directions clearly differ, as shown in Figure 5. As the indentation depth increases, the hardness values decrease. The hardness increases as the angle increases from \(0^{\circ}\) to \(90^{\circ}\) and is smallest in the ND direction. Hardness is affected by the plate texture during rolling and shows significant anisotropy. Some factors of uncertainty and error in nanoindentation tests include surface roughness and surface texture, among others. In this study, the maximum penetration depth of the sample was approximately 2000 nm; such a depth is sufficiently large to assume that the contribution of surface inconsistencies is negligible. The hardness values observed in the \(0^{\circ}\), \(45^{\circ}\), \(90^{\circ}\), and ND directions are listed in Figure 1:0  
Figure 1:1 Nanoindentation hardness-depth curves of \(45^{\circ}\) from three repeated experiments. Figure 1:2 Nanoindentation hardness-depth curves obtained during the loading of rolled AZ31 alloys in four directions. The indentation size effect refers to the variation in the indentation hardness (or indentation stress) as a function of indentation depth [11]. The effects of indentation depth on the mechanical properties of metals must be considered. Under the limit of infinite depth, [PERSON] et al. [12; 13] proposed a hardness model to investigate the indentation size effect of metal hardness as follows:  
Figure 1:3  
Figure 1:4  
where \(H_{0}\) is the hardness in the limit of infinite depth and \(h^{*}\) is the characteristic length.