# Stress Estimation Using the Acoustoelastic Effect of Surface Waves in Weak Anisotropic Materials
## 3 Numerical Simulations  
Numerical simulations were performed to verify the stress estimation performance in typical stress states when the proposed method was applied to anisotropic materials. For this, the anisotropy of a material is assumed to be orthogonal, and the acoustic anisotropy rate (\(\eta\)) is defined as the ratio of the difference between surface wave velocity in the direction in which the wave velocity becomes maximum (\(V_{max}\)) and the surface wave velocity in the direction in which the wave velocity becomes minimum (\(V_{min}\)) to the minimum surface wave velocity (\(V_{min}\)) as follows:  
\[\eta=\frac{V_{\max}-V_{\min}}{V_{\min}} \tag{16}\]  
The numerical simulation requires three directional surface wave velocities in unstressed and stressed states. For this, the model suggested by [PERSON] et al. [8] and [PERSON] [9] was used. For the simple analysis applied in the study of [PERSON], only one elastic constant, \(C_{11}\), was changed by anisotropy, and thus Equation (17) can be applied  
\[ \begin{array}{ll}\frac{V_{\theta}-V_{R}}{V_{R}}=\frac{1}{2\mu}A_{2222}\cos^ {4}\phi C_{11}^{\prime}+&(a_{\phi 0}+\sum\limits_{k=1}^{3}\frac{a_{kk}p_{k}}{\mu}) \frac{\sigma_{1}+\sigma_{2}}{\mu}\\ &+(a_{\Delta 0}+\sum\limits_{k=1}^{3}\frac{a_{kk}p_{k}}{\mu})\cos 2\theta \frac{\sigma_{1}-\sigma_{2}}{\mu}\end{array}  \tag{17}\]  
Here, \(V_{R}\) is the surface wave velocity in the unstressed isotropic medium, and \(V_{\theta}\) is the wave velocity affected by anisotropy and the acoustoelastic effect in the \(\theta\) direction. \(\mu\), \(A_{2222}\), \(\alpha_{ok}\), \(\alpha_{\Delta k}\) (\(k\): 0, 1, 2, 3), and \(\
u_{k}\) (\(k\): 1, 2, 3) are material constants of the isotropic material, and \(C_{11}\) is the anisotropic variation of \(C_{11}\). The left side of Equation (17) corresponds to the rate of change of the surface wave velocity, the first term on the right side reflects the effect of anisotropy on the change of velocity, and the other terms refer to the contribution of the acoustoelastic effect on the change of velocity In this model, \(\phi\) is the angle between the acoustic anisotropy direction and the surface wave propagation direction, and \(\theta\) is the angle between the principal stress direction and the surface wave propagation direction, as shown in Figure 2. The acoustic anisotropy direction is defined as the direction in which the velocity of the surface wave is maximum. The three-directional measurement technique was applied using the velocities of the surface waves propagating towards the \(\theta\) direction and \(-45^{\circ}\) and \(-90^{\circ}\) to the \(\theta\) direction. Simulations were performed for the six cases of the combinations of the different angle conditions shown in Table 1 and different stress conditions shown in Table 2. Table 1 shows the three cases for setting the anisotropy direction, principal stress direction and propagation direction. Case 1 is the case where all of the anisotropy, principal stress (\(\sigma_{1}\)), and surface wave propagation directions are the same. In Case 2, the anisotropy direction and the principal stress direction are coincident, but the surface wave propagation direction is different. Case 3 is a general case where all directions are inconsistent. Table 2 shows the given stresses for the two groups. Group A is the case where only uniaxial stress (\(\sigma_{2}\) = 0) is applied and \(\sigma_{1}\) is varied among 30, 60 and 90 MPa. Group B is the case where biaxial stress is applied while fixing \(\sigma_{1}\) at 90 MPa and varying \(\sigma_{2}\) among three stresses (\(\sigma_{2}\) = 30, 60, 90 MPa). \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Case 1** & **Case 2** & **Case 3** \\ \hline \(\varnothing\) & \(0^{\circ}\) & \(30^{\circ}\) & \(60^{\circ}\) \\ \(\theta\) & \(0^{\circ}\) & \(30^{\circ}\) & \(30^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Directions of the anisotropy, principal stress, and the surface wave propagation for numerical simulation.