# Stress Estimation Using the Acoustoelastic Effect of Surface Waves in Weak Anisotropic Materials
## 3 Numerical Simulations
Figure 2: Principal stress (\(\sigma_{1}\)) direction, and three-measuring directions in \(45^{\circ}\) increments, in which \(\theta\) is the angle from a measuring direction to the principal stress (\(\sigma_{1}\)) direction and \(\phi\) is the angle to the anisotropy direction, and \(V_{\theta}\), \(V_{\theta-45}\), \(V_{\theta-90}\) are surface wave velocities propagating in the \(\theta\), \(\theta-45^{\circ}\), and \(\theta-90^{\circ}\) directions, respectively. The isotropic elastic constants of aluminum in the COMSOL database, which are listed in Table 3, were used to calculate the surface wave velocity, where \(\lambda\) and \(\mu\) are Lame constants and \(l\), \(m\) and \(n\) are Murnaghan constants. The anisotropy rate \(\eta\) changed from 0 to 0.02 in 15 steps by varying the value of \(C_{11}\). That is, the anisotropy was varied up to 2%. Figure 3 shows an example of the calculation results for \(\eta\) = 0.02, and principal stress direction is coincidence with the anisotropy direction and principal stress magnitudes are \(\sigma_{1}\) = 90 MPa and \(\sigma_{2}\) = 0. The velocity distribution of the unstressed state has maximum values in the directions of \(0^{\circ}\) and \(180^{\circ}\), and a minimum value in the directions perpendicular to them. Also, it is possible to verify that the velocity distribution is slightly changed by the acoustoelastic effect when stress is applied. To demonstrate the difficulty of applying the isotropic theory to the anisotropic material first, we applied the surface wave velocity data set obtained for the Case 1A to the isotropic theory of Equations (7)-(9), prior to applying the proposed method. Figure 4 shows the stress estimation error of \(\sigma_{1}\), in which surface wave velocities in three directions of \(0^{\circ}\), \(45^{\circ}\), and \(90^{\circ}\) were used. As can be seen from the results, the stress estimation error is ridiculously large even for very weak anisotropy. Therefore, it is very difficult to estimate the stress of anisotropic materials using the isotropic theory. Next, to verify the performance of the proposed method, the surface wave velocity data set for all six conditions shown in Tables 1 and 2 was generated. Stress estimation was performed by using surface wave velocities in three directions of \(0^{\circ}\), \(45^{\circ}\), and \(90^{\circ}\). The errors between the estimated stress and given stress of \(\sigma_{1}\) were then calculated, where the results are shown in Figure 5. In all cases tested, the errors were less than 3%. In particular, in Case 1A, where all directions of anisotropy, principal stress (\(\sigma_{1}\)) and surface wave propagation are identical, no errors are generated in the stress estimation regardless of the stress. This is because Delsanto's model in Equation (17) used in this numerical simulation is consistent with our modified Equation (10). Actually, by rearranging Equation (17) for the uniaxial stress and angle conditions of Case 1A, it becomes equal to Equation (10) (see the Appendix A). Figure 3: Surface wave velocity according to the propagation direction for unstressed and stressed states in an anisotropic material. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(A\) [GPa] & \(\mu\) [GPa] & \(C_{11}\) [GPa] & \(I\) [GPa] & \(m\) [GPa] & \(n\) [GPa] \\ \hline
51 & 26 & 103 & \(-\)250 & \(-\)330 & \(-\)350 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material constant used to numerical verification. Meanwhile, in the case of A with uniaxial stress, the stress estimation errors are independent of the given stress, and only increase with the anisotropy rate. In the case of B with biaxial stress,  
Figure 4: Stress estimation errors of the isotropic theory with respect to the anisotropy rate (Case 1A). Figure 5: Stress estimation errors of the proposed method with respect to the anisotropy rate (**a**) Case 1A; (**b**) Case 1B; (**c**) Case 2A; (**d**) Case 2B; (**e**) Case 3A; (**f**) Case 3B. the stress estimation errors increase with both anisotropy and stress.