# Stress Estimation Using the Acoustoelastic Effect of Surface Waves in Weak Anisotropic Materials
## 5 Conclusions  
This study proposed a method to estimate stress in anisotropic materials, which involved modification of the equation for the isotropic theory. In this method, the ratios of the surface wave velocities in the stressed state to those in the stress-free state in three directions were utilized, enabling simultaneous estimation of the principal stresses and their directions. Numerical simulations and experiments were conducted for verification. The results of the numerical simulation showed that the stress estimation error occurred at a very small order when the acoustic anisotropy rate was less than 2%. Experiments were performed on hot-rolled and extruded aluminum plates that had different anisotropy rates of, 0.05% and 2.6%, respectively. The stresses applied by a tensile tester were estimated by measuring the ratios of the surface wave velocities in the stressed state to those in the unstressed state in three-directions. The estimated results showed good agreement with the given stresses in the range from 30 to 90 MPa. These results demonstrate that the stress estimation method proposed in this study can effectively estimate the stress in weakly anisotropic materials. The proposed method is simple, practical and is expected to be useful for monitoring changes in surface stress before and after machining such as punching or bending of plates. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Given stress [MPa]** & 24.95 & 33.24 & 41.50 & 49.78 & 58.06 & 66.34 & 74.61 \\
**Estimated stress [MPa]** & 21.56 & 33.43 & 42.20 & 48.58 & 53.87 & 61.57 & 69.89 \\
**Stress estimated error [MPa]** & 3.39 & \(-\)0.19 & \(-\)0.70 & 1.20 & 4.19 & 4.77 & 4.72 \\
**Estimated angle \(\theta\)**[\({}^{\circ}\)]** & 32.96 & 34.02 & 34.94 & 33.03 & 34.98 & 34.58 & 34.30 \\
**Angle Estimated Error [\({}^{\circ}\)]** & \(-\)2.96 & \(-\)4.02 & \(-\)4.94 & \(-\)3.03 & \(-\)4.98 & \(-\)4.58 & \(-\)4.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Estimated results and errors for the hot-rolled specimen (SB.2). Figure 14: Stress estimation results (**a**) Hot-rolled specimen (SA.2) and (**b**) Extruded specimen (SB.2). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Given stress [MPa]** & 30.40 & 40.48 & 50.57 & 60.66 & 70.75 & 80.83 & 90.92 \\
**Estimated stress [MPa]** & 28.90 & 36.42 & 47.01 & 61.46 & 66.54 & 77.82 & 85.78 \\
**Stress estimated error [MPa]** & 1.50 & 4.06 & 3.56 & \(-\)0.80 & 4.21 & 3.01 & 5.14 \\
**Estimated angle \(\theta\)**[\({}^{\circ}\)]** & 23.39 & 25.41 & 26.18 & 25.35 & 20.17 & 26.23 & 27.73 \\
**Angle Estimated Error [\({}^{\circ}\)]** & 6.61 & 4.59 & 3.82 & 4.65 & 9.83 & 3.77 & 2.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Estimated results and errors for the hot-rolled specimen (SA.2). **Author Contributions:** Conceptualization, J.J. and Y.-D.S. ; Methodology, J.J. and Y.-D.S. ; Software, Y.-D.S. ; Validation, J.J., Y.-D.S. and K.-Y.J. ; Formal Analysis, J.J.; Investigation, J.J. and Y.-D.S. ; Resources, K.-Y.J. ; Data Curation, Y.-D.S. ; Writing-Original Draft Preparation, J.J.; Writing-Review and Editing, K.-Y.J. ; Visualization, Y.-D.S. ; Supervision, K.-Y.J. ; Project Administration, K.-Y.J. ; Funding Acquisition, K.-Y.J. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was supported by the Nuclear Power Research and Development Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2013M2A2A9043241). **Conflicts of Interest:** The authors declare no conflict of interest.