## 4 Results and Discussion
### Stress Measurement  
Stress measurement tests were applied on the #A-4, #A-5, and #A-6 specimens and #B-4, #B-5, and #B-6 specimens. The calculated internal stress, using the transverse waves with polarization direction vertical to stress, was noted as \(\sigma_{vc}\), while the calculated internal stress, using the transverse waves with polarization parallel to stress, was noted as \(\sigma_{pc}\). The internal stresses \(\sigma_{vc}\) and \(\sigma_{pc}\) can be calculated based on Equation (3), in which the acoustoelastic coefficients had been calculated in Section 4.4 and the wave velocities of #A-4, #A-5, #A-6, #B-4, #B-5, and #B-6 specimens were measured using the same method as for #A-1, #A-2, #A-3, #B-1, #B-2, and #B-3 specimens. The average calculating results for \(\sigma_{vc}\) and \(\sigma_{pc}\) within \(-100\) MPa to \(50\) Mpa mechanical stress are shown in Figure 12, where the mechanical compressive values were equivalent to the actual internal stress \(\sigma_{a}\). The calculated stress \(\sigma_{lc}\) of the longitudinal waves method investigated in a previous study is also compared in Figure 12[2]. Then, the average relative error of calculated internal stress (the relative error for \(\sigma_{vc}\) was noted as \(E_{vr}\), and the relative error for \(\sigma_{pc}\) was noted as \(E_{pr}\)) can be found by comparing between the calculated internal stress (\(\sigma_{vc}\) and \(\sigma_{pc}\)) and the actual internal stress \(\sigma_{a}\), as demonstrated in Figure 12. The relative error \(E_{lc}\) of longitudinal waves method investigated in a previous study is also compared in Figure 13[2].  
In Figure 13, it can be seen that the \(E_{vr}\) varies in different stress states: the absolute values of \(E_{vr}\) can reach almost \(20\)-\(35\%\) in the stress range of \(-40\) MPa-\(30\) MPa; the absolute values of \(E_{vr}\) decrease to lower than \(12\%\) in the range of \(-100\) MPa-\(50\) MPa and \(35\) MPa-\(50\) MPa. The maximum absolute value of \(E_{vr}\) is \(35.4\%\) under the stress of \(20\) MPa. It can be seen that the \(E_{pr}\) varies in different stress states as well: the absolute values of \(E_{pr}\) can reach almost \(10\)-\(26\%\) in the stress range of \(-40\) MPa-\(30\) MPa; the absolute values of \(E_{pr}\) also decrease to lower than \(12\%\) in the range of \(-100\) MPa--\(50\) MPa and \(35\) MPa-\(50\) MPa. The maximum absolute value of \(E_{pr}\) is \(26.3\%\) under the stress of \(-10\)MPa. Comparing \(E_{pr}\) and \(E_{pr}\), it can be found that both of them are relatively large in the low-stress state and become smaller in the high-stress state. Moreover, \(E_{pr}\) is smaller than \(E_{vr}\) in the low-stress states, while no significant difference between them is observed in the high-stress states.  
Figure 12: The calculated internal stress \(\sigma_{vc}\), \(\sigma_{pc}\), and \(\sigma_{lc}\).  
Comparing \(E_{vr}\), \(E_{pr}\), and \(E_{lr}\) in the range of \(-\)70 MPa-0 MPa, it can be found that in low-stress states of \(-\)30 MPa-0 MPa, the absolute value of \(E_{lr}\) is obviously largest (85-37%), followed by \(E_{vr}\) (11-32%) and \(E_{pr}\) (17-26%); in high-stress state, there is no significant difference among them. The reason for the large difference in results errors in the low-stress state maybe due to the deviation in acoustoelastic coefficients, which indicate the stress-velocity sensitivity. A larger acoustoelastic coefficient and higher wave stress-velocity sensitivity suggest larger variation in the wave velocity for the same stress variation. This may be beneficial for stress measurement in low-stress states, as it is easier to detect the velocity variation. In this research, the acoustoelastic coefficient of the transverse waves polarized parallel to stress is the largest, indicating the highest velocity-stress sensitivity, smallest stress measurement relative error, and highest accuracy in the low-stress states.