## 4 Experimental Results and Analyses
### Results of the Measurement  
The length L and diameter d of the sample are measured six times with tape measure and spiral micrometer respectively. After taking the average and calculating its uncertainty, the results as shown in Table 2 are obtained.  
At the same time, Table 3 shows the data of specimen elongation \(\Delta L\) and corresponding tension when increasing and decreasing force.  
As the ultrasonic frequency \(f\) is equal to 400 kHz and the room temperature \(t\) is equal to 25.6 \({}^{\circ}\)C during the test, the acoustic velocity v and wavelength \(\lambda\) of ultrasonic wave propagation in the air can be obtained as follows:  
\[\left\{\begin{array}{l}v=331.45\sqrt{1+\frac{t}{273.15}}=346.63\;\text{m/s} \\ \lambda=\frac{v}{f}=8.67\times 10^{-4}\;\text{m}\end{array}\right. \tag{8}\]  
According to Table 3, \(\overline{\overline{F_{i+3}-\overline{F_{i}}}}=54.62\) N can be obtained by using the method of successional difference.  
The average value \(\overline{E}\) of the elastic modulus of the tested samples is:  
\[\overline{E}=\frac{8 FL}{\pi d^{2}n\lambda}=\frac{81\overline{\overline{F_{i+3} -\overline{F_{i}}}}}{3\lambda\pi d^{2}}=2.029\times 10^{11}\;\text{Pa} \tag{9}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Types** & **Values (10\({}^{-2}\) m)** \\ \hline length \(L\) & 148.7 \(\pm\) 0.3 \\ diameter \(d\) & 0.0626 \(\pm\) 0.0006 \\ \hline \end{tabular}
\end{table}
Table 2: The length and diameter of the sample.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Category** & \(\Delta L\) & \begin{tabular}{c} **Tension** \\ **Increasing** \\ \end{tabular} & \
\begin{tabular}{c} **Tension** \\ **Reducing** \\ \end{tabular} &
\begin{tabular}{c} **Average** \\ **Value** \\ \end{tabular} & \(\overline{F_{i+3}-\overline{F_{i}}}(N)\) \\ \hline
1 & 0 & 21.3 & 21.3 & 21.3 & \(\overline{F_{4}-\overline{F_{1}}}=54.65\) \\
2 & 0.5\(\lambda\) & 39.5 & 39.6 & 39.55 & \(\overline{F_{4}}-\overline{F_{1}}=54.65\) \\ \hline
3 & \(\lambda\) & 57.8 & 57.8 & 57.8 & \(\overline{F_{5}}-\overline{F_{2}}=54.6\) \\
4 & 1.5\(\lambda\) & 75.9 & 76.0 & 75.95 & \(\overline{F_{5}}-\overline{F_{2}}=54.6\) \\ \hline
5 & 2\(\lambda\) & 94.1 & 94.2 & 94.15 & \(\overline{F_{6}}-\overline{F_{3}}=54.6\) \\
6 & 2.5\(\lambda\) & 112.4 & 112.4 & 112.4 & \(\overline{F_{6}}-\overline{F_{3}}=54.6\) \\ \hline \end{tabular}
\end{table}
Table 3: Elongation \(\Delta L\) and corresponding tension data.  
Take the confidence probability P = 95%, and the relative uncertainty \(U_{Er}\) and total uncertainty \(U_{E}\) of the tested samples are [37; 38]:  
\[\left\{\begin{array}{l}U_{Er}=\sqrt{\left(\frac{U_{E}}{F}\right)^{2}+\left( \frac{U_{L}}{L}\right)^{2}+\left(\frac{2U_{L}}{d}\right)^{2}+\left(\frac{U_{L}} {\lambda}\right)^{2}}=6.5\%\\ U_{E}=\overline{E}\cdot U_{Er}=0.14\times 10^{11}\text{ Pa}\end{array}\right. \tag{10}\]  
The final elastic modulus \(E\) of the sample is:  
\[E=\overline{E}\pm U_{E}=\left(2.0\pm 0.2\right)\times 10^{11}\text{ Pa} \tag{11}\]  
The results shown in Equation (11) are in good agreement with the elastic modulus of 196-206 GPa carbon steel described in literature [37]. The relative uncertainty of measurement results is mainly contributed by the measurement errors of sample diameter and micro-shape variables. On the one hand, the sample diameter \(d\) is measured by a first-level spiral micrometer with an instrument error of 0.004 mm. The main reasons for the large error are the low precision of the instrument and the uneven thickness of the sample. On the other hand, the measurement error of \(\Delta L\) is mainly caused by the low resolution of Lissajous--Figure displayed by oscilloscope and the wavelength measurement error of ultrasonic wave propagating in air.