## 4 Validation of the Proposed Numerical Algorithm for the Inversion of Laplace Transformation
### Example 1
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Striker Rod** & **Semi-Infinite Rod** \\ \hline \multirow{4}{*}{Diameter, \(D\) (mm)} & 21.213 (\(A_{1}/A_{2}=0.5\)) & \\  & 30 (\(A_{1}/A_{2}=1\)) & 30 \\  & 42.426 (\(A_{1}/A_{2}=2\)) & \\ Young’s modulus, \(E\) (GPa) & 195 & 195 \\ Mass density, \(\rho\) (kg/m\({}^{3}\)) & 7850 & 7850 \\ Poisson’s ratio, \(\upsilon\) & 0.3 & 0.3 \\ Yield stress, \(\sigma_{0}\) (MPa) & 1275 & 1275 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the striker rod and the semi-infinite rod used in the model under the impact of different impedances. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{\(\mathbf{Z_{1}=0.5Z_{2}}\)} & \multicolumn{3}{c}{\(\mathbf{Z_{1}=Z_{2}}\)} & \multicolumn{3}{c}{\(\mathbf{Z_{1}=2Z_{2}}\)} \\ \cline{2-10} \multirow{2}{*}{**Cycle**} & **Time** & \(\mathbf{\sigma_{1}}\) & \(\mathbf{\sigma_{2}}\) & **Time** & \(\mathbf{\sigma_{1}}\) & \(\mathbf{\sigma_{2}}\) & **Time** & \(\mathbf{\sigma_{1}}\) & \(\mathbf{\sigma_{2}}\) \\  & **(μs)** & **(MPa)** & **(MPa)** & **(μs)** & **(MPa)** & **(MPa)** & **(μs)** & **(MPa)** & **(MPa)** \\ \hline
1 & 0–69.17 & \(-\)351.05 & \(-\)175.52 & 0–69.17 & \(-\)263.286 & \(-\)263.286 & 0–69.17 & \(-\)175.52 & \(-\)351.05 \\
2 & 69.17–138.34 & 117.02 & 58.508 & 69.17–138.34 & 0 & 0 & 69.17–138.34 & \(-\)58.51 & \(-\)117.02 \\
3 & 138.34–207.52 & \(-\)39.01 & \(-\)19.50 & 138.34–207.52 & 0 & 0 & 138.34–207.52 & \(-\)19.50 & \(-\)39.01 \\
4 & 207.52–276.69 & 13.00 & 6.50 & 207.52–276.69 & 0 & 0 & 207.52–276.69 & \(-\)6.501 & \(-\)13.00 \\
5 & 276.69–345.86 & \(-\)4.33 & \(-\)2.17 & 276.69–345.86 & 0 & 0 & 276.69–345.86 & \(-\)2.17 & \(-\)4.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Stress variation in rods without [PERSON]’s effect (Equations (48) and (49)) over the cycles at \(x=0\) m with different impedances in the striker rod and the semi-infinite rod under the initial impact velocity \(2v_{0}\). In Table 3 and Figure 9, the results of comparing stress wave propagation when varying the cross-sectional area of the striker rod for three cases (\(Z_{1}/Z_{2}=0.5\), \(Z_{1}/Z_{2}=1\), and \(Z_{1}/Z_{2}=2\)) are presented. The results obtained from the proposed numerical solution and the solutions with no [PERSON]'s effect reported in [PERSON] and [PERSON]'s paper [11] are in good agreement. When considering the influence of [PERSON]'s effect, the stress wave shapes oscillate around the without [PERSON]'s effect line until reaching a steady state, at which point the two results are the same. When comparing the three cases of varying cross-sectional areas, a common observation is that compressive stress is generated in the first cycle for all cases, with a significant variation observed in the second cycle. For \(Z_{1}/Z_{2}=1\), the stresses in both the striker and the semi-infinite rod are equal and amount to \(-263.286\) MPa. The stress returns to zero in the second cycle and continues to oscillate around this value. In the case of \(Z_{1}/Z_{2}=0.5\), the stress in the striker is amplified to \(-351.05\) MPa, while the stress in the semi-infinite rod is attenuated to \(-175.52\) MPa. Tensile stress appears in the second cycle, followed by compressive stress, alternating in this manner and gradually  
Figure 9: Stress wave propagation was observed at the contact surface (\(x=0\) m) with the different impedances of the striker rod (\(A_{2}=7.069\times 10^{-4}\) m\({}^{2}\), \(E_{1}=E_{2}=195\) GPa, \(v_{1}=v_{2}=0.3\), \(\rho_{1}=\rho_{2}=7850\) kg/m\({}^{3}\), \(2v_{0}=11.6\) m/s). approaching zero. This indicates that the two rods will separate after impact when \(Z_{1}<Z_{2}\). However, the constraint condition expressed by Equation (8) allows the existence of tensile stress at the interface \(x=0\). When \(Z_{1}/Z_{2}=2\), the stress in the striker rod is attenuated to \(-175.52\) MPa, and the stress in the semi-infinite rod is amplified to \(-351.05\) MPa. The stress in each cycle is compressive, decreasing gradually and approaching zero.