## 4 Validation of the Proposed Numerical Algorithm for the Inversion of Laplace Transformation
### Investigation of Stress Wave Propagation in a Semi-Infinite Rod at \(x=0.5\) m
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{3}{*}{**Cycle**} & \multicolumn{3}{c}{\(Z_{1}=0.5Z_{2}\)} & \multicolumn{3}{c}{\(Z_{1}=Z_{2}\)} & \multicolumn{3}{c}{\(Z_{1}=2Z_{2}\)} \\ \cline{2-9}  & \multirow{2}{*}{**Time (\(\mu\)s)**} & **\(\sigma_{2}\)** & **Time (MPa)** & **\(\sigma_{2}\)** & **Time (MPa)** & **\(\sigma_{2}\)** \\ \cline{3-9}  & & **(\(\mu\)s)** & **(MPa)** & **(\(\mu\)s)** & **(MPa)** & **(\(\mu\)s)** & **(MPa)** \\ \hline \multirow{3}{*}{1} & 86.46–155.64 & \(-\)175.52 & 86.46–155.64 & \(-\)263.286 & 86.46–155.64 & \(-\)351.05 \\  & 2 & 155.64–224.81 & 58.51 & 155.64–224.81 & 0 & 155.64–224.81 & \(-\)117.02 \\ \cline{1-1}  & 224.81–293.98 & \(-\)19.50 & 224.81–293.98 & 0 & 224.81–293.98 & \(-\)39.01 \\ \cline{1-1}  & 293.98–363.15 & 6.50 & 293.98–363.15 & 0 & 293.98–363.15 & \(-\)13.00 \\ \cline{1-1}  & 363.15–432.32 & \(-\)2.167 & 363.15–432.32 & 0 & 363.15–432.32 & \(-\)4.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Stress variation in rods without [PERSON]’s effect (Equation (49)) over the cycles at \(x=0.5\) m with different impedances in the semi-infinite rod under the initial impact velocity \(2v_{0}\). analysis in Abaqus [20], where \(L_{\rm min}\) is the smallest element dimension in the mesh and \(c\) is the wave speed in the material of the rod, ensuring that the simulation remains stable and accurate. Since the rods are axisymmetric, resulting in no friction force in the interaction of this model, the contact interaction between bodies is defined as frictionless in the Abaqus contact model. Figure 13 demonstrates the accuracy of the proposed numerical solution in simulating displacement, velocity, and stress in a rod, as shown by its comparison to FEM results. Particularly, in axial displacement, the proposed solution aligns closely with the FEM, indicating high accuracy in predicting displacement behavior over time. Despite minor discrepancies in velocity oscillations, the overall trend closely matches that of the FEM, underscoring the method's reliability. The stress response also shows a strong correlation with the FEM standard, solidifying the proposed method's potential as a valuable tool for dynamic stress analysis. In addition to its comparison with the FEM, the proposed numerical solution demonstrates considerable consistency with [PERSON] and [PERSON]'s solution [11], which--for Poisson's ratio--is equal to zero. This agreement also confirms the accuracy of the proposed numerical solution, reinforcing its potential as a reliable method for understanding the impact dynamics of rods with varying cross-sectional areas. Figure 11: Comparison of stress wave propagation in a semi-infinite rod observed at \(x\) = 0.5 m under the impact of different impedances of the striker rod (\(Z_{2}\) = 32,087.204 kg/s, \(A_{2}\) = 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, \(L\) = 0.2 m). In Figure 14, when considering the cases with the same impedance but varying cross-sectional areas \(A_{1}\) and \(A_{2}\), as well as differing \(\rho_{1}c_{1}\) and \(\rho_{2}c_{2}\), we can observe that the generated stress wave undergoes only one cycle, yet the duration of the stress signal varies. Figure 12: The 3D finite element model under the initial impact velocity \(2v_{0}\) using solid elements in Abaqus. Figure 13: Comparison of stress wave propagation in the semi-infinite rod observed at \(x=0.5\) m under the impact of the striker rod with different impedances by varying the cross-sectional area (\(Z_{2}=32,087.204\) kg/s, \(A_{1}=2A_{2}=2\)\(\times\)\(7.069\)\(\times\)\(10^{-4}\) 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, \(L=0.2\) m). The results of no Poisson’s effect are determined based on the research by [PERSON] and [PERSON] [11]. It is evident that as the modulus of elasticity \(E_{1}\) increases, the duration of the stress signal decreases.