## 3 Results and Discussion
### Equivalent Stress--Equivalent Plastic Strain Behaviour  
The values of equivalent von Mises stress and plastic strain obtained from numerical simulations were compared with the analytical solutions based on the strain rates which were obtained and presented in Figure 3. It is important to note that, as the strain rates presented in Figure 3 were not constant during plastic flow, the ranges of plastic strain rates discussed in Section 3.1 were selected for use in this section. The ranges selected from Figure 3 are shown in Table 4.  
Figure 5: Schematic illustrations of the parameters of the deformed Taylor impact test model. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Samples** & **C** & & **D** & & **E** \\ \hline Prescribed velocity & \(\mathrm{R_{f}/R_{o}}\) & \(\mathrm{L_{f}/L_{o}}\) & \(\mathrm{R_{f}/R_{o}}\) & \(\mathrm{L_{f}/L_{o}}\) & \(\mathrm{R_{f}/R_{o}}\) & \(\mathrm{L_{f}/L_{o}}\) \\
130 m/s & 1.356 & 0.843 & 1.371 & 0.782 & 1.362 & 0.801 \\
200 m/s & 1.630 & 0.710 & 1.662 & 0.667 & 1.647 & 0.664 \\
250 m/s & 1.874 & 0.612 & 1.897 & 0.581 & 1.878 & 0.600 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of deformed model parameter ratio for samples C, D, and E.  
To evaluate the capability of the numerical model to generate results equal to those obtained from analytical solutions, the instantaneous and average absolute errors at values of plastic strain up to a value of 0.4 were determined using the following expressions:  
\[\delta_{i}=\left|\frac{R_{A}-R_{N}}{R_{A}}\right| \tag{2}\]  
\[\delta_{m}=\frac{1}{N}\sum_{i=1}^{N}\left|\frac{R_{A}-R_{N}}{R_{A}}\right| \tag{3}\]  
where the symbol \(N\) denotes the total number of data points employed in the analysis, \(R_{A}\) and \(R_{N}\) denote the analytical and numerical values of flow stress, respectively, while \(\delta_{i}\) and \(\delta_{m}\) represent the obtained values of instantaneous and average absolute errors. The results obtained using these two equations are summarised in Table 5.  
Figure 8: Comparison of the equivalent von Mises stress and plastic strain curves for samples E obtained by numerical modelling at three imposed velocities, with those obtained from analytical solutions within the range of strain rates obtained in preceding section, under each simulation velocity. \begin{table}
\begin{tabular}{c c c c c c} \hline  & & & **Samples C** & & \\ \hline \multirow{3}{*}{Velocity} & \(|R_{A}-R_{N}|\) & \multicolumn{4}{c}{Instantaneous absolute error (\(\delta_{i}\))} & \multirow{3}{*}{Average absolute error (\(\delta_{m}\))} \\  & Strain & 0.1 & 0.2 & 0.3 & 0.4 & \\  & \(|R_{A1}-R_{N}|\) & 0.03 & 0.01 & 0.02 & 0.01 & \\  & \(|R_{A2}-R_{N}|\) & 0.01 & 0.03 & 0.02 & 0.03 & \\  & \(|R_{A1}-R_{N}|\) & 0.03 & 0.02 & 0.03 & 0.03 & \\  & \(|R_{A2}-R_{N}|\) & 0.05 & 0.01 & 0.05 & 0.01 & \\  & \(|R_{A1}-R_{N}|\) & 0.01 & \textless{}0.01 & 0.01 & 0.03 & \\  & \(|R_{A2}-R_{N}|\) & \textless{}0.01 & 0.03 & \textless{}0.01 & \textless{}0.01 & \\ \hline \multicolumn{6}{c}{Samples D} \\ \hline  & 0.1 & 0.2 & 0.3 & 0.4 & \\
130 m/s & \(|R_{A1}-R_{N}|\) & 0.004 & 0.02 & 0.03 & 0.02 & \\
130 m/s & \(|R_{A2}-R_{N}|\) & 0.04 & 0.02 & 0.01 & 0.02 & \\
200 m/s & \(|R_{A1}-R_{N}|\) & 0.02 & 0.01 & 0.03 & 0.03 & \\
200 m/s & \(|R_{A2}-R_{N}|\) & 0.05 & 0.05 & 0.03 & 0.03 & \\  & \(|R_{A1}-R_{N}|\) & 0.03 & 0.04 & 0.06 & 0.06 & \\
250 m/s & \(|R_{A2}-R_{N}|\) & 0.02 & 0.01 & 0.01 & 0.05 & \\ \hline \end{tabular}
\end{table} 
Table 5: Instantaneous and average absolute errors of the numerical solutions based on the analytical results at different imposed velocities for the three different forms of LPBF Ti6 Al4V (ELI). It is evident from the data presented in this table that the instantaneous and average absolute errors between the numerical values and analytical values of the model are generally less than 5%, which is an acceptable range.