## 3 Results and Discussion
### Stress Transfer Mechanism  
#### 3.4.1 Long Flange (\(L_{\rm f}>L_{\rm s}\))  
The schematic of the three waves when the flange is long (\(L_{\rm f}>L_{\rm s}\)) is shown in Figure 4. It depicts that the arrivals of the \(\bf R_{n}\) waves are delayed as the \(\bf F\) wave travels a longer distance. Consequently, an overly long flange length certainly leads to separation of the succeeding \(\bf R_{n}\) wave from the preceding \(\bf B\) wave. According to the one-dimensional theory [52; 55], the stress pulse length is \(2L_{\rm s}\) (600 mm) and the travel distance in the flange structure is \(2L_{\rm f}\). Therefore, if \(2L_{\rm f}>2L_{\rm s}\) (\(L_{\rm f}>L_{\rm s}\)), then the \(\bf F\) and \(\bf R_{1}\) waves will separate. In view of this analysis, the long flange length in this study refers to the case where \(L_{\rm f}>L_{\rm s}\). For the case where \(D_{\rm b}=10\) mm (Figure 4a), the \(L_{\rm s}\) value (300 mm) is \(30D_{\rm b}\). Then, a flange length (\(L_{\rm f}\)) of only \(30D_{\rm b}\) results in the juxtaposition of the \(\bf B\) and \(\bf R_{1}\) pulses, as shown in Figure 4a, where the \(\bf B\) and \(\bf R_{1}\) pulses for \(L_{\rm s}=30D_{\rm b}\) (green curve) are on the verge of separation. When \(D_{\rm b}=20\) mm (Figure 4b,c), the \(L_{\rm s}\) value (300 mm) is \(15D_{\rm b}\). Then, when \(L_{\rm f}=30D_{\rm b}\) (green curve), two fully separated pulses occur (Figure 4b,c). Next, the magnitudes of the \(\bf B\) and \(\bf R_{n}\) waves separated in the time axis are calculated. Note that in Figure 4, the \(\bf B\) and \(\bf F\) waves simultaneously form and propagate in opposite directions. To obtain the proportion at which the impact-generated stress separates into \(\bf B\) and \(\bf F\) waves, the overall magnitude of the impact-generated stress itself should first be determined. For this purpose, a one-dimensional equation [52] that was derived by considering the impact of a cylinder striker on a bar in the compression mode is employed:  
\[\sigma_{\rm b}=\frac{Z_{\rm s}}{Z_{\rm s}+Z_{\rm b}}\rho_{\rm b}C_{\rm b}V, \tag{4}\]  
where \(\sigma\) is the stress, \(\rho\) is the density, \(C\) is the sound of speed, \(z\) is the general impedance (\(A\rho C\)), \(V\) is the impact velocity, and the subscripts b and s denote the bar and striker, respectively. The magnitudes of the impact-generated stress obtained using Equation (1) when \(A_{\rm s}/A_{\rm b}=1.0\) (Figure 4b) and 0.5 (Figure 4c) are 303.24 and 202.36 MPa, respectively, as indicated by the dash-dot horizontal lines. The vertical positions of the drawn lines are consistent with the plateau height of the first tensile pulse of the case where \(L_{\rm f}=D_{\rm b}\). To estimate the proportions of the \(\bf B\) and \(\bf F\) waves in the impact-generated stress magnitude, the plateau heights of the \(\bf B\) wave in the green curves (\(L_{\rm f}=30D_{\rm b}\)) in Figure 4 are used as a reference (at this point in this discussion, the numbers marked for the plateau of the \(\bf B\) wave are unknown; they are marked in the latter part of this paragraph). In Figure 4, the plateau heights of the \(\bf B\) waves appear to be approximately half of the first tensile pulse when \(L_{\rm f}=D_{\rm b}\). Based on this observation, the impact-generated stress by the striker--the magnitude of which can be calculated using Equation (4)--is hypothesised to be equally divided between the \(\bf B\) and \(\bf F\) waves. Under this assumption, the calculated magnitudes of the \(\bf B\) and \(\bf F\) waves are 151.77 (\(A_{\rm s}/A_{\rm b}=1.0\); Figure 4b) and 101.18 MPa (\(A_{\rm s}/A_{\rm b}=0.5\); Figure 4c), respectively. The calculated half-magnitudes based on this hypothesis are marked on the first pulse (\(\bf B\) wave) of the green curves (\(L_{\rm f}=30D_{\rm b}\)) using dash-dot horizontal lines.