## 4 Theoretical Analysis  
[PERSON] previously proposed a theoretical study of the LSC liner collapse based on the arc deformation of the LSC liner [12]. This theory is based on the assumption that the flat  
Figure 3: Magnified flash X-ray images of LSC jet penetration into UHMW blocks showing 75 \(\upmu\)s (**Top**), 105 \(\upmu\)s (**Middle**), 165 \(\upmu\)s (**Bottom**) after initiation (detonation proceeds left to right). Red dotted lines represent cutting/penetration line and grouping of jet particles. LSC liner will deform in an arc during projection, meaning that the collapse-point angle cannot be identical to the original apex angle. In order to determine the non-linear liner collapse-point angle \(\beta_{t}\), the following schematic diagram was used (Figure 4). Where \(\alpha\) is the original liner apex half-angle, \(\beta_{t}\) is a liner collapse-point angle, \(V_{p}\) is the liner velocity, \(V_{py}\) is the vertical component of the liner velocity, \(h\) is half of the LSC base width, \(r(t)\) is the radius of the flyer liner arc, and \(R(t)\) is the vertical height of the arc center. Based on the diagram above, the following collapse-point angle equation is derived [12]. \[\beta_{t}=cos^{-1}\bigg{(}\frac{R(t)-V_{py}t}{r(t)}\bigg{)}\]  
In order to achieve the arc deformation of the flat liner, [PERSON] used a hydrocode simulation to realize the arc deformation. Later, the flat liner arc deformation was experimentally measured, and it showed favorable agreement with the hydrocode simulation in unpublished data [13,14]. According to the arc-deformed liner collapse theory, upon detonation, the two sides of the flat liner start moving toward each other to collapse. Due to the strong release of detonation gas along the edges of the liner, the edge of the liner moves slower than the center of the liner, and this causes an arc deformation [12, 13, 14]. This gradual arc deformation, followed by the collapse of the two arc-deformed liner sides, the collapse- point angle varies in a unique trend (Figure 5). As is schematically illustrated in Figure 5 above, when the two arc-deformed liners approached each other in collapse, the collapse-point angle initially decreased (second from left, Figure 5), but as the collapse progressed the collapse-point angle gradually increased. Ultimately this angle approached a point where the formation of the jet was no longer effective (right, Figure 5). The collapse-point angle can thus be divided into two areas, depending on whether it is larger or smaller than the original apex angle, or favorable or non-favorable for jet  
Figure 4: Schematic diagram of arc-deformed LSC liner collapse [12]. Figure 5: LSC liner collapse progress with arc deformation [13]. formation. This is an important characteristic of the LSC jet; the collapse point angle is an important parameter that controls the jet property [4]. The gradual change of the collapse- point angle provides a clue to the jet grouping described in the previous section. Jet speed variation will control the jet stretching, affecting particulation (high-/low-density jet), and eventually provides a clue to the difference in jet segment density in Jet Groups A and B. In order to illuminate the relationship of jet velocity variation and collapse-point angle change, the arced liner radius and calculated parameters were tabulated, and the following equations from the original Birkhoff theory were used to calculate the jet velocity (Figure 4), [12]. \[V_{1}=\frac{V_{p}}{sin\beta_{t}}sin\left\{90-\left(\frac{\beta_{t}-\alpha}{2} \right)\right\},\text{ and }V_{2}=V_{p}\left\{\frac{cos\left(\frac{\beta_{t}-\alpha}{2} \right)}{tan\beta_{t}}+sin\left(\frac{\beta_{t}-\alpha}{2}\right)\right\}\]  
where \(V_{j}=V_{1}+V_{2}\), and \(V_{s}=V_{1}-V_{2}\). The following data were collected from [PERSON] [12] (Table 2, Figures 6 and 7). Figure 6 shows a sudden change of the liner arc radius at around 4-5 \(\upmu\)s after detonation.