## 1 Introduction  
The problem of long-rod penetration has been extensively researched in the disciplines of protective engineering and armor and weapon design recently [1; 2; 3]. Unlike rigid projectile penetration, long-rod penetration will cause severe erosion at the contact area between the long-rod and the target, with a semi-fluid deformation mode. In the 1960s, [PERSON] [4] and [PERSON] [5; 6] proposed a theoretical model for the high-speed penetration of long-rods (the Alekseevskii-Tate model). This model assumes that the projectile tail remains rigid during penetration and that erosion occurs only near the contact surface between the projectile and the target. By introducing the projectile strength term \(Y_{\mathrm{P}}\) and the target resistance term \(R_{\mathrm{T}}\), the Alekseevskii-Tate model modifies the Bernoulli equation to take into account the strength properties. Subsequent studies have expanded the Alekseevskii-Tate model by considering the effective cross-sectional area [7], transient response stage [8], projectile head shape [9], target response region [10], and target material viscosity [11]. Calculating the appropriate \(R_{\mathrm{T}}\) is the main focus and challenge in the aforementioned models. Previous studies [6] have demonstrated that utilizing a cavity expansion model to derive \(R_{\mathrm{T}}\) can yield accurate results. [PERSON] [12] first proposed the cavity expansion model, which was primarily used to investigate the issue of normal penetration into semi-infinite targets. [PERSON] et al. [13; 14] conducted studies on the penetration of rigid projectiles into semi-infinite metal and concrete targets using the cavity expansion model. On this basis, [PERSON] and [PERSON] [15] investigated the shape of warheads and proposed a dimensionless head shape function. However, the penetration of finite-diameter targets violates the semi-infinite assumption in the traditional cavity expansion model, thus requiring the introduction of a finite cavity expansion model. [PERSON] et al. [16] first proposed the finite spherical cavity expansion model to study the penetration problem of layered geological materials, taking into account boundary and layering effects. Recent years have seen the finite cavity expansion model extended to metal [17] and concrete [18] targets, and applied to the study of penetration trajectories [19], perforation [20], and penetration of finite-diameter targets [21; 22]. However, the extended model still only applies to rigid projectiles and not to eroding rods directly. Experimental results [23; 24] suggest that a long-rod projectile penetrating targets of different diameters at high speed may experience a deviation of approximately 30% in penetration depth. To analyze the influence of the target diameter on high-speed long-rod penetration, [PERSON] [25] proposed the dynamic finite cylindrical cavity expansion model. It was found that the dynamic resistance is free from the target diameter. He then derived \(R_{\rm T}\) based on the average quasi-static cavity pressure instead of the semi-empirical formula in the Alekseevskii-Tate model, thus obtaining the earliest theoretical model of high-speed long-rod penetration into finite-diameter metal targets. However, it is necessary to derive the strength of the target material from the \(R_{\rm T}\) obtained from numerical simulations. Furthermore, the use of the dynamic cavity expansion model to calculate quasi-static resistance does not consider the influence of the lateral boundary expansion of the targets. [PERSON] [26] subsequently introduced the unified strength model to analyze different yield criteria. The results showed that different yield criteria can cause a maximum depth change of 15.93%. However, the selection of the yield criterion is dependent on the results of the penetration data. In summary, while there has been extensive research on the penetration of rigid projectiles, the research on the penetration of long rods into finite-diameter metal targets is insufficient.