## 2 Analytical Methods
### Multi-Stage and Multi-layers Penetration Modelling  
For better physical representation of soil-penetrator interaction, [PERSON] [29] developed a model of orthogonal layers or discs as an update of the cylindrical cavity expansion approach by representing the soil medium by soil discs or independent layers of small thickness (Figure 5). The so-called disc models or method of plane sections deliver more improved results over the previous models results because existing models are established on simple mechanisms that do not properly represent the penetration phenomenon and the interaction between the surrounding soil and the projectile. However, the stress field precedes the penetration front, thus, the next layer will affect the rate of penetration. \"Highly plastic volumetric deformations in the projectile nose vicinity are the  
Figure 4: Variation of penetration depth with projectile impact velocity in cohesionless soil, projectile trajectory (with and without rotation of projectile), and yaw angle (\(\alpha\)). Figure 3: Definition of angle of attack (AoA), oblique angle, yaw angle, and trajectory angle. principal mechanism that governs the penetration phenomenon\". The model is particularly useful to study the effect of projectile nose shape with projectile velocity. The disc models or model of orthogonal layers have not been further investigated by the researchers. The spherically symmetric cavity in granular media formed in response to the penetration of the projectile, from zero initial radius at constant low velocity, is assumed by [PERSON] and [PERSON] [44] to have three regions, namely cracked, plastic, and elastic. However, with the increase of velocity, the cracked region will gradually reduce and finally diminish, leaving only the plastic and elastic regions. They proposed separate mechanical properties and constitutive models for elastic-plastic and elastic-cracked-plastic region responses. The elastoplastic field dynamic expansion of a pressurized spherical cavity is due to [PERSON] and [PERSON] [68] and [PERSON] [69] for the compressible Mises solid. The researchers basically adopted material strength models as Mohr, Tresca, Mohr-Coulomb, or combined Tresca and Mohr-Coulomb yield criterion for heterogeneous materials. [PERSON] and [PERSON] [36; 70] introduce multi-stage penetration/perforation models as the major advance in impact penetration using analytical methods. They proposed five stages of penetration/perforation for a rigid projectile for ductile thin materials target, but the penetration model can be extended to semi-infinite soil media by considering three-stage penetrations, namely, near-surface penetration of projectile, bulge formation in target, and final penetration depth. \"The penetration stage continues till the plastic flow field reaches the rear of the target, at which point a spherical bulge begins to form, the uplifting of target material at the surface stops, and all target movement is in the direction of penetration\". Radially, away from the projectile, there is a transition from downward movement to horizontal movement. Once the projectile passes a given depth, the target continues to move downward, as momentum is shifted to the target. The target geometry in final stage is identified by the plastic flow in compressibility condition. A two-stage process for projectile-penetration process, cratering stage, and tunneling stage, as shown in Figure 6, is proposed by [PERSON] et al. [71] to analyze the impact response on hard target media. The cratering stage is usually approximately twice the depth of the projectile and the tunneling stage is approximately one projectile diameter in depth. [PERSON] [72] has viewed the impact penetration process as a two-stage process, transient state and quasi-steady state. The dynamics of the transient phase includes the initial impact with shallow penetration and is characterized by transient projectile-to-target interaction as crater is developed.