## 2 Analytical Methods
### Projectile Non-Normal Impact and Stability  
The obliquity of projectile and final penetration depth in target media affects the response of projectile penetration process. The obliquity introduces a non-axisymmetric resistance on the projectile surface. Non-normal impact is defined by \"angle of attack\" (AoA) and angle of obliquity. The angle of obliquity is the angle of the velocity vector and the vertical to the target surface, while AoA is the angle between instantaneous velocity vector and projectile axis of symmetry. A projectile can strike a target with oblique, pitch, and/or yaw angles and previous studies on the projectile trajectory in the soil target demonstrate that the projectile trajectory with an initial oblique angle could become curvilinear and could even travel towards the target surface [58]. The definitions of non-normal impact penetration are graphically shown in Figure 3. The study of non-normal penetration using the simplified analytical model of penetration with lateral loading (SAMPLL), which employs empirically based algorithms to treat the lateral loads resulting from angle of attack and trajectory angle of the projectile, is carried out by [PERSON] [59]. Ricochet phenomenon, associated with shallow penetration, occurs when the projectile's final velocity vector is directed away from the target and there is a threshold angle of impact, critical impact angle, above which ricochet cannot occur. Ricochet takes place in three stages, i.e., the entry phase, plough phase, and exit phase, and is affected by geometry, obliquity angle, impact velocity, mass, and target properties [60]. The critical impact angles for aluminum and lead spheres striking sand media, predicted by analytical and experimental procedures, are 23 and 52\({}^{\circ}\), respectively. [PERSON] [61] developed the analytical models for the penetration of targets vertically struck by projectiles using the formulation that assumes the average resistance of the target material consisting of cohesive static resistive pressure resulting from the elastic-plastic deformation of the target materials and the dynamic resistive pressure resulting from the speed effects. Comprehensive information on non-normal effects is limited, generally as it involves additional parameters that go beyond those essential for vertical impacts. [PERSON] [62] has summarized the research work on analytical modelling of non-ideal projectile impact on various targets. The effect of projectile trajectory and yaw angle on penetration in soil target studied by [PERSON] [62] is portrayed in Figure 4. Under certain circumstances, free surface effects may become important, such as long-rod projectile penetration, brittle target medium, large oblique angles of impact, and shallow penetration [63]. The analysis of general penetration dynamics problem including the non-normal penetration is carried out by [PERSON] et al. [64] considering the impact penetration as an inverse problem using semi-empirical relations that employ the dependencies between the initial velocity and the depth of penetration. The deep penetration study of hard projectile in soil target by [PERSON] and [PERSON] [65] concluded that the trajectory instability occurs at critical impact velocity, and the projectile trajectory returns to a straight line (linear motion) from the curvilinear motion near the characteristic convergent velocity. The trajectory instability and convergence of the curvilinear motion of projectile is driven by increase in impact velocity, projectile geometry, uniformity of grain size distribution within the soil mass, increase in grain size and porosity, and other striking parameters [66]. Analytical model for rigid projectile dynamics is proposed by [PERSON] and [PERSON] [67] to analyze the trajectory and stability of the projectile during deep penetration in an elasto-plastic medium.