## 2 Analytical Methods
### Granular Target Media  
The one the important and most researched soil media in impact penetration is granular or non-cohesive materials. The granular material behavior is quite different than the other penetrating soil materials. The strength and stiffness of granular material is dependent on scale, confining pressure, and fluid pressure within the void spaces. These materials are characterized by rigid plastic deformation and dilatancy during shear loading. The loading behavior of granular material depends on loading history and initial arrangement of the particles. The densification of granular system during the penetration process is incorporated in analytical model densification by adjusting the density or locking density in terms of volumetric strain, which depends on initial (bulk) target density and locked density or maximum elastic density. Length scale of penetration problem in granular media is selected based on whether to resolve the bulk and grain scale behaviors of penetration process. An extensive review of impact mechanics of granular material is due to [PERSON] et al. [46]. [PERSON] et al. [47]  
Figure 1: Description of soil behavior regions during cavity expansion. developed a predictive model for the process of penetration under quasi-static conditions using macroscopic laws or [PERSON] drag force theory, defining the drag force as a hybrid of viscous force and hydrostatic-like force. The viscous force emerges from the transfer of momentum between the grains and is proportional to the square of the projectile's instantaneous speed with reference to the granular medium. The hydrostatic-like force stems from the frictional plasticity of the granular matter and relies on the depth of the penetration. The thermal effect on projectile drag force during impact in granular medium is evaluated by [PERSON] and [PERSON] [PERSON] and [PERSON] (2018). [PERSON] et al. (2018) presents an analytical approach-based analysis of the influence of the basic properties of target (brittle materials) and geometrical factors of target and projectile, on formation of penetration resistance. [PERSON] (2018) proposed a steady-state hydrodynamic theory for analysis of penetration process including the effects of the projectile and target material strengths. The analytical models are also developed to predict depth of penetration as a function of impact velocity, soil moisture, and essential geometric and material properties of projectiles. [PERSON] (2018) has carried out investigation of projectiles penetration phenomenon in non-cohesive soil (Sand) and non-cohesive targets under dry, saturated, and compacted conditions. The results for soil (Sand) targets under dry, saturated, and compacted conditions with conical nose projectile penetration are depicted in Figure 2. The saturated condition of soil media reduces the projectile penetration depth. The error in the prediction of penetration depth for the projectiles of different nose shapes in different soil targets using the proposed analytical models lie within 10% when compared with experimental results. In shallow penetration, the granular medium dissipates much of its energy in the initial phase of penetration, while in deep penetration, the compression and shear of the granular medium result in energy dissipation [PERSON] et al. (2018). The absorption of energy in the shear is primarily due to friction and volume changes as the grains slip, roll, and climb over each other. The kinetic energy of impactor transferred to target during the penetration process is dissipated in form of tearing energy of strips or layers, total energy in soil volume change, frictional energy, motion energy of target media, and thermal energy. A projectile's infinite penetration into light-weighted \"expanded polystyrene particles (EPP)\" is detected when the projectile's weight exceeds a critical value. The role of initial velocity with various impact velocities in infinite penetration into a light granular material consist of EPP particles media is investigate by [PERSON] et al.