## 2 Analytical Methods
### Classification of Methods
[40] found that the effects of strain rates and remolding on the shear strength of the soil could not be considered for predicting the soil behavior of cohesive soil using the modified cam-clay (MCC) model. [PERSON] et al. [15] has found that the frictional bearing resistance is augmented during high-velocity penetration due to the combination of high strain rates and negative excess pore pressures. This is the most critical aspect to be implemented in modelling tools as mechanism by which the frictional bearing resistance contributes to the total penetration resistance is not well understood. The cavity expansion theory (CET) model was developed for improvement over previous models, with the assumption that when an impactor penetrates the target medium it creates a cavity, which give rise to the stresses on the neighborhood of the cavity. The developed stress in vicinity of cavity depends on constitutive model of target media representing various characteristics such as compression, shear, dilation, fracture, and other relevant properties. This cavity expands under the action of stresses generated into the target medium. The CET models for the projectile-penetration process completely describe the instantaneous local interaction between a target material and a projectile [28; 41; 42; 43]. The model proposes a relation between expansion of a cavity and projectile penetration in target material. CET model can be used to describe completely the projectile-penetration process, such as depth of penetration, deceleration-time history of projectile, time histories of velocity, stresses in the target material, stress variation with depth and radial distance, and forces at the projectile nose [44; 45]. The cavity expansion theory models are classified into three types based on description of cavity surface dynamics. Static (or sometimes called \"quasi-static\") models describe the stresses on a cavity's surface in static state. \"Quasi-dynamic models\" evaluate cavity surface stresses because of the constant rate of increase in cavity radius. Dynamic models consider cavity surface acceleration. An extensive survey of the research on penetration cavity expansion theory is summarized in the monograph of [PERSON] [28]. [PERSON] and [PERSON] [41] proposed a model of dynamic cylindrical cavity expansion (CCE) that assumes cavity expansion results in the formation of cracked, elastic, and commintuted response regions. In the CCE dynamic model, vertical penetration of a slender body of revolution (using cylindrical coordinates) is typically considered, and it is presumed that particles of the target material travel in the radial direction during projectile penetration. The target material is treated as a series of independent thin layers, and the cavity expansion induced by moving projectiles is modelled in each layer. This method enables the measurement of stress at the boundary of the hole in each layer and, thereby, the force acting on the projectile at every location on the projectile's lateral surface. The theory model of \"spherical cavity expansion (SCE)\" is commonly used in quasi-dynamic projectile-target interactions when a spherically symmetrical cavity expansion from a zero initial radius is at continuous velocity. The expansion cavity problem of penetration considering spherical symmetry conditions in compressible material under dynamic conditions divides the target material into three regions: Undisturbed, elastic, and elasto-plastic. The response of target material into different behavior regions is depicted in Figure 1. The boundary conditions for this model are radial stress continuity and plastic yield at the elastic/plastic interface and continuous speed at the cavity wall. [PERSON] et al. [42] compared the Forrestal model with the cylindrical cavity expansion penetration model [43] and concluded that gap between the prediction of these two models is minimal for impact velocity below 100 m/s and the discrepancy increases with the increase in impact velocities.