## Nomenclature  
\begin{tabular}{l l} Symbol & Description \\ \(A\), \(B\) and \(C\) & Model Constant (\(A\)/\(C\)\(v\) is static target resistance term, \(B\)\(v^{2}\) is drag term), \\ \(A\) & Projectile Area \\ \(B_{max}\) & Maximum hardness at crater bottom \\ \(D_{10}\)\({}^{\bullet}\) & effective grain size of soil, \\ \(E\), & Material Elastic modulus, \\ \end{tabular}  
\begin{tabular}{l l} \(F\) & net resistant force \\ \(F_{1}\), \(F_{2}\), \(F_{3}\) & respectively dissipative, dynamic, and static components of resistant force \\ \(F_{j}^{\prime}\) \(\&\)\(F_{\frac{j}{s}}^{j}\) & Normal and tangential forces acting on particle \(i\) by contacting particle \(j\), respectively; \\ \(\bar{F}\) and \(\bar{M}\) & Resultant force and moment on the projectile, \\ \(G\) & gravity acceleration \\ \(l_{i}\), and \(\omega_{i}\) (\(\omega\)) & Moment of inertia and angular velocity of particle \(i\) \\ \(K_{\phi}\) & coefficient (\(>\)1) \\ \(L\), \(D\) & Projectile length, Diameter \\ \(M_{R}^{i}\) & Rolling moment; \\ \(M_{T}^{i}\) & Twisting moment \\ \(P\) (=\(\rho_{s}\)\(g\)\(h\)) & hydrostatic-like pressure \\ \(P_{r}\) & Pressure \\ \(P_{st}\) & term of static resistance; \\ \(R_{t}\) & Target resistance to penetration \\ \(R(t)\), \(\bar{R}\) (\(t\)), \(\bar{R}\) (\(t\)) & Disc internal boundary radial values of displacement, velocity, and acceleration \\ \(S\) & Soil penetrability number representing soil properties, \\ \(Y_{p}\) & dynamic flow stress of projectile, \\ \(a\) & Hole radius, \\ \(c\) & cohesion of soil \(c\), \\ \(d\) & Penetration depth of a cone, \\ \(dS\) & Infinitesimal area element pointing normal to the intruder surface, \\ \(h\) & Heaviside function, \\ \(l_{1}\), \(l_{2}\) & first and second characteristic length \\ \(k_{\phi}\) & coefficient (\(>\)1) \(\alpha\) to the internal friction coefficient \(\varphi\), \\ \(m\) (\(m_{i}\)) and \(v\) (\(v_{i}\)) & mass and velocity of projectile (particle), \\ \(p(t)\) & Interaction pressure, positive in compression; \\ \(s\) & Extent of plastic zone in projectile, \\ \(t\) & Rise time of impact, \\ \(z\) & unit vector in z-direction, \\ \(\sigma_{y}\) & yield stress \\ \(\alpha\) and \(\beta\) & nose geometry factors \\ \(\rho_{s}\) & packing density of the granular matter \\ \(\eta\) & a parameter dependent on grain properties; \\ \(\rho_{p}\) & Density of projectile, \\ \(ac\) & [1+ (\(\eta/R_{c}\))] \\ \(\tau_{o}\) & principal stress difference at failure; \\ \(\lambda\) & Projectile aspect ratio \\ \(\varepsilon_{L}\). & volumetric locking strain; \\ \(\rho_{o}\) & initial density; \\ \(\rho_{L}\) & current locking density; \\ \(\rho\), & Initial or current \\ \(\rho_{*}\) & max. density reached during loading. \\ \(\sigma_{r}\), \(\sigma_{\theta}\) & radial and hoop components of the Cauchy stress \\ \(\bar{\sigma}\) & Angular momentum vector about the mass centre of projectile frame, \\ \(\bar{\sigma}_{n}\) & Normal stress, \\ \(|\tau_{n}|(=\mu|\sigma_{n}|)\) & shear stress on projectile surface, \\ \(\mu\) & coefficient of kinetic friction, or Factor of friction between projectile and soil, \\ \(\mu\) & or Tangent of angle of repose of granular material, \\ \(r_{i}^{j}\) & vector from the center of particle \(i\) to the contact point with particle \(j\); \\ \(\theta\) & apex angle of a wedge or a cone, \\ \(\varphi\) & angle of internal friction of soil, \\ \(\rho\) (\(\rho t\)), \(s\) & density of soil (or target) and, degree of saturation of soil \\ \(\theta\) & Semi-vertex angle of cone-nose projectile in degrees\" \\ \(y\) & Poisson's ratio \\ \(\rho_{*}\) & \(\rho_{*}\) \\ \(\sigma_{r}\), \(\sigma_{\theta}\) & \