Cavity-expansion approximation for projectile impact and penetration into sand

A one-dimensional problem of a spherical cavity expanding at a constant velocity from zero initial radius in an infinite granular medium, which has the first-kind self-similar solution, is considered. We are solving this dynamic spherical cavity-expansion problem to model rigid spheres penetrating into a granular media. Elastic–plastic deformation of the granular media is described in a barotropic approximation, using the high-pressure equation of state and Mohr–Coulomb Tresca’s limit yield criterion. The medium is assumed to be incompressible behind the shock wave front propagating through the unperturbed medium. The problem in this formulation was solved analytically. Besides, a generalized solution of the problem was obtained numerically, which involves transition of a continuous elastic–plastic wave into a plastic shock wave when pressure grows with the cavity expansion velocity. The comparison of the analytical and numerical solutions shows that a linearized analytical solution is a good approximation of the pressure along the boundary of the cavity as a function of its expansion, except for low velocities. The linearized rigid plastic solution can be used for analyzing resistance to a rigid sphere that penetrates into the granular media. The computational results are compared with known experimental relations for resistance to spherical projectiles penetrating dry and water-saturated sand. Good agreement between the numerical and experimental results is obtained without any correction factors.