Analyzing the Problem of a Spherical Cavity Expansion in a Medium with Mohr-Coulomb-Tresca’s Plasticity Condition

Abstract

An analytical solution of the one-dimensional problem of a spherical cavity expanding at a constant velocity from a point in a half-space occupied by a plastic medium has been obtained. Impact compressibility of the medium is described using linear Hugoniot’s adiabat. Plastic deformation obeys the Mohr-Coulomb yield criterion with constraints on the value of maximum tangential stresses according to Tresca’s criterion. In the assumption of rigid-plastic deformation (the elastic precursor being neglected), incompressibility behind the shockwave front and the equality of the propagation velocities of the fronts of the plastic wave and the plane shockwave defined by linear Hugoniot’s adiabat, a boundary-value problem for a system of two first-order ordinary differential equations for the dimensionless velocity and stress depending on the self-similar variable is formulated. A closed-form solution of this problem has been obtained in the form of a stationary running wave—a plastic shockwave propagating in an unperturbed half-space. The solution is a generalization of the earlier obtained analytical solution for a medium with the Mohr-Coulomb plasticity condition. A formula for determining a critical pressure (a minimal pressure required for the nucleation of a cavity, accounting for internal friction in the framework of Mohr-Coulomb yield criterion), which is a generalization of the earlier solution for an ideal plastic medium with Tresca’s criterion, has been derived. The resulting critical pressure was compared with a numerical solution in a full formulation at cavity propagation velocities close to zero in a wide range of the parameters of the Mohr-Coulomb yield criterion. The approximation inaccuracy of the introduced formula does not exceed 6% for the internal friction coefficient varying over the entire permissible range and the initial value of yield strength changing by three orders of magnitude. The effect of constraining the limiting value of maximal tangential stresses on the distribution of dimensionless stresses behind the shockwave front has been examined. Formulas for determining the range of cavity expansion velocities, within which a simple solution for a medium with Tresca’s plasticity condition is applicable, have been derived. The obtained solution can be used for evaluating resistance to high-velocity penetration of rigid strikers into low-strength soil media.