АППРОКСИМАЦИЯ НАПРЯЖЕНИЙ В ОКРЕСТНОСТИ ПОЛОСТИ, РАСШИРЯЮЩЕЙСЯ С ПОСТОЯННОЙ СКОРОСТЬЮ В СРЕДЕ С УСЛОВИЕМ ПЛАСТИЧНОСТИ МОРА - КУЛОНА

Abstract

A one-dimensional problem of a spherical cavity expanding at a constant velocity from a point in an infinite elastoplastic medium is considered. The problem has a first-kind self-similar solution. Elastoplastic deformation of the soil is described based on Hooke's law and the Mohr-Coulomb yield criterion. An analytical solution of the problem in the elastic region contacting with the plastic yield region has been obtained. To determine stress and velocity fields in the plastic region, a known algorithm, based on the shooting method, of analyzing a boundary-value problem for a system of two first-order ordinary differential equations, including the fourth-order Runge - Kutta method, has been realized. An effective algorithm of numerically analyzing an expanding cavity problem, earlier proposed in the works by М. Forrestal et al., makes it possible to solve the problem accurately enough for practical applications. A formula for determining the critical pressure - the minimal pressure required for the nucleation, accounting for internal pressure of a cavity in the framework of the Mohr - Coulomb yield criterion, has been derived, which is a generalization of the earlier published solution for an elastic ideally plastic medium with Tresca's criterion. The obtained critical value was compared with a numerical solution in a full formulation at the cavity expansion velocities close to zero in a wide range of variation of the parameters of the Mohr - Coulomb yield criterion. It is shown that the inaccuracy of the approximation of the proposed formula does not exceed 6% for the variation of the internal friction coefficient all over the admissible range, and for the initial value of the yield strength increasing by three orders of magnitude.