THE QUASI-STATIC EXPANSION OF A SPHERICAL CAVITY IN METALS AND IDEAL SOILS

Abstract

An infinite medium of ideal soil contains a single spherical cavity within which a slowly increasing pressure is applied. The analysis of stress in the resulting plastic-elastic system, in conjunction with the flow rule associated with the Coulomb law of failure, leads to an expression for the radial displacement of the soil. It is found, in particular, that the radius of the cavity increases indefinitely as the applied pressure approaches a finite limiting value. Since the work done at the cavity wall also increases without bound this pressure cannot be attained physically. When the angle of shearing resistance of the soil is zero the Coulomb law and flow rule are, under conditions of radial symmetry, formally identical to the yield criteria of Tresca and von Mises and their associated flow rules. A special case of the analysis is hence appropriate to the expansion of a spherical cavity in a non-hardening metal. Calculations have been carried out for copper and a comparison is made with numerical results obtained from an alternative solution due to Hill (1). It is shown that the inclusion of work-hardening does not alter the behaviour of the medium qualitatively. If, on the completion of loading, the pressure exceeds a certain limit further plastic flow takes place round the cavity on unloading. Equations are obtained from which can be calculated the residual stresses and displacement at any stage of the unloading process.