A graphical method for undrained analysis of cavity expansion in Mohr–Coulomb soil

Abstract

A rigorous analytical solution is developed for undrained cavity expansion problems in non-associated Mohr–Coulomb soil, based on a novel graphical analysis approach and on the Lagrangian description through tracking only the responses of a soil particle at the cavity surface. The mathematical difficulties involved with the flow rate calculation when the stress state lies on the corner/edge of two adjacent (Mohr–Coulomb) yield surfaces, for both cylindrical and spherical cases, are tackled by using the generalised Koiter theory for non-associated plasticity. In particular, through the unique geometrical formulation, the effective stress path pertaining to the cylindrical cavity problem can be very conveniently directly determined, and is found to consist of simple, piecewise straight lines in the deviatoric stress plane with the orientations dependent of the relative magnitude of Poisson's ratio and the friction angle. This thus renders possible the removal of the stringent intermediacy assumption for the vertical stress that is commonly adopted in previous formulations, and hence the development of a complete cylindrical cavity expansion solution in Mohr–Coulomb soil under undrained loading conditions. The stress and deformation responses of the cavity, including the typical pressure–expansion curves and limiting cavity pressure, are finally analytically obtained along with the Lagrangian form of the radial equilibrium equation in completely explicit forms. The closed-form solution provided in this paper is the first rigorous one of its kind, thus completing the analytical analysis of the cavity expansion problem with the classical Mohr–Coulomb model; this is deemed to be essential for the interpretation of in situ test results pertaining to cohesive-frictional soils.