Elasto-plastic solution to expansion of a spherical cavity in dilatant sand

The current solutions to spherical cavity expansion can not properly consider the peak strength and dilatancy of the sand, and thus there is some dispency between the solution and the practical situation. To obtain a rational solution to expansion of a spherical cavity in sand, a critical state model for sand is adopted to consider the effects of the peak strength and dilatancy of the soil on the cavity expansion. Based on the cirtical mode, the elasto-plastic constitutive tensor for the problem is derived according to the associated flow rule. By employing the large deformation theory and introducing an auxiliary variable, the problem becomes an initial value problem of a set of first-order differential equations based on the Lagrangian description. Under the elastic-plastic boundary conditions, a rigorous solution is obtained by solving the governing equaitons numerically. The effects of the peak strength and dilatancy of the sand on the expansion response are studied by comparing the results between the present solution and the modified Cam-clay model-based solution. The results show that the present solution can appropriately reflect the peak strength and dilantancy of the sand during cavity expansion and be reduced to the solution for non-dilanancy soils. Hence, the present solution can be widely applied to the geotechnical problems, such as the cone penetration tests and the pile installation.