The problem of cavity expansion from zero radius has no characteristic length and therefore possesses a similarity solution, in which the cavity pressure remains constant and the continuing deformation is geometrically self-similar. In this case, the incremental velocity approach first used by Hill (1950) to analyze cavity expansion in Tresca materials can be extended to derive a solution for limiting pressure of cavity expansion in other types of material. In this article, a rigorous semi-analytical solution is derived, following Hill's incremental velocity method, for the expansion of cavities from zero initial radius in cohesive-frictional soils with limited dilation. In particular, the radius of the elastic-plastic interface c is used in this article as the time scale and the solution for the limit pressure has been presented. Solutions are evaluated for a number of cases representative of a range of cohesive-frictional and dilatant soils. A comparison is also made between the solutions presented here and previous solutions for cohesive-frictional soils that have unlimited (on-going) plastic dilation. In particular, the influence of limited plastic dilation on the cavity limit pressure is identified and discussed.
Chemotaxis in shear flow: Similarity solutions of the steady‐state chemoattractant and bacterial distributions
