Rigorous Similarity Solutions for Cavity Expansion in Cohesive‐Frictional Soils

Abstract

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 [7] to analyze cavity expansion in Tresca materials may be extended to derive a solution for limiting pressure of cavity expansion in Mohr‐Coulomb materials. An analytical solution for cavity limit pressures in Mohr‐Coulomb materials was suggested by Carter et al. [2]. However, the solution of Carter et al. may only be regarded as approximate since the convected part of the stress rate was neglected in their derivation. By including the convected part of the stress rate, Collins and Wang [4] later derived a semi‐analytical similarity solution for cavity expansion in purely frictional soils. The solution of Collins and Wang [4] was, however, obtained from numerical integration as their solution could not be expressed in explicit form. In this article, a rigorous closed‐form solution is derived for the expansion of cavities from zero initial radius in cohesive‐frictional soils. The solution procedure adopted here follows the Hill incremental velocity method, which is different from that used by Collins and Wang [4]. In particular, the plastic radius c is used in this article as the time scale. Unlike the solution of Collins and Wang [4], it is shown that by using a series expansion the similarity solution can be expressed in closed form.