Solution for the Dynamic Elastically Compressible Power-Law Strain Hardening Cylindrical Cavity-Expansion Problem

Abstract

A closed form analytical solution for both the quasi-static or dynamic strain hardening cylindrical cavity-expansion (CCE) problems requires the plastic region to be incompressible. This is due to the fact that in both cases a hydrostatic stress state must be explicitly included in the strain hardening function, making an analytical solution intractable. Additionally, for the dynamic problem the elastic region must be compressible to avoid a logarithmic singularity at infinity. An alterative for the dynamic analytical formulation is to use the finite element method (FEM). With the FEM weak formulation, solutions for the radial stress at the cavity surface and the elastic–plastic interface interface velocity as functions of the cavity-expansion velocity can be obtained when both the strain hardening plastic and elastic regions are elastically compressible. In this study, a comparison is made using results from both analytical (strong formulations), and FE simulations for both strain hardening and perfectly plastic dynamic CCE problems. It is concluded that with increasing cavity-expansion velocities, the elastically compressible strain hardening FEM solution is less resistive than the corresponding closed form analytical solution with an incompressible plastic region. Additionally, it is shown that the elastic–plastic interface velocity asymptotes at the bulk wave speed for completely elastically compressible solutions. Furthermore, the completely elastically compressible solutions remain valid for cavity-expansion velocities beyond the critical cavity-expansion velocity associated with the closed-form analytical strain hardening solution.