Closed-form solutions for the hollow sphere model with Coulomb and Drucker–Prager materials under isotropic loadings

Abstract

Though the solution to the limit analysis problem of the hollow sphere model—with a von Mises matrix and under spherical symmetry—is well known, it is not available, to our knowledge, for both isotropic loadings (tension and compression) in the case of a Coulomb matrix and partially for a Drucker–Prager matrix. In the present Note, we establish in a unified framework, for this class of materials, closed-form solutions for stress and strain fields in a hollow sphere under external isotropic tension and compression. These analytical results not only give useful reference solutions, but can also be considered as a part of a trial velocity field in the hollow sphere submitted to an arbitrary loading. Comparisons with 3D finite element-based limit analysis approaches and with recent results in the literature are provided. In addition to the established analytical results, we present a rigorous evaluation of a recent Gurson-type macroscopic criterion corresponding to the Drucker–Prager hollow sphere under an arbitrary loading, by means of the previous 3D limit analysis codes.