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SUMMARYThis paper exposes an interior-point method used to solve convex programming problems raised by limit analysis in mechanics. First we explain the main features of this method, describing in particular its typical iteration. Secondly, we show and study the results of its application to a concrete limit analysis problem, for a large range of sizes, and we compare them for validation with existing results and with those of linearized versions of the problem. As one of the objectives of the work, another classical problem is analysed for a Gurson material, to which linearization or conic programming does not apply.

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.

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