Limit Analysis and computational modeling of the hollow sphere model with a Mises–Schleicher matrix

Abstract

Using the kinematic approach of limit analysis (LA) for a hollow sphere whose solid matrix obeys the von Mises criterion, Gurson (1977) derived a macroscopic criterion for ductile porous media. The relevance of this criterion has been widely confirmed in several studies and in particular in Trillat and Pastor (2005) through numerical lower- and upper-bound formulations of LA. In the present paper, these formulations are extended to the case of a pressure dependent matrix obeying the parabolic Mises–Schleicher criterion. This extension has been made possible by the use of a specific component of conic optimization. We first provide the basics of LA for this class of materials and of the required conic optimization; then the LA hollow sphere model and the resulting static and mixed kinematic codes are briefly presented. The numerical bounds obtained prove to be very accurate when compared to available exact solutions in the particular case of isotropic loadings. A second series of tests is devoted to assessing the upper bound and the approximate criterion established by Lee and Oung (2000) as well as the criterion proposed by Durban, Cohen, and Hollander (2010). As a matter of conclusion, these criteria can be considered as admissible only for a slight tension/compression asymmetry ratio for the matrix; in other words, these results show that the determination of the macroscopic criterion of the “porous Mises–Schleicher” material remains an open problem.