Finite Element Limit Analysis and Porous Mises-Schleicher Material

Abstract

By using the kinematic approach of limit analysis (LA) for a hollow sphere whose solid matrix obeys the von Mises criterion, Gurson (J. Eng. Mater. Technol. 99:2–15, 1977) derived a macroscopic criterion of ductile porous medium. The relevance of such criterion has been widely confirmed in several studies and in particular in Trillat and Pastor (Eur. J. Mech. A, Solids 24:800–819, 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 the 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 obtained numerical bounds 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 assess the upper bound and approximate criterion established by Lee and Oung (J. Appl. Mech. 67:288–297, 2000), and also the criterion proposed by Durban et al. (Mech. Res. Commun. 37:636–641, 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, our results show that the determination of the macroscopic criterion of the “porous Mises-Schleicher” material still remains an open problem.