Hyperelastic Models for Granular Materials

Abstract

Motivated by the need for a continuum mechanical description of particulate materials for nuclear safety analyses, several recently proposed hyperelastic models for granular materials are analyzed and compared with experiment data. As even the quasi-elastic regime of granular materials is non-linear, the hyperelastic forms considered here are all designed to capture the widely observed dependence of the elastic moduli on the square root of pressure. Building this sort of dependence in to the free energy results in some physically relevant behavior that is missed by other non-linear models, including stress-induced anisotropy and shear dilatancy. The granular elasticity (GE) model of Jiang and Liu additionally possesses a region outside a Drucker-Prager type yield surface in which the free energy is not convex, implying a lack of stable solutions there. This proves to be an over-constraint, as it limits yield angles to values lower than typically observed. Models due to Einav and Puzrin (EP), and Houlsby, Amorosi, and Rojas (HAR), lack this constraint, and thus provide greater flexibility; the EP model proves to best capture the sort of stress-induced anisotropy observed in experiments. All three models are implemented in the finite element code Abaqus, and used to calculate stress distributions in sand piles and silos, and the granular response function. The models agree qualitatively, but not always quantitatively, with experiments; paradoxically, the EP model proves to be the least accurate, producing an unphysically narrow and high peaked response function. They also possess shortcomings similar to those of linear elasticity. In silos with an applied surface load, they underestimate the observed "overshoot" of the saturated stress. For both sand piles and the response function, the stress profiles are insensitive to the values of the elastic constants, and as such are not able to account for the range of data observed experimentally. Incorporating some dependence on the pile formation history is likely necessary to describe these effects. In light of the findings, and the relative simplicity it affords, linear elasticity (despite its known shortcomings) is an appropriate choice for coupling to flow models in engineering analyses. ACKNOWLEDGMENTS Thanks are due to the many people who have shaped my experience at UW and at INL, without whom this work would not have been possible. My advisor, Mike Corradini, was charged with the difficult task of keeping tabs on a student over a thousand miles away, but has always been available and has continually been a source of new ideas and much needed direction. I am grateful to Phil Sharpe, my INL mentor, for encouraging me to pursue the technical problems I found most interesting, for his continued patience, and for bringing me to INL many years ago. It has been a pleasure to work with him and everyone else in the Fusion Safety Program at INL, and I look forward to continuing my work there. Lee Cadwallader deserves special thanks for his careful reading of this manuscript...

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