Geometrically exact bifurcation and post-buckling analysis of the granular elastica

Abstract

The nonlinear behaviour of a simply supported granular column loaded by an axial force is studied in this paper. The granular column is composed of rigid grains (discs) elastically connected by some shear and rotational springs. The in-plane buckling and post-buckling analysis in a geometrically exact framework is numerically investigated from a nonlinear difference eigenvalue problem. The granular column asymptotically behaves as an Engesser–Timoshenko column for a sufficiently large number of grains. An exact analytical solution of the buckling load of this discrete shear granular system is obtained from the linearization of the granular elastica problem. An asymptotic expansion is applied to the difference eigenvalue problem, to efficiently approximate the equation of the primary post-bifurcation branch of the discrete problem. Exact analytical solutions of the post-buckling branches are also available for the granular problem with few numbers of grains. Bifurcation diagrams of the granular elastica problem composed of few grains are numerically obtained with the simplex algorithm (for an exhaustive capture of all post-bifurcation branches). It is shown that the post-buckling of this granular column reveals complex behaviour similarly to the post-buckling of a generalized shear Hencky column (also called discrete Engesser elastica). Complex higher-order branches are exhibited, a phenomenon very similar to the discrete elastica problem. These branches reveal the specific nature of the discrete granular problem, as opposed to its continuum limit valid for an infinite number of grains