On the post-buckling of distributed microstructured system: The finite element elastica

Abstract

The buckling and post-buckling behavior of a simply supported lattice column with distributed microstructure is theoretically and numerically investigated. This problem differs from the nonlinear behavior of a lattice column composed of concentrated microstructure, as illustrated by the Hencky-bar chain system. It is shown that the problem based on a distributed microstructure is mathematically equivalent to the Finite Element formulation of the continuous elastica with linear rotation interpolation functions. The buckling load of this lattice column with distributed microstructure is analytically obtained by solving a linear difference boundary value problem. This linearized difference boundary value problem is asymptotically approximated by a continuum gradient elasticity boundary value problem with definite positive equivalent energy functional. The geometrically exact nonlinear behavior of this microstructured column with distributed microstructure is governed by a strongly nonlinear difference equation, which has not been reported in the literature to the authors’ knowledge. Bifurcation diagrams of the microstructured column composed of few cells are numerically obtained with the simplex algorithm. It is shown that the post-buckling of the lattice column with distributed microstructure reveals complex behavior similarly to the post-buckling of a lattice column with concentrated microstructure. The first post-buckling branch of the lattice system is compared to that of the gradient elastica problem and good correlations are found. The paper is concluded by a general analysis on the effect of concentrated and distributed microstructures at the macroscopic structural scale: Hencky-bar chain system composed of concentrated microstructure asymptotically behaves as a nonlocal (stress gradient) elastic column, whereas the lattice column composed of distributed microstructure asymptotically behaves as a gradient elasticity (strain gradient) elastic column.