A nonlinear model of shearable elastic rod from an origami-like microstructure displaying folding and faulting

Abstract: A new continuous model of shearable rod, subject to large elastic deformation, is derived from nonlinear homogenization of a one-dimensional periodic microstructured chain. As particular cases, the governing equations reduce to the Euler elastica and to the shearable elastica known as 'Engesser', that has been scarcely analysed so far. The microstructure that is homogenized is made up of elastic hinges and four-bar linkages, which may be realized in practice using origami joints. The equivalent continuous rod is governed by a Differential-Algebraic system of nonlinear Equations (DAE), containing an internal length ratio, and showing a surprisingly rich mechanical landscape, which involves a twin sequence of bifurcation loads, separated by a 'transition' mode. The latter occurs, for simply supported and cantilever rods in a 'bookshelf-like' mode and in a mode involving faulting (formation of a step in displacement), respectively. The postcritical response of the simply supported rod exhibits the emergence of folding, an infinite curvature occurring at a point of the rod axis, developing into a curvature jump at increasing load. Faulting and folding, excluded for both Euler and Reissner models and so far unknown in the rod theory, represent 'signatures' revealing the origami design of the microstructure. These two features are shown to be associated with bifurcations and, in particular folding, with a secondary bifurcation of the corresponding discrete chain when the number of elements is odd. Beside the intrinsic theoretical relevance to the field of structural mechanics, our results can be applied to various technological contexts involving highly compliant mechanisms, such as the achievement of objective trajectories with soft robot arms through folding and localized displacement of origami-inspired or multi-material mechanisms.