Abstract

We exhibit a planar truss structure that, once homogenised at a macro-level, can be regarded (i.e., will need to be modelled) as a third-gradient beam, meaning a one dimensional continuum whose deformation energy depends on the third derivative of its displacement. We call it a ZAPAB structure, for zigzaged articulated parallelograms with articulated braces. The idea of the construction is simple: we look for a planar mechanism formed by rigid bars and interconnecting terminal pivots (thus, a type of linkage) satisfying the following properties: (i) it has one degree of freedom characterised by one Lagrange parameter $u$; (ii) in it, a specific set of material points initially lies on a straight line; (iii) when $u$ varies, the positions of these points are equidistant and belong to circumferences whose radius depends on $u$; (iv) their distance is a function of $u$.

Suitably constraining the mechanism and assuming that the bars are elastic, the linkage becomes a planar ZAPAB truss structure. Since, in the configurations described in (ii), (iii) and (iv), no constituting bar is deformed, the class of zero-energy placements (floppy modes) for the ZAPAB structure, as a whole, consists exactly of circumferences with different radiuses. This implies that a planar one-dimensional continuum capable of describing as a whole the mechanical behaviour of ZAPAB structure must have a deformation energy depending on the derivative (with respect to its curvilinear abscissa) of the curvature.

We present a first piece of evidence for this statement, by using numerical simulations based on a novel code that efficiently calculates the minimum deformation energy for ZAPAB structures. This promising result motivates future investigations and proves how artificial is the (typically bureaucratic and Italian) split between structural mechanics and mechanics applied to machines design: Lagrangian mechanics is their common conceptual basis.

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