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
; (ii)
in it, a specific set of material points initially lies on a straight line; (iii) when
varies, the
positions of these points are equidistant and belong to circumferences whose radius depends on
; (iv) their distance
is a function of
.
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.
Keywords
third gradient beam, geometrically nonlinear, system of
trusses, design of metamaterials, ZAPAB structure