A nonlinear theory of prestressed elastic stick-and-spring structures

Abstract

The discrete modeling of a large class of mechanical structures can be based on a stick-andspring concept. We here present a stick-and-spring theory with potential application to the statics and the dynamics of such nanostructures as graphene, carbon nanotubes, viral capsids, and others. A key feature of our theory is its geometrical nonlinearity: we combine exactly defined strain measures with a general linear stress response; another, rarely found feature is a careful account of prestress states. A linear version is firstly proposed, where attention is restricted to study small displacements from an unstressed reference placement. Next, a theory linearized about a prestressed (preloaded or not) placement is displayed, which is based on a careful analysis of the tangent stiffness operator and its two parts, the elastic and prestress stiffness operators. Finally, two examples are proposed and solved; when an analytical solution is of prohibitive complication, numerical solutions are given, by the use of a specifically implemented 'stick-and-spring' code.

Experimental and Numerical Study of Wave Dynamics in Tensegrity Columns

“…We have performed an impulsive dynamic test on a physical model of a tensegrity column, on comparing the measured displacements to those computed numerically through two different models: a stick-and spring model accounting for bending deformation of the nodes [20], and a pure tensegrity model accounting for a pure-stretching response of all the members of the system [23]. The first model is intended to capture the experimental response of the analyzed physical model, while the second one is aimed at providing directions for the design of alternative systems featuring highly nonlinear dynamic response.…”

“…Such an observation leads us to employ a stick-and-spring (S&S) model [20] to describe the dynamic response of the phycal model under examination. A S&S structure is a collection of nodes, edges, and wedges: edges are imagined as node-to-node inflexible but extensible straight sticks, acting as axial springs when extended; wedges are imagined as complexes of two sticks sharing one end node, equipped by a angular spring reacting to relative rotations of the two wedge sticks in their common plane.…”

“…A S&S structure differs from a tensegrity structure because, owing to the presence of angular springs, edges are subjected to shear forces and bending/torsion moments, in addition to axial forces. In the remainder of this section, we summarize the treatment of S&S structures presented in [20].…”

“…[20] and [23]. An additive manufacturing (AM) technique based on Electron Beam Melting (EBM) is employed to build all the elements of the tested structure, except for the cables, which are subsequently added to the 3d-printed structure through a post-tensioning technique [18].…”

“…An additive manufacturing (AM) technique based on Electron Beam Melting (EBM) is employed to build all the elements of the tested structure, except for the cables, which are subsequently added to the 3d-printed structure through a post-tensioning technique [18]. The mechanical modeling of such a structure is first analyzed through the stick-and-spring model presented in [20], which takes into account that the nodes of the 3D-printed structure form bending-stiff connections between elements. It models the system by considering bars and bases to be inflexible, describing the cables as linear springs, and attaching angular springs to the nodes (mixed bendingstretching response).…”