Infinitely long isotropic Kirchhoff rods with helical centerlines cannot be stable

2020 ◽  
Vol 102 (2) ◽  
Author(s):  
Andy Borum ◽  
Timothy Bretl
Keyword(s):  
Kirchhoff Rods

scholarly journals Mechanics of tubular helical assemblies: ensemble response to axial compression and extension

2021 ◽  
Author(s):  
Jacopo Quaglierini ◽  
Alessandro Lucantonio ◽  
Antonio DeSimone
Keyword(s):  
Boundary Conditions ◽  
Elastic Properties ◽  
Central Region ◽  
Mechanical Response ◽  
Different Boundary Conditions ◽  
Kirchhoff Rods ◽  
Model Versus ◽  
The Individual ◽  
Opposite Chirality

Abstract Nature and technology often adopt structures that can be described as tubular helical assemblies. However, the role and mechanisms of these structures remain elusive. In this paper, we study the mechanical response under compression and extension of a tubular assembly composed of 8 helical Kirchhoff rods, arranged in pairs with opposite chirality and connected by pin joints, both analytically and numerically. We first focus on compression and find that, whereas a single helical rod would buckle, the rods of the assembly deform coherently as stable helical shapes wound around a common axis. Moreover, we investigate the response of the assembly under different boundary conditions, highlighting the emergence of a central region where rods remain circular helices. Secondly, we study the effects of different hypotheses on the elastic properties of rods, i.e., stress-free rods when straight versus when circular helices, Kirchhoff’s rod model versus Sadowsky’s ribbon model. Summing up, our findings highlight the key role of mutual interactions in generating a stable ensemble response that preserves the helical shape of the individual rods, as well as some interesting features, and they shed some light on the reasons why helical shapes in tubular assemblies are so common and persistent in nature and technology. Graphic Abstract We study the mechanical response under compression/extension of an assembly composed of 8 helical rods, pin-jointed and arranged in pairs with opposite chirality. In compression we find that, whereas a single rod buckles (a), the rods of the assembly deform as stable helical shapes (b). We investigate the effect of different boundary conditions and elastic properties on the mechanical response, and find that the deformed geometries exhibit a common central region where rods remain circular helices. Our findings highlight the key role of mutual interactions in the ensemble response and shed some light on the reasons why tubular helical assemblies are so common and persistent.

scholarly journals Variational principles for nonlinear Kirchhoff rods

2019 ◽  
Vol 231 (2) ◽  
pp. 625-647 ◽  
Author(s):  
Ignacio Romero ◽  
Cristian G. Gebhardt
Keyword(s):  
Variational Principles ◽  
Kirchhoff Rods

scholarly journals Static Kirchhoff Rods under the Action of External Forces: Integration via Runge-Kutta Method

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Ademir L. Xavier Jr.
Keyword(s):  
Reference System ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Kutta Method ◽  
Kirchhoff Equations ◽  
Runge Kutta Method ◽  
System Equations ◽  
Kirchhoff Rods ◽  
Runge Kutta ◽  
External Forces

This paper shows how to apply a simple Runge-Kutta algorithm to get solutions of Kirchhoff equations for static filaments subjected to arbitrary external and static forces. This is done by suitably integrating at once Kirchhoff and filament reference system equations under appropriate initial boundary conditions. To show the application of the method, we display several numerical solutions for filaments including cases showing the effect of gravity.

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