Abstract
Aiming at simulating elastic rods, we discretize a rod model based on a general theory of hyperelasticity for inextensible and unshearable rods. After reviewing this model and discussing topological effects of periodic rods, we prove convergence of the discretized functionals and stability of a corresponding discrete flow. Our experiments numerically confirm thresholds, e.g., for Michell’s instability, and indicate a complex energy landscape, in particular in the presence of impermeability.
Highlights
In this paper we extend the study of inextensible elastic curves by the first author [4] to inextensible and unshearable elastic rods
We find applications in different fields such as the modeling of coiling and kinking of submarine cables (Zajac [76]; Goyal et al [32,33]), cell filaments (Manhart et al [46]), computer graphics (Bergou et al [8]; Spillmann and Teschner [65]), and biomechanics [58]
We review the geometry of elastic rods in Sect
Summary
A rod is modeled by a curve y : [0, L] → R3 which corresponds to its centerline and an orthonormal frame F : [0, L] → SO(3) whose columns F = [t, b, d] are called directors. We will assume that the first column of F coincides with the unit tangent t(x) = y (x) y (x) , x ∈ [0, L]. Our analysis covers the case of closed rods where [0, L] is understood to be the periodic interval R/LZ. We will realize the latter by imposing suitable (periodic) boundary conditions at 0 and L. A rod is assumed to have some small diameter which can be considered infinitesimal; self-penetrations are not excluded at this stage A rod is assumed to have some small diameter which can be considered infinitesimal; self-penetrations are not excluded at this stage (see Sect. 6.6 below for a discussion on modeling impermeability)
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