Medial Model of Elastic Curve Contacts, from Clasps to Knits

Abstract

Materials constructed from interlocking and tangled fibers are challenging to model and understand due in part to the complex interplay between fiber shape and contact mechanics. Much of our intuition about contact mechanics comes from the Hertzian description, which is based on interactions between convex bodies. However, fibers often form clasp shapes, which involve saddle-shaped contact surfaces; the ability for two interacting bodies to conform to each other along a saddle surface confounds many existing mechanical contact models. Following an observation that contact surfaces between fibers tend to look like minimal, constant mean-curvature surfaces, we seek to test this hypothesis and provide a direct link to geometric modeling methods. To do this, we leverage the medial axis transformation, which generalizes the notion of a Voronoi tessellation to generating curves and surfaces. Through computational studies on an orthogonal clasp and a simple knit structure, we find that the medial axis transformation provides a reasonable model for contact surfaces between identical curves. We also show that a generalization of the medial axis transformation based on radical Voronoi tessellations can be used to well-approximate contact surfaces between non-identical curves. We apply this ``radical axis transform'' to curves of different radii and demonstrate a trend for surfaces to adopt finite mean curvature. Our results may point the way towards more sophisticated geometric methods for modeling tangles.