On the genesis of directional friction through bristle-like mediating elements

¹ CMAF-CIO – Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edificio C6, 1749-016 Lisboa, Portugal.
pgidoni@fc.ul.pt
² SISSA – International School for Advanced Studies, Trieste, Italy.
³ GSSI – Gran Sasso Science Institute, L’ Aquila, Italy.

Received: 10 February 2016
Revised: 14 February 2017
Accepted: 26 March 2017

Abstract

We propose an explanation of the genesis of directional dry friction, as emergent property of the oscillations produced in a bristle-like mediating element by the interaction with microscale fluctuations on the surface. Mathematically, we extend a convergence result by Mielke, for Prandtl–Tomlinson-like systems, considering also non-homothetic scalings of a wiggly potential. This allows us to apply the result to some simple mechanical models, that exemplify the interaction of a bristle with a surface having small fluctuations. We find that the resulting friction is the product of two factors: a geometric one, depending on the bristle angle and on the fluctuation profile, and a energetic one, proportional to the normal force exchanged between the bristle-like element and the surface. Finally, we apply our result to discuss the with the nap/against the nap asymmetry.