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.
Accepté le :
DOI : 10.1051/cocv/2017030
Mots-clés : Directional dry friction, rate-independent systems, wiggly energy landscape, with the nap/against the nap asymmetry, energy-dissipation principle
@article{COCV_2017__23_3_1023_0,
author = {Gidoni, Paolo and DeSimone, Antonio},
title = {On the genesis of directional friction through bristle-like mediating elements},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1023--1046},
publisher = {EDP-Sciences},
volume = {23},
number = {3},
year = {2017},
doi = {10.1051/cocv/2017030},
zbl = {1365.74021},
mrnumber = {3660458},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2017030/}
}
TY - JOUR AU - Gidoni, Paolo AU - DeSimone, Antonio TI - On the genesis of directional friction through bristle-like mediating elements JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1023 EP - 1046 VL - 23 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2017030/ DO - 10.1051/cocv/2017030 LA - en ID - COCV_2017__23_3_1023_0 ER -
%0 Journal Article %A Gidoni, Paolo %A DeSimone, Antonio %T On the genesis of directional friction through bristle-like mediating elements %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1023-1046 %V 23 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2017030/ %R 10.1051/cocv/2017030 %G en %F COCV_2017__23_3_1023_0
Gidoni, Paolo; DeSimone, Antonio. On the genesis of directional friction through bristle-like mediating elements. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 1023-1046. doi : 10.1051/cocv/2017030. http://www.numdam.org/articles/10.1051/cocv/2017030/
, and , Kinetics of materials with wiggly energies: Theory and application to the evolution of twinning microstructures in a Cu-Al-Ni shape memory alloy. Philos. Mag. A 73 (1996) 457–497. | DOI
and . Characterization of friction force dynamics. IEEE Control Systems 28 (2009) 64–81. | MR | Zbl
, Energetic formulation for rate-independent processes: remarks on discontinuous evolutions with a simple example. Acta Mechanica 227 (2016) 2805–2829. | DOI | MR | Zbl
V.I. Arnold, V.V. Kozlov and A.I. Neishtadt, Mathematical aspects of classical and celestial mechanics. Springer, Berlin (2007). | MR | Zbl
, , , and , Frictional adhesion: a new angle on gecko attachment. J. Exp. Biol. 209 (2006) 3569–3579. | DOI
and , Gecko adhesion: evolutionary nanotechnology. Phil. Trans. R. Soc. A 366 (2008) 1575–1590. | DOI
, , , , , , , , and , Evidence for van der Walls adhesion in gecko setae. Proc. Natl. Acad. Sci. USA 99 (2002) 12252–12256. | DOI
, , and , A new model for control of systems with friction. IEEE Trans. Autom. Control 40 (1995) 419–425. | DOI | MR | Zbl
B. Drincic and D.S. Bernstein, A sudden-release bristle model that exhibits hysteresis and stick-slip friction, in IEEE American Control Conference (2011) 2456–2461.
, and , Liquid Crystal Elastomer Strips as Soft Crawlers. J. Mech. Phys. Solids 85 (2015) 254–272. | DOI | MR | Zbl
and , Crawling motility through the analysis of model locomotors: two case studies. Eur. Phys. J. E 35 (2012) 85. | DOI
, , , and , Mechanics of hierarchical adhesion structures of geckos. Mech. Mater. 37 (2005) 275–285. | DOI
and , Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler. Meccanica 52 (2017) 587–601. | DOI | MR | Zbl
, and , Crawling on directional surfaces. Int. J. Non-Linear Mech. 61 (2014) 65–73. | DOI
, , and , Design of biomimetic fibrillar interfaces: 1. Making contact. J. R. Soc. Interf. 1 (2004) 23–33. | DOI
, and , Bioinspired directional surfaces for adhesion, wetting, and transport. Adv. Funct. Mater. 22 (2012) 2223–2234. | DOI
and , On the modeling and simulation of friction. J. Dyn. Syst. Meas. Control. 113 (1991) 354–362. | DOI
and , Role of seta angle and flexibility in the gecko adhesion mechanism. J. Appl. Phys. 116 (2014).
and , Frictional force microscopic detection of frictional asymmetry and anisotropy at (104) surface of calcite. Phys. Chem. Chem. Phys. 6 (2004) 129–133. | DOI
, , , , , , , , and , Friction anisotropy and asymmetry of a compliant monolayer induced by a small molecular tilt. Science 280 (1998) 273–275. | DOI
, and , Biomimetic ratcheting motion of a soft, slender, sessile gel. Proc. Natl. Acad. Sci. USA 101 (2004) 23–26. | DOI
, , and , Development of a biomimetic miniature robotic crawler. Auton. Robot 21 (2006) 155–163. | DOI
A. Mielke, Evolution of rate-independent systems, in Handbook of Differential Equations, evolutionary equations, edited by C. Dafermos and E. Feireisl. Elsevier (2005). | MR | Zbl
, Emergence of rate-independent dissipation from viscous systems with wiggly energies. Contin. Mech. Thermodyn. 24 (2012) 591–603. | DOI | MR | Zbl
A. Mielke, Variational approaches and methods for dissipative material models with multiple scales, in Analysis and Computation of Microstructure in Finite Plasticity. Springer, Berlin (2015) 125–155. | MR
A. Mielke and R. Rossi, Existence and uniqueness results for general rate-independent hysteresis problems. Math. Models Methods Appl. Sci.17 (2007) 81–123. | MR | Zbl
A. Mielke and T. Roubcíˇek, Rate-Independent Systems. Theory and Application. Springer, Berlin (2015). | MR
and , From discrete visco-elasticity to continuum rate-independent plasticity: rigorous results. Arch. Rational Mech. Anal. 203 (2012) 577–619. | DOI | MR | Zbl
, The motion dynamics of snakes and worms. ACM Siggraph Computer Graphics 22 (1988) 169–173. | DOI
and , A robotic crawler exploiting directional frictional interactions: experiments, numerics, and derivation of a reduced model. Proc. Roy. Soc. London A 470 (2014).
and , The effect of surface roughness on the adhesion of elastic plates with application to biological systems. J. Chem. Phys. 119 (2003) 11437–11444. | DOI
, , , , , Contact compliance effects in the frictional response of bioinspired fibrillar adhesives. J. R. Soc. Interf. 10 (2013) 20130182. | DOI
V. Popov, Contact mechanics and friction: physical principles and applications. Springer, Berlin (2010). | Zbl
V.L. Popov and J.A.T. Gray, Prandtl-Tomlinson Model: A Simple Model Which Made History, in The History of Theoretical, Material and Computational Mechanics-Mathematics Meets Mechanics and Engineering. Springer, Berlin (2014) 153–168. | MR
and , Thermodynamics of rate-independent plasticity. J. Mech. Phys. Solids 53 (2005) 655–679. | DOI | MR | Zbl
and , Cohesion-decohesion asymmetry in geckos. Phys. Rev. E 87 (2013) 032714. | DOI
R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer, Berlin (1998). | MR | Zbl
, , , , , , and , Adhesion and friction in gecko toe attachment and detachment. Proc. Natl. Acad. Sci. USA 103 (2006) 19320–19325. | DOI
and , Worm-like locomotion as a problem of nonlinear dynamics. J. Theoret. Appl. Mech. 45 (2007) 179–187.
Cité par Sources :
