Sun, Diankang and Daraio, Chiara and Sen, Surajit (2011) Nonlinear repulsive force between two solids with axial symmetry. Physical Review E, 83 (6). Art. No. 066605. ISSN 15393755. http://resolver.caltech.edu/CaltechAUTHORS:20110706112940606

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Abstract
We modify the formulation of Hertz contact theory between two elastic halfsolids with axial symmetry and show that these modifications to Hertz’s original framework allow the development of force laws of the form F∝z^n, 1<n<∞, where F is the force and z is the distance between the centers of the two solids. The study suggests that it may be possible to design physical systems that can realize such force laws. We let the halfsolids be characterized by radii of curvatures R_1 and R_2 and invoke a factor m>0 to describe any aspect ratio in the two bodies, all being valid near the contact surface. We let the xy plane be the contact surface with an averaged pressure across the same as opposed to a pressure profile that depends on the contact area of a nonconformal contact as originally used by Hertz. We let the z axis connect the centers of the masses and define z_(1,2) = x^(α)/R_(1,2)^(α1) + y^(α)/(mR_(1,2))^(α1), where z_(1,2)≥0 refers to the compression of bodies 1, 2, α>1, m>0, x,y≥0. The full cross section can be generated by appropriate reflections using the first quadrant part of the area. We show that the nonlinear repulsive force is F=az^n, where n≡1+(1/α), and z≡z_1 + z_2 is the overlap and we present an expression for a=f(E,σ,m,α,R_(1),R_(2)) with E and σ as Young’s modulus and the Poisson ratio, respectively. For α=2,∞, to similar geometrydependent constants, we recover Hertz’s law and the linear law, describing the repulsion between compressed spheres and disks, respectively. The work provides a connection between the contact geometry and the nonlinear repulsive law via α and m.
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Additional Information:  © 2011 American Physical Society. Received 21 November 2010; revised 13 March 2011; published 20 June 2011. S.S. thanks Prof. Robert W. Newcomb and a referee for valuable comments. D.S. and S.S. were supported by the US Army Research Office. C.D. was supported by the US Army Research Office and an NSF CAREER grant.  
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Classification Code:  PACS: 46.40.Cd, 43.25.+y, 45.70.n  
Record Number:  CaltechAUTHORS:20110706112940606  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20110706112940606  
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  24320  
Collection:  CaltechAUTHORS  
Deposited By:  Jason Perez  
Deposited On:  06 Jul 2011 20:16  
Last Modified:  08 Sep 2017 20:54 
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