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Nonlinear repulsive force between two solids with axial symmetry

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 1539-3755.

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We modify the formulation of Hertz contact theory between two elastic half-solids 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 half-solids 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 x-y 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)^(α-1) + y^(α)/(mR_(1,2))^(α-1), 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 geometry-dependent 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.

Item Type:Article
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URLURL TypeDescription
Daraio, Chiara0000-0001-5296-4440
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.
Funding AgencyGrant Number
Army Research Office (ARO)UNSPECIFIED
Classification Code:PACS: 46.40.Cd, 43.25.+y, 45.70.-n
Record Number:CaltechAUTHORS:20110706-112940606
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:24320
Deposited By: Jason Perez
Deposited On:06 Jul 2011 20:16
Last Modified:08 Sep 2017 20:54

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