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energyTL2010 - Tribol Lett(2010 37:453461 DOI...

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ORIGINAL PAPER Energy, Adhesion, and the Elastic Foundation Ira J. Hill W. Gregory Sawyer Received: 3 April 2009 / Accepted: 26 October 2009 / Published online: 13 November 2009 Ó Springer Science+Business Media, LLC 2009 Abstract A theory for elastic contact adhesion between a rigid sphere and an elastic foundation is developed. The theory derives relationships between the contact deforma- tion and the externally applied force. The derivation is based on elastic contact between a sphere and a thin linear- elastic foundation in which the strain energies are balanced by the work of indentation and the change in surface energy. Contacting regimes where there is either com- pressive strain energy or only tensile strain energy (pull-off regime) are both treated. The model is non-dimensional- ized and an order of magnitude analysis is performed in order to develop simpliFed closed form solutions; the simpliFed model is then evaluated and compared to the full solution. This theory Fnds that the adhesion force is sig- niFcantly larger for an elastic foundation in which the surface elements act independently as compared to more traditional solutions for elastic solids. The theory gives an adhesion force of F adh f 7 p R D c : Keywords Contact mechanics ± Adhesion ± Elastic foundation List of symbols a Half-width of the compressive contact zone d A Differential area within the contact surface d A = s d s d h b Contact half-width b adh Half-width of the contact zone at maximum negative force b 0 Half-width of the contact zone at zero externally applied load b trans Contact half-width at the transition condition for the pull-off regime C Stiffness of the foundation (pressure/unit displacement) d Deformation in the foundation at s = 0 d max Gedanken maximum deformation at s = 0 d trans Transition condition for the pull-off regime, d trans = 0 d s Deformation in the foundation as a function of the contact radial coordinate s D c Change in surface energy per unit area as a result of contact D c = c 1 ? c 2 - c 12 E Modulus of elasticity m Poisson’s ratio F Externally applied force: ( ? ) compressive; ( - ) tensile F adh The maximum tensile force or the force of adhesion U The Zenith angle, U = 0 along the axis of loading h Maximum height of the tensile region at the edge of contact h s Maximum height of the tensile zone at separation h trans Height of the tensile zone at the transition condition for the pull-off regime H dimensionless group representing the strength of adhesion P Contact pressure: ( ? ) compressive; ( - ) tensile h Angle around the contact h = 0 2 p R Radius of the contacting sphere s Radial coordinate from the center of the contact t Thickness of the elastic foundation U 0 Compressive strain energy U T Total strain energy I. J. Hill ± W. G. Sawyer ( & ) Department of Mechanical and Aerospace Engineering, University of ±lorida, Gainesville, ±L 32611, USA e-mail: [email protected]².edu 123 Tribol Lett (2010) 37:453–461 DOI 10.1007/s11249-009-9537-0
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U s Change in surface energy due to a Fnite contact area U max Gedanken maximum strain energy z
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energyTL2010 - Tribol Lett(2010 37:453461 DOI...

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