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Energy, Adhesion, and the Elastic Foundation

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Abstract

A theory for elastic contact adhesion between a rigid sphere and an elastic foundation is developed. The theory derives relationships between the contact deformation 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 compressive strain energy or only tensile strain energy (pull-off regime) are both treated. The model is non-dimensionalized and an order of magnitude analysis is performed in order to develop simplified closed form solutions; the simplified model is then evaluated and compared to the full solution. This theory finds that the adhesion force is significantly 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 \( F_{\text{adh}} \cong 7\pi R\Updelta \gamma . \)

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Abbreviations

a :

Half-width of the compressive contact zone

dA :

Differential area within the contact surface dA = s ds dθ

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

δ s :

Deformation in the foundation as a function of the contact radial coordinate s

Δγ :

Change in surface energy per unit area as a result of contact Δγ = γ 1 + γ 2 − γ 12

E :

Modulus of elasticity

ν :

Poisson’s ratio

F :

Externally applied force: (+) compressive; (−) tensile

F adh :

The maximum tensile force or the force of adhesion

Φ:

The Zenith angle, Φ = 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

θ :

Angle around the contact θ = 0…2π

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

U s :

Change in surface energy due to a finite contact area

U max :

Gedanken maximum strain energy

z :

Surface coordinate in the direction of loading: (+) into the film

(—):

Denotes a dimensionless/normalized variable

\( a = \sqrt {2Rd - d^{2} } \) :

Half width of the compressive zone in terms of the penetration d and the spherical radius R

\( b = \sqrt {2R\left( {d + h} \right) - \left( {d + h} \right)^{2} } \) :

Half width of the contact with adhesion in terms of the penetration d, the height of the tensile zone h, and the radius R

\( {\frac{{\pi b^{4} }}{4R}} \) :

Simplified solution for the volume of a spherical cap of half width b and a radius R based on small angle approximations

\( 2\pi R\left( {R - \sqrt {R^{2} - b^{2} } } \right) \) :

Surface area of a spherical cap of half width b and a radius R

\( C = {\frac{{\left( {1 - \nu } \right)E}}{{\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)t}}} \) :

Foundation stiffness for a Winkler foundation under conditions of plane strain

\( \sqrt {1 - \bar{b}^{2} } = \cos (\bar{b}) \cong 1 - {\frac{{\bar{b}^{2} }}{2}} \) :

Small angle approximations for the dimensionless contact half width that are used to develop the simplified solution

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, 32611, USA

    Ira J. Hill & W. Gregory Sawyer

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  1. Ira J. Hill
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Correspondence to W. Gregory Sawyer.

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Hill, I.J., Sawyer, W.G. Energy, Adhesion, and the Elastic Foundation. Tribol Lett 37, 453–461 (2010). https://doi.org/10.1007/s11249-009-9537-0

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