Elasticity_ Barber

456 29 Frictionless contact
PROBLEMS
1. Using the result of Problem 23.4 or otherwise, show that if the punch profile
u 0 (x, y) has the form
u 0 = r p f(θ)
where p > 0 and f(θ) is any function of θ, the resulting frictionless contact
problems at different indentation forces will be self-similar 11 .
Show also that if l is a representative dimension of the contact area, the
indentation force, F , and the rigid-body indentation, d, will vary according to
F ∼ l p+1
; d ∼ l p
and hence that the indentation has a stiffening load-displacement relation
F ∼ d 1+ 1 p .
Verify that the Hertzian contact relations (§30.2.5 below) agree with this
result.
2. A frictionless rigid body is pressed into an elastic body of shear modulus µ
and Poisson’s ratio ν by a normal force F , establishing a contact area A and
causing the rigid body to move a distance δ. Use the results of §29.1 to define
the boundary-value problem for the potential function ϕ corresponding to the
incremental problem, in which an infinitesimal additional force increment ∆F
produces an incremental rigid-body displacement ∆δ.
Suppose that the rigid body is a perfect electrical conductor at potential
V 0 and the ‘potential at infinity’ in the elastic body is maintained at zero.
Define the boundary-value problem for the potential V in the elastic body,
noting that the current density vector i is given by Ohm’s law
i = − 1 ρ ∇V ,
where ρ is the electrical resistivity of the material. Include in your statement
an expression for the total current I transmitted through the contact interface.
By comparing the two potential problems or otherwise, show that the
electrical contact resistance R = V 0 /I is related to the incremental contact
stiffness dF/dδ by the equation
1
R
(1 − ν) dF
=
ρµ dδ .
11 D.A.Spence, An eigenvalue problem for elastic contact with finite friction, Proceedings
of the Cambridge Philosophical Society, Vol. 73 (1973), pp.249–268, has
shown that this argument also extends to problems with Coulomb friction at the
interface, in which case, the zones of stick and slip also remain self-similar with
monotonically increasing indentation force.