Elasticity_ Barber

192 12 Plane contact problems
where Ċ is an arbitrary but constant rigid-body slip (or creep) velocity. Similarly,
the slip condition (12.78) becomes
(
p x (ξ) = −fp y (ξ)sgn V d )
dξ (u x1 − u x2 ) + Ċ . (12.90)
At first sight, we might think that the Mindlin traction distribution (12.85)
satisfies this condition, since it gives
d
dξ (u x1 − u x2 ) = 0 (12.91)
in the central stick zone and Ċ can be chosen arbitrarily. However, if we
substitute the resulting displacements into the slip condition (12.90), we find
a sign error in the leading slip zone. This can be explained as follows: In the
Mindlin problem, as T is increased, positive slip (i.e. ḣ(x)= ˙u x2− ˙u x1 +Ċ >0)
occurs in both slip zones and the magnitude of h(x) increases from zero at
the stick-slip boundary to a maximum at x=±a. It follows that d dξ (u x1−u x2 )
is negative in the right slip zone and positive in the left slip zone. Thus, if
this solution is used for the steady rolling problem, a violation of (12.90) will
occur in the right zone if V is positive and in the left zone if V is negative. In
each case there is a violation in the leading slip zone — i.e. in that zone next
to the edge where contact is being established.
Now in frictional problems, when we make an assumption that a given
region slips and then find that it leads to a sign violation, it is usually an indication
that we made the wrong assumption and that the region in question
should be in a state of stick. Thus, in the rolling problem, there is no leading
slip zone 17 . Carter 18 has shown that the same kind of superposition can be
used for the rolling problem as for Mindlin’s problem, except that the corrective
traction is displaced to a zone adjoining the leading edge. A corrective
traction
√ (a
p ∗ x(ξ) = 2fF ) 2 ( − c
πa 2 − ξ − a + c ) 2
= 2fF √
(a − ξ)(ξ − c) (12.92)
2
2 πa 2
produces a displacement distribution
d
dξ (u x1 − u x2 ) = fF A (
2πa 2 ξ − a + c )
2
; c < ξ < a , (12.93)
17 Note that this applies to the uncoupled problem (β = 0) only. With dissimilar
materials, there is generally a leading slip zone and there can also be an additional
slip zone contained within the stick zone. This problem is treated by R.H.Bentall
and K.L.Johnson, Slip in the rolling contact of dissimilar rollers, International
Journal of Mechanical Sciences, Vol. 9 (1967), pp.389–404.
18 F.W.Carter, On the action of a locomotive driving wheel, Proceedings of the Royal
Society of London, Vol. A112 (1926), pp.151–157.