Metals and Ceramics Division - Oak Ridge National Laboratory

where µ ad tan α i < 1.
When µ ad tanα i ≥ 1,
there is no relative movement between the two
asperities. The contact stress will cause elastic and then plastic deformation, and eventually will
either change the slopes to be low enough ( µ tan α < 1,
if αi < βi) to allow sliding or will break
ad i
the weaker of the two asperities. Note that when there is no penetration, it is the sharpness of
the blunter of the two asperities determines the contribution of the abrasion traction to friction.
When the blunter surface is smooth and flat (α is very small), µi will approach to µad, indicating
pure adhesion without mechanical grip. The average friction coefficient between the two
sliding surfaces, composed of numerous asperity contacts, can then be approximated as
Ff
∆ q,
min + µ ad
µ I = ≈
(4)
P 1− µ ad ⋅ ∆q
, min
where µ < 1 and ∆ = ∆ , ∆ ) . ∆q,h and ∆q,s are root mean square slopes of the
ad ∆ q,
min
q,
min min( q,
h q,
s
harder and softer surfaces, respectively. ∆q is a standard, measurable surface roughness
parameter for the root-mean-square slope of the asperities.
Case II. With penetration. If one surface is much harder than the other, its asperities will
penetrate the other one, as illustrated in Fig. 1(b). In this case, the relative motion between
asperities A and B follows the slope of the harder one (no matter whether it is sharper or
blunter). Using a derivation similar to that for Case I, the friction coefficient for Case II can be
obtained,
∆ q,
h + µ ad
µ II ≈
(5)
1− µ ⋅∆
ad
As shown in Eqn. (5), the sharpness of the harder asperity controls the abrasion factor: the
sharper the harder asperities, the greater the likelihood that they will penetrate and produce
higher traction. In an extreme case that the harder surface is perfectly smooth (∆q,h = 0), no
penetration will occur and the friction will be merely contributed by adhesion (µII = µad).
Considering Eqns. (4) and (5), we can allow for both cases as follows:
q,
h
⎛ H ⎞ ⎛ ⎞ ⎛ ∆ + ⎞⎛
⎞ ⎛ ∆ + ⎞⎛
⎞
s H s
q,
min µ ad
⎜
⎟
H s
q,
h µ ad
+ ⎜
⎟
H s
µ = µ ⎜
⎟ +
⎜ −
⎟ ≈
⎜
⎟
⎜ −
⎟
I µ II 1
1 (6)
⎜
⎟ ⎜
⎟
⎝ H h ⎠ ⎝ H h ⎠ ⎝1−
µ ad ⋅∆
q,
min ⎠⎝
H h ⎠ ⎝1−
µ ad ⋅∆
q,
h ⎠⎝
H h ⎠
If the harder surface is the blunter of the two, Eqn. (6) can be simplified to µ = µI = µII. Note
that the hardness and root mean square slopes can be measured; however, the adhesion factor µad
is not as straightforward to obtain because the phenomenon of adhesion is much less understood.
In one approach, Rabinowicz [3] claims that adhesion is high when contact surfaces have high
surface energy and low hardness. For certain crystal structures the adhesive contribution to
friction is roughly proportional to the ratio of the energy of adhesion to the hardness of the softer
surface. It is also observed that smooth, clean surfaces can experience severe adhesion. Future