Biomedical Engineering Reference
In-Depth Information
For adhesion via molecular bonds, the work of adhesion can be estimated as
"
#
2
¼
r
0
F
b
2
x
1
2
xu
2
max
1
2
x
F
b
x
W
ad
r
0
r
0
(8.19)
where
u
max
is the maximum deformation allowed for each bond, and
F
b
=x
is the
typical length scale of bond deformation. From (
8.18
) and (
8.19
), we can estimate
the dimensionless pull-off stress as
s
E
r
0
xa
p
p
s
Griffith
r
0
F
b
¼ C
G
(8.20)
4
=
p
. As the
adhesion size
a
is reduced, (
8.20
) predicts an increasing pull-off stress, approaching
infinity as
a
goes to zero. However, this trend cannot continue forever, since stress
cannot exceed the strength of molecular adhesion. This suggests a critical adhesion
size below which adhesion strength can no longer be described by the Griffith
theory. Similar arguments of transition between Griffith fracture and failure at
theoretical strength have been made for mineral pieces in bone [
76
] and fibrillar
adhesion in gecko [
28
].
On the other hand, the strength of molecular adhesion can be estimated as
It follows from (
8.20
) that the pull-off stress is proportional to 1
s
Bell
r
0
F
b
¼ C
B
g
1
þ g
(8.21)
where
C
B
is a prefactor. Thus a critical adhesion size can be identified by equating
the adhesion strengths in both Griffith and Bell regimes as
a
cr
E
=r
0
x
or
a
cr
1.
Here, we use the coupled elastic-stochastic model to investigate whether the
continuum theory based on the Griffith concept is applicable in the case of molecu-
lar adhesion. The normalized pull-off stress is plotted as a function of the
normalized cluster size or SCI in Fig.
8.5
for a rebinding rate of
g ¼
2. In the
plot, we include the predictions of Griffith's theory and Bell's theory by using
(
8.20
) and (
8.21
) with pre-factors
C
G
¼
3
7. It can be seen from Fig.
8.5
that the normalized pull-off stress is well predicted by Griffith's theory for large
adhesion size, corresponding to a large SCI
a
, and that reduction of adhesion size or
a
results in deviation from Griffith's theory and eventually adhesion failure at the
adhesion strength determined from Bell's theory.
Gao and Yao [
28
] have previously shown that, for adhesive contact between a
cylindrical punch and a substrate via van der Waals interaction, optimal adhesion
could be achieved by a combination of size reduction and shape optimization. At
small contact sizes, the shape of the contact surfaces does not play an important
role. At large contact sizes, optimal adhesion becomes sensitive to small variations
in contact surface shape, with the theoretical adhesion strength attained only if the
:
3,
C
B
¼
0
:
Search WWH ::
Custom Search