Biomedical Engineering Reference

In-Depth Information

Fig. 10.2
A fiber is brought into contact with a substrate. Depending upon the shape of the fiber

tip, the detachment process can occur either by (
a
) crack propagation (
singular shapes
) equivalent

to an infinite crack external to the contact area or by (
b
) uniform detachment (
optimal shapes
)in

which the stress at pull-off is uniformly distributed and equal to the theoretical adhesion strength

s
th
. The difference between the adhesive strength of these two failure modes vanishes as the size of

the fibril is reduced to below a threshold
R
cr
¼

8
E
Dg

ps

, which is taken as the condition for flaw

2

th

tolerant adhesion

material that allows the contact to fail not by crack propagation, but always by

uniform detachment at the theoretical strength of adhesion, a concept termed as
flaw

tolerance
[
7
,
25
,
26
]. According to this concept, in an ideal flaw tolerant adhesion

system, there should be no crack propagation and coalescence as the contact

interface is pulled apart by uniform detachment.

For a single fiber on substrate, Gao and Yao [
23
] investigated the condition for

flaw tolerant adhesion from the point of view of variations in contact shape. It was

shown that there exist two extreme classes of contact shapes: one class (singular

shapes) gives rise to a singular stress field at pull-off similar to that of an external

crack (Fig.
10.2a
) and the other class (optimal shapes) leads to a uniform stress at

pull-off (Fig.
10.2b
). For singular shapes, the pull-off force can be calculated

according to the Griffith condition [
10
]as

r

8

p

1
=
2

E
W
ad

R

P
crack
¼ pR
2

(10.1)

s
Þ=E
s
1
,

E
f
; E
s
; n
f
; n
s
being the Young's moduli and Poisson's ratios of the fiber and the

substrate, respectively. For a gecko sticking to a solid surface, we assume
E
s
E
f
,

therefore
E
E
f
=ð
1
n

where
W
ad
denotes the work of adhesion and
E
¼½ð
1
n

2

2

f
Þ=E
f
þð
1
n

2

f
Þ
. On the other hand, the pull-off force for optimal

contact shapes (Fig.
10.2b
)is

P
th
¼ pR
2

s
th

(10.2)

where
s
th
is the theoretical adhesion strength. Generally,
P
crack
is much smaller than

P
th
. However, as the size of the fiber is reduced, the value of
P
crack
increases towards

P
th
. At the critical size

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