Biomedical Engineering Reference
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Fig. 10.7 Comparison
between the fracture strength
s
max
n of a cracked fiber and the
n -th level adhesion strength S n
of the bottom-up designed
fractal hairs. If s
n >S n ,
adhesion failure is regarded as
the principal failure mode,
otherwise ( s
max
n <S n ) fiber
fracture is thought of as the
principal failure mode
max
10.3.5 Fiber Fracture: An Upper Limit on Flaw Tolerant
Adhesion Design
In the preceding discussions, we have focused on failure along an adhesion
interface and implicitly assumed that the fibers themselves do not fracture.
In practice, as the adhesion strength is enhanced by introducing hierarchical fibrillar
structures, the fracture of fibers eventually rises to become the dominant issue for
failure at the system level. In other words, a robust adhesion system must be robust
against not only adhesion failure, but also fiber fracture.
Consider a single fiber at hierarchy level n . A penny-shaped crack is introduced
in the center of the cross section as a possible internal flaw. Other configurations of
crack-like flaws, such as edge/corner cracks/singularities, can be considered with-
out affecting the basic idea. The maximum tensile stress that this fiber can sustain
can be determined from the Griffith's criterion [ 10 ] for crack growth as [ 39 ],
s
E f G f
R n
p
pR n = 2 a
max
n
s
¼
(10.20)
gða=R n Þ
where a is the crack radius, G f is the fracture energy and
¼
3
a
R n
1 0
:
5 a=R n þ 0
:
148 ða=R n Þ
g
p
1 a=R n
(10.21)
is a geometrical parameter. Considering a crack half the size of the fiber, i.e., a=R n
¼ 0
:
5, ( 10.20 ) can be further reduced to
63
q
max
n
E f G f =R n
s
¼ 1
:
(10.22)
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