Specific Adhesion of Soft Elastic Materials - Structural Interfaces and Attachments in Biology - page 154

Fig. 8.3 Normalized
interfacial traction
distribution for different
values of the stress
concentration index (SCI) for
an adhesion patch with all
bonds closed. The normalized
pulling force P=aF b r 0 ¼ 2

α=0

α=1
α=∞

10 1

10 0

-1 -0.5 0 0.5 1

x/a

ð 1

@ u
@r ¼

2 a
p

ru
r s

d s (8.8)

1

where u ¼ ux=F b ; r ¼ x=a; r ¼ r b =r 0 , s ¼ s=xF b r 0 ¼ ru , and

a r 0 x
E

a ¼ (8.9)

The effect of a can be immediately understood from the solutions to ( 8.8 )in
extreme cases. In the limit a ! 0, the solution is sðxÞ¼ constant for a constant bond
distribution within the adhesion domain a x a , indicating a uniform distri-
bution of interfacial traction independent of the bond location x . In this limit, the
interfacial traction is equally shared among all bonds. In the opposite limit when
a !1 , the solution to ( 8.8 ) becomes

P
pa

1
1 x 2

sðxÞ¼ p (8.10)

= a 2

This is the classical singular solution for a 2D external crack [ 60 ]. For the
intermediate range 0 <a<1 , the maximum traction generally occurs at the edge

of adhesion and the minimum at the center. In the case that r b ¼ r 0 , Fig. 8.3 shows
that the interfacial traction is nearly uniform for small a values, while crack-like
stress concentration emerges near the adhesion edge for large a . Therefore, we refer
to a as the stress concentration index (SCI) [ 40 ]. Equation ( 8.9 ) shows that a is
linearly proportional to the adhesion size, the bond density, and stiffness and
inversely proportional to the reduced elastic modulus of cell and substrate. All
these factors play a role in controlling the distribution of interfacial traction within