CELLS AND MEMBRANES - Nanoindentation with Biological Applications - page 332

central displacement, w 0 , film-shaft contact circle with radius, c , and the
associated membrane stress, σ . The deformed profile is governed by

4 w - 2 w = F

( r ) (10-15)
similar to Eq. 10-6. An exact solution to Eq. 10-15 c an be derived and
can be found elsewhere. Here we present only the asymptotic solution as
a guide to the experimentalists. The overall mechanical response is a
simple superposition of indentation, plate-bending, mixed bending-
stretching, and membrane-stretching, such that

κ ∇ σ b ∇ δ

2/3 1/ 3 1/ 3

1/ 3

9 1

2

3 a

2

1

2

4 a

2

1

2

w F

2/3

F F

1/ 3

0

16 R E

3

E E

4 b b

t

Indentation Plate Bending Membrane Stretching

(10-16)
Figure 10-6 shows a schematic of the mechanical response. The three
deformation modes take consecutive turns to dominate the membrane
deformation in different length scales. When the shaft comes to initial
contact with the film surface, the micro-scale ( w 0 << b ) response is
dominated by indentation with F w 0 3/2 , followed by the meso-scale

( w 0 ~ b ) deformation governed by plate-bending with F

∝

∝ w 0 , and

w 0 3 . Note
tha t Eq. 10-16 s hows only the asymptotic solutions and does not describe
the transition from one deformation mode to the next. The transition
from indentation to plate bending depends on the dimension and
geometry of the indenter and is difficult to derive exactly. The
deformation mode transition can be characterized by a crude gauging
factor

ultimately macro-scale ( w 0 > b ) deformation approaches F ∝

F a

2

2

λ = 12 ( − v ) (10-17)

κ b

Plate-bending dominates when λ < 85, the gradual bending-stretching
transition occurs in the intermediate range 85 < λ < 3 × 10 4 , and
stretching dominates when > 3 × 10 4 . Note that the intersection of

the pure bending and pure stretching asymptotes occurs at

λ

350 and
w 0 ~ 2 b . The asymptotes can be adopted as the constitutive relation of

λ