Biomedical Engineering Reference
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compressed by 2 parallel plates forming 2 planar contact circles at the
polar regions with radius,
a
; (ii) the top plate is then replaced by the
AFM tip, resulting in a dimple of radius,
a
. The dimple profile can be
found by linear elasticity
21
4
2
κ
∇
w
−
σ
b
∇
w
=
−
p
+
F
δ
(
r
)
(10-6)
=
Eb
3
/ 12(1
2
) the flexural rigidity,
with
κ
−ν
σ
the apparent membrane
stress within the dimple,
and
(
r
) the delta function. A few assumptions
are taken: (i) the cell wall deformation is dominated by membrane
stretching of the wall with bending moment a perturbation; (ii) the
deformation is assumed small and local such that increase in
p
and
R
are
negligible; (iii) equilibrium requires
F
= (π
a
2
)
p
; and (iv) σ is uniform
within the dimple. The external force is supported by the local membrane
stress and bending moment within the dimple. The external load induces
a local concomitant stress
δ
σ
m
superimposing on the intrinsic membrane
stress, such that
σ
=
σ
0
+
σ
m
, where
2
E
a
1
2
dw
a
(10-7)
σ=
rdr
rdr
m
2
dr
2(1
−ν
)
c
c
The global geometry of a truncated sphere resumes beyond the dimple
(
r
>
a
). The dimple profile is found to be
1
ω
=
{
C
[
I
(
βξ
)
−
I
(
β
)]
−
C
[
K
(
βξ
)
−
K
(
β
)]}
1
0
0
2
0
0
β
ϕ
ρ
2
−
log
ξ
−
(
1
−
ξ
)
(10-8)
2
2
β
2
β
with
I
i
the
i
th
order of the first kind modified Bessel functions, and
C
1
and
C
2
some functions of ϕ, ω =
w/b
, ρ
=
pa
4
/2κ
b
, ϕ =
Fa
2
/2
p
κ
b
, β =
(
σ
ba
2
/
)
1/2
and
2
=
β
0
2
+
β
m
2
. The dimple depth is given by
κ
β
ω
0
=
ω
(
ξ
=0).
2
with
f
(
x
) some
mathematically involved functions given earlier.
20
The relation
β
m
2
= 6 [
f
(
2
)β
It can be shown that
β
)-
f
(
βξ
)]/(1
−ζ
ϕ
(
ω
0
) can
now be found analytically by eliminating
β
m
. In case of pure stretching
linear at small
w
0
and cubic otherwise. In the linear region, the total
displacement traveled by the AFM tip is given by
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