Biomedical Engineering Reference
In-Depth Information
exerting a force on the spherical tip, they developed force-displacement
relationships based on several viscoelastic models such as Newtonian
liquid, Maxwell liquid, and Kelvin body. In particular, the final
constitutive relation based on the Kelvin body is listed as following:
4
E
2
T
i
1
/
2
3
/
2
F
=
R
h
+
πδ
R
(10-11)
2
r
3
ν
)
t
where E i is the Young's modulus of the interior of the cell, T is cortical
tension, and r t is the radius of the AFM tip. Viscoelasticity is then
incorporated into the model through converting the shear modulus
G=E/2 (1+
*
NM
ν
) into the frequency domain as
G
*
*
G
()
G
()
(10-12)
NM
i
t
with
1
/
2
π
(
ν
)
R
κ
=
T
(10-13)
T
2
r t
h
0
()
()
F
ω
1
ν
*
*
G
()
ω
=
i
ω
b
(0)
(10-14)
i
1/ 2
1/ 2
δω
4
Rh
o
where ω = 2π f is the angular frequency, h 0 is approximate depth around
an indentation point and i ω b* (0) is the correction for the viscous drag
force exerted by the liquid medium on the AFM cantilever.
3.2. Mechanical models for membranes
3.2.1. Shaft-loaded deformation
Most bio-membranes exhibit large and time-dependent deformation
pertaining to the mathematically involved rubber elasticity. We will
begin with classical linear elasticity. A thin film with planar elastic
modulus, E , and Poisson's ratio,
2 ),
thickness, b , and radius, a , being clamped at the periphery forms a
freestanding diaphragm. An external force, F , applied to the film center
via a ball bearing or a spherical capped shaft with radius, R , results in a
= Eb 3 /12(1
ν
, bending rigidity
κ
−ν
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