Biomedical Engineering Reference
In-Depth Information
stress due to deformation will
ultimately
supersede the residual stress.
The λ-value for the bending-stretching transition must be redefined
investigated by cyclic loading of the central load, and the materials
parameters can be obtained by expressing the elastic modulus in the
complex form,
E
=
E'
+
j
E”
, in the above model, assuming the simplest
standard model of linear viscoelasticity.
3.2.2.
Large-deformation model
When an ultra-thin biomembrane is investigated using the shaft-loaded
blister test, a large deformation model is needed in terms of rubber
elasticity. Without loss of generality, the Mooney-Rivilin model is
adopted for analysis, where the strain-energy function is given by
CW
(10-19)
where
I
= λ
1
2
+ λ
2
2
+ (λ
1
λ
2
)
−2
and
II
= λ
1
−2
+ λ
2
−2
+ (λ
1
λ
2
)
2
are the strain
invariants,
=
(
I
−
3
+
C
(
II
−
3
1
2
λ
2
are the meridian and circumferential stretch ratios
for the deformed membrane, and
C
1
and
C
2
are the material parameters
of the membrane. For a central load applied via a ball bearing, the
membrane deformation is governed by the Young-Laplace equation
λ
1
and
dT
1
1
+
(
T
−
T
)
=
0
(10-20)
1
2
d
K
T
+
K
T
=
0
(10-21)
1
1
2
2
with ρ = ρ(
r
) the radial coordinate parallel to the undeformed plane,
T
1
and
T
2
the corresponding tensile stress, and
K
1
and
K
2
the principal
curvatures. The equilibrium deformed profile gradient is governed by
1
/
2
2
d
ρ
F
=
λ
1
−
(10-22)
a
2
d
ξ
[
2
π
a
λ
T
(
λ
,
λ
)]
p
1
a
p
with
/
dr
the meridian gradient along the deformed membrane
profile and λ
a
= ρ/
r
. A numerical iterative method can be devised to
λ
p
=
d
ξ
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