Biomedical Engineering Reference
In-Depth Information
The diffusion coefficient for the solvent in the polymer is given as
23
/
L
1
3
ξ
ξ
K
(
)
−
sp
p
0
a
(6.6)
D
=
12
−
χ
+
sp
ξ
12
/
Ν
K
+
53
ξ
H
p
a
The hydrogen diffusion coefficient through solvent is formulated as
−
LKa
HK
−
+
ξ
sp
p
D
=
(6.7)
s
1
2
(
)
2
H
ξ
a
p
The nomenclature for material parameters used here is that of Flory (1953a, 1953b)
and Grimshaw, Nussbaum, Grodzinsky, et al. (1990).
The stress relationships for large deformation elasticity require the use of large
deformation strain quantities. The deformation gradient
F
(
t
) is defined as
F
(
t
) =
are the locations of the particle in the deformed and unstrained
states, respectively. In this problem, it is useful to factor the deformation gradient
into its unimodular part and a part representing isotropic swell:
∂
x
/
∂
X
, where
x
,
X
g
g
g
g
()
=
()
⋅
(
)
FF
t
t
α
I
(6.8)
uni
where
is the volumetric swell of the gel.
For a solvent concentration-dependent neo-Hookean type solid, the Cauchy
stress,
a
(
t
) = (det[
F
(
t
)])
and
α
1/3
3
S
, resulting from a given deformation is
()
=
()
()
⋅
()
T
−
St
Gc
I
−
EE
t
t
p
I
(6.9)
where
()
=
()
−
1
E
t
F
uni
t
,
(6.10)
is a Lagrange multiplier dual to the incom-
pressibility constraint on the swollen polymer. Because of the assumed incompressibility
of the gel/solvent system, the preceding equation presents stress only up to an unknown
pressure. Constitutive modeling of rubber-like materials is discussed with good clarity
in Segalman et al. (1992a, 1992b, 1992c, 1993) and Segalman and Witkowski (1994).
The incompressibility condition on the swollen polymer is simply a statement
that the volume of the material is not a function of the imposed pressure.
The conservation of momentum for the gel is
Note that
G
is the shear modulus and
p
=∇⋅
ρ
x
S
+
ρ
f
(6.11)
gg
gb
where
f
contains all local body forces, such as gravitational or electromagnetic forces.
b
