Biomedical Engineering Reference
In-Depth Information
of that constituent. If it is assumed that there can be no charge accumulation in
the mixture, this electroneutrality condition must be enforced with the following
constraint:
z
α
c
α
=
0
.
(17.5)
α
Multiplying the mass balance in Eq. (
17.3
) with
z
α
/M
α
, taking the sum over
all constituents and making use of the above electroneutrality constraint produces
div
α
z
α
j
α
=
0, or equivalently,
div
I
e
=
0
,
(17.6)
F
c
α
z
α
j
α
is the electric current density (the net rate of flow of electric
charge per unit area of the mixture) and
F
c
is Faraday's constant. Thus, the elec-
troneutrality condition produces a constraint on the current density vector field in
the mixture.
In most biological mixtures, the solvent is water and thus neutral (
z
w
where
I
e
=
=
0). The
charge on the solid matrix is described as a net fixed charge density
c
F
≡
z
s
c
s
. Thus,
for solids,
c
F
may be used in lieu of a molar concentration and associated charge
number.
17.2.3 Momentum Balance
Mixture models of biological tissues typically neglect the effects of the viscosity of
fluid constituents in comparison to frictional interactions between constituents. The
validity of this modeling assumption may be easily tested by performing compar-
isons of the orders of magnitude of these various terms, given representative values
of the relevant material properties. For example, in a solid-fluid mixture of artic-
ular cartilage, the non-dimensional number representing the ratio of internal fluid
friction (viscosity) to fluid-solid frictional drag is
δ
η/h
2
K
(Hou et al.,
1989
),
where
η
is the fluid viscosity,
h
is a characteristic length, and
K
is the solid-fluid
frictional drag coefficient. In articular cartilage,
η
=
10
−
3
10
−
3
≈
·
≈
Pa
s,
h
m, and
10
15
s
/
m
4
, thus
δ
10
−
12
, showing a very negligible relative contribution
≈
·
≈
K
N
of fluid viscosity.
With negligible viscosity, the Cauchy stress tensor
σ
in a mixture of intrinsically
incompressible constituents only includes two contributions: The hydrostatic pres-
sure
p
in the fluid, and the Cauchy stress
σ
s
in the solid, thus
σ
σ
s
, where
I
is the identity tensor. Under quasi-static conditions, and in the absence of external
body forces, the momentum balance for the mixture is given by
=−
p
I
+
div
σ
s
div
σ
=−
grad
p
+
=
0
.
(17.7)
A constitutive relation is needed to relate
σ
s
to the state variables adopted for a
particular analysis, such as solid matrix strain.
