Biomedical Engineering Reference
In-Depth Information
19.2.1.3 Kinematics of Superimposed Continua
Based on the fundamental assumption of superimposed continua, the TPM proceeds
from the idea that any spatial point
x
of the current configuration is simultaneously
occupied by material points of all constituents. However, each constituent follows
its own motion such that
x
χ
α
(
X
α
,t)
with
X
α
as the reference position of the
respective material point of
ϕ
α
and time
t
. This leads to the individual velocity fields
x
α
=
=
d
χ
α
(
X
α
,t)/
d
t
. The solid matrix is described by a Lagrangian formulation via
the solid displacement
u
S
=
X
S
as the primary kinematic variable, while the
pore-flow of blood and interstitial proceeds from a modified Eulerian setting via the
seepage velocities
w
ξ
=
x
−
x
ξ
−
x
S
with
ξ
={
B,I
}
. Assuming the velocities of the
liquid solvent and the interstitial fluid to be approximately identical,
x
L
≈
x
I
with
x
D
n
L
n
I
, the pore-diffusion velocity of the therapeutic solute
ϕ
D
reads
d
DI
=
≈
x
I
. This includes the possibility to define a seepage-like velocity
w
D
=
−
d
DI
+
w
I
.
19.2.1.4 Balance Relations
The set of governing equations for the numerical treatment within the finite ele-
ment method consists of the following balance equations, which are obtained from
partial mass and momentum balances (e.g., Ehlers,
2009
). Therein, materially in-
compressible constituents, no mass exchange between the constituents, quasi-static
conditions and a uniform temperature are assumed.
Concentration balance of the therapeutic agent
ϕ
D
:
•
=
n
I
c
m
S
+
div
n
I
c
m
w
D
n
I
c
m
div
(
u
S
)
S
+
0
(19.4)
•
Volume balance of the overall interstitial fluid:
=
n
I
S
+
div
n
I
w
I
+
n
I
div
(
u
S
)
S
0
(19.5)
Volume balance of the blood plasma
ϕ
B
:
•
=
n
B
S
+
div
n
B
w
B
+
n
B
div
(
u
S
)
S
0
(19.6)
=
α
ϕ
α
:
•
Momentum balance of the overall aggregate
ϕ
0
=
div
T
+
ρ
g
.
(19.7)
=
α
T
α
is the overall Cauchy stress, while
b
α
Therein,
T
=
g
characterizes
uniform constant gravitational force.
