Biomedical Engineering Reference
In-Depth Information
where the fourth invariant
I
4
,
c
/
s
can be expressed as scalar product of the right
Cauchy-Green tensor
C
and
Z
c
/
s
.
5.2.2 Balance Equations
Using classical non-linear continuum mechanics, a coupled problem of chemo-
mechanical SM contraction is formulated in terms of two primary field variables,
namely the placement
ϕ
(
X
,t)
and the calcium concentration
c(
X
,t)
. Consequently,
a chemomechanical state
S
of a material point
X
at the time
t
is defined as
S
(
X
,t)
=
ϕ
(
X
,t),c(
X
,t)
.
(5.4)
Spatial as well as temporal evolution of the primary field variables are governed by
two basic field equations: the balance of linear momentum and the diffusion-type
equation of excitation through calcium.
The balance of linear momentum in spatial form
+
b
div
σ
=
0
in
B
(5.5)
describes the quasi-static stress equilibrium. Herein
b
is the given spatial body force
per unit reference volume. The operator div
indicates the divergence with respect
to the spatial coordinates
x
, and
σ
denotes the Cauchy stress tensor given as
[•]
2
J
−
1
F
∂Ψ(
ϕ
)
∂
C
F
T
,
σ
=
(5.6)
depending on the deformation measures
C
and
F
as well as on a strain-energy func-
tion
Ψ(
ϕ
)
, see Sect.
5.2.3
. The mechanical problem is completed by essential and
natural boundary conditions,
=
t
ϕ
= ¯
ϕ
on
∂
B
ϕ
and
t
on
∂
B
σ
.
(5.7)
The surface stress traction vector
t
, defined on
∂
B
σ
, is related to the Cauchy stress
tensor
σ
via the Cauchy stress theorem
t
:=
σ
n
, where the outward surface normal
is specified as
n
.
The second field equation describes the calcium concentration inside the SM
tissue. The well-known Fick's second law
c
˙
=−
div
q
in
B
(5.8)
predicts how diffusion causes the concentration field
c
to change with time. Herein,
the diffusion flux vector
q
=−
d
(
ϕ
)
∇
x
c
(5.9)