Biomedical Engineering Reference
In-Depth Information
power conjugated thermodynamic forces to these fields. By requiring the internal
and external virtual powers to be equal and using the arbitrariness for the virtual
velocity fields, balance laws can be obtained for very general situations where a
classical free-body diagram will fail.
The virtual velocities in this model are taken to be
δλ
,
δε
ft
,
δν
,
δ
n
, and
δq
,
where
n
=
(n
A
,n
B
,n
C
,n
D
)
and
δ(
·
)
denotes the virtual velocity of
(
·
)
. Using these
velocities, we state the internal virtual power as
P
int
=
T
p
δλ
+
T
cd
δε
cd
+
T
ft
(δε
ft
+
δν)
+
T
q
δq,
(6.6)
where
T
p
is the stress in the parallel spring,
T
cd
is the elastic stress in the cross-
bridges,
T
ft
is the stress associated with the friction clutch, and
T
q
is the internal
force power conjugate with the calcium ion flux. The virtual velocity term
δε
cd
in
Eq. (
6.6
) is not an independent virtual velocity, as is clear from Eq. (
6.5
). It is merely
introduced for convenience since it describes the strain rate associated with cross-
bridge deformation. Furthermore, the term
δε
ft
+
δν
is the relative velocity between
the friction clutch disc and actin and is, therefore, the velocity power conjugate
with
T
ft
. Next, we define the external virtual power to be
P
ext
=
+
+
tδλ
t
ft
δν
t
q
δq,
(6.7)
where
t
is the external force applied to the system, and
t
ft
and
t
q
are the thermody-
namic forces power conjugate to
δν
and
δq
, respectively. The second term on the
right-hand side in (
6.7
) is the external power supplied to drive the friction clutch.
Setting Eqs. (
6.6
) and (
6.7
) equal and using the arbitrariness of the virtual veloc-
ity fields, we arrive at the force equilibrium equations
t
=
T
p
+
T
cd
,
cd
=
T
ft
,
t
ft
=
T
ft
,
(6.8)
and an auxiliary equation for the electrochemical equilibrium
t
q
=
T
q
.
(6.9)
The result above demonstrates the advantage of the virtual power method over the
classical force equilibrium approach. Equations (
6.8
) can be obtained from a force
equilibrium, but (
6.9
) cannot be obtained that way; it requires the electrochemical
problem to be considered separately.
6.2.4 Constitutive Equations
With the kinematics and balance laws defined, it is time to focus on the constitutive
equations. These equations are derived by applying the dissipation inequality which
states that the free energy
ψ
must satisfy
ψ
≤
P
int
,
(6.10)
