Biomedical Engineering Reference
In-Depth Information
and also
T
)
T
(
∼
σ
+
ρ
q
∼
=
ρ
∼
.
(11.10)
This equation is called the local equation of motion. Again, all variables in this
equation refer to the current configuration; the defined reference configuration (as
usual for a solid) is not relevant for this.
11.4
The local balance of mechanical power
It has to be stated explicitly that no new balance law is introduced here; only use
will be made of relations that were already introduced before. Again the element
is considered, that was introduced in Fig.
11.1
(volume in reference state
dV
0
,
current volume
dV
JdV
0
). The dot product of the local equation of motion
Eq. (
11.9
), with the velocity vector
v
is taken. The result is multiplied by the
current volume of the element, yielding
=
dV
∇·
σ
v
·
+
v
·
q
ρ
dV
=
v
·
a
ρ
dV
.
(11.11)
The terms in this equation will be interpreted separately.
•
For the term on the right-hand side it can be written:
d U
kin
,
v
·
a
ρ
dV
=
(11.12)
with
1
2
1
2
1
2
dU
kin
=
v
·
v
ρ
dV
=
v
·
v
ρ
JdV
0
=
v
·
v
ρ
0
dV
0
,
(11.13)
where
dU
kin
is the current kinetic energy of the considered volume element. With respect
to the material time derivative that is used in Eq. (
11.12
) it should be realized that
ρ
JdV
0
=
ρ
0
dV
0
is constant.
The term on the right-hand side of Eq. (
11.11
) can be interpreted as the change of the
kinetic energy per unit time.
•
The second term on the left-hand side of Eq. (
11.11
) can be directly interpreted as the
mechanical power, externally applied to the volume element by the distributed load
q
.It
can be noted that
dP
ext
=
v
·
q
ρ
dV
.
(11.14)
q
•
For the first term on the left-hand side of Eq. (
11.11
) some careful mathematical
elaboration, using the symmetry of the stress tensor
σ
, leads to
dV
tr
v
)
dV
∇·
σ
= ∇·
(
σ
·
(
∇
= ∇·
(
σ
·
v
·
v
)
dV
−
σ
·
v
)
dV
−
tr(
σ
·
D
)
dV
. (11.15)
