Biomedical Engineering Reference
In-Depth Information
For this, the definition of the deformation rate tensor
D
of Section
10.6
is used.
The first term on the right-hand side of Eq. (
11.15
) represents the resulting, externally
applied power to the volume element by the forces (originating from neighbouring
elements) acting on the outer surfaces of the element:
σ
= ∇·
dP
ext
(
σ
·
v
)
dV
.
(11.16)
This can be proven with a similar strategy as was used in the previous section, to
derive the resultant of the forces acting on the outer surfaces of the volume element.
Interpretation of the second term on the right-hand side,
−
tr(
σ
·
D
)
dV
,
will be given below.
Summarizing, after reconsidering Eq. (
11.11
), and using the results of the above,
it can be stated:
d U
kin
,
dP
ext
σ
−
tr(
σ
·
D
)
dV
+
dP
ext
=
(11.17)
q
and after some re-ordering:
tr(
σ
·
D
)
dV
+
d U
kin
=
dP
ext
,
(11.18)
with
dP
ext
=
dP
ext
σ
+
dP
ext
q
.
(11.19)
So, it can be observed that the total externally applied mechanical power is partly
used for a change of the kinetic energy. The remaining part is stored internally, so:
dP
int
=
tr(
σ
·
D
)
dV
.
(11.20)
This internally stored mechanical power (increase of the internal mechanical
energy per unit of time) can be completely reversible (for elastic behaviour), partly
reversible and partly irreversible (for visco-elastic behaviour) or fully irreversible
(for viscous behaviour). In the latter case all externally applied mechanical energy
to the material is dissipated and converted into other forms of energy (in general
a large part is converted into heat).
11.5
Lagrangian and Eulerian description of the balance equations
In summary, the balance equations of mass and momentum as derived in the
sections
11.2
and
11.3
can be written as
=−
ρ
∇·
v
,
balance of mass
ρ
=−
ρ
tr(
D
)
∇·
σ
+
ρ
=
ρ
˙
balance of momentum
q
=
ρ
a
v
,
