Biomedical Engineering Reference
In-Depth Information
local form reads
ʨ(
˕
,
F
,ˆ,
∇
X
ˆ,
a
0
i
,ʺ)
=
ʨ
int
(
F
,ˆ,
∇
X
ˆ,
a
0
i
,ʺ)
+
ʨ
ext
(
˕
).
(4)
2.3 Total Potential Energy
The total potential energy of a system additively combines the internal contribution
ʠ
int
, reflecting the action of internal forces, and an external contribution
ʠ
ext
=
ʠ
vol
+
ʠ
sur
due to volume and surface forces, i.e.
ʠ(
˕
,
F
,ˆ,
∇
X
ˆ
;
a
0
i
,ʺ)
=
ʠ
int
+
ʠ
ext
.
(5)
The internal energy contribution can be written as
ʠ
int
(
F
,ˆ,
∇
X
ˆ
;
a
0
i
,ʺ)
=
ʨ
int
d
V
.
(6)
B
0
while the external contributions, assuming 'dead' loads, are provided by
B
ʠ
vol
(
˕
)
=
ʨ
vol
d
V
=−
·
˕
d
V
,
(7)
B
0
B
0
T
ʠ
sur
(
˕
)
=
ʨ
sur
d
A
=−
·
˕
,
d
A
(8)
∂B
0
∂B
0
where
B
denotes the body force vector per unit reference volume and
T
characterises
the traction vector per unit reference surface area. In this regard, see, for instance,
Waffenschmidt and Menzel [
15
] where a double-layered thick-walled cylindrical
tube subjected to internal pressure is analysed on the basis of a total potential.
2.4 Variational Form
The boundary value problem is governed by the principle of minimum potential
energy
min
˕, ˆ
ʠ(
˕
,
F
,ˆ,
∇
X
ˆ
;
a
0
i
,ʺ),
(9)
