Biomedical Engineering Reference
In-Depth Information
and
∇
0
·
σ
+
ρ
0
q
=
0.
(13.12)
The equation of motion is reduced to the equilibrium equation. Only boundary
conditions are necessary to solve this set. Initial conditions are no longer appli-
cable. In every point of the outer surface of
V
0
three (scalar) relations have to be
specified. In this context, to find a unique solution for the displacement field, it is
necessary to suppress movement of the configuration as a rigid body. This will be
further elucidated in the following example.
Example 13.1
Consider a homogeneous body with reference volume
V
0
under a given hydro-
static pressure
p
. For the stress field in
V
0
satisfying the (dynamic) boundary
conditions and the equilibrium equations, it can be written:
σ
(
x
0
)
=−
p
I
for all
x
0
in
V
0
.
For the strain field, according to Hooke's law it is found:
p
3
K
I
for all
ε
x
0
)
=−
x
0
in
V
0
.
(
A matching displacement field is, for example,
p
3
K
x
0
.
It is easy to verify that the above given solution satisfies all equations. However,
because in the given problem description the displacement as a rigid body is not
prescribed the solution is not unique. It can be proven that the general solution for
the components of the displacement vector reads:
u
(
x
0
)
=−
p
3
K
x
0
+
c
1
−
c
6
y
0
+
c
5
z
0
u
y
=−
u
x
=−
p
3
K
y
0
+
c
2
+
c
6
x
0
−
c
4
z
0
p
3
K
z
0
+
c
3
−
c
5
x
0
+
c
4
y
0
,
with
c
i
(
i
=
1, 2,
...
6) yet undetermined constants. The constants
c
1
,
c
2
and
c
3
represent translations in the coordinate directions and the constants
c
4
,
c
5
and
c
6
(small) rotations around the coordinate axes.
u
z
=−
13.2.5
Linear plane stress theory, static
Consider a flat membrane with, in the reference configuration, constant thickness
h
. The 'midplane' of the membrane coincides with the
x
0
y
0
-plane, while in the
