Biomedical Engineering Reference
In-Depth Information
Herewith, the external load is specified using the constants
α
and
β
; the constant
α
β
for the 'bending'. With the above given boundary conditions the displacement of
the membrane as a rigid body is not suppressed; uniqueness of the displacement
solution is obtained, if it is additionally required that the material point coinciding
with origin O is fixed in space and if in the deformed state the symmetry with
respect to the
y
0
-axis is maintained. For the given problem an exact analytical
solution can be calculated. It is easy to verify that the solution has the following
form:
is representative for the 'normal' force in the
x
0
direction and the constant
E
b
x
0
1
y
0
u
x
(
x
0
,
y
0
)
=
α
+
β
y
0
x
0
2
b
1
E
u
y
(
x
0
,
y
0
)
=−
να
y
0
+
νβ
2
b
+
β
1
E
y
0
b
ε
xx
(
x
0
,
y
0
)
=
α
+
β
ε
yy
(
x
0
,
y
0
)
=−
E
b
y
0
α
+
β
ε
xy
(
x
0
,
y
0
)
=
0
y
0
b
σ
xx
(
x
0
,
y
0
)
=
α
+
β
σ
yy
(
x
0
,
y
0
)
0
σ
xy
(
x
0
,
y
0
)
=
0.
=
It has to be considered as an exception, when for a specified plane stress problem
an analytical solution exists. In general, only approximate solutions can be deter-
mined. A technique to do this is the Finite Element Method, which is the subject
of the Chapters
14
to
18
. Chapter
18
is especially devoted to the solution of linear
elasticity problems as described in the present chapter.
13.2.6
Boundary conditions
In the previous sections only simply formulated boundary conditions have been
considered. In the case of dynamic boundary conditions, the components of the
stress vector
p
(see Fig.
8.1
) are prescribed at a point along the boundary of
the volume of the continuum (in case of plane stress along the boundary of the
configuration surface). In case of kinematic boundary conditions the displace-
ment vector
u
is prescribed. Sometimes the boundary conditions are less explicitly
defined, however, for example when the considered continuum interacts with its
environment. In the following an example of such a situation will be outlined.
