Biomedical Engineering Reference
In-Depth Information
y
0
midplane
u
y
u
x
x
0
z
0
thickness h
mechanical load
Figure 13.1
Configuration with plane stress state.
direction perpendicular to that plane the domain for
z
0
is given by:
−
h
/
2
≤
z
0
≤
h
/
2. The thickness
h
is supposed to be small with respect to the dimensions 'in
the plane'. The loading is parallel to the midplane of the membrane, see Fig.
13.1
.
The midplane will continue to be a symmetry plane after deformation. It is
assumed that straight material line segments, initially perpendicular to the mid-
plane will remain straight (and perpendicular to the midplane) after deformation.
For the displacements this means
u
x
=
u
x
(
x
0
,
y
0
),
u
y
=
u
y
(
x
0
,
y
0
) .
(13.13)
The relevant strain components of the linear strain matrix
ε
for the membrane are
ε
xx
=
∂
u
x
x
0
ε
yy
=
∂
u
y
∂
y
0
∂
(13.14)
∂
u
x
.
∂
y
0
+
∂
u
y
1
2
ε
xy
=
ε
yx
=
∂
x
0
In general, from the other components
ε
zz
will certainly not be zero, while
ε
xz
=
ε
zx
and
ε
yz
=
ε
zy
will vanish. Actually, these other strain components are
not important to set-up the theory.
With respect to the stress, it is assumed that only components 'acting in the
x
0
y
0
-plane' play a role and that for those components (see Fig.
13.2
) it can be
written:
