Biomedical Engineering Reference
In-Depth Information
The stresses have to satisfy the equilibrium equations. For plane stress this
means that only equilibrium 'in the plane' (in the case considered therefore in
the
x
0
- and
y
0
-direction) results in non-trivial equations:
∂σ
xx
∂
x
0
+
∂σ
xy
∂
y
0
+
ρ
0
q
x
=
0
(13.20)
∂σ
xy
∂
x
0
+
∂σ
yy
y
0
+
ρ
0
q
y
=
0.
(13.21)
∂
Summarizing, it can be stated that in case of plane stress eight scalar functions
of the coordinates in the midplane have to be calculated: the displacements
u
x
and
u
y
, the strains
σ
xy
. For this objective
we have eight equations at our disposal: the strain/displacement relations (three),
the constitutive equations (three) and the equilibrium equations (two). In addition,
for each boundary point of the midplane two scalar boundary conditions have to
be specified and for uniqueness of the displacement field rigid body motion has to
be suppressed.
ε
xx
,
ε
yy
and
ε
xy
, and the stresses
σ
xx
,
σ
yy
and
Example 13.2
In Fig.
13.3
a simple plane stress problem is defined for a rectangular membrane
(length 2
l
, width 2
b
and thickness
h
) with linearly elastic material behaviour
(Young's modulus
E
and Poisson's ratio
ν
). The mathematical form for the
boundary conditions reads:
y
0
b
for
x
0
=±
l
and
−
b
≤
y
0
≤
b
it holds:
σ
xx
=
α
+
β
σ
xy
=
0
for
y
0
=±
b
and
−
l
≤
x
0
≤
l
it holds:
σ
yy
=
0
σ
xy
=
0.
σ
yy
=
σ
xy
=
0
thickness
h
y
0
x
0
y
0
b
σ
xx
=
α
β
+
2
b
O
σ
xy
=
0
2
Figure 13.3
A simple plane stress problem.
