Biomedical Engineering Reference
In-Depth Information
y
thickness
h
P
x
b
z
The left side (
x
) is loaded with a
distributed tangential load (the resultant force
P
is known) in the negative
y
-direction. The top and bottom surface of the plate are unloaded. A plane
stress condition is supposed. With respect to the stress field the following
assumption is proposed:
=
0) is clamped. The right side (
x
=
σ
xx
=
c
1
(
−
x
)
y
σ
yy
=
0
b
2
4
−
y
2
,
σ
xy
=
with
c
1
and
c
2
constants.
Determine
c
2
based on the relationship between
σ
xy
and
P
and subse-
quently, determine
c
1
by means of the local equilibrium equations.
13.5
In the environment of the origin of a Cartesian
xyz
-coordinate system it is
given that a (two-dimensional) stationary velocity field in an incompress-
ible Newtonian fluid (with density
ρ
) can be described as
v
=
α
(
−
y
e
x
+
x
e
y
) with
α
a constant.
Based on this velocity field the deformation rate tensor
D
can be calcu-
lated:
D
=
0
with
0
the zero tensor. With substitution into the constitutive
equation:
σ
=−
p
I
+
2
η
D
,
with
η
the viscosity and
σ
the stress tensor (and with
p
=
0 originated by
the applied boundary conditions) it follows that
σ
=
0
.
Then, by means of the Navier-Stokes equation it is found that the dis-
tributed load
q
(force per unit mass) necessary to realize the described flow
field is given by
2
q
=−
α
r
with
r
=
x
e
x
+
y
e
y
.
