Biomedical Engineering Reference
In-Depth Information
Combination of this equation with Darcy's constitutive law leads to:
∇
p
∇·
=
0.
(13.46)
This equation for the pressure
p
can formally be solved when for every boundary
point of the volume
V
one single condition is specified. This can either be formu-
lated in the pressure
p
, or in the outward mass flux
ρ
v
·
n
=−
κ
(
∇
p
)
·
n
. When the
solution for
p
is determined it is easy to calculate directly the mass flux
ρ
v
with
Darcy's law.
In the one-dimensional case (with
x
as the only relevant independent variable)
the differential equation for
p
reduces to
d
2
p
dx
2
=
0.
(13.47)
In this case
p
will be a linear function of
x
.
Exercises
13.1 Consider a material element with the shape of a cube (length of the
edges
). The cube is placed in a Cartesian
xyz
-coordinate system, see
figure.
z
y
x
All displacements from the bottom face of the element (coinciding with the
xy
-plane) are suppressed. The top face has a prescribed displacement in the
y
-direction, which is small with respect to
. The side faces are unloaded.
Assume that a homogeneous stress state occurs with
σ
yz
=
σ
zy
, the only
components of the stress matrix
σ
unequal to zero.
Why can this assumption not be correct?
13.2 Consider a thin rectangular piece of material (constant thickness
h
). The
midplane of the material coincides with the
xy
-plane of a Cartesian
xyz
-
coordinate system. The material behaviour is described by means of
Hooke's law (Young's modulus
E
and Poisson's ratio
ν
). The plate is
