Biomedical Engineering Reference
In-Depth Information
For a material point P within the subvolume the position vectors in the
reference state as well as the current state are given, respectively:
x
0
P
=
e
x
+
e
y
+
e
z
,
x
P
=
2
e
x
+
3
e
y
−
2
e
z
.
Another point Q within the subvolume appears to be in the origin in the
current configuration:
x
Q
=
0.
Calculate the position vector
x
0
Q
of the point Q in the reference state.
10.2 Within a subvolume of a material continuum the deformation tensor in the
deformed current state, with respect to the reference state, is constant. Con-
sider a vector of material points, denoted with
a
0
in the reference state and
with
a
in the current state. The angle between
a
0
and
a
is given by
φ
.
Prove that we can write for
φ
:
a
0
(
a
0
·
a
0
)(
a
0
·
F
T
a
0
·
F
·
cos(
φ
)
=
.
·
F
·
a
0
)
10.3 Within a subvolume of a material continuum the deformation tensor in the
deformed current state, with respect to the reference state, is constant. In a
Cartesian
xyz
-coordinate system the following deformation tensor is given:
F
=
I
+
4
e
x
e
x
+
2
e
x
e
y
+
2
e
y
e
x
,
with
I
the unit tensor. Within the subvolume two material points P and Q
are considered. The position vectors in the reference state are given:
x
0
P
=
e
x
+
e
y
+
e
z
,
x
0
Q
=
2
e
x
+
3
e
y
+
2
e
z
.
In addition, the position vector for the point P in the current state is given:
x
P
=
2
e
x
+
2
e
y
+
2
e
z
.
Calculate the position vector
x
Q
of the point Q in the current state.
10.4 The deformation of a material particle, in the current state with respect to
the reference state, is fully described by the Green Lagrange strain matrix
E
, with respect to a Cartesian
xyz
-coordinate system, with
⎡
⎤
430
310
003
1
2
⎣
⎦
E
=
.
Calculate the volume change factor
J
for this particle.
10.5 A tendon is stretched in a uniaxial stress test. The tendon behaves like
an incompressible material. The length axis of the tendon coincides with
