Biomedical Engineering Reference
In-Depth Information
u
∗
=
x
∗
−
+
λ
−
x
0
=
P
·
x
x
0
+
λ
−
+
λ
=
P
·
(
x
0
+
u
)
x
0
=
(
P
−
I
)
·
x
0
+
P
·
u
+
λ
−
+
λ
=
P
·
x
(
x
−
u
)
=
(
P
−
I
)
·
x
+
u
.
(9.38)
In the following, the attention is focussed on a fixed material point with posi-
tion vector
x
0
in the reference configuration, the position vector
x
in the current
configuration at time
t
and the position vector
x
∗
in the extra rotated and translated
virtual configuration.
For a scalar physical variable, for example the temperature
T
, the value will not
change as a result of an extra rigid body motion, thus, with respect to the same
material point:
T
∗
=
T
.
For the gradient operator, applied to a certain physical variable connected to the
material, it follows, based on the relation between
x
∗
, directly:
x
and
()
∗
∇
=
P
· ∇
=
P
·
F
−
T
· ∇
0
( ) .
()
(9.39)
Note, that the gradient is the same operator for the real as well as for the imaginary,
extra displaced, current configuration. However, the effect on for example the
temperature field
T
and the field
T
∗
is different. Eq. (
9.39
) shows this difference.
For the deformation tensor of the virtual configuration with respect to the
undeformed reference configuration it is found that
∇
0
x
∗
T
)
T
F
∗
=
∇
0
(
P
·
x
+
λ
=
∇
0
(
x
·
P
T
+
λ
)
T
F
T
·
P
T
T
=
=
=
P
·
F
.
(9.40)
Finally, the influence of the rigid body motion on the stress state will be deter-
mined. Assume, that the internal interaction between the material particles, with
an exception for the direction, will not change because of the motion as a rigid
body. For the considered material point the stress tensor
σ
relates the stress vec-
tor
p
on a surface element with the unit normal
n
of that element, according to:
p
=
σ
·
n
. Because the imaginary configuration is rotated with respect to the
current configuration, both vectors
n
∗
and
p
∗
can be written as
n
∗
=
P
·
n
p
∗
=
P
·
p
.
and
(9.41)
This reveals
p
∗
=
P
·
p
=
P
·
σ
·
n
=
P
·
σ
·
P
−
1
·
n
∗
=
P
P
T
·
n
∗
,
·
σ
·
(9.42)
