Biomedical Engineering Reference
In-Depth Information
·
dx
+
dx dt
=
dx
+
L
·
dx dt
z
υ
dt
dx
y
x
x
Figure 10.4
Change of a material line segment
dx
after a time increment
dt
.
where
v
=
˙
x
specifies the velocity of the material and
L
is the velocity gradient
tensor. The tensor
L
is purely a current variable, not in any way related to the
reference configuration.
Using the deformation tensor it can be written:
dx
=
F
·
dx
0
(10.57)
and therefore
d
F
F
F
−
1
x
=
·
d
x
0
=
·
·
d
x
,
(10.58)
accordingly resulting in the relation between the tensors
L
and
F
:
=
F
F
−
1
.
L
·
(10.59)
It is common practice to decompose the velocity gradient tensor
L
in a symmet-
rical part
D
and a skew symmetrical part
. The tensor
D
is called the
rate of
deformation tensor
and the tensor
the
rotation velocity tensor
or
spin tensor
.
The definitions are:
∇
v
T
∇
v
2
L
+
L
T
2
F
·
F
−
1
·
F
T
1
1
2
1
+
F
−
T
D
=
=
+
=
∇
v
T
∇
v
2
L
−
L
T
2
F
·
F
−
1
·
F
T
1
1
2
1
−
F
−
T
=
=
−
=
(10.60)
and so
d
x
=
(
D
+
)
·
d
x
,
(10.61)
with
D
T
=
D
(10.62)
