Biomedical Engineering Reference
In-Depth Information
The time rate computed by holding the places
x
fixed is called the local time rate.
In general, the material time rate is related to the local time rate by the following
operator expression that follows from (
2.29
),
x
fixed
þ
Dt
¼
@
D
v
r;
(2.30)
@
t
where
D
/
Dt
is the material time rate of change,
/
t
is the local time rate of change
∂
∂
and
v
determines the convective change of the quantity.
The second order tensor formed by taking the spatial gradient of the velocity
field
v
∇
¼
v
(
x
,
t
) is called the tensor of velocity gradients and is denoted by
L
, thus
2
3
@
v
1
@
v
1
@
v
1
4
5
@
x
1
@
x
2
@
x
3
¼
@
v
i
@
v
2
@
v
2
@
v
2
T
L
¼r
½
v
¼
:
(2.31)
@
x
j
@
x
1
@
x
2
@
x
3
@
v
3
@
v
3
@
v
3
@
x
1
@
x
2
@
x
3
L
is decomposed into a symmetric part
D
called
the rate-of-deformation tensor
, and
a skew symmetric part
W
called the
spin tensor
, thus
L
T
L
T
L
¼
D
þ
W
;
D
¼ð
1
=
2
Þð
L
þ
Þ;
W
¼ð
1
=
2
Þð
L
Þ:
(2.32)
The three nonzero components of
W
can be formed into an axial vector (1/2)
(
v
) which represents the local rotational motion and is called the angular
velocity or one-half the vorticity.
The rate-of-deformation tensor
D
defined by the second of (
2.32
) has the
component representation
∇
2
4
3
5
2
@
v
1
@
v
1
x
2
þ
@
v
2
@
v
1
x
3
þ
@
v
3
@
x
1
@
@
x
1
@
@
x
1
1
2
@
v
j
x
i
þ
@
v
i
1
2
@
v
1
x
2
þ
@
v
2
2
@
v
2
@
v
2
x
3
þ
@
v
3
D
¼
¼
:
(2.33)
@
@
x
j
@
@
x
1
@
x
2
@
@
x
2
@
v
1
x
3
þ
@
v
3
@
v
2
x
3
þ
@
v
3
2
@
v
3
@
@
x
1
@
@
x
2
@
x
3
The components of
D
along the diagonal are called normal rates of deformation
and the components off the diagonal are called shear rates of deformation. The
normal rates of deformation,
D
11
,
D
22
, and
D
33
, are measures of instantaneous time
rate of change of the material filament instantaneously coincident with the 1, 2, and
3 axes, respectively, and the shear rates of deformation,
D
23
,
D
13
, and
D
12
, are equal
to one-half the time rate of decrease in an originally right angle between material
Search WWH ::
Custom Search