Biomedical Engineering Reference
In-Depth Information
are related according to
⎡
⎤
0
−
ω
z
ω
y
⎣
⎦
∼
=
∼
(
∼
)
=
(
∼
−
∼
S
) with
=
ω
z
0
−
ω
x
.
(7.28)
−
ω
y
ω
x
0
The spin matrix
that is associated with the angular velocity vector
ω
(with
T
components
∼
) is skew symmetric:
=−
. With the uniform rotation as a
rigid body considered in this section, we find:
∼ ∼
T
T
∼
S
)
T
T
T
(
I
T
T
L
=
=
∼
(
∼
−
=
+
O
)
=
(7.29)
and for the associated velocity gradient tensor
L
:
L
=
ω
x
(
e
z
e
y
−
e
y
e
z
)
+
ω
y
(
e
x
e
z
−
e
z
e
x
)
+
ω
z
(
e
y
e
x
−
e
x
e
y
) .
(7.30)
7.7
Some mathematical preliminaries on second-order tensors
In the chapters that follow, extensive use will be made of second-order tensors.
This section will summarize some of the mathematical background on this subject.
An arbitrary second-order tensor
M
can be written with respect to the Cartesian
basis introduced earlier as
M
=
M
xx
e
x
e
x
+
M
xy
e
x
e
y
+
M
xz
e
x
e
z
+
M
yx
e
y
e
x
+
M
yy
e
y
e
y
+
M
yz
e
y
e
z
+
M
zx
e
z
e
x
+
M
zy
e
z
e
y
+
M
zz
e
z
e
z
.
(7.31)
The components of the tensor
M
are stored in the associated matrix
M
defined as
⎡
⎣
⎤
⎦
M
xx
M
xy
M
xz
M
=
.
(7.32)
M
yx
M
yy
M
yz
M
zx
M
zy
M
zz
A tensor identifies a linear transformation. If the vector
b
is the result of the tensor
M
operating on vector
a
, this is written as:
b
=
M
·
a
. In component form this
leads to:
b
=
(
M
xx
e
x
e
x
+
M
xy
e
x
e
y
+
M
xz
e
x
e
z
+
M
yx
e
y
e
x
+
M
yy
e
y
e
y
+
M
yz
e
y
e
z
+
M
zx
e
z
e
x
+
M
zy
e
z
e
y
+
M
zz
e
z
e
z
)
·
(
a
x
e
x
+
a
y
e
y
+
a
z
e
z
)
