Biomedical Engineering Reference
In-Depth Information
The trace of tensor
M
(associated matrix
M
) is denoted as tr(
M
)
=
tr(
M
) and
given by
tr(
M
)
=
tr(
M
)
=
M
xx
+
M
yy
+
M
zz
.
(7.38)
For the determinant of the tensor
M
with matrix representation
M
it can be written:
det(
M
)
=
det(
M
)
=
M
xx
(
M
yy
M
zz
−
M
yz
M
zy
)
−
M
xy
(
M
yx
M
zz
−
M
yz
M
zx
)
+
M
xz
(
M
yx
M
zy
−
M
yy
M
zx
) .
(7.39)
The deviatoric part of the tensor
M
is denoted by
M
d
and defined by
1
3
tr(
M
)
I
.
M
d
=
M
−
(7.40)
In matrix notation this reads:
1
3
tr(
M
)
I
.
M
d
=
M
−
(7.41)
Let
M
be an arbitrary tensor. A non-zero vector
n
is said to be an
eigenvector
of
M
if a scalar
λ
exists such that
(
M
−
λ
I
)
·
n
=
0.
M
·
n
=
λ
n
or
(7.42)
A non-trivial solution
n
from Eq. (
7.42
) only exists if
det(
M
−
λ
I
)
=
0.
(7.43)
Using the components of
M
and Eq. (
7.39
) will lead, after some elaboration, to
the following equation:
3
2
λ
−
I
1
λ
+
I
2
λ
−
I
3
=
0,
(7.44)
with
I
1
=
tr(
M
)
I
2
=
M
xx
M
yy
+
M
xx
M
zz
+
M
yy
M
zz
−
M
xy
−
M
yz
−
M
xz
I
3
=
det(
M
) .
(7.45)
Eq. (
7.44
) is called the
characteristic equation
and the scalar coefficients
I
1
,
I
2
and
I
3
are called the invariants of tensor
M
. In tensor form the invariants can be
written as
