Biomedical Engineering Reference
In-Depth Information
A chart relating the component notations of the matrix
c
(and its inverse
s
) to the
component notations for
C
and
S
is given in Table
6.1
. Table
6.1
also relates these
coefficients to the traditional notation for the representation of these tensors as
fourth-order tensors in a three-dimensional space, a notation that is not employed in
this text. The various components in Table
6.1
are either equal or differ by multiples
of
2 from each other. In the case of orthotropic symmetry it follows from Table 4.4
and (
6.19
) through (
6.21
) that
√
2
3
2
3
2
3
T
11
T
22
T
33
T
23
T
13
T
12
c
11
c
12
c
13
000
c
12
c
22
c
23
000
c
13
c
23
c
33
000
000
c
44
00
0000
c
55
0
00000
c
66
E
11
E
22
E
33
2
E
23
2
E
13
2
E
12
4
5
4
5
4
5
¼
;
(6.22a)
or
2
4
3
5
¼
2
4
3
5
2
4
3
5
;
T
11
T
22
T
33
c
11
c
12
c
13
E
11
E
22
E
33
c
12
c
22
c
23
(6.22b)
c
13
c
23
c
33
T
23
¼
2
c
44
E
23
;
T
13
¼
2
c
55
E
13
;
T
12
¼
2
c
66
E
12
;
and, in the case of isotropic symmetry, it follows again from Table 4.4 and (
6.19
)
through (
6.21
) that
2
4
3
5
2
4
3
5
2
4
3
5
T
11
T
22
T
33
T
23
T
13
T
12
l þ
2
ml
l
000
E
11
E
22
E
33
2
E
23
2
E
13
2
E
12
l
l þ
ml
2
000
l
l
l þ
m
2
000
¼
;
(6.23)
m
0
0
0
00
0
0
0
0
m
0
0
0
0
0
0
m
where the coefficients
c
11
and
c
12
are expressed in terms of the Lame´ moduli of
elasticity,
l
and
m
;
c
11
¼ l þ
2
m
and
c
12
¼ l
(note that
c
11
¼ l þ
^
2
m
and
c
12
¼ l
^
).
In equation (5.11N) the Greek letters
are also used to denote the viscosity
coefficients. This dual use for these Greek letters will continue throughout the text
as they are traditional notations in elasticity theory and in viscous fluid theory. The
reader should keep in mind that the significance of
l
and
m
will depend upon
context, viscous fluid or elastic solid. Developing the six scalar equations that come
from the matrix equation (
6.23
) in algebraic analogy with the transition from (5.8N)
l
and
m
Search WWH ::
Custom Search