Biomedical Engineering Reference
In-Depth Information
(5.7H), then (2) substitute the resulting expression relating the stress to the first
derivatives of the displacement into the three stress equations of motion (
6.18
). The
result is a system of three equations in three scalar unknowns, the three components
of the displacement vector. This algebraic simplification will be accomplished in
the case of an isotropic material after Hooke's law for isotropic materials is
developed in the next paragraph.
The stress-strain and strain-stress relations of the anisotropic Hooke's law,
(5.7H) and (5.24H), respectively, are developed next. It has been shown that the
tensor of elastic material coefficients
C
is symmetric and positive definite, as is its
inverse
S
, the compliance tensor of elastic material coefficients. The strain-stress
relations (as opposed to the stress-strain relations) are
¼ C
1
E
¼ S
T
; S
; ð
5
:
12H
Þ
repeated
The form and symmetry of
C
and
S
are identical for any material symmetry and,
for the material symmetries of interest, the form appropriate to the material
symmetry is given in Tables 4.4 and 4.5. The notation employed thus far for
C
and
S
, the notation that allows their representation as second order tensors in six
dimensions, is not the traditional notation. To obtain the traditional notation,
Hooke's law (5.7H) is expressed in its matrix format,
2
4
3
5
2
4
3
5
2
4
3
5
T
1
T
2
T
3
T
4
T
5
T
6
E
1
E
2
E
3
E
4
E
5
E
6
^
c
11
^
c
12
^
c
13
^
c
14
c
15
^
c
16
^
^
c
21
^
c
22
^
c
23
^
c
24
c
25
^
c
26
^
^
c
31
^
c
32
^
c
33
^
c
34
c
35
^
c
36
^
¼
;
(6.19)
^
c
41
^
c
42
^
c
43
^
c
44
c
45
^
c
46
^
^
c
51
^
c
52
^
c
53
^
c
54
c
55
^
c
56
^
^
c
61
^
c
62
^
c
63
^
c
64
c
65
^
c
66
^
and then converted to the traditional three-dimensional component representation
by employing the relations (A163),
2
4
3
5
2
4
p
c
14
p
c
15
p
c
16
3
5
2
4
3
5
T
11
T
22
T
33
E
11
E
22
E
33
c
11
c
12
c
13
p
c
24
p
c
25
p
c
26
c
12
c
22
c
23
p
c
34
p
c
35
p
c
36
c
13
c
23
c
33
¼
;
p
T
23
p
E
23
p
c
14
p
c
24
p
c
34
2
c
44
2
c
45
2
c
46
p
T
13
p
E
13
p
c
15
p
c
25
p
c
35
2
c
45
2
c
55
2
c
56
p
T
12
p
c
16
p
c
26
p
c
36
p
E
12
2
c
46
2
c
56
2
c
66
(6.20)
and introducing the matrix coefficients
c
ij
,
i
,
j
, 6, defined in Table
6.1
.Itis
easy to verify that the matrix equation (
6.20
) may be rewritten as
¼
1,
...
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