Biomedical Engineering – From Theory to Applications

454
In this case, the eigenvalues are:
and the matrix of eigenvectors is:
where:
X Biomedical Engineering – From Theory to Applications
1
λ 1=
C 11+C 12+C 33+ 2 2 2
C 11+C12-C 33 +8C
13
1
λ 2=
C 11+C 12+C33- 2 2 2
C 11+C12-C 33 +8C
13
1
λ 3=λ 6=
C11-C 12+C 44+ 2 2 2
C11-C12 -C 44 +16C
15
1
λ 4=λ 5=
C11-C 12+C 44 -
2 2 2
C11-C12 -C 44 +16C
15
1 1 1 1
sin cos cos 0 sin 0
2 2 2 2
1 1 1 1
sin cos cos 0 sin 0
2 2 2 2
cos sin 0 0 0 0
0 0 0 cos0 sin
0 0 sin
0 cos 0
0 0 0 sin0
cos
λ -C -C
λ1-C11-C12 sin
; cos
;
2
2
2
2C + λ -C -C
2C + λ -C -C
2 11 12
2
13 2 11 12
11 12 3
2 2
11 12 3 15
13 1 11 12
sin
C -C -λ
; cos
C -C -λ
(9)
C -C -λ +4C
C -C -λ +4C
Particular cases:
a. In the case of the cubic crystallographic structure:
The eigenvalues are:
11 12 4
2 2
11 12 4 15
C12=C13 ; C33=C11 ; C44=C11-C12 ; C15=0 (10)
S C C C 0 0 0
11
C12 C12 12
C11 C12 12
C12 C11 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
C44 0
0
0
C44 0
0
0
C44
(7)
(8)
(11)
λ 1=C 11+2C 12;
λ 2=λ 3=C11-C 12 ; λ 4=λ 5=λ 6=C 44
(12)