Biomedical Engineering Reference
InDepth Information
:
ð
3
:
211
Þ
Þ¼
w Q
C
Q
T
;
Q
K
1
Q
T
;
Q
K
2
Q
T
; ::::;
Q
K
N
Q
T
w G
;
K
1
;
K
2
; ::::
K
N
;
ð
The ''solution'' of (
3.211
) is given through the power expansion (each P = ?)
by
w
¼
w G
;
K
1
;
K
2
; ::::
K
N
ð
Þ
2
4
0
@
1
A
:::::
0
@
1
A
GG
::::
G
3
5
!
¼
P
P
P
::::::
P
P
P
P
P
ð
s
Þ
:
K
1
K
1
::::
K
1
K
N
K
N
::::
K
N
J
{z}
s
fold
scalar
{z}
k
times

{z
}
m
1
times

{z
}
m
N
times
k
¼
2
m
1
¼
0
m
2
¼
0
m
N
¼
0
ð
3
:
212
Þ
where the expansion coefficients arise in the form of families of isotropic tensors
ð
s
Þ
of sth grade, composed of M isotropic tensors of sth order I
j
!
:
¼
X
M
k
þ
X
N
k
¼
X
N
ð
s
Þ
ð
s
Þ
c
ð
s
Þ
j
J
I
j
;
s
¼
2
m
a
;
m
a
ð
3
:
213
Þ
j
¼
1
a
¼
1
a
¼
1
with c
ð
s
Þ
k
material parameters (m
a
= 0 for at least one m
a
with a = 1, 2,…, N)
(Silber 1986).
By means of an appropriate recursion relation in terms of the C
AYLEY
H
AMILTON
theorems (
3.212
) could principally be reduced to a polynomial in G. The form
(
3.212
) can be rewritten in a more convenient form
0
1
Þ¼
X
P
ð
2k
Þ
C
@
A
w
¼
w G
;
K
1
;
K
2
; ::::
K
N
ð
:::
GG
::::
G
ð
3
:
214
Þ
{z}
2k
fold
scalar
{z}
k
times
k
¼
2
ð
2k
Þ
where the material tensors of 2nd grade C
generated from dyadic aggregates of
second order direction tensors K
i
are defined by
2
4
0
@
1
A
:::::
0
@
1
A
3
5
Þ
:
¼
P
m
1
¼
0
P
m
2
¼
0
::::::
P
m
N
¼
0
ð
s
Þ
ð
2k
Þ
ð
2k
Þ
:
{z}
s
2
ð Þ
fold
scalar
K
1
K
1
::::
K
1
{z}
m
1
times
K
N
K
N
::::
K
N
{z}
m
N
times
J
C
¼
C
ð
K
1
;
K
2
; ::::
K
N
ð
3
:
215
Þ
Knowing that nonlinear constitutive stressstrain relations in polynomial form
(due to the C
AYLEY
H
AMILTON
recursion) can always be reduced to a quadratic
polynomial in the respective strain measure (
3.214
) may be reduced to the fol
lowing cubic form in G (P = 3)