Biomedical Engineering Reference
In-Depth 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 s-th grade, composed of M isotropic tensors of s-th 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 non-linear constitutive stress-strain 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)
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