Biomedical Engineering Reference
In-Depth Information
0
@
1
A
¼
C
ð
2k
Þ
Þ¼
P
3
ð
4
Þ
ð
6
Þ
C
w
¼
w G
;
K
1
;
K
2
; ::::
K
N
ð
:::
GG
::::
G
|{z}
2k
fold
scalar
|
{z
}
k
times
G
ðÞþ
C
GG
ð Þ
k
¼
2
ð
3
:
216
Þ
where
ð
4
Þ
:
¼
X
X
ð
6
Þ
:
¼
X
X
X
P
P
;
P
P
P
ð
8
Þ
ð
12
Þ
C
C
J
ab
K
a
K
b
J
abc
K
a
K
b
K
c
a
¼
1
b
¼
1
a
¼
1
b
¼
1
c
¼
1
ð
3
:
217
Þ
are material tensors of fourth and sixth order, generated from families of isotropic
ð
8
Þ
ð
12
Þ
tensors of eighth and twelfth order J
ab
which represent the direction
properties of the respective anisotropies in the form of fourth and sixth order,
respectively, tensor functions of the second order direction tensors K
i
.
Strain Energy Functions of Second Order (for Tensor-Linear Constitutive
Stress-Strain Relations). Restricting to tensor-linear constitutive stress-strain
relations (
3.216
) reduces to the quadratic term in G (P = 2) such that
and J
abc
ð
4
Þ
ð
4
Þ
:
¼
X
P
X
P
:
ð
8
Þ
Þ¼
C
G
ð ;
C
w
¼
w G
;
K
1
;
K
2
; ::::
K
N
ð
J
ab
K
a
K
b
a
¼
1
b
¼
1
ð
3
:
218
Þ
In the case of orthotropic materials, the direction effects are characterized by
three direction tensors (following Boehler 1975, 1979)
K
a
M
a
:
¼
e
a
e
a
ð
a
¼
1
;
2
;
3
Þ
a not summed!
ð
Þ
ð
3
:
219
Þ
with the direction vectors e
a
characterizing the respective anisotropy and the
properties
P
M
a
¼
P
M
a
f ur
a
¼
b
3
3
M
a
M
b
¼
e
a
e
a
¼
I
:
ð
3
:
220
Þ
0
f ur
a
6¼
b
;
a
¼
1
a
¼
1
Using (
3.219
), (
3.218
) transforms into (P = 3)
ð
4
Þ
Þ¼
C
w
¼
w G
;
M
1
;
M
2
;
M
3
ð
G
ðÞ
ð
3
:
221
Þ
ð
4
Þ
:
¼
X
X
3
3
and
:
¼
P
ð
8
Þ
ð
8
Þ
105
ð
8
Þ
c
ð
ab
Þ
j
C
with
J
ab
M
a
M
b
J
ab
I
j
:
j
¼
1
a
¼
1
b
¼
1
