Biomedical Engineering Reference
In-Depth Information
r
u
¼
A
1
b
u1
q
a
k
1
þ
A
2
b
u2
q
a
k
1
þ
e
o
b
u3
þ
B
k
b
u4
q
;
ð
41
Þ
r
z
¼
A
1
b
z1
q
a
k
1
þ
A
2
b
z2
q
a
k
1
þ
e
o
b
z3
þ
B
k
b
z4
q
;
ð
42
Þ
s
zu
¼
A
1
b
c1
q
a
k
1
þ
A
2
b
c2
q
a
k
1
þ
e
o
b
c3
þ
B
k
b
c4
q
;
ð
43
Þ
where
b
q1
¼
C
g2
þ
a
k
C
g3
;
b
q2
¼
C
g2
a
k
C
g3
;
ð
44
Þ
b
q3
¼
C
g1
þð
C
g2
þ
C
g3
Þ
a
1
;
b
q4
¼
C
g6
þð
C
g2
þ
2C
g3
Þ
a
2
;
ð
45
Þ
with the index q
¼
z
;
u
;
q
;
c corresponding to the g-index values g
¼
1
;
2
;
3
;
6,
respectively, and with
a
k
¼
C
22
=
C
33
;
ð
46
Þ
a
1
¼ð
C
12
C
13
Þ=½
C
33
ð
1
a
k
Þ;
a
2
¼ð
C
26
2C
36
Þ=½
C
33
ð
4
a
k
Þ;
ð
47
Þ
C
ij
denoting the
ð
i
;
j
Þ
component of the stiffness matrix
k
for the kth MLU.
It is worth pointing out that during the deformation path both strain and stress
components, and thereby also the strain-dependent MTU material properties, are
z- and u-independent.
Assuming the MLUs as perfectly bonded layers, the 3N unknown constants
A
1
;
A
2
;
B
k
(with k
¼
1
:::
N) are solved by imposing the following 3
ð
N
1
Þ
continuity conditions at the MLU's interfaces:
C
u
q
ð
r
k
Þ¼
u
k
þ
1
u
u
ð
r
k
Þ¼
u
k
þ
1
r
r
ð
r
k
Þ¼
r
k
þ
1
ð
r
k
Þ;
ð
r
k
Þ;
ð
r
k
Þ;
ð
48
Þ
q
u
r
and the three equilibrium incremental relationships:
Z
r
k
2p
X
N
q
2
s
zu
dq
¼
0
;
r
r
ð
r
i
Þ¼
p
;
r
r
ð
r
i
þ
h
a
Þ¼
0
;
ð
49
Þ
r
k
1
k
¼
1
the latter prescribing the average torque related to the tangential stress increment
s
zu
to be zero.
References
1. van Holde, K.E., Matthews, C.: Biochemistry. Benjamin/Cummings Publishing Company
Inc., Menlo Park (1995)
2. Fratzl, P.: Collagen: Structure and Mechanics. Springer, New York (2008)
3. Bruel, A., Oxlund, H.: Changes in biomechanical properties, composition of collagen and
elastin, and advanced glycation endproducts of the rat aorta in relation to age. Atherosclerosis
127, 155-165 (1996)
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