Biomedical Engineering Reference
In-Depth Information
m
F
;
LT
¼
m
F
;
TT
¼
m
M
;
G
F
;
LT
¼
E
M
=½
2
ð
1
þ
m
M
Þ;
ð
22
Þ
with E
eq
ð
e
F
Þ
as in Eq. (
12
) and where the subscript L refers to the direction
identified by t, whereas T to any direction orthogonal to t.
Once crimped collagenous fibers are reduced to equivalent straight fibers,
through the previous homogenization step and accounting for geometrical and
material non-linearities, standard arguments for fiber-reinforced composite mate-
rials can be employed in order to describe the macromechanics of soft collagenous
tissues.
7.1 Uni-directional Tissues: Tendons and Ligaments
A further homogenization step at the macroscale is carried out by employing the
mixture rule [
60
]. Accordingly, a uni-directional collagenous tissue is reduced to a
homogeneous medium with a transversally isotropic behavior, the isotropy plane
being orthogonal to t. Therefore, tangent equivalent elastic constants of the tissue
at the along-the-chord fiber strain level e
F
result in:
E
F
;
T
E
M
E
F
;
T
ð
1
V
f
Þþ
E
M
V
f
E
L
ð
e
F
Þ¼
V
f
E
F
;
L
ð
e
F
Þþð
1
V
f
Þ
E
M
;
E
T
¼
;
ð
23
Þ
1
1
V
f
G
F
;
LT
þ
1
V
f
G
M
V
f
G
F
;
TT
þ
1
V
f
G
M
G
LT
¼
;
G
TT
¼
;
ð
24
Þ
E
T
2G
TT
m
LT
¼
V
f
m
F
;
LT
þð
1
V
f
Þ
m
M
;
m
TT
¼
1
;
ð
25
Þ
where V
f
is the fiber volume fraction. Referring to the standard Voigt notation, the
tangent
~
stiffness
matrix
C in
the
material
coordinate
system
(t,n,k),
with
k
¼
t
n, is:
;
ð
e
F
Þ¼
L
ð
e
F
Þ
0
~
C
ð
26
Þ
0
M
where
2
3
E
L
ð
1
m
TT
Þ
E
T
m
LT
ð
1
þ
m
TT
Þ
E
T
m
LT
ð
1
þ
m
TT
Þ
ð
e
F
Þ¼
1
D
4
5
;
E
T
ð
1
jm
LT
Þ
E
T
ð
m
TT
þ
jm
LT
Þ
L
E
T
m
LT
ð
1
þ
m
TT
Þ
ð
27
Þ
E
T
ð
m
TT
þ
jm
LT
Þ
E
T
ð
1
jm
LT
Þ
E
T
m
LT
ð
1
þ
m
TT
Þ
D
ð
e
F
Þ¼
1
m
TT
2
ð
1
þ
m
TT
Þ
jm
LT
;
M
¼
diag G
TT
;
G
LT
;
G
LT
ð
Þ;
ð
28
Þ
with E
L
¼
E
L
ð
e
F
Þ
and j
¼
j
ð
e
F
Þ¼
E
T
=
E
L
ð
e
F
Þ
.
Accordingly, at the macroscale and in a global coordinate system, the tangent
homogenized constitutive law for the tissue results in:
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