Biomedical Engineering Reference
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Fig. 4 Left: strain rates associated with entropic (e
s
m
) and energetic (e
m
) mechanisms (normalized
with respect to e
m
) computed by the present approach. Right: collagen F
m
-
'
m
curves obtained by
experimental tests [
12
], MDS [
9
], classical WLC, and present model. Parameters: T
¼
310
:
15 K,
'
p
¼
16 nm,
'
c
¼
301
:
7 nm,
'
m
;
o
¼
1 nm, E
o
¼
5 GPa, E
¼
100 GPa, g
¼
10, e
o
¼
0
:
35
Moreover, in Fig.
4
b, the F
m
-
'
m
curve obtained in this case is successfully compared
with the experimental data proposed in [
12
] (available only for low molecular
forces), as well as with the numerical results in [
9
].
5 Mesoscale Mechanics: Fibrils
Collagen fibrils are modelled as long right cylinders. Let f be the fibril axis
direction, and A
f
a measure of the fibril cross-sectional area. From a computational
point of view, fibrils are thought as a collection of N
m
identical one-dimensional
molecules, with axis aligned along f and mutually interacting through N
cl
identical
cross-links, which connect pairs of adjacent molecules (see Fig.
5
). Molecules in
reference configuration are assumed to be long
'
m
;
o
¼ '
c
'
kinks
,
'
kinks
being the
length measure of molecular kinks.
Let n
s
be the average number of molecules along the fibril length and
'
f
;
o
the
reference fibril length. Disregarding axial offset among molecules, the following
relationship holds:
'
f
;
o
n
s
'
m
;
o
. Moreover, let k be an average measure of cross-
link occurrence for each molecule, such that N
cl
¼
kN
m
. In the case of a homo-
geneous traction and defining d as the sway of a cross-link along the fibril length, it
is consistent to assume that both molecular strain e
m
and cross-link stretch d are
space-independent quantities within the fibril domain.
Furthermore, denoting with D
'
f
the fibril length variation and with e
f
the fibril
nominal strain, the following kinematic assumption is enforced:
;
D
'
f
¼
n
s
ð'
m
;
o
e
m
þ
d
Þ!
e
f
¼
D
'
f
'
f
;
o
¼
e
m
þ
d
'
m
;
o
ð
8
Þ
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