Biomedical Engineering Reference
In-Depth Information
Fig. 6 Top left: fibril (E
f
) and molecular (E
m
) tangent moduli vs. fibril nominal strain e
f
(computed for kk
cl
¼
10 pN/nm). Top right: ratio between molecular and fibril moduli E
m
=
E
f
vs.
fibril nominal strain e
f
for different values of kk
cl
. Bottom left: fibril tangent modulus E
f
(in log
10
scale) vs. kk
cl
(in log
10
scale) for different values of fibril nominal strain e
f
. Bottom right:
molecular nominal strain e
m
and cross-link normalized extension d
='
m
;
o
vs. fibril strain e
f
for
different values of kk
cl
:(
) kk
cl
¼
1 pN/nm; (M) kk
cl
¼
10 pN/nm; (
) kk
cl
¼
100 pN/nm.
Constant parameters:
'
p
¼
14
:
5 nm,
'
c
¼
287 nm,
'
kinks
¼
22 nm, E
o
¼
1 GPa, E
¼
100 GPa,
g
¼
10, e
o
¼
0
:
35, A
m
¼
1
:
41 nm
2
;
T
¼
310
:
15 K, l
¼
1
a quantitative estimate of the fibril stiffness variation due to changes in cross-link
mechanics is provided, showing that fibril stiffness significantly varies both in
terms of absolute values and with respect to the molecule's. An increase in cross-
link occurrence produces an increase in fibril modulus up to a saturation level that
corresponds to the molecular stiffness. Furthermore, numerical results clearly show
the non-linear dependence of molecular and cross-link strain measures on both
fibril strain and occurrence/stiffness of cross-links.
These results at the mesoscale recover and justify also other significant
evidences. Since identical collagen molecules likely exhibit identical nano-
mechanical responses, the wide range of values for fibril/fiber modulus, generally
reported in the specialized literature as a result of microscale experimental
investigations (0.2-12 GPa [
2
]), can be justified via the proposed results as a
consequence of different occurrence and mechanical response of cross-links.
Moreover, since after the removal of fibril (fiber) geometrical crimp the
mechanical response of the fibrils' material (corresponding to cross-linked
collagen molecules) can be considered as representative of the elastic behavior of
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