Biomedical Engineering Reference
In-Depth Information
Fig. 5
Fibril model. Notation
that accounts for series mechanisms occurring between molecular and cross-link
stretching, and that is consistent with experimental evidences [
26
]. It is worth
pointing out that parameter k does not intervene in the compatibility equation (
8
)
because covalent cross-links occurring upon a given molecule are assumed to act
in parallel.
Assuming a linearly elastic behavior with sway stiffness k
cl
for each cross-link,
a measure of the nominal fibril stress along f is
r
f
¼
lr
m
¼
kk
cl
d
=
A
f
;
ð
9
Þ
with r
m
expressed by Eq. (
7
) and where l is the average measure of the ratio
between solid (occupied by molecules) and total cross-section. Accordingly, by
combining Eqs. (
8
) and (
9
), the fibril tangent elastic modulus along f results in:
1
1
E
m
ð
e
m
Þ
þ
A
m
kk
cl
'
m
;
o
E
f
ð
e
f
Þ¼
l
;
ð
10
Þ
where the contribution to the molecular elongation due to the fibril strain, that is
the function e
m
¼
e
m
ð
e
f
Þ
, is obtained by solving the following inter-scale equi-
librium differential problem:
E
f
ð
e
f
Þ
lE
m
ð
e
m
Þ
e
f
:
e
m
¼
ð
11
Þ
In the following, since fibrils within tissues are made up of densely packed
molecules and in agreement with evidences proposed by [
55
], the model parameter
l is set equal to one.
In Fig.
6
, numerical results obtained considering a uni-axial traction of a col-
lagen fibril are shown, highlighting the capability of the proposed approach to
predict the evolution of both molecular and fibril tangent moduli versus e
f
.In
agreement with recent theoretical evidences at the atomistic scale [
2
,
56
], lower
values of the fibril stiffness with respect to the molecular ones are successfully
reproduced, as induced by the stretching of inter-molecular cross-links. Moreover,
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