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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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