Multiscale Elastic Models of Collagen Bio-structures: From Cross-Linked Molecules to Soft Tissues - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 82

Fig. 3 Left: comparison among F m - ' m curves obtained for a collagen molecule by experimental
tests [ 54 ], classical WLC, and present model. Right: molecular (E m ), entropic (E s m ), and energetic
(E m ) tangent moduli vs. ' m for a collagen molecule. Parameters: T ¼ 310 : 15 K, ' p ¼ 14 : 5 nm,
' c ¼ 215 nm, ' m ; o ¼ 1 nm, E o ¼ 0 : 1 GPa, E ¼ 10 GPa, g ¼ 10 ; e o ¼ 0 : 65

For the uni-axial traction problem of a collagen molecule, the force pair
F m applied at the molecular ends can be computed in terms of the actual molecular
length ' m as F m ¼ A m r m , where the molecular stress measure results from:
r m ð e m Þ¼ Z e m
0

E m ð n Þ dn ;

ð 7 Þ

E m being expressed by Eq. ( 5 ).
In Fig. 3 , the F m - ' m curve obtained by means of the present approach is
compared with the experimental data proposed in [ 54 ] and with results computed
by the classical WLC model, that is by employing Eq. ( 1 ). As reported by many
authors [ 9 , 54 ], the WLC model fits well the molecular response experienced in a
low-force regime, but predicts an unrealistic molecular inextensibility at high
forces. On the contrary, the proposed approach exhibits an excellent agreement in
the overall force range herein addressed, accounting for the molecular compliance
during the entropic/energetic transition. In Fig. 3 b the molecular tangent modulus
is plotted versus the molecular extension, highlighting that E m E s m within the
entropic regime, and E m E m in the energetic one.
Another verification of consistency and soundness of the proposed model for the
entropic/energetic transition follows from Fig. 4 a: it shows the computed strain
rates e s m and e m (normalized with respect to e m ) plotted versus the molecular length
' m . Reference is made to the numerical analyses proposed in [ 9 ], wherein for a
collagen molecule with ' c ¼ 301 : 7 nm, MDS-based results predict that the entropic
regime dominates for ' m \280 nm and the energetic one for ' m [ 317 nm. This
evidence is fully recovered by present results obtained by solving Eqs. ( 6 ): e m ! 0
for ' m \280 nm and e m = e m ! 1 for ' m [ 317 nm. It is worth pointing out that
present equilibrium and compatibility conditions ensure that the pole at e s m ¼
1 = r ' 1 for the function E s m ð e s m Þ is never reached (e s m ! 0 for ' m [ 317 nm).