Multiscale Elastic Models of Collagen Bio-structures: From Cross-Linked Molecules to Soft Tissues - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 81

e m ¼ e s m þ e m : ð 2 Þ

Denoting with r ' ¼ ' m ; o =' c , the molecular tangent modulus E s m that describes the
entropic mechanisms can be introduced, in agreement with the WLC model [ 8 ], as

( )

E s m ð e s m Þ¼ q

r '
2 ½ 1 r ' ð 1 þ e s m Þ 3 þ r '

: ð 3 Þ

A m

Moreover, the molecular tangent modulus E m dealing with energetic mechanisms

is represented as [ 15 ]:

Er '

1 þ e g ð r ' e m e o Þ þ E o r ' ;

E m ð e m Þ¼ ð 4 Þ

where E ; E o ; g and e o are model parameters.
Expression ( 4 ) is able to grasp the experimental evidence that the molecular
mechanical response is characterized in the first stage by a pseudo-exponential
law, and then asymptotically tends to a linearly elastic behavior for high values of
' m =' c . The introduced parameters have a clear physical meaning and, thereby, their
values can be set by numerical atomistic computations or, when available, by
experiments: E and E o govern the asymptotic behavior of E m ð e m Þ for e m !1; g
describes the slope of the function E m ð e m Þ for e m ¼ e o , and e o is the molecular
strain contribution within the energetic regime at which E m practically attains its

mean value.
Therefore, the molecular tangent modulus accounting for the series elasticity
induced by entropic and energetic mechanisms results in:

E m ð e m Þ¼ E s m ð e s m Þ E m ð e m Þ

E s m ð e s m Þþ E m ð e m Þ ;

ð 5 Þ

and the strain contributions due to entropic and energetic effects, that is the
functions e s m ¼ e s m ð e m Þ and e m ¼ e m ð e m Þ , are obtained by solving the following
differential problem:

e s m ¼ E m ð e m Þ e m e m ¼ E m ð e m Þ e m

; : ð 6 Þ

E s m ð e s m Þ E m ð e m Þ

Accordingly, the description of the entropic/energetic transition directly
derives from the mutual competition of the stiffnesses associated to E s m and E m : when
E s m E m molecular mechanics is mainly governed by the entropic mechanisms
(E m E s m ), when E s m E m by the energetic ones (E m E m ), whereas when
E s m E m both mechanisms significantly contribute. As the analysis of Eqs. ( 6 )
reveals, present model of the entropic/energetic transition is based on equilibrium
and compatibility conditions, despite of the phenomenological approaches intro-
duced in [ 13 , 14 ].