Multiscale Elastic Models of Collagen Bio-structures: From Cross-Linked Molecules to Soft Tissues - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 86

the overall tissue [ 2 , 8 ], present results recover the tissue stiffening experienced at
the macroscale, confirming as this occurrence can be associated with the formation
of mature cross-links [ 57 , 58 ].

6 Microscale Mechanics: Fibers

A collagen fiber is modeled as a homogeneous beam with a circular cross-section
of radius r F and with a periodic planar centerline in both reference and actual
configurations (Fig. 7 ). Let the along-the-chord nominal strain measure be defined
as e F ¼ L = L o 1 ; L (respectively L o ) being the actual (reference) fiber period.
Moreover, let the Cartesian frame ð t ; n Þ be introduced, with the unit vector t
aligned with the actual fiber chord direction, and let x be the coordinate along t.
Accordingly, the centerline position vector results in r ð x ; e F Þ¼ f ð x ; e F Þ n þ x t,
where f ð x ; e F Þ denotes the centerline curve, whose slope with respect to t is defined
by the angle a ð x ; e F Þ .
Following a constrained Hu-Washizu variational approach, and disregarding
any shear and Poisson-related effect, the tangent equivalent along-the-chord fiber
modulus E eq results in [ 59 ]:

þ A F f 2 1 ;

i I F cos 2 a E eq ð e F Þ¼ E c I F cos a h ð 12 Þ

where E c is the tangent elastic modulus of the fiber material along the direction
perpendicular to the fiber cross-section, A F ¼ pr F ; I F ¼ pr F = 4, and symbol hi
denotes the curvilinear average operator defined along the curvilinear coordinate s
following the fiber centerline, that is

Z Lð e F Þ

1
Lð e F Þ

hi¼ ds ; ð 13 Þ

0

Lð e F Þ being the length over a period of the actual fiber centerline. By assuming the
fiber material behavior to be governed by the fibril non-linear constitutive
response, that is enforcing the constitutive inter-scale condition E c ¼ E f ð e f Þ , nano-
and mesoscale mechanical effects are straightforwardly included in the mechanical
description of a collagen fiber. Nominal strain e f associated with e F , that is the
function e f ¼ e f ð e F Þ , is computed from the following inter-scale compatibility
relationship:

e f ¼Lð e F Þ=Lð 0 Þ 1 ; ð 14 Þ

with

Lð e F Þ¼ Z L
0

df
cos a :

ð 15 Þ