Multiscale Elastic Models of Collagen Bio-structures: From Cross-Linked Molecules to Soft Tissues - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 93

Fig. 12 Model of the multi-layered aortic tunica media. Notation. The function h F ð q Þ has been
assumed in agreement with evidences proposed in [ 36 ]

Indicating with S ij (respectively, C ij and S ij ) the components of the 6 6

compliance matrix

k

ð q Þ ) expressed in the standard
Voigt notation, the homogenized tangent compliance results from [ 60 , 63 ]:

S (respectively of C ð q Þ and S

2
4

3
5

S 11 S 12 S 16

¼ h IL ð A k Þ 1 ;

S 21 S 22 S 26

ð 30 Þ

S 61 S 62 S 66

i ¼ð A k Þ 1 Z r k
r k 1

;

h

S 31 S 32 S 36

Q k ð q Þ dq

½ S 13 ð q Þ S 23 ð q Þ S 36 ð q Þ ð 31 Þ

Z r k

n o dq ;

S 33 ¼ 1

1
C 33 ð q Þ

1 S 31 C 13 ð q Þ S 32 C 23 ð q Þ S 36 C 63 ð q Þ

ð 32 Þ

h IL

r k 1

" # ¼ 1

h IL

dq ;

Z r k

S 44 S 45

S 44 ð q Þ S 45 ð q Þ

ð 33 Þ

S 54 S 55 S 54 ð q Þ S 55 ð q Þ

r k 1

where

2
4

3
5 ;

A k ¼ Z r k
r k 1

C 11 ð q Þ C 12 ð q Þ C 16 ð q Þ

Q k ð q Þ¼ Q k ð q Þ dq ;

C 21 ð q Þ C 22 ð q Þ C 26 ð q Þ

ð 34 Þ

C 61 ð q Þ C 62 ð q Þ C 66 ð q Þ

and where r k 1 and r k are the minimum and the maximum radial coordinate of the
kth MLU, respectively.
Since non-fibrous components within interlamellar space are essentially elastin
and muscle cells, and the stiffness of muscle cells can be retained as comparable to
the elastin's [ 64 ], as a first approximation no distinction between elastin and muscle
phases is assumed. Accordingly, the computation of ð q Þ via Eqs. ( 23 )-( 29 ) can be

carried out by applying the mixture rule and by letting:
E M

C

:

E e ¼ m M

V o
1 V f m e ¼ 1

ð 35 Þ