ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

v J
2
é l ˆ1
2 ù
å ê b b b 1b b 2b
b ú
(10)
ë û
b=
1
y = y ( ) + y ( l ) + y ( a ) + y ( a , l )
The shear energies are:
ˆ 1 b 1 b b
yb( a1 ) = m1 ( a1
2
- 2)
(11)
( ) 2
2 b b ˆ2
1 b b b
y b( a2, lb) = fm( l) ´ y ( a2) = m2 a2
-1
fm
( lb)
2
(12)
where
b
1
fm
( lb)
=
c -bb ( lb-lb)
1+ ab
e
(13)
is a sigmoid coupling function between along-the-fibre shear and tensile deformations,
while the volumetric energy is:
v 1 2
y ( J) = k(
J -2lnJ - 1)
4
and the energies stored in the two families of fibres are:
(14)
ìï
2
ïÀ b 2 2
b r0b
ï bm0( lb + - 3)
+ x0Àb
ln( l ) if lb
£ 1
ï
lb
ï
ï
æ ö
l
ç ÷
yb( lb)
= ï
í ç
÷
2 2
÷
ï L
ç
r 2
b lb 0b 1 lbr
÷
ï
ç 0b
b
r0b
b q
ç ÷
ïÀ
K ç 2 ÷ x
2
0 b ln( lb ) if lb
> 1
b ç + -
4L
ç
p L lbr0b
L
÷ + À
ï ç
÷
ï b
b ÷
ï ç ( 1 ) ÷
ç
-
ï
÷
ïî
çè L
÷
b ø
(15)
where
x
2 2 2 4 6
KqL
r0 L r0 L r0 r
b
b b b b - b b + 0b
0 =-
4 L 2 2 2
p Lb( Lb - r0b)
(6 9 4 )
Collagen fibres are allowed to sustain compressive forces and this is introduced through
b
a neo-Hookean law featuring a shear modulus m 0 .
The orthotropic hyperelastic model features 23 constitutive parameters:
0
b b b b
c
p = { K , q, k, m0, m1, m 2, Àb,
Lb, Lp, r0b,
ab, bb,
lb}
b=
1,2
(17)
This set of parameters is reduced to p by assuming that the matrix parameters associated
1 2
with each family of fibres are identical. p = p È p is further split into two subsets:
1
c c
2
n p = { K , q, km , 0,
ab , , Àn
, À , l , l }
0 m and 0 m
p =
0
0 n0 m
{ m1, m2 , Ln , L , , , , , }
0
0 p rn À L L
0 m0 m0 p rm
. The
0
first set of parameters was obtained from the literature (see [14]) and/or directly
estimated from visual inspection of the stress-strain curves of uniaxial tensile tests
performed on rabbit skin [12]. The identification of 2
p was performed using a
constrained non-linear optimisation procedure using a global hill-climbing algorithm
(GLOBAL OPTIMIZATION 8, Loehle Enterprises, Naperville, IL, USA) within the
MATHEMATICA ® 7 environment (Wolfram Research Inc., Champaign, IL, USA).
4. RESULTS AND DISCUSSION
The results of the identification are illustrated on Fig. 1 (detailed values of each
constitutive parameter are provided in [1]). The constitutive formulation is shown to
reproduce faithfully the anisotropic characteristics of rabbit skin (Fig. 1.). Overall, the
fit of the present constitutive model to the experimental data is better than that obtained
using a transversely isotropic worm-like chain model [16] which relies on network
properties of a one family of fibres model to induce orthotropic properties.
(16)