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

it is well known the non-linear nature of the elastic behavior of biological soft tissues. In
our study, we chose to use a structurally and functionally oriented, anisotropic,
nonlinear, hyperelastic model with finite deformations formulation based upon the work
of Eberlein et al. (2001). The isotropic behavior of both NP and AF’s ground substance,
was modeled with a Mooney-Rivlin law, in which the Helmholz strain energy density is
given by the following convex strain-energy potencial,
where 1
I and 2
gs C10(
I1
3)
C01(
I 2 3)
(2)
I are the first and second invariants of C (right Cauchy-Green strain
tensor) and 10 C and C 01 are material parameters.
The additional energy density coupled with the AF anisotropic behavior is modeled by a
Holzapfel law,
with
where 0
k and 0
1
f
1
2k
2
k1
e
k
2
2
2
I 41
k I 61
1
k1
e
I 4 f1 C
f1
and 2
k 2 are material parameters ( 2
2
2
1
I 6 f C
f
(4)
k is dimensionless), I 4 and 6
(3)
I are
invariants of , f 1,f are the unit vectors describing fibers’ directions (Fig. 3 c).
To assure that collagen fibers do not answer mechanically to compressive loadings, one
imposes that 1
C and 2
( 4 6
I , I ) .
a) b) c)
Figure 3 a) Axial section of IVD: Polar angle (α) definition, being α=0 o in the midsagittal ventral position (A) and
α=180 o in the midsagittal dorsal position (F); b) Linear regression line (solid) for all fiber angle data (n=324) - dashed
lines indicate the 95% confidence interval for the regression line and dotted lines show the 95% prediction interval of
fiber angles; c) Arrangement of collagen fibers in the reference (undeformed) configuration . (font of a) and b):
Holzapfel et al., 2005, pp. 128, 134; font of c): Eberlein et al., 2001, pp. 215).
The mechanical parameters selected from literature are shown in Table 1. For
additional information the reader is addressed to the work of Castro et al. (2012) within
these proceedings. In case of anisotropy, too different sets of parameters were found,
without a clear rationale justification for such differences.