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

Therefore, we have decided to carry out a new parameters’ identification based on the
minimization of a cost function, which quantifies and tries to minimize the
differences between the experimental curves and the ones predicted by the
constitutive equation, for the four regions experimentally characterized by Holzapfel
et al. (2005). It was found that if the set of parameters ( 2
k 1,k ) were reduced to 25% on
the circumferential direction (ventral to dorsal) and to 14.5% on the radial direction
(external to internal), then thus determined a set of parameters described in a good
agreement with the experimental mechanical behavior of the fibers, as can be seen in
Fig.5. As example of application of the above described constitutive model of the AF,
the model shown in Fig. 2 was submitted to a compressive displacement. Fig.6 shows
a compilation of the main results in terms of force and NP pressure evolution up to a
compressive displacement of 0.6 mm.
Force [N]
700
600
500
400
300
200
100
Force
NP Pressure
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0
0 0.2 0.4 0.6 0.8
Compressive Displacement [mm]
Pressure [MPa]
Data from Sato et al. 1999
Intradiscal
Pressure*
Spinal
Load*
0.535 MPa 800 N
(*) upright standing position
Figure 6. Evolution of the compressive force and intradiscal pressure with a compressive displacement
of up to 0.6mm; on right side some reference values from Sato et al. (1999) for the L4-L5 level in a
compressive loading equivalent upright standing position.
5. DISCUSSION
The material parameters proposed in the literature for the constitutive modeling of the
AF seems to have a quite not understandable variation of their values. In the model here
proposed, the identified constitutive parameters, coupled with both a circumferential
and radial linear evolution, seems to be in a very good agreement with the experimental
characterization carried out and published by Holzapfel et al. (2005), in such a way that
one can avoid the meaninglessness of the sets of parameters proposed by Eberlein et al.
(2001). The variation of the mechanical behavior of the fibers from ventral to dorsal and
from external to internal regions cannot be neglected, once it will drive the numerical
simulations of the IVD biomechanics. According to the experimental in-vivo study of
Sato et al. (1999), the spinal load and intradiscal pressure at L4-L5 level in an upright
standing position, are of around 800N and 0.535MPa, respectively. The preliminary
results of the compression of the IVD seem to be in a good agreement with these
experimental values, for an anatomically acceptable compressive displacement of
around 0.6mm.
6. CONCLUSION
A new constitutive model of the AF is proposed and briefly described in this work. The
major novelties are two: firstly, the fibers’ angles are defined as a function of the polar
angles, evolving from 23.2º at ventral region to about 43.6º at dorsal region; secondly,
the constitutive parameters of the anisotropic behavior introduced by the fibers are