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

z
y
x
z
z
x
(c)
y
x
(d)
Figure 1. 3D sketch of the IVD (a); sketch for computation of (b); sketch for
computation of (c); FE model (d) with tetrahedral, λ=2, (left) and hexahedral
mesh, λ=8, (right).
[MPa] [MPa] [MPa] [MPa] [MPa]
-
[MPa]
0.0033 67.02 0.036 0.069 2.16 0.093 53.23
Table 1. Finite element
model properties.
Where is the right Cauchy-Green deformation tensor, is the deformation
gradient tensor measure of the transformation, det is the Jacobian determinant, a
measure of the volume change. ̅ is the first invariant of the deviatoric deformation
tensor / and is the square of the fibre stretch.
, , , , , and , are the material coefficients for the NP and the AF,
see Table 1. The constitutive equations have been implemented in the ABAQUS finite
element package (ABAQUS 6.9.3, Hibbit, Karlsson and Sorenson Inc., Pawtucket, RI,
USA) with FORTRAN subroutines and tested through simple FE models. Linear fullyintegrated
hexahedral elements (C3D8) and second order fully-integrated elements
(C3D10) were used for the discretization of the system. The seed size was chosen by
dividing the disc height by a factor λ. For the hexahedral mesh, we have chosen λ = 8,
whereas λ = 2, 4, 6, 8 were used for the tetrahedral element models. We assumed the AF
to be comprised of eight alternating layers, with the fibre orientation described as in
Holzapfel et al. [15]. The study cases were: Axial compression (-1mm), flexion (5°),
torsion (5°) and lateral bending (5°). A pre-stress analysis was implemented prior to the
studies to compute the stress distribution induced by a swelling pressure of 0.1MPa.
4. RESULTS
(a)
The load-displacement curves obtained from the tetrahedral and hexahedral mesh
models showed a comparable disc response (Figure 2.a). The model with a coarse
tetrahedral mesh (λ=2) showed similar results to the hexahedral mesh in axial
compression (max. error 4.6%) and flexion (max. error 6.1%) when responses were
compared at the organ level. However, the coarse tetrahedral model started to diverge
when lateral bending greater than 2.5° was applied (max. error 17.6%). This was
improved by using a model with a finer tetrahedral mesh (λ=4). For torsion, both
y
x
(b)