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

The constitutive parameters were chosen based on literature data (Table 1) [8, 10-14].
The Holzapfel model (eqs. 10 and 12) was adopted to describe and model the
anisotropic behavior of the AF (due to fibers), and Mooney-Rivlin model to describe the
isotropic behavior (eq. 11). On the following expressions, I are invariants of the
a
MR
deviatoric strain tensor C , and k , k and i C are positive material parameters [10-12]:
( ) iso aniso W W =
( MR ) = C ( I −
MR ) + C ( I − 3)
ij
W C +
(10)
W
iso C 10 1 3 01 2
(11)
[ ]
1
2
2
k ( I4
−1) k ( I6
−1)
W aniso = { k4
[ e −1]
+ k6
e −1}
(12)
2k
The viscous behavior was modeled by coupling the hyperelastic behavior with N
Maxwell elements (serial spring-damper), through the following expression [15]:
K
⎛
⎛ Δt
⎞⎤⎞
⎥
⎟
⎟ K
⎝ ⎠
⎟
⎦⎠
N
visco
k k
UU = ⎜1+
⎜ ∑ ⎢1
− exp
k t ⎜
⎜−
=1 Δ
τ k
⎝
a τ ⎡
Fig. 2 Anterior-posterior cut of the FE model of a lumbar spine MS
Table 1 Material properties of the IVD components
MR
C10 (MPa)
⎣
Isotropy [12] Permeability [8, 13]
MR
C01 (MPa)
UU
*
K 0 (m 4 N -1 s -1 ) M
NP 0.12 0.03 7.5e-16 10.0
AF 0.18 0.045 7.5e-16 12.0
CEP 0.18 0.045 7.5e-15 10.0
AF
NP
Anisotropy [10]
k = k (MPa)
k 4 6
300 12.0
Viscoelasticity [14]
N=1 N=2
a 1 τ 1 (s) a 2 τ 2 (s)
5.6 0.132 1.7 11.765
(13)