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

where indexes U, P refer to the unknown displacement and pressure fields, respectively.
The major difference of this biphasic formulation is related with the introduction of K αβ
on the stiffness matrix and the dissipative terms 1 2 , , T T U on the right hand side of (eq. 2).
There additional terms are defined as follows:
∫ Ω
T *
Kαβ
= θ ∇ Ψ ⋅ K ⋅∇
ΨΔt
⋅ dΩ
(3)
0
0
→
0
→
0
( J − J ) ⋅ dΩ0
U Ψ ⋅ n
(4)
1
= ∫ Ω
∫
0
* → ( ~ →
K ⋅∇
0 p)
⋅∇
Δt
⋅ dΩ0
T = θ Ψ
(5)
Ω
* →
( −θ
) ( K ⋅∇
~ p )
→
1 ∫ 0 ⋅∇
Δt
⋅d
Ω
T =
n Ψ
2
FP
∫ Ω
= 0
⎛ 1
⎜
⎝ k
0
( p − p)
∂~
~ p ⎞
⎟
⎟⋅
dΩ0
∂pi
⎠
*
K introduces the permeability factor ( K ) in the stiffness matrix. The introduction of
αβ
permeability in the FE model will allow the fluid flowing. On the right hand side of eq.
2, since a totally implicit formulation was adopted (i.e., θ = 1 ), 2 0 = T . On the other
hand, while in case of a monophasic formulation FP weights the non-equilibrated
pressure, in case of a biphasic formulation one needs to introduce additional terms
to take into account the dissipative effect of the fluid flowing. The pressure
contribution is part of T , which allows the dismissal of 1
FP . The volume variation,
caused by the fluid flowing, is considered on U . It must be highlighted that the
index n refers to the beginning of the time increment, while the index 0 refers to the
initial configuration at t = 0 . Ψ are the shape functions for pressure interpolation.
The right-hand side of eq. 2 may be considered in the following form, where R is
the vector of the external forces:
⎡
⎢
⎣U
R ⎤ ⎡FU
⎤
−
+ T
⎥ ⎢ ⎥
1⎦
⎣ 0 ⎦
The permeability is assumed to be strain-dependent (Argoubi and Shirazi-Adl model,
1996), where M is a positive material constant [7, 8]:
K
K
J
0
Ω
0
(6)
(7)
(8)
* * M
= 0
(9)
In order to validate the biphasic formulation, Terzaghi’s theory of one-dimensional
consolidation was adopted [6]. The aim of this test is to evaluate the consolidation
process of a given material, as the fluid outflow takes place. The three different types of
u/p-c mixed elements were tested and validated.
The open source FE solver also allows some pre- and post-processing operations on FE
meshes, such as mesh refinement or imposition of local boundary conditions. Parting
from a model of a human VB and two IVDs first created by Smit [9], a full lumbar MS
model was developed (Fig. 2). This model includes all the relevant features of the IVD
as well as the adjacent structures. Both CEP and VB endplate were considered. The 27node
quadratic hexahedron was used.