ACME 2011 Proceedings of the 19 UK National Conference of the ...

2 MODEL
In an effort to keep an acceptable balance between the model’s complexity and the salient features it
is able to capture, an anisotropic hyperelastic biphasic swelling model is derived for the intervertebral
disc. As the proteoglycans undergo the same deformations as the solid matrix, the osmotic pressure
is constitutively determined from the deformation rather than a distinct phase. This offers the advantages
of requiring less material parameters than tri- and quadri-phasic models and keeps the number
of degrees of freedom relatively low while still accounting for the fluid flow, finite deformations and
electro-chemomechanically driven effects.
The biphasic swelling model is based on Ehlers’s theory [2]. The linear momentum and mass balance
equations are derived for both the fluid and solid phases and then combined together using the principles
of mixture theory for saturated and intrinsically incompressible materials. We subsequently obtain Eq.
(1) that reflects the coupled nature of the system (solid stress σ e coupled with the osmotic pressure ∆π
and the fluid pressure p in the momentum equation; solid velocity coupled with relative fluid velocity
in the mass balance equation).
� div (σ e − (∆π + p) I) = 0
div � v solid + w � = 0
The fluid flow is described using Darcy’s law in its classical form (Eq. 2), with a strain-dependent
permeability [5]. The constitutive model of the osmotic pressure is an isotropic generalisation of recent
experimental work [3]; because the osmotic pressure is directly related to the concentration of the
negative charges that are ”attached” to the solid phase, it is inversely proportional to the volume change
J = det (F) of the mixture (Eq. 3).
w = −k∇p with k = k0e M(J−1)
∆π = ∆π0
J
An additive split between the matrix and fibre contributions is adopted for the solid phase:
Wsolid(C, I4, I6) = Wiso(C) + Waniso(I4, I6), where the 4 th and 6 th invariants of the right-Cauchy-
Green tensor measure of the square of the fibre’s stretch (Iα = C : (a ⊗ a) where aα is the direction
of fibre α in the reference configuration). In a first approach, the solid phase is defined using a Neo-
Hookean material while the fibres are described using an exponential model [1], which accounts for the
fact that the fibres provide stiffness in tension only:
⎧
⎨
⎩
Waniso(I4, I6) = 0 if Ii ≤ 1, i ∈ {4, 6}
Waniso(I4, I6) = � � �
exp k2 (Iα − 1) 2�
�
− 1 if Ii > 1, i ∈ {4, 6}
α
k1
2k2
Material properties for this model were obtained from an iterative best-fit of Holzapfel’s experimental
results [4]. The 1D stress-stretch curve of a single lamellar sample are plotted on Fig.2. It is interesting
to notice that the fibre’s stiffness depends not only on the fibre stretch but also the polar angle (Fig. 2).
The set of coupled nonlinear equations is discretised in space (noting the Babuska-Brezzi condition)
and in time using a finite difference scheme. The consistently linearised problem is finally implemented
in a three-dimensional finite element scheme for high performance computing. An incremental solution
scheme is adopted.
(1)
(2)
(3)
(4)
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