Biomedical Engineering Reference
In-Depth Information
charges and hydration occurs. Understanding and prediction of failure of the IVD
asks for the combination of experiments and computational models. Although finite
element models of failure of the disc exist, the modeling has usually been restricted
to stress analysis studies (Kim,
2000
; Natarajan et al.,
2007
) or inserting contact
elements as a model for lesions (Little et al.,
2007
). Accurate modeling of the crack
improves the predictive behavior of those models. This is not only an issue in biome-
chanics, but there is a need for a good model for fracture in ionized porous media to
study geotechnical issues as well.
A macro fracture in a continuum is often of interest and therefore a discrete
fracture model is used. Herniation is not perfectly brittle. The macro-crack is pre-
ceded by a zone with small-scale yielding and micro-cracking. This process zone
is simulated by a cohesive zone model, where the decrease of strength in the zone
is lumped into a discrete line (in 2D modeling) and a stress-displacement relation-
ship across this line. Larsson and Larsson (
2000
) introduced a discontinuity in the
fluid by a regularized Dirac delta function. Armero and Callari (
1999
) assume only
discontinuous displacement. Steinmann (
1999
) extended these embedded disconti-
nuity models based on enhanced strains concepts without the restriction of locally
drained or undrained behavior by a new interface law based on Darcy's law. To
eliminate mesh dependency and artificial length scales, Babuska and Melenk (
1997
)
introduced a discontinuity in a mesh free way by adding an enhanced field on top of
the standard displacement (and pressure) field, in this case by a Heaviside function
using the partition of unity property of the finite element shape functions. The num-
ber of degrees of freedom at the nodes whose support is crossed by a discontinuity
is increased. Therefore, no new nodes are added during propagation. Belytschko
and Black (
1999
) introduced this method for a solid together with an asymptotic
enhancement of the displacement field at the crack tip. Wells and Sluys (
2001
)in-
troduced cohesive segments within the finite elements followed by Remmers et al.
(
2003
) who modeled the crack not as a single entity but as a collection of overlap-
ping cohesive segments. A practical benefit of the method is that standard discretiza-
tion is used and crack propagation is independent of the discretization. A downside
of this method is that it is difficult to implement in commercial codes. Gasser and
Holzapfel (
2006
,
2007
) have applied the partition of unity finite element method
to tissues, namely the fracture of an aortic wall and of bone. Réthoré et al. (
2007
)
considered shear banding by using partition of unity with crack tip singularity for
the solid phase and without crack tip singularity for the fluid phase, suggesting a
discontinuity in the pressure field in case of shearing, combined with Darcy's law
similar to the enhanced strain models. A diaphragm with low permeability at the
discontinuity is assumed.
The above work has shown that the partition of unity approach is promising for
crack propagation in porous media. Much discussion is still on the treatment of the
fluid phase, because there is no comparison to a benchmark solution. In this research
we analyze under which conditions a discontinuous enrichment of the pressure field
should be preferred above a continuous. For the modeling of the osmoelastic behav-
ior of the material, Lanir's plane strain model (Lanir,
1987
) for small deformations
is used. Lanir assumes incompressible constituents, namely the solid matrix and in-
terstitial fluid, and neglects the influence of ion flow. Lanir's model coincides with
