Biomedical Engineering Reference
In-Depth Information
Fig. 23.4 Schematic
representation of the fluid
flow at the crack surface with
parameter s the distance
along the crack
where x is the width of the pressure gradient and depends on the time t since dis-
continuity developed at that place, the bulk permeability K and aggregate modulus
of the material ( c =
2 μ + λ δπ/δ tr ( ε ) ), namely x 2
tKc . In other words, at
the crack tip, the pressure gradient is infinitely steep, while as one moves away from
the crack tip, the pressure gradient is softening. Equation ( 23.30 ) can be rewritten
in terms of an unresolved hydraulic permeability k d :
f Γ =− k d μ f + μ f =− k d μ f =−
hk d μ f .
(23.31)
23.2.3.2 Local Momentum Balance
A discrete crack is preceded by local damage. This micro-damage is lumped into
one constitutive relation (cohesive zone) and projected onto the crack (Remmers et
al., 2003 ). The model relates the decohesion, i.e. softening of the traction forces, as
result of opening of the crack. For the traction forces at each surface holds σ
n + =
·
t Γ =−
t Γ =−
n ,i.e.
σ e μ f
·
σ
π I ± ·
n ± =
t Γ ( x ,t), x
Γ ± .
+
(23.32)
The cohesive law holds locally and is described in the local coordinate system of the
discontinuity. The cohesive zone model is nonlinear and acts on the effective stress.
For the cohesive damage model an exponential law similar to Xu and Needleman
( 1993 )isused,seeFig. 23.5 . This law denotes the softening behavior after reaching
the critical stress state. Defined is critical length δ s = G c ult , with τ ult the ultimate
traction forces and
G c fracture toughness. The cohesive law is then
t s = τ ult [ u ] s
δ s e ( [ u ] s
δ s ) 2
.
(23.33)
Note that the surface underneath the curve is the fracture toughness
G c :
[ u ] s = τ ult δ s 2
0
r
2 e r 2 d r = τ ult δ s = G c .
t s d
[ u ] s =
t s d
(23.34)
−∞
−∞
A history parameter κ is introduced in case of unloading each time step the current
opening κ 0 and traction τ 0 are remembered. When the new opening is smaller than
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