Biomedical Engineering Reference
In-Depth Information
Fig. 23.3
A 1D representation of the effect of (
a
) the Heaviside function (case 1) and (
b
)the
distance function (case 2). The total field is the additive result of the standard field and the enhanced
field
a displacement jump is decoupled into
]
n
n
+
+[
]
s
t
+
,
[
u
]=[
u
u
(23.27)
with
n
+
=−
n
−
directed into the body and
t
+
is directed along the crack surface
opposite to the direction of propagation. We define crack surfaces
Γ
d
∂Ω
+
∩
=
Γ
d
and
Γ
d
∂Ω
−
∩
=
Γ
d
.
23.2.3.1 Local Mass Balance
Deformation around the discontinuity is strongly linked to fluid flow. Fluid flow
takes place at the surface of the discontinuity from the medium into the crack and a
flow along the crack when opening of the crack increases. When the crack is closed,
the normal fluid flow
f
c
over surface
Γ
d
determines the amount of fluid exchange.
When the crack opens, additional terms are included. Figure
23.4
shows a schematic
overview of the local mass balance. There is a balance between tangential flow and
normal flow
t
+
∂s
−
˙
∂
q
Γ
·
n
+
+
n
−
=
f
Γ
+
f
Γ
=−
q
Γ
·
q
Γ
·
[
u
]
n
,
(23.28)
t
+
=−
[
]
n
∂μ
f
Γ
k
K
d
q
Γ
·
u
∂s
,
(23.29)
where
s
represents the distance along the crack, with
s
0 the crack tip and
s
positive in direction of
t
+
. Tangential flow is assumed of the Couette type. The
fluid flows from the crack into the formation are calculated from Darcy's law. If the
pressure gradient is resolved (case 2), then the resolved pressure gradient is used. If
the pressure gradient is unresolved (case 1), then the pressure gradient is evaluated
from Terzaghi's analytical solution of linear one-dimensional consolidation:
=
K
μ
f
+
−
μ
f
Γ
K
μ
f
−
−
μ
f
Γ
x
f
Γ
(
x
,t)
Γ
(
x
,t)
=−
,
=−
,
(23.30)
x
