Biomedical Engineering Reference
In-Depth Information
the case of mode II cracking in Fig.
23.2
. The difficulty here lies in the fact that the
most critical area of the crack, namely the crack tip has a jump in chemical potential
that cannot be captured using continuous functions, and further away from the crack
tip a discontinuous description fails as the chemical potential becomes increasingly
continuous.
High gradients are either approximated by
μ
f
(
x
,t)
over the crack surfaces. In this case the decomposition
•
case 1: a jump
becomes
μ
f
(
x
)
μ
f
(
x
,t)
μ
f
(
x
,t),
=ˆ
+
H
Γ
d
(
x
)
˜
(23.21)
[
μ
f
and the difference in chemical potential at the discontinuity is given by
]
:
μ
f
(
x
,t)
=
μ
f
(
x
,t),
x
h
˜
∈
Γ
d
.
(23.22)
The gradient in the chemical potential becomes
+
hδ
Γ
d
μ
f
n
+
,
s
μ
f
s
μ
f
s
μ
f
∇
=∇
+
H
Γ
d
∇
(23.23)
or
•
case 2: a continuous enrichment of the chemical potential. In this case the decom-
position becomes
μ
f
(
x
,t).
(23.24)
The distance function is defined with respect to the coordinates of the crack
x
Γ
,
namely by
μ
f
(
x
)
μ
f
(
x
,t)
=ˆ
+
D
Γ
d
(
x
)
˜
2
(
x
n
+
,
x
h
D
Γ
d
=
−
x
Γ
)
·
∈
Ω,
(23.25)
with
h
the magnitude of the Heaviside jump. The gradient
∂
D
of the distance
function is given by
⎧
⎨
h
2
n
+
∈
Ω
+
x
s
Ω
−
=
H
Γ
d
n
+
,
∂
D
=∇
D
Γ
d
=
(23.26)
⎩
h
2
n
−
x
∈
s
u
u
)
T
/
2 holds.
The Heaviside function results in a discontinuous distribution with a jump at the
location of the crack and a linear distribution away from the crack (Fig.
23.3
a). The
distance function continuous distribution of the chemical potential with a nonlinear
distribution of the chemical potential away from the crack (Fig.
23.3
b).
where
∇
=[∇
u
+
(
∇
]
23.2.3 Local Behavior at the Crack
A local coordinate system (
n
,
s
) is introduced where
n
is in the direction of the
normal vector to
Ω
+
and
t
is orthogonal in the direction of propagation. This means
