Biomedical Engineering Reference
In-Depth Information
Fig. 23.2
When a shear band
develops, the displacement
jump becomes nonzero and a
large gradient in chemical
potential arises with a small
transition zone (
A
). With
time, the displacement jump
grows, the gradient in
chemical potential decreases
and the transition zone
widens (
B
)
be incorporated as a jump in the displacement. A jump in displacement is math-
ematically represented by the Heaviside function. We assume that the strain field
is defined everywhere even at the surface across which the jump occurs. Then the
displacement field
u
of the body can be additively decomposed into the continuous
part
ˆ
˜
u
and the enhanced part
u
(Remmers et al.,
2003
)
u
(
x
)
= ˆ
u
(
x
,t)
+
H
Γ
d
(
x
)
u
(
x
,t).
˜
(23.17)
The Heaviside function
H
Γ
d
is defined with the jump at the discontinuity as
−
Ω
+
,
h/
2
,
x
∈
H
Γ
d
=
(23.18)
Ω
−
.
+
h/
2
,
x
∈
The Heaviside function is acting on the smooth function
u
(
x
,t)
keeping the crack
surface continuous. The jump at the discontinuity
Γ
d
is given by
˜
[
u
]
and represents
the opening of the crack
u
(
x
,t)
=
h
u
(
x
,t),
x
˜
∈
Γ
d
.
(23.19)
The strain becomes
δ
Γ
d
˜
un
+
s
,
s
u
s
s
ˆ
˜
ε
=∇
=∇
u
+
H
Γ
d
∇
u
+
(23.20)
with
n
+
the normal at the crack surface directed inwards. The last term only has a
contribution in the variational description at the discontinuity. The second degree of
freedom that needs enrichment across the crack is the chemical potential, or in case
of a non-swelling medium, the pressure. Figure
23.2
shows in two steps what hap-
pens at the crack. On initial growth a small opening arises, i.e. small displacement
jump, and a steep gradient in the chemical potential occurs over a small transition
zone (case A). While the opening grows, the gradient in the chemical potential de-
creases in magnitude and spreads over a wider transition zone. This is illustrated for
